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Imaginary charge quantum electrodynamics : a running coupling analysis Ho, Andy C.T. 1998

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I M A G I N A R Y A  C H A R G E R U N N I N G  Q U A N T U M C O U P L I N G  E L E C T R O D Y N A M I C S ANALYSIS  By Andy C. T. Ho B. Sc. (Mathematics) Simon Fraser University M . Sc. (Mathematics) University of British Columbia  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS DOCTOR  FOR T H E DEGREE OF  OF  PHILOSOPHY  in T H E FACULTY OF G R A D U A T E STUDIES MATHEMATICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH  COLUMBIA  Aug 1998 © Andy C. T. Ho, 1998  In presenting this thesis in partial fulfilment  of the  requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or by  his  or  her  representatives.  It  is  understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  |l U*lieM*.irc.S  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  0 cX  -\  m%  Abstract  We construct a modified version of the renormalized tree expansion developed by Gallavotti and Nicolo, the loop regularized running covariance (LRC) tree expansion for expressing the connected Green's functions of the imaginary charge Q E D model in perturbation theory. B y integrating out a sequence of slice fields, the L R C scheme generates a flow of effective potentials V . Here we do not demand that the flow of V be gauge invariant s  s  but only that the Ward Identities hold at the end of the flow. From the flow of V , we obtain a flow of the couplings A of the local parts of V . Using s  s  s  a fixed point analysis in a suitable Banach space whose norm captures the asymptotic form of A , we determine the asymptotic behavior of X satisfying boundary conditions s  s  partially fixed by the Ward Identities. At each step of the flow, the slice covariance is transformed by shifting the local quadratic terms of V  s  to the Gaussian measure. In  this way, the corresponding A* is governed by a flow of an effective coupling  which  is ultra-violet asymptotically free around the origin. The U V asymptotic freedom of £  s  provides the stability of X so that the asymptotic form of A can be obtained from a s  s  primitive flow corresponding to only a few low order diagrams of the L R C expansion.  ii  Table of Contents  Abstract  ii  List of Figures  viii  List of Tables  x  Acknowledgment 1  2  3  xi  Introduction  1  1.1  Feynman functional approach to Q F T  1  1.2  Overview of the thesis  4  Preliminaries  12  2.1  dv in perturbation theory  14  2.2  Renormalization by counterterms  17  2.3  Renormalization group and asymptotic freedom  18  2.4  Gauge invariance & Ward Identities  21  2.5  Gaussian integrals and graphs  23  The G N Tree Expansion and Renormalization  29  3.1  Decomposition of fields  29  ,3.2  Flow of effective potentials  31  iii  4  3.3  Bounds on associated Feynman graphs  37  3.4  Renormalization by Taylor subtraction  41  3.5  Renormalized tree expansion  45  3.6  Running coupling constant tree expansion  49  Decomposition Of Field V i a Block-Spin Method  53  4.1  Block-spin transformation  53  4.2  r ' s bound in scalar theory  58  4.3  T of lattice Fermion  62  4.4  Components of T  65  4.5  Cancellation of poles  69  4.6 4.7 5  6  n  n  n  Existence of zeros of f (p)  72  n  Nonperturbative method (Lattice method)  80  L o o p R e g u l a r i z e d R u n n i n g Covariance  86  5.1  Running covariance  86  5.2  R C with neighbourly slicing  92  5.3  Loop Regularization  94  5.4  Loop-regularized Running Covariance  97  Two-Slice L R C  102  6.1  Bounds on running covariance and spinor loops  102  6.2  A-divergent graphs  108  6.3  Renormalized 2S-LRC on IQED  114  iv  7  8  Multi-Slice L R C  124  7.1  L R C coupling flow  124  7.2  The A —> oo coupling flow  126  7.3  Aspects of the calculation of the (3 functions  128  7.4  Summary of multi-slice L R C for IQED  L R C c o u p l i n g flow w i t h sharp slicing  142  8.1  Preliminaries  142  8.2  Asymptotic behavior of the sharp slicing primitive flow  145  8.2.1  Sharp slicing primitive flow in the Landau gauge  145  8.2.2  O D E analysis  147  8.2.3  The ' % = Z " condition of the coupling flow  154  8.3  9  137  2  R C scheme for other models •  156  8.3.1  (f> model  156  8.3.2  The g equation of a model with only one interaction vertex  4  . . .  159  L R C c o u p l i n g flow w i t h s m o o t h slicing  162  9.1  Preliminaries  162  9.2  The primitive  flow  166  9.3  Higher order  flow  171  9.4  9.3.1  The order N B functions  171  9.3.2  The order N flow and boundary conditions  180  The flow of E  s  . .  182  v  10 The Fixed Point Solution  188  10.1 The set up of the contraction map  188  10.2 Lipschitz continuity of the (5 functions  197  10.2.1 First order terms  197  10.2.2 Lipschitz bounds of B{£)  208  10.2.3 Fixed point argument for the first order flow  219  10.3 Higher order terms  226  10.4 The solution for the running couplings  237  10.4.1 The solution for 10.4.2  and  237  The solution for the full system  241  11 Ward Identities  243  11.1 Statement of Ward Identities  243  11.2 Root scale gauge variant local terms  248  12 Conclusions and Outlook  252  12.1 Summary of the L R C scheme for IQED  252  12.1.1 L R C scheme  252  12.1.2 Two-slice L R C  254  12.1.3 The multi-slice A -> oo coupling 12.2 Conclusion  flow  255 260  Bibliography  263  A Vestigial Gauge Invariance  265 vi  B Neighbourly slicing  273  C L R C running covariance  275  Cl  Photon field  275  C. 2 Spinor fields  278  D Bounds on Spinor Loops D. l  E  F  283  Proof of Lemma 6.2 of Section 6.1  283  D.2 Proof of Proposition 6.1 of Section 6.1  287  D. 3 Proof of Lemma 6.4 of Section 6.3  295  3 functions of the first order diagrams  299  E. l  Tools for computing the 3 functions  299  E.2  The (3 functions of the first order diagrams  307  E.2.1  Vertex diagram  309  E.2.2  Electron self energy diagram  311  E.2.3  Vacuum polarization diagram  316  E.2.4  Four photon legs diagram  322  E.2.5  Summary of coefficients  322  The contribution from expanding the slicing functions  vii  325  L i s t of Figures  2.1  A n initial momentum labeling of G  27  2.2  G with loop momenta p\ and p2  28  3.3  A tree r with forks / i , f and /  35  3.4  A labeling for G compatible with r  36  3.5  Subgraphs of G and their corresponding reduced graphs  37  3.6  The electron self-energy diagram  44  4.7  A 3x3 block of sub-lattice of (aL Zf  54  6.8  n-VP  6.9  L% (N  6.10  VjV--jmU+l)  2  3  n  112  u+l  +1  1:  N ) with n V^-^+i vertices and n > 4 n  112  Ki<n  113  7.11 The G V , 2P, 2F, 4P diagrams  127  8.12 The flows of ( near the origin  150  9.13 The diagrams of a order 1  flow  178  9.14 A n order 2 graph  178  9.15 A self-contraction of a Vr vertex  185  A. 16 The 4 photon legs diagram  267  viii  A. 17 The vacuum polarization diagram  271  B. 18 Example of h(x)  274  B.19 Only adjacent slices have overlapping supports  274  E.20 The vertex diagram  309  E.21 The electron self energy diagram  311  E.22 The vacuum polarization diagram  316  E. 23 The four photon legs diagram  322  F . 24 The electron self energy diagram  326  F.25 The one loop vacuum polarization diagrams  331  ix  L i s t of Tables  10.1 Bounds on the Lipschitz constants  223  10.2 Bounds on 5j <(0)  224  jP  x  Acknowledgment  I would like to acknowledge my enormous debt to Lon Rosen, who has supervised me, and who has given me advice, encouragement and support beyond all expectations of duty throughout the many years I have spent on the Ph. D program. I would also like to thank my parents for their support and encouragement through all these years.  xi  Chapter 1  Introduction  1.1 Feynman functional approach to QFT Since the discovery of the quantization of the radiation field by Dirac in the 1920's, quantum field theory (QFT) has became an indispensable framework for describing the fundamental laws of particle physics and nuclear physics. A salient feature of Q F T is that it provides a coherent treatment of the interplay between quantum mechanics and special relativity. Another feature of Q F T is that it allows a particle interpretation of the field operators. A Q F T is usually derived from a classical field using a quantization procedure. A classical field can be specified by a Lagrangian density  C($(x)) = £ ($(x))  - P($(x))  0  (1.1.1)  where x = (x,t)EWL ,  $ =  d  ($!,•••,$*),  £ ( $ ) is quadratic in the $;'s (including derivatives) and -P($) is usually a polynomial in 0  the $i's. The forms of £ ( $ ) is restricted by the symmetry requirement of the theory, e.g., the Poincare symmetry corresponds to a relativistic invariant theory. The same theory can be reformulated in terms of the Hamiltonian  =  +  1  (1.1.2)  Chapter 1.  Introduction  2  where H is obtained from the Lagrangian C through the usual Legendre transform. Quantization proceeds by re-interpreting the  as Hermitian operators on an infinite  dimensional Hilbert space satisfying the canonical equal time commutation relationships and the Heisenberg dynamics $ j ( f , r ) = e ^ {x,d)e- . itH  (1.1.3)  itH  j  A solution to the theory is specified by stating what the space of states ( the Hilbert space) is and by giving the manner in which <&i(x) acts on the states. In general, $i(x) is not a well-behaved operator, but rather it is an operator-valued distribution defined by fdx$i(x)f{x)  $i(/) =  where we can take / G C~(IR ), the space of smooth functions with compact support or d  / e &(K ), the Schwartz space of functions. d  In an axiomatic approach, a Q F T has the following standard properties: 1. There is a unique Poincare-invariant ground state |0 >. 2. States and action of $ on them can be reconstructed from time-ordered Green's functions (or Wightman functions) W {x n  u  •••,x )=< n  0|T  • • -<MO|0 >  (1.1.4)  where T orders the fields in the order of increasing time from right to left. 3. The Green's functions satisfy the Wightman axioms so that they are the Green's functions of a sensible physical theory (introduced by Wightman in 1956 [Wig56]). If the theory is a free one, i.e., Hj = 0, then a natural solution is to realize $ over the Fock space T which is an infinite direct sum of all n-particle spaces. But in general, the  Chapter 1.  Introduction  3  full Hamiltonian H + Hi cannot be realized on T where (1.1.4) may not be well defined. 0  So instead, in constructing a Q F T , one first constructs a regularized theory where a cutoff A or a regularization is installed so that the regularized Wightman functions are well defined. For example, (1.1.4) can be formally represented as a divergent integral. To regularize the theory, one may insert a cutoff A on the integration limits of the integral. One then tries to show in the limit of removing the regularization, lim W = W A  n  A—>oo  exists and determines a unique Q F T (usually, we only require the existence of a weak limit of W ). n  Around 1949, Feynman discovered a remarkable formula expressing the Wightman functions W (xi, • • • ,x ) in terms of a functional integral (sometimes it is call a path n  n  integral) [Fey49]: ST[^ (x )--^ (x )e S^^]v^ = .. i  il  W (x ---,x n  u  l  in  n  *  L  n  JT\e J L  (*( ))|  £  B  (1.1.5  £>$  where X>$ = fTj £j{dd<&i(x). In 1965, Symanzik proposed to consider the above with X  t —> it [Sym66]. In moving from the relativistic world into the Euclidean world, the relativistic path integral (1.1.5) becomes a Euclidean path integral for generating the Schwinger functions  S (Xl, n  • • • ,X ) n  =  —T-fcmx))l~~ IT  •  (  L  L  6  )  which is the basis of an E Q F T . The reason for going to the Euclidean world is that, by the positivity of J C, the oscillatory factor e J l  c  of (1.1.5) becomes an exponential  decay factor e~ J which makes the path integral easier to define and work with. If these c  Schwinger functions S satisfy the Osterwalder Schrader axioms [OS73][OS75], then by n  Chapter  1.  Introduction  4  analytic continuation, one can recover the set of Wightman functions. By introducing an external source field J, all the S (xi, • • • ,x ) can be expressed in terms of a single n  n  generating functional z\j] = j  -J cm*» + *J *°  e  $  V  where $  J  = E  IT  / $i(x)Ji{x)dx  6Ji (xi)  5J (x )  t  Sn (-^1  > ' ' ' j  •^n)  :  irl  (1.1.7)  Z[J]  n  Z[J]  j=o Subsequently, almost all current work in Q F T focuses on constructing the S using a n  Euclidean functional integral.  1.2  Overview of the thesis  Taking the functional integral approach, the central problem in Q F T is then to construct a measure  du($) = const. e - / * £ (  ( x ) ) d x  r»$  (1.2.1)  over the space of Euclidean fields $(x). The existence of a Gaussian measure dv where 0  C is quadratic in <&(x) is well understood. For a non-Gaussian du, usually it may only be constructed in perturbation theory.  However, there are a few cases where du can  exist beyond perturbation theory. In particular, Gawedzki and Kupiainen have developed a non-perturbative treatment for the scalar theory based on the renormalization group (RG) [GK86]. In their treatment, the domain of all possible $ in the functional integral is separated into small-field and large-field regions where the cut depends on the running couplings. In the small-field region, du is estimated by perturbation theory with renormalization. By using the positivity and largeness of the action / £ ( $ ( x ) ) dx, the  Chapter 1.  Introduction  5  large field region provides only a small contribution and thus does not affect the small field estimation. The central goal of current research in constructive Q F T is to develop a similar approach in constructing a non-perturbative treatment for an asymptotically free (AF) gauge field theory. Lately, the R G analysis of a lattice version of gauge field theory has been of great interest to many people. For example, Balaban & Jaffe have developed a gauge invariant R G flow of effective potentials for a pure gauge theory [BJ86]. In a parallel development, Gawedzki & Kupiainen constructed a R G flow for scalar theory using "block-spin averaging" on lattices [GK86] where they were able to construct an independent decomposition for a scalar field. Moreover, such a decomposition can be incorporated in the G N tree expansion method to obtain a renormalized flow of effective potentials and appears to be very promising for developing a gauge invariant flow for gauge theories. The main model that we study in this thesis is imaginary charge Q E D (IQED). The reason for studying IQED is that it is an A F gauge model. As suggested from the work of [GK86], one must work out a thorough analysis of the coupling flow in perturbation theory before making the endeavor into the non-perturbative analysis. Our analysis on the IQED model is confined only to perturbation theory of a fixed order. We make no attempt in trying to show that the 8 function of the coupling flow is a convergent sum of contributions from all orders of diagrams. I have made some attempts to adapt the "block-spin" approach to construct a decomposition for the Fermi fields of the Q E D model. I was hoping to employ the G N scheme, which was developed by Gallavotti and Nicolo in 1984 for the renormalization of the 0 theory [GN85a][GN85b], with the "block-spin" decomposition to the IQED 4  model to produce a gauge invariant flow of effective potentials. But my work fails to give the necessary decay bounds required by the G N tree expansion scheme. Nevertheless,  Chapter 1.  Introduction  6  I would like to present some of the work that I have done on the lattice approach in Chapter 4. I hope that some reader can find a way to improve the decay bounds so that the "block-spin" scheme can be fitted into the G N scheme for producing a gauge invariant flow for the Q E D and IQED model. After the failed attempt in working with the lattice approach, we abandoned the idea of constructing a gauge invariant flow. Instead, we went back to the G N scheme under the simple momentum shell scale decomposition. We require only that the Ward Identities of the final potential at the end of the flow can be recovered when the U V cutoff is removed. In this thesis, we investigate the perturbative analysis of IQED by applying the loop regularized running covariance (LRC) tree expansion scheme (see Chapter 5 for the construction of L R C ) . In terms of the Euclidean fields A^, tp, tp, the basic Lagrangian density of Q E D is given by C  =  int  ^F  Here  2  F  2  + tp(-ip + m)tp +  — F F  F  etp4tp.  (1.2.2)  =d A — 8 A  A = A,Y, ?> = d»r, (1.2.3) where 7 has a representation as the 4x4 anti-hermitian Dirac matrices, which satisfy M  7  Y  +  7  Y  = -2c5 . F  (1.2.4)  The basic Lagrangian density of the IQED model is obtained by replacing e by ie in (1.2.2), and we do so for the rest of this thesis.  Chapter 1.  Introduction  7  For a renormalized expansion for IQED, we need to further reparameterize the I Q E D Lagrangian density £ . For a scheme using momentum cutoffs for both the Fermi and photon covariances (the corresponding fields have an ultra-violet (UV) cutoff M  u  where  M > 1 is a scale parameter), £ = C + C\ is parameterized with 7 coupling constants 0  X — (Af, Af, • • •, Af) and a gauge-fixing parameter 77: u  C  = - [A- {-A)A + r]{l-r])-\d-  d  =  Af + A ] + # n - $)</>,  l  0  zAf +^  '(1.2.5a)  2  + i> [m(Xi - 1) - i(Af - 1)0] ^ [(Af - 1)A • {-A)A  + A f (d • A) ] + A f A ,  + M X A 2U  U  (1.2.5b)  2  4  2  5  where we have rewritten 1/4 F as 1/2 [A • ( — A ) A — (d • A) ] and have added a photon 2  2  and a gauge-fixing term 1/2 (1 — rj)~ {d • A) , where 0 < 77 < 1.  mass term 1/2 A  l  2  The reason for introducing the M  2U  2  in (1.2.5a) is that we would like all couplings to  be dimensionless (the couplings remains invariant under scaling transformation of the momenta of the fields). The presence of the 1/2 A term in C serves as an infra-red (IR) 2  0  cutoff. Since all calculations are done in momentum space, let us express the terms of C correspondingly. C  =  -jAD^A  C  =  zAfVl + (Af - 1)V + (Af - 1)14 + (Af - 1)V  0  T  (1.2.6)  + ^ S ^ , 3  2  +Af M V  (1.2.7)  + A f V + Af V  2U  b  6  7  where Du  =  ^ 7 7 [aL + bT]  T  —  A  1  _ ^  a = p' + l  P  t  T  /  b  —  P / l P l /  M 9  Chapter 1.  Introduction  8  Su  p-  u  1  =  -^jirn+rf)  if  =  VpTfi  (1.2.9)  is a cutoff function corresponding to the U V cutoff of the fields; and Vi  V  =  v  =  2  =-±Ap TA,  y  =  4  (1.2.10)  I^,  V = A\  2  6  ^v>,  7  Since (cL + d T )  - 1  = c^L +  the photon propagator, obtained by inverting D p<u  _ 1  <u  p  d^T,  is : H _  \  = iTp " fr^)+5 L  (1  T  +  -  211a)  Two commonly used gauge fixings are rj = 0 (Feynman gauge) D  ^  =P ~  U  1  ^ +p  (1-2-llb)  and 77 = 1 (Landau gauge)  Similarly, the Fermi propagator Su, which is obtained by inverting S^ , has the form 1  Su = (£ {m+jf)-\  (1.2.12)  u  In general, the perturbation theory of using cont. e~ I °^U§ C  as the free Gaussian  measure contains notorious divergences when we remove the U V and IR cutoffs. The common treatment is to introduce a reparametrization of the couplings in the Lagrangian as in (1.2.5b) where the terms involving the A's would serve as counterterms for cancelling the divergences in a renormalization procedure (see Section 2.2).  Chapter 1.  Introduction  9  Motivated by the approach of Gawedzki and Kupiainen [GK86], we consider the perturbation theory in which we decompose the fields $ = (A, ip, ip) into a sum ]C^=-i ® (roughly speaking, the fields  at scale s correspond to momenta of order M ) and we s  integrate out the fields scale by scale from s = U down to s = 0. This produces a running coupling constant (RCC) expansion in which the coupling constants at scale s, X , are s  determined as a function of the coupling constants at higher scales. Schematically, X = B{X \---,X ). s  The values X  u  s+  (1.2.13)  u  at scale U are the "bare" or "input" coupling constants, and the values  A at the root scale r are the "physical" or "output" coupling constants. For the details r  of this procedure, see Section 3.6. The goal of this analysis is to show that for suitable boundary conditions (i.e. specifying A f or A£), the flow equation obtained from (1.2.13) with B retaining only terms from a fixed perturbation order have a unique solution with controlled asymptotic behavior as U —> oo. The flows defined by (1.2.13) can then be realized perturbatively by the solutions of the fixed order equations. To analyze a fixed perturbative order flow of (1.2.13), we first consider a primitive flow corresponding to (1.2.13) using only the lowest order terms of B (see Section 9.2) . —*  For a given boundary condition, it is fairly straightforward to determine the solution A  s  of the primitive flow equations A = B {A ,---,A ). s  S+1  (1.2.14)  U  L  We then would like to establish that the flow (1.2.13) is stable in the sense that when we include the higher order terms of B — B\ the solution A = K + o(A ). s  s  s  A t first, in  our analysis, we were unable to determine the stability of the flow by using the usual R C C scheme. Our difficulty was that we were attempting to realize A F as the statement  s  Chapter 1.  Introduction  10  that A f —» 0 - it doesn't! Inspired by Gell-mann and Low's paper on the R G flow of the effective charge in Q E D [GML54], we modify the R C C expansion into the running covariance (RC) expansion by shifting quadratic local terms into the sliced covariance at each scale of the flow. In this way the running couplings Aj of the local quadratic terms will appear approximately as 1/Af in the flow equations. In the R C scheme, we discovered that the flow (1.2.13) is roughly governed by the effective coupling  '  C  =  '  (L2 15)  wM^WY  It is the flow of C which is A F , and this leads to the stability of the flow (1.2.13) (see s  Section 8.2). To make the argument of the stabilization rigorous, we employ a fixed point argument on a suitable Banach space whose norm captures the asymptotic behavior of the couplings. From the fixed point treatment, we also show that the bare electron kinetic coupling A f has the same asymptotic behavior as the bare vertex coupling Af, i.e. lim  = e,  £/—>oo A  (1.2.16)  2  where e is interpreted as the physical charge of the theory. This allows us to establish the desired Ward Identity with charge e in the limit U —> 00 (see Chapter 11). (1.2.16) is the analogue of the physicists' " Z\ — Z " condition. 2  Because we require our model to satisfy Ward Identities, we apply a loop regularization on the Fermi loops to avoid putting in a U V cutoff on the Fermi covariance. This leads to the loop regularized running covariance (LRC) expansion constructed by incorporating the technique of shifting quadratic counterterms into the sliced covariance in a loop regularization set-up. In the L R C for IQED, we discovered there are two levels of divergence corresponding to the loop regularization parameter A and the U V photon momentum cutoff parameter U (see Section 6.1). Correspondingly, the renormalization is also divided into two levels where the top level renormalization involves A occurring at  Chapter 1.  Introduction  11  the top scale and lower level renormalization involves the remaining scales. In the limit A — > • oo and setting fictitious external fields to zero, the resulting coupling flow is effectively the same as the coupling flow of the R C scheme. Furthermore, by using suitable restrictions on the support of external source fields, after the A-renormalization at the top scale, there is no renormalization required for the (7-divergence at the subsequent lower scales (see Section 6.3). Basically, the restriction on the external source serves as a U V cutoff which prohibits large external momenta. In order to probe the system with arbitrary high external momenta, one needs to resort back to renomalization.  Chapter 2  Preliminaries  In this chapter, we would like to discuss a few preliminaries before presenting the G N tree expansion in Chapter 3. We first describe how the functional measure dv can be constructed as a perturbation of a free measure du . This is done by expanding the mo0  ment generating functional of du as a perturbative series in the interaction coupling A, such that the coefficients are determined by evaluating Gaussian integrals with respect to the free measure dv . 0  Because of the notorious divergence of the coefficients of the  perturbative series, we need a renormalization procedure in order to render finite coefficients for the perturbation series. We then discuss how the divergence can be remedied by the use of counterterms. In a renormalized theory, all parameters are reparameterized by a single parameter A, called the physical interaction coupling (in practice there may be more than one). There is an arbitrariness in assigning a value to the physical coupling. Physicists have derived a dynamic equation for A called the renormalization group (RG) flow equation to describe the value of A for different momentum scales. We will give a brief historical account of the R G equation in Q E D and the classification of a renormalizable Q F T as an ultraviolet asymptotically free theory (UVAF) or an infar-red asymptotically free theory (IRAF). Finally, we present some rules on the Gaussian integration and its graphical representation as Feynman graphs. The idea of the R G was greatly advanced by Wilson in the 1970's [Wil70][Wil71][Wil72][KW74].  12  Chapter 2.  Preliminaries  13  In his work on the fixed source meson model [Wil70], Wilson's strategy was to decompose the problem into many parts with each corresponding to a different momentum scale, and to solve the problem one part at a time. In his perturbative approach, the momentum-space continuum is replaced by well separated momentum slices and the Hamiltonian corresponding to the largest momentum slice is treated as the unperturbed Hamiltonian with the terms for the lesser slices as perturbations.  By integrating out  the field variables at the highest momentum scale, one reduces an effective Hamiltonian for n momentum slices to an effective Hamiltonian for the remaining n-1 slices. The new effective Hamiltonian is identical to the original one except that the meson-nucleon coupling g is renormalized (modified). By iterating the procedure, one obtains a discrete flow (sequence) of effective Hamiltonians and thus a flow of g, such that each effective Hamiltonian is obtained by the elimination of a momentum scale. Following the development of his R G approach, by thinking about the degrees of freedom of fields, Wilson developed the idea of partitioning the phase space into unit volume such that the momentum space is marked off in a logarithmic scale. By translational invariance, for each momentum shell, the position space is partitioned into blocks of the same size. In this picture, he learned that a cut off at some finite value of the momentum scale is needed in order to make sense out of his R G analysis. This gives the basis for the formulation of field theory on a lattice; one needs to define phase-space cells covering all the cut off momentum space and a position-space lattice in which the field variable $ would be defined. Wilson's R G idea motivated the development of the G N tree expansion scheme, which is the basic tool of our perturbative analysis. The intuitive idea of the G N scheme, which we will elaborate on in more detail in Chapter 3, can be already seen in Wilson's work on R G and lattice theory. In Wilson's R G approach, there is a dynamical flow of effective  Chapter 2. Preliminaries  14  potentials giving the effective behavior of the theory at different length (or momentum) scales. The G N tree expansion procedure has an analogous feature of having a flow of effective potentials. From the flow of the effective potential of the G N tree expansion, one can obtain a dynamical flow of the interaction coupling which determines whether the theory is U V A F or I R A F .  2.1  du in perturbation theory  Consider a Euclidean quantum field theory (EQFT) in R with Lagrangian density, d  k  C = Co + A • Ci t = CQ + "^2 \j Cintj , n  (2.1.1)  j=l  a local polynomial in the fields of the theory $ = ( $ i , •••,$#) and their derivatives, where Co is quadratic, Ci  nt  stants.  has degree > 3 and A = (Ai, • • •, A^) are the coupling con-  Co also depends linearly on physical parameters Q such as masses and field  strengths. The measure is formally expressed as : du{<5>) =  const, e" / * £ (  ( x ) ) d x  V<$>  (2.1.2)  where V$ denotes the integration over all fields and const, is the normalization constant such that / du = 1. We can also rewrite the measure in terms of its free measure du : 0  di/($) = const.  -I^n (^(x))dx  e  t  'o,  dU{  (2.1.3a)  where dv (§) Q  = const. / _  e  £ods  2?$.  (2.1.3b)  Chapter 2.  Preliminaries  15  The free measure duo can be understood through its generating functional Z [J} = const. 0  where J(x) = (J (x),J (x)), L  Je *dv ($).  (2.1.4)  J  0  J $ is defined in (1.1), and the source field J(x) has  N  the same characteristics as $(x).  In particular, if $i(x) is fermonic then so is Ji(x).  Integration of monomials in the fields with respect to dv^ can be computed by taking appropriate functional derivatives of Z [J]. 0  Using integrating by parts, Jc (x)dx  = ^C' ®,  (2.1.5)  1  0  where C~ is a first or second degree differential operator. Using (2.1.5) and making the l  change of variables $ —>• $ + CJ in (2.1.4), we get Z [J] = e^ .  (2.1.6)  JCJ  0  Given a monomial in the field M = $jj(xi) • • • $ (x ), in  then the Gaussian integration of  n  M with respect to du is given by 0  SJ^ixi) A  8J {xi) h  5J (x )' in  7=0  n  i  . \JCJ  (2.1.7)  e  SJ (x ) in  n  7=0  When Fermi fields are involved, these derivatives are Grassmannian and we need to be aware of additional ± signs. Using (2.1.7), Gaussian integration of the monomials can be expressed elegantly as a sum of graphs (Euclidean Feynman graphs) with vertices labeled by the coordinates of $i(xi) in M and a line connecting two vertices xg and yi corresponds to a covariance Cij(x£,ye). then Cij(x,y)  In the case that $i(x) and  are fermions  = —Cji(y,x) and it becomes necessary to specify the direction of a line  connecting two vertices.  Chapter 2.  Preliminaries  16  For a nontrivial theory with nonzero interaction, the full understanding of the measure du(Q) is very much a mystery. Here we only want to inquire about dv{$>) in terms of perturbation theory. As in the free case, we study dv(&) through its moment generating functional Z[J] Z[J]  =  j  e*  di/($)  =  y ^* -/X-£nt(*(z))ds  J  e  e  d l / Q  ($) .  (2.1.8)  The expectation of a monomial in $ can be computed by taking functional derivatives of Z[J]. We say dz/($) exists in perturbation theory (let k = l in (2.1.1) for simplicity) if Z[J] exists as a formal power series in the coupling constant A whose coefficients are computed by evaluating Gaussian integrals with respect to du . Letting 0  V($) = -\JdxC {${x)) int  ,  from (2.1.7), we may write Z[J]  =  :T  =  E* Zn{J}.  Z0[J]  (2.1.9)  n  For each order in A, the coefficient Z [J] is represented by sums of Feynman graphs n  corresponding to Gaussian expectations. The Green's functions (Feynman graph) obtained from (2.1.9) can be expressed in terms of connected Green's functions (connected Feynman graphs). We introduce the external effective potential V ($ ) which is the generating functional for the connected e  e  Green's functions (amputated by the free field propagator). Let us define the external field <fr = CJ and consider the expression: e  Z[J] z [J] 0  _  i^c- ^ 1  [  „$ c- <& e  l  Chapter 2.  Preliminaries  17  j ^c-^-\*c-^  =  const. -\**c-^  =  const. J e - ^ - ^ - ^ - ^ e - ^ ^ W V Q .  e  -xJc m <$>  e  e  in  V  (2.1.10)  Making the change of variable $ —> <3> + $ and defining V ( $ ) = — J \-£ ($), e  int  we define  the effective potential V as the log of (2.1.10), e  =  l 0 g  [  fe^W  o  (  $  )  =  log[£(e^+* ))] ,  =  l o g £ - [ £ ( V ( $ + <r)%,  e  0  (2.1.11)  n=l  where [  2.2  ]o denotes dropping constant terms independent of $ . e  Renormalization by counterterms  A closer inspection of the perturbative expansion shows that the coefficients of the power series (2.1.11) are actually infinite because some of the Gaussian integrals are divergent. Two problems can arise which cause the Gaussian integrals to diverge: 1. Ultra-Violet (UV) problem Local singularities of the covariance C(x,y) may cause integrals to diverge . For example, the covariance C = (— A +m )~ 2  1  behaves like ^ _ ^ _ as \x — y\ —> 0. 2  2. Infra-Red (IR) problem In the case of a massless theory, integrals may diverge because of problems associated with infinite volume when C(x,y) does not decay fast enough as \x — y\ —> oo. Fortunately, in many cases, these divergences can be removed by renormalization. The basic idea in renormalization is to introduce a reparameterization of the Lagrangian.  Chapter 2.  Preliminaries  18  Let the Lagrangian of the model be of the form £(Af,Af)=£ (Af) + Af£ 0  (2.2.1)  i n t  where A f and A f are the parameters of the model. By a reparameterization of (Af, A f ) , i.e. expressing A f and A f as a function of a new parameter A, one wishes to write £ ( A f (A)) + Af (X)C 0  int  = £(Af (A), Af (A)) + SC(X),  (2.2.2)  where SC are local counterterms which have the same form as those in the Lagrangian. The parameterized (Af, Af) are called the bare parameters and are expressed as formal power series (fps) in A; in particular, the parameterized interaction coupling oo  Af(A)~A + £  aX.  (2.2.3)  n  n  71 = 2  We call A f the bare coupling and A the physical coupling. The coefficients a are infinite n  and are suppose to cancel the infinities. Now the renormalization problem of V ($ ) is r  e  e  to show that SV = — J t5£($) dx can be chosen such that  has a well defined fps in A. In summary, a renormalization scheme of V ($ ) in perturbation theory consists of e  e  introducing regularizations or cut-offs to render the Gaussian integrals finite, quantifying the sizes, locating the sources of the would-be infinities, choosing the counterterms SV = — J SC to cancel the would-be infinities, and then proving that the coefficients of the renormalized V ($ ) are finite, uniformly as the regularizations are removed. R  e  2.3  Renormalization group and asymptotic freedom  In a renormalization scheme, the pertubative expansion of V ($ ) is parameterized by e  e  the physical coupling A. If A is parameterized by another parameter A then we can  Chapter 2.  Preliminaries  19  reparameterize the theory in terms of A and call A the physical coupling. The physicists have defined the term renormalization prescription R as a prescription of assigning a value to A. The physicists refer to the set of transformations that relate the renormalization prescriptions as the renormalization group (actually, these transformations do not always possess a group structure). The idea of the renormalization group was first discussed by Stueckelberg & Petermann [SW53] and by Gell-Mann & Low [GML54]. In the Gell-Mann & Low paper, a family of parameters  is introduced as the electron charge in the theory of Q E D with  —iSfj, as the interaction coupling. e supposedly relates the behavior of the theory at arM  bitrary momentum scale \i. Gell-Mann & Low found that e obeys a differential equation M  of the form ^  =  ( 2  - 3  1 }  where j3 is an fps in A. The experimentally measured electron charge e is supposed to be the low momentum scale limit of e^, e = lim e , M  and the bare charge eo is the high momentum scale limit of e^, e = lim e„ . 0  (J,—>oo  ^  As a consequence, the bare charge must be of fixed value ( it can be infinite). In general (2.3.1) is called the R G equation. It reveals some important information about the large or small momentum behavior of a Q F T . We can treat (2.3.1) as a dynamical system by setting // = e* and (2.3.1) becomes X'(t) = e  t  ^  = P(X(t)).  (2.3.2)  Chapter 2.  Preliminaries  20  The solution X(t) of (2.3.2) must go to a fixed point Xf (a zero of fj) or infinity in the limit t —>• ± o o . Thus the behavior of fj around the fixed points provide a basis for a very important classification of many theories in terms of their low or large momentum behavior. If for any A(i ) = A near A/ 0  0  lim X(t, An) = A/-  then Xf is called a U V stable fixed point. Similarly, if lim X(t, An) — Xf t-»-oo  J  it is called a IR stable fixed point. In perturbation theory, A = 0 is a fixed point, and the only place we can compute fj is near A = 0. A theory is called U V asymptotically free if A = 0 is a U V stable fixed point and IR asymptotically free if A = 0 is a IR stable fixed point. In practice, we can only compute the bare coupling by inserting a cutoff N. Define A( ) as the bare coupling at the cut-off N so that w  lim A > = X (JV  f  (2.3.3)  oo  \W  =  A + £ a ^ A "  (2.3.4)  71=2  the coefficients  are infinite as N—> oo. From (2.3.4), we see that it is desirable to have  a U V A F theory since we can choose A ^ small and keep A(/x) small so that perturbation theory is valid.  More importantly, we end up with a positive physical coupling, i.e.  X(pL = 0) > 0. On the other hand, I R A F theory is undesirable since to end up with a non-trivial theory, i.e. X(fj, = 0) ^ 0, it seems necessary to take A ' ^ —>• oo as N —>• oo and then the validity of perturbation theory is completely in doubt since X(fi) does not stay near the origin throughout the flow.  Chapter 2. Preliminaries  2.4  21  Gauge invariance & Ward Identities  In this section, we describes briefly the gauge fixing and Ward Identities [War50] of Q E D . Q E D is a gauge theory which means that the classical Lagrangian of the theory is invariant under a group of gauge (local) transformation. The Lagrangian density of Q E D is given by (1.2.2). By integration by parts we can replace L by p  C = ± A „ [D-j\ A , P  D-] = d\  V  u  - d»d„  (2.4.1)  and write Lj as C = ipS- ip,  S~ = -ip + m.  l  1  F  (2.4.2)  In momentum space, D- M l  = V [<W -  f-\  2  5" (p) =j/ + m.  Vj  J  (2.4.3)  It is not too hard to verify that D~l is non-invertible while S~ is invertible with inverse x  S(P) = ^  (2.4.4)  Since e  -  i e x s  -l ie e  X  g-l  =  +  e  ^  the Lagrangian (1.2.2) is invariant under the gauge transformation $^e~ i),  i)^e ip,  iex  A^A-edx-  iex  (2.4.6)  Because of the gauge invariance, the differential operator D' of (2.4.1) is non-invertible; 1  put differently, the measure corresponding to D~ does not have Gaussian decay in all l  directions. The standard procedure is to modify C with a gauge fixing term P  1/2  {l-n)-\d-Af  Chapter 2.  Preliminaries  22  where d • A = d^A^. Thus, in momentum space, as in (1.2.8), we write D-l{p)  =  D^ip)  =  p [V + 2  (1 -  7 7 ) - % , ] ,  (2.4.7)  +(1-77)7^].  In particular, in Feynman gauge with 77 = 0, D  =  ^ p2  and in Landau gauge with 77 = 1, pi  The original gauge invariance is broken, but in Q E D the Ward Identities (WI) still hold. It is important that the renormalized perturbation theory respect the W I because it implies that the theory is perturbatively unitary and that the S-matrix is independent of the choices involved in gauge-fixing [tHV72]. The identity underlying the W I in Q E D is (2.4.5). A form of the W I can be expressed in terms of the generating functional for the connected Green's functions (2.1.11)[FHRW88] K($ )  = log ^ J ( ^ + ^ X * + * ) dv {$)  (2.4.8)  Z = j ( +^)(*) du (<!>)  (2.4.9)  e  e  e  0  where y  e  0  du {A)  =  const.  e  dv (ip,ip)  =  const.  e-I i Vil)V^  =  - J eipAijj dx,  0  0  V($) SV  — counterterms.  -I " VA c  dx  c dx  (2.4.10)  23  Chapter 2. Preliminaries  By (2.4.5), we have  dis (e *tp,e- $) = ie  iex  0  e- I^ du {tP,tP).  (2.4.11)  tp -4 e *?/;,  (2.4.12)  e  M  0  Hence under the change of variables  tp -4 e ip, iex  16  it is easy to see that we can arrive at the following W I ( a non-linear form ): V ( , fj, B) = K ( ( l + eSfi )e- V, iex  e  V  X  *?e *(l + e? S), B + p )ie  X  X  (2.4.13)  In our construction of IQED in perturbation theory, we wish to show the renormalized model under the running covariance tree expansion scheme respects the Ward identities with e -4 ie. Note that the WI depends on dv^,^). Thus the W I is broken when one puts in an UV-cutoff U using the momentum shell decomposition. There are various techniques for maintaining the WI in a G N tree expansion of Q E D . We have chosen the loop regularization scheme as presented in [FHRW88]. We call the tree expansion which uses the running covariance with loop regularization a loop regularization running covariance (LRC) tree expansion.  2.5  G a u s s i a n integrals a n d graphs  It will be helpful to elaborate more on the rules of Gaussian integration and their graphic representations since they will be used later in the perturbative analysis of nontrivial models. Denote (2.5.1) as the expectation of the monomial M = $ i • • • $ „ . We write M as a product of local monomials M  1 ;  • • •, M where each Mj has argument X j . We regard each Mj as a local p  vertex with the fields of Mj as legs emanating from the vertex.  Chapter 2.  Preliminaries  24  Gaussian Integration Rule 1 £{M  • • •, M ) =  U  {-lT Val{G)  (2.5.2)  {G)  p  G where the sum over G is over all graphs formed by contracting $  x  ••• $  n  in pairs in all  possible ways. A contracted pair of fields <3>(x) and <&(y) produces a covariance C(x,y) which we represent as a line joining the vertex at x and the vertex at y. ir{G) is the number of commutations of Fermi fields required to move each field in M next to the field with which it contracts. The product of covariances Val(G) corresponds to the lines in G . Each G is formed by making a particular choice of joining all the legs of the local vertices M i , • • •, M . p  Define the "connected" or "truncated" expectation by (V  rr,  £ (M T  U  • • •, M ) = p  ° _  log£(e ^ - * >>) x  d  x  + +x  (2.5.3)  M  A=0  Gaussian Integration Rule 2 From (2.5.3) we have : £ (M  • • •, Mp) = £  T  U  (-iy^Val(G),  (2.5.4)  G where the sum G is over the connected graphs in (2.5.4), i.e. graphs where any two vertices are joined by a sequence of lines. Rules 1 and 2 are statements about "vacuum expectations" and the graphs associated with these expectation are "vacuum graphs" with no legs from the clusters left uncontracted. £(Mi($  If we take an expectation of monomials involving external fields, say  + $ ) , • • •, Mp($ + $ ) ) , then the resulting Feynman graphs will not be vace  e  uum graphs but will be graphs with uncontracted external legs corresponding to external fields.  Chapter 2.  Preliminaries  25  Gaussian Integration Rule 3 £ ( M i ( $ + $ ) , • • •, M ( $ + $ ) ) = 52 (-l) ^Val(G) T  e  e  n  p  (2.5.5)  G  where the sum over G is over connected graphs. We denote the set of uncontracted legs by A(G) and the set of lines of G by C(G). Then the value of G is Val(G)=  [f[dx  II  =\  eec(G)  v  v  C(x ,y ) t  & (x ),  t  x  x  (2.5.6)  AeA(G)  where xg, yi are the vertices Xj, yj at the ends of t, and x\ is the vertex from which the leg A emanates. The Fermi fields in the product IlAeA(G) ®\{ \)  occur in the same order  x  as in M. Gaussian Integration Rule 3 in Momentum space Next we would like to describe Val(G) of (2.5.5) in the momentum space representation. Using the translation invariance of the covariances, each covariance of G can be written as C(x  tl  y ) = C(x e  e  , 0) = J  y e  C(%),  where C is the Fourier transform of C. As for each ^A(^A)  =  /  ^-/  X x  '  q x  (2.5.7)  we have *(9A).  (2.5.8)  We think of each Fourier variable as a momentum variable. For a Feynman graph with momentum labeling, there is one momentum (internal momentum variable) per line and one (external momentum variable) per leg, flowing in the direction of the arrow given by the coordinate space Feynman graph. We denote the set of vertices of G as V(G). B y substituting (2.5.7) and (2.5.8) into (2.5.6) and integrating out the coordinates variables, it is easy to see that, at each vertex x , we get a momentum conserving delta function v  where its argument is the sum of the momenta flowing in and out of the vertex v. Furthermore, since Val(G) is translational invariance, there is one more delta function  Chapter 2.  Preliminaries  26  for conserving all the external momenta.  To express the arguments inside the delta  functions more concisely, we define E, the incident matrix, by +1  if £ —> v ( i.e., the direction (arrow) of the line £ flows into the vertex v)  E e = s —1 if £ i— v ( i.e., the direction (arrow) of the line £ flows out of the vertex v) • Vt  0  if the line £ is not connected to the vertex v  Thus (2.5.6) becomes y<G)  = j  n  n  AeA(G) \  1  y f e  eeC(G)^ >  n  n  ieC(G)  H  (2-5.9)  AeA(G)  where q\ are the external momenta of the legs and kg are the internal momenta of the lines, 5G = 5 ( £  q)  * ((E k)  x  v  \AeA(G) / veV(G) V X —> v are the legs that are attached to the vertex v.  +J2<b)> \->v /  By integrating out the delta  functions except the one that conserves all the external momenta, i.e., <5(XaeA(G) Qx), one can reduce the number of integration (internal) variables to just |£p(G)| = | £ ( G ) | - | V ( G ) | + l ,  (2.5.10)  where Cp(G) is the set of loops of G . Thus we may rewrite (2.5.9) as  I  AeA(G) \*' > l  AeA(G)  iec (G) y > LI[  P  eec(G)  AEA(G)  where we rename some of the internal momenta as pi, the loop momenta, while the other internal momenta, the P^s, are linear combinations of the q\s and the p/'s. There is no canonical way of choosing the loop momenta. Usually, one can do the momentumlabeling of the lines of a Feynman graph G in the following way. First, one labels a selected line for each loop of G by a loop momentum. Then one labels the external legs by external momenta so that they are conserved (sum to zero). Finally, one can obtain the momenta for the rest of the other lines in terms of loop momenta and the external momenta by the conservation of momenta at the vertices of G.  Chapter 2.  Preliminaries  27  Example 2.5.1:  k  3  Figure 2.1: A n initial momentum labeling of G  Consider the Feynman graph G with an initial momentum labeling given in Figure 2.1 obtained from S (V ($ T  int  + <S> )( ) V ($ + $ )(* ) V ($ + $ ) ( x ) V ($ + $ ) ( x ) ) e  e  Xl  mt  e  2  e  int  int  3  4  (2.5.12)  where V {$) = e J  fyipdx.  int  (2.5.13)  From (2.5.7) and (2.5.8), in momentum space, Val(G) =  / n - | % II7S4 *G  n^(*i)  (2.5.14)  where 5  G  = S(  qi  + q ) % i -ki + k )6{k + k 2  A  5  3  k )S(k - k - k )6{k! + q - k ). A  2  3  5  2  2  (2.5.15)  Let pi = & be the loop momentum for the larger loop and p = k be the loop 4  2  5  momentum for the smaller loop. Integrating out the delta functions (except S(qi + q )), 2  we have h = qi+Pu  k =pi, 2  k =p2-pi. 3  (2.5.16)  Chapter 2.  Preliminaries  28  Figure 2.2: G with loop momenta pi and p  2  Using the loop momentum labeling of G given in Figure 2.2, we have d qi 4  Val(G) = J  ^ + ) JWft.fc) ^ f a W f c ) &  (-- ) 2  5  17  where (2.5.18)  j d pi tr 4  [Pi+m  1  .j  M I  V2 .  .  -7 l  p' +<£ l  1  +m  ^ + m  ^  + m-r  Here we suppress the spinor indices of the 7 matrices and sum over repeated indices.  Chapter 3  The G N Tree Expansion and Renormalization  3.1  Decomposition of fields  The G N tree expansion was first invented by Gallavotti & Nicolo [GN85a][GN85b] and was further extended by Rosen, Feldman, Hurd, and Wright [FHRW88] to general models involving massless particles and gauge symmetries . This approach is based on making scale decompositions of the fields or of the covariance: oo  oo  s=—oo  s=—oo  where C has length scale M~ ,M > 1 being a fixed scale parameter. s  S  In effect, this  decomposition resolves the U V and IR singularities of C. By successively integrating out the fields $ (from high to low), one obtains a tree expansion for du . The tree expansion s  ren  scheme allows one to choose counterterms without the problem of overlapping divergences or the usual combinatorial complexities. From the control of the tree expansion, one can show a renormalized graph is finite and obtain a sharp estimate on its size. To obtain a flow of effective potentials we can decompose each field $j into a sum of independent fields oo  *i=  E  (3-1-1)  h=—oo  where the $|'s are independent free fields with length scale M~ (momentum scale M ) , s  s  M > 1 being a fixed scale parameter. More precisely, that the <£?'s are independent of  29  Chapter 3.  The GN Tree Expansion and Renormalization  30  each other means that j $!$»cfc, = 0  (3.1.2)  0  for h ^ s. That $ | has length scale M  s  means the sliced propagator  <??.(*, 2/) = /  ^)dv  Q  satisfies lb ^C*(p)||oo  <  a  c  MV'-^HPl)'  (3.1.3a) (3.1.3b) (3.1.3c) (3.1.3d)  where P" \a\  Pi  1  =  •Pd n > ad  a>i +  3P =dpi 0  01  w  ••• Pd d  pd  w  (3.1.4)  h a, d  c is a positive constant and de is the dimension of the line :  dt =  d—2  ii £  is a Bose line (Ce is a Bose propagator)  d—1  if £  is a Fermi line (Ce is a Fermi propagator).  The effect of decomposition of the covariance is to resolve the singularities of  (3.1.5)  Ct(x,y).  Roughly speaking, a covariance satisfying (3.1.3) behaves like C ~  ciM e~ °\ ~ \  s  sdt  C2M  x  y  In particular as \x — y\ —> 0, C (x,y) ~\xe  y\~  dt  ~  ]T  (M ) e- '^- \. 8  dt  cM  v  (3.1.6)  Chapter 3. The GN Tree Expansion and Renormalization  3.2  31  F l o w of effective potentials  From the decomposition (3.1.1), a natural way to regularize the theory is to put a cut-off on the index s. For simplicity, here we will consider only massive theories where m / 0 so that s ranges only from 0 to +oo. Thus for a UV-cutoff U , the regularized version of (2.1.11) is  V where $  e  u e  m  =  u  (3.2.1)  \og(l[£ s=0  s  is in Schwartz space. With the removal of the local singularities in Ce (x,y),  we see that V ( $ ) has a well denned fps in A. For future convenience, we introduce the c/  e  e  following notation: s=0  s=0  U  £>k  —  n £s  (3.2.2)  etc...  s=k+\  Given a cutoff U we define a flow of effective potentials V , k = U, U — 1, u  k  •1 by the  following recursive relations (suppressing the superscript U): Vn = V  (3.2.3) (3.2.4)  Note that the flow represents successive Gaussian integration from k = U to k=0, and that  Vl! =  V. e  For each V -\, k  w  e  make the following cumulant expansion: V _ fc  2  =  log(£ (e *)) v  fc  OO  =  I  £ ^ > g ( 4 ( e ^ ) ) U  ]  -TPOO  1  Y- £Z( v ,---,v )  =  k  =i y-  P  * — » — '  k  ,  p arguments J  0  0  (3-2.5)  Chapter 3. The GN Tree Expansion and Renormalization  For the expectation -\ £\T(Xi,  • • •, X )  32  , we represent it by the following fork:  p  T £  s  ( X j , . . . , Xp)  =  By iterating (3.2.5), and using the above representation, a tree of forks is built up. For the expectation [£( ^£j(Xi,  • • • ,X )]  k  P  , we represent it by the following sub-tree:  =  X|  Xp  s  k  *  •*  fork  =  connected expectation £ ^  highest scale of output field  The succession of simple expectations £(k, ) s  setting  k)  function of input field VP  after £j(Xi,  • • •, X ) p  in the above means  = 0, for k < j < s after making the contractions at fork / . When p = 1, we  have the trivial fork  (j) s  which corresponds to  $J=0;  k<j<s  (3.2.6)  Chapter 3. The GN Tree Expansion and Renormalization  33  We would like to give a more detailed description of the trees in the above expansions of effective potentials and their corresponding values in terms of their associated Feynman graphs.  Some arboreal terminology is needed here for describing the trees and their  values. A tree r is a connected tree graph (no closed loop) consisting of a set of vertices connected by branches; the distinguished end-vertex at the bottom is called the root; each end vertex at the top is called an end point = e. Every branched-vertex having one branch going down and pf > 1 branches going up is called a fork. We denote T(T) as the set of forks and end points of r .  is partial ordered such that for f  J-(T)  u  f  2  G  ^F(T),  fi > h if fi lies above f . 2  Given a r, each / G J-(T) carries a "scale-label" denoted as Sf. Let s be an assignment of labeling J-(r) with scales such that s = {sf | / G J~(T)} belongs to the set H (r) r  = {s\r<s <U;s f  f  l  >  s  h  iff f  x  >  f} 2  where r is the root scale and - 1 < r < U. We denote 7 r ( / ) as the vertex immediately below f and g(f) as the set of forks or end points immediately above f. Given / G F(T), we define 77 to be the subtree of r such that the root of Tf is  7r(/).  For a given r and s, the value V(r,s) is defined inductively: V(e, s) (3.2.7)  v(x .r) fi  Chapter 3. The GN Tree Expansion and Renormalization  34  where we consider £ as a null tree consisting of only one vertex and /i, • • •, / are forks or p  end points immediately above f. Note that V(r/,s) depends only on the scales Sf, f > f. The operations £  in the above formula ensure that the highest scale of the fields in  > S f  each V(r,s) is Sf before contracting with others fields at f. Using the above notation, we can represent the iterated cumulant expansion of the effective potentials as : For -1 < r < U  K =E T  E  V(T,a)  (3.2.8)  S6-H (r) r  By applying the Gaussian integration rules given in (2.5.5), each V(r/,s) can be expressed as a sum of connected Feynman graphs  Q{TJ).  Hence, we can inductively  express V(r,s) in terms of Feynman graphs which can be constructed in terms of their subgraphs {G/}/ ^-( ) . Each associated labeled graph G (r) consists of a set of vertices s  e  T  V(G), a set of labeled lines C (G) and a set of external legs (half-lines) A ( G ) . Each S  vertex v G V(G) corresponds to a monomial input at the end points of r. Each external leg A G A(G7) corresponds to an uncontracted field $ (x\) <r  attached to the vertex x\. A  line labeled by s/ corresponds to a contraction at scale s/ and has value  C^ \ Sf  We may picture the formation of an associated labeled graph G(r) from a tree by the following scenario. Local monomials as vertices are fed in at the end points of r. The monomials slide down along the branches and meet at the immediate forks below the end points where subgraphs Gf are being formed by contraction of legs. We view t  these subgraphs as generalized vertices by shrinking all the lines of G/  r  Then the same  process of these vertices sliding to the next immediate forks below and contracting legs to form bigger subgraphs is repeated. The end product is a labeled graph with each line labeled with a scale given by the scale of the fork at which it was formed. We call  g j  = Gj mod {<?,!*(/) = /}  a reduced graph at / obtained by shrinking the lines of G f to a point.  Chapter 3. The GN Tree Expansion and Renormalization  35  Given a r and s, we denote the value of a labeled graph as G  s,u  Gs  U  = j  K^ (x)U {x ) u  e  e  and  dx  (3.2.9a)  where n  K°> (x) = ( - 1 ) ^ c A* u  G  C{ (x,, ) / ( l  %  (3.2.9b)  £e£(G)  is the kernel of G ' , s u  x = (xi, • • •, x ), and II (x ) = YlxeA(G) ® {xx)e  e  e  v  Here the combina-  toric factor CQ is the number of ways of assigning the legs from each vertex to its lines, v is the number of vertices of G , where x and ye are the coordinates of the endpoints of £. e  Expressing (3.2.9a-b) in momentum space using (2.5.7), as in (2.5.11), after integrating out the delta functions, (3.2.10a) AeA(G)  AeA(G)  AGA(G)  where q is the set of external momenta and the kernel K is an integral over loop momenta P/'s, K(q) =  c A" / G  J] C'wiPe) iec(G)  II 7 ^ , iec (G)  (3.2.10b)  P  where each Pe is a linear combination of loop and external momenta. Example 3.2.1:  Figure 3.3: A tree r with forks / i , f  2  and /  3  Consider the following graph G of Q E D corresponding to the tree r in Figure 3.3. A possible labeling for G compatible with r is in Figure 3.4.  The corresponding labeling  Chapter 3. The GN Tree Expansion and  Renormalization  36  Figure 3.4: A labeling for G compatible with r in momentum space is the same as the graph in Example 2.5.1. Suppose that the sliced covariances have the form D (p) = P (P)D(P), s  S (p)  S  s  B  = P (P)S(P)  (3.2.11)  S  F  where S(p), D(p) are given in (2.4.4), (2.4.7) respectively, and the p , X e {B,F}, x  are  the slicing functions for restricting the support of the covariance in momentum space. The corresponding Val(G )  (in momentum space) is  s  d Qi 7 & % i + ft) AfM)K^MiA2)Af {q ) 4  Val(G)  = J  2  (3.2.12)  2  where *W (gi>92) 2  =  c A  f dp  4  4  G  2  P2  [L^M  Sf i  l  tr  (3.2.13)  Sf  2  J dp  - (l-r/)^^)] 3  I  [pi+m'  ~J i 1 -J , p/ + i +i m '' p +m !  x  x  l  l  /  _/ -J ^-pj^ +, m "  Finally, we express V ( r , s) in terms the values of the associated graphs of r: V  where n(r) = fl/e^Cr) / V  a n  ^  = ^T)  (3.2.14)  E  d (3.2.9) becomes Vr = E  E  E  G'  (3.2.15)  Chapter 3. The GN Tree Expansion and Renormalization  37  s  s  Figure 3.5: Subgraphs of G and their corresponding reduced graphs By taking r = -1 in the above equation, we obtain an expansion for V in which each e  topological graph that contributes to V has been decomposed into a sum over labeled e  graphs associated with different trees and different labelings. The expansion is organized in such a way that each G can be naturally estimated and renormalized inductively in terms of its nested subgraphs Gj. For each G , the subgraphs G f are partially ordered s  S  by containment; thus there can be no overlapping divergence in Gf (See an example in Figure 3.5).  3.3  Bounds on associated Feynman graphs  In order to study the convergence problem, we need to establish bounds in terms of the UV-cutoff U on the value of the associated graphs in the expansion. These bounds will naturally explain why we need renormalization and why it works. Subgraphs contributing to the divergence can easily be identified and renormalized by doing "Taylor  Chapter 3. The GN Tree Expansion and Renormalization  38  subtractions". For a fixed order of perturbation, there are only finite number of r's in (3.2.8) and of Cr's in (3.2.15) contributing to the expansion. The convergence problem is to show that for a given r and G(r), G  -  U  Y °'  S U  seW-i(r)  = Y  [^  ,  C  /  ( x ) n  e  ( x )  (3.3.1)  cix  ?G«-I(T)  converges uniformly as the cutoff U —> oo. Here x stands for all the vertices of G , both the external vertices occurring in n (x) and the internal vertices occurring only in e  When working in coordinate space (x-space), we shall estimate G  S,U  K'. s u  using the "pinned  L -norm" of its kernel 1  11^111= /  K*' (x)  dx ---dx _ .  u  l  v  (3.3.2)  l  x =0  J  v  The reason why we pin down the last coordinate of K  in defining the norm is because,  s  by translational invariance of K , we can write s  G> S  U  =  j dx ---dx _ K°' (x ,---,x - ,0) v  1  1  where \&(x) = llAeA(G') ® ( '\)e x  G  U  where  j dy ^{x  u  1  =  Y  q = (qi,---,q ) n  v  1  x  + y, • • •, x v  X  + y, y)  When working in momentum space (p-space), [ % i + " - + 9n) / K*> (q, p)dp n ( q ) dq u  e  are the external momenta, and  the corresponding norm is the sup-norm on K ' (q s u  il  p)  p are  (3.3.3)  the loop momenta, and  without the delta function S(qi +  \-q ), n  11^1100 = sup / q  ^(q,p)  dp.  (3.3.4)  J  Note that a more general norm is needed when the IR end is also included in the scale decomposition (see §6 of [FHRW88]).  Chapter 3. The GN Tree Expansion and Renormalization  We can.bound  ||/^ ||i s  39  by using (3.1.3c-d) (suppressing the superscript U) and  ||A^||oo  by using (3.1.3a-b). Let M(G) be a minimal subset of lines of G where M.(G) connects all the vertices of G. In bounding  we first pull out the lines in C{G)/M.(G)  the integral by || • H ^ ' s and then bound each of the leftover lines in A4(G) by || •  in ||i's.  Thus by (3.1.3c-d),  < c  \\K%~  n  i i ^ i i o o n iic'iii  £eC(G)/M(G)  c n  <  ecM(G)  n  M  dtSl  eec(G)  Dually, in bounding  H-K^Hoo,  M~ .  (3.3.5)  dst  . i€M(G)  we select the lines in C(G)/M.(G)  to be loop momentum  lines each carrying a single (independent) loop momentum pi, i — 1, • • •, A, where A = | £ ( G ) | = L-v  + 1, L = \£(G)\,  v = \V(G)\, and we let the other v - 1 lines be the  lines in M.(G). We first bound the lines in Ai(G) by || • ||oo's, and then the loop lines by || • l l ^ s . Thus by (3.1.3a-b),  Halloo < c n ii^iioo teM(G)  <  c n  n  ii^iii  ieC(G)/M{G)  JJ  M  dlSl  £eC(G)  M~ . dst  (3.3.6)  £eM(G)  Using (3.3.5) in x-space or (3.3.6) in p-space, we can formulate a theorem which will enable us to identify divergent G's in terms of the U V degree of divergence of Gfs D(G )=  d -d(v(G )-l)  f  e  tec(G ) f  where dt is given by (3.1.5) and v(Gf) = |V(G/)|.  f  (3.3.7)  Chapter 3. The GN Tree Expansion and Renormalization  40  Theorem 3.1 (Dyson-Weinberg power counting theorem) \\K*\\ <  c  n  M ^f  =  c  JJ  M ^ f~ ^\  D  (3.3.8)  s  D{G  s  (3.3.9)  s  f€T(r)  where || • || stands for either the x-space norm in (3.3.2) or the p-space norm in (3.3.4) and for the lowest F, S ^ F ) = 0.  Proof: From (3.3.5) or (3.3.6), using (2.28) of [FHRW88], we have \\K \\ < c M ^ e a o ^ ' - E / e r o o r f M P / - ) K  (3.3.10)  1  By regrouping the sum in the above exponent, we get E  d  e  h  ieC(G)  m  -  dh (p -l) f  f€T(r)  =  f  .  f  E  h  /G^(r)  =  £  de-d(p -l) f  l:f(t)=f h D(g ). f  f  (3.3.11)  /e^(r)  To get the last equality, we use the fact that gj has pf generalized vertices and the sum zZe-.f(e)=f ranges over the lines of gf. To finish off the proof, we apply the following lemma with cif = hf,bf = D(gf) and / i the root of r.  Lemma 3.1 (Summation by Parts Lemma) Let cif and bf be functions  defined on the forks of a tree r. For fi a fixed fork or the root  ofr,  E / > / i  (f a  -  a  h) f b  =  E («/ - Mf)) f B  f>h  (-- ) 3  3  12  where Bf = 5Z/>/^/Proof: See the proof of Lemma 2.1 of [FHRW88]. From (3.3.6), it is easy to see that I^II-K^H converges uniformly in U if and only if each D(Gf) < 0. When D(Gf) > 0 we must renormalize the subgraph Gf.  Chapter 3.  3.4  The GN Tree Expansion  and  Renormalization  41  Renormalization by Taylor subtraction  The Dyson-Weinberg Power Counting Theorem shows that it is necessary to renormalize the tree expansion if there exists one subgraph G / ( r ) with D(Gf) > 0. The renormalization is defined inductively starting at the top of a tree. In our renormalization scheme, we define an external degree 5(Gf) to play the role of D(Gf) in the unrenormalized expansion. We introduce a localization operator L for picking out divergent parts and apply the renormalization operator R = l - L to remove divergent parts so that for each G, S(Gf)  < 0 at each fork / of its associated tree. These subtractions can be implemented  by adding counterterms SV  U  to the original potential V ($- ). u  to be of the same form as the local monomials of V (^- ^), u  u  U  Because 5V  is required  U  there will be extra graphs  coming from the renormalized tree expansion in which modified potentials V  u  +  5V s u,  are fed in at the top end points instead of V . u  Because we will present the renormalization tree expansion in the x-space context as described in §2 of [FHRW88], we would like to describe the L and R operations under the x-space context. Dual to the convenience in computing the coefficients of the R C C flow equations in p-space, we would also like to describe L in p-space.  But we will  not translate the x-space context of the renormalization tree expansion into its p-space counter-part here. In our renormalization scheme, when working in x-space, a G j (here we suppress the scale assignment s ) has the general form  where x = (xi, • • • ,.x ), and v  n/(x) = a  a i x  l  $ f ( x ) ••• 1  dz$Z {x ), k  v  k = s  7r(/  )  (3.4.1)  Chapter 3. The GN Tree Expansion and Renormalization  where o% = Yli=i ( )  > \\ a  =  £i=i i-> a  a  n  42  d the x-derivatives  arise from renormaliza-  tion operations we are about to define. Let Nf be the total number of derivatives on the legs of Gf. We define the dimension of II dimlJ = 52 dimQij + Nf,  (3.4.2)  where d-2 2  dimQj  if $j  is a Bose field  if $;  is a Fermi field.  (3.4.3)  (Note that dim<&i = d^/2 where £ is a line formed by contracting $j with another field of the same type). For simplicity, we assume that the couplings of the theory are dimensionless so that 5(Gf) = D (G ) d  - N (see (page 215) of [Ros90]).  f  f  We now define the localization operator L and renormalization operator R = I — L. Select one of  the localization vertex Xf of Gf and let x(tf)  = (Xiitf),  - • • ,X (t )), v  Xj(tf) =Xf+t (Xj  f  - Xf)  f  (3.4.4)  with 0 < tf < 1. The local parts of II/ and Gf are 0  if 6{G ) < 0 f  £11/(x) =  (3.4.5)  ^7/(*('/)) n  if S(G ) > 0 f  tt-Q  and LG  f  = J K (x) (LII/(x)) dx. f  (3.4.6)  The renormalization of a Gf is then defined as RG  f  = (1 - L)G . {  (3.4.7)  Chapter  3. The GN Tree Expansion  and  Renormalization  43  By Taylor's theorem if 6{G ) < 0  G  f  f  RGj ^7? Jo dtf(l - tfYf~  l  J K (x)(A  • d)^f Il/((x(t/))rfx  f  if  S(G )>0 f  (3.4.8)  where fif = 5(Gf) + 1 and A • d = Y,j{ j — f) • d - such that the S's do not act on the x  x  x  A's. The basic idea behind these schemes is that for 6(Gf) > 0, by applying the Taylor subtraction, additional /if derivatives are introduced on the legs of Gf.  Thus after  renormalization, the new external degree  (  5f  if L is applied at f  1  Sf — [if = —1  if R is applied at f  From the Taylor subtraction, we also obtain the factor A^ . f  .  (3.4.9)  According to our bounds  (3.1.3c) or (3.1.3d) on covariances and the general form of Kf, A^f produces a good factor M~^i f. s  But the legs with the rrif derivatives contracting at the next fork (note  these legs could contract at forks lower then ir(f)) typically would produce a bad factor M^f ^.  Thus we get a net improvement of M ^ ^ ^ ^ ) ^ in the power counting to  SiT  -  -  convert the bad (unrenormalized) power counting factor into a good one. We now describe the L operation in p-space. In p-space, Gj = f K (q) fi/(q) <5 dq f  where <5 = S(qi + q  q  h q ). Since when taking Fourier transform, the multiplication by v  Xi becomes —id ? and d ? becomes multiplication by ipf, it is not too hard to see that p  x  in localizing Gf in p-space, each resulting local term is obtained by inserting external momenta to 11/ and taking derivatives on the kernel Kf.  More precisely, by Taylor  Chapter 3. The GN Tree Expansion and Renormalization  44  expanding Kj(q) with respect to q, localizing G up to S(Gf) order means keeping up to 5(Gf) order terms in the Taylor expansion of Kf(q): LG  = j (LK (q))  f  for 8(Gf) > 0,  Il (q) £ dq  f  f  (3.4.10)  q  where LK (q)  =  f  K (0) + £ ft • 3 tf,(0) + f  (3.4.11)  ft  l<i<t;  E  6 ( G f )  ) ((9 -^)---(C-^;)K (O)) 1  /  1  )  |QI|+-+K|=<5(G/)  and the 9".' do not act on the qf Example 3.4.1:  Figure-3.6: The electron self-energy diagram Consider the electron self-energy diagram in Q E D with momentum labeling given in Figure 3.6. Ignoring scales and combinatoric factors, in coordinates-space, LG  =  LJ  =  jdxi  =  j K(0,x )  7p( ) Xl  2  u  2  ip{x ) dx dx 2  if)(xi) ^dx K(0,x ) )  x  (3.4.12)  2  ip{xi) + jdxi4>{x )  S  2  2  where K(xi,x )  (K(x x ))  2  dx j tp(xi)tp(xi) dxi + j x K(0,x ) 2  = D(xi,x )S(xi,x )  2  d ^{x )  >  2  2  dx J V>(xi) d »ip(xi)  i  2  2  (^jdx x%K (0,x ) j  L  2  x  x  dxi,  is the product of a Bose line and a Fermi line. In  2  p-space, using the momentum labeling in Figure 3.6 LG  =  J ~^{ ) (LK( )) qi  qi  xP(q ) S( 2  qi  + q ) d dq 2  Ql  2  (3.4.13)  1  Chapter 3. The GN Tree Expansion  and  Renormalization  =  J ~^{q) K(0) <4>(-q) dq + J$(q)  =  K(0)  45  {q"d >)k(0) 4>(-q) dq ql  J $(q)j>{-q) dq + d ,K(0) j i>(q)q^(-q)  dq,  q  where K(q) = J  D(p)S(q-p)dp.  We check that (3.4.13) is indeed (3.4.12) in p-space. Taking Fourier transform of (3.4.12) (ignoring the l/(27r) factors), we have 4  Jdx K{0,x ) 2  Ji>(x )tP(x )  2  =  J e ^ D( )S(p )  =  J D(p )S(-p )  lipi+p  x  2  Pl  dp  x  x  (3.4.14)  x  d dp  X2  Pl  dx  1  x  J e ^ ^ ^{q )^{q ) l  2  +q  x  J kliM-li)  dq dq dx  x  2  x  dq, = K(0)  2  x  J $(q)i>(-q) dq,  and Jx^K(0,x )  dx j $(x )d ^{x )  2  2  x  x  =  j x^ ^D( )S(p )d d  =  H ) j D(pi) (-d_ ^S(p ))  =  j D( )  dxi  x  j (d ,e^ )  +  Pl  2  Pl  d ,S(-p ) p  x  e*™${q )]>(q )  Xi  P2  x  p  Px  . (3.4.15)  S(  2  Pl  + p )d dp  dp j i(qM(-qi) x  x  2  Pl  2  dq dq dx  2  x  2  (-i) j $(qi)(-q$)$(q )6{qi  + q )dq dq  = d »K(0)  dq.  2  dq  x  x  q  j ^(q)q^(-q)  2  3.5 Renormalized tree expansion We briefly describe how to implement the Taylor subtractions given by the L and R operations given by (3.4.5)-(3.4.8) into the tree expansion to cancel the divergences so that each graph G obtained from the expansion has the corresponding G  u  uniformly  bounded in U . (For more details, see §2 of [FHRW88]). A renormalized tree r is a tree defined as in the unrenormalized expansion except that it has an extra label o~f = R or C at each fork f, and the set of scale assignments is modified as: V.r{s, B) = {s | s (/) < Sf <U 7r  if  Tf = R ; 0 < Sf < s^f)  if  Of = C }  (3.5.1)  x  2  Chapter 3.  The GN Tree Expansion  and  Renormalization  where, here, S^F) = r for the lowest fork F.  For a given a, the value V(r,a,s)  determined by the following rules. If X , • • •, X x  46  is  are monomials arising from V , or from  p  an R-fork or C-fork, then  =  X(s > k)  (1 - L) £J  ($<*), • •  -,X (&>j) p  (8<k)(-L)£] (x (^') --. X (^')) ,  X  1  1  1  P  *J=0; s<j<k  By carrying the above definition inductively down the trees, then the counterterms are  &V  U  =  E  E  n.t.T  ff:a =C  E F  V (T,C7,S) u  (3.5.2)  seri (r,S) c  where the sums are over non-trivial trees (n.t.r) (since there is no Wick ordering here, the n.t.r are just fork-less trees without any nodes in between the top and bottom end points), a's with op — C for the lowest fork F, and scales s in the set ric{r,a)  which  is defined as in (3.5.1) except that the root scale S^F) is taken to be U instead of -1. Clearly SV  U  in the  is a fps in the interaction coupling A whose coefficients are local polynomials  fields  .  We define the renormalized effective potential (3.5.3)  V U* )=[loge<u{e > »)] . e  v  r  +sv  0  and as in (3.2.8) we have  K e n = Vu($ ) e  +  E E  n.t.r  a:a =R F  E  seH(r,a)  V (T,0,S) U  (3.5.4)  Chapter 3.  The GN Tree Expansion  and  Renormalization  47  As in (3.2.15) V(r, a, s) can be expanded as a sum of graphs. Each unrenormalized graph G € Q(T) gives rise to a number of renormalized graphs G  ren  choice of m / in (3.4.5) and to how the derivatives S  tf  V {T,3,3) = where the value G ^ is similar to that of G s  s,u  r  n  act in (3.4.5) and (3.4.8). We write  £  U  6 Q(T, a) according to the  G%  (3.5.5)  in (3.2.15) except that R and - L operations  are applied to each subgraph as stipulated by a and there are resulting integrals over the interpolating parameters t = (i/)/e^( )T  Now it remains for us to show that the renormalized expansion (3.5.4) converges uniformly in the cutoff U for any finite order of perturbation. The proof of the convergence can be stated as a corollary to the following theorem.  Theorem 3.2 Let G  be a graph contributing to V (T,O,S)  s  u  ren  in (3.5.5) with r = -1.  Then C  s  < c JX M * ' * ' - M / ) ) / (  ren  (3.5.6)  where for the bottom fork s (F) = 0. n  Proof: See the proof of Theorem 2.5 of [FHRW88] In [FHRW88], the bound (3.5.6) is called the "Spring-Loaded Bound" because at an R-fork 5f < —1 and sj - s^f) > 0,  (3.5.7)  and at a C-fork 0 < S < d and s - s f  f  <f)  <0.  (3.5.8)  Thus, when we sum the bounds of (3.5.6) over the scales sp, we only have to consider the marginal C-forks for which <5/ = 0. Using the stiffness of exponential spring (exponential decay) of R-forks and non-marginal C-forks, and the fact that at each marginal C-fork,  Chapter 3.  The GN Tree Expansion  and  Renormalization  when summing over the scales from 0 to s  48  we get a factor of s , the sum of the bounds  F:  F  of (3.5.6) over the scales S f for a tree with K marginal C-forks above the bottom R-fork F is bounded by c  52  F  S  M  <  W  .  + 2) M  =CK\{r  K  5FT  (3.5.9)  sp>r (Note that the left hand side converges uniformly in U.) From the above, we have the following corollary which gives the convergence of the renormalized tree expansion: C o r o l l a r y 3.1  (UV-Renormalizability)  action and positive masses. Let G  be a graph contributing  S  malized tree expansion (3.5.4)-  Consider an EQFT  REN  with dimensionless to V(T,O,S)  inter-  in the renor-  Then  E  \ ren\ < c G  [G) 0  K\(r + 2) M K  5FT  (3.5.10)  se'H {T,cj) r  where c  0  is a constant independent  in T, and S  F  — S(G ) REN  of U, G, and r, K is the number of marginal  satisfies (3.5.7) or  C-forks  (3.5.8).  The G N tree expansion that we have described can also be applied with some modifications to models involving massless fields where the kernel of a graph in the unrenormalized expansion can have a non-integrable infrared singularity at oo as well as a UV-singularity at coinciding arguments (see §6 of [FHRW88]). As in the U V case, we decompose the covariance C with index s ranges from — oo to -1 to cover the IR end as well as the U V end. To regularize the theory, one imposes both a UV-cutoff U and an IR-cutoff I to the scales s. To set up the renormalized scheme, just as in the U V regime, one needs to introduce R- and C- operations at the forks of r which are slightly modified in the IR end (see (6.16) of [FHRW88]), to develop a renormalized tree expansion, and to prove the resulting graphs are finite, uniformly as U —> oo and as I —> — oo.  Chapter 3.  3.6  The GN Tree Expansion  and  Renormalization  49  Running coupling constant tree expansion  From the previous section, we have that  V f ($ ,A)  =  e  r  n  [  ] exists  \og£(e ^~ '^) v+sv  u+ipe  0  at least in perturbation theory and it can be expressed in terms of the tree expansion or of a flow of effective potentials Ken,IT,  Ken,t/-i, • • •, Ken,-i = Ken- Here we would  like  to modify the tree expansion so that, at each scale k, instead of explicitly having the counterterms as part of the effective potential, they are absorbed as part of the original potential by a shift in the coupling constants. Explicitly, we have Ken,*'  =  [log£  =  L[log£  =  - A  f  e  - V  (e -"' v  fc+1  f  c  f c + 1  +  fe+1  )]o  (eW )] + 1  + R[\ g£ (e —  *+*)]<>  v  0  0  R[log£ (e ->  k+1  (3.6.2)  «+i)]o  v  k+l  (3-6.1)  where R , L are the operations defined in (3.4.5) and (3.4.8) and = -52Xi  V = -\-V  V? = J dx Ci($^ )  jdxd  .  k  (3.6.3)  where the sum over i terms correspond to the potential given by the Lagrangian density in (1.2.5a-b). We call the expansion obtained by iterating (3.6.1) the "running coupling constants" (RCC) tree expansion. From the flow of the  Kent's  w  e  obtain a flow of the  coupling constants A . fc  We call A the coupling at scale k, \ fe  pling.  Ken,*:  is a function of $ -  fc  u  the bare coupling, and A  and the coupling constants  the physical cou-  - 1  X . Thus the u  A , • • •, f c  (UV)  renormalizability of a theory (in perturbation theory) is equivalent to this: for a fixed physical interaction coupling constant A, there exists Ken Af  as a fps in  A,  a s  a  fP  s m  $  e  a n  d A, and each  i.e. oo  V (* ^ ) U  r en  e  U  =  (3-6-4) ra=l  ce  Chapter 3.  The GN Tree Expansion  and  Renormalization  50  = IX»*> B  Af  (3-6.5)  n  ( in particular, \  u  = A + a f A + • • • ) such that l i m ^ o o K% exists. We remark that 2  as U —> oo the coefficients a f diverge. On the other hand the actual R C C ' s Af may n  converge as U —> oo. This is a key advantage of the R C C tree expansion. In terms of the tree expansion, we have the following proposition: Proposition 3.1 V , ren  k  = -A* • V* + £  where J2 .t. r 'means sum over nontrivial  (^  £  V  ( -6-6)  T  3  trees and  n  = {$ I Af) < f < s  s  u  ;k =s }  .  n{F)  (3.6.7)  V(T, S) is defined inductively, if k = s ( ) then n  £  V(e, s) = -X  k  •V ,  (3.6.8)  K  otherwise V(r , f  s) = ^[R£j  f  (£ V(r ,s), >Sf  fl  • • •, £ V(r , >Sf  fp  5)) ]  0  (3.6.9)  where fi G £>(/)• Thus, in the R C C tree expansion, each fork has a R label and no C label. Moreover, the couplings attached to the local monomials that are fed in at the end points of a tree are indexed by scales associated to the fork immediately below the end  Chapter 3. The GN Tree Expansion and  Renormalization  51  points. Example 3.7.1: Using the same graph and tree from Example 3.2.1 The corresponding Val(Gf ) enr  d qi  is  4  Val(G)  = j A%( )  [RK, ( ,q )}  gi  lll2  qi  A^(q )6(g  2  2  - q) U  1  (3.6.10)  2  where K^A ,q ) qi  = <* (\[ ) W) h  2  2  f dtp*  2  S  PF  tr  -J  ./xi  [p + m 2  (px)  = .( d J  vv  x  [  R  K  s  h  {  p  i  )  ]  l/ + m 2  ~ oT {jti\  P Pl  , _ P l _  r  m  2  -\ \L {p ) Pi  4  /3  PF p! + 4 i +  Yj—77jj —V-, + m  w  (3.6.11)  2  From (3.6.1)-(3.6.2), we have -\ -V k  L[\og£ (e ^ +i)}  =  k  v  k+1  k  0  (3.6.12)  = -A  f c + 1  •V + k  This gives a system of discrete flow equations : \  k  =  A  f c + 1  + fps involving in \  =  A  f e + 1  + f3 (\ , X , • • •) k  k+1  k + 1  ,\  k + 2  - - - in terms of deg > 2,  (3.6.13) (3.6.14)  k+2  The flow equation given by (3.6.14) is the analog of the R G flow equation of (2.3.2) and (3 is the analog of the /3-function, i.e. k  ^  = /3(A)  —»  A  f c + 1  -X  k  = f3 . k  Chapter  3.  The GN Tree Expansion  and  Renormalization  52  Here, (3 is only defined perturbatively. As in the continuous R G equation, A = 0 is a h  fixed point. The theory is U V A F if X  u  ->• 0 as U -» oo for an initial fixed A . The -1  theory is I R A F , if X diverges as U —> oo; roughly speaking, this corresponds to X —> 0 u  as s —¥ —oo.  s  Chapter 4  Decomposition Of Field V i a Block-Spin Method  4.1  Block-spin transformation  In this chapter, we would like to discuss the block-spin method for constructing a field decomposition which may be employed by the G N expansion. In order to make good use of the block-spin decomposition, one needs to establish that the resulting n  slice  th  covariance r  n  has the usual exponential decay (see (4.2.4) below). Gawedzki and K u -  piainen have demonstrated that the block-spin method produces the required bound in scalar bosonic theories [GK86]. In their work of showing the required bound, they applied the contour shift method which requires that there exist an e so that when f  n  (f  n  in momentum space) is analytically extended from its real domain to a strip of width e, it is bounded uniformly in n over the extended domain. Here we attempted a similar approach in trying to establish the required bound in a decomposition for a Fermi covariance. However, our attempt has failed because f „ fails to have a bound uniform in n on any e strip above its real domain. Let us first present the block-spin lattice decomposition of a field. Given a E Q F T , we may consider it as a limit of a lattice cutoff theory with an action S defined on a hypera  cubic lattice A = (aZ)  d  where a is the spacing of A. For example, in scalar theories, a  typical action has the following form Sa  = a E c k v ^ ) ) + \m^(x) 2  d  xeA  1  1  53  + A P(<f>(x))  (4.1.1)  Chapter  4. Decomposition  Of Field Via Block-Spin  Method  54  where P is a polynomial of degree > 3 and (p(x + ae^) - (f)(x)  (4.1.2)  a with  a unit vector in the p direction. In the limit a —> 0, we recover the continuous  action S = J dx lid^ix))  2  + -mcj> { ) + A l  2  X  (4.1.3)  P^x))  The idea of the lattice block-spin decomposition is to decompose a lattice field into a "smeared out" local average and a local fluctuation. By rescaling and iterating the same process, we obtain the desired decomposition of the field and its corresponding covariance. Let us introduce a local averaging operation for a lattice field. Let L be an odd integer and define A = (aL Z) . n  n  We partition A into hyper-cubic blocks of lattice such that  d  n  each block has volume L x (aL ) d  n d  and the center of each block belongs to A . n  Example:  ? I ?  ?  T  Figure 4.7: A 3x3 block of sub-lattice of  Consider A  n + 1  (aL Z) n  2  as a sub-lattice of A", let T  71  = {</>" | <j) : A ->IR}. n  n  (4.1:4)  The block-spin average transformation (4.1.5)  Chapter  4. Decomposition  Of Field Via Block-Spin  Method  55  is denned by (Qn4 )(x)  = c  n  £  4> (y) = 4> (x) n  (4.1.6)  n+1  \y-x\< f !  for all x G A  where c is a positive constant to be chosen below (in (4.1.15)).  n + 1  It is convenient to work on the fixed unit lattice Z . For now, we set a = 1 (we will d  reinstate a at later stage) and introduce the rescaling transformation S :F* 1  ->• T  1  (4.1.7)  n  defined by (S4>)(x) = Ll-^(Lx)  for all x G A " . The canonical factor L^  (4.1.8)  is chosen so that / ( c ^ ) dx is invariant under  1  2  the rescaling operation. Intuitively, we want the </> to decompose as a sum of its block-spin average 4> n  n+1  and  a local fluctuation p , i.e., n  4> = 4> n  n+1  Next we define 4> = S (j) . Applying S~ n  n  n  n  cf) = ( S - V n  where p  n  = S~ p . n  n+1  ) ( x ) + p {x) n  A then S~ cj) l  (4.1.10)  At a second glance, we see that (4.1.9) (as well as (4.1.10)) is  n  1  1  (4.1.9)  to the above equation yields  mathematically unsound because </»" and S~ (j) if x  + p.  n+l  n+1  do not belong to the same space, i.e.,  does not make any sense.  Gawedzki and Kupiainen suggest the following modification: introduce a "smearing" operator A : F° —> F° so that given <> / G F° and its corresponding covariance C,  A = CQ'iQCQY , 1  (4.1.11)  Chapter 4. Decomposition  Of Field Via Block-Spin  Method  56  where Q = SQ. Using A, we write (j) = $ + Q where $  = AQ<f>  (4.1.12a)  = </,_$ = (/_  e  AQ)4>  (4.1.12b)  so that <®,Q>dpc=Q where <,>d^  ^  s t  c  n  e  m n  (4.1.13)  e r product with respect to the Gaussian measure du. - Notice C  that the above equation implies g and $ are independent with respect to the covariance C  Similarly, we can apply (4.1.12a-b) to each 4> . Notice that for 0 6 JF , n  Q:T°-*  0  T°.  (4.1.14)  Explicitly, {Q<f>){x) = L-i-  4>{Lx + y).  1  (4.1.15)  Furthermore, we have the following relations ^n+l  =  n+l^n+l  S  +  n =  S  Q S'  l  ((f> ).  n  (4.1.16)  n  It is easy to see that SQ S~ n  1  =Q  and  n  Q ~ S = SQ . n  l  n  (4.1.17)  Hence <j) = SQ<f> = Q4> . n+1  n  n  (4.1.18)  Iterating the decomposition (4.1.12a-b), we get the following independent decomposition if  = A Qcj) + Q n  n  n  (4.1.19a)  Chapter 4. Decomposition  Of Field Via Block-Spin  A  n  g  n  Method  57  =  GnQ'iQGnQ')- = 0^0-1,  (4.1.19b)  =  (I-A Q)4>  (4.1.19c)  1  n  n  where  G =< <j> , <j> > , =< Q <l>, Q <t> n  n  n  n  n  >»  d c  d  = Q C(QT  (4.1.19d)  n  c  is the covariance for </>. As a result, we have the decomposition, n  <f>=T.X  + *  n  N  +  1  ,  (4.1.20a)  n=0 where =  n X  $  = A A ---A . ,  N  n  0  1  n l6  = A A --.A ^ n  0  1  (4.1.20b)  n  By taking N —> oo, we have the desired decomposition oo  0=Ex"  (4.1.21)  n=0  where the sliced fields are independent of each other as we now show. Let us first define the covariance C  n  =<  >» d  = C(Q ) G~ Q C. t n  c  1  (4.1.22)  n  From the facts that QA  n  AoA^-.An.,  =  I,  (4.1.23a)  = CiQ^G' , 1  (4.1.23b)  and (4.1.19d), we have <§ \x >d» n+  n  c  =  <<S> \$ > , -<<S> \$ > » n+  n+  n+1  d c  = CiQ^G-^Q =  n  C \ n+  —C \ n+  d  c  < 0", p > , G- Q C l  d c  = 0.  n  - C  n + l  Chapter  4. Decomposition  Of Field Via Block-Spin  Method  58  Thus, we may decompose C according to the decomposition (4.1.20a): f  =  n  C — C i = covariance of x n  (4.1.24a)  n+  N  J2f  C =  n  n=0  4.2  +C .  (4.1.24b)  N+1  F 's bound in scalar theory n  From the block-spin decomposition of a lattice field, we obtain a corresponding decomposition of the covariance C given by (4.1.24b). A slice covariance r f  = C(Q*) [G- - Q'G-^Q] B  B  n  Q C,  1  has the form (4.2.1)  n  where G  =  Q S{Q )  Q${x)  =  L~(»)/  n  n  (4.2.2a)  t n  $(Lx + y),  2  x£A°  (4.2.2b)  eB°  y  A  =  (aL Z)  B°  =  {y e A I  n  n  (4.2.2c)  d  < L/2}.  0  (4.2.2d)  We would like to inquire about the exponential decay bound of T . In the case of scalar n  bosonic fields, [GK86] has shown that the r n has the following bound.  Theorem 4.1  Let t  n  be a slice covariance given by (4-2.1) with  C{x,y)  = =  (~) \2irJ  [ dpw{p)- e - JM  d  2  a  =  iv[  +  m  x  y)  (4.2.3a)  a  ^l-cos(op^) »  M  l  2  a  7T  7T  a' o  (4.2.3c)  Chapter 4. Decomposition  Of Field Via Block-Spin  Method  59  Then \T (x,y)\  <  n  a\L fn  C l  e~ ^,  d  (4.2.4)  c  where c c are constants independent of a and n. l5  2  Here, I give only an outline of the proof. The reader can refer to [GK86] for details. Their strategy is first to rescale r „ as T = k  n  n  f  (&')" = &*C(Q*) [G- - Q'G^Q] n  n  n  (4.2.5a)  x e A" .  (4.2.5b)  1  Q C(&T>  where &$(x) = L -  ( d i m  *)/ $(Lx), 2  1  Then T is factored into n  r „ = A X{A y, n  (4.2.6a)  n  where An  =  & C(Q*) G^  *  =  G  n  n  n  -  (4.2.6b)  1  (4.2.6c)  GnQ^-^QG,,.  They then establish exponential decay for An and X separately with \A (x,y)\<c e- ^  \X{x,y)\<  £  n  3  c e~ ^ £  4  (4.2.7)  where c , c are positive constants independent of a and n. The desired bound (4.2.4) is 3  4  then obtained by combining the above bounds and rescaling back to the unit lattice. To establish the exponential decays, one first converts An and X to their corresponding pspace expressions using the following p-space representations of Q and Gf. In momentum space, from (4.2.2b) and for $ a scalar field, (Q»(p)  =  L"(")/  2  £  Y. <f>( + vY~ Lx  iVV  (4-2-8)  Chapter 4. Decomposition  Of Field  L - ( « ) /  =  Via Block-Spin  52 l ( ( )  52  2  Method  Lx  60  + y)  e i i { L x + y )  ^  L-^)i 52m^s { ),  =  2  L  P  where  "  = 1  li/^Kf  n  e  — z ^ - j —  =n  1  L  -1  —m—-EE  =  n  - d t -  (4-2.9)  Therefore, (Q>)(p) = L " ( " ) /  2 SL  ( p ) 0(|).  (4.2.10)  From (4.2.10), /  JM„  dpf(p)Qg(p)  =  L"(")/  2  /  dpf(p)s (p)g(p/L)  (4.2.11)  L  JM  a  =  L^  d  i  m  ^  1  f  2  dk f{Lk)  =  L-(")/  2  g(k)  s (Lk) L  LM  J  a  52 [ dk f{V) 8 {k>) g{k),  jeJ  L  1  where J"  if  =  | b'„|<L"/2},  { j e Z d,  =  *,veA-.  (4.2.12a)  (4.2.12b)  Therefore,  (Q7)>)  = L -  (  "  )  /  E/(^'KM-  2  (4.2.13)  The idea behind this strategy is to apply the Contour Shift Lemma and display a cancellation of poles for X(p).  It is easy to check from (4.2.3b) that in p-space, the  Fourier transform of (4.2.2a) G  n  has a pole at p = 0. It is then easy to see that both  terms in X = G  - G Q G~] QG t  n  n  rl  n  (4.2.14)  Chapter  4.  Decomposition  Of Field Via Block-Spin  61  Method  have a pole at p = 0. Furthermore, X also has poles at p^ = (±27r/aL, • • •,  ±2n/aL),  coming from Q* operating on QG in p-space. But these poles are actually removable. n  At p = 0, there is cancellation of poles between the G and G Q G~\ QG .  A n d at  t  n  P(i) = (±2ir/aL,  • • •, ±2ir/aL),  n  the zero of G Q G~l QG l  n  coming from the G~l  t  n  l  n  x  would  remove the corresponding pole coming from the G . This allows one to display uniform n  bounds for A , X in M x i[0,e/a}. The exponential decay can then be obtained by the n  a  Contour Shift Lemma.  Lemma 4.1 (Contour Shift Lemma) Suppose F(p,x) P f l  in an open set containing  \Re \ Pll  < | , \Im \  is analytic  < e/a.  PlJi  Furthermore,  \F{p,x)\<c, uniformly  in each argument  (4.2.15)  in p, x, where c is a positive constant, and it is periodic in the real component  of each p,  F(Pi, • • • , - - + iq„ •••) = F(Pu • • • , - + iq„, • • •)• CL  CL  (4.2.16)  Then F  {  X  )  =  {^K)  1  L  (P> ) ' P  P  x  (-- )  eW Xd  4  2  17  satisfies  \F{x)\<ce~^.  (4.2.18)  Proof: Choose ji such that  > \x\/d. Suppose Wlog, we shift the p^ contour in (4.2.11) Imp  lx  = e  - - < R e a  P  l  l  < - . a  (4.2.19)  Using the analyticity and periodicity of F(x), we have 1^)1  =  \F(p + Alt  <  ce'^.  iee ,x)\\e*-*- *>\dp e  lt  JM  a  Q.E.D.  (4.2.20)  Chapter 4. Decomposition  4.3  Of Field Via Block-Spin  Method  62  T of lattice Fermion n  In this section, we would like to carry over a similar analysis for finding an exponential decay bound for T as done in the scalar bosonic case. There are many variations of the n  fermonic propagator S(x,y).  We take S(x,y) = -  a,  to be the following :  [ilvsm(ap )  + 2sin (ap /2)| + m 2  u  J/  (4.3.1)  v  where = {£f  S(x,y) {1(1,1^}  = 2 5 ^ , and M  a  j  dp e**-v\S(p)  M  is defined in (4.2.3c).  (4.3.2)  The term 2sin (ap^/2) is added to 2  eliminate the extra singularities of S at p = ( ± 7 r / a , • • •, ± 7 r / a ) . It vanishes in the limit a —> 0 (pointwise in p) . We wish to establish the exponential decay  r  <c a  {L ) - e- ^ n  5  l  d  (4.3.3)  c  uniformly in n where c , c are positive constants independent of n,a,x,y. 5  6  As in the  scalar case, we want to establish the bound by the contour shift strategy . A parallel investigation on the exponential decay property of f  n  was done by [BOS91]  where they used an exponential weight block-spin transformation defined by the a —> oo limit of (T p)(Z exp  X)=f  dH^P e^-W'X-W  p(4>,#  (4.3.4)  Using the Renormalization Group Transformation T , by assuming that the exponenexp  tial weight term would act almost like a delta function, they were able to establish a decomposition for S(p) where the corresponding slice covariance has the desired exponential decay (4.3.3) independent of n but dependent on the weight parameter a. In their analysis, r  is composed of terms involving factors of ( S " ) ^ ) where 1  n  S  n  = a' ! 1  - Q S{Q ) n  t n  -1  (4.3.5)  Chapter 4.  Decomposition  Of Field Via Block-Spin  (see (3.11) of [B0S91]) is the n  th  Method  63  transformed covariance (scaled by &) and ot — ot\_^_ n  (see Lemma II.l of [B0S91]). Since l i m ^ o o Q S{Q ) {±ir/a, n  ±ix/a)  t n  n  = 0, the bound  (4.3.3) is no longer uniform in n in the limit a —>• oo. For finite a , what prevents  S (p) n  n  from having zeros (so that T has the bound (4.3.3)) is the a~ I l  n  term which vanishes in  the limit a —» oo. Because x , "0, Xi 4>  independent variables, (x~Q'4 iX~Q' P)  a r e  }  Hence, it is in question that l i m ^ o o T a  x  would provide the decay to force x = Q4>,  Q  is ° t definite in sign. n  = T since it is not clear why the exponential  exp  weight e (*~Q^' ~ ^  l  d X = Qi -  a n  This  3  tempted us to probe the matter more directly with block-spin transformation defined by the delta function. From (4.2.5a), (4.2.10), (4.2.13) and , we have T (x,y) =  j  n  dp e ^T (p,x,y),  (4.3.6)  ip  M  n  where =  t (p,x,y) n  H (p,x)G-\p)Hi(p,y)-\H (p,x)L-^ n  n  E leJ  e-W^SLfr  ,  + 27rJ/aL) #(p + £/L, y)  (4.3.7)  1  where H (p,x) n  G {p) n  =  (L y -  =  (L )-  n  n  d  n  J"  1  2  2 d _ 1  s  s^tf)  Stf),  (4.3.8a)  Sip )  (4.3.8b)  1  sm(L ap„/2) n  =  11 —7-,  P( )  =  L p — 2irj/a, n  jx,a  52 i V )  SL*(P) n  and J ,  d  T^-,  4.3.8c  for some j € J"™ so that |p( )| < ir/a, n  (4.3.8d)  pP are defined in (4.2.12a-b). In applying the contour shift lemma to T , perin  odicity for T (p, x, y) is obvious, we only need to verify the boundedness of T uniformly n  n  Chapter 4. Decomposition  Of Field Via Block-Spin  Method  64  in n. It turns out that even though that there is a cancellation of poles at ap = 0 and P{i) = 0 as in the bosonic case, our analysis also fails to yield the desired exponential  a  decay bounds uniformly in n. The source of the problem is the fact that the lattice Dirac operator  -Yl^Hw»)  G{p) =  a  (4-3.9)  ^  in momentum space vanishes not just at ap = 0 but also at ap = ( ± 7 r , • • •, ± 7 r ) . B y rationalizing S^ip)  — G(p) + H(p), where H{p) = - £ s i n ( a p / 2 ) + m,  (4.3.10)  2  M  fi  a  we have b  {  P  )  ~  Dip)  Dip)  +  ( 4  - 3  1 1 }  where * denotes complex conjugate and Dip) = S-\S- Yip).  (4.3.12)  l  We now describe briefly the difficulty which we shall investigate in the remainder of this chapter.  Because the block-spin transformation in Fourier space retains  odd, even symmetries and periodicity, and the block-spin transformed H(p)/D(p) Q(L~  n  + aL m), n  As a result, T  this suggests that G = iQ^SQ  71  n  is  has zeros near ap = ( ± 7 r , • • •, ± 7 r ) .  can not be bounded uniformly in n since we have yet found a way to  n  show that there are cancellations of poles coming from the zeros of G . Thus we cannot n  perform the Contour Shift Lemma to T to obtain the desire exponential decay bound n  on the sliced covariances. In Section 4.4, we provide the expressions for the components of T . n  In Section 4.5, we show the cancellations of poles at ap = 0 and ap^  = 0 by  rationalizing the RHS of (4.3.7). In Section 4.6, we show the existence and locations of the zeros of G (p). n  Chapter 4. Decomposition  4.4  Of Field Via Block-Spin  Method  65  C o m p o n e n t s of T  n  In analyzing the boundedness of T , it is convenient to write H n  n  and G of (4.3.6) as a n  sum of components. First, by rationalizing S{jp>) using (4.3.1), we get  stf) = { [ s r W } " [STV) = a ^Xl^ 1  1  2  (4A1)  where  [S*}-\p)  =  - Y  [-i-YvSm(ap„)  + 2 sin (a^/2)] + m  Dn{p)  =  5 > i n ( a p „ ) + [am + 2 £ s i n ( a p „ / 2 ) ]  (4.4.2a)  2  2  2  (4.4.2b)  2  Substituting (4.4.1) back into (4.3.8a) and (4.3.8a), we write H (p,x)  =  n  a Y.- lvK , (p,x)+  T (p,x),  i  l >n  (4.4.3a)  n  V  = o ^ - i ^ W + W ^ ) ,  G (p) n  (4.4.3b)  where  Kv, (p,x) n  T (p,x) n  =  (L )~ ~  =  (L )~ ~  n  n  d  d  1  l  Y  e^*'*s ntf)*^Q  ,  e^5 .(ry)  f.(  L  Y  L  2 E  -  S i n 2 ( a  jej  =  2 )  +  a  m  ,  ( .4.4b) 4  D {p>)  n  WJp)  (4.4.4a)  n  £ ^(Y^^'f^'"" ,  (4.4.4d)  n{P )  jeJ  L>  n  J  and p ' is defined in (4.3.8d). 7  In analyzing the boundedness of | f | in n  Se = {(Pi,-• • ,Pn +ik,-•  • ,Pd) I PV€[-IT,IT],  1 <u <d,\k\<e}  ,  (4.4.5)  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  66  by the symmetries of TZu,n, 7~n, V„>ri, Wn, it suffices to restrict our analysis of Tn to the domain S'e/a =  | Re >o}.  {p£S  (4.4.6)  Pl/  e/a  Furthermore, Wlog, we choose the <i -component of p for the continuation into the t/l  complex plane, i.e., we take p = d in  S/. £ a  We would like to introduce a couple of lemmas for showing that \(L )~ SLn(p)\ n  bounded uniformly in S' /  e a  is  d  and for finding bounds for the components of TZ , T , V UtU  n  ,  u>n  Moreover, from the lemmas, we can easily see that TZ^n, T , V^ , W „ can only be  W.  n  n  n  singular at p = 0 where the source of singularity comes from the term  Lemma 4.2 Let z = x + iy, x  6  1/D (p°). n  [0, TT], \y\ < e, and e is sufficiently small. Then  (1-)  C12 \z\ < | sin ^| < c i \z\ if x < n/2 3  C12 \n — z\ < | sin z\ < c  13  \n — z\  if x > n/2  (4.4.7a) (4.4.7b)  Cu x < Re sin z < C i x  if x < n/2  (4.4.7c)  cu (7r — x) < Re sinz < c i (n — x)  if x > n/2  (4.4.7d)  5  5  (2.) cie  ci  8  k-  z\/2 <  (n -x)/2<  I cos(z/2)| <  Re cos(z/2)  <  cn \n - z\/2 c  19  (TT - x)/2  (4.4.8a) (4.4.8b)  (3.) for j an integer and aq — z, C20  -  sin(aq/2)  n  1+  where the Ci's are positive  sin(aqi/2)  constants.  <  C1 2  (4.4.9)  Chapter 4. Decomposition  Of Field Via Block-Spin  Method  67  Proof: We obtain ( 4.4.7a-d) and ( 4.4.8a-b) from the following identities. For x,y real, s'm(x + iy)  = sin(x) cosh(y) + i cos(x) sinh(y)  | sin(x + iy)\  =  sin (:r) + sinh (y)  =  cos(x) cosh(y) — i sin(x) sinh(y)  =  cos (x) + cosh (y),  2  cos(x + iy) | cos(x + iy)\  2  2  2  2  2  and for 0 < x < n/2, —x < n  \  — — X ]  2  <  J ~  sinx  < x  cos x  <  2 (n  X  —  ~ n \2  As for (4.4.9), from the above, we have 1 sin(ag/2)| <^ 2  2  + y)  (4.4.10)  2  and x±2jn\ 2L  2  sin(a V2)| > c 2  9  22  n  J  ( y \2L  n  (4.4.11)  Thus r  _  n  sin(ag) sin(aqi)  \aq\ n \aq/n + 2j\ ~  The upper bound follows from a similar argument as well. Q.E.D.  1 +  (4.4.12)  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  68  L e m m a 4.3 For j e J , j ^ 0 and 0 < Re ap < ir, define k(j) e J n  n  d  j  if Re apl" < TT/2  N-j„  if Re apl" > TT/2  v  then for p  by  (4.4.13)  real,  d  (4.4.14)  L D (pi)>c MJ)\ , Zn  2  n  2  and for complex p , d  L D {p>)\ n  <  \L D (tf)\  >  ln  2n  n  c  Ykl(j) + (L- \j\ n  25  2  (4.4.15a)  + aL my n  c \k(j)\ ,  (4.4.15b)  2  26  where the Ci's are positive constants.  Proof: (4.4.14) and (4.4.15a) follow from (1.)  of lemma 4.2. Using these two results, we get  (4.4.15b) by the following result on the reciprocal of an analytic function: if f(z) analytic in \z — z \ < R, sup| _ |<# f(z) < M and f(z ) 0  \z - z \ < r = {\m\/M)RjA  z  2o  0  = m, then l/f{z)  is  is analytic in  and \l/f{z)\  < 2/\m\ for \z - z \ < r. For each fixed j ^ 0,  p„, v ^ d, we apply this result to f{p )  = D {p>) with R = u/2a center at a p with p  0  d  0  n  d  real. For p real, applying (4.4.14) d  m = f(p ) d  Q.E.D.  = L D(p>)>c 2n  \k(j)\ . 2  24  d  Chapter 4.  4.5  Decomposition  Of Field Via Block-Spin  Method  69  C a n c e l l a t i o n of poles  We demonstrate the cancellation of poles arises from the singularities of H (p,x) n  G (p,x).  From the components of H (p,x),  n  G (p,x)  n  in (4.4.4a-d) and Lemma 4.3, we  n  see that the only possible singularities of H (p,x)  and G (p,x)  n  zeros of D (p°).  and  n  would come from the  To find these zeros, we write  n  Dntf)  = G ^ [ s i n ( a ^ / 2 ) + (Q , ) ], 2  (4.5.1)  2  n 2  where  4(1 + am + 29n) 9  n  =  £sin (ap>/2). 2  Let a d/2 = q + ik, then P  sm (ap /2L )  + (e£, )  =  [sm(ap /2L )  + i& ]  =  [sm{q/L )  cosh(k/L )  +  [sm(q/L )  cosh(k/L )  + i ( 9 ^ - cos(q/L )  2  2  n  2  d  n  d  n  n  Thus D (p°) n  [sin(ap /2L )  n  n  -  n  nfi  d  i® ] nft  i(e£ + cos(q/L ) n  >2  smh(k/L )} n  smh(k/L )}.  n  n  2  (4.5.3)  = 0 when q = 0,  sinh(k/L )  = ±e  n  n t 2  (4.5.4)  Consequently, both H (p, x), G (p) have first order poles at the point p satisfying (4.5.4). n  n  At a closer look on F (p,x,y),  these singularities cancel each other out. Let  n  E'7,V„, (p)4-W (p) ,  (4.5.5a)  =  G* ( )G (p)=G (p)G (p),  (4.5.5b)  =  {kej  G* (p) = n  f (p) n  J?  n  n P  n  n  n  n  n  | MO}  .  (4.5.5c)  Chapter 4. Decomposition  Rationalizing t (p,x,y)  Of Field Via Block-Spin  Method  70  of (4.3.7), we get  n  (4.5.6) =  H (p,x)y^G:(p)H (p,y) n  H (p,x)L'^- s (p) l  n  n  L  /n+l(P(l))  where p(i) is defined in (4.3.8d).  Taking out the / = 0 term from the sum Y^iej  1  a n  d  combining it with the first term, we have F (p,x,y)  (4.5.7a)  n  lri (p,x) n  fn i( )  [UP)  L[P)  +  E  H (p,x)L-™-hdp)^HH n  =  /n+l(P(l))  H (p,y) n  Pw  e-  W a L )  i  2rd ^.(P+^)^(p+^,2/) aL  leJ  Ql,n(p) ~ Q2,n(p)  where  Q\,n{P)  H (p, x)G* {p) n  ZieJl  n  l(P  s  + gf)<?n(p + S)) G Mi))Hi(p,  V)  n+  fn(p)fn+l(P(l)) (4.5.7b) 2d-i„  /^ n+ib(i)) G  v - ^  _.vo,» M., n  2nL^,  ,  2nl (4.5.7c)  We now show that we can removed the poles of H (p,x), n  We separate out the j = 0 term from the j-sums in H (p,x) n  G (p,x)  in Q i , n and Q2,n-  n  and G (p,x), n  so that we  can extract the terms responsible for the singularities. Let us write G (p) n  H (p,x) n  0»(PO  (-- )  = MP'O+EW),  (4.5.8b)  =  g (p°) n  +E  4  5  8A  Chapter 4. Decomposition  Of Field Via Block-Spin  Method  71  where gntf)  =  (L )~ ~ s (p )S(p ),  h {pP)  =  (L )~  n  n  2d  1  2  j  (4.5.9a)  j  Ln  N  D _  V  2 7 R  -  7 3 : /  '  s n(p )S(p ). j  A  (4.5.9b)  j  L  Then  UP)  = gnWniP ) + E 0  9ntf)9n&)  + gntfKtf)  E  +  E  9ntf)  9ntf)  (4.5.10) =  I L - ^ S K P  0  )  +  £  ^ ( P V L O ^ ) [[5 ]- (P )5*(p'') + ^ J f f l - ^ ) ] ,  + 7J(p°)[ £ L - ^ P O W ] [ E n  1  0  1  j  ^v^'W)]  ]  T2nr  (  0  o,  where  SV) =«  •  (4.5,1)  Similarly,  + E e ^/Ui2  3dn Si  (^>i(p )5( y)[5]- (p ) 1  0  0  ?  + [ £ e ^/ L-^ (^)5(^)] [ £ L - s (p»')5V")] L2  Since the sums in Q i  > n  FL  M n  SL  2  and Q2,n do not contain the I = 0 term, it is easy to see that we  can remove the poles by multiplying factors of the type D° (p) n  = L D (p°). n  to both the numerator and the denominator of Q i , F(p,x,y)  (4.5.13)  2n  = An A , A  n 2  n  -  and Q ,n- Thus we write 2  Ai,3,  (4.5.14)  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  72  where -4^i,l  D°(p)H (p,x)G (p) n  (4.5.15a)  n  D° (p)fn(p) n  E  L ' ^  slip +  ^ ^ ) )  " ( P  g  g ) ^ i ( P d ) ) ^ n ( P , v)  +  (4.5.15b) A n,3  -2d-l  E  e  -*(2T'/ai)l/ ( S L  +  p  27r  // ) ( p ) aL  (4.5.15c)  5i  £>S (P(i))gn(p,g)C?^. ( (i))^(p+ g,y) •P£+l(P(l))/n+l(P(l)) 2  +1  1  Note that in (4.5.15b), for Lp + ^  =  P  if Z 7^ 0 then we get cancellation between  D ° ( p ( i ) ) and the singularity of G ( p + ^f)G'* (p(i)), and if Z = 0 then we get can2  +1  n  cellation between D° {p^)  +1  and G* (p(i))i7*(p,y).  +l  Similar cancellations are done in  +1  (4.5.15c). Now the next source of possible singularities of T (p,x,y)  is the zeros of f (p).  n  n  Only  if we can show that either / (p) has no zero or there are cancellations of zeros, then we n  can be sure that T {p,x,y) n  4.6  is analytic in  S/. £ a  Existence of zeros of f (p) n  From Section 4.5, we found that the poles of G  and H  n  singularities for f . n  n  do not actually pose any  But there is another source of singularities which we believe that  they can not be removed as the poles of G and H . These non-removable singularities n  n  come from the zeros of (4.6.1) given by the following proposition.  Chapter 4.  Decomposition  Of Field Via Block-Spin  P r o p o s i t i o n 4.1 For a < L there exist n's, p G S' /  E a  with pd =  < 1, aL  n  n  ~  1  Method  sufficiently  such that f (p) n  73  small, £ independent  of n,a,  = 0.  Before proving the above proposition, we state a couple of observations which suggest that f {p) should have zeros near ap = ( ± 7 r , • • •, ± 7 r ) . n  1. Since V„, K^I/,  has odd symmetry under p —> —p ,  n  v  V „  has even symmetry under p  W  has even symmetry under p —> —p  K)  —p ,  v  n  (4.6.2a)  u  (4.6.2b)  v  v  (4.6.2c)  v  and each of the above is periodic with period 27r/a in Re p . Thus v  VuM\ ^  =0  v  and  Vi,  tn  (4.6.2d)  as a function of p has an odd symmetry about ap =  V„,n(7T  v  v  i.e., V „  IT,  ) n  ("7T—  p) v  =  +p„).  2. Since for p away from the poles of 1/D°(p), each sine term of the form sm(api") or sm(api"/2)  is Q(L~ ),  thus  n  w  e  s e e  t n a t  W„(P) - ( L T ^ E ( L T 4 V ) M  =  0(^"  n  ^  s  i  n t f  + aL m).  /  2  )  (4.6.3)  n  Let us investigate more thoroughly on the behavior of V„, and W . Away from the n  poles of 1/D°(p),  W  n  n  and V„, are analytic functions at each p e S' / . Since the pole of n  v  E  a  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  74  1/D^(p) is near p = 0, we divide S' / into two regions where one contains the pole and e a  the other contains the zeros of V . Let v>n  S;  = 5< u5<  / a  (4.6.4)  1  V 2  T / 2 l  where S<„,2 = {pe S'  e/a  and  5 < /2 ^ ,  7r  s  | \Re ap \ < TT/2}  (4.6.5)  v  the complement of S< /2 relative to S' / . When considering upper bounds n  £  a  on \V^ \ and |W„| for p e S< /2, one should consider bounds on |£)£(p)V„ (p)| and n  n  )n  |D°(p)Wn| instead . From the observations, let us write V„,  =  n  W  (TT - ap„) A (p) ,  (4.6.6a)  = L~ B (p) + aL mC (p).  (4.6.6b)  v>n  n  n  P r o p o s i t i o n 4.2 For e- sufficiently 1.  n  n  n  small:  forpeS' , e/a  c  <ReA (p),  27  Utn  ReB {p),  Re C {p)  n  (4.6.7)  n  2. / o r p e S<*./, 2  c <|D°(p)A,, (p)|, n  29  3. for p e  |/J°(p)5„(p)|,  |D°(p)C„(p)| < c  3 0  (4.6.8)  5^/2,  c i < |4,,„(p)|, 3  where the Ci's are positive  constants.  |B„(p)|,  \C (p)\ n  <c  3 2  (4.6.9)  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  75  Because the proof of the bounds is laborious, we only demonstrate a sketch for V  u>n  over S< j .  The other bounds follow by similar techniques.  n 2  B y viewing V , {p) as a v n  function of p with other components being fixed, V„ is analytic over the given domain. u  )Tl  By analyticity over a compact domain and choice of smallness of e, it is suffice to show that, for p G S< /  and p real,  n 2  c (7r 33  ap ) < V (p) < u  v>n  c (7r 34  - ap„),  (4.6.10)  where the Q'S are positive constants. Let  us rewrite V„ as )Tl  V„ (p) >B  =  (L")-  Pto)vu,ntf),  2 d + 2  (4.6.11a)  -N<j <N;kjiu K  where N = (L — l)/2. We further simplify v (p>) into n  Utn  N  v , {p>) = (2N + I)"" sin (ap„/2) 2 3  £  2  u n  b(p>)  (4.6.12)  iu=-N  where V) Next we show that YYJ„=-N HP*)  c  a  = ^ f f f n  •  (4.6,3)  ^ written as a sum of non-negative terms where e  each term has ap = TT as its only zero. Let us rearrange v  J^^JI-NHP*)  s  o  that the first  2N terms is written as a difference of two positive quantities. Using _  a p , - ( j „ + l)27r _  ~  27V + 1  ~  =  -a((2yr/a) -  p) , iv  v  (27r-ap„)+j„27T 27V + 1  1  '  Chapter 4.  Decomposition  Of Field Via Block-Spin  the first N terms  Method  76  E KPO = -EV')>  (4.6.15)  j„=0  ju=-N  where for K ^ v, q = p , and q = (2ir/a) — p . Thus K  K  v  Y  v  b(pi) = b(p ) +  Yb(pi)-b( i).  N  N  (4.6.16)  q  >=0  3u=-N  We would like to extract a factor p — n/a from b{p>) — b(q ) and b(p ). Since 3  N  v  ap  7T  N  = IT  —  ap  v  (4.6.17)  27V + 1'  we have tan((7r-o  N l P  ) / ( 2 i V + l))  P</  (4.6.18)  D (p»)  }  n  Let <> / = ap>/2,,  d = aqi"/2.  (4.6.19)  and write =  cot(fl-cot(fl)  1  +  Z) (p?")  D {qi)  n  COt(</>)  - cot(0)  D (pi)  n  D {q ) ~ Dnjp )' [ D {pi)D {qi) j  + cot(0)  3  n  n  (4.6.20)  n  n  We extract a factor p — IT/a from cot((/>) — cot(0) and D (q )  — D (jpi).  j  v  n  n  From the  difference apt"  d-(j) =  2  (IT  2  y>  sin(</>) sin(0)  n  =  (4.6.21)  '  sin(c/>) sin(0) sin(0 — (/>)  - D (p>)  v  sin(0) cos(</>) - sin(</>) cos(0)  cot((/>) - cot(0)  Dntf)  - ap )  sin((7r —  ap )/L ) n  u  (4.6.22)  sin(</>) sin(0)  [sin (2^)-sin (20)]+4[sin (^)-sin (0)] 2  2  4  +2 [sin (0) - sin (</>)] | am + 2 2  2  4  sin (apj."/ ) 2  2  Chapter 4. Decomposition  Of Field Via Block-Spin  Method  77  Further simplifying the first two terms of the above yields [sin (29) - sin 2  2  (20)]  sin (0)] 4  + 4[sin (0) 4  =  4[sin (0) cos (0) - sin(</>) cos(0)] + 4[sin (0) - sin (</>)]  =  4[sin (0) -  2  2  2  2  2  4  4  sin (0)]. 2  (4.6.23)  Using an interpolation integral, sin (#) - sin(</>) = 2(9 - <f>) sin(i# + (1 - t)<j>) cos(t9 + (1 - t)<j>) dt Jo 2  2  =  ^ ~n^  [  P  L  s i n  (  2 t g  + ( ~* M 2  1  d  (4.6.24)  t  Note that since , for 0 < j < N — 1, v  0 < <\> < 9 < TT/2,  (4.6.25)  and (j) — 9 only when ap — ix. Thus since for 0 < x < IT/2, cotx is strictly decreasing v  and sin x is strictly increasing and both are positive, it is easy to see that (4.6.18) and 2  (4.6.20) are strictly positive except for ap = IT where V„ (p) v  From (4.6.16), we break V  v<n  tn  = 0.  into three parts corresponding to the two terms in  (4.6.20) and b(p ). Since each part is non-negative, we can bound V N  v<n  by bounding each  part separately. We demonstrate only the existence of an upper bound like (4.6.10) for the part corresponding to the cot(^) — cot(#) term of (4.6.20). The lower bound and the bounds of the other parts can similarly be achieved using Lemma 4.2 and Lemma 4.3. The part of V„  >TV  2  V  - <&N-*#, N  corresponding to the cot(^) — cot(#) term is  (  L-™^rUv)Y  L  ~ " 2  S I N  \^ \sm(^msm( ir/2)))\  u{P  0  q  W ) 2  )(L- sm((ir-ap )/L»)\ D (pi) J (4.6.26) n  lJ  n  Since in Lemma 4.3, the bound (4.4.15a-b) excludes the case when j = 0, let us consider the j = 0 term separately. Now by Lemma 4.2, and (4.4.15a-b) of Lemma 4.3 , (4.6.26)  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  78  minus the j' = 0 term has the upper bound  =  c  36  (TT - ap,,),  where k(j) is defined in (4.4.13). From (4.4.2b), L~ D (p ) 2n  0  n  =  L~  sin (ap./L") + [am + 2 £ s i n ( a p , / 2 L ) ] ^  >  C3 £(ap )  2n  2  7  i y  2  2  > c  n  (4.6.28)  2  7T /4 2  37  V  since one of the p„'s is > 7r/2. Thus it follows that the j = 0 term is also bounded above by  c (7T 3 8  ap„).  Q.E.D. From (4.6.1) and (4.6.6a-b)  /n(p) = E < „ ( P ) ( T - ap,) + (L- B (p) 2  + aL mC (p)) .  n  n  n  (4.6.29)  2  n  V  From Proposition 4.2, we expect to find a zero of f  for p near (TT, • • •, TT) and L ~ " ,  n  aL m n  sufficiently small. Let us prove Proposition 4.1 now. Proof of Proposition 4.1: Let  Qk  =  (TT,-• • ,TT,TT + ik)  a„ (fc)  =  Al (q ),  >n  >n  k  (4.6.30a)  b (k) = B (q ), n  n  k  c (k) = C (q ). n  n  (4.6.30b)  k  where |fc| < e. From the symmetries of V„  )Tl  and W  n  and (4.6.6a-b), it is easy to see that, as an one  variable real function in p„ with other variables being held fixed, each of A , v>n  has even symmetry at ap„ = TT. By the even symmetries, it is easy to see that  B,  C  n  n  a ^ (k), v  n  Chapter  4.  Decomposition  Of Field Via Block-Spin  b {k), c (k) are of real values. Evaluating f (Qk), n  n  2  d  >n  k + (L~ b (k) 2  79  since V„ (gfc) = 0 for v ^ d, we get  n  /«(?*) = -a ,n(k)  Method  + aL mc {k)) .  n  n  n  2  n  (4.6.31)  Given an e > 0, by choosing n sufficiently large and aL sufficiently small, using the n  bounds in Proposition 4.2, we see that f (q ) n  < 0. Since f (qo)  e  > 0, By the Intermediate  n  Value theorem, there exist an k such that f (Qk) = 0. n  Q.E.D. We remark that a-d,n{k)  and k is of the order 0(L~  + aL m).  n  n  Also, there are more zeros at  (TT — Si, • • •, TT — S -i,TV + ik)  (4.6.33)  d  for Si are sufficiently small. For these points, ,,,  ^J(L- b (k) n  n  fc =  + afrmcnik))*  :  and k is of the order 0(L  n  + aL m n  +  77^  E^ aljk)5i d  :  4.6.34  + S) where S = ^Y^^dM-  Not long after our failed attempt in showing the sliced Fermi covariance of the blockspin set-up has the usual exponential decay, it has come to our attention that Pereira [PP97] has discovered an unorthodox averaging which renders the desired bound for the sliced covariance. In his work, imaginary terms are incorporated in the averaging to break the odd symmetry of G . This allows G to be bounded away from zero and establishes n  n  the desired bound by using the Contour Shift Lemma.  Chapter  4.7  4.  Decomposition  Of Field Via Block-Spin  Method  80  Nonperturbative method (Lattice method)  We conclude the presentation of the block-spin method for slicing up covariances with a brief discussion of nonperturbative methods (showing the flow of the effective potential converges nonperturbatively). In Chapter 3, we have studied the flow of the effective potential 14 using the cumulant expansion (3.2.5) and such an expansion only makes sense in the case of a small potential 14 nonperturbatively, there are cases where nonperturbative convergence or at least some sort of stability bounds (bounds for the exponential of the effective potentials) can be achieved . Balaban & Jaffe have proven ultra-violet stability for the nonabelian Higgs model in space-time dimension d=2 or 3 using a nonperturbative renormalized gauge invariant flow of the effective action defined on a lattice where BS averaging is used in deriving the field decomposition [BJ86]. By a similar method, Imbrie has also treated the abelian Higgs model in space-time dimension d=2 or 3 with success [Imb86]. Gauge invariance at each step is crucial since it guarantees a gauge invariant result at the end of the flow. Gawedzki and Kupianinen have also studied these lattice methods using BS averaging where they have developed a nonperturbative flow for treating the scalar theory [GK86]. In the all above cases, the basic procedure seems to be the following. We separate the domain of all possible $ in the functional integral defining the effective potential (or action) into small field and large field regions where the cut may depend on the running couplings. In the small-field region, we can expand using perturbation theory with renormalization and obtain a flow of the effective potential. By using the positivity and largeness of the action, we argue that the large field region provides only a small contribution and thus does not affect the small field estimation. To give the flavour of this approach to the reader, here we give a brief description of Gawedzki and Kupianinen's set up of a flow of the effective action defined on the unit  Chapter  4.  Decomposition  Of Field Via Block-Spin  Method  81  lattice and its application to the lattice hierarchical $ 4 model. Gawedzki and Kupianinen set up a flow of the effective action defined on a unit lattice given by the iteration of the following transformation. Let  be the lattice action of a model, we define a  transformation T by the following: (TS){ip) = - log I P $ 5(Q$ - V) e" *'  (4.7.1)  5(  where ip is a field belonging to a coarser lattice A') and Q is the block-spin averaging 1  (see [GK86] for the definition). Notice T effectively integrates out degrees of freedom of the lattice action with momenta between a , the lattice spacing, and p = a - 1  _ 1  L  - 1  .  Next, we separate the RHS of the above equation into its local free (quadratic) part and the remaining part we called the interaction part. By partially expanding e~  in (4.7.1),  Sl  one can generate a nonperturbative flow of the effective action or a nonperturbative flow of its local free part and effective potential (the interaction part). Gawedzki and Kupianinen have developed a nonperturbative flow of renormalized potentials as described above for the lattice hierarchical $ 4 , a simplified version of $ | [GK86] in which the action has no (nonlocal) kinetic energy term.  The hierarchical  covariance is given by G(x, y) = ( l -  ^-dyi (2-d)(N( ,y)-i)  L  L  x  ( 4 ? > 2 )  where N(x, y) is the smallest positive integer N s.t. [L~ x] = [L~ y] and the interaction N  N  is Vm  = - £ v($(x)) = - £ X  X  \p  2  Z  :$ : + A 2  ^  :  &  :  .  (4.7.3)  ,  where : W(<3>) : is the Wick ordered W(<&). The hierarchical model has IR-behavior and UV-behavior similar to $ 4 . It is much simpler to study the hierarchical model than the real one since the V remains local after each BS transformation T. Let du, (£) r  be the  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  82  Gaussian measure derived from the BS-decomposition of the field. Since V is local, the n  recursion (4.7.1) reduces to a recursion for a function of one variable: tv($) = - log jdpr{0  (4.7.4)  e x p ( - L ^ ( L - ^ $ + 0)( 2  2  The hierarchical recursion retains most of the interesting features of the real one. Due to the local nature of (4.7.2) there is no wave function renormalization in the hierarchical model. The basic idea of showing that the flow of v  n  = t v given by (reftvphi) can be n  approximated by perturbation theory is that, using analyticity of tv in a strip, one can extract the <& and <& terms of tv from a Taylor expansion at $ = 0. The coefficient of 2  the $  2  4  and $ terms can than be shown to have good approximation by the second order 4  perturbation theory of the model. Take d = 4. If | $ / L + f | < 0 ( A ~  1/4  ) and u is of order A , it follows that \v\ < 1. 2  2  Thus we make the following partition: <3> is in the small-field region if |$| < B\~ l  l A  where  B is some large constant. Let x be a partition function such that x(0 = x ( l ^ l < O ( 5 - A - / ) ) 4  1  (4.7.5)  4  and express tv as tv  x  +  tv ±  (4.7.6)  x  where tnx  (4.7.7)  to x  =  x  Jdp(0  (1-x) exp(-L u($/L + 0 4  .  (4.7.8)  With the cutoff x, using the positivity of v, for |$| < . B A / , we can show that tv x is - 1  4  x  a small nonperturbative contribution, since tv ± x  = Q(exp(-cA- / )) < 0(A ), V n , 1  2  n  (4.7.9)  Chapter 4. Decomposition  Of Field Via Block-Spin  due to the small probability of |£| > of tv in a strip | i r a $ | < A ~  1 / 4  0(A ) 1/4  i  n  Method  83  the Gaussian d/j(£). Using the analyticity  , we Taylor-expand tv at $ = 0 : (4.7.10)  where (d/d&)v'> (0) = 0,i < 6. By Cauchy's theorem, for | $ | < £?A~ , 1/4  6  -($)  Thus the contribution of (tv) ±  (4.7.11)  < 0 ( e x p ( - c A - / ) ) , i = 2,4,6. 1  2  to the coefficients in the Taylor expansion is nonpertur-  x  batively small. Furthermore, by Taylor expanding tv (denoting u = L u ( $ / L + £)), 4  x  tv ($) x  1 = const.+ < u > - - < u  1 r > +- J dt(l - t) < u l  2  2  x  3  x  >£ , t  (4.7.12)  where <  w  > x = jf J wxdp,  < w > , = -A; j x t  we  t  u X  dp.  (4.7.13)  The first two terms in (4.7.12) give our standard perturbation theory, except that the Gaussian integrals are cut-off by x- Since JCdMZ)  ~ /rx^r(0|<O(exp(-cA- / )), 1  2  (4.7.14)  tv can be well approximated by our standard perturbation theory. From a second order x  perturbation theory (see [GK86]) and the above, it is not hard to derive an approximation of the coefficients in (4.7.10) in terms of the /3 functions of the perturbation theory: Lp 2  +  2  0(\ ,p X), 2  2  (4.7.15)  A'  A - & A + O(AVA),  (4.7.16)  v'  0(A ),  (4.7.17)  O(A) •  (4.7.18)  «>6  2  2  Chapter 4. Decomposition Of Field Via Block-Spin Method  84  Gawedzki and Kupianinen apply the above procedure inductively to establish the following nonperturbative results. There is an interval I  C [—CnA , CnA ] of values for 2  n  2  the bare mass p? in (4.7.3), such that (a .) exp(—v (<$>)) is analytic on \Im$\ < B(X )" ^ n  1  n  < exp(-A |$| + A„(/m<I>) + L>) ,  \exp(-v {§))\ n  where D = 0(1) (b .) n  and satisfies  4  n  1/2  4  2  (4.7.19)  constant.  For | $ | < S A - / , 1  4  «"(*) = ^ with (d/d&)v$ (0) 6  2  :$ : + ^  : $ : +r? : $ : + ^ ( $ ) ,  2  4  6  n  6  (4.7.20)  = 0, i < 6, and  <  1/41  (4  CoA , 2  < <  ^>6 1  (4 (4  C2A„,  and 1 , 1 1 - + c_n < A„ < - + c n +  A  , (4.7.24)  .  A  A n immediate corollary of the above result is that for A < A , there exists a p rit{X) 0  C  in (4.7.3), such that Jim exp(-t u($))-4 1 n  (4.7.25)  uniformly for $ on compacts of QI This implies that the model is I R A F and it is a trivial free Gaussian theory. Although we obtain a nonperturbative flow of the potential, many parts of the analysis actually rely on the analysis of the renormalized flow of the potential coming from  Chapter 4.  Decomposition  Of Field Via Block-Spin  Method  85  perturbation theory. The basic elements of the above discussion, the perturbative and the small-field-large-field analysis as well as the analyticity arguments, can be carried over in a natural fashion to the real $ 4 model. Moreover, we hope such techniques can be extended to gauge theories involving Fermi fields, perhaps with different type of averaging (e.g. a Wilson average on gauge group elements which are defined on bonds instead of sites of the lattice [ B J 8 6 ] ) .  Chapter 5  Loop Regularized Running Covariance  5.1  Running covariance  From their multiplicative renormalization scheme, the physicists derived a R G flow equation for determining the U V and IR behavior of the perturbative Green functions. In the Q E D model, the physicists' flow equation seems to depend only on the photon self-energy diagrams because of the Zi = Z " condition. Although we have some reservations on u  2  the physicists' derivation of their flow equation, we would like to follow the spirit of a multiplicative renormalization to the covariance in which we modify the covariance by shifting the corresponding slice of all the quadratic local parts of selected primitive diagrams and counterterms in the tree expansion to the sliced covariance at each scale of the decomposition. We shall call this scale by scale shifting of sliced covariances the "running covariance (RC) scheme". A further reason for having the R C scheme is that it allows the stabilization of the flow equation of the running couplings in the case when some running coupling A* of the quadratic terms are unbounded as the scales s — > oo. These unbounded couplings appear approximately as 1/Af in the flow equations of the R C scheme. We leave the details of the analysis of the flow of the running couplings in Chapters 9-10. In this section, we demonstrate the technique of shifting the sliced covariances in a R C scheme. From iterating the shifting procedure described in Lemma (5.1) in conjunction with the R C C renormalization expansion described in Section 3.6, we can establish a flow 86  Chapter  5. Loop Regularized  Running  Covariance  87  of renormalized effective potentials similar to that of the R C C scheme except the lines of the resulting graphs are the running covariances H to be described in the below. In s  the next section, we will determine the explicit form of the running covariances in a R C scheme for IQED that uses a neighbourly slicing on the fields. After that, the last two sections introduce a loop-regularized running covariance scheme (LRC) for I Q E D where one can avoid having an U V momentum cutoff for the Fermi covariance in regularizing the Fermi loops. We first would like to briefly explain how the R C scheme works. In applying the R C C tree expansion, a flow of a running potential is generated recursively by making sliced Gaussian integrations. The Gaussian integrations can be represented by Feynman diagrams obtained from contracting local vertices which represent local monomials of fields. In the tree expansion with a R C scheme, rather than allowing all local terms to be contracted explicitly in the tree expansion, selected local quadratic terms of the running potential are treated as part of the slice covariance rather than of the potential. Using the modified slice covariance, Gaussian integrations are performed to obtain the running potential for the next scale. The running potential is then renormalized by extracting relevant local parts of the Feynman diagrams.  The selected quadratic parts of these  extracted terms are now ready to be used for the next shifting of the covariance and scale-by-scale renormalization process. Using the shifting of covariance, the resulting coupling flow of the R C C tree expansion separates into two systems: an internal one and an external one. The internal system consists of equations for the couplings of non-quadratic local terms and unshifted quadratic local terms, while the external system consists of equations for the multiplicative factors used in modifying the sliced covariances. Because the shifted quadratic counterterms never enter the tree expansion explicitly, the corresponding Feynman diagrams for the (5  Chapter 5. Loop Regularized Running Covariance  88  functions (which are the RHS's of the flow equation) of these equations consist of only local non-quadratic vertices and unshifted local quadratic counterterm vertices. But the lines of these diagrams are replaced by modified lines representing the running (renormalized slicing) covariances. We now describe the basic step of integrating out the field at a scale 5 in a R C scheme. Given a decomposition of covariance G = J + H with corresponding field decomposition $ = B + £, where £ is the sliced field at scale s with covariance H which is to be integrated out, we shift a quadratic part  = / ^ (p)K (p)^(p)  dp of the effective  t  e  potential into the sliced covariance by means of the following lemma. Here (•)* denotes the transpose of (•). Lemma 5.1  Suppose (HK Y  = K H,  e  logfe W-l* '*dn (Z) v  then  e  tK  =  H  1^ ^ --BtReB + logc  +E n=l  L  log  n!  Je^^d^iO  ,  (5.1.1)  A=0  where $ = B + £,  H  =  (H^ + Ke)- = [l + K H)- H  B  =  (l + HK )- B  K  e  Furthermore,  1  = H{l + HK )-  l  1  e  e  = (l-HK )B  1  e  (5.1.2b)  e  = {l + K H)K .  the covariance  e  (5.1.2a)  (5.1.2c)  e  of B is  J=(l  + HK )- J(l 1  e  + K H)- . 1  e  (5.1.3)  Chapter 5. Loop Regularized Running Covariance  89  Proof: First of all, (5.1.3) is a direct consequence of (5.1.2b). Next, from the decomposition $ = B + f, the LHS of (5.1.1) can be written as pm  r  log J e  ^ 71 =  x  v  ^  e - ^  K  ^  t K  °  ^ d i i  B + B t K  (5.1.4)  ( 0  H  1 L  A=0  We can shift the second exponential term into the covariance with a square completion. Formally,  Using (5.1.2a-c), we simplify the exponent in the RHS. From (5.1.2a) and the hypothesis, {HK y = K H, we have e  e  (HK y = [HK y{\ + (KHy)-  =  1  e  e  + HK y  =  l  K H(I  e  £  K H. £  Hence C{H~  + K )£, + CK B + B K£  l  l  e  e  = (£ + HK By  H  e  (f + HK B)  - B K HK l  e  e  e  B.  (5.1.5) Thus by making the change of variable £ —> £ — HK B, e  |F(0  =cci  e-i«'^+f « +^««d/i (0 tif  fl  H  where c = [det(l + i f K ) ] e  (B'K^l-HKe]  _ 1  B t / f  - ^ J F(EB  d/z*(0.  HK B) e  is the normalization factor. Hence (5.1.4) becomes ^  B)+logc+Y  log  Je ^- ^dp {C) xv{  (5.1.6)  kK  k  71=1  A=0  The quadratic term B K [l - HK ] B = B* (1 + [HK y)K l  e  yielding (5.1.1). Q.E.D.  e  e  B = B (1 + l  e  #  e  # K e  B = B KB l  e  (5.1.7)  Chapter 5. Loop Regularized Running Covariance  90  Let us demonstrate the first step of the iteration of combining (5.1.1) and renomralization in the running covariance scheme. Using the notation in (5.1.1), we start the iteration at scale s = U by splitting the original covariance G = C-  = p- C,  u  the original full covariance, into J = C  = p C  < u  and H = H  <u  = p C,  u  where C is  u  with K  u  =  e  K  u  being the original quadratic counterterms. In the Q E D model, the corresponding C and K  have respective kernels,  u  \  D{p)  0  0  0  0  S{p)  0  -5*(p)  0  C( ) = P  K (p)  K%ip)  0  0  o  =  u  0  V  \  0 -Kgipy  K AP)  o  U  ) (5.1.8a)  where D(p), Sip) are defined in (2.4.7), (2.4.4) respectively and K%ip) = iX^-l)p  +  2  M i\^-M2U  Here we don't shift the id-A)  2  From Cip) and K ip),  K ip)=mi\ i-l)  2U  u  + iX -l)p .  l  u  F  (5.1.8b)  term (see the discussion after (5.4.11) for an explanation).  it is straightforward to see that iH K Y  u  J  2  u  = KH.  u  U  U  By ap-  plying (5.1.1), integrating out the shifted [/-slice £ with covariance H and renormalizing, the summation on the RHS of (5.1.1) becomes  E  |  log  1 ~  Je^^dp^)  -  -B*6K B U  A=0  n=l L  °° 1  + 5ViB) + £  ^5|(V(B)")  T  (5.1.9)  where the the first two terms are local counterterms extracted from the renormalization, the last term is the renormalized effective potential, and £ j ( - ) is the connected expectation w.r.t. dpfji^). We add the quadratic local terms ^B^SK B to further modify 11  K. e  Now setting  K"-  1  C  < u  = K -5K  =K +  =  U  U  e  K H K -5K ,  U  J = il + H K )U  1  U  p C <u  il+  U  U  U  (5.1.10a)  K H )~\ U  U  (5.1.10b)  Chapter 5. Loop Regularized Running Covariance  we t h e n s p l i t C  91  into  <u  <(u-i)  [1 + H K )~ u  c  H  u-i  u  < -VC  l  ;i + HKy u  u  (1 +  U  P  K H )~\ U  u  (5.1.11)  KHY  p ~ C (1 +  l  U  l  u  u  l  for t h e s h i f t i n g i n the n e x t scale s = U — 1. C o n t i n u i n g t h e a b o v e process leads t o t h e following iteration:  K*-  =  1  s-l  H  K -5Kl  +  s  KlH Kt s  1 + H K )- p - C a  8  1  a  (5.1.12a)  (1 +  i  e  K'H )3  1  (5.1.12b)  and  H  s  C  <s  =  (1 +  K H )~ H\  =  (1 + H K )~  s  s  l  (5.1.13a)  e  s  s  p C  l  <s  e  (1 +  KIH )3  1  (5.1.13b)  M o r e o v e r , b y a n i n d u c t i o n a r g u m e n t , i t is easy t o see t h a t the r u n n i n g t e r m s h a v e t h e following m a t r i x forms  H (P) S  { 1  H*(p)  = \  w h e r e each set (5.1.13a-b).  H%(p)  0  0  0  0  H {p)  0  F  0  F  0  0  0  0  H'M K^ , x  Kip)  s  -(H' Y(p)  H%{p)  [  N  o  H , x  C }, s  x  f  <s  P  F  , X G {B, F},  o  o  o  0  c () =  -(H (p)Y o  \  7  KB(P)  V  C£ (p) s  0 -(Ki (p)y F  Kl (p)  0  F  0  0  0  0  0  -(C< ) (p) s  ^  Cf{p) J  0  also satisfy t h e e q u a t i o n s ( 5 . 1 . 1 2 a - b ) ,  ]  Chapter 5. Loop Regularized Running Covariance  5.2  92  R C w i t h n e i g h b o u r l y slicing  Next we would like to express the running covariance in terms of the running couplings. To make the procedure manageable, we use "neighbourly" slicing (see Appendix B for an explicit slicing) which is a partition of unity for which only adjacent slices have overlapping supports (this includes sharp slicing as well ). Also, we would like to study the flow of the running covariances component-wise. subscript X in {H ,  K  s  x  x  , H, x  C }.  Let K  s  be the kernel of the shifted local (photon  s  x  We let R = Cx and drop the  or Fermi) quadratic terms in the shifting scheme which has the form K  s  = Y X jKj  (5.2.1a)  a  3  and Af where K  s  = A* - 8] (or K ~  1  s  = K  l  s  - 8K ),  (5.2.1b)  S  are given explicitly in (5.4.6) and (5.4.16) below.  L e m m a 5.2 Using a neighbourly slicing, the solution of (5.1.12a-b), (5.1.13a-b) is H  s  =  p R (l + p  K  s  =  K  s  RK )~  s+1  s+l  (5.2.2a)  2  u e  s  + R52  P\ f  (5.2.2b)  jt  t=s+l  and p R s  H  "  6 <  -  where  an d p  u+l  = K  u + 1  = 0.  1 + if R IC +  =  =  (if^RK-r  [(1 + R A T - +  1  )  2  -  1]'  ( 5 2  (  5  '  -  2  3 a )  '  3  b  )  Chapter 5. Loop Regularized Running Covariance  93  Proof: For s = U, (5.2.2a-b), (5.2.3a-b) are just the initial terms  So assume (5.2.2a-b) and apply the iteration (5.1.12a-b). Because the slicing is neighbourly, pK 8  = p [K + p R(K ) ]  8  8  s  8+l  8+l  .  2  (5.2.5)  Hence HK s  s  =p  R (l + p  s  e  K y s+i  R  s+l  2  [K + p s  s+1  R (K ) ] s+1  (5.2.6a)  2  and K HK s  s  = p R{J ) .  s  e  s  e  s  (5.2.6b)  2  From (5.2.6a), (5.1.12b) becomes H8  = p~  1  s  R (l + p RK )'  1  s  8  2  .  From (5.2.6b) and (5.2.4), (5.1.12a) becomes  u K8  1  K + R52  =  PV)  s  =  K8  +  1  t=s+l U  -SK"  2  + p R{J ) s  8  (5.2.7)  2  RY,P (J ) t  t 2  t=s  From (5.1.13b), (5.2.6a) and the fact the slicing is neighbourly, we have (5.2.3b). Substituting (5.2.2a-b) and (5.2.6a) into (5.1.13a), and using the fact that (p + p ) s  the overlapping support of p and  p ,  8  p R(H )8  8  1  =  (l + H K )  =  (l + p  =  l + p RK  8  s  8+1  (l + p  8  R  s+1  8  = 1 on  s+1  K f 8+l  RK )  s+1  s+l  + p R [K + 8  + 2p RK s+l  2  8+1  8  +  p R(K f\ 8+l  s+l  (p -rp )p (RK ) 8  s+1  s+1  s+l  2  Chapter 5. Loop Regularized Running Covariance  1 + p R K + 2p s  s  R K  s+1  s  +p  s+1  1+ p R K +p s  94  (R  s+l  [(1 + R K f  s+l  K )  s+l 2  - 1].  s+l  Q.E.D We shall call the term V (X)  (5.2.8a)  S  ~ D>(\y  where D {\) s  = (1 - p - p ) s  s+1  + p (l + RK ) s  +p  S  s+l  (1 + R  K ),  (5.2.8b)  s+1 2  the running slicing for the sliced covariance H . The running slicing serves as the mods  ified slicing which has a dependence on the running couplings X . Under a R C scheme, s  the coefficients of the R C C flow are computed using Feynman diagrams with lines corresponding to these modified covariances.  5.3  Loop Regularization  We would like to incorporate the loop regularization of [FHRW88] in the running covariance scheme and call the combined scheme a loop-regularized running covariance (LRC) scheme. We first briefly describe the set up of the loop regularization in this section. Here we adopt the notation used in [FHRW88].  Following the set up in §3 of  [FHRW88], a loop regularized IQED is a model of IQED where the Lagrangian is modified by inserting additional terms as monomials in the fictitious spinor fields ^j,^fj, j = 1, 2, 3, where ^  is a Fermi field and \& , ^3 are Bose fields with propagator 2  <  (5.3.1)  >= 5(M,-)  where M ( A ) = m + 2A , 2  2  2  M (A) = m + A , 2  2  2  2  M (A) = m + A . 2  3  2  2  (5.3.2)  Chapter 5. Loop Regularized Running Covariance  95  For convenience, we define M (A)  (5.3.3)  = m,  2  2  and tf  =  (tf ,---,tf ),  $  =  (A,tf ,tfo,---,tf3,tf ),  0  tf  3  0  = (tfo,---,tf ),  (5.3.4a)  3  (5.3.4b)  3  where tf = ip and tf = tp. 0  0  The modified Lagrangian (cf. (1.2.5a-b)) depends on an U V cutoff U on the photon field and a loop regularization parameter A: C = C + d  (5.3.5)  0  where Co =  + (l- )-\d-A)  \[A-(-A)A  +  2  V  V  A }+J2^ (M -ip)^ , 2  J  j  (5.3.6)  j  3=0  Ci  =  ]Ttf, i=o  [zAf' 4-^A ' -l¥+(A ' -l)M ] A  /  A  /  2  +\ \(^'  A  J  ~ I)A • {-A)A + M \f A 2U  K  tf,-  A  4  (5.3.7)  + A ' (<9 • A) + A^'M ] . UA  2  2  4  6  Here, the Af' have a dependence on U and A. A  From the expanded Lagrangian, the corresponding free covariance C\ of $ has the following non-zero components : (CA)II  =  D  (C )  4 5  =  - ( C ) £ = SfMx)  (C )  6 7  =  (C )l = (C )  A  A  where S(Mj)  (C ) A  23  = -(C )l A  = S(M ) 0  (5.3.8)  A  A  A  89  = (C )l = S(M ), A  2  and D are defined in (2.4.4) (with m being replaced by Mj) and (2.4.7)  respectively. Given a graph G of the G N tree expansion, a loop regularized graph G  A  Chapter 5. Loop Regularized Running Covariance  96  represents the sum of graphs with the same structure as G , summed over types j = 0,1, 2, 3 for each loop of G with a - sign for each loop of type j=2 or 3 (no sum over types for the spinor lines in G whose type is determined by the type of the attached external field if,-)Let N be an UV-cutoff of the ^ fields; for a given UV-cutoff U of the photon fields and A , G\ converges uniformly in N . This means an UV-cutoff for the  is actually  unnecessary, (see Lemma 3.1 of [FHRW88]). Thus in our set up of the L R C scheme, the initial slicing of $ has no U V cutoff for the ^ fields. This means the U  th  slice of the  ^-covariances corresponds to scales [U, oo). More explicitly, for h <U, S {M )  = p S{Mj),  h  D =p D  h  j  h  . (5.3.9)  h  and s  u  [u,oo) ( ^  =  p  S  D  u  Mj  =p D  (5.3.10)  u  where p is the neighbourly slicing functions defined in (B.2a) of Appendix B . h  Without the cutoff on the Fermi covariance, Ward Identities can be preserved provided the coefficients of the counterterms are chosen independent of the index j of the spinor fields  ^ . The choice of j-independent counterterms can be implemented through the  modified localization operator L  A  described in (3.31) of [FHRW88]. For graphs G with  no fictitious legs or with LG = 0, L G = LG. For a graph Gj with fictitious legs of K  type tyj and ^tj, let G be the graph which is identical to Gj except that the spinor line connecting the external spinor legs is replaced by a real Fermi line. Then we define: (L G )(A^ ,^ ) A  J  ]  J  =  j:{ f)  where L$G consists of the monomials of degree 8.  J  (L G)(A^^ ), S  3  (5.3.11)  Chapter 5. Loop Regularized Running Covariance  5.4  97  Loop-regularized Running Covariance  In this section, we introduce a L R C scheme for the IQED model and write down the explicit form of the resulting running covariance of the scheme.  Also, we state some  imposed conditions on the running couplings in order to ensure that K  s e  in (5.2.2b) and  V '(A) in (5.2.8a) are well defined and uniformly bounded in U and A. First, K s  s e  can  be ensured to be well defined and uniformly bounded in U and A by requiring that the external sources have compact support and the norm of the term 1 + p  R K  t + 1  t+l  in  (5.2.4) stays away from zero. As for V (X), we need to examine its explicit form as a s  function of the running couplings Af. We make these imposed conditions explicit by imposing that the slicing p at scale s, 0 < 5 < U, has the following support, s  (1 - e)M  2s  <p < M . 2  (5.4.1)  2s+2  The derivation of the running covariance and the corresponding imposed conditions on the running couplings are not done here but in Appendix C. We set up a L R C scheme of I Q E D by first loop-regularizing the model as described in the previous section. In the loop-regularized model, the corresponding sliced covariance is obtained by decomposing (5.3.8) into C  S  using the neighborly slicing p . s  A  We then  apply the R C scheme of Section 5.1 to modify the slicing p of the components of C s  A  into V (X) using (5.2.8a-b). s  We separately state the components of the running covariance of the L R C scheme. For the photon field, the corresponding external local quadratic Bose term at scale U is \ A K U and the corresponding K  s  = (A- - 1)  + M™Xl  (5.4.5)  and R in Lemma 5.2 are K  s  =  [(A* - 1) p L + M 2  2s  X] , s  5  (5.4.6)  Chapter 5. Loop Regularized Running Covariance  p + 1  98  ^ (1 - r/) + p  2  '  2  where L is defined in (1.2.8) and T = 1 — L. Let the angle brackets <  ># denote the Gaussian expectation with respect to the  covariance H. The local parts of selected Feynman diagrams G (e.g. vacuum polarization diagrams) that are shifted into the "external" quadratic term AK^A L  V3  < G >fr=5^  A(p L)A,  L  2  < G >„=  Vs  Note the reason for introducing the factor M  are given by A.  (5.4.7)  2  in the coefficient of the A  2s  2  counterterm  is that by counting the naive degree of divergence, the coefficient 5\ of an A -\oca\ term 2  is Q)(M ). 2s  By explicitly having the factor M ,  the corresponding running coupling X  2s  s 5  becomes dimensionless. From (C.1.5a-b) of Appendix C, the running photon covariance is H  S B  =  H  + H^  S L  (5.4.8a) i+p  i  where  **W = i  V  ^ .  +  V h  >  A X )  =  1  +  P  s  p  p  s  P k  +  .+i[(^+i +  +  +  ^  i ) 2 _  ( 5  1  ]  4 9 a  (5.4.9b)  Y  and Y  =  K + Xl,  A - = ^ _ ^  5 )  +  p  2  ^ = w - *) (if-*) > ^ = ^ r $ -  (5.4.9c) (5A9d)  )  Chapter 5. Loop Regularized Running Covariance  99  In the Feynman gauge where r\ = 0, ~ Vj>A\) L + V (X) T H' = ^p +, 1»' , S  B  BT T{  M  (5.4.10)  1  2  B  and in the Landau gauge, where rj = 1, V AX)L S  B  H% = -ffj,-.  (5.4.11)  Note that because we prefer to use the Landau gauge in L R C for IQED, in the corresponding tree expansion, a (<9 • A) local part of a subgraph at scale t contracting at a 2  lower scale s < t produces a zero kernel since the resulting lines from the contraction contain a product of two orthogonal projection operators T and L defined in (1.2.8). Thus there is no need to shift the (d • A) counterterm in the running potential for making the 2  coupling flow stabilize. In order to have a cleaner formula of the running photon slicing, we prefer not to shift this term into the running photon covariance. From Section (C.l.7a) of Appendix C, in addition to require the external photon source have compact support, we also require A* > \\'\  (5.4.12)  From the above imposed conditions, the running slicings V (X), BL  V BT(X) S  have the  following bounds.  n.(A) V  s  <  (X) <  ^  <  l  ,^;y-l>.  +  1 + (1 -  (5„3a)  V)M~ *2  2  where J = X + Xl S  (5.4.14)  s  3  As for the spinor fields tf, tf, the external Fermi term at scale U is tfK^tf  = tf ((\1-l)M  +( \ % t f ,  (5.4.15)  Chapter 5. Loop Regularized Running Covariance  100  where M is the diagonal matrix with diagonal elements being the Mj defined in (5.3.2) and (5.3.3). The corresponding Kj  =  R  =  3  The local parts L < G  [X -1)TI+(\\-1)M  (5.4.16)  s  2  3  feZ+M,)- . 1  of selected Feynman diagrams G (e.g. the electron self-  energy diagram) that are being shifted into the "external" quadratic term ip K  F  tp are  given by L  V2  < G >= k  5-  < G -=  l  2  <J i * M * .  >H  4  (5.4.17)  where 8 and 84 are ^-independent. 2  From (C.2.1)-(C.2.8) of Appendix C, we have = VpjRj  H  FJ  = p'iDfi-iRj  (5.4.18)  where for s = U, Df = l + p A";  (5.4.19a)  u  for s < U, D]  =  1 + p A] + p B° -  A]  = RlKjRj,  K]  =  A^Mj + A ^ + ^ - A ^ M ^ ,  R)  =  (p + M )-\  s  2  s+1  (5.4.19b)  +l  B =R (k ) s  4  j  s  j  + 2R K ,  2  2  J  j  (5.4.19c) (5.4.19d)  A? = A ? - 1 .  2  s  j  (5.4.19e)  Also we impose the conditions, A^  >  1-e,  (5.4.20a)  A^  >  1,  (5.4.20b)  A  >  max(M  c / + 1  ,4M  c / + 1  sup \~X \), S  2  0<s<U  (5.4.20c)  Chapter 5. Loop Regularized  Running  Covariance  101  where 0 < e < 1/2. Because for Vpj with j = 0, we do not have the large A factor as in the fictitious terms, we further require the following bounds -4  > \u \ r  -  >  S  s  1  \u \r + s  s  where u = \\ — \\ and (r )~ s  (5.4.20d)  s  1  s  2  2\u X \r + s+1  s +1  (u )  s+1  s+1  2  = p +m. 2  2  2  (r ) , s+1  2  (5.4.20e)  Also, the running spinor slicing has the  following bound. For j =fi 0, from (C.2.21) of Appendix C, Halloo <  (5.4.21)  For j = 0, from (C.2.28) of Appendix C, I|PF,OIIOO<T7.  (5-4-22)  Chapter 6 Two-Slice LRC  6.1  Bounds on running covariance and spinor loops  From the set-up of an L R C scheme for IQED described in chapter 5, we need to choose the top scale couplings in (5.3.6) so the subsequent running couplings of the running potentials remain finite and satisfy (5.4.12) and (5.4.20a-e) as we remove the regularization cutoff. Since there are two parameters, the momentum cutoff U of the photon covariance and the loop regularization parameter A, for setting the regularization in an L R C scheme, an appropriate choice of the top scale couplings should be functions of U and A. The first criterion for a choice of the top scale couplings is that they provide the necessary counterterms for the removal of the would be divergence if the U V cutoff is removed. In our analysis of the L R C model for IQED, we would like to determine the forms of the top scale couplings as functions of U and A so that the resulting coupling flows stabilize and stay near the origin. We first determine the dependence of the top scale couplings on A and leave the dependence on U to Chapters 8-10. In a loop regularized perturbative model, the U V divergence is separated into two types. One type is the A-divergence coming from the spinor loops as the loop regularization parameter A —> oo and the second type is the [/-divergence coming from loops involving photon lines as the photon line cutoff U —> oo. In an L R C scheme for IQED, the renormalization of the divergence can also be separated into two levels corresponding respectively to the A-divergence and to the [/-divergence. 102  The first level removes the  Chapter  Two-Slice  6.  LRC  103  A-divergence by choosing counterterms corresponding only to graphs with A-divergence at the top scale. The second level renormalizes graphs containing the U V divergence at the subsequent scales as U —>• oo (after taking A —> co). It turns out that if appropriate external sources with compact supports are used, then the second level of renormalization is not required. As a result, a multi-scale expansion is not necessary; a two slice decomposition is sufficient to construct a perturbative model with finite coefficients in the perturbative series. In this chapter, using an L R C scheme for IQED where the fields are only decomposed into two neighbourly slices, we construct a perturbative model where renormalization is required only for the A-divergence at the top scale. In this two-slice version of L R C , p[u+i,u']^  pWM = Yli= P i l  w  n  e  r  e  l s  a  u s e  d as the top photon slicing with U V cutoff U'  and one does not have to make a multi-scale expansion when integrating out the fields corresponding to the bottom slicing p^°' ^ (Note that U is not the U V cutoff parameter u  in this set-up). Rather, one can integrate the whole slice corresponding to  at once  without renormalization and obtain a finite limit for the A-renormalized graphs (which are renormalized at the top scale but not at lower scale) when removing the cutoffs by taking A —> oo and U' —> oo. After removing the A cutoff, any A-renormalized graph is bounded in terms of powers of M . u  Since here U is held fixed when removing the U V  cutoff U' of the photon field, there is no U V divergence for these non-fully renormalized graphs. Because the two-slice version does not allow the use of external sources with arbitrary high momenta (external momenta P with \P\ > M  u+1  are not allowed), the  U V divergence is actually restrained by the " U V cutoff' of the sources.  This is just  saying that one can not probe the U V regime without a "full" renormalization. Let us denote the two-slice L R C model as 2S-LRC. This chapter is divided into three sections. The first section presents the set up of the 2S-LRC for I Q E D in Landau  Chapter 6.  Two-Slice LRC  104  gauge and states various bounds on the running covariance and spinor loops subject to some imposed conditions on the running couplings. The second section describes the Asingularities of the non-renormalized graphs. The third section describes how to remove these A-singularities by renormalizing at the top scale using only the F  counterterm  2  corresponding to the F local part of the scale U+1 vacuum polarization (VP) diagram. 2  By employing appropriate external sources with compact supports, in taking A —>• oo and setting fictitious legs to zeros, the resulting A-renormalized graphs have no top scale photon lines and fictitious lines. Since there are no top scale photon lines, these graphs are bounded by powers of M  uniformly in U'.  u  We denote p^  — p[ > '} and p  +1  u+1  u  = p^°' ^ respectively as the top photon slicing and  u  u  B  the bottom photon slicing. As for the spinor fields, the top slicing is p f and the bottom slicing is p  = p^°' \  =  pl + '°°) u  1  We apply the L R C to IQED with only two slices  u  F  + 1  corresponding to the above slicing of the fields. The starting effective potential at the top scale has the form,  EA7 ^ + 1  (6.1.1)  1=1  where A; = Aj — 1 for 2 < i < 4, A, = A; for i = 1 or i = 5, and Vi are defined in (7.1.2). We do not include the (d • A)  2  and A counterterms by setting A f 4  + 1  = Af  + 1  = 0 here  because the (d • A) and A local parts of the graphs from the expansion do not contain 2  A  any A-divergence (UV divergence coming from taking A —» oo; see the calculation of the local parts of the vacuum polarization and four photon legs graphs in Appendix E.2). In this model, we apply the Landau gauge in the photon covariance where 77 = 1 and only shift the Vf's with 2 < i < 4. We wish to find the resulting graphs generated from the two-slice expansion that are A-divergent. In finding these A-divergent graphs, we first examine the spinor loops. We need bounds on the spinor lines in order to find bounds on the spinor loops. Let us state bounds on the sliced covariance in the usual  Chapter 6. Two-Slice  LRC  105  L R C with multi-scales. Similar bounds on the sliced covariance in 2S-LRC follow from the bounds in the multi-scale case. Here we imposed the conditions that  O(lnA), and for Af ^ A3  (6.1.2)  with 0 < s < U + 1 (6.1.3)  In the two-slice model A- = Af for 0 < s < U. We will see later in Section 6.3 that by choosing A ^  as the coefficient (which is of O(A);  + 1  part of the VP  U+1  s e e  Proposition 6.1) of the F  2  local  diagram plus a constant independent of A, and all the other couplings  at scale U + 1 as constants independent of A, then the resulting coupling flow satisfies (6.1.2) and (6.1.3). We also assume the imposed conditions (5.4.12) and (5.4.20a-e) so that the running covariance are well defined. In the following all 0 '  s  a  r  e  with respect to A and unspecified scale s is assumed to  be in the range 0 < s < U + 1.  Lemma 6.1 Suppose the conditions in (6.1.3), (5.4-12) and (5.4-20a-e). 1. (6.1.4)  O(i) if 3 = 0  O(l)  (6.1.5)  2. Further suppose the condition in (6.1.2). t  O(anA)" )  , O(l)  1  ifs = U + l if s < U  (6.1.6)  Chapter 6. Two-Slice LRC  106  0((lnA)- ) x  oo  \B n  { O(l)  2 / 5 = [7 + 1  (6.1.7)  ifs<U  Proof: Note the bounds on the running covariance follow from the bounds on the running slicing. For the spinor case, (6.1.4) follows from (5.4.21) and (5.4.22). As for the photon case, setting 77 = 1 and 7 = r+^Ag, and using (6.1.2) and (6.1.3), s  the bound in (5.4.13a) simplifies to  r  i  = 0((lnA)- ) 1  I 7 T r  ifs = £/ + l  V%<{  Q.E.D. £r-77 = 0 ( l )  Let us denote L%j  1  (6.1.8)  ifs<[7  as a pure scale U+1 spinor loop with n type-j scale U+1 spinor  lines and L ^ J as a mixed scale spinor loop with n type-j spinor lines with at least one of the scales at s < U. Note that for a nonzero trace loop L , n must be an even number n  (confer Furry's Theorem in [FHRW88]). We will see that there are not too many of these spinor loops that are A-divergent.  L e m m a 6.2 Suppose all the external momenta feeding into the following spinor loops are confined in a compact domain. 1. Forn > 2, ^n,j  0(1)  if 3  =  0  00  0(A" )  if j ? 0  O(l)  if 3  2  (6.1.9)  2. For n > 6,  ll-^nj^loo — I  0(A" ) 2  =  0  if3 7^0  (6.1.10)  Chapter 6. Two-Slice LRC  107  Proof: see Appendix D. Next we consider the spinor loops L  and L% .  + l  +1  2  Since only the A-divergent parts  are of significant, we determine these parts only up to O ( l ) - E  y e n  though, because the  effective potential at the top scale in the L R C setup satisfies the Ward Identities, the gauge variant local parts of these loops are formally zero, the contribution of a single slice is not necessarily zero. But because of vestigial gauge invariance, we will see that these one slice gauge variant terms contain no A-divergence and are O ( l ) - Thus in considering the Vj-local part of these loops, we only give an explicit form on the O 0 A ) P t of the n  F local part of 2  a r  VP . U+1  Proposition 6.1  1. 3  VP  U+1  = A[Y sign(j) L%f ) A = /hF* + fo(d-A) + A I £ sign{j)RL ^ \j=o / \j=0 l  2  ) A,  u  2  (6.1.11)  where  (\U+1\2 \2 ) A  fo l ^ l l o o  =  O(i)  i O(l) 0(  =  if3=0  {  •  0(A )  if  O(l)  if 3 =  _ 1  (6.1-13)  3*0  2. ^ |  1  | | o o  0  —  { 0(A~ ) 1  (6.1.14)  if3 + 0  Proof: See Appendix D. - From Lemma 6 . 1 , Lemma 6 . 2 , and the above proposition, it is not too hard to see the only graphs that can contain A-divergence are graphs containing VP subgraphs. In the next section, we will give a more explicit description of the A-divergent graphs.  Chapter 6. Two-Slice LRC  6.2  108  A-divergent graphs  In the following, using the results from Section 6.1 on the bounds of the spinor loops and the running covariance, we determine which type graphs contain A-divergence. Furthermore, since we need the running couplings in the 2S-LRC model to satisfy conditions (6.1.2) and 6.1.3), we wish to show that there exits the couplings A f  + 1  in the effective  potential (6.1.1) so that the running couplings A f at the next scale are constants independent of A plus error terms which are o(l). More specifically, we show that the only graph that needs A-renormalization at the top scale is the VP  graph and by choos-  U+1  ing a counterterm which corresponds to the F  2  local part of VP , u+l  the subsequent  A-renormalized graphs at the lower scales are finite when removing the cutoffs. Before we proceed further, we would like to introduce the following notations. 1. G% is a graph obtained from a tree  s  where all forks f > f are R-labeled forks, and G' = j dQdPK (P,Q)H^(p).  (6.2.1)  s  A  i 2. | | G | | = || / dQ\K (P,Q)\ a  A  |U = sup J dQ\K (P,Q)\. s  (6.2.2)  3. Halloo = M o o  where Hi is the covariance corresponding to the line L  (6.2.3)  Chapter 6.  Two-Slice LRC  109  4. L is a spinor loop containing n spinor lines. N  5. Q(G) is the set of loop momenta of G not belonging to a spinor loop of G. 6.  Vol(Q(G))  is the volume of the domain of integration of the momenta  Q(G).  7. C(G) is the set of lines of G not belonging to a spinor loop of G. 8.  C  9.  GL{G)  is the set of scale U + l photon lines of  {G)  T+1 B  is the subset of spinor loops of  G  G.  excluding the loop  L  + 2  L  loops; a is the  index labeling the loops in Q . L  10.  Q U+L(G) L  is the set of  L  +  spinor loops'of  1  2  G; a2  is the index for  Q u+i(G). L  11. A subgraph (not necessarily a subgraph obtained from a fork) of G is called internal if all its lines and legs carries internal momenta of G . 12. A subgraph (not necessary a subgraph from a fork) of G is called external if all its non-loop-lines (lines not part of a loop) and legs carry only external momenta of G . The following proposition allows one to narrow down the search of A-divergent graphs. P r o p o s i t i o n 6.2 Suppose G  S  contains more scale U+l photon lines than L 's +l  A  2  it has a fictitious line not belonging to an L , +1  2  or  then 0  (6.2.4)  Proof: A l l lines are estimated by  ||^||oo  except for lines in L^  +l  loops. These loops are estimated  Chapter 6.  Two-Slice  LRC  110  by Lemma 6.2 and Proposition 6.1. For example, if G  S  lines than L  + 1 , 2  A  contains more scale U+1 photon  s then  \\G%\\ <  c (lnA)'^  <  Vol(Q)  + l h | £ r i  \ \  J]  II^IU  0((lnA)- ).  (6.2.5)  x  The other case can be argued in a similar manner. Q.E.D. The above Lemma substantially reduce the number of graphs that are potentially A-divergent. Specifically, it allows us to deduce that A-divergent graphs must contains the following generalized vertices. 1. For n > 1, let  n -  VP  u+l  =  J dp A(p) K^ip)  A(-p).  (6.2.6)  where -  K™  = L^H^jr)  ••• L ^ H ^ L ^ *• ' n—l copies  (6.2.7)  2. For n > 0, let u+i  U+1  = Note that  VQ_ U+I VP  I  dpdr^p-r)i{r)Hl (r)K ^ {r)^{-~p) +l  v  n  p  is just an ordinary vertex.  U+1  u  +  u  u+i  1  +  1  u  +  )  (6.2.8)  Chapter 6. Two-Slice  Lemma 6.3 If G  LRC  is not a generalized vertex V _ u+i or a graph consisting only  +1 A  of the V _ u+i's n  111  n  vp  connected by Fermi lines then  vp  lim \\G \\  = 0.  u +l A  (6.2.9)  A—>oo  Proof: Note that each V _ u+i has the same number of scale U+1 photon lines as the number of n  L2 's  vp  and each internal n — VP subgraph has one more scale U+1 photon lines than its  +l  L2 s.  Hence if the reduced graph G /V - u+i,  +l,  obtained by replacing each V - u+i  +l  A  with an ordinary vertex v in G , +l  A  n  vp  n  vp  contains fictitious lines or scale U+1 photon lines  then it satisfies the hypothesis of Proposition 6.2 and thus lim | | G J [ | | = 0 .  (6.2.10)  +1  A—>oo  Q.E.D.  Corollary 6.1 The only (nonrenormalized) are the graphs: n — VP ;  L^ {Ni,  U+1  n-chained V^-ypU+i,  +1  G  u +l  • • •, N ) n  A  that can have l i m ^ o o | | G A | | ^ 0 + 1  with n V ^ u+i Ni  vp  vertices and n > 4;  1 < i < n (see Figures 6.8-6.10).  Proof: Follow directly from Lemma 6.3. From the above corollary, for n > 2, since an n — VP  u+l  has photon lines which carry  only external momenta, it does not have a K-local part (localizing at p=0). Thus it is not renormalized at scale U+1. It will be shown later that a graph G  e  n — VP &s u+1  A  containing an  a subgraph is zero in the limit A —> oo provided we impose some restrictions  on the external sources. Consequently, the only A-divergent graph at the top scale that requires renormalization is VP  U+1  and we have the following result.  Chapter 6.  Two-Slice LRC  112  Figure 6.8: n -  VP  u+l  Chapter 6.  Two-Slice LRC  113  •  U+l  •  •  U+1  Figure 6.10: V ._ u+i, N  U+l  1< i < n  vp  From (6.1.12), let us write h{VP  )  = b\nA + C + o(l)  =b  +  (6.2.11)  3  vp  where  (Af ) " (Al#' +1  6  =  2  C  ( 6  ' ' 2  1 2 )  C is a positive the constant, and C is a constant independent of A but with a possible, 3  dependence on U.  Proposition 6.3 By choosing Af  + 1  =  K  =  b\nA + C + K ,  lt  \ 3  u  + l 2  = K  2  3  — K4,  = K  5  (6.2.13) (6.2.14)  where the Ki are independent of A, then lim Af = Ki A->oo  (6.2.15)  Chapter 6.  6.3  Two-Slice LRC  114  R e n o r m a l i z e d 2 S - L R C on I Q E D  In this section, we complete our assertion to the claim that A-renormalized graphs in a 2S-LRC on I Q E D do not require further renormalization, i.e., the A-renormalized graphs have finite limits as we remove the cutoffs A —• oo followed by C/'  oo. What is left  to show is that the A-renormalized graphs in Corollary (6.1) have zero limit as A —>• oo, that is, given a G obtained at the root scale containing n — VP s u,  A  or  V _ u's n  vp  lim \\G%\\ = 0.  then (6.3.1)  A—»oo  In other words, by Corollary 6.3 and the above, in the limit A —¥ oo, a non-vanishing G is not allowed to contain any unrenormalized L ^ ' s , U scale photon lines, or fictitious e  lines. Actually (6.3.1) might not be true in some cases where an n — VP  U  is external.  subgraph  In order for (6.3.1) to holds for these cases, we need to imposed some  compactness requirement on the external sources. We would like to introduce more notations. 1. Each external source has support V  e  C {p | p < (1 - e)M } 2  (6.3.2)  2U  where e is the constant defined in the slicing functions p . s  2. For a region V | | / ( p ) | | = sup|/(p)|.  (6.3.3)  P  pEV  3. Let (G) denotes the graph G with all its legs amputated. In particular, u  ( -VP ) u+1  n  +  U+l  1  U+l  U+l  = U+l  =  U+l  L \p){H \p) u +  u +  2  B  u  +  1  L \p)T-' u + 2  u  +  1  (6.3.4)  Chapter  Two-Slice  6.  LRC  115  and  0  u+i  (V,n—vp' U+1  U+1  u +  B  2  U+1  U+1  n  {H \p)L \p)) u +  u+i  (6.3.5)  4. For a given G , we group the non-renormalized L ' s of G into the following two 2  types of subgraphs with legs amputated. (a) A maximal k-chained n — VP  of a graph G is a sequence of n —  u+l  VP s u+1,  chained together by scale U photon lines. The sequence is obtained by amputating all the legs of a subgraph G -n-vp max  of n — VP s  and two V - u+i's  u+1,  V -vp + u  with the following arrangement. Each  vp  is not a part of a (non-renormalized)!/^ , the n — VP s  l  1  n  V _ u+i's n  n  which is a chain consisting  and the  u+1,  are connected by scale U photon lines with the two V ^ u+i's  vp  n  — VP ) ...^  situated at both ends of the chain. We denote Max(n maximal k-chained n - VP .  as a  u+l  nu  Graphically, a Max(n  u+l  are  vp  nk  - VP ) ...  is the  u+1  nu  tTlk  following diagram. U  (V . u+,)vAy-(n -VP ) n  U  U u+1  vp  2  •  •  vA^(n -yp ) v / V X u+1  •  V . -)  k  n  1  = (V _ )H^(n ni  VP  k  vpU+i  - VP ) u+l  2  • • • H {n ^ u  B  k  VP )H^(V _ v ),  -  u+1  nk  vp  +1  (6.3.6) where k > 2; n , n x  V _ u+i = VP  u+l  1  vp  k  > 0; for 1 < i < k, rii > 2; £*=o n; > 0(note that  are all renormalized at scale U+1). The reason for calling  it maximal is that, by the definition of a Max(n — VP ) ... u+1  nu  >nk  where the two  Chapter 6. Two-Slice  LRC  V _ u+i's n  116  are not part of a (non-renormalized)^ " ), it cannot be contained 4  vp  - VP ) ... n  in a longer Max(n  where k > k.  u+1  nu  t  1  k  (b) In a similarly manner, we define a maximal k-chained V _ u+i n  vp  (an external  subgraph with legs amputated) as the sequence obtained from the a chain like a G ax-n-vp  except that there is only one V _ u+i  m  n  chain. We denote  Max(V _ u+i) n  VP  n\,—,nk Graphically, a Max(V _ u+i) ... U n  (y . n  VP  nit  = iV _ v i)H (n vp  +  B  a  •  maximal k-chained  •  • ••H {n  u+l  2  s  B  YH=O ^  To get control on bounds of graphs G\ containing V _ u+i n  > 0.  and n - VP  U+1  vp  lower scales, it suffices to consider bounds on maximal k-chained n — VP  u+l  vp  vp  (6.3.7)  U+1  k  x  ni  n  v/v-\n k -vp u + 1 )  •  - VP ),  u  where k > 1; n > 0, ; for 1 < i < k, rii > 2;  k-chained V _ u+i.  V _ u+i.  is the following diagram. U  ;nk  - VP )  u  ni  a  )vyv^(n 2 -vp u + 1 )  vpU+1  situated at one end of the  vp  contracted at and maximal  We consider the following two lemmas on bounds of n —  VP  U+1  subgraphs.  Lemma 6.4 1. Fort<U  Let n > 1 and n > 2 be positive integers. x  2  -1, (i)  (n - VP )H  = 0  u+l  B  2  (")  (V  n i  _^  ) ^ = 0.  + 1  (6.3.8) (6.3.9)  2. (i) (ii)  A  lim ||(V _ ni  t)pt  / i)||  lim | | ( V _ , n i  A—>CO  +  p C  00  = l  / )^|| + 1  o o  (6.3.10) = 0.  (6.3.11)  Chapter  Two-Slice LRC  6.  117  3. i)  lim \\{n -VP )H \\ < u+l  1,  u  2  B  o0  • (6.3.12)  A — > o o  lim 11 (n - VP + )H%\\  (ii)  u  = 0.  l  2  x  (6.3.13)  A — > o o  Proof:  See Appendix D L e m m a 6.5  1. (a) For (n^n^) /  (0,0),  Urn || Max(n - V P  u + 1  )  n i !  ...  IU = 0,  ! r i f e  (6.3.14)  (b) For (m,n ) = (0,0), fc  IU <  Um || M a x ( n -  •  (6-3.15)  2. Um || Max(n  ||  - VP ) ... nu  vp  nu  tTlk  nk  e re  = {P\P =  Y1 iPi a  u+1  nu  ink  ; \Pi\<r ; <k = 0 or ± 1}, e  is a positive constant. If m r < (1 — e) / M l 2  e  — VP ) ...  is re-  domain  V where r  (6.3.17)  V  ^. Suppose that the external momentum feeding into Max(n stricted to the  (6.3.16)  x  \\ e = 0  Max{V _ u+i) ..., n  \\ = 0  u+1  e  11 Max{n  - VP ) ... u+1  nu  >nk  u+1  (6.3.18)  then  \\^ = 0.  (6.3.19)  Chapter 6. Two-Slice  LRC  118  Proof: Part (3-4) follows from the fact the support of p  is disjointed from the  u+l  V s. e,  For Part 1(a), Wlog, we suppose n\ ^ 0. Using the fact that fc—I  || Max(n-VP ) ...  < {{(V^u^H"^  u+1  nu  tnk  11(V„ _^i)11  JJ \\(n -VP )H \\ u+1  u  i  00  fe  w  +  (6.3.20) from (6.3.11), (6.3.12), (6.3.10) of Lemma (6.4), we have (6.3.14). If ( n n ) = (0,0), 1;  fc  then the first term of (6.3.20)  IKKj-opU+O-H^Hoo =  Halloo <  ^jj  M  2 U + 2  "  (6.3.21)  Since the kernel of an ordinary vertex is 1, the last term ||(Vo_ i/+i)||oo = 1- Thus by up  (6.3.12) and (6.3.10) of Lemma 6.4, we have (6.3.15). For Part 2, if(ni,n*) ^ (0,0), then (6.3.14) of Part 1 implies (6.3.16). For the case (ni,n ) = (0,0), fc  || Max{n-VP ) ,., u+1  ni  || < H ^ I U n Wi^-VP^ )^^  ||(n _ -VP  1  nk  a  f c  1  p + 1  )^|| . 1  (6.3.22) By (6.3.21), and (6.3.12), (6.3.13) of Lemma 6.4, we have (6.3.16). Q.E.D. Next we would like to state the proposition that lim _>oo | | C | | = 0 if it contains an n— A  VP . U+1  A  The strategy of the proof is that we factor the kernel of G\ into sets of Maximal  chains, L f ' s , and the remaining lines. We can first select a subset of internal chains + 1  corresponding to the loop momenta (excluding L ) +l  2  that flow through the internal  chains. These selected internal chains are estimated by 1-norms. The remaining objects are estimated by sup-norms and the volume of the domain of the integration over loop momenta not belonging to the L ^ ' s . + 1  Chapter 6.  Two-Slice  LRC  119  In proving the proposition, we adopt the notations defined in Section 6.22 and introduce more in the following. Let us drop the subscripts -,n nu  k  from the Maximal chains.  Given a G , 1. N (G)  is the set of internal Max{n - VP )  of G;  2. N^ (G)  is the set of external Max{n - VP )  of G;  l  U+1  max  U+1  ax  3. Vmax(G) i the set of (external) Max(V _ u+i)  of G;  s  n  vp  4. Q\(G) is the set of loop momenta (excluding L Maxin  -  + l 2  loops)of G that run through the  VP Ys; u+l  5. we arrange the loop momenta so that for each q — VP )(q)  Max(n G  the set of these selected internal Max(n  i s  6. Q (G) =  - l/P  c/+1  —  VP ); U+1  )(g)'s;  1  is the set of scale U photon lines of G not belonging to a Max(n —  7. C^ (G) +l  VP ) u+1  Max(V - u+i); n  vp  8. C(G) is the set of lines of G not belonging to a spinor loop, a Max(n a  we select a  Q(G)/Q (G);  2  or a  Qi(G),  where q is the momentum feeding into Max(n  u+l  Lax( )  N  e  — VP ) u+l  or  Max(V _ u+i). n  vp  Proposition 6.4 lim IIGlll = 0, A-K30  if G  e A  satisfies one of the following:  1. it contains a fictitious line; 2. it contains an internal Max(n  —  VP ); U+1  (6.3.23)  Chapter 6.  Two-Slice  LRC  120  3. it contains a (external)  Max(V _ u+i); n  vp  4- it contains a scale U+l photon line not contained in a Max(n  — VP )  or a  U+1  Max(V _ u+i); n  vp  5. it contains an external Max(n — VP )  and m external sources where the support  u+l  of each of the source is a subset of the domain  V  = {p\p  e  < ( l / m ) ( l - e)M }.  2  2  (6.3.24)  2U+2  Proof: Part 1 follows from Proposition (6.2). We now suppose that G\ contains no fictitious line. Part 2-5 follows from the following bound and Lemma 6.5  IIGAII  <  c  II  II  \\Max(V _ i)\\ n  vpU+  De  " m a x  i  voi(Q ) 2  (A )-I^ I 3  J+I  +I  y  \\Max{n-VP )\\ u+l  be  m a x  n nwioo aeg  imioo  (.ecic  u +l  L  II  n B  ]J \\Max(n  WMaxin-VP^ )]^ 1  -  V  r J  P  c  /  +  1  ) (  9  ) | |  1  .  (6.3.25)  Q.E.D. Note the above Proposition is not adequate enough for removing the graphs involving external Max(n  — VP  t/+1  )o,n ,-,n _ ,o in the limit A —> oo since the number of external 2  fc  1  source can be arbitrary high. A remedy for the inadequacy is to use bounded external source and bound the external Max(n  — VP  t/+1  )o n ,-,n _i,o by an 1-norm. )  2  fc  Chapter  6. Two-Slice  LRC  121  Theorem 6.1 Suppose each external source <$>\ has bound Halloo < c .  (6.3.26)  e  where c is a positive constant. If G\  contains a fictitious line or a scale U photon line  e  then  H m | G | = 0.  (6.3.27)  A  where \G%\ is the "absolute value" of the graph with external sources  included.  Proof: Let {(j>t}i<i< be the set of external source for G\. We consider two cases. m  1. Suppose that N^  is empty, then  ax  \G \ e  A  <  \\Gl\\  m niltflloo  (d Y e  1=1  <  l | G | | (c )  (r ) - -  m  A  4 m  e  (6.3.28)  4  e  which, by Part 1-4 of Lemma 6.4, goes to zero in the limit A - » co. 2. Suppose N^  0.  ax  Let Max(h  — VP) be an element of N!^  ax  and N^  ax  =  N* /{Max{h-VP)}. max  m  \G \ e  A  <  \\G /N* \\ A  max  II  Halloo  i=l  (^ ) e  4m  n Me  8  \\Max{n-VP ^)\U u  max  [  dP \Max(h-VP)(P i  i  + P)|,  (6.3.29)  where G / A ^ a x * the reduced graph obtained by deleting all the external s  A  Max(n  — VP ) U+1  in G , Pi is the external momentum of <j>\ and P is a linear e  A  combination of other external momenta.  In the last factor of the above , we  Chapter  6.  Two-Slice  LRC  122  make a translation, Pi = Pi + P and bound the resulting translated integral by \\Max(h  — VP)\\i.  Inserting this bound in (6.3.29), we have  \G%\ <  \\G jN \\ e  (c ) (d ) -  e  m  max  JJ fife J  e  4m  8  e  \\Max{n-VP )\\  \\Max(n  u+l  oa  - VP)\\ .  (6.3.30)  x  max  From the bound in (6.3.25) and Lemma (6.5), it is easy to see lim \\G jN* \\  (6.3.31)  e  max  A—>oo  is bounded by a constant (which may be zero). Now if (6.3.31) is not already zero then the \ \Max(h — VP)\\i  term would make \G\\ zero in the limit A -» oo.  Q.E.D. The result of this chapter can be easily extended to similar models where the employed external source fields have compact support but without the restriction that the support must sit inside the ball of radius M . u+1  In the less restrictive models, one must do a multi-  scale expansion on the bottom slice corresponding to the slicing p ' ' ^ . In the remaining 0  chapters, we will present a L R C model with "full" renormalization where the top slice is A-renomalized by only a F -counterterm and the bottom slice is renormalized by the R C 2  multi-scale tree expansion scheme developed in Chapter 5. From this multiscale L R C , we would like to analysis the resulting coupling flow of the expansion. For the multiscale L R C mention in the above, one immediate consequence of Theorem 6.1 is that in computing the /3-functions of the G  +l A  limA_><x>  coupling flow, we may discard any  that contains a fictitious line or a scale U+1 photon line. The following corollary  describes what kind of G  u +l K  can survive the  Corollary 6.2 If G% is non-vanishing  limA_>oo-  in the limit A  —>  oo then any G  + 1 A  of G\ must be a graph consisted of only ordinary vertices connected by Fermi  subgraph lines.  Chapter  6.  Two-Slice  LRC  123  We conclude this chapter by setting up a 2S-LRC model for IQED that only require renormalization at the highest scale forks of the corresponding trees of the expansion. Let VQ be the root scale potential of the 2S-LRC model for IQED with top scale couplings Af  + 1  Af " 4  1  =  Ki,  i^3  =  blnA +  K,  (6.3.32)  3  where Ki are constants independent of scales and A, and the external source satisfying the condition (6.3.26) and (6.3.2). In this model, there are no renormalizations at the lower scale forks.  Corollary 6.3 Let G  r A  be a graph  lim  ofV . r  0  lim \G \ < K (U). r  A  G  where KG is a finite constant depending only on U and the perturbation  (6.3.33) order of G.  The above corollary implies that only the top scale renormalization is required for obtained well defined renormalized graphs with no cutoff.  Chapter 7  Multi-Slice L R C  7.1  L R C coupling flow  In Chapter 6, we determined how the top scale coupling A f  + 1  of the 2S-LRC IQED  model depends on the loop regularization parameter A. By the choice of A f  + 1  stated in  Proposition 6.3 of Section 6.2, the resulting A —» oo graphs of the expansion of the two slice model do not contain top scale photon lines and fictitious lines. Although the two slice model is simple, it lacks the flexibility of allowing external sources with arbitrary high momentum. In order to remove this deficiency, we extend the two slice model into a multi-slice model where the bottom slice .p^°' ^ is further sliced up into [7 + 1 slices, and u  after integrating out the scale U + l shifted slice as described in the two slice model, the R C scheme described in Chapter 5 is applied to the remaining slices. The dependence of A f  + 1  on A for the multi-slice model remains the same as described in Proposition 6.3.  In order to determine the dependence of A f  + 1  on the U V cutoff parameter U, we need  to analyze the A —>• oo limit coupling flow of the multi-slice model. This chapter is divided into 4 sections.  In this and the next section, we describe  the set-ups of the coupling flow and the A —> oo limit coupling flow for IQED obtained from the L R C with multi-slicing. In the third section, we describe some aspects of the (3 functions of the the A —> oo limit coupling flow. In the last section, we present a complete summary of the multi-slice L R C expansion for IQED. From the L R C set up, a flow of running potentials V ($- ) and a flow of running s  124  S  Chapter  7.  Multi-Slice  LRC  125  external quadratic terms Q- K $S  8  are generated from iterating the procedure described  S  in Lemma 5.1 and the subsequent discussion: shifting of covariance; integrating out the scale s + 1 sliced field £  s + 1  (transformed by the shifting); renormalization by modifying  the coefficients of the local terms of V (Q- ) S  and &- K $- .  S  s  s  Recall that in the running  s  e  coupling tree expansion scheme described in Section 3.6, one obtains a flow of the running couplings by adding up the coefficients of the local terms extracted from the graphs of V (<S>- ) as depicted in (3.6.12). In L R C , a flow equation is obtained in a similar S  S  way except that the graphs of the running potential V ( $ - ) contain no local quadratic s  S  vertices since the quadratic terms go down the next scale via the shifting transformation defined in Lemma 5.1. In the L R C for IQED, the local terms that we wish to keep track of are: 3  i  =  v  3  £ ^ ¥ %  ^2 = 5 ; * ^ ,  j=0 V  3  =  3  v = E  j=0  F,  V = A,  2  i  -  (7.i.i)  j=0  V = (d-A) ,  2  2  5  W  4  6  V = A\ 7  Vi with i = 2,3,4, 5 are "external" local terms that "flow down" the scales via the shifting transformation (which includes renormalization of these local vertices) and the remaining local terms flow via the Gaussian integrations with respect to the shifted sliced covariances and the renormalization procedure. From adding up the local parts at scale s, we get  -\ V° S  =  - A T  1  ! ? +  £  L sG  where Q is the set of graphs obtained from the scale s potential s  (7.1.2)  V  y ($ s  <s+1  ).  The corre-  sponding flow equation is then  AJ  +  1  -AJ = §  (7.1.3)  Chapter  7.  Multi-Slice  LRC  126  where St = E  Pi(G)  (7-1-4)  Geg  s  and Pi{G) are the coefficients in the localizations LG  = P!(G)Vi.  Vi  (7.1.5)  We call the Jj's the @ functions of the coupling flow.  7.2  The A —> co coupling flow  In this section, we describe the A —> oo coupling flow obtained from the multi-slice L R C for IQED in the Landau gauge. As in the two slice model, the scale U+l  (top scale)  couplings are chosen to have the following form Af X  =  Ki(U),  =  b\nA +  + 1  +1 3  (7.2.1) K (U), 3  where 6  =  "  C  ( W  ( 7  - 2  2 )  C is a positive constant and Ki(U) are functions of the scale U but independent of the loop regularization parameter A. The term b ln A comes from the coefficient of the V  3  local part of the scale U + l VP diagram (see Proposition 6.1 of Section 6.1). Note that for IQED, b is positive since ( A f ) + 1  2  < 0. The reason for choosing the Landau gauge  is that the corresponding equations of the coupling flow seem to be easier to analyze than when choosing other gauges. From here on, we let the gauge parameter 77 = 1 (see (1.2.5a)). From the analysis of two slice model in Sections 6.1-6.2, at the top scale, only the V P diagram needs renormalization. In the limit A -+ 00, by Corollary 6.2, the subsequent  Chapter  7.  Multi-Slice  LRC  127  GV =  4P =  )  \  2P =  2F =  Figure 7.11: The G V , 2P, 2F, 4P diagrams subgraphs obtained at scale U + 1 contain no scale U + 1 photon lines and no fictitious spinor lines.  Hence in computing the (5 coefficients of the A —>• oo coupling flow, it  suffices to consider only the coefficients of the local parts K defined in (1.2.10). Also, let us denote G V , 2P, 2F, 4P respectively as the generalized interaction vertex diagram, the two photon legs diagram, the two Fermi legs diagram, and the four photon legs diagram (see Figure 7.11). These are the only graphs of concern in setting up the A —> oo coupling flow. We denote Q as the set of these graphs obtained from the trees s  The corresponding flow equation is then A*  + 1  - A? = St  (7.2.3)  where  E Pt(G)  st = Geg  s  (7.2.4)  Chapter  7.  Multi-Slice  LRC  128  where Pf(G) is the coefficient of the Vi local part of G. As suggested by the naive degree of divergence of graphs that contain V , V local 4  5  parts, the photon mass coupling X and electron mass coupling A 4 should be dimensionful s  5  couplings which respectively should carry a factor M  2s  and M . s  However, because of  the vanishing of local terms with coefficients that are integrals of odd integrands on a rotational symmetric domain or of integrands containing the trace of an odd number of 7 matrices, A 4 remains dimensionless throughout the flow (see the calculation of the K-local term of the E S E diagram in Appendix E). Thus we single out Af as the only dimensionful coupling. In anticipating the M  2s  equation at scale s + 1 by M  2s  A*  7.3  + 1  -A*  factor in the A 5 equation, we rescale the  and get =  — ( M — l)Ag 2  +1  + M~ 8 . 2s  s +l b  (7.2.5)  Aspects of the calculation of the /3 functions  We would like to discuss some aspects of the calculation of the @ functions of the A —>• 00 coupling flow obtained from the L R C scheme, we first show that the A —» 00 graphs from the root scale potential of the multi-slice L R C are effectively like the graphs obtained from a running covariance (RC) scheme without loop regularization except that some Fermi lines of the scale U subgraphs of the L R C has the slicing function modified as [U,oo)_  p  Let us observe the running lines from the non-trivial A —> 00 graphs of the multi-slice L R C for IQED. From Corollary 6.2, after taking A —> 00 and setting external fictitious legs to zero, even though the scale of the running covariance goes from 0 to U + 1, no graphs contain scale U + 1 photon lines. The Corollary also states that the non-trivial A —> 00 graphs contain no fictitious lines. From Proposition 6.3, A f = K$(U) where  Chapter  K (U) 3  7.  Multi-Slice  LRC  129  is independent of A and Af = A f  (7.3.1)  + 1  From the above observations, we show that effectively, the scale U photon lines are like the scale U photon lines of a non-loop-regularized R C model, and that the scale U + l and U Fermi lines can be combined as scale U Fermi lines of a non-loop-regularized R C model. We first consider the scale U photon lines. For a scale U photon slicing function Vg, we can split it into two parts: V% =  V~B,L+sh  +  where the first part is V% restricted  ^B,R  to the left overlapping 5ff and the sharp region S^ of p  and the second part is V%  u  h  restricted to the right overlapping region S  of p . u  R  Let p , respectively p , denotes p s  s  L  s  R  restricted to the left overlapping region S[, respectively the right overlapping region. From (5.4.9a) and the fact that p^ + p  u+l  = 1, 1-  V  —  u  '  B R  where j  s  o  u + 1  (7  H L  (i-pD(f+i)+r(f +i) +i  2  is defined in (5.4.9c). Thus it is easy to see that, since j  {  u + 1  I 0) ]  = O ( l n A ) where  A is the loop regularization parameter, following a similar argument as from (D.3.23) to (D.3.27) of Appendix D, lim | | P £ ( p ) | | i = 0. Thus' after taking the limit A -+ oo, we can disregard the V  (7.3.3)  H  P t °f a r  BR  a  ^B-  We now consider the scale U and scale U + l Fermi lines. In a L R C model for IQED, one can incorporate the scale U + l subgraphs into the scale U graphs so that effectively, the scales s of the running Fermi slicing functions range from 0 to U. Let us describe the incorporation in more detail. Now since (non-trivial) scale U + l subgraphs contains only Fermi lines, when summing over the scales on these subgraphs, effectively, each Fermi  Chapter  Multi-Slice  7.  LRC  130  line of the subgraphs represents the sum H^  +H  +1  where Vp  +1  U F  =  + V )R  (7.3.4)  U  F  and Vp are the running Fermi slicing functions, and R(q) = (m +  We  7  show that Vp  + Vp has the same form as V  +l  except the slicing function p  +1  u  is replaced by p  F  u  + p[  c/+1  '°°] = pl M.  Vp together. From (7.3.1) A f = A f , and so for  +1  + 1  F  simplicity, we set A f = A f  We illustrate our point by explicitly adding the  u  two running slicing functions V , + 1  of Vp  = A and we write the Fermi slicing functions (see (C.2.31a))  as _  •pu+i  P ^+1.00)^-1)  1 +  '  i + p (x-i)  +  u  i +^)(\ -iy u  2  P  Since the slicing is neighbourly, to justify our claim, it suffices to add the two in the overlapping region. Let Xs% be the characteristic function of the right overlapping region S% of V . V  F  By (E.l.24) of Lemma E.2, we see that on S g , V  u  +l F  + Vp =  ^f-  V  which bridges the corresponding sharp regions of Vp and  Vp . +1  Let us discuss the calculation of the f3 functions in somewhat more detail. For convenience in analyzing the coupling flow, we shall discard "irrelevant terms", using only the dominant part of the running covariance. In picking out the dominant terms, we make the following assumption on the order of the couplings.  Xt = 0(s ), a  a>0,  AX = o(X ), s  s  1 < * < 7.  (7.3.5)  Also, we assume the bounds A^ >  |A*|,  Al > 1 - c,  A*>1,  (7.3.6)  Chapter  7. Multi-Slice  LRC  131  where e < 1, so that the running slicings are well defined (see Appendix C). Later, we will show the solution of the flow is consistent with the above assumptions. Since we are interested only in the dominant behavior of the flow, we discard terms that are 0(-^~ ) as  with a > 0 because they have no effect on the dominant behavior of the flow. Moreover, these terms are not divergent when summed over scales. We shall call terms that are Q(M~ )  irrelevant.  as  We first consider the dominant part of the running photon slicing covariance. From (5.4.8a) of Section 5.4, in the Landau gauge where 77 = 1, the photon slicing covariance has the form H- = 3m  (7.3.7)  B  where V  S B  the running slicing defined in (5.4.9a) and L is the projection defined in (1.2.8).  From the form of V%,  it is easy to see that  1+ 7  s  where 7 is defined in (5.4.9c). In calculating the j5 function of low order diagrams, we s  would like to approximate 1 + 7 by 7 = \ s  s  s 3  + Af for extracting leading terms. From the  fact that  X  X  s  1+ 7  \  s  s  7's  M 2 S  ) A'+Y^" '  X  s  1  (1 + 7 ) 7 S  0( - »; r' " ). (M2  =  1)(A  (A!  S  1)  )  (7-3.8)  where x is the characteristic function of the support of p , in the calculation of the (5 s  s  functions, we may replace the 1 + Y by j  in V% defined in (5.4.13a) with the caution  s  that there is an error of Q((M  2  - l)\ (*y )~ ) s  s  fr°  2  m  b  e a  c h photon line.  Next we consider the Fermi running covariance which has the form H  S  = V R S  F  F  (7.3.9)  Chapter  Multi-Slice  7.  where R(p) = (tf+m)  LRC  132  and Vp is the running Fermi slicing function defined in (5.4.18).  1  By assuming the order in (7.3.5), from (C.2.31a) of Appendix C, n  n E  s  n  s  s  * - j B V +  =  &  >  "-  (7  3 10)  where B  =  s  (1 - p - p 5  s + 1  ) + p A* + p s  (A^ ) ,  s + 1  1  (7.3.11)  2  - W § v &) ' E (p)  =  s  and u  s  u (p R) + 2u s  s  Xp-  s+1  1  = m(X\ — X ,). We show that E s  s  (p  s + 1  --  (7 3 12)  R) + {u ) s+1  2  (p  R ),  s+1  2  is irrelevant. Observe that, for a graph G ,  each insertion of a term p R in a Fermi line of G lowers the degree of divergence D{G) s  by 1. Thus two insertions of p R in a 2P diagram (defined in Figure 7.11) reduces s  the corresponding D(G) = 0. This means, in a zero order localization of the resulting diagram, there can only be a V local term with coefficient of o(M ).  But since the A  2s  5  equation is scaled by M~ ,  5  this local term is irrelevant. From the above observation, we  2s  see that only the 2P and 2F diagrams (defined in Figure 7.11) with an insertion of one p R can produce relevant terms. Thus by assuming the orders in (7.3.5), effectively in s  our calculation, we may use p>s _  P_  5  P  O  (1 - p - p°+ ) + p° X +  s  s  l  (A^ ) '  s  1  2  2  ^  I  o\  '  provided that we include the following extra diagrams obtained from replacing the running slicing function of some Fermi lines of the relevant diagrams by the relevant part of the second term p E (B )~ s  s  s  of (7.3.10). More explicitly, let  2  H  S  = -mV^'R  (7.3.14)  2  EF  where ,E,S __ F  ~  P  S  S  P  (A^ -  A*) + 2 p*+! (BO*  A$  + 1  (AS :  + 1  -A£  + 1  ) '  (7.3.15)  Chapter  7.  Multi-Slice  LRC  Given a diagram < G >  133  where each Fermi line corresponds to a H , let < G >  B  E  denotes a diagram where one of the Fermi lines corresponds to a H  and each of the  EF  remaining Fermi lines corresponds to a H  B  with Vp defined in (7.3.13). To account for  S F  the contributions corresponding to the term E in (7.3.10) , for each 2P or 2F diagram s  G, we include the relevant local terms of all < G >  in the set up of the coupling flow.  E>H:  Actually, of the extra diagrams < 2P >  E  and < 2F > jj, only the latter has  B  E  relevant local terms. We first demonstrate in a simple case of a one loop < 2P >  E  ^  using a sharp slicing where the slicing functions p are characteristic functions. The sharp s  slicing versions of Vp and V '  S  E  (7.3.13) and in V '  S  E  are obtained by setting p  = 0 and B = A | in V  s+l  s  of (7.3.15). Consider the following one loop <2P >  E  S  B  F  of  and its local  parts from a zero and a first order localizations.  <^ 2  ( A ^ - A l O l  >U  (A§)W  K ^{p,m)  =  s  m J  J  dpA<°(p)K^(p)A< (-p), s  2  dqp (q)  p {p + q)tr[f  s  s  1  R{p + q) Y R(q) }2  (7.3.16)  Let us write m q + m 1  (7.3.17)  q + m  2  2  2  In the zero order localization, K' o m  = mj  dq p (q) tr [f s  R(q) Y R{qf  } •  Since odd integrand contributes zero, the relevant part is obtained by replacing a R by the first part of (7.3.17). Thus ^ , o = 3m j  dqp (q) s  tr Y  m R(q> q* +  R(q) Y  Now since the A equation is scaled by M 5  _ 2 s  , K^ V s  u0  s 5  = 0(1).  is an irrelevant term. The kernel  of the first order localization is KUM  = P°  M  J  d ( 1  P ^) S  T R  [T" R(qh R(q) a  i R(q) ' v  2  •  Chapter  7.  Multi-Slice  LRC  134  Since the trace of an odd number of 7 matrices is zero, the relevant contribution, obtained by replacing all R(q) with 4/{q + m ), is zero. 2  2  For a general < 2P > fi, we use the same argument as the one loop case to show that E  there are no relevant terms. Note that, firstly, the degree of divergence D(< 2P > fi) E  of < 2P > fi is 1 and so there can only be relevant terms from zero or first order localE  izations. Let us first consider a 2P that has no subgraph which requires renormalization. Then, there are an odd number of R's with an even number of 7 matrices placed in between the i?'s in the kernel of < 2P >  E  fi. In a zero order localization, the kernel is  an integral with its integrand containing a trace of an odd number of i?'s with an even number of 7 matrices placed in between the R's. The leading terms of the integral are then obtained by replacing a R with the first part of (7.3.17) (replacing all the R by the second part of (7.3.17) would yield an odd integrand), and thus are O ( l ) by power counting. Thus a zero order localization yields only a V 5 term with coefficient of Since the A equation is scaled by M~ , 2s  5  o(M ). 2s  this local terms is irrelevant. In a first order  localization of the kernel of the diagram, the resulting integral now contains a trace of an even number of R's with an odd number of 7 matrices placed in between these R's (the derivative operation of the localization procedure introduces an extra 7 ) . But since the relevant part of the integral is obtained by replacing each R with the second part of (7.3.17), the resulting trace is zero. In the case where a 2P contains renormalized subgraphs, since from the renormalization procedure, each derivative operation Pid  Pi  on  a R produces an extra pair of 7 matrices and maintains the same power counting and oddness argument, the zero and first order localizations of < 2P > fi E  To account for the contributions from the < 2F >  E  contributions with L  Vi  remain irrelevant.  ^'s, it is better to combine the  < 2F >fi in the following way. Since D(< 2F > fi) Ej  = 0, there  can only be relevant terms from zero order localizations contributing to the (5 function  Chapter  Multi-Slice  7.  LRC  o~ of A . Thus, we consider L 4  4  < 2F > with each running Fermi covariance replaced by  V4  H  = H  a  where V  S  and V '  S  +tF  + H  = V R - mVp R  S  F  S  EF  s  (7.3.18)  2  F  are defined in (7.3.13) and (7.3.15) respectively. Since  S  F  135  F  q -m 2  2  ~  R  W =  /  ,  2  (g  2  2  + 2m<JL 2  + m  ^ 2  )  m-<$  ,  2  R(q) =  ,  2  g  2  2  +  2  ,  m  2  we have  H V t e )  -  "C  m(T  g  2  + m  +  ^  ^ - n^H-  + (g  2  + m  2  Next we show that the relevant part of L y with using only part of H .  2  )  g  2  2  +  (7.3.19) m  2  < 2F > can be computed effectively  4  As in the argument of the irrelevancy of < 2P >  s  +F  it  E  suffices to consider an 2F that has no subgraph which requires renormalization. Since L  < 2F > is obtained from the zero order localization, it is easy to see that the kernel  Vi  of L y  4  < 2F > can written as a sum of terms J2j Kj where each Kj contains an odd  number of H ys. +  This means by power counting and the fact that terms with odd  degree in the internal momenta can be discarded, Kj can be written as a sum of parts where each relevant part of Kj is obtained by replacing a H+ (q) by F  m(V  s F  +  Vp ) s  (7.3.20)  q + m 2  and the rest of the other  F  2  (g)'s by  (-- ) 7  q  z  Since Vp + V '  +  m  is linear in the A 's, the above implies that L  S  F  4  3  21  2  Vi  < 2F > has coefficients  linear in the A 's. Subsequently, since the effective Vp defined in (7.3.13) involves only 4  A and no X\, except for the A equation, all other equations of the coupling flow have 2  4  no A appears in them. 4  Chapter  Multi-Slice  7.  LRC  136  Next, we show that some Vi local parts of some diagrams can be dismissed even though they have D(G) > 0. Since compactly supported external source field A (p) are e  used in the generating functional, the Feynman diagrams that contain lines carrying only external momenta p with \p\ bounded do not require renormalization because p (p) = 0 s  for sufficiently high s. For example, let us consider , VP > =  < V  X  R S  W  •  (7.3.22)  From the calculation of the V local part of < VP > in Appendix E , it is easy to see 5  that the relevant zero order local part of (7.3.22) is b X\ M  < v,A  2s  2  5  > - where b is H  3  5  defined in (E.2.76a). Since u  £  M  2  < «, A >  S  2  ks  |  (7.3.23)  s=r  =  dp dq i) {p) 52 e  i i ^ s  7°  9  2^ ^  n  (let  q  2  the restricted support of p (q) and A (q) being compactly supported allow (7.3.23) to be s  e  bounded (in norm) uniformly in U. Thus, in the set up of the coupling flow, we drop the diagrams of the form < GV, 2P >. Lastly, we consider graph containing the V . 6  Since V = A p T 2  6  A and TL = 0  where T and L are the projections defined in (1.2.8), we see that any graph obtained from contracting a V$ with any vertices including itself is trivially zero. Thus there is no relevant terms from the  functions containing Ag. We state our observation regarding  the dependence of A and A in the fi functions in the following proposition. 4  Proposition 7.1 functions  6  The fi function  5\ is linear in A and contains no Xe- For the fi 4  5 with i ^ 4, Sf contain no A nor Xes  4  Chapter  7.4  Multi-Slice  7.  LRC  137  S u m m a r y o f multi-slice LRC for IQED  We summarize the multi-slice L R C expansion for IQED in the Landau gauge. 1. Basic Lagrangian: 3  £  =  Y^jW+Mj^J+Ml+P^LA  (7.4.1)  3=0  - {Af  +1  K+ X M V u +l  +\% M V +1  2U  +  2U  2  2  + \V V  1  3  +  +1  5  - 1)V + ( A ^ - 1)V  6  4  \V V } +l  7  where 3  3  3=0  3=0  V  =  V  =  M (A)  =  m ,  Ml {A) T  = -  m + A , A _ P^P"  3  7  2  L  ^  3  -A p L A, 2  3=0  V = -A ,  V =  2  5  6  -Ap TA, 2  -A\ M (A)= m + 2A ,  2  2  ~ V  2  2  p 2  '  2  2  M (A)= m + A . T -  (7 A 0\  V - ~p^T-  [7 A.2)  2  2  P t i P u  2  2. Running terms Running fields: - scale s transformed "external" fields u+i  <> | < = i=s+l JJ ^ s  where  and Ti is the scale i shifting transformation defined in Lemma 5.1;  (7.4.3)  Chapter  7. Multi-Slice  LRC  138  scale s decomposition of the field:  = $<«-i + £  (7.4.4)  s  where £ is the scale s shifted slice field. s  Running external quadratic term $ - K <3>-: s  s  5  e  U+l  K  =  s e  K  K + RJ2 t^+i  p'iJ ?,  S  The nontrivial components of K  s  t+l r>  t ,  J =  1  t  (]St+l\2 [  1 + P  R  t+l  and F t (w.r.t.  —-• K t+l  the 9 components of $ - ) s  are : K'  n  K' 2 j+  =  K  = (\l-l)p L  + M° X ,  2  B  2  5  = .signU) (K° ) sign(j)  =  TTp^>  R  b  (R  = R, B  s  +W  Rj+2,j+s =  U  = K' = (X - l)p!+ (X\ -  T  J+3  R  s  j  )  2  jt  = Rj,  T  j+3J+2  R = tp!+M )- ,  (7.4.5)  1  3  where 0 < j < 3, sign(0) = sign(l)  l)M  3  = - 1 , and sign(2) — sign(3) = 1. Com-  pactly supported external sources are used to control the possible divergence from the term £ t = s +i p*(J*) . +  2  • Photon covariance:  l+p  2  1 - (p + p ) s  s+1  .2  where L is defined in (7.4.2).  + (p 7 + 1) + 9 + (7*+ + l ) s  s  S  /  \  2s  M  1  1  2  Chapter  7. Multi-Slice  LRC  139  • Effective Fermi covariance: Hp V  =  s  Vp Ro  t  =  (1 -  F  s P  -p )  +p \ +P  s+1  s  S  (\ 2 )  S+1  S  2  +l  2  where RQ is defined in (7.4.5). • Running potentials V : S  — top scale potential  u+i(Q<u i)  v  [Af+ y +A 1  +  1  =  /+1 2  2t/  r  7 + 1  2  + Af y + +1  +\V M V +1  M V + (A3  2U  6  5  \  - l ) V 3 + (A^ -l)y +1  V]  ($^  U +L 7  7  + 1  4  )  where the V^($) are defined in (7.4.5); -  top scale shifted potential  u+i^<u ^u i  v  +  x" v^ +^ )+\% v (^ H ); +l  + )  u  +1  +l  u  (7.4.6)  U+1  6  =  — scale s running potential -I  OO  n=l =  where  <P  S  K  <p  s  5  e  +  n  -  T>(<|<s)  is the scale s + 1 shifted slice field, £ ^  expectation w.r.t.  dp^ , 3+1  and V (Q- ) S  S  s + 1  ( - ) is the connected  is the scale s shifted running  potential where 00  1  n=l n  with r < s < U; -  in the limit A —> oo, graphs of the root scale potential V  R  scale photon lines and fictitious lines.  contain no top  Chapter  Multi-Slice  7.  LRC  140  3. Imposed conditions on parameters • Forms of the top scales couplings: Af  =  + 1  Ki{U)  -  b  Af  + 1  = Mn A + K (U), 3  (7.4.7)  ^  +l)2  c  -  b  i^3,  C  ( x ^ r  where C is a positive constant and Ki(U) are functions of the scale U but are independent of A. • Conditions on the running couplings:  X  = 0(s ),  A*  >  s  a  where A A | = A |  + 1  \X \, S  5  -  a>0,  AX = o(X ), s  l<t<7  s  X >l-e,  (7.4.8)  X >1,  s  S  2  A  X. s  4. A —>• oo coupling flow equations Af = A f  + 1  = UU)  z#3,  .  X =K (U). U  3  S  Let  =  the set of relevant generalized vertex diagrams obtained at scale s  Q  =  the set of relevant 2 photon legs diagrams obtained at scale s  Q  =  the set of relevant 2 Fermi legs diagrams obtained at scale s  Q  GV  2P  2F  Q\p =  the set of relevant 4 photon legs diagrams obtained at scale s. (7.4.9)  Chapter  Multi-Slice  7.  LRC  141  For 0 < s < U - 1, AA;  =  6' =  £  Pi (G)  AA5  =  <$| =  £  ^  +1  +1  GeS  A A3  =  AA  =  4  =  (G)  2 F  $+i(G)  £  +1  A  Gee,^  AA5  S + 1  + i  £  S' =  (7.4.10)  +l  = -5  Af  M  1  1  + M~ 5 2s  s +1 5  = -5  Af  M  1  + M~  Y  2s  Gee** AA*  =  S'  AA?  =  5 =  +1 6  s+l  =  Y  Pt\G)  Y, G e ^  1  ^7 (G) +1  1  where 01(G) is the coefficient of the V^-local part of the graph G. Aspects of the A —>• co /3 functions: • the effective scales of the running lines are from 0 to U ; • in the calculation of the /Vs, except for /3 , the Fermi lines are replaced by the 4  effective Fermi covariance; • pi with i 7^ 4 are independent of A and A6, and /? is linear in A and inde4  4  4  pendent of A . 6  We will make further analysis of the A —>• co coupling flow to determine the dependence of A f  + 1  on U in Chapters 8-10.  Chapter 8  LRC coupling flow with sharp slicing  8.1 Preliminaries In this chapter, we wish to study an O D E analogue of a primitive (i.e. only low order diagrams are retained) A —•) oo L R C sharp coupling flow obtained from (7.4.10) with the P functions computed using sharp slicing functions. From the analysis of the O D E , we find a crude estimate of the asymptotic behavior of the actual coupling flow with smooth slicing. Later in Chapter 10, we will justify the estimation by showing the error can be bounded by terms which are of smaller order than the O D E estimates. Because of the shifting of the covariance, one finds that the coefficients attached to the graphs in the L R C expansion of IQED can be roughly expressed in terms of the primary variable (generalized coupling) r = (^i) - ( A ^ A i + Af)-  f« i n  2  s  ^  •  }  In perturbation theory, it is desirable to keep £ small through out the flow of the running s  potential. Thus we would like to choose boundary conditions on the running couplings so that the flow of £ is U V free, i.e., l i m ^ . £ = 0. s  s  0 0  In Section 8.2, we analyze a one variable O D E in £ obtained from a primitive flow s  of IQED with sharp slicing. We discover that, by choosing the value £ of £ at the root r  scale r sufficiently small, the flow of £ of IQED is U V free. From the solution of £ , one s  s  then can recover the O D E estimates of the Af's. In Section 8.3, we apply the running 142  Chapter  LRC coupling flow with sharp  8.  slicing  143  covariance technique to the negative charge 4> perturbative model and demonstrate that 4  the resulting primitive O D E analogue in the generalized coupling is easier to analyze than the corresponding O D E with no shifting of covariance. We then briefly describe how one can extend the technique of using the shifting of covariance in some models in which the coupling flow can be characterized by a flow of a generalized coupling in the form of (8.1.1). In this section, from the L R C scheme for IQED in the Landau gauge with A —> oo, using the sharp slicing  (8.1.2)  otherwise with the property p p> = <% p\ l  (8.1.3)  we will set up a A —> oo primitive coupling flow using diagrams that contain up to two photon lines. The use of the sharp slicing provides an easy analysis of a crude picture of the behavior of the flow. We compute the p functions of the IQED by first using real Af and then make the switch Ai to iX\ in the flow equation (7.4.10) afterward to obtain the appropriate sign for the terms in the P functions. Note that the multi-slice L R C with smooth neighbourly slicing described in Chapter 7 is not well defined for Q E D for following two reasons. Since, for Q E D with real A f , the + 1  coefficient of the V local part of the scale U+1 VP diagram is negative (see Proposition 3  6.1 of Section 6.1), the choice of the X$  +l  in (7.4.7) is negative. Thus the corresponding  photon slicing function is not well defined in the left overlapping region of the slicing function p , u+l  i.e., the denominator (8.1.4)  Chapter  8.  of V  has a zero for p  +l  LRC coupling How with sharp  sufficiently small.  u+1  B  slicing  144  It is only well defined if the slicing  function is sharp and A is sufficiently large. The second problem is the corresponding shifted photon covariance violates the positivity requirement of a Gaussian measure. Even though L R C is not well define for QED, we still call the resulting coupling flow obtained from computing the j3 function using real Af a Q E D coupling flow. Since the IQED flow can be obtained from the Q E D flow by switching A to iXi, we do not distinguish the x  two system in the set up until we need to consider the solution of the flow. Using the sharp slicing defined in (8.1.2), from (5.4.8b) and (5.4.18), the running photon and Fermi covariances for scales 0 < s < U become Hi  =  H  = p'iD")- ^  8  where R~ (p) = m  7* = W ,  (8.1.5a) (8.1.5b)  1  F  L is the projection defined in (7.4.2), and  1  D  — ^  =  1 +  p  U  ^  ,(AS-i)  + 2  m  +  A  ^  ( ^ - i y  (8.1.6a)  (A|-A;w  +  f i  p + m z  z  The corresponding effective sharp Fermi slicing functions are obtained by setting  p  s+l  0 and B = X , in Vp of (7.3.13) and Vp + Vp' of (7.3.20); s  s  s  n w , -n*,. Vp + Vp  =  £  (8-i-7)  PL , P ( 4 ~ X°, (A|) S  A  A  2  2)  =  P^A| (A|) ' 2  Chapter 8.  8.2  LRC coupling flow with sharp slicing  145  Asymptotic behavior of the sharp slicing primitive flow  8.2.1  Sharp slicing primitive flow in the Landau gauge  We set up a sharp primitive flow correspond to a primitive flow of the coupling flow defined in (7.4.10) using only the low order diagrams described in the below. The (3 functions in the sharp flow are computed using the sharp running photon covariance of (8.1.5a) and the effective sharp Fermi running covariances in (8.1.7). Let us introduce the following notations for the relevant diagrams G V , 2P, 2F, 4P defined in Figure 7.11 of Section 7.2. For the one loop relevant diagrams, the vertex diagram, the vacuum polarization and the electron self-energy diagram are denoted by V , V P and E S E respectively. For relevant diagrams with n loops, GV^ \ n  2P<~ \ 2F^ \ 4PW denote the n loop G V , 2P, 2F, 4P n  n  diagrams. The relevant (3 functions of the primitive flow are the following:  6  = Pi(GVW),  h  =  fc(VP),  S = (3 (ESE),  =  fo(VP),  5, = MVP),  1  S  5  5 = (3 {2F^), 2  4  2  (8.2.1)  4  5 = /M4P ). (1)  7  We put in the two loop diagrams in the Ai and A equations because in the Landau 2  gauge, L  Vl  < V > and Ly < ESE > are irrelevant (see Section E.2 of Appendix E). 2  In the following, we write down an estimate of the pi functions of the primitive flow. We also assume the conditions (7.4.8) for the running couplings.  These assumptions  allow the running slicings to be well defined (see Appendix C) and extracting leading terms from the j3 functions (see the Appendix E). The detailed calculation of these (3 functions is done in Appendix E .  Chapter 8. LRC coupling how with sharp slicing  146  The A functions of (8.2.1) are: b\2  5M  51  s\5 (Af)  (-M) (r) 4  (8.2.2)  (1 + * U A ' ) )  2  (Af) (1 + E ( A ) ) (Ai) (r) (Af) Ag (Af) h (AI) 3(A|) (Af) 4  S  &22  5M  fi  s  3  622  2  2  5M  2  2  2  2  °4  5M  Al7  51  (Af) (A!) (Af) (AI) (Af)  s  2  5M  5  2  2  A  5M  6  5\  4  AT  5M where b , 6 , & i , 6 3  4  2  a r e 22  2  (AIT  constants independent of X and s  /?|, Af are described below.  If A? = A,-+ o(l),  l<j<2  (8.2.3)  where A i and A are constants, then each of As, A> | A 7 has the following form 2  A i - Af Af where  A - A 2  2  A  i  /  A  \  (8.2.4)  ^2  is a constant. In the calculation, there is a factor 5M = M — 1 coming from 2  the size of the support of the slicing function p . l  We divide each 5- by this factor  in anticipating making a switch to its O D E analogue. Each of E (X ), s  bl2  E (X ) s  b22  is of  0(5M A | / ( 7 ) ) and comes from replacing 1 + 7 by 7 in the photon running slicings s  (see (7.3.8)).  2  s  s  Chapter 8. LRC coupling flow with sharp slicing  8.2.2  147  O D E analysis  Our strategy for investigating the U V asymptotic behavior of the sharp primitive flow is to study the U V asymptotic behavior of the corresponding O D E analogue, where respectively, the difference operator A/8  where 8  M  = M — 1, and the discrete index 2  M  8M S are replaced by the derivative operator d/ds and the continuous index s. In switching to the O D E analogue, (8.2.2) become d\i  1 (Ai)  ds  7  d\2  022  ds  4  2  2  A?  ds dX ds dX  (8.2.5c)  q\2 A6(Al,A ), OA 2  2  4  (8.2.5b)  3  2  \2 A  b X  4  (A ) '  A?  4  (8.2.5a)  (A ) '  1 (Ai)  7  dXz ds dA  2  5  2  1 (Ai) 7(A )  2  (8.2.5d)  2  2  5  -A + /3 (Ai,A )  (8.2.5e)  Ae(Ai,A ),  (8.2.5f)  /? (Ai,A ),  (8.2.5g)  5  6  ds dX  5  2  2  7  ds  7  2  where 7 = A + A , 63, 64 are constants defined in (E.2.75), and 3  5  "A?  A (Ai,A ) =  b  Ae(Ai,A ) =  b,  5  2  2  /?r(Ai,A ) = 2  5  b  7  X  2  A?" A  (8.2.6)  2  A?_A|  X  2  A. 2  2  X\  A\  A  A  4  4  where b , b and b are constants defined in (E.2.75). Note that the terms corresponding 5  6  7  to the F ' s of the discrete flow become zero because they are 0((8M) ) 2  as 8  M  —> 0.  Since the subsystem (A A , A , A ) has no dependence on other couplings, it is con1;  2  3  5  venient to study the subsystem first. In analyzing the subsystem, we first drop the term  Chapter  A  8.  LRC coupling flow with sharp  slicing  148  2  /3e(Ai,  A ) in the A equation and justify its dismissal after finding estimates for the 2  3  solution of the resulting subsystem. In analyzing the subsystem, we follow the wisdom of the physicists by introducing the variables a EE A ,  WEE A .  2  (8.2.7)  2  We study the flow of the a, UJ, and 7 instead of A i , A , and A . From (8.2.5a)-(8.2.5c) 2  3  and (8.2.5e), the corresponding O D E in the new variables is dot — = ds  , cn 2 a bi2 - 5 — ^  .  2  UJ  x  2  ^ ds dy  = 2ujb -^,  — ds  =  6 7  =  - A + /? (A ,A ).  22  ^  3  2  2  a  A 5  05  a; 7  5  5  1  n  n  (8.2.8b)  j  UJ  . 8.2.8a  (8.2.8c)  -  UJ.  (8.2.8d)  2  The U V asymptotic behavior of the Aj's can easily be obtained once we find the asymptotic behavior of the new variables. Observe that from (8.2.8d) and  A!~/? (Ai,A ) 5  2  ~  (--)  0  8  2  9  and the rest of the system, asymptotically, is independent of the A equation. 5  Using (8.2.9), we rewrite (8.2.8a)-(8.2.8c) as  I£  = 26  I£  =  2b  22  \u> 7 /  (-"-)',  (8.2.10b)  \u> 7 /  UJ ds  LtL 7 as  (8.2.10a)  12  a ds  „  ft ^. 3  UJ 7  (8.2.10c)  Chapter  LRC coupling How with sharp slicing  8.  149  From (8.2.10a-c), one sees that the system naturally yields a flow of the variable C= — . UJ  (8.2.11)  7  We would like to interpret £ as the product ( Z i / Z ) ( l / Z ) in the physicists' multiplicative 2  3  renormalization scheme. The physicists derive the flow of the running charge square a from a flow equation of the renormalization multiplicative factor (l/Z ) 3  of a. In their  set up, Zi = Z and there is no A contribution in 1/Z because of the Ward Identities; £ 2  5  3  corresponds to 1/Z and only the F local part of the 2P diagrams contributes to 2  3  l/Z . 3  It turns out that the flow of £ determines the flow of all the running couplings. Differentiating £ by s yields 1 dC £ ds  ]_da _ l_duj_ _ 1 dj a ds co ds 7 ds  =  (8 2 12)  We express the flow of £ as an autonomous system with one variable by substituting (8.2.10a-c) into (8.2.12). This yields the (primitive) C equation of Q E D : ^  ds  =  02 =  &£ + &£ , 2  (8.2.13a)  3  -h,  & = 2 (6 - b ). 12  22  (8.2.13b)  As for IQED, since Ai —> iXi, we have the change £ —> — £ in the £ equation:  ^ds = -A>£ + &£ . 2  3  (8.2.14)  Since the R H S of the £ equation can have arbitrary higher powers of £ if we consider higher perturbations, we are interested only in the case when the origin is a strictly stable fixed point. Also, we forbid £ to be negative since this means 7 < 0 and the photon Gaussian term in defining the sliced photon Gaussian measure has a positive exponent, which makes the corresponding Gaussian functional integral i l l defined. We determine the U V stability of £ near the origin by making a phase space analysis of  Chapter 8.  LRC coupling flow with sharp slicing  150  (8.2.13a) and (8.2.14). From the explicit calculation of b (see (E.2.75) of Appendix E), 3  we have b < 0; thus /? > 0. From the sign of the term j3 C [ l + {P3/P2) (], we sketch 2  3  2  2  the solutions of (8.2.13a) and (8.2.14) near the origin (restricted to ( > 0 ) in Figure 8.12. QED:  fa/02  > 0  -P3/P2 > 0  Figure 8.12: The flows of C near the origin  From Figure 8.12, we conclude that only the case of IQED has a strictly stable fixed point at the origin. In this case, where £ flows towards the origin, the dominant U V behavior is given only by the quadratic term. Thus we may solve for an approximate  Chapter 8. LRC coupling flow with sharp slicing  151  (UV asymptotic) solution by dropping the cubic term in the £ equation. Subsequently, for IQED, f o r O < C < |A>/A|, r  C~,  v C c (s - r) + 1  (8-2.15)  C  r r  (  3  where c = —b > 0, 7" is the root scale, and — 1 < r < s. 3  3  From (8.2.15), we find the asymptotic behavior of the running couplings of IQED. Let 4  -  x  Then C = e /Y, C ~ ( r/Y) 2  e  s  d  a n  r  din ds  1 da;  d i n a;  1 d7 7d7  . x (  ; c  - ,  3  7  ^  1  +  C  3  (  2b C - 2  s  _  1 r  )  (  e  x(s) ,  2  ;  /  r  )  .  (8-2.16)  „, ~ , 26 C - 2 6  =  din 7 ~ds~ =  22  (8.2.17)  2  12  ds =  S  s o  OJ  1 da ct ds a; ds  W  /e 22  2 N 2  :x(s) , 2  Y7'  e 3C-c -,(s). 2  c  3  This implies that a\  2b  el  \u) J  c  Y  12  r  3  '  J  Hence a  ~  a  ^  ~  /2 6 2 e \ w exp I —— — [a;(s) - 1] J ,  r  exp  ^  [z(s) - i f ) , 2  2  7 ~ 7 x(s) r  = c (s - r) e + Y • 2  3  (8.2.19)  Chapter 8.  LRC coupling flow with sharp  slicing  152  This suggests that both a, to approach a non-trivial fixed point: oo  _  l-  / \  a-  =  }}ma(s)  o;°°  =  lim u(s) = OJ  ^12 r2 exp \2 ^ ^ j ,  „ „ „ (o  r  = a  r  e  exp (2 — ^  T  (8.2.20)  ].  From (8.2.19), (8.2.7) and (8.2.5d), for C U V A F , we obtain the following U V asymptotic behavior of the Aj's with i — 1, 2, 3, 4. (We are still using the continuous analogue of the discrete flow. To express the discrete flow, one needs to replace the continuous index s by the discrete index s S .)  For IQED,  M  A l  „  A ^ x p ^ ^ [x(s) - 1] j  A  2  ~  X exp ^  ^  A  3  ~  c (s - r) e + X ,  2  2  3  ~ ~  AA 4  ,  r  r  3  64  By  (8.2.21)  /C3  A ^4 R  setting Al = A  ' ftl)' e x p  A2  = ' ftr)' A  e x p  ''  (8 2 22)  It is easy to see that, for i = 1, 2,  £  ^  i=l,2  2 j»W  = A  *  C  3  + OM,) ). 2  (8.2.23)  7  This justifies the dropping of the /? (Ai, A ) term in the A equation. From (8.2.5f-8.2.5g), 6  2  3  (8.2.9), (8.2.6),  We  A  5  ~  - 2 h (bu -b )  A  6  ~  h(b  22  x2  et x(s)  - b ) lnx(s), 22  (  A ~ 67(^12 - b ) lnx(s). 7  22  summarize the U V limit of the couplings of IQED in the following table.  g  2  2  4  )  Chapter 8.  LRC coupling how with sharp slicing  153  U V fixed points of (UVAF) IQED zero fixed point  nonzero fixed points  \ oo \ oo  C°°, A °° 5  divergent couplings \s  \s  \s  3> 4> 6>  A  A  A  \s 7  A  Remarks 1) It is desirable to pick the boundary conditions of the couplings so that one can determine for what set of values (or the asymptotic values) of X , the resulting coupling u  flow is stable and C remains close to the origin. For example, in (8.2.21), by choosing the boundary conditions at the root scale for A,, 1 < i < 4, we can estimate for what asymptotic values of Af, the resulting flow of Af is stable and £ stays close to origin. s  2) Even though A is a zero fixed point, the actual coupling of the photon mass coun5  terterm is of Q(M /U) 2U  and hence not a zero fixed point.  3) If we consider the primitive flow of Q E D , from the phase diagrams, we cannot determine the asymptotic behavior of £ from using just the leading quadratic term of the (3 function of £. It seems not possible to determine the stability of the coupling flow without carrying out more detailed calculation of the higher order contributions. 4) In the running covariance scheme, the internal lines are modified by 1/7, 1/UJ. One can view this modification as having the vertex with each of its legs being renormalized by V \ / 7 (f° 7 > 0), V v ^ r  a n  d the coupling charge being renormalized by yfa.  This  is the multiplicative scheme of the physicists being iterated in a scale-by-scale fashion. Thus, it is £, rather than cn, that corresponds to the "effective" running charge square of the physicists' R G flow. Moreover, it seems that the asymptotic behavior of a is not as important as £ since the renormalized perturbative Green functions are not expressed alone in terms of a alone but rather through ( and others finite quantities. Hence the exponent of the highest powers of ( used in the ( equation should be the natural parameter for defining the perturbation order of the model.  Chapter 8. LRC coupling Row with sharp  8.2.3  slicing  154  The "Zi = Z " condition of the coupling flow 2  Because of an U V cutoff U introduced in the scale decomposition for regularizing the divergence in the tree expansion, the Ward Identity of Q E D is no longer an exact identity for the generating functional of the tree expansion. However, in the limit of removing the cutoff, i.e., U —> oo, one would expect that the exact Ward Identity should be restored. The physicists assume that the Ward Identity holds in the multiplicative renormalization scheme, even in the presence of a cutoff ( e.g., dimensional regularization). They argue that, as a consequence of the Ward Identity, there are no photon mass local parts for any Feynman diagram and that the multiplicative factors Zj's for the renormalization satisfy the relation Z — Z . We do not wish to take such assumptions on faith without 1  2  a rigorous justification. Nonetheless, we would like to implement the physicists' idea of "Zi = Z  2  in Q E D . In the coupling flow of the tree expansion scheme, we interpret the  "Zx = Z " condition in IQED as e\ = X[(l + o(l)) where s  2  2  X  u  lim -rjj = e.  (8.2.25)  (7-foo A f  To study the "Z = Z x  2  ;  condition in the view of the O D E flow, we consider the flow  of e(s) = a(s)/u(s) in the Landau gauge. We wish to determine for what value of e(r) where r denotes the root scale, the primitive flow of s(s) has e°° = e . Let m be the order 2  of the perturbation (in () used in the primitive flow and b\ and b n  2n  be the (3 coefficients  for the higher order vertex and ESE diagrams. From the primitive (ODE) flow (8.2.17) (in the Landau gauge), we have 1 de  _  e ds =  1 da  1 du  a ds  u ds  ^12 - 622) C + 2  (6  - M C + • • • + (61m 3  13  -  b )C ] = M  2M  K(C).(8.2.26b)  Let K(s) - K(r) = J K(C(*)) dt so that s  r  e(s) = e{r)e ^ - . K  s)  K{r)  (8.2.26c)  Chapter 8.  LRC coupling how with sharp slicing  Since ((s) = 0(l/s),  155  it is easy to see that e°° = e(r)e- \  Thus  K(r  e(r) = in order to establish Z\ = Z  e°°e ^ K  for the primitive flow.  U  2  Remarks  1. Note that if the initial condition e(r) = e , then it is easy to see that we must 2  have K(C) = 0. Thus b\  n  = b  for all n < m; otherwise, e°° would shift away  2n  from e . In this instance, the "Zi = Z  condition of IQED in perturbation theory  2  2  is equivalent to b\  n  = b  2n  for all n. The condition that (8.2.26b)=0 has stronger  implication than Ai = e A ( l + o(l)): it actually implies that Ai = e X at every 2  2  step of the flow. 2. So far, our analysis of the the Zi — Z u  2  condition has been done in the Landau  gauge. To establish criteria for the "Zi = Z  condition in other gauges, one might  2  try considering the equation A d(\ j\ ) 2  x  Ai using ^  = Pi(X) and ^  2  ds  =  1 d\  1  Ai ds  A ds  1  d\  2  2  = P (X) computed with the gauge parameter n ^ 1. 2  Chapter 8.  8.3  LRC coupling flow with sharp slicing  156  R C scheme for other models  8.3.1  (f> m o d e l 4  We would like to apply the R C scheme to </> and analyze the primitive O D E analogue of 4  the running coupling flow. We will first determine the running covariance of the sliced Gaussian integration.  Using the running covariance, we write down a primitive flow  and the corresponding O D E analogue. From the O D E analogue, we will determine the corresponding £ from which the primitive (ODE) coupling flow can be solved. Let us adopt the following notations for the R C scheme for c/>: 4  1. The running potential is  v = w s  + (M X 2S  S 2  -  )v  m  s 2  +  (A3 -  i)v ; 3  where (8.3.1) 2. The diagrams used in the primitive flow are the  Self-Energy Diagram = S E =  and  Vertex Diagram =  V =  Chapter 8.  LRC coupling flow with sharp slicing  157  3. The free scalar covariance for (p is 4  fl=-TT—a-  (--) 8  3  2  4. The external term for generating the running covariance is K\ = M \  - m + (A* - l)p .  2s s  (8.3.3)  2  2  5. The couplings Af satisfying the flow equations Af" = Af + 6?  (8.3.4)  1  where  M~ S V 2s  s  2  s 2  S V S  3  S 3  i  s  =  L  <V> .=  =  M~  =  L  <SE>B.=  =  K, +  A3.  Vl  &  2s  V3  b  1  6  M  L  ^ V  (8.3.5)  1  <SE>fi =b 6 -^V , {  V2  3  b  9  6  M  ^ V  2  M  2  3  From (5.2.3a) of Lemma 5.2, the iterated running covariance is V (X)  o  s  H = s  s  V (X) = S  p +m 2  2  v  where as usual, with the assumptions 7* = 0(s ), a  equality up to 0(M~ ). S  ;  7  s  v  (8.3.6) '  a > 0 and 7 > C , we denote = as s  From (8.3.4)-(8.3.5), the corresponding O D E analogue of the  primitive coupling flow under the running covariance scheme is  Chapter 8.  LRC coupling how with sharp  Setting g = Xi/'j  2  slicing  158  and assuming that A - b g 7 = 0(g ), 2  2  1 dAi  =  Ax ds dg  1 dj  h g,  1 dAi  We interpret g as the Z\jZ\  = h g +  2d7  = h g +  3  0(g )-  2  7 ds  (8.3.8)  0(g ),  2  7 ds  = 9 Ai ds  ds  we have  3  2  3  of the physicists' multiplicative renormalization scheme.  From the g equation, it is easy to see the driving term comes from the vertex diagram. The solution to the above system is 1  9 Ai 7 A  Ai  h [s + 1) + 1  -g  r  I-  7 exp  {-gr  (8.3.9)  61 (s + 1) + 1 1  &2 2  A3  In  -g'  h (s + 1) + 1  (P ) 7 r  2  r  b (S + 1) + 1)2  7 exp  exp  &! L - 5  X  61  b (s + 1) + 1  - 1  r  61 (S + 1) + 1 A,.  x  a > 0; thus, for g to stabilize around the origin, we need g < 0 which means  Ai < 0. For g U V A F , we have Ai = 0 ( l / ) s  a n  d A = 0(V 2  s 2  ) which are in complete  agreement with the results which I have obtained previously in my M.Sc. essay. In the previous set up, I applied the ordinary R C C scheme and dropped the flow of A in order 3  to determine that the system would stabilize. On the other hand, under the R C scheme, we can enjoy the advantage that the coupling flow can be run without dropping A 3 since it has no effect to the relevant behavior of the flow. Moreover, we see that A flows to a 3  non-trivial fixed point \ 00 A  for the U V A F case.  3  y  e  —63/61  (8.3.10)  Chapter 8.  8.3.2  LRC coupling Row with sharp  slicing  159  The g equation of a model with only one interaction vertex  We would like to generalize the technique of solving the coupling flow of a R C scheme. Here we consider models with only one interaction vertex term in the Lagrangian of the form A JTi(x)dx,  where TL(x) is a product of fields and their derivatives A\ • • • A . r  In  applying the R C scheme, each dimensionful coupling Aj of a dimensionful local counterterm Vi is scaled according to the naive degree of divergence di of the (5 function Si, i.e. we express the corresponding local counterterm as XfM Vi. diS  photon mass counterterm is expressed as X M A . s  2s  For example, in Q E D , the  We would like to determine the cor-  2  5  responding "g" equation since it supposedly governs the flow of all the couplings involved in the model. Observe that the g's of Q E D and (/> correspond to the renormalized charge 4  in the physicists' multiplicative renormalization scheme. Let I/71 be the corresponding multiplicative factor for the covariance of Ai.  The  effective coupling g of our model has the form  I1<=1  y/Ti  where A is the interaction coupling. The reason for calling g the effective coupling is the following. Since a line connecting two vertices has a factor of 7;, we may credit a ^/Yi to each of the two legs that contracted to make the line.  In this way, if a local  vertex v of a graph G has all legs contracted, then the A of v and  credited to  the legs of v together contributes a g to G . (In Q E D , we did not want to have squareroot factor in the equation, thus we were considering C = Af/[(A2) (A + A )] instead of 2  3  9  =  5  A i / [ A 2 \ / A 3 + A ].) Differentiating g by the scale s yields 5  dg ds  =  1 dX X ds  ^-v i 2  1  dji  7J ds  (8.3.12)  This is the g equation that would determine the asymptotic behavior of the flow of the couplings.  Chapter 8.  LRC coupling how with sharp  slicing  160  We claim that the RHS of (8.3.12) is a polynomial in g for a finite order of perturbation. Let G be a Feynman diagram consisting of n (interaction) vertices, C(G) be the set of lines of G , and A(G) be the set of external legs of G. The net product of the A's and l/7i's of G is (8.3.13) From the discussion in the preceding paragraph, it is easy to see that if G is allowed to have all its legs contracted, then the net product from the A's and the l/^/77's is just g . n  Thus (8.3.13) is just g with the 1/^/77 s associated to the uncontracted legs removed, n  i.e., = 9  II  n  nUeC{G)li  fceA(G)  STi-  (8-3-14)  where & denotes a leg of G . For the case that G is a generalized vertex diagram, H  v ^ = ^ ;  &eA(o  (8-3.15)  y  hence, \  X n  =A^- .  (8.3.16)  1  Since dX/ds is obtained by localizing generalized vertex vertices, we see that  (l/X)(dX/ds)  is a polynomial of g. As for a generalized quadratic vertex with two external legs A , 2  the coupling term is of the form g  n  7,.  Let 7$ be of the form  the set of couplings for the counterterms Ki = J2j ^ijVij for modifying the covariance of Ai. Since dXij/ds  u s e  d  where { } m  the shifting process  is obtained by localizing generalized  quadratic vertices, we see that for a dimensionless A^-, (l/7i) (dXij/ds)  is a polynomial  of g. As for a dimensionful coupling A^-, the corresponding discrete flow equation is M^Xn d  s  s  = M ^ ^X it di  s+  S  l  is  + S i+ s  l  Chapter 8.  where  LRC coupling how with sharp slicing  is the naive degree of divergence of  161  It is easy to see that the above  dimensionful flow can easily be converted into the following dimensionless one. A X-  M J8  s  DI  M ij - 1 d  s  sJrl  M >j - 1  13  d  Thus the corresponding O D E analogue is of the form dXn ds  -X  + cj g  n  ij  i  + ---.  (8.3.17)  Hence A^- ~ c 7, g , and they have no effect on the g equation since dX^/ds ~ 0. Hence n  we see that (l/ji)(dji/ds)  is also a polynomial in g.  Chapter 9  L R C coupling flow w i t h s m o o t h slicing  9.1  Preliminaries  In the previous analysis of the primitive flow obtained from the L R C scheme, we use a sharp slicing (characteristic functions) for making a decomposition of the free covariance. The sharp slicing allows a simple analysis in which the running covariances H  s  (8.1.5a-b)  take on relatively clean forms. However, when applying the L R C scheme to the tree expansion, the slicing cannot be sharp because of the bounds (3.1.3a-d) on the sliced covariances.  Thus we wish to study the L R C coupling flow for IQED again with the  sharp slicing replaced by a smooth neighbourly slicing in which only adjacent slices have overlapping supports. Let Af with 1 < i < 7 be the discrete flow of couplings satisfying the equations in (7.4.10) with the /3 functions computed by using graphs of a finite perturbation order. We wish to determine the asymptotic behavior of Af as U —> oo for a given boundary condition which is chosen so that resulting flow of £ = A ^ A ^ A f + Af)) stabilizes and 5  stays near the origin. We use the O D E flow of Chapter 8 as a guide for determining the dominant behavior of Aj near the origin. In the analysis of the flow under the smooth slicing, for Af with 1,2,3,4, we will provide a solution which is written as a dominant part, satisfying a primitive flow exactly, plus an error term which is higher order in 1/s. Unlike the sharp analysis where an approximate solution of the discrete flow is obtained heuristically from the O D E solution, the higher order in 1/s of the error term is obtained  162  Chapter 9.  LRC coupling How with smooth  by using a rigorous contraction argument.  slicing  163  The dominant behavior of the remaining  couplings Af with i = 5,6, 7 can also be shown to have 0 (  m s  )  bound by the contraction  argument. We briefly describe the layout of this chapter. In this section, we introduce a few preliminaries and notations for later convenience. In Section 9.2, we state the /3 functions of the A —> oo coupling flow using one loop diagrams : V , VP, ESE, 4 P ^ (defined in beginning of Section 8.2). We then set up and solve a primitive flow obtained from the "one loop" order flow. In Section 9.3, we define a perturbation order in terms of the natural parameter (. We then would like to describe the set-up of a higher order flow. In the last section, we set up the flow equation for the difference E = X — A where A s  s  s  s  is the primitive solution A described in Section 8.2. s  We would like to give more on the preliminary setting for the (A —>• oo) coupling flow with smooth slicing. The smooth slicing {p } used here is the one defined in Appendix s  B.  Here, unless it is stated otherwise, the scale s is understood to be in the range  — 1 < s < U. Also, we consider only the Landau gauge with n = 1. Using the sharp analysis as a guide and from the same reasoning for imposing the conditions in (7.3.5) and (7.3.6), we impose the following conditions on the discrete flow of Af :  K  = OU),  A5 = o(i),  K  = O00,  K = 0(s ),  X  6  =  0 ( M ,  A  2  >  1 - e,  s  a  (9.1.1) a>o,  A* = 0 ( m s ) , 7 = 5  A3  x = o(l), s  5  AAf = o(Af)  + Af > C , 7  A* > 1,  where both e and C are some positive constants with e < 1. Later, we will see that for 1  some given boundary conditions, the solution of the primitive flow is consistent with the above conditions.  Chapter 9.  LRC coupling how with smooth  slicing  164  In the flow with the smooth slicing, there is a complication in computing the j3 functions coming from expanding the slicing function p (see Appendix F). In expanding s  p , the lowest order E S E diagram has a nontriyial relevant contribution to the A equation. s  2  As seen in the lowest order flow under sharp slicing, it is the absence of the relevant contribution from the vertex and E S E diagrams and the V P diagram having the "correct sign" that allows an U V A F solution in the IQED model. At first, the presence of the nontrivial term seems alarming because it does not have the "correct sign" and it might undermine the U V A F behavior of the flow. But a careful analysis (see Appendix F) of these terms obtained from expanding the slicing functions shows that the nontrivial contribution is actually of a higher order in £ than the leading contribution coming from s  the V P diagram. Thus the nontrivial contribution actually poses no threat to the U V A F behavior of the system. For the "one loop" order diagrams, the corresponding (5 functions are integrals where the integration domain are momenta restricted to the support of the slicings p p  s+1  s _ 1  ,  p, s  and their intersections. Thus it is convenient to separate p into the following three s  parts. From (B.3a), we write p = p + p s  s  +p  s  L  s  sh  with support S  s  R  = ((1 — e)M , 2s  M ), 2s+2  where the corresponding supports are S  L  =  ((1 - e)M ,  S  R  =  ((1  S  S  2s  M ),  S  2s+2  s  = [ M , (1 -  e)M },  2 s  sh  2s+2  -e)M ,M ). 2s+2  2s+2  In computing these integrals, it is often convenient to rescale S to S by making a change s  1  of variable q —> M q in the integration momenta. When scaling back to the scale s=l, s  we drop the superscript 1 in the above notations. Correspondingly, we also separate the running slicing V  s  into three parts, V , V , V . S  s  L  sh  R  They may be further subscripted by an  F or a B to correspond respectively to the Fermi slicing or the photon slicing. We remark on the 8M ~+ 0 limit of the coupling flow difference equation. Because the  Chapter 9. LRC coupling How with smooth  slicing  165  expansion of the slicing function contributes relevant terms to the flow, it does not seem possible to obtain a differential equation from the 6M —> 0 limit of the difference equation. The reason is that these terms contain derivatives of p . Because of the presence of the s  derivative, one no longer can extract a factor 8  M  by simply bounding the integrand. We  illustrate by the following example. Consider the coefficient of a marginal term as an integral I obtained from expanding a slicing function on its overlapping regions SL and SR.  After making the changes of variable as described in Appendix E (see Section E . l ) ,  let the form of I be the following. I  =  I +  )  ir F r), M  „(l - Mr)),  X  dr F(,»(r),l -  u  A ) ™ '^  P . W . A l ^ ' ^ M f "  P  * '  ( r )  W )  '  A )  A )  (9.1.2)  where F is a product of running slicing functions and XR is the characteristic function of SR.  Scaling the ^-integral into a SVintegral and making the change of variable PL = x,  the above integral becomes f dx F(x,0, ) Jo 1  X  d V { x  :°' dx  X )  + F ( l - x,x, A)  dV(l - x,z, dz  It is not obvious that the integral above is 0(5 ) M  a factor 6M-  A)  dV(y,x, dy  A) y=l-x  unless the integrand explicitly contains  Chapter 9.  9.2  LRC coupling flow with smooth  slicing  166  T h e p r i m i t i v e flow  From the O D E analysis in Chapter 8, the relevant terms of (3 functions are estimated by orders in powers of (  = (A;) /(A|)7 where 7  8  2  S  s  = A 3 + X . From the asymptotic s  5  estimates (8.2.21), (8.2.24) of A , we see that it is convenient to estimate the relevant 5  terms of the (3 functions by order in powers of 1/7 . In this section, we set up and solve S  a primitive flow of (7.4.10) which is obtained from using only the 0 ( V 7 ) S  OiK/Y)  a n c l  terms in the one loop (3 functions. The solution of the one loop flow is then expressed as the solution A. of the primitive flow plus an error term which can be shown to be o(A ) s  s  using a contraction argument. Here we only present the solution of the primitive flow and leave the contraction argument to Chapter 10. The one loop (3 functions of the flow equations (7.4.10) are computed using the one loop diagrams: V , V P , E S E , 4 F  ( 1 )  (defined in the beginning of Section 8.2).  All the  calculations are done in Appendix E and Appendix F. The calculation in Appendix F concerns the contributions from expanding the slicing functions. The relevant contributions from Appendix F are 0 ( V ( Y ) ) 2  a n  d they contribute only to /3 . From the 2  summary of the f3 coefficients (E.2.75)-(E.2.79) of Appendix E and Proposition F . l of Appendix F , we write the (3 functions in the following compact form. Let us denote 7 by A* for the convenience in indexing and use the notations o = Af/A , and E = A i / A  s  s  2  2  introduced in Appendix E . Pl(V)  =  0,  P s(VP)  =  S  P'(VP)  =  5 {b,[(a ) -E ]+J },  S  where S  2  j [h (a )  s  = M  M  = S  S  s  M  2  M  P (ESE)  2  2  2  M  J* ,  (9.2.1)  2  + J>] - j [(a*) - S ] j , b  a  2  2  {3 (4PU) s  =  5 {b [(a ) -Z ]+J°}, s 4  M  A  7  — 1, the coefficients 6, are defined in (E.2.75) of Appendix E , and the  Chapter 9.  LRC coupling how with smooth  slicing  167  various terms in the RHS of the equations are described in the below. For i = 2,4,  1  E  —  T  s  M  r  J i  for  1 - —  ~  A A ' ^ . + tf,  (9.2.2)  .7=1,2,7  E  8M  AAJ<% + #  (9.2.3)  i=l,2,7,4  i = 3, 5, 6, 7 1 8M  E  (9.2.4)  AAJG:  .7=1,2  where  o for  ,  (7 ) s  2  m  ,  o  (9.2.5)  i — 3, 5, 6 and £ = 1,2 s\2'  Gij and for i  =  O  = O U;  A?  (9.2.6)  rAj  = 4,7,  O  =o  (*') 7 Af s  Since from the sharp analysis, we expect 7  s  V)  (9.2.7)  = 0( )> the terms involving A A f in s  (9.2.2-9.2.4) may be regarded as terms of 0 ( V ) - We set up a primitive flow by uss 2  ing only the leading terms in one loop (3 functions where the terms in (9.2.2-9.2.4) are dropped.  The reason for picking these terms is that the solution A of the primitive s  subsystem (Af, A^, A 3 , Af) stabilizes meaning that any further addition of (higher order) terms obtained from the localization procedure to the (3 functions can only yield a solution with deviation of o(A ) from A . Now by dropping the terms (9.2.2-9.2.4) in (9.2.1), s  5  the (3 functions of the primitive flow become  Chapter  9. LRC coupling how with smooth  slicing  At = o,  168  $ =o  (9.2.8)  6. S b (cr ) - | [(a*) - E" 3 5  M  2  2  3  <T ) S  A  S  <5M  4  AI M  2 5  64  A4  <W M  b [(a ) - E ] s  2  "  2  Ae  2  2  2  5  8M h [(cr ) - S ] s  2  2  Al = SM b [(a ) - E s  4  4  7  Because /?i and A2 are trivial, we set Af = A and A?, = A where A , A are constants. x  2  x  2  Subsequently, since E = A / A , , 2  2  AI = A = A? = S  6  (9.2.9)  0.  By denoting e = E , the non-trivial A functions become 2  2  A3  =  (9.2.10)  h 5 e  2  M  X  s  A  bi 8  e — /yo 2  M  In the IQED flow, we make the switch A —> zAi in the flow equations (7.4.10), so the x  corresponding squared charge e ->• - e 2  2  . For convenience, we adopt the convention that  the coefficients e - are positive, so that each negative coefficient is written explicitly as a 3  minus sign multiplying a positive coefficient £j. Thus we let c  3  =  -63  >  0,  £ = 8  c  4  =  —64 >  0,  e = 8  3  4  c e > 0,  (9.2.11)  c e > 0.  (9.2.12)  2  M  3  2  M  4  Using the above notation and (9.2.10), the primitive flow for IQED reduces to  Chapter 9.  LRC coupling how with smooth  AA3  =  AAf  =  AAf  =  e  slicing  169  (9.2.13a)  3  (9.2.13b) -S X .  (9.2.13c)  s +l  M  5  Since the A and A equations are independent of A , let us leave the solution of A 3  5  4  4  until after we have solved for A and A . From the fact that $ = 0 for i = 1,2,6,7, 3  5  (9.2.13a) and (9.2.13c), it is clear that by choosing a boundary condition for A and A at 3  5  the root scale r, and for Ai where i = 1,2,6,7, the constants Ai independent of s, then \{  =  Ax,  Af  =  M- ( - >A£, 2  x =e (s-r)  A* = A , s  r  3  s  r  r  5  3  Af = A ,  (9.2.14)  A*=A ,  6  where A 5 , A^ are the boundary values at r. (M~( ~ ^\ ,  + Xl,  s  2  7  Here we do not consider the solution  A , A ) as the primitive solution for (Af, Af, A ) since they are not the domi6  7  7  nant part of the flow when higher order terms are included in the j3 functions. In the above, the solution of A and A are determined in terms of the boundary 3  5  condition chosen at the root scale r. If we had chosen to fixed the boundary condition of A and A at scale U , then the corresponding solution of A and A 5 is 3  5  3  \i[ = \V + s (U-s), 3  X = S  5  M^ - hl U  S  Since we want the flow of A to respect the conditions (9.1.1), we need to specify the boundary condition very precisely. In particular, for A5, because of the linear term in the A5 equation, Af has to be chosen with "exponential" precision in order for Af to flow to a finite value as U —> 0 0 . Thus if we wish to include higher order terms in the flow, then it is more convenient to chose the boundary condition of A and A5 at the root scale r. 3  Chapter 9. LRC coupling flow with smooth  slicing  170  To end the section, we compute a primitive solution for the A 4 coupling for a given boundary value A 4 . From (9.2.13b),  From (9.2.15), dropping the irrelevant terms  M~ ( 2  s _ r  in the denominator,  )A5  *=*"K,(. i% j+  The  <--» 92w  +  '-  solution of (9.2.16) is, for s > r  <927 i)  ^ ^ - M ^ r  where A 4 is the boundary value at s = r. Extracting the leading term, we get A*  where K  =  A^exp ( -  =  'Kz * ( K3  t= +i r  \ In f l  K + A  £  *  + e _(s-r)\% , ^( K ^ ~ ) " + O ((*. z  +  e (t-r) z  lTjr  £  r  t  , _ e (s-  (  )  +  r))^)  3  = X . From (E.2.75) of Appendix E , r  3  3  e  4  _  6 4 SM  e  h 5M e  £3  ,  n  (  2  2  »-3mfc  I  =  ( 2 7 R  (i + O ^ ) ) ? 9  )  4  I n M 3 (2TT) (M - 1) 4  n M  2  _  1  2  _2k \nM n  (9.2.18)  Chapter 9.  9.3 9.3.1  LRC coupling how with smooth  slicing  171  H i g h e r order flow T h e order N (3 functions  We wish to show that the primitive solution (Af, A£, A | , A ) in (9.2.14), (9.2.18) of the 4  subsystem (Af, A|, A|, Af) stabilizes, that is, any higher order flow of the subsystem obtained as a perturbation of the primitive system with additional higher order diagrams has a solution with the same asymptotic behavior as (Af, K , A | , Af). Since the O D E s  2  analysis in Chapter 8 suggest the effective coupling C C  =  J ^ i l ! _  (AI)  2  f  r  [  9  3  i) j  where 7 = A 3 + A 5 , would be a natural parameter for defining a perturbation order for the coupling flow of IQED. In the following, we define a perturbation order for the coupling flow of IQED in terms of C. Provided that for i = 1, 2, 5, Af stays bounded, we will see that the contribution to (3 from each relevant diagram has order given by a power of s  C . From the primitive flow, we saw that 1/7* = 0 ( l / ) 3  s  a n  d Af = O ( l ) , * = 1)2,5.  Heuristically, since higher order contributions to the flow would only add terms of higher order in 1/s, a higher order flow should stabilize. Rigorously, we apply a fixed point argument to show that, for the subsystem, the solution Af with 1 < i < 4 of a higher order flow is a perturbation of Af with error Ef such that | E / | / | A f + Ef \ = o(l). The fixed point argument also shows that the flow of (A^A^Ay) stabilizes when the order of perturbation includes the two loop diagrams. It is convenient to set the primitive solution of (A?;,Ag,Ay) as (A  s 5  = 0,A^ = 0,A? = 0)  in our set-up even though they may not be the dominant terms of higher order flows. After defining the perturbation order in terms of (, we set up a order N coupling flow.  Chapter 9.  LRC coupling flow with smooth  slicing  172  Then, by expressing Af = Af + E- in the order N flow, we obtain a system of flow equations of the errors E- where the corresponding (3 functions are expressed in terms of A£ and E£ with  s<h<U.  We now define the perturbation order of a flow in terms of the order of the contributing Feynman diagram (defined by the number of vertices).  We will first show that a  contributing term (31(G) from a graph G to the (3 function of Aj can be estimated in terms of powers of £. We will then define the order N flow as a flow obtained by using the /3? (G) that are of O(C)  where  n<N.  Recall from the discussion on the aspects of the j3 functions in the Section 7.3, relevant graphs contributing to the flow do.not contain any VQ vertices and the relevant terms that contain A 4 only contribute to the A 4 equation. Moreover, only the GV, 2P, 2F, 4, (3f do not contain A 4 , we first  AP diagrams have relevant local parts. Since for i /  consider the estimate of the /3*(G)'s with i ^ 4. For convenience, we also exclude the /3f((?)'s that contain Af. We show that, for a given relevant generalized vertex G of order n  x  (defined as the number of Vi-vertices), where G contains no V7-vertices, the (3  contribution satisfies fl{G)  = 0(F  (\'))  (9.3.2)  GTNI  where  F v, W = C G  ni  ^ ( A )  =  C  l / 2  ( n i  ~  7,  1 ) / 2  A!,  n >2 x  m>3  F F,n (A)  = C  = C  7 ,  2  i p,n (A) ?  4  1  1  n i / 2  2  n i / 2  A ,  n > 2  2  x  n >6 x  (9.3.3)  where £ is defined in (9.3.1). The reason for having the restriction on n for the various x  diagrams is that: for the V s , 2F's and 2P's, the corresponding lower bounds of riy is the minimum vertices required for constructing the diagrams; for n = 4, F (\) x  IP  is actually  0 ( C 7 ) because of vestigial gauge invariance (see Section E.2 of Appendix E). After 3  2  Chapter 9.  LRC coupling how with smooth  slicing  173  obtaining the estimate (9.3.2), it will be easy to extend it to the / ? ! ( £ ) ' s and the Pf(G) that contain A . 7  Recall from the running local terms (3.6.12), a typical (5 contribution to the coupling flow coming from a Feynman graph G associated with a tree r is the coefficient of the local generalized vertex given by £  L G(h)=  Y  l  Pi(G,s,h)V  (9.3.4)  t  where Li is localization operator and h is the scale assignment of the lines of the graph, U {T)  = {h\U>h >h >h  8  h  fl  = s for h > h > F ; f f  F  u  2  e F(r)},  (9.3.5)  F is the lowest fork of r and F[r) is the set of forks of r. A contribution Pi(G,s,h)  with i ^ 4 is usually an integral over the loop momenta  P = (Pi> • ' iPn) of the graph G and has the form -  0i(G,s,h) where K (h, G  —*  = J dPA (X,h,P)K {h,P), G  (9.3.6)  G  P) is the kernel of the localized graph which depends on the scale assignment —*  h and AG(A, h, P) is a product of the running slicing functions and the \{'s.  Let us write  the product in the following form. A ( A , h, P) = A^VG, A, h) V(C , A, h)  (9.3.7)  G  G  where  A ^ V G A M  =  n  A  I">  ( - - ) 9  3  8  vevo  v(c ,x,h) G  V, S  F  =  n  'PF n 1  vi*,  V% are the covariance respectively denned in (E.2.5), (5.4.9a), and V  G  Vi-vertices of G, £ F G<  is the set of  (respectively £G,B) is the set of Fermi (photon) lines of G, and h , v  Chapter 9.  LRC coupling How with smooth  slicing  17 A  hi are the scales assigned to the vertex v and the line L From [FHRW88], we have the following bound on 0i(G, s, h) with the term A (X, h, P) being removed. G  Theorem 9.1 Let G be a graph in the RCC renormalized tree expansion associated with a tree r whose bottom fork F is labeled R or L, or is unlabeled. Then J  dP \K (h,P)\  < c  where h^^p) = 0 and c  G  G  G  JJ  M - i f- <»\ 5{ G ){h  (9.3.9)  h  is a positive constant independent of scales.  We are now ready to show that Pf(G) = Q(F (X )).  Let us denote E £  s  Gtni  e W  by E^>  s  and 12n>aPi(G,s,h)  i^5  { M- Zfi> Pi(G,s,h)  (9.3.10)  i =5  2s  s  Proposition 9.1 Assuming (9.1.1-d) and that G contains no V-j-vertices, then \fr(G,s,h)\  <  \P i(G)\  jTIi  s  0  <  S  where FQ  C F (X )c M(G,h)  is defined in (9.3.3) and  C  Gtni  G  C F (X )  (9.3.11b)  s  G  0  (9.3.11a)  Gtni  is a positive constant depending on the conditions  C  0  in (9.1.1-d), ~JJ  M{G,h)=  M f^ f- ^\ s{G  h  h  (9.3.12)  and c Zn> M(G,h) G  C >{  s  i^5  G  ( c M- Zn> M(G,h) 2s  G  s  (9.3.13)  i =5  is a positive constant independent of s.  Proof: First of all, (9.3.11b) can be obtained by summing (9.3.11a) over h. Next, we bound the  Chapter 9.  LRC coupling flow with smooth  term A (X,h,P). G  slicing  175  Let A (X,h,P) G  A (X,h,P) G  (9.3.14)  AG(A, h) '  where (9.3.15) and y  h  = A3 + A£. From the definition of j  s  1  XSh  7"  min  1+ 7  I inf p es 2  h  K  in (8.1.6a), -1  •2h  - , mi 1 +p ^es 1 +p 2  h  < 2  2  P  (9.3.16)  /  where e is the overlap parameter and Xs is the characteristic function of the support of h  the slicing function p . By the fact that h  A ^ < 1 ,  (9.3.17)  (1 + M < 1 ,  where x /> is the characteristic function of the support of p , it is easy to see that h  5  A (X,h,P) G  where  |£G,B|  < 2  | £ g  -  (9.3.18)  b |  is the number of photon lines of G. Thus from the conditions in (9.1.1-d),  we have A (X,h,P) G  < <  (9.3.19)  2^A (X,h) G  2'  £g>b  <  ' A ( A , h) G  C A (X,s), 0  G  where  n K n  A (A )= G  ) S  ^  n  ^  (9.3.20)  From counting identities  n = \V \ = \C \ x  G  G>F  + l^J- = 2|A  | + |A  G)B  GiB  (9.3.21)  Chapter 9.  where \A \  LRC coupling how with smooth  176  |AG,B| respectively are the number of Fermi legs and photon legs of G ,  and  G>F  slicing  we have A (s, h) = F G  ( A ) and S  G > N I  A (\h,P)  <C F (X ). s  G  0  G>ni  From the bound on A ( A , h, P) and Theorem 9.1, G  \Pi{G,s,h)\  <  J  dPA (\,h,P)\K (h,P)\  <  C F (X )  <  c C F , (\ )  G  J  s  0  Gtni  dP  0  \K (h,P)\ G  M(G,h).  s  G  (9.3.22)  G  G ni  Q.E.D.  For a given graph G, let the number of vertices of G (the order of G) be denoted by n i ( G ) . Now by (9.3.11b), (9.3.3), and letting N(G) = M G ) / 2 ] where [a] denotes the integer value of a, then for the graphs GV, 2F, 2P, 4P, and for i ^ 4, B £ p _  =  0  (  (  r  )  W  ( ^ )  Q((C ) s  =  iV(2P)  0((C )  fl(4P) (7  )  S  5^2  )  N  {  G  V  )  >  mm  x  o(( »)^))  =  C  >  N(2F)  >1  ), N(2P) > 1  JV(4P)  )  A^(4P) > 3  0(C ) 3  (9.3.23)  N(AP) = 2 '  For the case of /3|(G), by the discussion in starting in the paragraph containing (7.3.18) in Section 7.3 regarding that /5|(G) is linear in A , each relevant term of A|(G) 4  is obtained by one of the replacements described in the paragraph after (7.3.18) where the V  and V '  k  F  F  are now the ones defined in (E.2.5) and (E.2.6). As a result of the  replacement, the integrand of Af (G) is linear in V  +V  S F  with no constant term. From  ,S F  (E.2.28) +1\s+l  V + Vp S  s  F  = \\{V ) + V 2  F  F  F  B  R  + n  (i-Pi)-Pl  + 1  (AI  R  + 1  )  2 (  9  3  2  4  )  Chapter 9. LRC coupling how with smooth  where B  s  = 1- p - p s  + pX + p  s+l  s  s  s + 1  2  slicing  177  ( A 2 ) , and terms subscripted by R, respectively + 1  2  L, are terms restricted to the right, respectively left, overlapping region. By (9.3.24), (9.3.17), and the conditions in (9.1.1),  v + vp s  where x  s  i s  x  A  ,s  =  s  F  4  [i + (xi + x R) s  (W  O  (9.3.25)  (i)],  the characteristic function of the support of p . From the above, it is easy to s  see that a Pl(2F)  has the form I  dPA , (\,h,P)K (h,P) e G  (9.3.26)  G  where A (X, h, P) is of the form defined in (9.3.7-9.3.8) and each Ae (\, h, P) is obtained G  iG  from A ( A , h, P) by replacing a V  h  G  l F  with V +V ' . e  Since we assume A£ = 0(s )  ht  F  a  F  with  a > 0, we need to borrow a little bit of exponential decay from M(G, h) (9.3.12) to make  J2 ^M(G,  c  G  h) < C A, G  h>s  and obtain the bound \ v i ( m \  <  (X,h,P)\  E  G  K (h,P) G  h>s  <  C C -fA (A, ) 0  G  G  A  (9.3.27)  S  2  where A ( A , s) is defined in (9.3.20). From above, it is easy to see that G  Ai  ='0«cr  (2F)  ).  (9.3.28)  If there were no V counterterm vertices, then we would define an order N flow with 7  N > 1 as a flow constructed from using graphs GV, 2F, 2P, AP with the following restrictions on their ni(G)'s: ni(GV) ni(4P)  <  2N + 1,  =  {  m(2P)<2N,  2N  N>3  0  N <3  rn(2F)<2N, (9.3.29)  Chapter 9.  LRC coupling flow with smooth  slicing  178  Example 9.3.1:  y  VX  y  2  "^\^  ^^xr^^ ^  " ^^^^ X,  1/X  F  V  = X] I (X\y) =  2  *  F = X*/ X = C Y 2  YP  Xl  ESE  2  = ^ / <»-Y) = C \ 2  Figure 9.13: The diagrams of a order 1 flow The order 1 flow uses only the V diagram (m(V)=3),  the E S E diagram (ni(ESE)  = 2),  and the V P diagram (rii(VP) = 2). Example 9.3.2:  \  t  s\  G =  / /  s  V/  Figure 9.14: An order 2 graph  Consider the order 2 graph G with ni(GV) = 5 associated to the tree r with hp = s, hf = t in Figure 9.14. Let  LxG(M) = (A;) (Ai) / 3  where LiK (Q) s,t  2  (^(T))  2  (PMA2))  is the localized kernel.  2  (74(A )) L i i ^ ( Q ) 2  2  V,  (9.3.30)  Upon summing over t, we have an order 2  Chapter 9. LRC coupling how with smooth  slicing  179  /^-contribution  p\{G, s) = E ( A D ( A i ) / ( n ( 7 ) ) ( n ( A ) ) {V {\ )fL K^(Q) 3  2  2  2  dQ.  t  2  F  2  l  (9.3.31)  t>s  By assuming (9.1.1-d),  ^ where c  G  and C  G  i  (  g  '  s  )  -  i  c  ^ i ^ i  g  "  M  (  t  "  s  )  -  C  G  (  c  s  )  2  A  -  are positive constants independent of s.  To extend the definition of order N flow including graphs with V -vertices and 7  we first extend the expression A G ( A , h, P) in (9.3.7) to include A 7 and Vp + Vp  Pl(G),  factors.  ,s  Let us introduce some notations for convenience. Given a graph G containing n number 7  of Vy-vertices and ri\ number of Vi-vertices, we would like to partition the set of photon lines CQ,B of G by the following. For each V -vertex of G, there is a factor A * and at 7  7  least one photon line contracted at the same scale se attached to the vertex. For each V7-vertex with scale sg, we select one photon line with the same scale which attaches to it. The set of these selected lines is denoted by £G,B,U and the set of the remaining 7  photon lines not in CG,B,U is denoted by C ,B,d 7  G  Let us write the extended version of  B  (9.3.7) in the following form.  A ( A , h, P) = A i ( V , , A, h) A ( V , , A, h) V(C , G  G  n i  7  G  n 7  G  A, h)  (9.3.32)  where  A^V^AK)  =  v(c ,x,h) = G  n  n  i">  A (V ,„ ,A,K) =  A  7  n  B  V  G  r">  (9-3-33)  A  7  PB  X <  Y.eeca, CP"/ + Vf**) r  Yleec , w G F  V^ h  i =4  Chapter 9.  LRC coupling Bow with smooth  slicing  180  By including the V -vertices, the counting identity (9.3.21) becomes 7  = 2 | £ , | + |AG,B|  n + 4n = 4n + \C , \ + x  7  7  G F  G  (9.3.34)  B  where n\ is the number of ordinary Vi-vertices of G, and n is the number V -vertices of 7  7  G. From the identity, it is easy extend the result of Proposition 9.1.  Proposition 9.2 Assuming (9.1.1-d), then \P°(G)\  < C  C F  G  0  (X )  (9.3.35)  a  Ginun7  where C is a positive constant depending on the conditions in (9.1.1-d) and Q  [  (T%Y  [ (^) where F  2  F , (\)%  z= 4  G ni  7  0((Af/A^)  P (ESE)  i^4  Gn  is defined in (9.3.3). From the O D E estimate, we expect A to behavior like  GjTll  and ESE  F A\)  7  2  Ins).  Hence, from Proposition 9.2, and the fact that for the order one V  diagrams, p\(V) = 0 and p (ESE)  = 0 (see the calculation of p (V)  2  x  = 0 and  = 0 in Section E.2 of Appendix E), we expect fi(GV) Af ff(2F) A? ff(2P) ryS  O ((C)  ( n i _ 1 ) / 2 + 2 n 7  O ((CT 0((C ) s  l / 2 + 2  ni/2+2n7  ' 0((CT ff(4P) 2  (r)  = <  0(C  7  " (M ) ,  i = 2,4  (lns) ),  i = 3,5,6  7  n 7  ny  l/2+2n7  3+2n7  (ms)" ) 7  (lns)" ) . 7  O(C " 0ns) ) 2  (9.3.37)  (Ins)" ),  7  n7  ni > 6 ni = 4 • ni = 0  9.3.2 The order N flow and boundary conditions From (9.3.37), we define the order N perturbation for the coupling flow of the multi-slice L R C for IQED. We would like to treat each V vertex as a 4P^ 7  diagram constructed  Chapter 9.  LRC coupling how with smooth  slicing  181  from 4 ordinary vertices in which an extra factor 0 ( ( ) ) is attached. A n (ny, n ) graph m  s  7  is defined as a graph G consisting of ny Vi-vertices and n Vy-vertices. We define the order 7  N A -> co coupling flow equation for IQED as the system of equations in (7.4.10) with the P^s computed using (ny,n )  graphs satisfying the following conditions: for n = 0,  7  7  < 2N + 1  ny I  <2N  G is of type GV ;  (9.3.38)  G is of type 2P, 2F, 4P  for n > 1, 7  ny + 4(n - 1W  < 2N + 1  G is of type GV  < 2N  G is of type 2P, 2F, 4P  .  7  The reason for having n — 1 instead of n is because of the O 0 ( ) n  7  7  s  n 7  (9.3.39)  ) factors.  An order N flow {A"} ,.!^ is a solution satisfying the order N flow equation with a 5  given boundary condition. From the discussion of the choice of boundary conditions for the primitive flow in Section 8.2, since we require the flows to stabilize and to stay near the origin (stays finite as U —>• oo), without precise estimates of the P functions of higher order diagrams, it is convenient to choose the boundary conditions at the top scale U for Ai and A and at the root scale for the remaining couplings. Thus we only consider order 2  N flows with boundary conditions of the form: = Ai,  Af = A  A3  =  K,  A^ = K  A  =  K,  A£ =  5  3  5  2  (9.3.40) A  K, 6  A = K r  7  7  (9.3.41) where A  1 ;  A , and Ki with 3 < i < 7 are constants independent of the U V cutoff 2  parameter U. From the fixed point argument in Chapter 10 and the Ward Identities in Chapter 11, we can determine some conditions on the values of A i , A , and Ki so that 2  the resulting flow has the same asymptotic behavior as the primitive flow under the same boundary condition.  Chapter 9.  LRC coupling how with smooth  slicing  182  The flow of E*  9.4  For finding a solution A of the order N flow equation where X is a perturbation of the s  s  primitive solution A* described in Section 8.2 and satisfies (9.1.1), it is convenient to study the flow of the difference E  = X — A . In this section, we describe the set up of  s  s  s  a subsystem of the flow of E . s  From Proposition 7.1, the /? functions of the subsystem Af, i = 1,2, 3, 5, 7 do not contain Af and Ag. Thus we would like to first study the subsystem Af, i = 1,2,3, 5, 7. Let us denote the subsystem as A = (Af, A|, A|, A|, A ) and its corresponding primitive solution 5  7  as A = (Af, A£, A | , Af, A ) . We would like to write the flow of X as a perturbation of s  s  7  the primitive flow A . For convenience, we take the components of A to be the solutions s  s  obtained in (9.2.14) with the modification that the exponential decay term X M ~ ( r  2  5  s _ r  )  is dropped so that A = 0, i.e., s  5  A  s  = (A A , e l5  2  (s + 1) + X ,0,0).  (9.4.1)  r  3  3  More precisely, for a choice of ( A A , A 3 , Af, A ) satisfying (9.1.1), and A ^ 0, A / 0, 1;  A3+A5  2  7  :  2  > 1, and for a given order N, we would like to show there exists a flow A satisfying s  the order N flow equation of (7.4.10) such that Af  =  A i + E[,  X  =  A + £|,  s 2  E[ = o{l) E =o{l), s  2  2  XI  = £ ( s + l) + A + £ ,  Xt  =  El,  X  =  E,  7  s  3  s  7  (9.4.2)  3  3  El =0(8),  E =o(l), s  5  E  s 7  =  0(\ns),  where ( A A ) is the asymptotic value of (Af,A ) as s —> 0 0 , ( A , A 5 , A ) are the initial 1 ;  2  3  2  values of (A , X , A ) at the root scale r = - 1 , El = 0, El = X , and E  7  s  3  r  5  7  5  7  = A . We shall 7  Chapter 9.  LRC coupling Bow with smooth  slicing  183  apply a fixed point argument in Section 10.4 to show that the Ef satisfy (9.4.2). In the following, we rewrite the flow equation in terms of the error variables Ef. Let X = A + E s  s  s  where E  = (E ,E ,E ,E§,E )  s  S  S  S  is the error to the full solution at a  S  3  particular perturbation order N, and where if subscript i is omitted, then the quantity stands for the whole vector. Also, let any quantity that depends on scales s with its index s omitted denote the whole sequence of the quantity.  For example, A means  {(Af, A|, A 3 , Af, X )}. The equations of a higher order flow have the form 7  AX  S  = fi {A + E).  (9.4.3)  s+1  For our convenience, we shall introduce the notion of 1/s-order for describing a decomposition of the fi functions into dominant and sub-dominant terms. From here on, we call a fi contribution fii(K) a higher 1/s-order term whenever, under (9.1.1),  The  OQ / ) n5  5  fii(K)  =  0(l/s ),  i = l,2  fii(K)  =  O(IA),  2  (9.4.4a)  i = 3,7,  fi (K)  = 0(\n s/s).  5  (9.4.4b)  °f 05 comes from the term (9.4.9) below obtained from a self-contraction  of a V vertex. 7  We shall call a flow system obtained by adding higher 1/s-order terms to (9.2.8) a higher order flow system. We will omit the prefix 1/s in saying 1/s-order whenever there is no confusion with the order defined in terms of £ s  We extract the primitive part B {A ) s+l  fi {A s+l  + E) = B (A ) S+1  of fi (A + E):  s  s+1  + 6 B (A ,  S  S+1  E ) + B {A,  S  s  E  S+1  E) + A (A S+1  + E)  (9.4.5)  where < 5 ^ ( A , E ) = B (A +1  E  S  s  s+l  s  + E ) - B (A ), s  S+1  S  (9.4.6)  Chapter 9. LRC coupling Row with smooth  Bf ,  slicing  184  i = 1,2,3,5 are the lowest 1/s-order (3 functions given by (9.2.8), B  +1  s +l 7  = 0,  Bf  +l  are the higher 1/s-order (3 functions coming from the order one diagrams given by (9.2.29.2.4), and A (A  + E) are the (3 functions coming from higher order diagrams. By the  S+1  fact that A A* = B - ( A ) , + 1  (9.4.7)  S  applying the difference operator A to A = A + E , the flow equation for the E is s  AE*  = 5 B° (A , +1  s  s  E ) + Bt (A,  S  s  E  s  E) + A (A  +X  + E).  S+1  (9.4.8)  For convenience, we now drop the A in the argument of the f3 functions. We denote the RHS of (9.4.8) as (3 (E).  Now  S+1  contains terms which are of the form A A ^ G f j  (see (9.2.1-9.2.5)). For these terms, we replace each A A * and  AAf  1  by -5 {h M  M  2  [(a ) 8  - e} + £ |  2  2  + 2  ) + (3 (E) S +2 5  + 1  by a f3t {E) +2  for i = 1,2,3,  where a = (Af/A|) , so that s  2  the corresponding terms have an order of a higher power in I / 7 . In these replacements, 5  the / 3 - ( £ ) are given by (9.4.5) in which A A f + 2  recursively replace A A - by  = A f - A f  2  3  2  (we do not want to  for h > s + 1, beyond h = s + 1).  1  Let us rewrite the higher order flow using the notation in (9.4.5)-(9.4.6) and replacing AA|  + 1  by the appropriate f3f (E)  in (3 (E).  +2  Also we would like to write down some  s+l  order one terms more explicitly. From the way we defined the order one flow (see (9.3.39)), we include the following term corresponding to graphs that contain one Vj vertex in the order one contribution. Let Ic,(E ) be the f3 contribution of the graph G obtained from 7  2  a self-contraction of a V vertex as displayed in Figure 9.15. 7  P (E )=c E  s  5  7  5  7  [  drV (j)  where V^y) is the running photon slicing using l+j  s  for the definition of 7).  (9.4.9)  s  instead of 7 = A 3 + A 5 (see (5.4.9c) s  Chapter 9.  LRC coupling flow with smooth  slicing  185  Figure 9.15: A self-contraction of a V vertex 7  Next we consider terms of the form (a ) - e or (a ) 5  2  2  8  - e where o = (X{/X )  2  4  8  s  2  2  (see  (9.2.1)). These terms are the remainder parts of VP and AP diagrams after discarding the leading terms due to vestigial gauge invariance (see the calculation of the VP and AP diagrams in Appendix E). For these remainder parts, we would like to express them using the following variables. Let  E{  For example, let us consider the term 5  8  A  ~*  M b [(o ) 2  M  ^2  U  Ai'  8  (9.4.10)  2  — e ] from the /?f of (9.2.1). Using  2  2  5  the variables in (9.4.10),  '(M + Ei)  2  5  M 6 2  M  s\2  6 Mb  =  5  2  M  5  L(A + £ ! )  — e  2  2  (1 + Ut)  2  5  M b e 2  M  5  (9.4.11)  _ ( l + [/|)2  Let us introduce the following notations for expressing terms like (9.4.11) more concisely. Let  X (E) 8  (1 + ut)  2  =  + o(u )  (9.4.12)  (1 + ^ f ) - 1 = Oiut) + 0(ui) 4 (1 + ui)  (9.4.13)  - 1 = o(ut)  8  (1 + Ul)  2  Y {E) S  =  4  and £3  =  c  3  5  e, 2  M  e EE 65 M e , 2  5  2  e =b S 7  7  e, 4  M  (9.4.14)  Chapter 9.  LRC coupling how with smooth  slicing  186  where c = —b , and b , b , b , b are the constants defined in (E.2.75-c) of Appendix E . 3  3  3  5  e  7  Using (9.4.12), we can write (9.4.11) as 6  M  b e X  2  M  2  {E).  s  5  Now the higher order flow equation (9.4.8) can be expressed as AE*  =  AE  =  S 2  ~p[ \E)=  A[ {E)  +  (9.4.15)  +l  P^(E)=Y:  P° (E) +2  G +\E) S  i=l,2  +G +\E)  [-6 (E*  +H {E)  +  S  2  AE*  =  AEl  = Pt (E)  s+2  M  s +l  +2  5  P {E)\  +  S +2 3  A (E) s +l 2  p +'(E)=e X (E)+Y: s  + P* (E)  + e X (E))  +2  2  Gf  s+l  (E) p*+ (E) + A  s  3  2  3  (E)  s +1 3  i=l,2  =  +1  + £  -5 [E +e X (E)} s +l  M  P? (E)  Gtf(E)  s+l  5  5  + It'iE)  +2  +  Ap-\E)  i=l,2  AE  =  S 7  ~p {E) s +l 7  =  e  Y (E)  +  S+1  7  £  G f{E) a  7  Pl {E) +2  +  A (E). S +1 7  t=l,2  From (9.3.37) and (9.1.1), the P\ can estimated by order of powers of 1/Y 7  s  = Af + Af. Since we would like to express A* as a perturbation of A  s  where  (9.4.1) as  described in (9.4.2), we would like to estimate the Pl in terms of the "primitive" part V  s  of 7 . From (9.4.1), A = A + e (s + 1) and A = 0. Let s  3  3  T where K  3  s  =  5  A + Af = A + e (s + 1) 3  3  (9.4.16)  3  = A > 1 and e is defined in (9.2.11). Since A 3 = V + E s  3  3  3  3  and X = Eg, it is 2  easy to see that Y = T +E+ S  s  3  E. s  5  (9.4.17)  By assuming (9.1.1), the expected orders of the higher order terms from order one diagrams are given in (9.2.2-9.2.4) and the higher order terms from liigher order diagrams  Chapter 9.  LRC coupling How with smooth  slicing  187  in (9.3.37). From the primitive solution, if Ef obey (9.4.2) then  Af =  A O(l),  A* = A 0 ( 1 ) ,  x  7  2  s  = r O(l)s  (9.4.18)  Hence we expect that (9.4.19)  =o ( ^ r ) ,  fc{E) + 8 Et M  M(E)  =  o(±y  = O(^)-  ®{E)  Note that for the order one V and ESE  diagrams, Pi{V) = 0 and 0 {ESE)  that the dominant term of /3|(F) + 5 E^  is not X (E)  M  iactorV (E) s  B  E = s  7  0(\nF /T ). s  s  2  s  = 0 and  but I§(E) since it contains the  Chapter 10  The Fixed Point Solution  10.1  The set up of the contraction map  In this section, we present the set-up of a contraction map K, for showing that the EiS of the perturbation A = A + E s  s  s  from the primitive solution A  3  (9.4.1) have the  expected behavior given by (9.4.2). The strategy of the contraction argument is to replace some of the Ei's in the flow equation (9.4.15) by given "external" terms <fj so that the resulting modified equation becomes a set of linear equations in the E^s with non-linear external terms depending on £ . After this replacement, each  is now only linear in  Ei and does not contain any other E for j ^ i; hence the solution Ei{£) of each scalar 3  non-homogeneous linear equation obtained from the replacement has a simple explicit expression analogous to the quadrature of a linear O D E . In this way, for a given boundary condition E  B  of E, we rewrite the original equation as E = JC(£, E ) which we treat as a B  fixed point problem. We consider the map K which takes the sequence £ to the sequence E generated from the modified equation, where the domain of /C is restricted to a ball B$ of radius 5 residing in a Banach space of sequences with an appropriate norm || • ||. We then show that, for suitable boundary conditions, /C is a contraction on B$ satisfying a Lipschitz bound and that JC(B$) C B$. The latter condition can be fulfilled by showing that ||/Cj(0, E )\\ B  is sufficiently small. At the end of the section, we present a lemma  which shows how one can obtain a Lipschitz bound for each Ki(£,E ) B  Lipschitz bound for the /3 functions Bi(£).  in terms of the  Similarly, bounds for ||/Cj(0,E )\\ can also B  188  Chapter 10. The Fixed Point  Solution  189  be obtained in terms of bounds for ||5j(0)||. Later in Section 10.2, finding Lipschitz bounds for the Bi(£) involves extracting the "primitive" parts of A ( A , h, P) of (9.3.32). These primitive parts are expressed in terms G  of the primitive parts A  A , V , V respectively of Ai, A , 7 = A?, + Af, 1 + 7 where 7 s  l 5  s  5  2  2  is defined in (8.1.6a). We would like to describe T , V in more detail. Recall at the end s  s  of Section 9.4, we introduced the primitive part V of 7 where V is defined in (9.4.16). s  s  s  To express 7 as a perturbation of V , we introduce the variable s  s  W° = ^ ,  (10.1.1)  where E° = E + Eg. From (9.4.17), s  z  7 The  running photon  V (1+ s  Ei + E? + ) = r (i  slicing V% of (5.4.9a)  (10.1.2)  + w ).  s  5  3  s  is actually  expressed  in terms of  1 + 7 = 1 + Af + Af instead of 7 = Af + Af, where by' rescaling p ->• M p s  s  s  H = / 3 V ) (AS - 1),  Af = fl{p )  Af,  2  (10.1.3)  and  / V) = j  AV) =  ^  ,  (10.1.4)  and e is the overlap parameter. Thus as for 7 , we define  where 1 — e < p < M 2  p  5  2  s  W* = | ;  (10.1.5)  where f  s  E;  MP ) 2  =  mP ) T - K(P ) {Kz + e (s + 1))  =  fs(P )  2  s  2  3  2  m )E 2  +  P  = I-/I(P ) = ^  S 3  +  2  F  2  f°(p )Eg 2  -  (10.1.6)  Chapter 10.  and  The Fixed Point  Solution  190  == A 3 > 1. Expressing 1 + Y in terms of W ,  K  s  3  1+  = £ + E° = f (l  S  S  7  + W ).  s  (10.1.7)  s  Note that for the overlap parameter e sufficiently small and the slicing parameter M > 1 sufficiently close to 1, say e < 1/3 and M  < 3/2, then  2  / a V ) < M~ ,  1/2 < fl(p )  2s  < 1,  2  From (9.3.37) and (8.2.15) with C = e /y r  2  r  1/2 < fl{p )  < 3/2.  2  = 0(e ),  (10.1.8)  for i = 1,2 and an order N graph  2  G, we expect  =  0  ((£)")•  ((C)") = 0  (10.1.9)  Because of the factor Aj, it is convenient to use the variables t/j = Ef/ki  with i = 1,2,  introduced in (9.4.10) and write  ( *» -S) = (rr® f f  K = ui un, +  where e = 2  We  2  2  e 2  2  k\lk\.  set up a Banach space B for £ in the following way. We remind the reader that  we are not including the E and E at this stage. Let us denote A  6  E° and  the whole sequence {E } s  = {UIUIEIEIE> )  by E. Correspondingly, let S  = ( £  where each £j is a sequence {£•}• U[  =  E  S 5  =  O  —  u  £  2  , £  3  , 8  b  , £  7  (10.1.12)  ) ,  From the expected orders (9.4.19), we expect  O(^), /In  (10.1.11)  7  ^2=0(^7).  r \ 5  ,  ^  = o(inr-).  EI = 0(lnT ) s  (10.1.13)  (  Chapter  10. The Fixed Point  Solution  191  Note that because of the J | term (defined in (9.4.9)) of  fil {E)  in (9.4.15), the expected  +l  order of F | is O ( I n P / P ) instead of O 0-/T ). We define a norm for each component s  of £ according to the expected estimates (10.1.13) by the following. For F a sequence where each F  is a function of £ and p with 1 — e < p < M ,  s  {F }U s  =r  2  2  2  (recall the root  scale r = -1), let \\F\\ ,m=  sup  n  sup  l-e<p <M 2  Note that since K  where k , k , k 3  5  7  s  _1  n  5  )  m  |F |.  (10.1.14)  S  s  11^211(2)  ll^lll(l)  =  |N|l,0  ll^3||(3)  =  ^3 H ^ l l o . - l  ||^7||(7)  =•  k e~ ||^7||o,-i  = W&Wlfi h e~ H l ^ l l l - l  (10.1.15)  2  11^511(5) =  2  7  are small positive constants to be determined later. The reason for 2  5(r ) ,  s  3  having the factor e~  £2 ~  (r ) (l+ l n T  = X > 1, 1 + ln T > 1 for s > - 1 . Let r  3  r<s<U  2  in the definition of || • ||( ) is that we think of £1 ~ 7  5(r ) s  _1  and  from the "primitive" part of (3 (see the (3 of (9.2.8)) , we expect 7  ^ - e ^ O ^  7  +O ^ ) - ^  2  InF.  t=s  A similar reason applies for having the e~ factor in the definition of || • ||( ). We will see 2  5  later in Sections 10.2-10.3 that the Lipschitz bounds of the (3 functions E>i(£) are to be controlled by the smallness of e , £\, £ , £ , £7/V . 2  s  2  5  The purpose for having k , k and k 3  5  7  available is that they help to control the Lipschitz bounds on terms that are insufficiently small. We express B as a direct sum of five spaces of sequences with norm = max  11^111(1) > II^IL) , ||^3||(3) > I N | ( 5 ) , I N | ( 7 )  •  (10.1.16)  Also, we denote B = {£ | \\£\\ < 5}, s  (10.1.17)  Chapter 10.  The Fixed Point  Solution  192  the 5 ball centered at the origin in B. Next, we convert the flow equations (9.4.15) into the following system of linear equations by replacing E by £ in the /3 functions except for the linear term in (9.4.15). AE*  =  B} (£),  AE*  =  -5 E°  i = 1,2,3,7  +1  +  +1  M  (10.1.18)  B° (£), +1  5  where B [  +  \ £ )  2 \£)  B  +  =  (  2  li=i,  U  i  l  9  )  2  + G'£(£) [ ~ S M ( £ t + H° (£) +1  +  (  = T { E G$\£) A  Bl \£)  1  =  +  2  + e X *(£))  + p>+*{£) + {%»{£)]  s+  5  A + (£)} s  e X° (£)+Y: + 1  l  Glf{£)(5t \£) + +  3  A?\£)  i=l,2  Bt {£) +1  =  n  +  l  {£)-s 5 X  s  \£)+Y  +  M  b  G y(£)fir (n s  2  5  +  A (£) s +l 5  1=1,2  B \£) s+  =  e Y \£)+Y: s +  7  G f{£)^\£) s  7  +  A \£). +  7  i=l,2  Consult with the discussion starting at (9.4.5) for the definitions of the various terms in the above. (Do not confuse the above B's with the B's in Section 9.4.) As mentioned in the analysis of the primitive flow in Section 9.2, because we wish the coupling flow to respect the conditions (9.1.1), unless one has very precise estimates on the (3 functions, it is more convenient to specify the boundary conditions at the root scale r for divergent variables (as s —> oo) and at the scale U for o(l) variables. We prescribe the following boundary conditions for E(£) satisfying the equations (10.1.18). For Ei with i=3,5,7, we prescribe W  == {El, El, E ) = (0, XI, X ), r  r  7  7  (10.1.20)  Chapter 10. The Fixed Point Solution  193  and for U\ and U , we prescribe f/f = U 2  = 0. From the prescribed conditions, the  2  solution to (10.1.18) is, for r < s < U  u £  U[ =  B{{£)  (10.1.21a)  B{(£)  (10.1.21b)  j=s+l  Ui  =  £ j=s+l  El  =  Y,Bl{£)  (10.1.21c)  3=0  Eg  =  J2M- ^ -^B (£) 2  S  +M~ ^E  J  2  5  (10.1.21d)  r b  3=0  s  E  s  = Y. i{ ) B  7  + 7-  £  (10.1.21e)  E  3=0  Note that from the prescribed boundary conditions, (10.1.21a-10.1.21b) are sums running from s + 1 to U while (10.1.21c-10.1.21e) are sums running from 0 to s. Thus for each E , we obtain a map defined by (10.1.21a-e) r  K : (£, E ) —»• (JC /C , AC , K , JC ) = (U U , E , E , E ). r  1}  2  3  A  S  u  2  3  5  7  (10.1.22)  For a function / on the ball B$ and £;£ G Bs, we define the difference operator V/{£,£)  = f (£)-/{£).  (10.1.23)  To show that /C has a unique fixed point, we show the following. 1. For any £ , £ belonging to Bs, /Q satisfy a Lipschitz bound of the form ||Z>£i(£,f)||(0 <5i \\£-£\\, where 5{ are small constants. 2. K{B ) C B . S  5  (10.1.24)  Chapter 10. The Fixed Point  Solution  194  For the second condition, since  \\JC(£,E )\\  <  r  \\K{£,E )-  JC(0,E )\\ + \\)C(0,E )\\,  r  r  (10.1.25)  r  using the contraction property of IC, it suffices to show that ||/C(0, E )\\ r  can be made  arbitrarily small with an appropriate choice of parameters. Next we present a lemma which shows that Lipschitz bounds for /Cj = Ei with Ei given in (10.1.21a-10.1.21e), respectively bounds for ||/Cj(0, £ ' ) | | can be expressed in terms of Lipschitz bounds for r  Bi(£)  of (10.1.19), respectively bounds for ||5j(0)||.  From the lemma, we then show  AC has a fixed point by demonstrating that, by choosing appropriate parameters, there exists small Lipschitz bounds for Bi{£) and small bounds for ||5j(0)||. Before we state the lemma, let us introduce the following bound to be used in the proof of the lemma. For s > 0 and M > 1, d  M~ T S  = -M~  s  ds  "lnM  S  [ln M V + e ] =  -M' e  s  (  K  3  (10.1.26)  s  3  3  where 6M — M — 1, and V and e are defined respectively in (9.4.16) and (9.2.11). Since 2  s  3  K  > 1 and c > 0, (10.1.26) is negative for sufficiently large s. Hence it is easy to see  3  3  that there exists a constant CM depending only on M , such that for s > j > 0 p(P')-  1  < C.  (10.1.27)  M  For example, let M — 1.1. Since ^ d  M  ~  s l s  OM  Vc e  2  3  + (s 5 )) - 1 > s l n M - 1, / M  " < 0 for s > 11. Thus, for 11 < j < s, M " T  S  < M~ T . j  j  (10.1.28)  For the remaining scales  0 < j < s < 10, M-l*-fl  r (T )s  3  Hence, here we may chose CM = 12.  1  < -  K  K  3  + }}  z  ez  + £  3  K  3  + e  3  < 12. -  (10.1.29)  Chapter 10. The Fixed Point  Solution  195  Let (mi.nx) = ( m , n ) = (2,0), (m ,n ) 2  Lemma 10.1  2  3  = (m ,n )  3  7  7  = (1,0), and ( m , n ) = (1,-1). 5  5  Suppose  WVBiisMU.m < M M \ \ s - i \ \  (10.1.30a)  \\Bi(0)\\ , <8?6 .  (10.1.30b)  mi  ni  M  Then VKi{£,£) ll^(0)||  (i)  <  C  5  <  C 6°  Bi  \\£-£\\,  t  Bi  (10.1.31a)  + Cl,  (10.1.31b)  where C  Bi  c e  ;, 1 = 1,2  (10.1.32a)  2  3  CB  3  c  2 he, CM  GB  7  c e  4  7  3  3  ci  0, i = 1,2,3,  Cl  k c K 5  M  \E;\  3  C =  k  7  r  7  (l + l n K ) e ' 2  3  and CM is defined in  k  CB  5  \EJ}\  (10.1.32b)  (l + \nK ) e ' 2  3  (10.1.27).  Proof: From (10.1.21a-b) and the hypothesis (10.1.30a), for i=l,2  <  c e  z  \£-£\\.  3  Let d = k and d = k e~ . 2  3  3  7  7  From (10.1.21c) and (10.1.21e), for i=3,7  ^ ( l + l n r ) - \VK\{£,£)\ 8  1  < ^ ( l + l n r ) " ^ \\£ - £\\ 8  1  £ j=r+l  <  dj c e' 3  ;Si\\S-£\\.  5M  Chapter 10. The Fixed Point  Solution  196  From (10.1.21d), (10.1.27), and the hypothesis (10.1.30a),  ^ s \ - 1i i n  cw /  c11 ^  s wc  s  pa+inrr !^^^)! < 6 s \ \ £ - £ \ \ g , ^ <  6 C  \\£-£\\  M  5  M  ±  M  2 ( s  ^ m + inr-?  T J  i  2S C 5  l  n  r  s  M-l-n^L  j=r+l  <  +  M  \\£-£\\.  M  Thus we have (10.1.31a). The proof for (10.1.31b) is similar except we further have to show M -  2  ^ E l  and E of (10.1.21d-10.1.21e) satisfy the bounds \\M- ^E \\ r  2  < Q,  r  7  5  {5)  11^711(7) < C . It is easy to check that for i = 5, r  The case for i = 7 is similarly done. Q.E.D. From the above lemma, it is easy to see that one can show that K(£, £) is a contraction by showing (10.1.30a) with 5ie~ , 5 e~ , k 5 e~ , 2  2  2  small.  2  3  3  k 5 C e~ , 2  5  5  M  and k 5 e~  4  7  7  sufficiently  Chapter 10. The Fixed Point  10.2 10.2.1  Solution  197  Lipschitz continuity of the 0 functions First order terms  We would like to establish some Lipschitz bounds on the B{(£) of (10.1.19) which are to be used in Lemma 10.1 for proving that the map /C defined in (10.1.22) is a strict contraction. Note that each E>i is finite sum of terms N  (10.2.1) n=l  where N is bounded by a constant multiple of N, the order of.the perturbation defined by (9.3.39). Suppose that each of the K  has a Lipschitz bound similar to (10.1.30a)  ijn  with constant  5i which can be made arbitrary small by choosing e = A / A and S, the 2  2  2  >n  radius of the ball B$ defined in (10.1.17), sufficiently small. Then by linearity, Bi has the Lipschitz bound of (10.1.30a) with a small Lipschitz constant. Because the Lipschitz bound on the 0 functions can be studied term by term, we organize the terms in Bi(£) in (10.1.19) into the following four classes of terms for convenience. We first divide the terms of Bi(£) into the following three classes. Referring to (10.1.19), let  (£) = Bt {£)  + B-  +l  (10.2.2)  (£) + At  where B {£)  0  B (£)  £ X (£)  B \£)  s Y \£);  s+1  s +1 3  s + 7  (10.2.3) S+1  3  s+  7  B (£) s +1 5  = -e  5  S  X {£) s+l  M  +  I  S 5  +  1  (£)  Chapter 10. The Fixed Point  B C \£) S  G  i$V)  B  =  Solution  198  0  =  (10.2.4)  E  G$\e)K \E)  j =3,5,7  +  i=l,2  B${£)  =  T"(E A  2  Gff(£)  ft *(£) +  [i=l,2  [-M*l + e X +2  (£))  5^(5)  =  =  + # (£)  s+2  5  +2  + $ (£)]}  ;  +2  i = l,2,  (10.2.5)  Ai (£) i = 3,5,7. +1  The i?*^ ^) are the higher order terms coming from higher order diagrams, and  B- (£),  1  Btfi~{£)  are terms from the order one diagrams.  functions of order one diagrams in which A A (9.4.15) and (9.2.2-9.2.4)). Because 0t {E)  S + 1  Bf^(£)  +1  are terms coming from 0  has been replaced by the 0f (E)'s +2  (see  can be split into first order and higher order  +2  terms, let us write # + 2 ( £ )  + A\+2{£)  B° {£) +2  =  where As+2(£) are the higher order terms and Bf (£)  (10.2.6)  denotes the terms from (9.2.1)  +2  and J | defined in (9.4.9). Correspondingly, we further divide the terms of Bf^ (£) l  into  two classes. Let Bi£ {£)  = B$ (£)  1  + B& . (£),  1  (10.2.7)  1  A  where B[%\£)  =  0,  £  G$\£)  B! (£)  j =3,5,7  +2  (10.2.8)  i=l,2  =  E A  2  Gff(£)Bt (£) +i  li=i,2  +Gff(£)  [-6 (£t  2  M  + s X (£)) s+2  5  + B' {£) +2  6  + B (£)}} s +2 3  ;  Chapter 10. The Fixed Point  B'+ \ (£)  =  B&AV)  = i  G  A  Solution  199  B°%\{£)=Y G$\e)A?*{£)  0,  j = 3,5,7  i=l,2  [  E G$ (£) 2  A?\£)  + Gft{£)  JA^(£)  (10.2.9)  + A ^(£)}  j .  s  We split our task of finding the appropriate Lipschitz bounds for the above four classes. We will first formulate a few lemmas for obtaining Lipschitz bounds for B? (£) +1  BiG (£)l  and  The Lipschitz constants for other higher order terms will be dealt with in the  next section. Let us introduce more notations' It is clear that if both £ and £ belong to Bg defined in (10.1.17) then so is each £(t) of the interpolation £{t) = £ + t{£ - £),  (10.2.10a)  0 < * < 1,  and that V f {£,£)= where dsf(£)  [ Jo l  ±f(£( ))dt= at  ^ Jo  t  denotes the sequence {dsrf(£)}  d f (£)•(£-£) £  d f(£(t))-(£-£)dt  (10.2.10b)  e  of partial derivatives and  = Y,d f (£)(£?-  £?)•  £f  i,s'  For a function F of £ , we denote the sup of the norm of F over the range of the interpolation £(t) as \\F(£,£)\\ , =  sup | | F ( £ ( t ) ) | |  m n  m > n  .  (10.2.11)  0<4<1  We state a lemma concerning using the differentiability of a function to establish the Lipschitz continuity with respect to || • | | , . Let K(£) be some sequence of differentiable m  n  functions of £ such that for each index s, K {£)  only depends on £ ', s < s' < U. Here  s  s  differentiability means the partial derivatives dsK (£) s  exist.  Chapter  10. The Fixed Point  Solution  200  Lemma 10.2 Suppose that for  E II  s'>0  ILi,m  [ Sf ](€,£) d  ,K  2  where a, is some positive constant. Then for p\,p  > 0 or —p = p\ > 0,  2  \T> K(£,£)||  (10.2.12a)  < Oi  2  p < £ ai \\£i IPl)P2•— £i  mi+Pl!m2+  (10.2.12b)  2  Proof: By (10.2.10b),  (r ) s  <  (1 + In r ) s  (r ) (i + s  m i  E  inr r s  2  ro2+P2  f  1  \ V K (£, £) I s  dt £ £ | c v /P_\  < £ <  m i + p i  mi,m  s'>s  ^  -  P 1  VT ' / S  2  K {£{t)) ( r ) ( i + i r ) s  s  p i  s  n  / l + lnr'V '  P 2  \ef-£f\  2  I 1 + ln V ' J s  \\£i  £i\\pi,. ,P2  i.  i \ lpi,P2-  Note that in the case — p  2  = p\ > 0, we use the fact that for z > 1, (1 + \nz)/z  is a  decreasing function so that for z < z' z{l+\nz') z'(l + lnz)  ~  '  Q.E.D. Recall the errors variables W, W defined respectively in (10.1.1) and (10.1.5). Let us define their corresponding "external" replacements in the map K defined in (10.1.22) as  £ +£ 3  fz(p ) + h(p )£z 2  5  2  +  h(p )£ 2  5  (10.2.13)  where / | , / , / , and f are defined in (10.1.4) and (10.1.6). Because the photon running S  3  S  s  5  slicing V (E) of (5.4.9a) and its approximation V (E)\ ~ S  B  B  l+  obtained from replacing  1 + 7 by 7 are functions oiW and W, the /3 functional E>i{£) of (10.1.21a-e) are functional  Chapter 10. The Fixed Point  Solution  201  of £i, W , and VV. In taking derivatives of V {E) S  B  in (10.2.10b), we  and V (E)\ ^ B  l+  7  would like to use W and VV as the "error" variables instead of £ and £ . 3  5  Let us study the Lipschitz bounds of W and VV. We would like to set the following conditions on the parameters of the set up of the fixed point argument: e K  3  2  = A /A , 2  2  = A 3 , k , k , k , 5, M, e, where the fcj's are the parameters defined in definition of 3  5  7  the norms (10.1.15), 8 is the radius of the ball Bs defined in (10.1.17), M is the slicing parameter, and e is the overlap parameter . Some of these conditions will be further refined at later part of Section 10.2.3. Let e  2  h 1  <  min((10 b M )  <  1/2,  <  M < 3/2,  2  - 1  5  , k ),  K  5  k < 1/2,  3  e < 1/2  5 < k /2 < 1/4  k <l,  5  > 1,  7  3  e < 1/3.  2  (10.2.14) (10.2.15) (10.2.16)  where b is denned in (E.2.75-c) of Appendix E . 5  Lemma 10.3 llW-VVll!,-!^- ^-^!.  (10.2.17)  l i v v - ^ l l i , - ^ ^ -  (10.2.18)  1  1  ! ^ - ^ ! ! .  Proof: From (10.2.13),  r (1 + l n r ) | W - W s  <  s  (1 + l n P ) k  3  1 + lnT  5  1  _ 1  s  s  (10.2.19)  l ^ - ^ l + l ^ - ^ l So So I ~f~ L  '  3 1  ' kT  s  5  e (l + l n T ) 2  8  Chapter 10.  The Fixed Point  From (10.2.14-10.2.15), K  Solution  > 1, e /k  < 1, and k < 1/2. Thus from (10.2.19)  2  3  5  <  202  3  1 11^-5311(3) + K  (10.2.20)  ^ ||5 _5 ||(5) 5  5  /C5 A 3  3  From (10.2.13) and the definition o f f in (10.1.6),  T  s  (1 +  InT )- 1W -VV s  <  (1 + l n T ) -  1  <  (1 + l n T ) -  1  5  5  s  1  '/aT 3" 1 cs  cs  3  1  / T  S  J t>  cs  5  °3  c  cs\  °5  5\  c  fi  1^3-^1 +  (10.2.21)  ^ 5 - ^ 1 J3  Also, from (10.1.4) and (10.2.16), it is easy to check that / | / / | < 2. Hence,  ||W-VV||l,-l  <  ||5  ^ K  3  -f ||(3) 3  +  3  T  ^  K A 3  lift  (10.2.22)  -5 ||(5) 5  5  Q.E.D. Since we wish to apply Lemma 10.2 to obtain Lipschitz bounds for the B(S)'s  of  (10.2.2), we need to establish bounds on the derivatives of the B(£) with respects to the £j's,  W , and VV. Let us first consider the bounds on the derivatives of the factors of  B(£).  We consider five types of these factors. The first type is the ft's, £ ' s , ft's, and 2  ft's. The second type is the running slicing functions of the forms (E.2.5), (5.4.9a), and (5.4.9a) with 1 + 7  s  (defined in (5.4.9c)) being replaced by 7 = Xi, + Af . Let s  x denotes a A?, or 7 , or 1 + 7 s  s  s  (10.2.23)  Let us drop the superscript s from p and write a running slicing function as s  V{x ,x ) s  s+1  D(x ,x ) s  s+1  (10.2.24)  Chapter  10. The Fixed Point  Solution  203  where +1\2  D(x , x ) = 1 + p(x -1) + R PR{{X Y s  s+1  s  S+1  X  (10.2.25)  ~ 1)  PR and XR denote respectively p restricted to the right overlapping region SR and the characteristic function of SR, and p  = (1 - p ).  R  Let D(x , x ) s  R  be denoted by D , and  s+1  S  D restricted to the support of pR be denoted by DR. The third and fourth types are respectively of the forms Z  EE {Ax*)-  [V(x ,x )  1  x  s  -V(x ,x )\  s  3  PR PR (X +  a+l  x )  s  s+l  D (x ,x )D (x ,x ) s  s+1  s  R  s  R  PR X  PR DR(X , S  DR(X ,  X +^)  S  X) S  S  PR  x (p s  p  + p x)  R  R  s+1  R  (the last line is obtained by the fact that D (x ,x ) s  2  s\-l EE (Arc?)  "  P  S  (10.2.26)  PRX  S  s  R  PR  X)  R  'PR +  X  s  and  PR  [l + p (x + -l) s  S  X  = X (PR + p x ))  s  R  Z  DR(X ,  X + ) S  +v (x>y  X  s  R  PR X"  PR  S  D (x ,x )  s  R  + DR(X ,  l +  l  R  p (x'-l) R  (PR)  2  [l + p (x°-l))  (l +  R  p (x<+i-l)) R  (PR)  2  (pR + PR x ) (p  + p  s  R  (10.2.27)  x )' s+l  R  They come from expressing the G - j ( £ ) / ? - ( £ ) ' s (see (E.l.7) and (E.l.29) of Appendix +1  E) using differences instead of interpolation integrals. The fifth type is the relevant part of  Z  3  EE =  VV(j ,j ) s  V(l + f , l + p (pVY  ~  s+l  Y )-V(y ,j )  + p (y  +1  s+l  R  + 1+ 7  7J(1 + 7 ,1 + 7 S  S + 1  s  S+1  s+1  )737  )/J(7 ,7 + ) S  S  1  S+1  )  Chapter 10.  The Fixed Point  Solution  204  where V is defined in (10.2.24),  and p e S = (1 — e < p < M ). Z comes from the term # | ( £ ) arised from replacing 1  2  2  2  Z  1 + 7 by Y in the photon covariance (5.4.9a). Discarding irrelevant terms, s  Zs  =  K f (p)V(Y,Y )V(l  + f,l  +1  Z3  Af /lfb)^(7 ,7  +  1  s  s + 1  +Y )  (10.2.28)  +1  )D {l  + YA  R  + i  + AS /ir(p)^(i+r.i+y ) y +1  s + 1  )  +1  Pfl  >r+1)  where / | ( p ) = (p - l ) / p and p e S i . Note that since p e S i , e/(l - e) < \fz \ < 2  2  2  2  3  5  /M . 2  M  3  Even though / |  =  3  we cannot extract a factor 5 from H because the 2  Q(SM),  M  relevant part of H contains the term V' which makes it O(^M) 2  F  relevant terms of E S E  2  (See Appendix F for the  from expanding the slicing function and the later part of Section  9.1). Let p/D  us first consider the derivatives of the second type factors which are of the form where D is defined in (10.2.25). Let a product FG of two sequences F, G be  defined by (FG)  = FG  S  d p/D x  = — p/D  S  and an inverse { ( F ) } of F be denoted by F~ .  S  s  (d D/D),  - 1  l  we would like to consider bounds on \d D/D\.  x  x  Since  From the  definition (10.2.23) of x , and the decompositions s  A = A (l 2  2  + U ), 2  = r ( l + W ),  s  s  7  f  5  = f (l s  + W ), s  let us correspondingly write x = z(l + y) where  (y,z) =  (VV, T) or (VV, f )  for the Bose case  (£ ,A )  for the Fermi case  2  Also, let both D(x ,x ) s  s+1  2  and 1 + y be denoted by D . s  s  (10.2.29)  Chapter 10.  The Fixed Point  Lemma 10.4  Let £, £  Solution  205  G Bg where 8 satisfies the inequality in (10.2.15). I d tD -(y,y) v  < 4.  D  (10.2.30)  Proof: Note that D only depends on £ and £ s  s  s + 1  . Hence it suffices to consider only the cases  for t = s, and t = s + 1. The case where D = 1 + y is obvious. From (10.2.15) and the fact that £ G B$, we have: Fermi case:  —> 1—-— z ~ K  - > 1 — 25 > ~ - 2  L L J  s  3  x Bose case : — > 1 z ~  (A)  1 + lnT  M  W  s  (10.2.31) > 1 - 5 A;," > 1  i,-i  (A)  where W denotes a W or W . We get the last line from the fact that, for z > 1, (1 + In z)/z < 1. From the derivatives d sD  s  8  x  D  Z  s  P  z  D  X  s  + \D  X  s  s  s+l d s+\D D  S  dyS  p  s  s  x  z  D  s  s  s+l 2p  x  R  z  2 zs+l  s + l  D with \px /D \ s  s  < 1 and \p (x ) /D \ s+1  2  PR  rS+l  s  (X ) S+1  2  D  s  < 1, we have the desired bounds (10.2.30).  s  R  Q.E.D Next we consider terms from the third and fifth type factors (10.2.26-10.2.28). Let F {y)  PR  s  z  s  s  PR z {l  + y )  s+l  G (y) s  z  D (z {l s  R  H (y)  s+1  + y ),z ^(l s  1  s  z  (10.2.32)  PR + PR z (l + y )  +  s  y )) s+1  PR  z {l + y ) PR + PR z (l + y )' s  s  s  s  Chapter 10. The Fixed Point  Solution  206  Note that the factors (10.2.26-10.2.28) are composed of the F ' s , G % z  z  and H 's.  We  z  would like to consider following bounds on (10.2.32) and their derivatives. Lemma 10.5 Let £, £ £ B$ where 5 satisfies the inequality in (10.2.15). one of the F ,G ,H Z  Z  \\zf \\ <2 z  y  (10.2.33)  Q  I k d tf (y,y)  | | < 16  z  (10.2.34)  0  where z is defined in (10.2.29). Proof: From (10.2.29) and (10.2.31), 1 + y = x /z s  s  1*7,'I <  > 1/2. Hence  s  < 2.  l+  2/  s  and we have (10.2.33). As for (10.2.34), we consider the following bounds. z  s  (PR)  s  y  *  2  \d sF  z  S  (p + p zs(l R  + y*))  2  R  1  PR X  [i + y°y  PR + PR X  S  < 4. S  It is easy to check that < 1.  -1 —  Hence it follows that 3y3  Z  p  G^  R  p  XZ S  R  S+1  (DR)  2  z  s  z (l s+1  s  y  z  =  z  PR x"  + y )(l +  y )  s  s+1  2 (PR)  X Z  2 z  PR  2  z \d s+iG (y)\ s  z  defined in (10.2.32).  S  Z  Let f be  s  ^+1(1 + ^ + 1 ) 3  S+1  D  PR  S+1  S+1  D R  S+1  DR  R  (X )  (* )  2  < 16.  2  < 4,  Chapter  10. The Fixed Point  Solution  Lastly, z \d H (y)\ s  s  yS  z  <  -  <  + y (PR + PR z (l + y )y s  l  < <  (1 + V ) 1  Q.E.D.  2  (i + y ) 2 s  (l+y ) S  8.  s  PR PRX s  <  1  PR PRZ s  2  2  (PR + PR X ) S  2  +  PR  (i + y ) s  PR PR + PR X  S  s  PR PR + PR X  PR X S  PR + PR X  S  +1  2  PR + PR z (l s  + y) s  Chapter 10. The Fixed Point Solution  10.2.2  208  Lipschitz bounds of B{£)  Let us consider Lipschitz bounds on functions that are of the form K{£) = P(i,2, )( ) where  (10.2.35)  £  p  and Q(£) are defined as the following.  - P ( i , 2 , 5 ) ( £ ) , P(7)(£)  -P(i,2,5)(£)  Q( )-  (7)(£)  £  5  is a product of £is, £ ' s , and £ ' s : 2  P(1,2,5)(<?)  =  S  5  (10-2.36)  ^,2)^)^(5)^) rip  h){£) = II  s < < U ii = l,ov 2  p  Sj  .7=1  k(£)  =w n \  p  7  p  S l  + lnr))'s:  is a product of {£ /(l  P(7)(£)  s< <u.  ^£)s  = n rn^j'  s  - >s  u  -  (  1 0  - 2  3 7  )  Q(£) is an integral of the following form. Q(£) = j dQ  (Y[P)(Q) Q(£,Q)  T(£,Q)  G(Q),  (10.2.38)  where : 1. Q = (qi, • • •, q ) where the g^'s are loop momenta of the integral; r  2. n  7  Q - ( £ , Q ) = U j=l  (1 _ | _ l p s ^ )P n  (  £  B  n huh'  SJ n -n B  7  6=1  S  b  n  F  B f=l  S  U  '  (ia2  f  F  where Dp and D are defined in (10.2.25) respectively with x = A | , and x = 7 S  s  B  or 1 + 7 , and s s  <  s&,  Sf, Sj  < U;  s  s  39  Chapter 10. The Fixed Point  Solution  209  3.  i f I I  = i f fZfa)  T°(£,Q)  t=l  <2=1  where s < s , s , s , s < U, and each f  s  q  f  u  v  z  G' *(W)  (10.2.40)  Z  u=l  is a F | , G ^ , H defined in (10.2.32); z  4. / where  S^JJ )(Q) p  Q T (£, S  M  (10.2.41)  G  denotes the support of the product of the slicing functions from  Q) and 5  S  dQ\G(Q)\<5 C ,  = M - 1. 2  M  Using Lemma 10.2 - Lemma 10.5, we wish to establish Lipschitz bounds for the components of (10.2.35). Let us denote || • | | as || • 11o,o where || • | | 0  m)Jl  is the norm defined  in (10.1.14). We first consider the case of (10.2.35) with Q{£) = 1 and P( )(£) = 1. 7  Recall that e  s  = Af/A|,  ki are parameters defined in the norms || • ||(;) in (10.1.15), and  8 is the radius of the ball B§ defined in (10.1.17). L e m m a 10.6 Let P(i,2,s), P(i,2) , and P^  be the products defined in  <  i v p ^ i s ^ ) ^  \\e-e\\.  (10.2.36).  (io.2.42)  2.  \VP (£,£)\\i,-i  <  (6)  e  |2>^(i,2^)(f,f)||i,-i <  n  2 n 5 5  e  8  n &  -  1  \\£-£\\.  (10.2.43)  2n5^n +n -l 5  ^-i,  p  B  B  [n + n ] \\£-£\\. P  5  (10.2.44)  Chapter 10. The Fixed Point Solution  210  Proof: Since|H|i,_ <|H| 1  1 > 0  ), by Lemma 10.2 with (mi, m ) = (n — 1,0)  <l/lC  2  p  and (pi,p ) - (1,0), 2  \\VP , (£,£)\\  <  1,-1  (l 2)  \\VP , (£,£)\\  T  {l 2)  (10.2.45)  npfi  1 A  \£-£\  j  3  np-1,0  It is easy to see that  ft ,. ps  „ C  _  where n is the number of repeated £•] in P( )> S  1  a n  1)2  <  n  '  (10.2.46)  )  2  d f ° £ £ #<5> r  (10.2.47)  n^.111,0^^-. 1  j=l,j&  np-1,0  By counting the number of terms in P* )'  a  n  (  lt2  l (10.2.45-10.2.47), we have (10.2.42).  For 2. of the lemma, we apply an argument similar to the proof of 1. of the lemma. We apply Lemma 10.2 to \\VP (£,£)||i,_i  with ( m i , m ) = (0,0) and [p p )  {5)  2  u  2  = (1,-1).  Since 1 + l n P e ||ft||  1+ l n P e  2  |  and (1 4- mz)/z  t  s  |  2  (5)  -  S  P fe  d  '  (10.2.48)  < 1 for z > 1, we have II  -  i  fe \ 2  n  From (10.2.49) and Lemma 10.2, using the fact that  \£5 -  ft||l-1  it is easy to see that we have (10.2.43).  = T~Hft - ft 11 (5) fe  5  _  1  (10.2.49)  Chapter 10. The Fixed Point Solution  211  From (10.2.42) and (10.2.43), using differencing by part,  |2>P(l,2,5)M)J|l,-l  ||©P(l, )(f^)||l,-l + r n| | P ( l , 2 ) ( 5 ) | | o | | 2 > P ( 5 ) ( f , f ) | | l -  < <  l|P(5)(£)||0 2 n Sn  <  RHS of (10.2.44).  e  5  a  5  >  p  x RHS of (10.2.42) + — ^ x RHS of (10.2.43) Q.E.D  Next we consider the case of (10.2.35) with  P(i,2,5)(£)  = 1 and Q{£) = 1.  Lemma 10.7 , 2 \ "7  l-l  \VP (£,£)\\ <n {7)  0  7  ^ l l f - S H .  (10.2.50)  Proof: We apply a similar argument as in proof of 1. of Lemma 10.6. vpfa(e,i)\  <  Y, (l +  <  - E (l +  m  dP, d.  \£^  r<)^%,£)  -£ '\ 7  (1 + l n T ' ) s  BP  \nr<)?-W(£,£)  ll^-^ll-  (10.2.51)  Since m £° (1 + \nT )d nPU£) = n TT — i ^ (7)1 ; . ^ i + inr-i where n is the number of repeated £ in Pfo(£), and 7  St  £  (10.2.52)  =  l  7  < 1 + ln I >  iN|(7) < e 5 2  A)  A;  7  7  (10.2.53)  '  by counting the number of terms in the product Pfo {£)  E  P, (l + l n r " ) ^ ( £ , 5 )  /e <5\  Thus from (10.2.54) and (10.2.51), we have (10.2.50). Q.E.D.  2  n7_1  (10.2.54)  Chapter 10. The Fixed Point Solution  212  Next we consider the case of (10.2.35) with  denote a Q(£) (defined in (10.2.38)) with a f , f ZF  (defined in (10.2.40)) by Q (£).  or, G factor removed from  ZB  z  T(Q,£)  Let  fx  N  = n +n, ,  NQ  =  F  = 1 and Pcj)(£) = 1. Let us  -P(i,2,5)(£)  F  N =n + n  z F  32 (N  B  B  + h,  ZtB  (10.2.55)  B  (10.2.56)  + k^Ng)  F  where the n's are defined in (10.2.39-10.2.40).  Lemma 10.8 Suppose there exists a constant CQ satisfying the following bounds: for all £ E Bs,  ||Q(£)IL,n  2  \ ~ Qfz(£)\\n ,n z  <  1  1  2  (10.2.57)  < $M CQ, 0~M CQ,  where z is defined in (10.2.29). 1. (10.2.58)  "•1 + 1,712-1  2. For Q(£) contains only Fermi running slicings, i.e., N = 0, B  \\VQ(£,£)\\  < 32 S C N \\£ - £\\.  ni+hn2  M  Q  (10.2.59)  F  Note that the norm in the second case is different from the first. Proof: Since D ( £ ) s  £ 1 1 ^  only depends on £ and £ s  , by Lemma 10.4 and Lemma 10.5,  s + l  ds rD f +  W,£)||  ,Sf+r  t  f=l  t  r=0,l  + £ll(^)- Q^(ft^)lkn 1  q=l  <  D  £ \\^  d s rfz(£,£)\ £  +  r=0,l  5 C M  2  f F  Q  [8 n + 32 n }. F  z>F  (10.2.60)  Chapter 10. The Fixed Point Solution  213  Similarly, by Lemma 10.4 and Lemma 10.5, we have 1 1 ^ 0 ( ^ ^ ) 1 1m,n  E  w  2  + Kyy < 5 C [8 n + 32 (n  K  w  M  Q  B  +n )  z>B  B  Using the above and applying Lemma 10.2 and Lemma 10.3, \VQ(£,£)\\  ,  712 —  ni+1>  <  8(n + 4n )  5M CQ  r  2  2  2  \\W -  Skeins + 4 ( n , + h )\ \\£ - £\ z  B  B  3  <  5 CQ 32 [N + k^N ] \\£-£  <  SMCQNQ  M  w  F  2  | | W - W||i,_i +  + K  2  + 4n , - \\£2 - f ||( ) + l + lnK F  -  \\£ - £ \l^i  ZtF  F  n S m C q  1  F  B  \\£-£\\.  The proof for Case 2 is similar to the above except that there is no (1 + \nK )  1  3  factor in the third line of the above because the norm for VQ is of type || • | |  n i +  i  j n 2  .  Q.E.D. We now show that for ( n i , n ) = {N ,—n ) 2  B  in (10.2.57) where  7  N  B  and  n are de7  fined respectively in (10.2.55) and (10.2.39), then for NB > n , we can choose CQ = 7  C  2  /A%  Nb+Nf  G  F  where C  G  is defined in (10.2.41).  Lemma 10.9 Suppose N  > n.  B  7  \\Q(£)\\N ,-ni  <  B  \ ~ QfA£)\\N ,-n z  (10.2.61)  < SM c ,  l  B  $MCQ,  7  Q  where  0  2  CQ ~~AT  Nb+Nf  F  (10.2.62)  Chapter 10. The Fixed Point Solution  214  Proof: Recall the definitions oix ,  z , and y in (10.2.23) and (10.2.29). From (10.2.25) and  s  s  s  (10.2.31), each running slicing p /D t  (10.2.33), each j\, Q  7  T (£,Q)Q (£,Q)  Sj,Si > s.  (T )  S  (1 + ln  Thus it is  t  > n,  B  where  in (10.2.40) is also bounded by 2/z .  S  easy to check that, for N  in (10.2.39) is bounded by 2/zK From  s  factor of T {£,Q)  z  S  oiQ {£,Q)  t  S  NB  m  ONB+NF  ps/i  I  ] „ r s , \ NB-n7  ™  T )"7 s  From (10.2.38), (10.2.41), and (10.2.63), since  Sj,Si > s,  we have  ll<5(^)lliv ,-n < B  The bound for \\z~ Qf (£)\\N ,-n  can be shown by a similar argument.  l  z  B  (10.2.64)  7  7  Q.E.D. We now consider more specific Lipschitz bounds on K(£) defined in (10.2.35). Let (mi,m ) 2  d  B  where N  B  =  (1,0), (0,1), or (0,0),  =  N  B  (10.2.65)  - n  7  and n are defined respectively in (10.2.55) and (10.2.39). We would like to 7  consider the following norms of V K{£, £): \\VK(£,£)\\ _ . mi+lt  (10.2.66)  m2  with following specification d >0,  C  B  where d is defined in (10.2.65), n B  K  =e C  (10.2.67)  2NK  Q  Q  > 0, CQ is the constant given in (10.2.62) and CQ is a  constant independent of the parameters e , K , ki, S. The parameters 2  3  N ,n ,n ,mi,m2 B  7  p  Chapter 10. The Fixed Point Solution  215  satisfy one of the following conditions: 1. ) NB  =  0, n > 1, m = 0  2. ) d  >  2  >  2, mi = 0  B  4.) n  7  (10.2.68)  x  p  3.) d = l, B  mi = 0  5.) n = 1, m = 0, m = 1. x  7  2  Note that each of the terms of E>i{£) from (10.1.19) fits the above specification. Let us state some useful bounds to be used for determining Lipschitz bounds on (10.2.66).  L e m m a 10.10  Suppose ds > 0.  1. For ds > 2; or for ds > 1 and m\ = 0, \M\m  1  l-N ,n -m  +  B  7  <  2  ^ B - f m i + l ) (1 + l  n  ^  2  '  (10.2.69) 2. For n = 1 and mi = 0 and m 7  = 1; or /or n > 2 and mi = 0, 7  2  2 IlilUi-JVB.mz+riT  <  w/iere (mi,m ) = (1 + m i , — m ), (mi, — m ), 2  2  2  (10.2.70)  or (mi, 1 — m ). 2  Proof: From the fact that d = N B  B  II  — n and 7  ,,  l-i,i  <  sup z>l  lnz  1+ Z  < 1,  the cases:  d  B  > 2,  d  n = 1, mi = 0 and m = 1 , 7  2  B  > 1 and mi = 0  n > 2, m = 0 and m = 1 7  x  2  Chapter 10.  The Fixed Point Solution  216  > 2, m = 0 and m = 0, using the fact that  are clear. For the case n  x  7  ||l||_i,  2  =  2  >-<  sup^  3/2,  Z  2>1  we have  ^•\\mx-N ,m-m2  5:  B  11111 —djs — (?i7—2),ri7—2  3  <  -1,2  | | l | -2,2  <  -1,2  <  2 K 1  B 3  Q  6 2A f  Q.E.D. We consider the \\V K { £ , S ) \ \ \  + M U  -  with CQ, - / V , n , n , b  M 2  7  p  m i , m  2  satisfying the  specification (10.2.67)-(10.2.68).  Lemma 10.11 \\VK(£,£)\\ _ <5 1+mu  m2  (10.2.71)  \\S-£\\,  VK  where  <W <  —  rHTT^  2  [*> n  and NQ, CQ are defined respectively in (10.2.56),  + n s  +  +  717  6  <d  (10.2.72)  N  (10.2.67).  Proof : Using differencing by part, VK(£,£)  =  P  =  P(i,2, )Q(£)  W  )  Q { £ )  Suppose NB = n  5  7  VP (£,£)  +P  VP {£,£)  + P {£)  {7)  {7)  = 0, n  p  VP Q{£,£)  {£)  W)  {7)  {7)  \Q{£)  VP  > 1 and m i = 0. Since P(j)(£)  {  W  )  (£,£)  +  P , , (£)VQ(£,£)] {1 2 5)  = 1, we only have to deal  with two terms. \\VK(£,£)\\ _ 1+MU  M2  <  ||Q(f)||o||2?P(i,2, )(ftf)||i 5  1  r o a  + ||P(i  >2l  5)(f)||o,-m  2  VQ{£,£)  l+mi,-m2  Chapter 10. The Fixed Point Solution  217  From Lemma 10.6, the first term ||<9(f)||o||2?P(l,2 5)(f,f)||l,I  <  m2  \VP  (£,£)\\  {l>2t5)  SMC  h0  (1 + l n K ) 2  Q  m  3  $M C (n + n ) ^ ^ 2  Q  <  5  P  5  +n  k?K?-  (l + l n K ) ™  1  ~  np+n5  e  l  \e-e\\.  2  3  (10.2.73) From (10.2.59) of Lemma 10.8, the second term |P(l,2,5)(£)||o,-7n  2  |-P(l,2,5)(£)||o  <  VQ(£,£)  32 5 C Np M  32 6  M  <  CQ N  \\£-£\\  Q  (^+n ) n +n  2 e  F  K S  p  5  K?k?(l-rlnK ) ' m  £-£\  3  (10.2.74) Collecting the two bounds (10.2.73) and (10.2.74) and using the fact that K  > 1, we  3  have (10.2.71). Suppose the parameters satisfy one of the following cases: ds > 2, n  d  B  > 1 and mi = 0  = 1 and mi = 0 and m = 1,  7  n > 2 and mi = 0.  2  7  The corresponding norms can be bounded by a sum of three terms. \\VK(£,£)\\i _ +mu  m2  <  \\P i, , Q(£)\\ {  2 5)  l+mi , - m 2  \\VP (£,£)\\o VP , (£,£)\\i,_i Q(£) Imi,—m +l  +\\P(7)(£)\\o  (10.2.75)  {7)  ih2 5)  2  + ||P(7)(£)||o ||P(1,2,5)^)||0  \\VQ(£,£) 11 l + m i , — m  2  We consider each of the three terms in the RHS of (10.2.75) separately. From Lemma 10.7, l|P(l,2,5)Q(£)|| l+mi,—7Ti2 \\VP (£,£)\\ {7)  <  ||P(l,2,5)(£)||o  ||l||l+mi-JV ,7i7-m B  Q  2  N ,-n B  r  I |2>P(7) (5,5)110  Chapter 10. The Fixed Point Solution  218  g2n £n +n5 6  SCn  p  <  M  Q  5 ~e n7  7  \l+mi-N ,m-m B  < llll  2n7  \\£-£\  /07  2  (jnp+^+nr-l^CriB+nT+nx:)  8M CQ HJ l+TOl—  l  N ,m— 7712 B  -"•3  P L.n 7  ft  5  ft  ||£-£||.  1,717  S  (10.2.76)  From Lemma 10.6, •  11^011.  11^>, ,„,„-„«| P P  <  HP(7)(f)||0  Hl|L,-„.,„  0(f)  + 1— T712  5M C  p  v  5  n )e ^ 2  TfTi -l,n p  +  5  ^5 "  +n7+n  5  ,m-rri2  B  ^ " -  1  -"-3  p  I l + m i -N  2 n 5  5  SM CQ (n +  <  M)  (n + n ) e  q  B  7  \VP (£,£)\\  N ,-m  I l+mi-N ,m-m,2  &  W  B  2\ " 7  <  (£,£)||  ( 1 > 2 ) 5 )  n  -  +n5+n7  1  in  5  7  (10.2.77)  From Lemma 10.8,  PQ(5,5)  l^( )(f)||o | | P ( i , ) ( f ) | | o 7  <  2>5  11^(7)^)110 11^(1,2,5)^)11 0 1111 | m i — N ,m+l B  <  For d  B  B  g  B  B  1+N ,-717-1  C JV ||5-5||  llJ-l|mi+l-JV ,n7-m  > 2, or for d  2  —m,2  fe  £7  £ ^  l+mi,-m  M  C Q A^Q  e  2("«+"7+nj ) n +n +n r  r/"p i.n i.n 5  2  > 1 and m  x  0  g  p  B  7  (10.2.78)  ^I  IP  7  = 0, from (10.2.76)-(10.2.78) and the bounds of  Case 1 of Lemma 10.10,  ||PA(£,5)||  1+MI  ,_ 7712  S  CQS""^-  ^+m n )  1  M  —  k  7 7  e  K$ ~ lp  1)+dB  ~  +  il+mi)  K  ^  +  n  ^  +  K {n  +5 N )  l  3  (!  7  + l n  Q  ^m  K  2  < 6VKThus we have (10.2.71). The remaining cases can be bounded similarly by using Case 2 of Lemma 10.10 (the factor 2 in (10.2.72) corresponds to the factor 2 in the bound (10.2.70)). Q.E.D.  Chapter 10. The Fixed Point  10.2.3  Solution  219  F i x e d point argument for the first order flow  In this section, we wish to show there exists a fixed point for the map K.(£, E ) defined r  by (10.1.21a-e) with the B(£)'s  restricted only to the first order terms = B' (S) + B' (£),  B , .t(£) i  where B?(£) and B? (S) tG  i  1  (10.2.79)  itG  are defined in (10.2.3) and (10.2.8) (the JC(£,E )  here corre-  r  sponds to a first order flow). We show the existence of the fixed point by showing that IC(£, E ) is a contraction with respect to the norm defined in (10.1.16) over the domain r  Bg and K,(Bg) C Bg, where Bg is the ball defined in (10.1.17). From Lemma 10.1 in Section 10.1, we desire to have small Lipschitz bounds for the i?j(<f)'s for showing that )C(£,E ) r  has a fixed point in Bg. More explicitly, we need the following bounds for the  S^s in Lemma 10.1: 6ie~  <  2  c  3  k 5 e~  < c  k 5 e~  <  k 5 e~  < c  2  3  3  5  A  7  (10.2.80) (10.2.81)  3  2  5  fori=l,2  7  (2 C )'  (10.2.82)  1  M  (10.2.83)  3  where the fcj's are defined in the norms || •  in (10.1.15), e = A^/A ,, c = —b , b is 2  2  3  defined in (E.2.75). If we take the radius 8 of the Bg to be Q{e ), 2  says that for a K^ (£) t  3  then Lemma 10.11  satisfying the specification (10.2.67)-(10.2.68) in which for i = 1,2,3,5, for i = 7,  n + n +n +n 7  5  v  K  >2  (10.2.84)  n + n + nj< > 3 or n > 2, 7  p  7  then the corresponding Lipschitz constant 5 K is 0 ( ) e 4  V  Hence for sufficiently small e , we can make 5 2  VK  satisfied.  3  for z ^ 7 and 0 ( ) e 6  f ° i = 7. r  small so that (10.2.80)-(10.2.83) can be  Chapter  10. The Fixed Point  Solution  220  Since not all the terms of the first order (5 functions the first order terms in  satisfy (10.2.84). Let  B ist[£) it  that do not satisfy (10.2.84) be denoted by K(£).  B xst{£) it  choosing appropriate value for the kiS defined in the || •  norms, for K(£),  By  we can  again make the corresponding Lipschitz constants 6X>K from Lemma 10.11 sufficiently small so that the map  is a contraction. We isolate K(£) from other first order  JC(£,E ) r  terms by extracting leading terms from X  and Y  s  defined in (9.4.12). From (9.4.12), let  s  us write X (£)=X (£)+X (£), s  s  Y°(£) = Y:(£) + Y °(£),  s  a  b  (10.2.85)  b  where X°(£)  =  *'W  =  2 ^ ~ ^ , (1 + U$) 4  Y  X (£)  = X (£)-X (£),  s  2  s  b  n\"3'  (10.2.86)  s  a  Y (£)  =  s  b  Y°(£)-Y:(£).  The terms that do not satisfy (10.2.84) are: Kl{£)  =  Kl{£)  =  K°{£)  =  e X (£)  of B»{£)  s  3  a  (10.2.87)  I (£)-S e X (£)ofBI(£) s  s  5  M  5  a  e Y:(£)o{B° (£) 7  7  where the e's are defined in (9.4.14) and 7| is defined in (9.4.9). Let us study K{£) in more details. Recall the parameters for the K(£) in Lemma 10.11: (mi,m ); (n ,n ,n ,n ,nK, 2  5  7  N , N ); and CQ. For each of Ki(£), let us specify the cor-  p  7  B  F  responding parameters and the Lipschitz constant 5VK of Lemma 10.11. K {£): 3  (m m ) u  = (0,0);  2  (n ,n ,n ,n ,N ,N ) 5  7  p  K  B  F  = (0,0,1,1,0,2), 8VK < 8 | c - 6 / 3 | e 5 2  3  6  M  C  Q  = 4|c - 6 /3|;  [1 + 64 5].  3  6  (10.2.88)  Chapter 10. The Fixed Point  221  Solution  K (£): (mi,m ) = (0,1); 2  5  i) I (n ,n ,n ,n ,N ,N ) 5  7  p  K  B  = (0,1,1,0,1,0),  F  CQ = |c | 5  where C5 is the coefficient from J | (see (9.4.9));  ii)  5 e X (£) M  5  a  (n ,n ,n ,n ,N ,N ) 5  7  p  K  B  = (0,0,1,1,0,2),  F  C  Q  =b  5  where 6 is defined in (E.2.75) of Appendix E ; 5  S  VK  <2S  M  e  ip+4o k  2  5  (1 + 64 5)  (10.2.89)  7  K (£): (mi,m ) = (0,1); 2  7  (n , n , n , n , N , N ) = (0, 0,1, 2, 0, 4), 5  7  p  K  B  C  F  Q  = 4 |6 | 7  where 6 is defined in (E.2.75); 7  < 8 |6 | e 5  [1 + 128 S\.  4  7  M  (10.2.90)  From (10.2.88), (10.2.89) and (10.2.90), in order for the £<(£)'s to have small Lipschitz constants satisfying (10.2.80)-(10.2.83), we rake the following choice for fc , k , k , and 5. 3  5  7  Let =  min  C3  16\c -b /3\'2j' 3  k  7  = min  C3  1  k = min 5  6  M  5 < min ^C,? e , ^ 2  16|6  ,8C ((|c |/A; )+ 4 6 ) ' 2 r 5  7  5  (10.2.91)  7  where C 5 is a positive constant to be determined later from the bounds of ||-Bj(0)||  mi)ni  so that the condition JC(Bs) C Bg can be satisfied. Note that the above choice of ki only  Chapter 10. The Fixed Point  Solution  222  depends on the coefficients b , b , b , b from the first order diagrams (see Appendix E 3  5  e  7  for the derivation of these coefficients). Recall the solution E{£) in (10.1.21a)-(10.1.21e) for defining the map the contribution K  Let  K,(£,E ). R  denotes the part of  E (£) KB  E{£)  corresponding to  of B(£).  B  Proposition 10.1 For the terms Ki defined in (10.2.87), by choosing S and ki as in £ 6 B,  (10.2.91), for£,  5  l  ^  ^  l  l  w  ^  Q  +  C  2 i  e  )||5-£||,  where Ci is a positive constant independent of e . 2  Proof: From (10.2.88) and (10.2.91), and applying Lemma 10.1, we have VE {£,£) K%  w  <  8e | 2  -& /3|[l + 6 4 C e ] ^ | | £ - £ | | 2  C 3  6  5  C3  < (I + C e )||£-£||, 2  3  where C = 512 |c — b /3\ C$/c . 3  3  e  3  The other cases can be proven by a similar manner.  Q.E.D. Let us consider the first other terms from  Bi^st[£)  defined in (10.2.79).  We first  consider the factors of the terms of B-(£) defined in (10.2.3). • From (9.4.14), the e<'s are • The X  b  0(e ). 2  and Y defined in (10.2.86) has n = 2 b  p  We also observe that the i 7 | ( £ ) of B (£) 2  factors of B {£) s  iG  p?+h is a B i  s +1  has n = 1 and n 5  K  = 1. Next we consider the  defined in (10.2.7-10.2.8). These terms are of the form G l3°+\ s  itj  defined in (10.2.6) or -5 {£1  +2  M  after (10.2.1)); and the G 's itj  + e X {£)) s+2  5  are defined in (9.2.1-9.2.7).  where  if % = 7 (see the discussion  Chapter  10. The Fixed Point  Solution  223  From (9.2.6-9.2.7), it is easy to see that for iy£ 7  0(e ) 2  G1,3  (10.2.92)  Q(e ) fori = 7 4  • For 2 = 1,2, 3, 5, each P?~[Ji is Q(e )  or has n = 1.  2  7  Using the above observation, we can check that each term from E>i^st{£)  (after splitting  the X and Y according to (10.2.86)) that is not a A ; defined in (10.2.87) satisfies (10.2.84). From Lemma 10.11, we give a list of the corresponding 0( ")  °f t  e  n e  5T>K$ for the first  order terms that are not Ki in Table 10.1. Table 10.1: Bounds on the Lipschitz constants  m  : B  K{£)  T  • B  2  • Q(e ) 4  K(£)  :  G\jP$  B  t  ITG  e Xl  :B  3  0(e )  4  X : s  b  B  7  0(e ) 6  <  1  4  0(e )  4  : BUG i ± 7 GhPjfi 0(e )  E$b~M XI : B$ £7  3  :Br  G  0(e ) 6  Next we consider the bounds for the terms from B t(0). Observe that by setting i)P  £ = 0, terms involving explicit ft factors are zero. Thus it is easy to see that -B;(0) = 0. As for the terms GjjPj^}t(0), we consider the following estimates: • a Pj\}t{0) contributes a factor of Q)(e ); 2  • for j = 1, 2, a Pj*h(0) contributes at least a factor of Q(l/F); • for j = 7, the interpolation form of G^-(O) shows that it contributes a factor of Q ( l / ( r ) ) (see (9.2.5-9.2.7). 2  Chapter 10. The Fixed Point  Solution  224  From the above estimates, for B^p^O), we give a list of estimates for the norms ||i?i,p*(0)|| with (mi,ni) as defined in Lemma 10.1 in Table 10.2.  Table 10.2: Bounds on 13^(0)  Bi(0)  11^(0)112,0 = 0 ti^5,i-(0)lli,-i =  ||£ ,i-(0)||2,o = 0(e ) 4  ||53,i-(0)|ko =  2  0(e ) 4  l|£7,i-(0)||i, = O(e ) 6  0(e ) 4  0  From the estimates in Table 10.2 and (10.1.31b) of Lemma 10.1, it is easy to see that  I(0)|| where E s {£) itl  t  ( i  ) <  e + Cl  B  (10.2.93)  2  C o .t tl  denotes Ei{£) defined (10.1.21a-10.1.21e) with £ ; ( £ ) = B .t,  Cl are  itl  defined in (10.1.32b) and C °,\ is a positive constant independent of e . To find bounds 2  B  for the nontrivial C\ and C so that fZ(Bs) C B$, we need to set bounds for the boundary r  conditions El and E  7  of (10.1.18). Let us chose the following bounds for Cs, \El\, and  \E \: r  7  C$ > 8 CBO,!**,  \E \ < r  5  C  (l +  s  \nK )e  4  3  \E \ < r  8k K 5  3  C  '  M  r  Cs (l + lnK ) 3  e  4  8 k  7  (10.2.94) From (10.1.32b) and (10.2.94), we have Ce  2  ci<  s  8  Ce  2  C  r 7  <  s  8  (10.2.95)  We conclude this section by stating the map JC(£, E ) defined in (10.1.22) with using r  only the first order contributions, i.e., setting the Bi(£) in (10.1.21a-10.1.21e) to  B i*t(£). it  mij  Chapter  10. The Fixed Point  Solution  225  Proposition 10.2 Suppose fC(£,E )  is the map defined by (10.1.21a-10.1.21e)  r  E>i(S) =  For sufficiently  Bi^st(£).  small e , by choosing Cs, E\, 2  and E  7  with  satisfying  (10.2.94), for £,£ eBs with 8 = C e , then 2  s  1. \\VIC(£,£)\\  <  \\)C(0,E )\\ at  (10.2.96) (10.2.97)  <  r  where Cx>B,i  \\£-£  is a constant depending only on Cs;  2. K{B ) C B . S  6  Proof: Let £,£ e B . From (10.1.31a) of Lemma 10.1, Table 10.1, and Proposition 10.1, s  \\V }d(£,£)\\  l|5-5||  < (I + C W , i " ) e2  {i)  By choosing e satisfying 2  CT?B,I"  E  * <  (10.2.98)  \  (10.2.99)  we have (10.2.96). (10.2.97) follows from (10.2.93) and the choice of bounds for the £ [ ' s in (10.2.94). From (10.2.97), (10.2.98) and the bound for e in (10.2.99), 2  \\K.(£,W)\\  <  ||/C(5,^)-/C(0,^)|| + imo,^)||  <  (\  <  + CVB,I>* e ) C e + ±C 2  2  S  Ce. 2  5  Thus \\JC{£, E )\\ < 8 and it follows that K{B ) C B . r  S  Q.E.D.  s  e  2  S  Chapter 10.  10.3  The Fixed Point  Solution  226  Higher order terms  In this section, we extend result of Proposition 10.2 for the map fC(£, E ) r  (10.1.22) with the higher order terms Bf {£) A  and Bf . {£)  defined in  (defined in (10.2.5) and  G A  (10.2.9) of a fixed order included. We apply the same strategy as in the proof of Proposition 10.2 by showing that the corresponding Lipschitz bounds and bounds of the higher order terms are also of Q(e )  for i ^ 7 and are of 0( )  4  f ° i = 7.  e6  r  We now study the Lipschitz bound of a contribution j3f{G)(£)  to Bf {£) A  correspond-  ing to a graph G. Let us state the various notations used in Section 9.3 regarding the lines and vertices of a graph G . Let \S\ the size of a set S. VG,UI  =  the set of V\ vertices of G,  n = |VG  V ,m  =  the set of V vertices of G,  n =  CQ  =  the set of lines of G  CG,B  =  the set of photon lines of G,  C ,F  =  the set of Fermi lines of G,  G  G  £-G,B,m  =  £G,B,d  =  B  x  |VG  7  7  n n  F  B  > T I I  = =  J T 1 7  |  |  |£G,B|  \CG,F\  the set of selected photon lines attaching to the V vertices (see (9.3.33) 7  CG,B/CG,B,m  Note that \C \ = n  B  G  + n, F  \C ,B,n G  7  \ = n , and \C ,B,d 7  G  B  \= n  B  - n  7  > 0.  Recall from Section 9.3 (see (9.3.6-9.3.39)), a typical 0 contribution (3f(G) to the coupling flow has the form ft{G)  =  Y,Pi{G,s,h)  (10.3.1)  h>s  where (10.3.2) A (X,h,P) G  A i ( V , , A,h) A {V , , G  n i  7  G n7  A, h) V(C , G  A,h)  (10.3.3)  Chapter 10. The Fixed Point  A  Solution  227  A, h), A (VG,n , A, h), V(C ,X,h)  i(V(?,ni,  7  7  are defined in (9.3.33) and E^> denotes s  G  We would like to bound (3-(G) by a similar argument as done in the proof of Proposition 9.1. Here we do not assume conditions (9.1.1) since the running couplings A = (A A , A , A , A ) are replaced by (A + £) = (Ai(l + £{), A ( l + ft), T + <f, ft, ft) where 1;  2  3  5  7  2  3  £ £ Bs and T is defined in (9.4.16). Also, we do not estimate 1 + 7 (defined in (5.4.9c)) by 7 = A + A in the photon running slicing functions VB defined in (5.4.9a) and we 3  5  keep better account of the factor C of the bounding constant in (9.3.35). 0  Let the corresponding /^(GQ's defined in (10.3.1-10.3.2) with the replacement A —¥ (A + £ ) be denoted by Pi(G,s,h)(£) and P?(G)(£) with £ e B . 5  We set the radius S  of Bs as Cs e where Cs is a constant to be determined later. We would like to further 2  introduce the following notations for the account of the C corresponding to Pf{G). 0  T(£,h) =  II  ^  i'ec , ,n G  C  G  =  B  II ft  7  r \  (10.3.4a)  iec , ,d G  B  B  Y,M{G,h)  c  G  (10.3.4b)  h>s  =  2  n  B +  C  n ^ F  K  G  = ^B+n 2  / Cx \ =. 2 '  C  G  C [j-j  ,  G  C°  K  where c , M(G,h), G  ^  (10.3.4 )  = 2 C,  (10.3.4d)  S_y  7  C  G  (10.3.4e)  K  n  C  nB  PG  if nx is odd  ^  ^  n r  n +2nB+nF  n  F  — if ni is even 2 + 2n ,  (10.3.4f)  7  and f are defined in (9.3.9), (9.3.12), and (10.Irrespectively.  From the form of P?(G)(£) in (10.3.2), let us write  PtiG){£) =  E4 , (£) G ni  h>s  Kl {£) G  (10.3.5)  Chapter 10. The Fixed Point  Solution  228  where (10.3.6)  Kl (S) G  A ((A G  where A (V , 7  GtTl7  £)XP)  +  A ((A  j  dP  =  A?A (V , , 7  +£),h,P)  G  =  K (h,P).  (10.3.7)  G  (A + £), h) V(C , (A + £), h)  G n7  G  X, h), V(C , X, h) are defined in (9.3.33). The reason for writing G  (3-(G)(£)  in the above form is that, later in finding Lipschitz bound for fi-(G)(£), we would like to apply an argument for finding Lipschitz bound for A (V , 7  of K^ (£) G  (A + £), h) V(£ ,  GtTlT  G  (A + £ ) , h)  which is similar to the proof of Lemma 10.11 . Let  KI { ) = T.KIG{S). £  (10.3.8)  G  h>s  Recall the p functions Bi(£) for the map }C(£,E ),  for i=l,2, the P functions are scaled  r  by Zi where Zi = A^  1  for i=l,2 and Zi = 1 for i = 3,5, 7.  Proposition 10.3 For £ e Bs with 5 = Cse ,  4^(£)i \z K? (£)\ t  G  h G K  n  <  2>  <  C  (10.3.9a)  ni  A  g  e ^ °  ~  G  M  {  ° J  H  )  (10.3.9b)  |r(5,/i)| (P) no  e  2n  (10.3.9c)  K  (10.3.9d) where various terms of the RHS are defined in (10.3.4a-10.3.4f).  Chapter 10. The Fixed Point  Solution  229  Proof: (10.3.9a) is obvious from fact that we set 5 < 1 (see (10.2.15)). (10.3.9c) can be obtained from (10.3.9b) by the following. From (10.1.6) and (10.1.8), T /T h  h  < 2. From the definition of V in (9.4.16), for h > s > - 1 , s  l + lnT'  l + lnT  1  p/i  ps  —  '  Hence, using the fact that \S \ < (Cs/k ) 7  p/i  4  — ps '  h  B  TT —  /  1  e (1 + \nT ) and n  7  < 1  1  s  TT  TT  TT  1  I 7 I  > n, 7  TT TT  JftT T T  <  "  eec r <  ( ) TS  <  {  e  h  nB  GB  ^  r  B  From (10.3.9b), the definition of C , C G  (  1  0  3  1  0  )  in (10.3.4b-10.3.4c), and the bound (10.3.10),  Ka  we have  \*iKla( )\ £  ^  Y.\^ IG{£)\ K  h>s  <  £ C  <  C  A  G  ^ G M {  G  : K )  +  AG  ^  -  p-j^  C  g  E  M  {  Q  t K  )  h>s  '  •  (10.3.11)  We now show (10.3.9b). As in the proof of Proposition 9.1, from (10.3.7) and Theorem 9.1, we have \Klo(£)\  <  \A (£,h)\  <  \A (£,h)\c  J  G  G  G  dP\K (h,P)\ G  M(G,h),  (10.3.12)  Chapter 10. The Fixed Point  Solution  230  where  A (s,h) = A G  n  eec \„ G  A"  n  '  7  A  eeCo, ^  A (l + ^ )  B  1 + W*< J  2  1  n  r(5,/i)  n  (10.3.13)  For £ e B , from the bounds (10.2.31) and the definition of C A in (10.3.4c), s  g  n <  Ar  = c,A  G  *  IT _  J  _  ni  (10.3.14)  Ar '  From the identity ri\ = \£G,F\ + \AG,F\/2, we have n  F  = ny for i = 3,5,7, and n  F  = rii — 1  for i = 1, 2. Hence from the definition of n y in (10.3.4f), it is easy to see that (10.3.15)  Ar From (10.3.12-10.3.15), we have  \z K? (£)\ t  JG  <  c  C  AG  M(G, h) e >< n  G  \r(£,h)\  (10.3.9d) can be obtained by setting £ = 0 in Pi(G, s,h)(£) argument as the above. Q.E.D.  (10.3.16)  and applying the a similar  Chapter 10. The Fixed Point  Solution  231  We now turn to finding Lipschitz bounds for the Zi (3-(G)(8). || • | | m i , - m  Let us consider the  norms of V (3f(G) where the range of values of ( m , m ) are described in 1  2  2  (10.2.66).  Proposition 10.4 Suppose  8,8  G B with 5 = Cse , and (3-(G)(8) as defined in 2  5  (10.3.5).  1.  \ i D 0i(G)(8, 8)||l+mi,-m  11^  < VP  z  e  2  (10.3.17a)  f  —  G  where £vp  G  and N  Q  — C,  2  is defined in (10.3.29)  (10.3.17b)  C  s  and C$ ,  n  G  are defined in (10.3.4d),  v  (10.3.41)  respectively, 2. M|(G)(0)||  1 + m i  ,_  m 2  (10.3.18a)  <e%  G  where P°c =  £  C  k  e  2  n  K  (10.3.18b)  >  UK are defined in (10.3.4e), and C°^ is the constant independent of Cs defined in G  (10.3.4d). Proof:  From (10.3.5),  V p (G)(8,8) s  t  =£  [V 4 (8,8) GNI  Kl (8) G  + 4GJ8)  V K* (£,£)] . G  (10.3.19)  Chapter 10. The Fixed Point  Solution  232  Hence, from (10.3.9a)  1+7711,-7712  <  VEl (£,£)Kl (S)\\ .  II* E  >ni  G  (10.3.20)  l+mu m2  h>s  +2" \\ V 1  Zi  J2 Kf (£,£)\\ _ . >G  1+mi>  (10.3.21)  m2  h>s  Let us find a Lipschitz bound for each of the two terms in (10.3.20-10.3.21). For first term (10.3.20), we first consider the factor £ y  G n  {£)•  It is easy to see that for rii ^ 0 and £,£ e B , since for h > s, [X — — t £i \ v  6  v  < 2 and  l/r " < i/r , h  s  VEi (£,£)\  <  Gni  n  i  2  n  i  "  1 r  f  g  |  1  .  (10.3.22)  Hence,  -7712 /l>S  <  m 2" " | | ^ A ^ ( £ ) | | 1  1  G  m i  ,_  m 2  ||£-£||.  (10.3.23)  From the counting identity (9.3.34), it is easy to check that for a higher order G used in the coupling flow,  d = d -n B  where  | A  G  , B |  B  7  = n + 3n - \A , \ > 2 l  7  G  (10.3.24)  B  is the number of photon legs of G. Hence, from range of values of (mi, m ) 2  given in (10.2.66), the bound in (10.3.9c) of Proposition 10.3, and (10.3.24)  i  z  Ki,G(£)\\mi,-m  2  C K  <  e  2nv  Kg  <  d B - m i ^  C  K G  e  2 n v  .  +  l  n  K  ^ m  2  (10.3.25)  Chapter 10. The Fixed Point  Solution  233  Collecting the bounds in (10.3.23) and (10.3.25), since Cp  G  = 2 C ,, ni  K(  the first term  (10.3.20) has Lipschitz bound m  e "v  C  2  PG  2  \\£-£  (10.3.26)  As for the second term (10.3.21), we apply the same argument as in the proof of Lemma 10.11 for finding a Lipschitz bound for K(£) of (10.2.35) to the term  = 2^ A C A (V ,„ , (A + £), h) V(C , (A + £), h).  Al (£)  n  G  A\ (£)  Zi  7  G  G  7  has a form similar to the K{£) of (10.2.35) with P{£) = 1, T(£,P)  G  = 1, and  (n , rip, N , N ) = (0, 0, n , n ). 5  B  F  B  F  From the proof of the bound (10.3.9b), it is easy to check that we can pick the corresponding CQ of Lemma 10.11 as  \VAl {£,£)\\ ^ G  l+m  <  m2  (n + SN ) \\£ - £\\ (10.3.28) 7  Q  where  N = 8(n + k n ) l  Q  F  3  (10.3.29)  B  and k is defined in (10.2.91). From (10.3.28), z  (r ) s  1 + m i  (i + inr ) s  m 2  2"  \ziY,VK£ (£,£)\  1  G  h>s  < h>s  <  £  /  dP\\VAl (£,£)\\ . G  1+mu m2  \K (h,P)\ G  h>s  <  (10.3.30)  Chapter 10. The Fixed Point  Solution  234  Since 8 = C^e , from the definitions of Cp , n  in (10.3.4d), (10.3.4f) respectively,  2  G  v  (10.3.30) can be simplified to e ^-D n _ 7 + e NQ 2  C  PG  \\£-£\\.  2  Cs  (10.3.31)  From the bounds (10.3.26) and (10.3.31) , we have (10.3.17a). (10.3.18a) follows from (10.3.9d) of Proposition 10.3. Q.E.D. Since for Zi(3-(G)(6)  of the higher order f3 functions  B (£), iiA  > 2 for i ^ 7  n  (10.3.32)  v  > 3  i=7  from (10.3.17a) of Proposition 10.4, the corresponding Lipschitz bounds for are  C)( ) E4  for z 7^ 7 and 0 ( ) for i = 7.  Zi(3-(G)(£)  Similarly, since setting 5 = 0 implies a  e 6  nonzero ft-(G)(0) has its corresponding n = n , it follows from (10.3.32) and (10.3.18a) K  v  of Proposition 10.4 that (3?(G)(0) are 0(e ) for i ^ 7 and 0 ( e ) for i = 7. From the 4  6  Zi  bounds on the higher order terms of Bf (£),  it follows that the terms from  A  have similar bounds (see (10.2.7-10.2.9) for the description of  Bf . (£) G A  B- . (£)). G A  We summarize our results on the Lipschitz bounds on the higher order terms by the following corollaries. Let K (£) Bi  be a term of B (£)  or B . (£)  s  s  iA  iG A  with £ £ Bs-  Corollary 10.1 For £,£ G Bs with 8 = Cse , and e sufficiently small, 2  2  \\VK (£,£)\\ ^ Bi  m  \\£-£\\  2ai  ||A^(0)|U, where  < Ce  ni  K  < C° e <, 2a  n i  K  (10.3.33) (10.3.34)  = 2 for i ^ 7, 07 = 4, CK, C° are positive constant independent of e , and C^  is also independent of Cs-  2  K  Chapter 10. The Fixed Point  Solution  235  For a given order N flow defined in (9.3.39), Let E ,N{8) A  denotes the higher order  contributions to E{8) defined in (10.1.21a-10.1.21e) corresponding to terms of  B {8) i<A  that are of order up to N. From Lemma 10.1 and Corollary 10.1, we get  and B . {8) s  iG A  EA,N{8).  the following bounds for  Corollary 10.2 For 8,8 e Bs with 8 = Cse  2  VE (8,8)\\  <  A>N  ll^(0)|| where CVA,N  and CA°,N  and e sufficiently  small,  2  e \\8 - 8\\,  (10.3.35)  2  C A,N V  C o, e  <  (10.3.36)  2  A  N  a-re positive constants depending on N with CT>A,N further de-  pending on CsFrom the results of Corollary 10.2, we wrap up the fixed point argument for showing existence of a solution to the subsystem (Ai, A , A , A , A ) satisfying the boundary con2  3  5  7  dition (Af = A  X  u  1 }  = A , A = K >\,  XI = El, A = W )  r  2  2  for a given order N flow, where E  3  and E  r b  7  Z  7  (10.3.37)  7  are the boundary values of £ f ( £ ) and E {8)  of  7  the map JC(8, E ) defined in (10.1.22) and the ranges of E and E T  T  5  7  From the bounds of ||£P*(0)|| in (10.2.93) and the bounds of | | £  are to be chosen below. A i i V  ( 0 ) | | in (10.3.36), let  us choose C >8 s  (C o,i + C o, ) • B  A  (10.3.38)  N  where Cs is the constant for defining the the radius 8 of the ball Bs, i.e 8 = Cs e . Let 2  the boundary values of El and E  r  C  5  ^ where k  5  (10.3.38).  5  -  l  and k  7  7  (l + l n i v ) e  satisfy the following bounds: C , ( l + lnif ) e  4  3  8k K C 5  3  M  4  3  '  8^  •  -  (  m  3  -  3  9  )  are defined in (10.2.91), CM is defined in (10.1.27), and Cs satisfies  Chapter  10. The Fixed Point  Solution  236  Theorem 10.1 Let K be the map defined in (10.1.22) for a coupling flow of order N from the LRC of IQED.  Suppose the ball Bs defined in (10.1.17) has radius 5 = Cs e  where e = ( A / A ) . Then for For e sufficiently 2  2  2  small, K  2  > 1, E\ and E  T  3  7  2  satisfying  (10.3.39), K is a strict contraction satisfying IC(Bs) C Bs-  Proof: We basically repeat the argument in the proof of Proposition 10.2. Let £,£ G BsFrom (10.3.35) and (10.2.98), \\VJC (£,£)\\ s+1  {i)  <  + (CVB,I«  +C  )  VA>N  e) 2  \\£ - £ \ \ .  (10.3.40)  By choosing e satisfying 2  (CDB,I«* + C )  e < ^,  (10.3.41)  \\VK(e,i)\\ < ^\\£-£\\  (10.3.42)  2  VAtN  then  and K is a contraction. From (10.1.31b) of Lemma 10.1, (10.3.39), (10.2.93), (10.3.36), and (10.3.38) \\JC (0,E )\\ <^-.  (10-3.43)  r  t  {l)  From (10.3.42) and (10.3.43), \\K,(£,E )\\ r  Hence K{B ) C B . S  Q.E.D.  s  < \\JC(£,E ) r  - K,(0,E )\\ r  + ||/C(0, £ ) | | < C e . r  2  6  Chapter 10. The Fixed Point  10.4 10.4.1  Solution  237  The solution for the running couplings The solution for A| and Af  To make the solution of the full coupling flow complete, we determine X and Af for S  A  given boundary conditions Af and A chosen at the root scale since A^ and Af diverges 4  as s —>• oo. From the the fixed point solution A* of Theorem 10.1, X\ and Af satisfy the following difference equations.  A A : = PI e b 2  /?|(A ,A*) 4  AAf  =  A^(A*) + ^ (A ,A*),  Bi(A.)  + 1  4  4  = 6  (10.4.1)  M  (3^  =  •  '1 + E{ 1 + ElJ  -b~ be,e  2  M  (10.4.2)  where A* is the primitive solution of A*, and ^ ( A , A*), A\(A*) are the higher order terms. 4  From the discussion in Section 9.1, A\ is linear in A . Furthermore, since A ( 1 , A » ) and 4  AQ(\*)  are  4  0 ( C ) where ( = ( A i / A ) / 7 , from the solution A*, it is easy to see that 2  2  2  A*(1,A*) and Af (A*) are of  0(l/(T ) ). s  2  Let us first determine Af first. Since the j3 function (10.4.2) is independent of Af and is a function only of A*, for a given Af, the solution of Af can be readily written down as (10.4.3)  A5 = AS + 5 ] # ( A , ) . t=o  Since the leading term of /?|(A*) is 5 bee  2  'l + E(  M  and the remaining terms are of Q(l/(T ) ), S  2  - 1  s  M  e  4  it is easy to see that 2  IAS — A5| <  t=0  o (vn  < CRr e  ln(P) ln(r°)  (10.4.4)  Chapter  10. The Fixed Point  Solution  238  where Cp is a positive constant depending on N. e  We now consider the A 4 . From Section 9.2, the primitive solution A 4 is given by (9.2.17)and satisfies the equation A A 4 = A°+ Bl (A,). l  (10.4.5)  +1  From (9.2.18), A\ has leading behavior  tf (-J  ,  4  where K4 is a positive constant, a = 4  (10.4.6)  ~ 9m/2 and m is the "electron mass" of the  64/63  model. We would like to show that (10.4.6) remains the dominant behavior of A 4 when higher order terms are included. Our strategy is to show that r| = ^  = 1 + (1).  (10.4.7)  0  Let us consider the difference equation of r . From (10.4.1), 4  Art  = K 'A(A^)+  + (A^^AAl  1  X\-4 ^1v + ^  _ _|_ ( . A ^ -  1V4  4  1  + M - ^AS+lt r ^ A )  (10.4.8)  From (10.4.5), the first two terms of (10.4.8) add to zero. Setting r = 1 + E4, we have 4  the following difference equation for E±.  AEl  = (A^)- ^ (A 1  +1  4  (1 + E ) , A , ) 4  (10.4.9)  Since we expect El = o(l), for convenience, let us set E% = 0. From the forms of the (3 function described in Section 9.3.1, A\ (\*) +1  £ h>s+l  is sum of terms where each is of the form  JdPA (\ ,\*,h,p)K (h,P) G  4  G  (10.4.10)  Chapter 10. The Fixed Point  Solution  239  where if \X\\ are only of order in powers of s then the exponential decay from K (h, G  P)  allows the following bound. A (X ,X*,h,p)K (h,P) <C \A {X ,X*,h  T,jdP  G  4  G  G  G  ,1)|.  4  (10.4.11)  h>s  where h  denotes setting all components of h to s. Hence by assuming that E\ = o(l),  from the the leading behavior of A in (10.4.6), 4  |(A4) ^4 (A _ 1  + 1  4  ,A*)| <  C p  *  S M e  *  (10.4.12)  where C@ is constant depended on the order N. From this fact, we see that ii El = o(l), 4  then the RHS of (10.4.9) is of e  4  O ((H )-2  Hence we expect E = e s  4  2  O ((r ) )s  -1  We can establish that El = o(l) by a fixed point argument similar to the one in Sections 10.1-10.3. Let Bs be the Banach space  Bs = {£±  where || • | |  1 ) 0  |  | | £  4  | | i , o < Q e  2  (10.4.13)  } ,  is the norm defined in (10.1.14) and  1  l  s  1  4  4  t=s+l  1,0  We consider the map E : Bs —> Bs defined by E 4  EKSt)  = 0 and  4  =  (10.4.14)  J2 ( A - ) - 4 ( A , A , )  -  C >2  ( A - ) - A ( A ( 1 + <? ),X).  E  1  1  4  4  4  4  (10.4.15)  i=s+l  Since A\(A  4  (1 + £ ), A*) is linear in £ , from (10.4.12), it is easy to see that 4  4  \VE (£ ,£ )\ S  4  4  4  <  E ( A n - ^ A)  \£± — £4111,1  1  <  t=s+l e C V E i  £4 —  (10.4.16)  Chapter 10. The Fixed Point  where CVE  4  Solution  240  is a positive constant depending on N. From above, one can easily establish  the Lipschitz bound \\VE (£ ,£ )\\ s  4  4  4  < - H^-^Iko-  (10.4.17)  l  lfi  From the (10.4.14), | | £ ( 0 ) | k o < C e /2. Thus from (10.4.17), it is easy to see that the 2  4  map E±(£±)  8  is a well defined contraction with a fixed point in Bg. This establishes our  desired result (10.4.7). Note from the fixed point solution X\, A 4 = K  (1 + e F(K )) 2  4  as e —> 0. Since F(K ) 2  A  is continuous in K4, for K  4  A  with F(K ) 4  = 0(1)  e S4 = (1/2, K^) with KI a fixed  constant, for e sufficiently small, a subset of S4 can be chosen so that XKK4) satisfies 2  the condition 1 < A 4 < k\ where k\ is a constant.  Chapter 10. The Fixed Point  10.4.2  Solution  241  The solution for the full system  Let us write down the solution of the coupling flow for the full system. Let Q  S N  be the set  of graphs defined in (7.4.9) restricted to the order N graphs defined by (9.3.38-9.3.39) (see Section 9.3.2). Consider the order N coupling flow equation obtained from the multi-slice L R C for IQED: ^  =  £  /?••"«?)  (10  . .18) 4  ^G V\N  G  +  ^  G  ^  =  £  PT\G)  K  =  E  Pt (G)  A  A  N  ^ N  G  A A*  =  e  =  AX*  =  A A  +1  -S  X  s +1  M  E  5  + M~  P (G)  £  2s  S +1 5  P ? \ G )  PT (G)+1  ^  G  N  with boundary conditions: Af A  r 3  XI where A , K , K , K 4  5  T  e  s  =  =  Ax,  >  1,  =  K  5  Ao/ = A ,  (10.4.19)  2  K >X >l, r  4  A  Af = A ,  X = K, r  6  7  7  are finite constants. Recall that  7  £ (s + 1) + 3  A3,  e = c 5 3  3  e, 2  M  e = 6 5 4  4  e. 2  M  Chapter 10. The Fixed Point  Solution  242  Theorem 10.2 For e = (Ai/A ) 2  sufficiently small, there exits a positive  2  2  constant  Cg depending on N such that for  |*.| <  |*| <  +  8k K C 5  where k , 5  k,  and C  7  3  o.4.20)  (1  8k  M  7  are respectively defined in (10.2.91) and (10.1.27),  M  then there  is a unique solution X satisfying (10.4-18), (10.4-19), and (10-4-20) with the s  following  asymptotic behavior. For r < s < U, X{  =  M0. + E1),  \ E  S  \ < ^ ,  A* = A ( I + E | ) ,  m <  XI  \E \<C e  2  =  A (l + E ), s  s  3  C  s  s  3  3  e  (10.4.21)  2  (1 + l n P )  2  s  k T' ' 3  Ce  2  XI  =  Al{l  + Efi,  Ce  \E \< S  s  4  (1 + lnP )  A  5  d  k T° 5  \K\  <  \Xl\ + C e  ln(P)  2  s  Cse  (1 + l n P )  4  k  7  where E\ = 0, A  3 = r  A4 = ^ n ( i - ^ )  s  t=o and K  v  1  _ 1  7  is a constant depending on X . r  4  Note that A f  4  + 1  = A f for i ^ 3 and A f  + 1  = b ln A + c + A f where 6 and c are constants  given respectively as the coefficient of the InA term and the O ( l ) term in (D.2.24).  Chapter  11  W a r d Identities  11.1  Statement of W a r d Identities  In this section we show that the L R C expansion of IQED respects an identity which underlies the Ward Identities. Furthermore, we show that by not using gauge variant counterterms in the model, the resulting root potential satisfies the Ward Identities. Let us introduce the following terms and notations for the L R C set up of the running potentials for IQED.  1. (11.1.1)  S  where M is the 4 x 4 diagonal matrix with diagonal entries M - defined in (5.3.2)3  (5.3.3). 2.  * =  (* ,*1,*2,*3), 0  $ = (A*,*)-  3.  3=0  243  Chapter 11. Ward Identities  244  4. = Af+Vx + ( A ^ - 1)V + ( A ™ - 1)V + ( A f 1  + 1  3  2  - l)V . 4  5. V£  = M  + 1  2 C /  A  U + 1 5  V + A? V + Af V . + 1  + 1  5  6  7  6. VJJ+1 — gi  + * V  v  7. V  U + 1 r  log-A-  (* ) = r  /  % i]($)e^ +  ( $ + $ , )  .  where  (<S>)e u Z^  = J dP  U  v  (11.1.2)  [0>u+1]  rfF[o [r+i]($) is the Gaussian measure w.r.t. the quadratic terms denned in (5.3.6), )  and fields superscripted by r denote external sources. Recall that the L R C set up does not require a momenta cutoff for the bare spinor covariance. That is, in momentum space, the decomposition of the spinor covariance p[o,u+i] s = S , has the following slicing u+i o,u+i] = £ i  p[  p  (  1  L  1  3 )  i=0  where for 0 < i < U, p = p , p l  l  u+l  = pl >°°\  and the p are the neighbourly slicing  u+1  l  functions defined in (B.2a). It is easy to see that S satisfies the identity e-^S"  1  e  iex  = S-  1  +  epX  (11.1.4)  Chapter 11. Ward Identities  as in (2.4.5). Under the change of variables  A'  =  A,  A ' = A, T  r  it is easy to check using (11.1.4) that we have the following. 1. For i ^ 2, V ($' + $ ') = Vi($ + $ ) r  r  (11.1  r  <  and V ($' + 2  _(^r  =  g-ie (_-^ye*  +  X  ^  +  =  V ($ + $ ) + e Vxidx, tf + tf ,tf +  =  V ($ + $ ) + e V {dx,  r  #)  r  r  2  r  x  2  + * ', r  +  * ')r  2. V j + ($ + $ ) /  r  1  1  r  y ^ ( $ ' + $ ') + (A r  =  e  a+1 2  -l)Vl(^x,*' + * \ * ' + * ')r  r  (11.1 3. dP ($) u+1  = cdP (&)e (W>*'l eVl  u+1  where c is a constant. Let us consider the root potential V  u+1 r  under the change of variables  (11.1  Chapter 11.  246  Ward Identities  From (11.1.6) and (11.1.7), V (V)  = log-^—  u+l  r  [ d P u ^ e M V ^  x exp (e [Vi(0x,  +  1  (11.1.8)  *') + ( A ^ - l)Vi(0x,  + * ',  1  + ^ ')]) •  r  r  The exponent of the second exponential term in (11.1.8) can be simplified into  eAf+Vx^x, =  + * ',  + * ') - e [Vi(<9x,  r  eAf V (ax,*' + * +  r  1  + * ',  r  + * ') - Vi(<9x,  r  *')]  r  \*' + ^ VeV'i(^X>^ >*'^ '',* ') r  ,  T  r  (11.1.9)  where  = vi (d , * \ $') + vi(a , r  x  x  * ') + vi (a , * , f ' ) r  r#  x  From (11.1.9), since V ( $ + dx) = V ($) where $ + 9% = ( f , 3  Vu£(®  , A + dx),  3  + $ ') + e A f + ^ a x ,tf'+tf ',tf'+ tf ') r  r  r  = v^+H^' + ^ ' + ax) +^  c/+1  (9x,  (n.i.io)  $ ') + ( e A ^ - A f ) VI(9x, r  1  + ^ ' ,tf'+ tf '),  +1  r  r  where  w^ (<9x,  $)=v  +1  r  ($ + $ ) - v  u +l g v  r  ($ + $ + ax).  u +1 g v  (n.i.ii)  r  Thus the RHS of (11.1.8) becomes log ^  / dP (&)  x exp (W  u+l  (dx,  u+l  exp ( V f f i  1  ^ ' + <T' + 9x) - cVi(9 ,  $ ') + ( e A ^ - A f ) r  * ')) r  X  1  + 1  Vi(<9x,tf'+tf ',tf'+tf "')). r  7  (11.1.12)  Chapter 11. Ward  Identities  247  (11.1.12) would give us the desired Ward Identities if the last two terms of (11.1.10) are zero. Making the latter term zero can be achieved by having the "Zi = Z " condition 2  X  u+1  e =  TZ7TT-  (H.1.13)  The vanishing of the former term requires having no gauge variant counterterms in Vy+{. Let us state our observation in the following proposition.  Proposition 11.1 By setting e = A f / A + 1  terterms  Vff+l  in  and not allowing gauge variant  7 + 1 2  in the LRC set up, then the root potential V  u+l r  coun-  satisfies the Ward  Identities V ($ ) u+1  = l o g - i - /dP (<S>) exp ( V ^ C * + <F') - Vi(dx, W, %  r  r  u+1  , (11.1.14)  where W' = (A ',$> ',q ') r  r  = {A + d , e W,  r  r  iex  X  e~ W). iex  (11.1.15)  Thus in order to obtain the Ward Identities (11.1.14), we seem to need to run the L R C without any gauge variant counterterms. From the analysis of the coupling flow of L R C set up in Chapter 9, the fixed point argument requires that boundary conditions for X  gv  — (A5, A , A ) to be selected at the root scale r. Moreover, the fixed point argument 6  7  does not indicate by choosing A ^  + 1  = (0,0,0), the corresponding X  would have a  finite limit for its components as U —> 0 0 . However, in the next section, by the Ward Identities, we show formally that the gauge variant local terms at the root scale sum to zero if A ^  + 1  = (0,0,0).  Chapter 11. Ward Identities  11.2  248  Root scale gauge variant local terms  By using a similar argument as in Lemma 4.2 - Lemma 4.4 in [FHRW88], we demonstrate that for an L R C expansion with A f / = ( A f , A f , A f ) set to (0,0,0), the resulting 1  +1  + 1  + 1  perturbative sums of the gauge variant local terms at root scale are zero. Let us borrow some notations used in Section 4 of [FHRW88]. 1.  =  V«AA\%,%)  Vr\.__ ^ . i:  where % are the components of tf =  (11.2.1)  >a  $ , $> ). r  e  r  2  3  2. LVo,  (11.2.2)  EA'VJ  =  r  \xpl=ipV-0,i>0  where each \ \ denotes a formal sum of contributions from all graphs at root scale  r and VI = V ($ ). r  j  3. H(x) is the dimension zero local part of ipe e(Sfix iex  + $xS) ~ ^ e  (  iex  s e e  Section 4  of [FHRW88]) We state a couple of lemmas which are analogous to Lemma 4.2 and Lemma 4.3 of [FHRW88] and which are proved in basically the same way.  Lemma 11.1 (analogue of Lemma 4-2 of [FHRW88]) Suppose VVf^ has gauge in1  variant form such that A f smooth polynomially V (A ,V ,$> ) u+l  r  r  r  r  + 1 ;  Af  + 1  , Af  + 1  are all zero. Let e = A f  /Af  + 1  .  Then for any  bounded function x{ )> x  =  dx,(l + SX)e W,q e- {l  V (A + u+1  r  iex  r  + j V e- (X r  where X = e$x-  + 1  iex  +  XSX)e m  r  iex  r  iex  + XS)) (11.2.3)  Chapter 11. Ward Identities  249  Lemma 11.2 (analogue of Lemma 4.3 of [FHRW88]) variant form.  Then for any smooth polynomially  LVoAA ^ ^ ) r  r  r  = LV {A  bounded function +e j ^  + dx,e ^ ^ e- )  r  tex  0tr  r  r  Suppose  iex  X  has gauge in-  x{x),  * + ^#(x)- (H-2.4)  Because there is a shifting at each scale of the flow of the effective potentials, the external legs at scale s are transformed by B  = l-H K  s  where K  s  and H  s  external field $  S  (11.2.5)  S  are defined in (5.2.2b) and (5.2.3a). Thus at the end of the flow, the becomes  r  u l  = I]B * s=Q  r  s  (11.2.6)  r  Lemma 11.3 LMiV)  = Vi($ ).  (11.2.7)  r  Proof: From the formulae of H  s  and K , it is easy to see that we can write s  u l[B  s  = l + B,  (11.2.8)  s=0  where B is a sum of terms in which each term contains the p 's as factors. Let K ( $ ) be s  r  of the form j dp&(p)  K(p)  &(p).  Then from (11.2.8), LiVi{$ ) r  = Vi{V) + Li J dp <F(p) [BK + KB + BKB] (p) $ (p). r  (11.2.9)  Chapter 11. Ward Identities  250  Since the localization of a graph can be defined as taking Taylor terms of the kernel of a graph localized at momenta p = 0, and since the term B and all of its derivatives are zero at p = 0, the second term of the RHS of (11.2.9) is zero. Q.E.D. Let us write out the terms involving Vi, 5 < i < 7 from the RHS of (11.2.4):  J(A  r  +d )  2  X  + K j[d • (A + d )} + X 71[A + dxf r  r  2  r  X  (11.2.10)  Comparing the formal sum of both sides of (11.2.4), since x is arbitrary, it follows that Ag, Ag, and X are formally zero. Let us state our observation in the following Theorem. r  7  Theorem 11.1 Suppose the top scale couplings of a LRC expansion for IQED has the form Af  =  A  X\>  =  b\nA + K (U),  Af  = 0  + 1  +1  + 1  A^ = A ,  (11.2.11)  1  l 5  2  XV  +1  3  =  K (U), 4  z = 5,6,7, (11.2.12)  where bin A is the coefficient of the V local part of the VP diagram with lines as the scale 3  U+1  shifted spinor covariances and  6  "  C  (A »«)  2  2  C, Ai and A are constants independent of scales and the loop regularization 2  and K$(JJ),  K (U) are functions 4  of the scale U but independent of A. Then the resulting  root scale couplings X , Ag, X are formally zero. r  5  7  parameter A,  Chapter 11. Ward Identities  251  The above theorem suggests we must choose the boundary condition in Theorem 10.2 as Af  =  Ai,  AS >  1,  \  0,  r 5  =  Af = A , 2  K >X >l, r  4  XI = 0,  A  A = 0, 7  in order to find the asymptotic behavior of the running couplings of a L R C model for IQED that respects Ward Identities.  We remark that from Theorem 10.2, we cannot  conclude the Afj*" of the fixed point solution of the order N flow is zero since the theorem 1  only provides bounds on the components of A ^ . We can not apply the uniqueness of + 1  the solution to argue that A ^ order N perturbation.  + 1  is zero since we have not shown that X  r gv  is zero in an  Chapter 12  Conclusions and Outlook  12.1 12.1.1  Summary of the LRC scheme for IQED LRC scheme  We summarize our analysis of the L R C expansion for IQED. We have constructed a tree expansion scheme, the loop regularized running covariance (LRC) scheme, for expressing the connected Green's functions for the imaginary charge Q E D model in perturbation theory. The L R C scheme is based on the loop regularized Gallavotti-Nicolo (LGN) tree expansion introduced by [FHRW88]. The L R C scheme inherits the properties of L G N that the model naturally preserves the identity (11.1.4) that underlies the Ward Identities (11.1.14), and that the lines of the Feynman graphs are sliced up according to a momentum scale slicing for making a scale-by-scale renormalization. In addition to the inherited properties, L R C has the extra features that at each step of the flow of the running potential V , the local quadratic terms of V s  s  are shifted to the scale s slice  covariance via the shifting transformation (5.1.1), and the coefficients of the local terms Vi (7.1.2) with 1 < i < 7 are added up as the running couplings Af. By shifting the local quadratic terms at each scale s of the flow, the sliced lines of the Feynman graphs are replaced by running sliced lines where the slicing functions p of the slice covariances are s  replaced by the running slicing functions V  s  defined in (5.2.8a) of Section 5.2. A more  detailed description of the running spinor and running photon covariance is presented in  252  Chapter  12. Conclusions  and  Outlook  253  Section 5.4. Using the L R C scheme with neighbourly slicing defined in Appendix B, we studied the flow of X = {Af}J=1 in the Landau gauge for determining the asymptotic form of s  A as s —> oo. The resulting renormalized connected Green's functions of the expansion s  are well defined in perturbation theory and respect the Ward Identities in the limit of removing the regularization. The main analytical tool that we employed for analyzing X  s  is a fixed point argument in a suitable Banach space whose norm captures the asymptotic form of X . s  We chose the Landau gauge in our set up because the corresponding relevant V\ local part of the vertex diagram and the relevant V local part of the electron self-energy 2  diagram vanish (see the calculation of these local part in Section E.2 of Appendix E) and the kernel of the divergence term V is orthogonal to the corresponding photon free e  propagator. The vanishing of these local parts and the orthogonality seem to make the coupling flow easier to analyze than in other gauges. The reason for shifting the local quadratic terms to the slice covariance is that, without the shifting, we were not able to obtain a coupling flow that stabilizes in the sense that adding an arbitrary number of extra diagrams to the flow equation does not change the asymptotic form of the solution. We are aware that in [Hur89], Hurd was able to show that in the G N tree expansion for Q E D using only a momentum cutoff for the photon and Fermi covariances but no loop regularization, the Ward Identities can be recovered in the limit of removing the cutoffs. However, in our context, we were not able to execute his technique to show that all the graphs are properly renormalized.  Chapter 12. Conclusions and Outlook  12.1.2  254  Two-slice LRC  In the L R C scheme for IQED, there are two types of U V divergence in the limit of removing the regularization. The first type is the A-divergence coming from the top scale spinor loops containing only top scale spinor lines (see Section 6.2) as the loop regularization parameter A —> oo. The second type is the [/-divergence coming from loops that contain photon lines as the momentum cutoff parameter of the photon covariance U —> oo. B y using compactly supported external sources (in momentum space), and invoking vestigial gauge invariance (see Appendix D), only the top scale vacuum polarization (VP) diagram contains A-divergence (see Corollary 6.1 of Section 6.2 and the subsequent discussion). Consequently, renormalization in the L R C scheme naturally splits into two levels. The top level renormalization involving A only requires Af  -1-1  containing a term correspond-  ing to the coefficient of the V" local part of the top scale V P diagram, and lower level 3  renormalization only requires that Af be functions of U . For clarity of presenting the top scale renormalization, in Chapter 6, we developed a simple version of the L R C where there are only two slices in the decomposition of the free covariance. The slicing of the model is defined in terms of the neighbourly slicing {p }^L as follows. For the photon field, p s  0  PB  =  +1 B  P^°' ^ U  I S  slicing is p  t n  u +l F  e bottom photon slicing, where = pl '°°} u+1  = p[ ' ]  is the top photon slicing and  u+1 u>  As for the spinor fields, the top  U < U'.  and the bottom slicing is p  u F  = p'  0,17  '. In the two slice model,  from the choice of the slicing, only the parameter U' —> oo while the parameter U stays finite when removing the regularization. In this way, by choosing appropriate top scale couplings X  U + 1  and using suitable domain for the compact support of the external source,  the lower scale graphs do not require renormalization. For the two slice model, the top scale couplings are chosen to have the form Af  + 1  =  Ki  z^3,  (12.1.1)  Chapter 12. Conclusions  and  Outlook  =  A3  255  M n A + A-3,  where the AYs are constants independent of U' and A, and the term bin A serves to cancel out the A divergence of the top scale V P diagram. By using uniformly bounded external sources and restricting the support of the external source to be inside the ball B(r)  centered at the origin with radius r < M , 2U  the A —> 00 renormalized graphs  contains no top scale photon lines nor fictitious lines (see Corollary 6.2 of Section 6.3 ). Also note that by taking the remaining cutoff U' —» 00, U stays finite. Hence, in the limit of removing the regularization, the only possible source of divergence remaining is from the "external" quadratic terms K  s  (see (5.2.2b) for the definition of K ) which flow s  down the scale via the shifting transformation. From the form (5.2.2b) of K , s  it is easy  to see that the divergence of the external terms are controlled by allowing only external sources "with low momentum" restricted to the inside of the ball B{r).  As a result, by  Corollary 6.3, there is no need to have renormalization for the lower scale graphs.  12.1.3  T h e multi-slice A -+ 00 coupling flow  The two-slice model lacks the flexibility of allowing external sources with arbitrary high momentum. In Chapter 7, we extended the two slice scheme to a multi-slice scheme where the lower slice is further decomposed into U + l slices, and further renormalizations are required for the removal of the [/-divergence as U —> 00. In the multi-slice model, the AYs in (12.1.1) might have possible dependence on U. For determining the dependence of A f  + 1  on U, in Chapters 8-10, we studied the A -+ 00 coupling flow for the remaining  scales from 0 to U. In Section 7.2, we set up the A —» 00 flow equation expressed in the form A A ? = <y?(A)  Chapter 12. Conclusions and Outlook  256  where 5i are called the (5 functions which are computed as sums of the coefficients of the local parts of Feynman graphs obtained at the scale s forks (see (7.2.4)). The reason that it suffices to consider the A - » oo flow is that, as described in Section 7.3, the A —> co graphs are effectively like the graphs obtained from a running covariance (RC) scheme without loop regularization. In the same section, we made a couple of observations regarding the dependence of the (5 functions on the couplings. The first observation is that, by the oddness and power counting argument described in the section, 8\ is linear in A  4  and that the other Si's with i ^ 4 have no dependence on A . The second observation is 4  that, by choosing the Landau gauge, the kernel of the free photon propagator is proportional to Lfj, (defined in (1.2.8)). Since the kernel of V is proportional to the projection v  6  T^y (defined in (1.2.8)) which is orthogonal to L^ , contracting a V vertex in a graph v  6  gives zero. Hence, for having a cleaner form of the running photon slicing function, we chose not to shift V in the shifting procedure and left it in the running potentials. From 6  the orthogonality, A6 is basically an auxiliary variable that does not appear in the /3 functions. From these observations, we saw that it is more convenient to first study the subsystem containing only the flow of A, with i ^ 4,6. ODE  analysis  In the study of the A —>• oo coupling flow, we made a heuristic analysis of the coupling flow by considering the O D E analogue of a primitive flow of the sharp slicing coupling flow in Chapter 8. A primitive flow is the solution to the coupling flow equation where the /? functions only retain diagrams of low order. The sharp slicing coupling flow is obtained by replacing the smooth neighbourly slicing functions of the running covariances with the sharp slicing functions of (8.1.2) in the calculation of the (5 functions. A n O D E analogue is then obtained by dividing the discrete flow equations with M — 1 and taking the limit 2  M  —> 1 , where M is the slicing parameter of the smooth slicing. From considering +  Chapter 12. Conclusions  the subsystem X  sub  and  Outlook  257  = (Ai, A , A , A , A ) of the O D E , we discovered that the O D E flow is 2  3  5  7  governed by an O D E flow of the generalized coupling (12.1.2)  (A|) (A| +Ag) 2  where the solution of the couplings can be recovered from solution of ( . For the flow of s  C satisfying a sufficiently small initial condition, C =  (12.1.3)  0(1/8)  as s —> oo. The O D E analysis indicates that it is the flow of £ that is ultra-violet asymptotically free (UVAF) rather than the flow of the interaction coupling Ai. Moreover, since the (3 functions (of the O D E equations) are functions of £, the existence of a primitive U V A F flow around the origin implies that when higher order diagrams are included, the O D E flow stabilizes provided we choose sufficiently small boundary condition for £ . Discrete flow and the fixed point analysis By using the O D E flow as a guide and imposing the conditions (9.1.1) on the coupling, the (3 functions can be estimated by powers of ( . In Section 9.3, we defined a perturbation s  order for the A -> oo coupling flow in terms of the orders of the /3/(G)'s in powers of C  s  (see Proposition 9.2).  In Chapters 9-10, we studied the A —> oo coupling flow in  the context of an order N perturbation. We wanted to determine under what boundary conditions, the corresponding flow of ( stays around the origin. As in the O D E case, we s  proceeded by first considering a primitive flow. We then analyzed the order N flows as perturbations of the primitive flow. In Section 9.2, we analyzed a primitive flow A corresponding to the (3 functions in s  (9.2.1) by explicitly solving for the solution. For the boundary condition A A-  =  Ki  3<  2  i< 7  (12.1.4)  Chapter 12. Conclusions  and  Outlook  258  where r = —1 is the root scale and the AYs are constants independent of scales, the corresponding primitive flow is A?  =  Ax,  A* = A ,  Al  =  K  Af  =  V^A AT < - >  IT fl  4  2  Af =  2  s  ^ \  £ 3  ( s - r ) + K,  —)  ,  e (t-r) + K) tf , K = K, 3  r  (12.1.5)  3  3  5  A* = A .  e  7  where e and e are defined in (9.2.11) and (9.2.12). In the subsequent sections, we then 3  4  analyzed the higher order flows by expressing an order N flow X as a perturbation of the s  primitive flow A where s  X = A + E. S  S  (12.1.6)  S  An order N flow equations (9.4.15) of E is obtained by substituting (12.1.6) in the order s  N flow equations for X . For convenience, we set A = K  = 0, and for the corresponding  s  6  7  order N flow of E , we first considered the subsystem s  with boundary condition E : B  U?  = 0,  U? = 0,  El  =  z = 5,7.  XI  E =0  (12.1.7)  r  3  In Sections 10.1-10.3, we showed that for sufficiently small e = ( A i / A ) , Af, and Ay, 2  2  2  there exists a unique stable solutions of E  s  solution of the subsystem X  sub  condition Xf . ub  satisfying E . B  Hence there exists a unique  = (A A , A , A , A ) satisfying the corresponding boundary l5  2  3  5  7  The existence of a unique solution of E satisfying E  showing that, under the conditions on E  B  B  is obtained by  specified in Theorem 10.1, the map JC (10.1.22)  on the Banach space of sequences Bs (10.1.17) with norm (10.1.16) is a contraction and  Chapter 12. Conclusions  and Outlook  259  )C(Bg) C Bg. The map /C is defined by mapping an element 8 of Bg to a sequence where the terms K(8)  of the sequence satisfy the the boundary condition E  s  B  fixed point equation obtained by replacing the non-linear terms B i(E)  K(8)  and the  with B i(8)  n  n  in  the flow equation of E (see (10.1.22)). In Section 10.4.1, from the solution of X  and choosing boundary condition of A  sub  4  and A at the root scale, we obtained the solution of Ag and X\. Finding a solution for 6  X required a fixed point argument similar to the fixed point argument used in finding s  4  the solution X .  Collecting all the results on the components of the order N flows, for  sub  the boundary condition (12.1.4) satisfying the conditions stated in Theorem 10.2, there exists a unique solution of the order N coupling flow which is of the following form. For r  <s<U,  X{ = A (l + E ),  \  s  1  l  \ < ^ ,  S E  A*  =  A (1 + £ ) ,  | £  A*  =  T"(l + E' ),  \E \<C e'  S  2  2  2  (12.1.8)  ' | < ^ £ n -i- i n r i s N  r  s  3  3  s  kT  s  3  £  4\~  l  M  , ™  '  , ™, ^ C e  N  2  5  C,e (l + lnP) 4  <  k T" 5  \K\  <  l^l+C.e  2  I n Q  Cge* (1 + l n P ) "  | A t I  hr  where T = K + e 5 (s s  2  3  M  '  + 1), e = ( A / A ) , 8 2  2  x  2  = M - 1, E 2  M  r z  = 0; Cg is a constant  depending on TY; k , k , and CM are respectively defined in (10.2.91) and (10.1.27). 5  7  We remark that the fixed point argument does not depend on using the perturbation order defined by (9.3.38) and (9.3.39). The result of Theorem 10.2 is still valid for the flow equations obtained from using a finite set Q of graphs in the calculation of the  Chapter 12. Conclusions and Outlook  260  (3 functions where Q contains at least the four lowest order diagrams: vertex diagram, vacuum polarization diagram, electron self-energy diagram, and one loop four photon legs diagram. Ward Identities In Section 11.1, Proposition 11.1 shows that in order for the root potential of the L R C scheme to respect the Ward Identities, the top scale couplings A f  + 1  with i = 1,2,5,6, 7  satisfy Af where e = A i / A . 2  + 1  = Af e  + 1  ,  Af  + 1  =0,  5 < i < 7.  (12.1.9)  From Theorem 11.1, by having the top scale couplings satisfying  condition (12.1.9), the Ward Identities imply that the root scale couplings X\ = 0 with 5 < i < 7. Thus for an order N flow, we must further restrict the boundary condition (12.1.4) by having A[ = 0 with 5 < i < 7.  12.2  Conclusion  We have set the task for determining the asymptotic forms of the couplings flow of the multi-slice L R C for IQED by analyzing the asymptotic form of the solution of an order N flow satisfying the equation obtained by using only a finite number of graphs in the calculation of the (3 functions. In order for the model to respect the Ward Identities, we have chosen the order N flow to have boundary condition: Af  =  Ai,  A?3  >  1,  A-  =  0,  Af = A  2  (12.2.1)  \\>l, 5<  i < 7.  The resulting flow has the asymptotic form (12.1.8) showing that these A —> oo coupling flows stabilize; and for 1 < i < 4, the asymptotic behavior of Af is the same as the  Chapter 12. Conclusions  and Outlook  261  primitive solution Af given in (12.1.5); for 6 < i < 7, |Af| is bounded by 0 ( |  | is bounded by O 0  m s  )>  a n  d  / )-  n s  S  In order that the model respects the Ward Identities, the top scale couplings with i = 1, 2, 5, 6, 7 are chosen as Af  =  + 1  A  A?  = A  + 1  2  5 < i < 7. The result of our analysis of the order N flow suggests that the asymptotic form of the remaining two top scale couplings should be A where e  2  J + 1 3  =  b\nA + 0(\n(e 5 U)),  Af = Q ( A ? ) = Q((e 5 U) / )  2  = (Ax/A ) , 5 2  2  = M  2  M  (12.2.2)  9m 2  2  M  M  - 1, and m is electron mass of the model.  Recall  in Appendix D, the coefficient 61nA is calculated by using the full spinor propagator. Hence by combining the contributions from all the scales, the leading part of the V local 3  part of the V P diagram should be bin A + O ( l )  a n  d not 6 In A + A 3 + c e 5 (U 2  3  M  + 1) as  one might think. In the formal system of the L R C for IQED, from the Ward Identities, we have the property: by setting X + u  l  g  = (0,0,0) where A „ EE ( A , A , A ) , then X  r  because we were not able to show that X  5  5  6  7  gv  = (0,0,0). But  = (0,0,0) in an order-by-order fashion as done  r gv  in [FHRW88], we were not able to argue that the unique solution of the order N flow with boundary condition (12.2.1) must have X  gv  of the solution equal (0,0,0). Thus further  work is required to establish the above property in an order-by-order fashion. From the L R C expansion for IQED, the asymptotic form of the coupling flow should allow more accurate estimates of the renormalized graphs and hence the n-point connected Green's functions of the model. In finding a non-perturbative solution for IQED, the result of the  Chapter 12. Conclusions and Outlook  262  study of the finite order coupling flow lays the first step of the path for finding bounds of the path integral. There is a hybrid of the L R C scheme that I would like to mention briefly here. In the hybrid scheme, only a selected finite set of local terms of low order graphs are renormalized by the running coupling constants and the remaining local terms are renormalized by the ordinary counterterms as featured in the G N expansion scheme. In this way, the coupling flow of the expansion has its /3 functions corresponding to only a finite set of graphs. This means the solution of the running coupling flow is well defined as opposed to the L R C set-up where the coupling flow can only be realized perturbatively. The study of the perturbation model of IQED using the L R C scheme is still incomplete since the infra-red (IR) part of the model is left unaddressed. In the L R C context, the flow of the potentials should be extended into the infra-red end by removing the existing IR cutoff in the photon covariance, and continuing the slicing of the free photon propagator and renormalization at the IR end. Another interesting problem worth investigating is ' the physical mass which can be defined as the decay rate ln\G (x)\ 2  (12.2.3)  x  where G {x) is the renormalized two point Green's function of the model. The asymptotic 2  estimates of the running couplings should help in estimating  G (x). 2  Bibliography  [BJ86]  T . Balaban and A . Jaffe. Constructive gauge theory. In Fundamental problems of gauge field theory (Erice, 1985), volume 141 of NATO Adv. Sci. Inst. Ser. B: Phys., pages 207-263. Plenum, New York, 1986.  [BOS91]  Tadeusz Balaban, Michael O'Carroll, and Ricardo Schor. Properties of block renormalization group operators for Euclidean fermions in an external gauge field. J. Math. Phys., 32(ll):3199-3208, 1991.  [Fey49]  R. P. Feynman. Space-time approach to quantum electrodynamics. Rev. (2), 76:769-789, 1949.  Physical  [FHRW88] Joel S. Feldman, Thomas R. Hurd, Lon Rosen, and Jill D. Wright. "QED: a proof of renormalizability", volume 312 of Lecture Notes in Physics. SpringerVerlag, Berlin, 1988. [GJ87]  James Glimm and Arthur Jaffe. Quantum physics. Springer-Verlag, New York, second edition, 1987. A functional integral point of view.  [GK86]  K. Gawedzki and A . Kupiainen. Asymptotic freedom beyond perturbation theory. In Critical Phenomena, Random Systems, Gauge Theories, Part I, II (Les Houches, 1984), pages 185-292. North-Holland, Amsterdam, 1986.  [GML54]  M . Gell-Mann and F. E . Low. Quantum electrodynamics at small distances. Physical Rev. (2), 95:1300-1312, 1954.  [GN85a]  G . Gallavotti and F. Nicolo. Renormalization theory in four-dimensional scalar fields. I. Comm. Math. Phys., 100(4):545-590, 1985.  [GN85b]  G . Gallavotti and F . Nicolo. Renormalization theory in four-dimensional scalar fields. II. Comm. Math. Phys., 101(2):247-282, 1985.  [Hur89]  T . R. Hurd. Soft breaking of gauge invariance in regularized quantum electrodynamics. Comm. Math. Phys., 125(3):515-526, 1989.  [Imb86]  John Imbrie. Renormalization group methods in gauge field theories. In Critical Phenomena, Random Systems, Gauge Theories, Part I, II (Les Houches, 1984), pages 295-358. North-Holland, Amsterdam, 1986.  263  Bibliography  264  [KW74]  J. Kogut and K. G . Wilson. The renormalization group and the e expansion. Physics Reports, 12(75-200), 1974.  [OS73]  Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. Comm. Math. Phys., 31:83-112, 1973.  [OS75]  Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. II. Comm. Math. Phys., 42:281-305, 1975. With an appendix by Stephen Summers.  [Pok89]  Stefan Pokorski. Gauge field theories. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, .1989.  [PP97]  Emmanuel Pereira and Aldo Procacci. Block renormalization group approach for correlation functions of interacting fermions. Lett. Math. Phys., 42(3):261-270, 1997.  [Ros90]  Lon Rosen. Renormalization theory and the tree expansion. In Constructive quantum field theory, II (Erice, 1988), volume 234 of NATO Adv. Sci. Inst. Ser. B Phys., pages 201-233. Plenum, New York, 1990.  [SW53]  E . C. G . Stueckelberg and G . Wanders. Thermodynamique en relativite generate. Helvetica Phys. Acta, 26:307-316, 1953.  [Sym66]  K . Symanzik. Euclidean quantum field theory. I. Equations for a scalar model. J. Mathematical Phys., 7:510-525, 1966.  [tHV72]  G . 't Hooft and M . Veltman. Combinatorics of gauge fields. Nuclear B50:318-353, 1972.  [War50]  J. C. Ward. A n identity in quantum electrodynamics. 78:182, 1950.  [Wig56]  A. S. Wightman. Quantum field theory in terms of vacuum expectation values. Phys. Rev. (2), 101:860-866, 1956.  [Wil70]  Kenneth G . Wilson. Model of coupling-constant renormalization. Phys. Rev. D (3), 2:1438-1472, 1970.  [Wil71]  Kenneth G . Wilson. Renormalization group and strong interactions. Rev. D (3), 3:1818-1846, 1971.  [Wil72]  K . G . Wilson. Phys. Rev. D (6), pages 419-426, 1972.  Physical  Phys.,  Rev. (2),  Phys.  Appendix A  V e s t i g i a l Gauge Invariance  Because of the gauge invariance of the Lagrangian of the model, in a perturbative expansion of an external source generating functional for a gauge model of a Q F T , the corresponding generating functional would possess Ward Identities (WI), which formally imply certain counterterms to be zero. In particular, the QED4 model without cutoff forbids counterterms X  of type A , A , or (d • A) . 4  gv  2  2  However, because of the presence  of a cutoff for regularizing the divergence of Feynman diagrams, the W I are lost and we can no longer demand these local parts of a Feynman diagram to be zero in the tree expansion scheme. As a result, we have to consider the flow of these local terms in the R C C tree expansion (with no shifting of covariance) of the QED4 model. But in the case where the X  gv  is marginal, i.e. its degree of divergence is zero, then "vestigial gauge  invariance" still renders X  gv  bounded uniformly in U .  We state the "vestigial gauge invariance" in the following lemma. We also exhibit a couple of examples where, for Euclidean invariant cutoff, we explicitly show that the A -local part of the 4-photon legs diagram and the (d • A) -local part of the vacuum 4  2  polarization diagram are bounded uniformly in U .  265  Appendix  A. Vestigial Gauge  Lemma A . l  Invariance  Suppose that LiG  266  is a marginal local term of G which is formally  u  because of WI but which is nonzero because of an UV cutoff ((M~ p) u  LiG  u  € <S(IR ). d  zero Then  is bounded, uniformly in U.  Proof: The coefficient of LiG  u  has the form (3(LiG )  = JR{p)C,{M- p)  u  u  dp  (A.l)  where R is a rational function of p, with (for marginality) deg(R) = deg (numerator) deg (denominator)  —  = —d. Without the cutoff, by the WI, / R(p) dp is zero under inte-  gration by parts. Hence, we can integrate by parts to obtain P(LiG ) u  = J Q(p)M- C(M- p) u  u  dp,  (A.2)  where £' is a first order partial derivative ( or linear combination of such) of £ and Q is a rational function with deg Q = 1 — d (see examples below). Making the change of variable q = M~ p, u  f3(UG ) u  = J M^ Q(M q)C'(q) u  u  dq.  (A.3)  Now lim M^ Q(M q) u  = S(q),  u  (AA)  U—>oo  where S is a homogeneous rational function of deg 1 — d. Hence lim P(LiG ) u  (7->oo  = f S(q)C(q)dq,  (A.5)  J  which is finite ( at co because of the ('(q) and at 0 because deg(S) = 1 — d). As a continuous function of U bounded at co, (3(LiG ) u  Q.E.D.  is uniformly bounded.  Appendix  A.  Vestigial Gauge  Invariance  267  Example A . l :  Figure A. 16: The 4 photon legs diagram Considering the 4 photon legs diagram in Figure A. 16, the corresponding marginal local term is  X ^ ^ J A ^ A ^ A ^  (A.6)  with r X  M ™ M 4  / ±  =  d  1  4  p  trj{-  Y  j = \ V "f"  where p- (p) U  =  e  ~  (A.7)  U  m  is an UV-cutoff. There are many choices for the UV-cutoff, for examples,  u  P~ (p)  P- {PY  j  M  2 U p 2  for an exponential decay slicing and p- (p)  = 1 — h(M~ ~ p )  u  2U  2  for  2  a neighborly slicing where h(p) is defined in (6.1.1). Wlog, here we use the exponential decay cutoff p^ (p) u  = - ~ r. M  2U  Let C{M~ p)  2  = (p^ {p))\  u  e  and < • >= J dp tr(-).  u  A  neat way to compute X is to use the identity - d R(p) Pa  = R{p)p R(p).  (A.8)  l  Pa  and to integrate by parts. Thus (A.7) becomes 1 ~ —  where  1  r/+m° ^ p  _J£"M2£»lM3/i4 _  pf+m  1  1 p!+m  1  ;  J£-/J2/J3/J1/J4 _|_ y 7 * l M 2 W M  >  ^  Appendix  A.  Vestigial Gauge  Invariance  =  <  268  v  fi  P+m  y^d  3  p+m  M2  c >  M1  (A.10) where  ^  /  +  m  ^  ^  +  '  jrY-i.  ^  m  ^  ,  ^ 3 » 2 ?Vl  2  ^  >  (A.11)  s  By (A.10) and ( A . l l )  Y C I M W W  ^  / I  f  lyMi/i /i3/i4 2  A  A  A  M-M ft-Hi ^  A  3 ^  A  A  A  A  —  4  y ^ /il Al A»3 ^4 1  —  /1  /1  A  A  /1  2  1 v/Jiwww f A  A  J SI in.-ft-HI ^ V  3  (A.12)  ^Vt-  Now trpi^  V4  _ 4  P/i4  (A.13)  7  where D = p + m . Thus integrating by parts on Z 2  2  =  / i l / i 2 / i 3  ' , t 4  ±jdp^d ^d d ^{M- p) u  p  j  =  A  =  4 y.dp  dp  - 5 M3M4  Pn  2  p  P P/*4 W  9 d C{M~ p) u  Pli2  £>  _ 2(5  M3/M  p  Pn  2  2(5n j, Pfj, 2l  /i2  3  L>  + 5^ p^ )  A  2/i4  3  D  2  2  d C(M- p). u  Pfii  D  3  (A.14) Let  M- c'(M- ) = a c /  f /  P  Pfli  c(M-^),  Appendix  A.  Vestigial Gauge  Invariance  269  then  'i^jmnPiii  +  °"/i2/i3P/i4  ^tJLiinPiiz)  +  47J/x P/i P/i4  _j_  2  r>  3  D  2  3  M- C(M- p). U  (A.15)  U  It is easy to see that ( ^ / X 3 M 4 ^ 2  Q(P)  +  =  ^ 2 / 4 3 ^ 4  +  ,  ^AMP/^)  ^P^jP^Pjlj  D  (A.16)  D  2  3  has deg(Q) = —3 and 5(g) = lim M Q(M q) 3U  (°~/t3/M°/J2  = -  u  +  ^ 2 / ^ 3 ^ 4  ^/J2^4?w)  +  ,  ^?M2?M3^M4  3  f7—>oo  (A.17) For p- (p)  = e  u  M  2 U p 2  , £ is Euclidean invariant and (A.18)  and X " " " " 1  2  3  4  becomes  8  3 | dp 2 M- V -  2  -2  4  M  2ry  4C  4M  ^^3tljPlJ-lP^2  +  2V  °"^2M3P/tiP/i  £>  4p P P P  °"/U2/i4^/tlP/i3  +  4  ftl  /t2  £)  2  jU3  (A.19)  /Li4  3  From (A.19), by the identities (E.l.l) in Section E . l and the change of variable g = 2M~ p, u  we have  X  ™  M  I^  4  4 ^ ^  p(L G )A' , u  =  i  (A.20)  7  where 3g l(q  2  4g  2  + (2M- m) ) u  2  (q +  2  c/  2  3  U  2  3  (9g g -g ) + 3 g ( 2 M - m ) ] . 2  2  (A.21)  (2M- m) )  2  (g + ( 2 M " m ) )  4  1  2  2  2  1  c/  2  Appendix A.  Vestigial Gauge Invariance  270  Changing to polar coordinates: (<?i,<72) ->• (r cos 9, r sin 9)  and letting mrj =  (£ cos 0, £ sin <£)  (93,^4)  2M~ m, u  8 ^  "( ' » = 3 i  G  2  /-oo  ^oo  -(r +t ) 4 2  e  2  r  ^  +  +  ( A  '  2 2  '  where A  =  f d9[9cos 9sm 9-cos 9}=9-Jo  B  =  3 /  2n  2  2  — = — 4 2  i  1  d# cos cos 6=^-. 0 = d9  4  J  (A.23)  22  «/ 0  Changing to polar coordinates:  (t,r)  ^  —» (s cos cr, s sin a), and following with another  change of variable: s —> s, yields 2  roo  /3(L G )  = 8n [l + ml]  U  2  7  /' /2  e~  7r  s2  ds s ^ 5 - ^ - / da cos a sin a Jo {s + mfjY Jo r°° e~ s 5  z  2/T  s  2  3  J7T^Y'  = T ^ ^ l  ds  ( A 2 4 )  which is positive for all U and has limit 2TT r°°  lim (3(L G ) = — / 7  U->oo  Hence, f3(L G ) u  7  ds e'  s  2TT  = —.  6 JO  (A.25)  O  is positive and bounded uniformly in U.  Example A.2: Consider the (d • A)  2  term in QED .  The vacuum polarization diagram in Figure  A  A . 17 has value =  j^y\I  ^  dpA  U[p)  *£(PMF(-P).  ( A  2 6 A  where the kernel  KM  =  j  d q  p~ ^+^ u  p~ ^) u  tr  [-f (p+1) ^ R  u  1>  (- ) A  26b  )  Appendix  A.  Vestigial Gauge  Invariance  271  p+q  Figure A.17: The vacuum polarization diagram and  = i + ra.  We read off the relevant local-(d • A)  part from the second order Taylor expansion  2  R(p + q) = R(q) - R(q)p!R{q) + R(q)j/R(q)j/R(q)  -  (A.27)  This yields  j ~  ~ J^y  j dp [| dqC:{M- q)tr u  A,{p)A {p)  >f Rjf Rjf R-f R^  u  J P ^ MMP) d  K  A  where R = R(q) and C = (p- ) .  (A.28)  We project out the local-(<9 • A)  2  u 2  part by the following.  Since C is Euclidean invariant, then the kernel K in (A.28) can be written as  K = Ap T 2  +  Bp L 2  (A.27)  where L and T are defined in (1.2.8), and  A  =  pKp p* (A.28)  B  The coefficient of the local-(<9 • A)  =  2  K^, (P) 2  3p  2  part is just A .  A  (A.29)  Appendix A. Vestigial Gauge Invariance  272  We perform an integration by parts using the identity (A.8) to obtain A = ±r[ dqp (d £) v  C tr (r/i?) .  tr yR? - 2-1dq  q  (A.30)  4  Hence (A.31) which is the desired form (A.2). If we wish to "evaluate" A, we proceed exactly as in (A.9)-(A.12). The result is given in (A.15): A=  3p<  h  D(q)  AT  D(q)  2  3  K -u dq '  (A.32)  u  It is clear from (A.32) that we need Euclidean invariance, i.e., £ = ((M~ q ), 2U  in order  2  that a be a constant, in which case 8 3p4  A =  8  2  h  -3  D(q)  +  Ml  +  D{qf  2  8 _(q + M- m ) 2  For p^ (M' p) u  u  =  e  - P  2  /  M  2  U  ,  2U  2  2  M- C(M- q ) 2U  D{qf  2U  2  M- C(M- q ) 2U  2U  2  + (q  4q\ 2  +  CV)-  M- m ) 2U  2  3  (A.33)  we have  -2(7„2\ _  „-2M-  2U 2  C(M- q ) = e 2U  4(p-q)  4  2  ~H D(q)  h  3  p p-q  P  Q'(M~ q ) 2U  2  -2e~ ~ 2M  =  2  By an almost identical calculation as in (A.21)-(A.24) with  2ir  u  3  ml) 2 ^3 '  (s + Hence A is positive and bounded uniformly in U .  mu = V2M~ m,  e s s  ds  2Up2  (A.34)  Appendix B  N e i g h b o u r l y slicing  In choosing a smooth slicing for the R C scheme. We wish to choose a smooth slicing which closely resembles a sharp slicing (see Section 1 of Chapter 7) so that the running terms can be well approximated by the sharp sliced running terms with only a small deviation. Also, for the convenience of keeping the analysis relatively clean, we arrange that only adjacent slices have overlapping supports. We can construct such a smooth neighbourly slicing satisfying the above criteria by the following partition of unity. Let h(x) be a C°° monotonic function obeying 0  h(x) =  (1  for  x < 1 - e, 0 < e <• 1 -  for  x > 1  M~  2  (B.l)  with 0 < h(x) < 1 for 1 — e < x < 1. Since the free photon covariance is given a unit mass, there is no need to have IR decomposition (negative scale). Let, for s > 0 p (x) = h{x/M ) s  2s  -  h(x/M ), 2s+2  (B.2a)  and p°{x) = 1 -  h(x/M ). 2  (B.2b)  It is easy to see that, for x > 0, oo  (B.2c) Let Su(f)  denote the support of a function f(x).  It is easy to see that (B.3a)  273  Appendix B. Neighbourly slicing  274  h(x)  1  Figure B.18: Example of h(x).  s+1  s  P(x)  P (x)  j 1/2  A  J (1-e) M  (1-e) M  Figure B.19: Only adjacent slices have overlapping supports.  Su{p ) n Su{p>) ± l  Su(p ) n Su(p ) s  s+l  =  0iff|i-j|<l, ((l-e)M , 2s+2  (B.3b) M ) 2s+2  = I, . s  s+1  (B.3c)  (See Figure B.18 and Figure B.19 .) It is easy to check that the above partition of unity (B.2c) provides a decomposition of the covariance satisfying the bounds given in (3.1.3a-d).  Appendix C  L R C running covariance  Cl  Photon field  In the following, using Lemma 5.2, we derive an explicit form of the running covariance H for the L R C scheme on IQED described in Chapter 5. From the derived form, we s  impose conditions on the running couplings so that V (A) in (5.2.8a) and KI in (5.2.2b) s  are well defined and uniformly bounded in U and A. We make these conditions explicit by imposing that the slice function p at scale s, 0 < s < U, has the following support, s  (1 - e)M  <p < M ,  2s  2  (C.l.l)  2s+2  where e < 1 is some small positive constant. (See Appendix B for an explicit example of a slicing satisfying (C.l.l).) We also assume the external sources have compact support. Let X , = (XI-1)f (p),  (C.1.2a)  X = X h(p).  s  s  3  s  5  5  where v  -  2  AW  =  M  2s  ftiP) = ^  (d.2b)  By the fact that L = L, T = T, LT = 0, 2  2  we have RK  S  =  p +l 2  L +  [l-ri)  (l-r))+p275  (A* - 1) p L + M X ) 2  2s  s  5  Appendix C. LRC running covariance  276  (**•+')' = (^ + ^)'^+( "_y^)'r, (1  (C.1.3b)  (1  where K and R are defined in (5.4.6). Let s  * =  +  ^  Using the fact that [a L + b T]'  = a'  1  f ^ -  {  ^  L + b~ T, and substituting (C.1.3a-b) into  1  l  (5.2.3a), the running photon covariance H = H[ + H^, where s  H  =  K  W  j  ^  %  = W >7  ^  (C.l.Sa)  ^  .  (C.1.5b)  and  1 - {p + p  =  w>  s  =  1  ^./;  +  Also, the denominator 1 + p  s + 1  +  s + 1  1  R K  (  ) + p {j s  )  t  f  l  )  + 1) + p-+i(7«+i + 1)2  s  2  (C1.6a)  .  (c.i.6b)  in (5.2.4) is  s+1  1+ p  (j L + A*T) .  s+1  (C.1.6c)  s  We wish to impose bounds on A 3 and A 5 so that the denominators in (C.1.6a-b) and (C.1.6c) stay above zero. On the support of p (p ) defined in ( C . l . l ) , M p s  ^^w-i|)(i-rr^) * (1  + M-2«  -  2  2s  1^1 K  l  < IAS -11 ~  (1  ~ 1 and  - jTjr^s)^-!-^)  1+'M-2.'  -" +' M ~< Kl<„ - (1 -  2  (  C  ' ' 1  7  b  )  )1A|  (1 - ri)M-  2s  2  1  5 1  r))M-  2s  + 1'  (C1.TC)  Appendix C. LRC running covariance  T h u s the  277  conditions  XI  are sufficient t o g u a r a n t e e 1 + p j s  0,  >  Af>0,  (C.1.8)  > 0. In the case o f c h o o s i n g t h e L a n d a u g a u g e , we  s  d o n ' t n e e d t o r e q u i r e A5 t o be p o s i t i v e . T h e c o n d i t i o n  (C.1.9)  XI > \X \, S  5  is sufficient. Let 7  s  = Af 4- X . F r o m t h e above r e q u i r e d c o n d i t i o n s , i t is easy t o see t h a t s  5  n  <  w L K  1  '  -  1+  7  <  i  t 1  < s  -  i  i  7  +  T--('+y> 7 (1 5  s  ^  ^  f 1  +  + Y) -  fl_  '  l  (d.lOa)  n)M~ '  2s 2  Appendix  C.2  C. LRC running  covariance  S p i n o r fields  For each component Hj = VjRj of the spinor covariance, let us write V =p {DS)- , a  1  j  where {OTS = U, Df  =  l +  for s<U,  D)  =  1 + p A] +  A]  =  R, Kj,  =  (1 + Rj K° )  B+ s  l  p Af; u  p B° ;  s  s+l  - 1.  +l 2  and Kj and Rj are defined in (5.4.16). Let us introduce more notations here. 1. For k = a + bp , let us denote 1  k* = a — bp",  = ^  RE(k)  = a,  A- — k* AH(k) k  = 2  = kk* = a + b p .  2.  A-  where Af = Af — 1.  = H  EE (/<-')*/<-;  2  +l  2  2  Appendix  C. LRC running  covariance  Let us rewrite Aj and B*  279  in terms of the above notations. For convenience, we suppress  +1  the indices j and s except at places where they are necessary for avoiding confusion. K  =  A M  + A p + (A - A ) M j /  A  =  RR* [(iT )* K] = R K,  B  =  {RK + l ) - 1 = {RK)  2  (C.2.6)  2  4  2  2  4  1  (C.2.7)  2  2  + 2RK = R K  2  4  We wish to impose conditions on A- so that \\D/p \\  2  + 2R K.  (C.2.8)  2  and ||(1 + p A)/p \\  s  s  s  stay away  from zero and | | p D | | is bounded uniformly in U and A (here || • || is the Hilbert Schmidt _ 1  norm). We start by first considering lower bounds for RE{D) and 1 + p RE{A).  From  s  (C.2.7) and (C.2.8), 1 + p RE(A)  =  1 + pR  RE{D)  =  l+(p  =  1 + [p R  =  {1-p  s  s  RE{K )  2  S  RE{A) + p  s  s  2  s+1  RE{K )  +p  S  - p )  s+l  +p  RE{B))  s+1  [R RE{[K } )  s+1  4  + p {l + R  s+l  s  2  [l + R RE{{K } ) 4  s+l  S+1  + 2R  2  2  S+1  RE{K )) S  + 2R RE{K )]  2  RE{K ))  2  .  s+l  (C.2.9)  In trying to make the quantities in consideration non-negative, we first impose that A^  >  1,  (C.2.10a)  A  >  M .  (C.2.10b)  u+1  We consider the cases j = 0, and j ^ 0 separately. For the case j ^ 0, we take advantage of the fact that A can be chosen arbitrary large. From (C.2.6),  = R RE{[K} ) 4  2  =  ^ R ([A M + \ p } 4  2  2  4  2  M {M -p )~ 2  2  (p2 +  M  - (A - A ) M V ) 2  2  2  2 ) 2  4  Mp  2  2  "  2  A  4  -  2  (p2 + M 2 ) 2  2  A 2  -  (  C  >  2  -  1  2  )  Appendix C. LRC running covariance  280  From (C.2.10a-b), dropping the first term in (C.2.12) and using the fact Mj > A , 2  - + R RE([K] ) >  2  (  p  2 +  M  2  )  •  2  (C.2.13)  Similarly, by dropping non-negative terms, we have ^ + 2R RE(K)  >  i  \ + &RF(ft\  >  (  l + i2J*£?(lO  >  2  2  ^ A 2  (  p  s  f  (C.2.14)  -|W)  2 +  M  2  )  (Af + 1 ) M +  ( p 2 +  M  2  )  2  •  (C.2.15)  Since A —>• oo before [/ —> oo, we may impose that A  2  >  1-e,  (C.2.16)  A  >  max(M \4M u+  sup |A||),  u+1  (C.2.17)  0<s<U  where £ is a constant satisfying 0 < e < 1/2.  From (C.2.15), and the above imposed  conditions, p {l + RE(A)) s  (AS + 1 ) M  >  p  >  P j-  l  M? +  2  M  2s  (C2.18)  s  From (C.2.11) and the imposed conditions, (C.2.13) and(C.2.14) are non-negative. Thus by dropping the p  ( l + RE(B))  s + 1  term in RE{Df)  and using (C.2.18),  RE(D A  >  p (l + RE{A)) > p ^  (C.2.19)  l + p RE{A)  >  p {l + RE(A))>  (C.2.20)  S  s  s  s  s  p ^ s  (C.2.19) implies that  Halloo < | - < A  2.  (C.2.21)  4  (C.2.20) together with using external sources with compact support implies that K  s  well defined and \\K \\ is bounded uniformly in U and A. s  e  e  is  Appendix C. LRC running covariance  281  As for the case j = 0, we use the fact that p Rl  ~ M  s  r  (p) = Jl% = llyji? + m\  u = (X\ - X ) m.  r = p r, s  to control bounds. Let  2 s  s  8  s  2  By rewriting p RK  S  s  = p X + u p R, 8  s  (C.2.22)  s  s  2  we have p RKs\-l )  \\p (l + 8  s  8  1 + PX S  1+  S  =- p R l + PX ,  (C.2.23)  s  s  2  s  2  Here in addition to the imposed conditions in (C.2.10a) and (C.2.16), we require \u \ r < -. - 4 s  (C.2.24)  s  From the imposed conditions (C.2.16) and (C.2.24), since e/(l — e) < 1, ||// (l + p i2 X ) - || < s  s  1  l-el-(e/4(l-e))  (C.2.25)  3(1 - e)'  Similarly, by (C.2.22),  V (X)  = p'iB'y^l  S  +  iB^E }- , 8  B  =  [1 - p - p )  E  =  u (p R) + 2u  s  s  s  s  + p {X + l) + p  s+1  s  (C.2.26)  1  s  s  (A*  s+l  2  s+l  (X  +1 2  + 1) (P  s+l  +1  + l) , 2  R) + (u ?  (p  s+l  s+l  R ). 2  If we require that \u \r s  + 2\u X \  s  s+l  s +1 2  r  s+1  + (u ) s+1  2  (r  s + 1  )  2  (C.2.27)  < -,  2 then ||P (A)|| S  <  (1-dl^H/IB'l))-  tf\B'\-  x  < p IB ]' \ l s  8  1  1  1  14°  r + 2\u X \ 8  8+1  s +1 2  r  s+1  + (u ) s+1  2  (r  s + 1  )  2  -l  \B  S  _2_  -  X l-(e/2(l-e)) "Al  <  ^ - < 4 . 1— £  8  1  (C.2.28)  2  (C.2.29)  Appendix C. LRC running covariance  282  We would like to estimate V (\) by a simpler form i n the case that s  K = Q(s ),  K = 0(0,  a  (C.2.30)  a > o.  A s s u m i n g these orders, we express the running slicing of the Fermi covariance as a sum of a dominant term and an exponentially small (in s) term. Since r  s  easy to see from (C.2.26) and (C.2.30), that \\E (p)\ \ = Q(M- s ) s  S  2a  = Q(M~ ), S  it is  and so (C.2.31a)  where E  s  pE s  Pi  =  1  s  ~ l + p (E ) s  s  g  2  (B°Y  u  E (B )~ s  s  f-E ^ v B° 8  1  l  (C.2.31b)  and \\E {p)\\ =  0(M- s ).  s  Since the part E  s  2s  4a  is irrelevant, we can approximate V (X) by s  (f_  pE  B~ ~  (B ) '  s  s  s  s  2  (C.2.31c)  Appendix D  Bounds on Spinor Loops  D.l  Proof of Lemma 6.2 of Section 6.1  Here we imposed that conditions  A" 3  and for Af ^ A ^  1  +1  = 0(lnA),  (D.l.l)  with 0 < s < U + 1  Af = O ( l ) .  (D.1.2)  For a mixed scale spinor loop, since at least one of the lines is at scale s < U, the domain of integration is restricted to lie inside a compact domain. Let q be the loop momentum of L^j, lines of L ^  u  Vol(q) be the volume of the domain of q, and i be an index for the  . It is easy to see that  \\L' f\\ao< n  Vol{q)  niTOIIoo.  (D.1.3)  i=i  The result of part 1 is then follows from the the bounds on the spinor lines in Lemma 6.1. As for a pure scale U + l spinor loop, since there is no U V cutoff for the spinor lines, the volume of the domain of the loop integration is unbounded and we can not apply the same argument as above. But in the cases where the number of the lines n > 6, there are enough factors of the R,'s to bound the integral. More precisely, from (6.1.4) of Lemma  283  Appendix D. Bounds on Spinor Loops  284  6.1,  iWflloo <  n  J dqH  +  i=l  7  ~  <  [ dqf[  flWVpjWoo i=l  \Rj(q + )\  (D.1.4)  Pi  i=l  where each p; is a linear combination of the external momenta feeding into the loop. By scaling q —> Mjq, we get the desired estimate. (D.1.4) = M -  f dq  2  |1 +  4+  M - V i l  -  (D.1.5)  1  i=l  J  Now the result of Part 2 follows from the orders of the M,-'s. Q.E.D. As shown in the proof of Part 1 of Lemma 6.2, we see that an integral pertaining to the graphs in a L R C on IQED with bounded integration domain has no A-divergence. Moreover, it is of Q(A~ )  where m is the number of fictitious spinor lines in the integral.  m  Recall that the support of the spinor slicing p f ( p ) is [(1 — e)M + 1  that integrand with 1 — p f  + 1  2  ,  2U+2  oo). This implies  factors would render the corresponding integration domain  to become a bounded region. Thus in finding estimates for A and B, we can replace each neighborly slice p f  by unity and attach an error of Q(A~ ).  + 1  m  We substantiate the  above claim by the following lemma. For convenience, we drop the subscript F in p f expression of the scale U+1 spinor covariance H^ . +l  Hf  +1  where K^f  1  =  p (l.+  =  p R \  . Let us write down a more explicit From (5.4.18), (5.4.19a-e) we have p R K% )- R  u+1  u+l  + 1  u+1  1  1  j  j  (D.1.6)  u + 3  and Rj are defined in (5.4.16), and R?  +1  = [(1 + pWll+^Mj  + (1 + p \ )p\-\ u+l u +1 2  (D.1.7)  Appendix D. Bounds on Spinor Loops  where \f  +l  = Af  + 1  - 1, Thus Hf  is like Hf+  +1  respectively by 1 + p X^ u+l  285  and 1 + p \% .  +1  with the M,- and ^ being modified  1  u+1  Let  +1  [Fj(p)](P) =  JdQ[K (p)](P,Q)  (D.1.8)  3  be a function of a set of external momenta P = {pi, • • • ,p } m  where each pi is restricted  to a compact domain and Q = {<?i, • • •, q }- The kernel Kj is a product of lines which n  depends on a slicing function p. By suppressing possible 7 matrices placed in between the lines, we write n  rik  = n n&(p)](?*+pi) k=l 1=1  [Ki(p)](p,Q)  (D.1.9)  where e (p)=pR (p), j  A,(p) is defined in (D.1.7) with p  u+1  (D.1.10)  j  being replaced by p, and pi are linear combinations  of the external momenta.  Lemma D.l O(l)  if 3 = 0  0(A" ) d  if  (D.l.ll)  j*0  where d = Y^k kn  Proof: Let us introduce an interpolation between Fj(p) and ^ ( 1 ) . For 0 < t < 1, (D.l.12) Now dt ^0  =  ^  dt  dF[ dt  j dQ  (P)  5p  (P  U+1  ~  1) ( W - 1 3 )  Appendix D. Bounds on Spinor Loops  To consider the derivation SKj(p)/Sp,  286  we first consider the derivation on a line £(p). Let S£ (p) 3  6p Sp  (D.l.14)  '  Using the facts that  ~  -RAP)  5p  ,c5R-\ )  ~ ,  P  §  (D.l.15)  Rj(p)  p  and m ( p ) ) -  ui '  1  K  =  5p where Kft  (D.l.16)  +  e j  is defined in (5.4.16), we have  l  lj{p)  R ^ - p R ^ K l f  =  1  R^p).  (D.l.17)  Since Kj(p) is a product of lines, by the product rule, we have  (D.l.18) k,l  (k,l)^(kj)  where (-)( '') denotes a term with momentum q + p and Vol(Q) fe  k  t  domain of Q restricted by the respective (1 — p Y ' ^s. u+1  k l  is the volume of the  From the definition of the lines  £j and £j, it is easy to check that both ||^-||oo and ||^||oo have the following estimates  i  O(i)  if = o  [ 0(A" ), 1  (D.l.19)  if j 7^0  Thus from (D.l. 18), it follows that  O(l)  if J" =  Q(A- )  if j 7^ 0  d  Q.E.D.  0  (D.1.20)  287  Appendix D. Bounds on Spinor Loops  D.2  Proof of Proposition 6.1 of Section 6.1  We compute the coefficients of the 1/j-local parts of VP . u+l  VP  U+1  The formal expression of the  is (\U+l\2  <V P > ^  3  1  =^f - / l  +  1  d Af[p)  Y,M9n(j)K?£(p)AZ (-p), u  P  where sign(0) — sign(l) = 1, sign(2) = sign{2>) = —1, and K?£(P)  dotr [f  = J  H^\p  The kernel of the projection of the a type-j VP  u+l  + q) Y H? \q)  }.  +  (D.2.1)  diagram onto F and (<9 • A) has the 2  2  form {LK^HP)  dq (p(q) ) U+1  = J  2  tr [y l  & p ^ d q ^ i q ) )  r}  Rf \q)+  (D.2.2)  Let us introduce the following convention that we employ for convenience. A l l sum over the index j has a sign factor sign(j) which we do not include explicitly. that since p  u+l  has no U V cutoff, (LKj) ^ ,  integrands of {LKj)  2  2illv  l  Note  is not well defined without summing the  over the index j before integrating the q variable. Let us allow  ourselves to manipulate the integrands of these j integrals individually without explicitly implementing an UV-cutoff. For a rigorous treatment, we actually need first to have the domain of the loop integration to be restricted to a finite region (e.g., a hyper-sphere S% with radius N) and then to take the restricted domain to IR (N —>• oo) after making the 4  manipulation on the integrand. We will suppress the index j and scale U + 1 at some places for clarity in manipulating expressions. We extract the coefficients of the local parts by employing the same technique as done in the discussion on vestigial gauge invariance in Appendix A . We let {LKj) ^(p) 2  = p^A^  +B M ,  (D.2.3)  Appendix D. Bounds on Spinor Loops  where  288  and L „ are defined in (1.2.6). As in Appendix A , (summing over p and v) M  A  p„ (LK) ^u,j Pv  =  3  (D.2.4)  2  (LK)  (p)  _ Aj_  2tlilid  3p We would like to show that Aj = Q(A  )  (D.2.5)  3  2  where m is the number of fictitious lines in  m  the loop, and that  £  Bi = 0(lnA).  . .2.6) (D  3  We now consider the coefficient Aj. Aj.  We apply Lemma D . l and set p  u+1  = 1 in  Subsequently, by the Ward Identities, the corresponding integrand of Aj becomes  an exact derivative. Thus, accounting for the Q)(A~ ) error where m is the number of m  fictitious line on the loop, we have  A  3  =  ' O(l)  if J = 0  0(A" )  (D.2.7)  i f "  2  Note that by applying the same argument of using the vestigial gauge invariance (Ward Identities + Lemma D.l) as above, it is easy to see that the coefficient of any gauge variant local term is also O ( l ) Next we consider the coefficient B. In the following, we suppress the j's and U + l's for ease of notations. By setting p = 1 and integrating by part,  KumM  = j  -  dqtr 1»  \{p<rPad d „R) qa  tr r(pA,R)  q  1" (PAM  + 0(A" ) m  + 0(A- ) m  (D.2.8)  where here R =  (A M +A 4  2  (D.2.9)  since (D.2.10)  Appendix  D. Bounds on Spinor  Loops  289  To evaluate the trace in (D.2.8), we first consider the term p d R. a  qa  Using the fact  that  Rp!=p!R* + 2X {p-q)  R\  2  (D.2.11)  we have  -p d R a  =  qa  X RpjR 2  =  XR  [ +2X (p-q){Rr yR?]  =  XR  =  A R  2  J  2  2  [p +2X (p-q)(X M  2  -  J  2  2  4  X i)R  2  2  [(2A A (p • G/) M) + (R-y-  4  2X (p • q 2  2  4  2J  =  l  V  X R {a+k),  (D.2.12)  4  2  where  a =  2X X {p-q)M A  k =  (D.2.13)  2  R- p-2X (p-q)q. 2  (D.2.14)  2  2  In terms of a and k, the trace term in (D.2.8) is  XR 2  tr [ " (a + y )  8  7  (a+ff)}  / i 7  =  XR  =  XR  =  XR  =  -8X R  (2a + k )  =  -8X R  [8X X {p-q) M  =  -8X R  2  2  2  (a tr[YY]+tr[Yf/Yffi)  8  2  (-16a + tr [7" (7"A; - 2k k) ])  8  2  2  il  (-16a + k tr[YY  8  2  2  8  2  2  2  8  2  2  2  2  [R-Y  8  2  -8A p 2  2  2  2  + R-y-AX R- {p-q)  2  2  + 4(p-q) X 2  2  2  (2X M 2  2  2  32A A (p • q) M 2  2  ( A M + A c- ) 2  + 2})  2  2  2  4  2  (A M + A ? ) 2  2  2  2  + AXi{p-q) q  2  +  2  X q -R- )} 2  2  2  2  4  '  (D.2.15)  Appendix D. Bounds on Spinor Loops  290  From (D.2.8) and (D.2.15), and the identity / dq(p • q) f(q ) 2  = ^ J' dq q f(q ),  2  2  (D.2.16)  2  we have  [LK) ^{p) 3p  4A f dq 3 J [\lM + \ q ) AX X M r dqq 2  2  2  2  2  2  A  2  2  2  2  +  2  ^ 1  f  3  J (\ M 2  +\  2  (A- ) m  + 2  2 q  y  0 +  U  [  ( D 2 17)  A  [ U  - Z  U )  By further making a couple more changes of variable, first to spherical coordinates with radial variable r and then x = (A^/A )/ , 2  { L K  ] -/ 3p 2  -2  = ^  P )  r +  3A|  2  I* + O ( A - ) ,  3A , 2  (D.2.18)  where 00  dx x ( M + x) 2  =  J b  and  M  2  [°° r^ \*  2  (D.2.19)  X  Jo (M + x) 2  4  '  v  is the angular contribution of the integral (D.2.17) in spherical coordinates. Note  that I  a  is a divergent integral without summing over j.  Let us compute the integrals I  a  and I . b  We now reinstate the index j back in our  discussion.  i  J  =  roo /  d  ~\ x  772  Jo  3  =  lim ln  Upon summing 7" over j , we have  roo i  /  M  +X  M [M 2  + N\  2  ^  3  0  3  Jo  - 1.  d  x  ~\ 7T77 (M + x) 2  2  (M + X 2  D.2.20  Appendix  D. Bounds on Spinor  -=  .  In  Loops  291  [m2 1_ yY ) 2  /  (m + 2 A ) m J 2  2  2  ' l  n  2  (  A  )  +  1  ^  (  (m + A ) 2  2  m  +  2  2  A  2  )  m  2  2  A  2  ;  2 ln(A) - ln (2m ) + o(l).  (D.2.21)  2  As for Ij, integrating by part, we get  r,  M  2  L  o  D  X  X  (M  \ .  =  2  ( -  + x)  2  X  X  2  V 3(M +x)  A  2  = Upon summing  M  3  3 ( M + x) 2  3(M +x)  2  2  \-  (D.2.22)  over j, we have 3  £/j-=0.  (D.2.23)  i=o  Tracing through (D.2.5), (D.2.7), (D.2.18),(D.2.21) and (D.2.23), we have ^  4Cn(Af  + 1  E gj= " , U  ) , h ( A ) 2  1  A  S  ln(2m ) +  " V  2  ,  ,  '  o(1)  --  (d 2 24)  s  n  n  n  A  s  In writing down /? , we remind the reader that there is extra minus sign needed because 3  of the anti-commutativity of the spinor fields. We now consider the renormalized part of the L%  +1  with n= 2 or 4. In the following,  L(-) denotes the localization of (•) onto the V^-local parts and R(-) = (1 — L)(-) denotes the corresponding renormalized part. Again, here we suppress the index j and scale U + l and only have them be conspicuous where they deem necessary. Let C(q, P)K (q, n  the kernel of a L (P) n  where ({q,P)  is a product of the p 's u+l  P) be  from the lines, and (by  suppressing the Ai's, combinatoric constants, and the gamma matrices multiplied to the lines) K {q, P) is a product of lines with loop momentum q. We write n  K (q,P) n  = f[£(q + ) i=i Pi  (D.2.25)  where pi are linear combinations of the external momenta P and £ = R defined in (D.l.7).  Appendix D. Bounds on Spinor Loops  292  Because of the presence of the slicing term  P ) , the L operation on the kernel is  not taking Taylor terms expanded at P = 0 from the integrand £(g, P)K (q, n  as in the VP  U+1  K (q,P) n  P). Rather,  case, the kernel of the local part is the Taylor terms taking from just  alone. More explicitly, LL (P) n  = Jdq C(q, 0) L(K (q, P)),  (D.2.26)  n  where L(K (q,P))  =  n  ± j^ dkK  tP)  k=0  and 5 is the degree of divergence of L%  +1  (D.2.27)  0 , 1  t=0  (here 8{L±) = 0 and 8(L ) = 2). Since each 2  t-derivative dijq + t ) — Pi  =  ~  ^d{R-\q + tpj) -  -R{q + tpi)  R{q + tpi)  f  —  = -(l + p\ )e( + t ) £ e(q + t ), 2  L{K (q,P)) has n  q  Pi  (D.2.28)  Pi  the form  E  k I I 4/,  fc=0  where kl indexes the lines of LL  n  (D.2.29)  p  i=l  and P^ are products of linear combinations of the  By add and subtract the same term /  dq ((q, P ) L(K (q, P)) to RL (P) n  n  iS.  P  = (1 —  L)L (P), n  we write RL (P) n  = I (P) + 7 (P) X  (D.2.30)  2  where h(P)  =  jdqaq,P)R(K (q,P)),  J (P)  =  Jdq(aq,P)-<:(q,0))L(K (q,P)).  2  The term h(P)  (D.2.31)  n  n  (D.2.32  is easy to bound. First, we write  aq,P)-aq,0) = jo  dt ^^.  1  d  (D.2.33)  Appendix D. Bounds on Spinor Loops  Since the derivative dp (x)/dx,  293  where here p (x)  u+1  is viewed as a function of a real  u+1  variable x, is bounded, we have sup t where  '\dC(q,tP)\  (D.2.34)  = C  (  dt  is a positive finite constant which has a dependence on U. Let  support of the derivative dp (x)/dx.  Since  u+l  ((1 — e)M , w  M ),  V u+\ p  V u+i p  be the  is contained in the open interval  it is easy to see that the volume Vol(q) of the domain of q restricted  2U  is finite. By bounding the lines £ i with sup-norms and  by the support of dC(q,tP)/dt  k  using Lemma 6.1, we have \h, {P)\ 3  <  fdq\(((q,P)-C(q,0))\  =  C Vol(q) c  = <  \LK (q,P)\ nd  J2 W  II  A;=0  (=1  Halloo  ifj = 0  O(l) 0(A" ) n  (D.2.35)  if J 7^ 0  As for the Ii(P) term, we first have  (D.2.36) By applying Lemma D . l , we get  Iij(P)  = fdq  Rk (q,P)  +  nJ  ' 0(1)  if J = 0  0(A~  ( n + m )  )  (D.2.37)  if3*0  where K (q,P)  =  n  and that each of the £'s in K (q,tp) n  K {q,P)\ n  pU+i=1  (D.2.38)  is now the R in (D.2.9).  After taking the t-derivatives, RK (q,P) n  can be written as a sum of terms as the  following. RK (q,P) n  = J2(RK)  (q,P)  {n>m)  (D.2.39)  Appendix D. Bounds on Spinor Loops  where each  (RK)^ )(q,P) m  294  has the form p\  P(n,m)  and  P(n,m)  / J0  n+l+5  dt(l-t)  KQ  S  • i i=i  (D.2.40)  + tPi)  is a product of the pj's and A2's. By power counting and making the scaling  q —> Mjq, since 4 — (n + 5 + 1) = —1, we have  rfg|(i?^) , | (Ti m)  <  M/"  '  ( n +  + 1 )  |P , | ( n  dt dq TT | A + A ^ + t M 7 - V ) r  m )  4  •  =  7 o  ^  2  1  i  i=l  0(Mr ).  (D.2.41)  x  Thus  O(i)  Q.E.D.  ifi = o  (D.2.42)  Appendix D. Bounds on Spinor Loops  295  D.3 Proof of Lemma 6.4 of Section 6.3 Part 1 is obvious from the property of neighborly slicing. For convenience, we drop the subscript B from the photon slicing and form factors. For Part 2 (i), we first write V_ u, ni  vp  = (L^  +  +  1  H  u  +  1  (D.3.1)  r.  Subsequently, it suffices to show that each term of the power has its sup-norm equals to l + 0((  m  A ) ) . Let us write down the detailed expressions of the various terms involved - 1  in the bounds. L  + l 2  H  s  =  KpL  =  1 -P T~^ l+p  V  =  u+1  2  +RL"  (D.3.2)  +1  p  S  (D-3.3)  L  2  u+i  0  Sr-:1  (D.3.4)  l + P^W -!)^  u  P  v  i  =  +  -  i ) ^ r + P  u  +  U  +  l  m  i  +  i  -1)  G&)+1]  2  -1}  ( D  ' ' 3  5 )  where R is the renormalized operator and L is the projection defined in (1.2.6). By (D.3.2-43) and using the estimate of RL  in (6.1.13) and the value of b  +1 2  in  vp  (6.1.12), we have  I % ^ A  +  l  V  u  +  l  n  +O(0nA)- )  (D.3.6)  1  p  3  where  n =J ^ .  (D.3.7)  r  From (D.3.4), we have  a  "  +  i  "  k  +  i  -  =  !  +  S  •  - i i ) ^ , =  1  +  °  (  <  i  n  A  r  ,  )  -  (  a  3  -  8  )  Appendix D. Bounds on Spinor Loops  296  Thus since MnA + C + 0((lnA)- ) 1  vp  J  l + OianA)- ),  3  A3 7 4  61nA + C  1  + Ar  3  (D.3.9)  1  3  we have L"  + l  H  = l + 0((lnA)" ).  u + l  (D.3.10)  1  For Part 2 (ii), we write 0^ -v + )H u  l  = (V( -i)-v u+ )  u  H  l  P  ni  P  ni  u + 1  L,2  (D.3.11)  H .  + l  u  From (D.3.10) and (D.3.2-44), we have ll(V  _^  n i  + 1  )^||oo  \\L™H™\\^  <  \\H L™H U+1  {\v ) V V +1  <  (X^)  2  U  p  u  2  + 0((lnA)" ). (D.3.12) 1  2  Inserting (D.3.5) and (D.3.4) into (X^ ) V V +1  u  (I+P ¥  2  +1  (X^ ) V V {1+p )  2  2  U  U  p /(l+p ) , 2  2  we have  2  p  U  2  2 2  (1 + (A^  1  - l)p ^n )  p  u  2  p  1 + pU{\}> - l)n  + pP+HK^  ~ IK  1  p  + I] - 1}' 2  (D.3.13) Since the above involves the product p p , u  (where pf  is p  +1  u+1  (X^ ) V V +1  2  U  U  we have p  u+l  = pf  u+l  +1  and p  u  restricted to the intersection of the supports of p  = 1 — pf  +1  and p ), and  u+1  u  p  2  (1+P )  2 2  1(1 + (Af  <  +1  PI  P \^ fn U +  + 1  L  - IK n )P + 1  p  M- - (1-e)- + 2U  2  1  2  1 + (1 - P L )M U  +1  +l  2 p  ~ IK + PL {[(A + 1  7 + 1 3  - I K + I] - 1} (D.3.14)  0{(\nA)~ ).  2  (D.3.15)  l  From (D.3.12) and (D.3.15), we have  M- ~ vpb, + 0((lnA)- ) = 0((lnA)- ). (1 - e) (Af"") 2U  \\(V - u+i)H ni  vp  2  1  |oo _  1  1 2  1  (D.3.16)  Appendix D. Bounds on Spinor Loops  297  For Part 3, by the fact that (n - VP )H u+l  = (L H ) *-  u  u +1  u+l  n  L H L H  2  u +l  2  2  u+1  u +l  2  (D.3.17)  u  2  and (D.3.10), it suffices to prove it for the case n = 2. 2  For part3 (i), again we expand L W L ^ H ^ L ^ H ^  +  and apply the estimate (6.1.13) of R L  1  2  < \\b p L H  b pL  +  (RL ) \U  2  u+1  \\H  <  V(A  )  2  u+l  . RL \  u  U +1  p  2  u +l 2  u  / + 1 3  2  vp  H  U+1  + 2||6, p L H H  2  vp  + l 2  2  + 2-  (Af+^M ^ 2  1  + 0((lnA)- )1  <  (l + 0((lnA)' )) | | ( A ^ 1  We further simplifying the  {\  + 1  ^ l | o o + Q((lnA)- )p.3.18) 1  yv v  u +l  2  )V^  in (D.3.18).  \\{\\ ) p*H H \\ J+1  + 1  u+l  <  u  00  3  u+l  (1  u  4 P  + P  )  2  2  From (D.3.14), {\  u  + l z  YV ^yv  p  u  (1+p ) 2  4  2  P M )n U +1  1 - Pi  +1  1+W  +1  +I  2  2  L  - l)pt n +l  {[(A3  1 + (1 - p ){X - l)n + P L u  p  u  L  z  p  p  - l)np + I] - 1} 2  (D.3.19)  < l + CXQnA)- ). 1  Thus  < l + O(0nA)- ). 1  I I L ^ ^ + ^ H o o  (D.3.20)  For part 3 (ii), from the proof of part 3 (i), it is easy to see that 2  '\{\ i I  + 1  ) v 2  u + l  (1+p ) 2  2  v p u  4  +  0((lnA)" ). 1  (D.3.21)  Appendix D. Bounds on Spinor Loops  298  We now show that the || • | | i term is zero in the limit A — > oo. From (D.3.19), we have  (\% ) V V (1+p ) +1  2  u+1  2  p\  u  A  (D.3.22)  <  2  To estimate the integral (D.3.22), we make the change of variable r = p . The domain of 2  r = (1  integration of the variable r is [r,,,^] where  = 0 and p (n)  p (r ) u+1  e)M , r\ = M 2U+2  —  . Note that  2U+2  = 1. Now  u+l  0  0  dr r (1 - p ) u+l  (D.3.22)  <  ci r JT 1 + (A ™ - l ) ^  < 2c  +  1  3  0  f  r  i r i  2+  Jr, a  (i^)  (D.3.23)  (X^-l)p ^u  L  where c\ is a positive constant. To show the above r-integral is zero in the limit A —> oo, we choose a point r < e(A) < r\ satisfying 0  p  c/+1  (e(A)) = ( l n A ) - / , 1  (D.3.24)  2  and spilt the integral into two parts according to e(A).  Dropping constant factors,  (D.3.23) becomes  dr r  a  2 +  f  -  l)pU+  p  u  +  l  (e-r ) 0  +  1  u  l  (ri - c) 1 + (lnA) / ' 1  2  dr  1  + Je  •„ 2 + (Af" " - l)p + 1  < By the continuity of  r  dr  e  2 + (A  7+1 3  -  (D.3.25)  ,  lim e — r = 0. 0  A->oo  (D.3.26)  Hence lim A—too  Q.E.D.  (l+p )' 2  = 0.  (D.3.27)  Appendix E  (3 functions of the first order diagrams  E.l  Tools for computing the (3 functions  We list the following identities which will be handy in the computation of the integrals pertaining to the (3 coefficients. 1. If g is a function of p and A is a four-vector then 2  J dpgp^ J dpg{A.pY  =  ^ „ j  dpgp ,  (E.l.l)  2  = (A ) J dpgp\. 2  2  2. trYYF^ L  l  I  I  I  I  1  =  £il/i2A»3M4  -4F ^ I  1  W  ,  /il/ilA»3M3  (E.1.2) J  /il/^2MlM2 ~  2  Ml£*3M3/il J )  where repeated indices are summed over.  3. /  = V,  299  (E.1.3)  Appendix E. (3 functions of the first order diagrams  4. Let R{p) =  m)' ,  300  then  1  -Pad R(p)  (E.1.4)  = R(P)P R(P)I  Pa  We would like to establish a couple of handy lemmas for computing marginal diagrams. For a first order marginal diagram, the coefficient of its zero order localization is usually of the form c J T (q)K (q) s  (E.1.5)  s  G  where T (q)  is a function of the the running slicing functions  s  K (q)  {/°'}t> and {X\} , s  t>s  and  is a rational function of degree —4 in q. After changing to spherical coordinates  s  and letting r = q , the relevant part of the integral breaks up into a sum of terms each 2  of the form I  K  =  c kuI s  Is  =  [ -T(X ,X ,p ,p ). Js s r  p  G  p  s  s+l  s  (E.1.6)  s+l  p  where S * is the support of the neighbourly slicing p defined in Appendix B, and &n is s  p  (  the value of an integral involving only the angular spherical coordinates.  (E.1.6) in the case that  We would like to extract dominant terms up to o(A|) from AAf = Xi  +1  - Af is o(A|). Let us write T(A , A S  s + 1  , p, s  )  = u(X , ,p )  s+1 P  s  s  + p (X , X \p ,  s+1  s  P  s  s+  p )  s  s+1  (E.1.7)  where u(X ,p ,p ) s  and p  s  s  = T(A ,A ,p*,p  s+1  is the corresponding error. p  s  s  s  s+1  )  (E.1.8)  can be expressed as the following interpolation  integral. p (X , X \p ,p ) s  s  s+  s  s+1  = ^(Ar  1  - AJ) PKA , X \p , ) S  s+  s  s+1  P  (E.1.9)  Appendix E. (3 functions of the first order diagrams  301  where pt(X ,X \p ,p ) s  s+  s  =[  s+1  dtj^T(X ,X s  + tAX ,p ,p° ).  s  s  s  +1  (E.1.10)  Thus, correspondingly, J » can be split into two parts p  I =[ pa  ±  Js  pS  x ,p\p s  r  u {  s + l  ) + Y:&Xl  f  ^  -pl{X\X \p\p ). s+  (E.1.11)  s+l  Js s r p  To get an estimate on pf, we consider the following lemma. From (C.2.31a-c), we may suppose that the running slicing function Vp, V% have the following form. Let  / ( X S  '  X S + 1 )  =  I +^(^-I)+VH(^  + 1  ) -I)" 2  ( E  '  L 1 2 )  where {x } is a sequence of numbers. We further subscripts x by i to denote a member s  {xf} of a set of sequences indexed by i. Let X R be the characteristic function with support S  as the right overlapping region of p . The following lemma estimates the resulting error s  of T from replacing f(x ,x ) s  Lemma E . l  s+1  by  f(x ,x ). s  s  1. Suppose that xf > 0 and A x f = o(xf).  —fap—-  X E  Then for 0 < t < 1  °IWFJ'  (E113)  2. Suppose that E = Hi/(xf, xf + t A x f ) . Then s  £o(lliV  ^faf r  = x> - I V  (E.1.14)  X,  Proof: It is clear that Part 2 follows from Part 1. As for Part 1, dropping the subscript i from Xi,  5f 5x  s+l  2p p x s  [1 + p {x s  s  s+1  s+l  - 1) + p {{x ) s+l  s+l  2  - !)]<  (E.l.15)  302  Appendix E. (3 functions of the hrst order diagrams  Since p < 1, the [• • •] term in the denominator of (E.1.15) can be bounded below by p x s  s  or p {x ) . s+1  s+1  Thus, since Ax  2  s  = o(x )  s  s  From the interpolation form (E.1.10) of pi, the assumption that Axf = o(x|), and Lemma E . l , it is not too hard to see that the first term of ( E . l . l l ) dominates the second term of ( E . l . l l ) .  Thus let us focus on the first integral of ( E . l . l l ) and denote it as  I . Next we split the integral Ipa into three parts respectively corresponding to the left pS  overlapping region S  = ((1 - e)M ,  S  where p  s  M ) , the sharp region S* = [M , (1 -  2s  L  2 s  = 1, and the right overlapping region S  S  ps _|_ ps+i _ -p L  = ((1 — e)M ,  tu  denote p restricted to SI, S* and S  g  2s+2  M )  2s+2  R  s  e  e)M ]  2s  h  S  h  where  2s+2  by respectively p p , and s  R  s  L  sh  p . Splitting Ip* as described above, we get s  R  dr,  -- [ Js  IpS  -r(x ,x ,pi,o) s  s  r  s L  + [ —T(X , X , 1,0)+ [ -T(X , Js\ r Js% r sh K By rescaling r —>• M~ r in the first integral, then since S  s  S  X ,p , s  s  R  (1 - p )).  (E.1.16)  s  R  2  / -f{Pi)=f -m-M, Js  s L  r  Js  s R  (E.1.17)  r  (E.1.16) = / —T(X , X , 1,0)+ / - (T(X , X , p , (1 - p )) + T ( A , X , (1 - p ), 0)). Js° r Js' r S  s  S  s  s  s  R  sh  S  s  s  R  R  R  (E.l.18) Equivalently, by rescaling r -> M r in the third integral, then p (M r) 2  s  2  R  becomes (1 — p {r)) s  L  and (£.1.16)= / -T(X ,X ,1,0)+ Js° r S  h  S  [ -(T(X ,X ,p ,0) J' r s  s  s  L  +  T(X ,X ,(l-pl),p )). s  s  s  L  S L  (E.l.19)  Appendix E. f3 functions of the hist order diagrams  303  The term T(X , X , p , (1 - p )) + T ( A , A , ( 1 - p ),0) can actually be simplified s  s  s  s  R  S  S  R  R  further. Since in (E.l.19), the argument A  s + 1  of T is replaced by X , in studying I s , we s  p  may further suppose that the running slicing functions V  are of the form  s  V  where {x }  ^  x  )  l + ^(x.^-l)+p^((^) -l)'  =  2  ( E  '  L 2 0 )  is a sequence of positive numbers. This form allows further simplifications  s  when restricted to each of the three regions Si, S* and S . h  SI, S* and S h  sh  s  Clearly, Vg =  by respectively V[ V , and V . s  R  Let us denote V restricted to  R  R  h  When restricting  p h/ s  xS  s  to Si,  and when restricting to  n  =  1 + p (x° R  S, R  - 1) + (1 - p )((x°) s  - 1)  2  R  x>{p> + (I-  =  For / i ( r ) a function with support 5 £ , let fi(r) S  Let  f (M r)  X = X(S ),  (E.1.23)  s  X^  =  x(^),  (E.l.22)  with support £ £ .  be a set of sequences indexed by i, V\ EE V (xi), support of p ,  •  2  R  =  R  2  S  R  s  = ./^(Af ~ r) where the support of  / ( r ) is S . Similarly for /#(r) with support S , let /i?(r) L  p )x°)  R  s  R  and  S  S  L  L  x ^ ^ x ^ t / ^ ) ,  where x(S) is the characteristic function with support S.  Lemma E.2 Suppose that xf > 0 and A x f = o(x\).  1. XR  (E.l.24)  x\ +  xi  HR  vt + V\f  (E.l.25)  x\ X ^ =  VU  +  +  X|A^i  Q  /M _  ( R 1 26)  Appendix E. (3 functions of the hrst order diagrams  304  2. n  dr Li=l  i=l  s  ->  AT  i=l «  j  x  where the' index j in Yj Ax? runs over the set of distinct  in  Xi  i  x  /  n  i  \i=l i . (E.l.27) x  ri?=i Vf  and e is the  overlap parameter.  Proof: Part 1: Since the two forms in (E.l.24) and (E.l.25) are symmetrical, it suffices to prove one. Let us consider (E.l.24) and drop the label s and i in p = PL and p  L  p , and xf. Let us denote s  = (1 — p). From (E.l.21) and (E.l.22), when restricted to S  R  x(px + p )  1 4- p(x - 1)  x  1  ±  p - + px 1  x(px + p -) 1  (E.1.28)  X  As for (E.l.26), from the form of V[ in (E.l.21), it is easy to check that  (l + p i ( x + 1  a + 1  -l))(l + Pi (x -l)) + 1  ^  8  s  W  W '  v  ' '  Thus from (E.1.28) and (E.l.29), V+V?  1  =  V  s  +V  s  L  sh  +V  +V  s +l  R  L  X Ax S  s  R  L  s f t  X  X  s  +  P£ + x  x  Part 2: From (E.l.26) of Part 1, (E.1.29), and V (x) s  n^+pffi-n^1)  i=l  i=l  ^ ( l  Vx'  s  O (A ) • \x  (E.1.30)  = p O (l/x ), s  s  Appendix  E. (3 functions of the first order diagrams  n  LL  2=1  r  i=l  j  s  A s  x  -j  j  /  n  i N  n ^i + x t E ^ o f n i  s  i=l  / n  npe'j+xis-^ofni  xf  n  As  n  xk i=l  305  r  (E.1.31)  X  i=l  j ^0  \i=l  i/  x  Now by scaling back, -M  7 7**  r  2  rt — = 21nM, r r  < J(i-e)  r  Hence we are left to show the first term of RHS of (E.1.31) is zero when integrated with 1/r.  /  dr r  n ^ L - n ^ L  .i=i  (E.1.32)  i=i  The second integral is identical to the first when scaling back by r —>• rM .  Thus the  2  difference of the two is zero. Q.E.D. Suppose now A*>0,  7 >0,  AA*=o(A*),  s  A  s 7  = o( ).  (E.1.33)  s  7  Let us apply Lemma E.2 to marginal terms of the form  I  G  =  c J dqr (q )K (q ) s  G  2  2  G  =  cj G  T (r)^  (E.1.34)  s  where (E.1.35)  K (q ) is a rotational function in q of degree —4, V% = V (j), Vp V (A ), 2  s  G  V {x) s  s  2  = f{x ,x ) s  s+1  where / is defined in (E.l.12).  and  Appendix E. functions of the hrst order diagrams  306  Lemma E.3 1 \ ™ / 1\ s  -  IG  c 2 ln M  „ 1 + eE  n p  —  G  [AY AX (—4-+  lG  (E.l.36)  where Ej = 0(1), £ is the overlap parameter, and = denotes equality up to irrelevant G  terms (Q(M~ ) as  terms with a > 0).  Proof of Lemma E.3: Following the discussion from (E.l.5) to ( E . l . l l ) and applying Lemma E . l ,  dr ' . . r s  [AY  AX \ S  2  s  (E.1.37)  where  Y = (V + V +l) S  s  B  nB  B  (f  s F  + 73^)  n F  - {V' J;) (ni )"', +  nB  1  B  V (i) and V (X ) are of the forms in (E.l.20), and 8  S  B  F  2  Applying Part 2 of Lemma E.2 to / dr u (r)y, we have (E.l.36). s  Q.E.D.  (E.1.38)  Appendix E. j3 functions of the first order diagrams  E.2  307  The P functions of the first order diagrams  We compute the coefficients of the P functions pertaining to one loop order graphs of the L R C on QEDi  in the Landau gauge. The neighborly slicing {p} used is the one defined  in Appendix B. Let  . biVl  . =  L. < V > -^-z ,  b V  . =  L <VP> —?—  T r  , bV  v  T r  2  2  L < ESE > = -^—= ,  OM  3  3  . L <ESE> b Vi = —— V4  ,  4  b V 7  7  M  j  ,  V  0~M  L  < 4P >  Vr  =  . . (E.2.1b)  ,  O  OM  5  n  OM  V3  b V5 =  _ „ . (E.2.1a)  V2  b  6  ,  T/ = 6  (E.2.1c)  UM  ,  (E.2.1d)  OM  where 8M = -A^ — 1 and = denotes equality up to irrelevant terms. In our calculation, 2  we discard contributions from the gauge variant local parts that are zero when summed over scales. Here Vp(p) and V {p) respectively denote the slicing functions for the Fermi and phos  B  ton covariances with momentum p. In calculating the P coefficients, we often encounter integrals which are rotational invariant. For these integrals, we would make the following change of coordinates: q —>• spherical coordinates; in the radial integral, r  2  —> r. For  example, / dqf(q ) 2  = JdtoJ  r drf(r ) 3  2  = y / r drf(r)  (E.2.2)  where k  n  = j dtl.  (E.2.3)  In the following, we compute and determine the signs of the coefficients bi up to irrelevant terms.  Here we assume the imposed conditions (5.4.12), (5.4.20e), so that  the running covariance satisfies the bounds in (5.4.13a) and (5.4.22). In addition to the  Appendix E. (3 functions of the first order diagrams  308  stated imposed conditions, we also require that, for 1 < i < 5,  = o(A|)  AA? and  A| = O (  s 0  (E.2.4)  ) where a > 0. The last condition is needed so that the Fermi running  slicing Vp (defined in (C.2.1) with j = 0) can be estimated by  B  1  =  1 - p> - p°+i + p°\s2 +  p (X ) S+1  S +1  ^ ' ' '  2  E  2  2  5  )  with an relevant error —mVp' R, where R is the Fermi propagator (m + #0 , _1  s  Vf  s  = (BT  [(K -  2  (P f + A ^ ( A f - A^ ) s  +1  1  p p  +1  s  s+1  ].  (E.2.6)  (see (C.2.31b) of Appendix C and the discussion after 7.3.9 of Section 7.3 ) . As mentioned in the discussion starting at (7.3.18) of Section 7.3, except for V local terms, this error 4  contributes only irrelevant terms. Also, for the case of V local part of the one loop 4  ESE,  the relevant part is obtained by replacing the running Fermi line by  H' (q) = V R(q) + Vp R (q) s  +iF  where R(q)  (E.2.7)  2  F  = (m + 4)~ , Vp and Vp'  are defined as in the above. We also estimate V%  s  l  by  1 -p -p s  with an error of 0 ( S M X  S  /(Y) ) 2  5  s+1  P +p Y +p A  2 s") S + L  (Y  + 1  )  2  ( see (C.1.10a) of Appendix C ).  ' '  Appendix E. (3 functions of the hrst order diagrams  E.2.1  309  Vertex diagram  Figure E.20: The vertex diagram  J^hy  <V>=  J P ^ (p)K(P,r)A< (r)^< (-p-r) d  d  s  <S  (E.2.9)  s  where - q)YR(p - q + r)Y,  K' (p,r) = J dqT (p,q,r) ^M-fR(p s  0  (E.2.10)  and T (p,q,r) = s  (X[) P {p-q)V {p-q 3  a  + r)V (q)  a  F  + E(*)^ A  (E.2.11)  B  F  2  [P (p -q)V (p-q  + r) V {q)  s  F  F  B  t>s  + (VF(P - <?) V iP -Q + r) V%{q) + V (p - q) V (p - q + r ) ) V (q) s  F  + E(M)  [P ( -q)V (p-q  3  P  F  t>s q  F  B  r)V M t  + r) V {q) + V {p - q) V (p - q + r)) V (q)] .  + (V (p - )V (p-q F  +  t  F  s  F  s  F  B  F  F  B  Zero order localization We show the relevant contribution of L  Vl  A^(0,0) =  Tf%,0,0) Jdq-± + q 2  < V > is zero.  iR{-q)rR{-q)i  (YR(-qh°R(-qhn  (E.2.12)  Appendix E. fd functions of the first order diagrams  310  Let us first separately consider the two terms in [• • •] of the above. Using the identities (E.1.3), Y{<i + m)Y(4 + m)Y  (E.2.13)  (q + m ) 2  7  2  2  [4l°4 + m[j, 7 ] + m j ) 7^  M  2  a  (q + m ) 2  2  a  2  Y (q j - 2q 4 - 2mq + m 7 ) 7^ 2  a  2  a  CT  a  (q + m ) 2  2  2(q + m )Y 2  2  -Aq {4-2m)  2  a  (q + m ) 2  2  2  and + m)j (4 + m) a  iR{-q)YR{-q)4  (E.2.14)  (q + m ) q 2  2  2  2  ,2 U<7  -mq [r,4}  + ™ 4Y4 (q + m ) q  f-flf  2  2  _  2  2  2  7 (<7 + m )q + 2mq q — 2m q 4 <J  2  2  2  2  2  a  a  (q + m?) q 2  2  2  From above, by scaling and dropping irrelevant terms, we have 7^(0,0)  = =  Jdqr (M q,0,0)^ s  J d  s  "2  CT  7  AqJ  7  (q )  q  r \ M % o , o ) - ^ 7  q  From (E.l.l), it easy to see that K {0,0) = 0. Thus b i = 0. s  a  2  2  4q„4  0  (E.2.15) 2  Appendix E. j5 functions of the hrst order diagrams  E.2.2  311  Electron self energy diagram  Figure E.21: The electron self energy diagram  < ESE  I dp 4> (p) K (p)  >=  <s  ^ (-p)  s  (E.2.16)  <s  where  K'ESE(P) = jdq  Y^l"  [f (Q,P S  - Q) R(P ~ O) ~ mT (q,p- q) R (p - q)] Y 2  E  (E.2.17) and T'{q,p-q) =  ( A J ) n ( p - « ) n ( « ) + E W ) fa(P " Q) V {q) 2  J  B  + V (p - q) V {q)) s  F  B  t>s  (E.2.18) T%(q,p-q)  (E.2.19)  = .T'( ,p-g)|^ ,  where V , Vp and Vf  g  S  s  B  ?  are defined in (E.2.8), (E.2.5) and (E.2.6) respectively.  Zero order localization Let us write the zero order localization as a sum of two terms corresponding to Vp and Vp'  s  in (E.2.17). K {0) ESE  = K'{0) + K' (0)  (E.2.20)  E  where YT (q,-q)R(-q)rs  >(q,-q)R(-c q  1  (E.2.21)  Appendix E. (3 functions of the first order diagrams  312  ^ (q,-q)R (~ 2  —m f dq —J 1+  E  YT {q,-q)R\-q)Ys  E  (E.2.22)  Let us compute the terms K (0) and K (0) separately s  E  dq T (q,-q) s  K (0) s  =  J  (1 + q )(q + m ) 2  2  2  dq T (q, -q) s  •  3(4  (l + q )(q + m ) 2  J  2  2  (E.2.23)  - m).  By using the fact that an integral with an odd integrand in a rotational invariant domain vanishes, T (q,-q) s  (£.2.23)  =  -3m  kY 2  —3m kn f dr  where T (r) = T (q, — q)\ 2 s  and kn is defined in (E.2.3).  s  q  (E.2.24)  =r  We now compute K (0). E  . «  r , -m —m T J (q,-q) j dq +q s  (0) = =  =  E  7  2  —m  j  T%(q,-q) ( l + o )(o + m ) 2  2  2  2  r%(q,-q)  —m  = —m  dq  (1 + q )(q 2  j to  2  n  Y(4  + )  Y(-Q  2  +  m)  +  m)  2  2  r (q,-q) s  E  (1 + q )(q  3mk  2 M  2  2  3(q 2  2  2  -  +  ^  J  2m4 +  m) 4' 2  -q — 2  m )Y 2  2m4 + m )4 2  2m4 — m ), 2  f dr  (E.2.25)  From (E.2.24) and (E.2.25),  K'ESEM =  / y TO + T ^ ) ] .  (E.2.26)  Appendix E. (5 functions of the first order diagrams  313  Let us simplify the term T (r) + T (r). From (E.2.18) and (E.2.19), T (r) + T%(r) is s  s  s  E  linear in V  S  in  + V ' . From (E.2.5) and (E.2.6), it is easy to see that V S  F  + V ' is linear  S  F  S  F  F  A4.  Let us extract leading terms by making the substitution X  —> X in X ( r ) + T (r).  s+1  For a term K(X , X ) S  which is a function of A and X ,  s+1  s  s  s  E  let us denote K(X , X ) by K^.  s+1  S  s  By denoting V  = V'  + T F  +  F  V$' , A  we have ?U  =  (T (r) + T ^ ( r ) ) |  =  (AD  s  2  A 5 + 1  ^  A 5  [(n,F,->+n A ^){v B^+n i^) +  +  s  L  To determine Tt>, let us first simplify V  s +F  - n ,h^n i^\ • (E.2.27) +  +  by gathering terms with a A4 factor into one  group and the remaining terms into a second group. From (7.3.13) and (7.3.15), n  +  v?  = .{ y  1  — V  s  +  ^W  f l :  ^  « W W  +  +  - M " ) |  1  V  s  -4-  (E.2.28)  where  VLF  = K(v f  +K  s  F  s  0,F  R  — —  V  s  r  2 p  s  s  s + 1  A^  + 1  (E.2.29)  F  B - p X -2 s  V  v  +1  p {X )  s  s+1  2  s +1  2  2  B  1  F  l-{p  + p )B  s  p {X )  s+1  s+l  s +1  2  2  (E.2.30)  s  Let us determine the expressions of VQ _> and F  _+.  F  Using the fact p — (1 — p ), s  s +1  R  L  p£(i-p£)-pr(i-pi )(Ar) 1  0,F  V{  P£(I-PI) [(1 - pi) + pm)  +1  1 2  Pi 2  + 1  [(1 _ p f ) - ^ + 1  3+U  (i-pr)  x Y  s +1 P L  s + 2  (E.2.31)  Appendix E. (3 functions of the hrst order diagrams  314  Hence V  A' - A \  s  (E.2.32)  +  L  L  M),F,-  where  p£(i-pi)  A",  (E.2.33)  KI - P D + p i A i r  For the term V% , 4 F  s \2 (V F)  2p  Vi  s + 1  \  s  + l 2  + AA* v%  W  s  s+l\s+l 2 P \ + i  L  s  (E.2.34)  2  B  s  Hence, V  s  9 B  \  s  P L ^Y S  +  s  +1  PL  +  + p A|  (l-p£)  PR  EJA _|_  +K+  A§  A§[^ + ( 1 - ^ ) A 5 ] 2 / c+1 _j_ Xs/t _|_ X i ?  L  Pi  A§  ; i - P i ) + p Ai L  Xsh+R  A§;  VPfl + ^  Pl  PL  (A!) +  .(I-PD +  2  We show that the term A — A S  V F,^, +!  = =  PLAS;  -  \  ^  1-PR (l-p)\)A|],  5 ,  2  i  + 1  V ( 1 - P L )  +  PL  + 1  A|  contributes zero in the localization. From (E.2.27),  s +l  L  T.  2  B» Pl  since Y_> is linear in  a  L  let us consider the expression (Ai) \{A'  - A  2  " (A i) s  2  L  A V ^ S  S  L  B  + 1 L  + A ^)(V ^ S  B  A V % s +l L  s  B  + V l^) s + B  -  i s +s i+1 - T js s + i  A V +l L  B  (E.2.36)  For a marginal integrand, the above term is irrelevant, i.e.,  (E.2.37)  s  Appendix E. j3 functions of the hrst order diagrams  315  Hence, from (E.2.27), (E.2.32), and (E.2.35) dr s  (E.2.38)  A?  r  s  After extracting the leading term, we write the contributions from the remaining terms as e  £1=1,2,4,7  0(AAj/Aj) where A = 7  7  and e is the overlap parameter. Hence it  is easy to work out that s\2 (A?)  '  1+ e  AA*  Af  (\s  s  1+  5  M  O  +  AA? AA2 r~~~ + \  \  S  2  A +  7  n/"  S  —0(i)|  7  4  A  S  A  (E.2.39)  -7  where the 1 + 5 O ((7 ) *) factor comes from replacing 1 + j by 7 in V . s  s  s  B  M  First order localization  K[(p) =  L,„rP«(d „R(p-q)\ »)r  j d q ^ f ^ -  p=  v  T M) S  'Po  (E.2.40)  0.  We get the last line by a simple inspection that the integral of the second last line is identical to (E.2.12). Thus it is irrelevant. We conclude that  b  2  =  0  -3mk \nM\  s  b  n  4  (2TT) 8M 4  i +  so M  4  s\2 (X[)  (Ai)  2  1  H  AA? r~ Af  AA?  AAi  A~ —T~~ ~\~ —T~~ ~\~  A7 ryS  S  O(i) (E.2.41)  Appendix E. (3 functions of the first order diagrams  E.2.3  316  Vacuum polarization diagram  q  p+q  Figure E.22: The vacuum polarization diagram  When we compute the p? contribution of the V P diagram, we need to multiply the contribution by a minus sign because of the anti-commutativity of the Fermi fields. That is, to get the V P diagram, we need to move a tp through an odd number of Fermi fields for the Gaussian integration.  <VP>  H  KM  = - ^ ~ J  dpA< (p)K^(p)A?(- ), s  P  = J dqT (q,p)tr[rR(p s  + q)YR(q)}-  (E.2.42)  where  T (q,p) = (X{) PMP + 9)n(g) + E W ) s  2  2  n(P  + Q) PM+VUP  + Q) PM-  (E.2.43)  t>s  From the formal Ward Identities, it seems that the gauge variant part of VP should add up to zero. But because in the R C scheme, the running couplings are not constant parameters and running slicing terms do not sum to unity, we cannot discard their contribution to the coupling flow. However, suppose that for i = l or 2, A? = Ai + o(l)  (E.2.44)  where Aj are constants. Then for the gauge variant local parts of VP and 4P, we can at least remove the leading terms obtained from replacing (Af, A|) by ( A A ) . 1;  2  Appendix E. fd functions of the first order diagram's  Recall that VP  317  has an A local part whose coefficient is O ( l ) with respect to the  U+1  2  loop regularization parameter A —> oo. However, it is Q>{M ) with respect to U —> oo 2U  since it equals the contribution from the remaining slices. Let us determine the exact relevant contribution of this leftover term. Let  *  ™  WW) where {p , p } u  u+l  =  i-^C^(Af)  =  1- p  (  K  2  -  4  5  )  + 'pP(\%) + p ^ + H A f ) ' U  - p+  u  u  l  2  is a two neighbourly slices decomposition of unity (recall that  \f  +1  Af for i / 3).  Proposition E.l LL ^  = (-pf + 0(A )) V  U  V5  (E.2.46)  - 1  2  5  where /?f f (r) u  =  J t (r)dr  (E.2.47)  =  ( A f ) [(V )  u  2  T  F  2  + 2V V U  ]  U +1  F  F  (E.2.48)  Proof: Recall that we argued that the A local part of L f ( p 2  that L {p +1  2  u+1  + 1  c / + 1  ) is O ( l ) by using the fact  = 1) = 0 from Ward Identities and L \p ) U + 2  U+l  - L \l) u + 2  = 0(1).  (E.2.49)  Since the terms involving the j ^ 0 spinor lines are 0 ( A ) , it suffices to consider the _ 1  terms of the LHS of (E.2.49) that involve only Fermi lines. Furthermore, in considering  Appendix E. f3 functions of the first order diagrams  318  only the relevant part, we may replace the Fermi running slicings by the slicing defined in (E.2.45). Thus to show (E.2.47), it suffices to show that  1 (Af)  Since {p , p } u  2  =  {v y + 2v v  =  (P +v  u  u  F  u  +  u +l  F  F  (v  )  u +l F  2  y.  (E.2.50)  u +x  F  F  is a two neighbourly slices decomposition of unity, we only need to verify  u+l  the above on the overlapping region. In the overlapping region, let us denote p = p  c / + 1  ,  p = p , and A = Af. From (E.2.45), since p + p = 1, u  2  V  +V  U  u  F  +l F  A (p + pA ) 2  2  + p+  P pA  (E.2.51)  A '  2  2  Q.E.D. Zero order localization Using the identities in (E.l.l-E.1.2), we have j dqr-(q,0)tr[r  R(l) l" R(q)  (q + m ) T (g,0) 2  2  2  + m) f  (-4  t r  (-4 + m) }  s  (q  2  Zq^qu - 5^ (q + m ) 2  +m) 2  T (g,0) (q + m ) s  -45,  2  -5^ 2k  2  v  2  2  9  9 ^ 9 2  q - -q + m  l  JdrT (r), s  n  where T  s  =  {\{) (V ) 2  2  F  Let us also denote (E.2.52) by T ( A , X ) S  A  s + 1  S  s+1  +  2{\ ) (V V + ). S +1 1  2  S  1  F  F  L  (E.2.52)  so that T (x,y) denotes (E.2.52) with A and  respectively being replaced by x and y.  s  s  Appendix E. /3 functions of the hrst order diagrams  319  Assuming (E.2.44), let us extract the leading terms of T (X , X ). S  S  Let Y (p)  s+l  s  =  T ( ( A i , A ) , (Ai, A )). Let us write s  2  2  T (X ,X ) S  S  =  S+1  E.  T (p)+El  +  s  r  T (X ,X ) S  S  E  (E.2.53)  s bJ  xl(Af)  -T (A ,A ) =  S+1  S  S  S  AA|  2  +  A A |  Af  A  O(l)  2  (E.2.54) T (A*,A )-T (p) = S  S  s  (Af)  s x  A?  2  L(AI)  O(i).  A5.  2  Let us show by setting (Af, Af) = ( A i , A ) , the leading term A | ( A ) = / drT (p) of s  2  A | ( A ) = /drT (r) S  s  sums to (E.2.47). Using similar notation for denoting replacement  of arguments in T , let us denote Vp(A , A ) by V . Also, as from (E.1.21)-(E.1.22), let s  s  2  V  s  2  = P sh/^2. S  sh  Vi  =  Pl  (E.2.55)  ( l - p i ) + p!A ' 2  V rR s  = —  PR  A (p%A 2  + (1 - p )  2  Ay  R  2  Let us sum the T (p)'s from s = 0 to s = U. s  Proposition E.2 By setting (Af, Af) = ( A i , A ) , 2  f ; r »  = A [{V f 2  +  u  F  2V Vp !  (E.2.56)  T+1  F  s=0  where Vp andV  are defined in (E.2.45).  +1 F  Proof: E  T  s=0  » = ( i) A  2  E T O  + TO + ( n ) + 2 ( n n 2  2  s=0  2  + 1  )  Since V\ — 0, rearranging the sum, we get u-i  J2r°( ) = ( A 0  s=0  P  2  x:(n)  2  +(n + ^ ) + 1  (E.2.57)  s  s=0  +(Ai)  2  [Vl?  +  [V f + 2 u  R  (V V ') U  U +  R  L  (E.2.58)  Appendix E. [3 functions of the hrst order diagrams  320  From (E.2.55) and a similar identity as (E.2.51), it is easy to see that  V  r  +  V  (E.2.59)  " V  l  Hence by setting Af = Aj with i = 1, 2,  u  77-1  A? [ ( ^ ) + 2 ( 2  53 Xs/i+R + X s / l s=0  =  Aj [(V f  + 2V V  v  T  F  F  '  Q.E.D.  .  T+1 F  Thus from Proposition E . l and Proposition E.2, the leading contribution Z ) f  =0  A f (A)  of £!L A f (A ) exactly cancels out the "leftover" relevant part (E.2.47) of the coefficient s  0  of the V local of VP .  By discarding the leading term A f (A), from (E.2.53), it is easy  U+1  5  to see that  b = (At) 5  (Al)  2  5  o  +o  'Af - A x ' ,  Ax  'Af, - A  s\2  (AH  2  A?  (Al)  2  AAJ L  AA* '  Af  Ai  O(l).  (E.2.60)  Second order localization The kernel of the projection of the V P diagram onto V and (d • A)  has the form  2  3  =  K%M  j  ^ T  S  tr  M )  7" \{PaP dqadq R{q)) 7" R(q). a  a  (E.2.61)  where T is computed in (E.2.52). Let s  K^(P)=AA'T », where  and  (E.2.62)  + B'L ,)  I  IU  are defined in (1.2.8). Using the trace of K  (p)  s  A  =  B  =  s  s  Because (d • A)  2  because (d • A)  2  m  2  pKp  (E.2.62), we have (E.2.63)  p* K^, (p) 2  3p  2  A  s  (E.2.64)  3  is inert to other graphs, we will not calculate A in detail. However, s  is gauge variant, by using the same argument as in the removal of the  Appendix E. (5 functions of the first order diagrams  321  leading term in b , we may discard the leading term of A obtained by setting Af to Aj. s  5  It is easy to see the remaining relevant part is  A = s  5  M  s\2 (Af)  (^)  0  Since we will show that B  'A -Ag'  +0  2  AA?  +e  Af  1  AA? 2  +  AS  (E.2.65)  = 0(1)> (E.2.65) implies that the sign of B  s  O(l)  does not depend  s  on A8. We now compute the coefficient B . From (E.2.61), s  1  J dq T (q, 0) tr [ " -(p„p dq dq R( )) 7" /%)] • s  3p  3p  2  2  7  a  a  a  (E.2.66)  q  To compute the trace in (E.2.66), we first compute  7  M  \{PaPadq dq R{q)) 7" a  \paP dq dq ('fR{q)'f)  a  a  1  a  a  a n  ,-24-Am. q + m  1  1  (E.2.67)  2  2p  (q + m ) 2  2  2  (g + m ) 2  2  2  (g + m )  3  2  2  2  J'  Hence  4]/ (p • q) 8(p-q) W + 2m) (g + m ) (q + m )  2p  2  tr [• • •]  tr  2  2  2  2  (g + m ) 2  3  2  2  2  2  2  ~4 + m\  (g + m )  3  '8(p-(z)  4(p • g) - (g + 2m ) ( q 2  2  2  2  +  2  - 2p  2  q +m J 2  2 y  2  (E.2.68)  m  2  Using the identities (E.l.l) and above, we have  3p 4 f  2  .  T (g,0) s  (<7  2  +  m  2\3  ,  g + 2  2m (q + 2m ) 2  2  2  v  (g + m ) 2  2  '  (E.2.69)  Applying Lemma E.3, up to an error of the form (E.2.54), (E.2.69)  =  ifcnlnMJ^.  (E.2.70)  Appendix E. (3 functions of the first order diagrams  322  Hence (recall that we need an extra minus sign),  b b  5  =  3  =  £ +o  s\2  kn  (Af)  2(2TT) (A!) 4  'A - A x  b5  2  -Ak I n M (Af)  Af  'A -A^  O  2  (E.2.71a) A - A  2  n  2  6SM (2TT) (A§)2 4  2  (E.2.71b)  where the E's are of the form AA?  AA  s  Af  E.2.4  2  s  H  A  (E.2.72)  O(i).  2  Four photon legs diagram  Figure E.23: The four photon legs diagram  To compute the coefficient of the A  4  local term of < 4P > diagram, we use the same  argument as in the calculation of the (d • A)  2  term in the V P diagram. Using vestigial  gauge invariance, we can reduce the dominant term of the coefficient to 5M (Af) (A!)  4  4  O  Al  ~ + o (—  Af  AS  (E.2.73)  Appendix E. /3 functions of the hrst order diagrams  E.2.5  323  Summary of coefficients  Smooth slicing We summarize our calculations. Let cr  =  h  =  £=A  X2 s  Ji  -2k  =  -3m  3 (2TT) (M -  ln M  n  (2TT)  M  4  1)'  2  k  2  (E.2.75)  - 1  2(2TT)  4  M  2 _  y  1  Xah+fl-RVp.Vfe,  where K p,v and K V  2  InM  n 4  h  (E.2.74)  2  6  APtVl  &7 = ^ 2 ~ 7 J T  Xah+RK.AP,V , 7  respectively are the relevant part of the V% and V localized 7  kernels (excluding the running slicing functions) from the V P and the 4P diagrams. Ao ^ ,-,  bi  = 0,  b  3  = M^) (i +£  b  4  = 6  b  5  =  b  6  =  7  =  (E.2.76a)  ——E  b  ryS  ) - be f [(<7 ) -£ ] 3  2  s  b  3  Xt (a*)  2  2  b  h [(a ) - E ] + 2  (a ) E s  2  (E.2.76d)  2  b5  h[(a ) -Z }+(a ) E  h6  (E.2.76e)  + [o- YE  h  (E.2.76f)  s  b \(a ) s  (E.2.76b) (E.2.76c)  4  s  2  A  2  2  s  2  s  7  where, for the coefficients obtained from the Fermi loops, the errors E are of the form AAf1  2  Eb  2  = £  AAf  AAf,  Af  X, s  AA? 2 AS  Af  and for the coefficients b and b  +  O(l),  . (E.2.77)  4  A  S 7  OW +^  O  S  U ^ O ( ^ ) (E-2.78)  Appendix E. ft functions of the first order diagrams  6  AA? , A A 2 , \ X S  . l A  2  A  , A '  AA4  s  324  7  \  s  O/S  7  i  A  (E.2.79)  Sharp slicing In the case of using sharp slicing defined in (8.1.2), for the low order diagrams considered here, all lines must contract at scale s and there are no terms involving the overlap parameter e. Hence the coefficients for sharp slicing can be obtained by setting e = 0 in (E.2.76a-f).  b =0  0,  2  (E.2.80a) (E.2.80b) (E.2.80c) (E.2.80d) (E.2.80e)  b  b (a ) s  7  7  4  S  4  .  (E.2.80f)  Appendix F  T h e c o n t r i b u t i o n from e x p a n d i n g the slicing functions  In this appendix, by assuming the conditions (9.1.1-d) on the couplings, we show that there is no 0 ( 1 / 7 ) contribution to the (3 coefficients from expanding the slicing functions S  in the one loop diagrams with the set-up described in Section E.2 of Appendix E . In the < V >jjs and the < ESE >^s diagram, because the naive degree of divergence is E  zero, expanding a slicing function p (p — q) = p (M~ (p — q)) can only leads to irrelevant s  1  s  contribution. Hence, there are only two diagrams, the < VP  >^  s  and < ESE  >^  3  diagrams, that we need to examine for possible relevant contribution from expanding the slicing functions when localizing. Because of no Wick ordering, there are self-contracting lines in some Feynman diagrams. In particular, for the /? functions at scale s , we have to include one loop E S E and V P diagrams where one of the two lines is of scale t > s. For these diagrams, after localizing, the slicing functions are of the same momentum (same argument). Thus only the ones with t — s + 1 are non-zero since the slicing is neighbourly. We will show that, from expanding the slicing function, there is no relevant part from the V P diagram while there is only an 0 ( V ( 7 ) ) contribution from the E S E S  2  diagram. In the renormalization procedure of the tree expansion scheme, counterterms are obtained from localizing generalized vertices into local vertices. Because each generalized vertex is "sliced up" by an insertion of slicing functions into the covariance, there are  325  Appendix F. The contribution from expanding the slicing functions  326  p-q  Figure F.24: The electron self energy diagram additional "sliced up" local vertices arising from localizing the slicing functions. In general, the counter terms corresponding to these local vertices are irrelevant. We illustrate this statement with the example of the E S E diagram in QED^ under the ordinary tree expansion scheme. The kernel of the ESE diagram in Figure F.24 is  K(p, 5, t) = CGj dq p (p - q) p (q)  ^ ^ ^ M  s  F  B  where CQ is the usual combinatoric factor and p and p s  F  B  (F.l)  are the Fermi and Bose slicing  functions at scales s and t, respectively. The counterterms required (equivalently, the contributions to the A 4 and A flow equations) have kernels K(0, s,t) and 2  dq /4(tf)y  + g  (F.2)  2 7  + d ^p {-q)-q + m -$ + m_ s  —Q  +m  yP  F  according to the requirement of locality (i.e. that the counterterms be a polynomial in p).  We examine the second term (the term arising from the "expansion of the slicing  function "). When summed over all scales s and t, this term contributes to the original action a kinetic energy term with kernel  c Pa J dq p G  f  ^  ^  r  d^p?{-q)-±  <i + m  (F.3)  Appendix F. The contribution from expanding the slicing functions  327  Since the term d p (—q) restricts the support of the integrand to be in the scale U u  qa  F  shell, it is easy to see that the integral is bounded uniformly in U. From this example, we argue heuristically that there is no need to expand the slicing function in the set-up of the flow of the couplings under the ordinary tree expansion scheme since these local factors are summed into irrelevant counter terms. However, the same argument does not apply in the L R C scheme because the sum of the running slicings do not add up to a cutoff function. Thus we need to examine the possible relevant contributions to the flow coming from expanding the slicing functions. We now describe the relevant V local part of the E S E diagrams in the L R C scheme. 2  The running slicing Vp and V  are the one defined in (E.2.5) and (E.2.8) respectively.  B  From (E.2.17) of Appendix E , the kernel of the < ESE >jj  s  K  = [dq ^M^ {q,pJ l + <r  q) R(p ~ q)Y  s  ESE  diagram is (F.4)  where T (q,p-q) s  + 'Z(>\)  = X{X[V {p-q)V (q) F  2  B  (V {p - q) V {q) + V {p - q) V (q)) s  s  F  B  F  B  t>s  (F.5) From (F.4), by expanding V (p - q) at p = 0, extracting the first derivative term, we F  obtain the following relevant local term (we replace denominators m + q , 1 + q by q 2  2  2  and the resulting error is only irrelevant) K>,  =  -  Jdqf°(  =  - J  dqf (q ) s  2  2 q  )2p.qY^Y^  2p-q q  q  2  (F.6)  [q f  2  2  where f (q ) = (X{) (V )'(q ) V (q ) + s  2  2  2  F  s  2  B  (AJ+ ) 1  2  ((V )'(q ) V 2  F  s  (q ) + (V )'(q ) V (q ))  +1 B  2  +1  F  2  2  B  (F.7)  Appendix F. The contribution from expanding the slicing functions  and f'(q ) 2  Applying the 7 algebra, and using Euclidean  denotes the derivative df(q )/dq . 2  328  2  symmetry, we have l'  =-  K  / ^ V )  ^  (F-8)  J q  where C\ a positive constant. Applying the usual change of coordinates as described in Section E . l of Appendix E , we get A;, where S is the support of p (r), s  s  s  t  1  f  dr T (r)  (F.9)  s  is defined in (E.2.3).  2  = 0(1/(7 ) )- Since (p° )' = 0, from (F.7) S  =  s  -C2P  c = k^ci and  s  We show that / . dr f  =  (K)  2  2  ah  {(V JV  +  s  F  BtL  (V JV ) F  B:R  +(K ) {(nj n:i + (n% )"p , ) • +1  l  2  B R  We would like to write T as a sum of two terms y4-p(A ) and Bf (X , s  s  s  s  first term is obtained by having all the A  5 + 1  AX ) where the S  being replaced by A in T and the second s  s  term is the error as a result of the replacement. It is easy to see that the first term Af. {X ) = (x[) {[(v s  2  s FtR  + r j(v F  + v , )} + [(v jv , s  B t R  B  L  s  F  B  L  - (n, )'n,J}• L  (F.ll) where V is defined as in Section E . l of Appendix E . For convenience, let us denote s  A = 7  The second term Bf (X , s  s  S 7  = A* + A*.  (F.12)  A A ) can be written as the interpolation integral S  E i=l,2,7  where Ei  ^r-E , hP  i  (F.13)  (F.IO  Appendix F. The contribution from expanding the slicing functions  329  From (E.l.24) of Lemma E.2 in Appendix E ,  (n R+nj  = (h' = o-  t  (F.i5)  Thus ( F . l l ) = (X{) [{V )'V  -  2  FtL  BiL  (P yP \ FtL  BtL  .  (F.16)  By scaling the second term back to the S , it is easy to see that S  L  dr (A?) [(V )'V  J  - (P JV , ]  2  sm  FtL  BtL  F  = 0.  B L  (F.17)  We now consider Bf (X , A A ) . Since for i = 1,2,7 S  s  s  f ^ o ( i ) , we are left to show that (F.14) = Q)(l/j ).  Let us consider the partials derivatives in  s  (F.14). r)T  iw  x{  A ^  s  =  ^  2a;a;+1  +  ^i )'^ .*) 1  (R19)  = (AD ^(^^)'n^+(Ar ) ^(^n, 2  1  2  ) n-2 ,  f i  A^^rni )'^  (F.20)  ^^^(ni )'  (F.21)  1  +(Af ) + 1  2  1  +( *i ) A  +1  2  We apply the following estimates and formula of derivatives to show (F.14) = 0 ( V 7 ) S  V R) FM  = O (^)  .  * W ) = O (£)  •  (F.22)  Appendix F. The contribution from expanding the slicing functions  2. Let Vi = V  B  330  and V = V . 2  F  d  2p^Pl A|AJ +1  Af y+i  rR  d  <  IT  (^AJ + ^ (Af ) ) + 1  Af  +1  + 1  2  2  P£ P£ A? + 1  <9Af  +1  i l L  dVp;l <9p£  (l + ^  _  l+  -l))  + 1  *'  ,  + 1  (Af  + 1  (F.23) A  Af  " (Af + A A f )  2  rf, (AS -l)-^+ (AS -l) +1  +1  (1 + p f W  +1  1  +1  (F.24)  1  (1 + P l ( A ^ - 1 ) ) ' + 1  + 1  2  P^A| + (1 - ^ ( A f ) 1  2  - p [AS - ( A f ) ] 1  (p^AI + a - p ^ X A ^ ) ) 1  (Af ) 1  2  R  2  2  2  ( p ^ + (l-p| )(A^ ) ) ' 1  2  2  J  We demonstrate the calculation of one term and the rest can be handled in similar fashion. Considering the absolute value of the second term of (F.21), since the sign of g ^ ( r ) is uniform for r € S , from (F.23) s  <  1  1  r s+l\2 (Af 5  }  7  s  7  M (l-e)  s  1  1  2  M (l-e)  Y  2  (Af ) 1  YX  s+1  1 2  (Af )  1  2  M (l-e)  s  A!  dj  2  2  W)  U  1  f  '  r )  rM  2  dr(V J s  JM M (l( l - e )  F  2 2  V  s  r  PR=0  F,R PR = 1  i +O  i +O  /Ay  V 7  s  i +O  ' A Y  (F.25)  Appendix  F. The contribution  from expanding the slicing  functions  331  It is easy to see that the other terms have similar orders. We state the result of the above discussion in the following proposition.  Proposition F . l Let in < ESE  b iV<i p  be the contribution from expanding the slicing  function  >, s\2 P  A|  AAf  S  , AA^_ , A  +  Af  7  +  H(X ,X )  7  S  S+1  (F.26)  where  H(X ,X ) = Q(1). S  S+1  (F.27)  Next, we examine the contribution of the one loop V P diagrams from expanding the slicing functions.  s  s  s  s+1  s+1  Figure F.25: The one loop vacuum polarization diagrams  Proposition F . 2 The one loop VP diagrams of Figure F.25 have no relevant contribution from expanding the slicing  function.  Proof: We first look at the V P diagram with all s lines. The kernel has the form K (s,s) VP  = c  f T (q,p + q) t r ^ ^ p s  G  + q)^ R(q)\ u  dq.  Appendix F. The contribution from expanding the slicing functions  332  where r (q,p + q) = (Xt) V (p + q)V (q) + i:(^) s  2  s  F  2  F  V (p + q) V {q)+P (p + q) V {q) (F.28) s  F  F  F  F  t>s  Let  t V ) = (K) n(q) + (K 2  ) n \q) v (q).  +1  2  2  +  F  (F.29)  In the expanding the slicing factors, we get the relevant contribution c  P J [f (q )}' 2  G  s  2  tr[ R(q)^R( )} %  dq = c  q  p j [T (q )]' 2  G  s  2  (F.30)  Making the usual change of coordinates, (F.30) = c  M p 2s  G  2  f dr4- [t (r)l = 0, Js dr s  s  1  2  J  (F.31)  since V (r) is zero at both ends of 5 . s  F  S  Q.E.D From the above analysis, we conclude that for the one loop diagrams, there are no 0 ( 1 / 7 ) relevant contributions to the (3 functions from expanding the slicing functions. S  

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