I M A G I N A R Y C H A R G E Q U A N T U M E L E C T R O D Y N A M I C S A R U N N I N G C O U P L I N G A N A L Y S I S 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 R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S M A T H E M A T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A 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 0 cX -\ m% DE-6 (2/88) Abst rac t 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 QED model in perturbation theory. By integrating out a sequence of slice fields, the L R C scheme generates a flow of effective potentials Vs. Here we do not demand that the flow of Vs be gauge invariant but only that the Ward Identities hold at the end of the flow. From the flow of Vs, we obtain a flow of the couplings A s of the local parts of Vs. Using a fixed point analysis in a suitable Banach space whose norm captures the asymptotic form of A s , we determine the asymptotic behavior of Xs satisfying boundary conditions partially fixed by the Ward Identities. At each step of the flow, the slice covariance is transformed by shifting the local quadratic terms of Vs 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 Xs so that the asymptotic form of A s can be obtained from a 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 xi 1 Introduction 1 1.1 Feynman functional approach to Q F T 1 1.2 Overview of the thesis 4 2 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 3 The G N Tree Expansion and Renormalization 29 3.1 Decomposition of fields 29 ,3.2 Flow of effective potentials 31 iii 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 4 Decomposi t ion O f F i e l d V i a B lock -Sp in M e t h o d 53 4.1 Block-spin transformation 53 4.2 r n ' s bound in scalar theory 58 4.3 T n of lattice Fermion 62 4.4 Components of Tn 65 4.5 Cancellation of poles 69 4.6 Existence of zeros of fn(p) 72 4.7 Nonperturbative method (Lattice method) 80 5 Loop Regular ized R u n n i n g Covariance 86 5.1 Running covariance 86 5.2 RC with neighbourly slicing 92 5.3 Loop Regularization 94 5.4 Loop-regularized Running Covariance 97 6 Two-Sl ice 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 M u l t i - S l i c e 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 137 8 L R C coupl ing 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 2 " condition of the coupling flow 154 8.3 RC scheme for other models • 156 8.3.1 (f>4 model 156 8.3.2 The g equation of a model with only one interaction vertex . . . 159 9 L R C coupl ing flow w i t h smooth sl icing 162 9.1 Preliminaries 162 9.2 The primitive flow 166 9.3 Higher order flow 171 9.3.1 The order N B functions 171 9.3.2 The order N flow and boundary conditions 180 9.4 The flow of Es . . 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 and 237 10.4.2 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 flow 255 12.2 Conclusion 260 Bibliography 263 A Vestigial Gauge Invariance 265 vi B Neighbourly slicing 273 C L R C running covariance 275 C l Photon field 275 C. 2 Spinor fields 278 D Bounds on Spinor Loops 283 D. l 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 E 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 F The contribution from expanding the slicing functions 325 vii Lis t of Figures 2.1 An initial momentum labeling of G 27 2.2 G with loop momenta p\ and p2 28 3.3 A tree r with forks / i , f2 and / 3 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 (aLnZf 54 6.8 n-VPu+l 112 6.9 L%+1(N1: Nn) with n V^-^+i vertices and n > 4 112 6.10 VjV--jmU+l) 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 Lis t of Tables 10.1 Bounds on the Lipschitz constants 223 10.2 Bounds on 5jjP<(0) 224 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($(x)) = £0($(x)) - P($(x)) (1.1.1) where x = (x,t)EWLd, $ = ($!,•••,$*), £ 0 ( $ ) is quadratic in the $;'s (including derivatives) and -P($) is usually a polynomial in 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 = + (1.1.2) 1 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 ) = eitH^j{x,d)e-itH. (1.1.3) 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 $ i ( / ) = fdx$i(x)f{x) where we can take / G C~(IRd), the space of smooth functions with compact support or / e &(Kd), the Schwartz space of functions. 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) Wn{xu •••,xn)=< 0|T • • - < M O | 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 H0 + Hi cannot be realized on T where (1.1.4) may not be well defined. So instead, in constructing a QFT, 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 A = Wn A—>oo exists and determines a unique Q F T (usually, we only require the existence of a weak limit of Wn). Around 1949, Feynman discovered a remarkable formula expressing the Wightman functions Wn(xi, • • • ,xn) in terms of a functional integral (sometimes it is call a path integral) [Fey49]: ST[^il(xl)--^in(xn)eiS^^]v^ Wn(xu---,xn = L- . . * (1.1.5 JT\eLJ £(*(B))| £>$ where X>$ = fTj X£j{dd<&i(x). In 1965, Symanzik proposed to consider the above with 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 Sn(Xl, • • • ,Xn) = —T-fcmx))l~~ • ( L L 6 ) I T 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 elJc of (1.1.5) becomes an exponential decay factor e~ J c which makes the path integral easier to define and work with. If these Schwinger functions Sn satisfy the Osterwalder Schrader axioms [OS73][OS75], then by Chapter 1. Introduction 4 analytic continuation, one can recover the set of Wightman functions. By introducing an external source field J, all the Sn(xi, • • • ,xn) can be expressed in terms of a single generating functional z\j] = j e-J cm*» + *J *° V $ where $ J = E / $i(x)Ji{x)dx : (1.1.7) Sn (-^ 1 > ' ' ' j •^n) IT 6Jit(xi) 5Jirl(xn) Z[J] Z[J] j=o Subsequently, almost all current work in Q F T focuses on constructing the Sn using a Euclidean functional integral. 1.2 Overview of the thesis Taking the functional integral approach, the central problem in Q F T is then to construct a measure du($) = const. e - / £ ( * ( x ) ) d x r » $ (1.2.1) over the space of Euclidean fields $(x). The existence of a Gaussian measure dv0 where 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 devel-oped 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 averag-ing" 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 de-composition for the Fermi fields of the QED model. I was hoping to employ the G N scheme, which was developed by Gallavotti and Nicolo in 1984 for the renormalization of the 04 theory [GN85a][GN85b], with the "block-spin" decomposition to the IQED 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 QED 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 LRC) . In terms of the Euclidean fields A^, tp, tp, the basic Lagrangian density of Q E D is given by int ^F2 + tp(-ip + m)tp + etp4tp. (1.2.2) F2 — F F F = d A — 8 A A = A,Y, ?> = d»r, (1.2.3) where 7 M has a representation as the 4x4 anti-hermitian Dirac matrices, which satisfy 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. C = Here Chapter 1. Introduction 7 For a renormalized expansion for IQED, we need to further reparameterize the IQED Lagrangian density £ . For a scheme using momentum cutoffs for both the Fermi and photon covariances (the corresponding fields have an ultra-violet (UV) cutoff Mu where M > 1 is a scale parameter), £ = C0 + C\ is parameterized with 7 coupling constants Xu — (Af, Af, • • •, Af) and a gauge-fixing parameter 77: C0 = l- [A- {-A)A + r]{l-r])-\d- Af + A2] + # n - $)</>, '(1.2.5a) d = zAf + i> [m(Xi - 1) - i (Af - 1)0] ^ (1.2.5b) +^ [(Af - 1)A • {-A)A + M2UXU5A2 + Af (d • A)2] + Af A 4 , where we have rewritten 1/4 F2 as 1/2 [A • ( — A ) A — (d • A)2] and have added a photon mass term 1/2 A2 and a gauge-fixing term 1/2 (1 — rj)~l{d • A)2, where 0 < 77 < 1. The reason for introducing the M2U 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 A2 term in C0 serves as an infra-red (IR) cutoff. Since all calculations are done in momentum space, let us express the terms of C correspondingly. C0 = -jAD^A + ^ S ^ , (1.2.6) CT = zAfVl + (Af - 1)V2 + (Af - 1)14 + (Af - 1)V3 (1.2.7) +Af M2UVb + Af V6 + Af V7 where Du1 = ^77 [aL + bT] T — A _ ^ P t / T — P / l P l / M 9 a = p' + l b Chapter 1. Introduction 8 Su1 = -^jirn+rf) ( 1 . 2 . 9 ) if = VpTfi p-u is a cutoff function corresponding to the U V cutoff of the fields; and Vi = v2 = ^v>, y4 = I ^ , ( 1 . 2 . 1 0 ) V 6 =-±Ap2TA, V7 = A\ Since (cL + d T ) - 1 = c^L + d^T, the photon propagator, obtained by inverting D _ 1 is : p<u p<u H _ \ = iTpL" + fr^)+5T- ( 1-2 1 1 a ) Two commonly used gauge fixings are rj = 0 (Feynman gauge) 1 +p D ^ = P ~ U ^ (1-2-llb) and 77 = 1 (Landau gauge) Similarly, the Fermi propagator Su, which is obtained by inverting S^1, has the form Su = (£u{m+jf)-\ (1.2.12) In general, the perturbation theory of using cont. e~ I C°^U§ 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 ® s (roughly speaking, the fields at scale s correspond to momenta of order Ms) and we 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, Xs, are determined as a function of the coupling constants at higher scales. Schematically, Xs = B{Xs+\---,Xu). (1.2.13) The values Xu at scale U are the "bare" or "input" coupling constants, and the values A r at the root scale r are the "physical" or "output" coupling constants. For the details of this procedure, see Section 3.6. The goal of this analysis is to show that for suitable boundary conditions (i.e. specify-ing Af 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 As of the primitive flow equations A s = BL{AS+1,---,AU). (1.2.14) 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 s = Ks + o(As). At 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 Chapter 1. Introduction 10 that Af —» 0 - it doesn't! Inspired by Gell-mann and Low's paper on the R G flow of the effective charge in QED [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 RC scheme, we discovered that the flow (1.2.13) is roughly governed by the effective coupling C' = wM^WY (L2'15) It is the flow of C s which is A F , and this leads to the stability of the flow (1.2.13) (see 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 Af has the same asymptotic behavior as the bare vertex coupling Af, i.e. lim = e, (1.2.16) £/—>oo A 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 2 " condition. Because we require our model to satisfy Ward Identities, we apply a loop regular-ization 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 effec-tively the same as the coupling flow of the RC 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 du0. This is done by expanding the mo-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 dv0. Because of the notorious divergence of the coefficients of the perturbative series, we need a renormalization procedure in order to render finite coeffi-cients 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 QED 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 decom-pose 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 free-dom 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 RG 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 IRAF. 2.1 du in perturbation theory Consider a Euclidean quantum field theory (EQFT) in R d with Lagrangian density, k C = Co + A • Cint = CQ + "^2 \j Cintj , (2.1.1) j=l a local polynomial in the fields of the theory $ = ($i , •••,$#) and their derivatives, where Co is quadratic, Cint has degree > 3 and A = (Ai, • • •, A^) are the coupling con-stants. Co also depends linearly on physical parameters Q such as masses and field strengths. The measure is formally expressed as : 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 du0: du{<5>) = const, e" / £ ( * ( x ) ) d x V<$> (2.1.2) di/($) = const. e-I^nt(^(x))dxdU{ 'o, (2.1.3a) where dvQ(§) = const. e _/ £ o d s2?$. (2.1.3b) Chapter 2. Preliminaries 15 The free measure duo can be understood through its generating functional Z0[J} = const. JeJ*dv0($). (2.1.4) where J(x) = (JL(x),JN(x)), J $ is defined in (1.1), and the source field J(x) has 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 Z0[J]. Using integrating by parts, Jc0(x)dx = ^C'1®, (2.1.5) where C~l is a first or second degree differential operator. Using (2.1.5) and making the change of variables $ —>• $ + CJ in (2.1.4), we get Z 0[J] = e^JCJ. (2.1.6) Given a monomial in the field M = $jj(xi) • • • $in(xn), then the Gaussian integration of M with respect to du0 is given by SJ^ixi) 5Jin(xn)' A i .e\JCJ 7=0 (2.1.7) 7=0 8Jh{xi) SJin(xn) 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). In the case that $i(x) and are fermions then Cij(x,y) = —Cji(y,x) and it becomes necessary to specify the direction of a line connecting two vertices. Chapter 2. Preliminaries 16 For a nontrivial theory with nonzero interaction, the full understanding of the measure du(Q) is very much a mystery. Here we only want to inquire about dv{$>) in terms of perturbation theory. As in the free case, we study dv(&) through its moment generating functional Z[J] Z[J] = j eJ* d i / ($) = y e ^ * e - / X - £ n t ( * ( z ) ) d s 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/($) exists in perturbation theory (let k = l in (2.1.1) for simplicity) if Z[J] exists as a formal power series in the coupling constant A whose coefficients are computed by evaluating Gaussian integrals with respect to du0. Letting V($) = -\JdxCint{${x)) , from (2.1.7), we may write Z[J] = :TZ0[J] (2.1.9) = E*nZn{J}. For each order in A, the coefficient Zn[J] is represented by sums of Feynman graphs corresponding to Gaussian expectations. The Green's functions (Feynman graph) obtained from (2.1.9) can be expressed in terms of connected Green's functions (connected Feynman graphs). We introduce the external effective potential Ve($e) which is the generating functional for the connected Green's functions (amputated by the free field propagator). Let us define the external field <fre = CJ and consider the expression: Z[J] _ i^c-1^ [ „$ec-l<& z0[J] Chapter 2. Preliminaries 17 = const. e-\**c-^ j e^c-^-\*c-^ e-xJcinmV<$> = const. J e - ^ - ^ - ^ - ^ e - ^ ^ W V Q . (2.1.10) Making the change of variable $ —> <3> + $ e and defining V($) = — J \-£int($), we define the effective potential Ve as the log of (2.1.10), = l 0 g [ f e ^ W o ( $ ) = l o g [ £ ( e ^ + * e ) ) ] 0 , = l o g £ - [ £ ( V ( $ + <r)%, (2.1.11) n=l where [ ]o denotes dropping constant terms independent of $ e . 2.2 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 +m2)~1 behaves like ^ _ ^ _ 2 as \x — y\ —> 0. 2. Infra-Red (IR) problem In the case of a massless theory, integrals may diverge because of problems associ-ated 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 £ ( A f , A f ) = £ 0 ( A f ) + A f £ i n t (2.2.1) where Af and Af are the parameters of the model. By a reparameterization of (Af, Af) , i.e. expressing Af and Af as a function of a new parameter A, one wishes to write £ 0 (Af (A)) + Af (X)Cint = £(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 A f ( A ) ~ A + £ anXn. (2.2.3) 71 = 2 We call Af the bare coupling and A the physical coupling. The coefficients an are infinite and are suppose to cancel the infinities. Now the renormalization problem of V r e($ e) is to show that SV = — J t5£($) dx can be chosen such that has a well defined fps in A. In summary, a renormalization scheme of Ve($e) in perturbation theory consists of introducing regularizations or cut-offs to render the Gaussian integrals finite, quantify-ing 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 Ve($R) are finite, uniformly as the regularizations are removed. 2.3 Renormalization group and asymptotic freedom In a renormalization scheme, the pertubative expansion of Ve($e) is parameterized by 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 & Peter-mann [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. eM supposedly relates the behavior of the theory at ar-bitrary momentum scale \i. Gell-Mann & Low found that eM obeys a differential equation 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 eM , and the bare charge eo is the high momentum scale limit of e ,^ e0 = lim e„ . (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 informa-tion about the large or small momentum behavior of a QFT. 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 —>• ±oo. 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(i0) = A0 near A/ 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(w) as the bare coupling at the cut-off N so that lim A(JV> = Xf (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 21 2.4 Gauge invariance & Ward Identities In this section, we describes briefly the gauge fixing and Ward Identities [War50] of QED. QED 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 QED is given by (1.2.2). By integration by parts we can replace Lp by CP = ±A„ [D-j\ A V , D-] = d\u - d»d„ (2.4.1) and write Lj as CF = ipS-lip, S~1 = -ip + m. (2.4.2) In momentum space, D-lM = V2 [<W - Vjf-\ 5"J(p) =j/ + m. (2.4.3) It is not too hard to verify that D~l is non-invertible while S~x is invertible with inverse S(P) = ^ (2.4.4) Since e - i e x s - l e i e X = g-l + e ^ the Lagrangian (1.2.2) is invariant under the gauge transformation $^e~iexi), i)^eiexip, A^A-edx- (2.4.6) Because of the gauge invariance, the differential operator D'1 of (2.4.1) is non-invertible; put differently, the measure corresponding to D~l does not have Gaussian decay in all directions. The standard procedure is to modify CP with a gauge fixing term 1/2 {l-n)-\d-Af Chapter 2. Preliminaries 22 where d • A = d^A^. Thus, in momentum space, as in (1.2.8), we write D-l{p) = p2[V + (1 - 7 7 ) - % , ] , (2.4.7) D^ip) = + ( 1 - 7 7 ) 7 ^ ] . In particular, in Feynman gauge with 77 = 0, D = ^ p2 and in Landau gauge with 77 = 1, pi The original gauge invariance is broken, but in QED the Ward Identities (WI) still hold. It is important that the renormalized perturbation theory respect the WI because it implies that the theory is perturbatively unitary and that the S-matrix is independent of the choices involved in gauge-fixing [tHV72]. The identity underlying the WI in QED is (2.4.5). A form of the WI can be expressed in terms of the generating functional for the connected Green's functions (2.1.11)[FHRW88] K($e) = log ^ J e ( ^ + ^ X * + * e ) dv0{$) (2.4.8) where Z = j e( y+^)(*) du0(<!>) (2.4.9) du0{A) = const. e-Ic"dxVA dv0(ip,ip) = const. e-IcidxVil)V^ V($) = - J eipAijj dx, SV — counterterms. (2.4.10) Chapter 2. Preliminaries 23 By (2.4.5), we have dis0(eie*tp,e-iex$) = e-eI^Mdu0{tP,tP). (2.4.11) Hence under the change of variables tp -4 e iexip, tp -4 e16*?/;, (2.4.12) it is easy to see that we can arrive at the following WI ( a non-linear form ): Ve(V, fj, B) = K ( ( l + eSfiX)e-iexV, *?eie*(l + e?XS), B + pX)- (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 WI 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 QED. 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 Gaussian integrals and 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 as the expectation of the monomial M = $ i • • • $„ . We write M as a product of local monomials M 1 ; • • •, Mp where each Mj has argument X j . We regard each Mj as a local vertex with the fields of Mj as legs emanating from the vertex. (2.5.1) Chapter 2. Preliminaries 24 Gaussian Integration Rule 1 £{MU • • •, M p ) = {-lT{G)Val{G) (2.5.2) 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 , • • •, Mp. Define the "connected" or "truncated" expectation by rr, (V £T(MU • • •, Mp) = ° _ d x log£(ex^+-+x*M>>) (2.5.3) A=0 Gaussian Integration Rule 2 From (2.5.3) we have : £T(MU • • •, Mp) = £ (-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 asso-ciated with these expectation are "vacuum graphs" with no legs from the clusters left uncontracted. If we take an expectation of monomials involving external fields, say £(Mi($ + $ e ) , • • •, Mp($ + $ e ) ) , then the resulting Feynman graphs will not be vac-uum graphs but will be graphs with uncontracted external legs corresponding to external fields. Chapter 2. Preliminaries 25 Gaussian Integration Rule 3 £ T ( M i ( $ + $ e ) , • • •, M p ( $ + $ e ) ) = 52 (-l)n^Val(G) (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[dxv II C(xt,yt) &x(xx), (2.5.6) v=\ eec(G) A e A ( 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) ®\{x\) occur in the same order 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(xtl ye) = C(xe - y e , 0) = J C ( % ) , (2.5.7) where C is the Fourier transform of C. As for each we have ^ A ( ^ A ) = / ^ - / X x ' q x *(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). By 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 xv, we get a momentum conserving delta function 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 EVte = s 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) —1 if £ i— v ( i.e., the direction (arrow) of the line £ flows out of the vertex v) • 0 if the line £ is not connected to the vertex v Thus (2.5.6) becomes y<G) = j n n y f e n n (2-5.9) A e A ( G ) \ 1 eeC(G)^ H> ieC(G) A e A ( G ) where q\ are the external momenta of the legs and kg are the internal momenta of the lines, 5G = 5 ( £ qx) * ((E k)v +J2<b)> \ A e A ( G ) / veV(G) V \->v / I X —> v are the legs that are attached to the vertex 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 AeA(G) \*' l> AeA(G) iecP(G) y LI[> 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 momentum-labeling 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 Example 2.5.1: 27 k 3 Figure 2.1: An initial momentum labeling of G Consider the Feynman graph G with an initial momentum labeling given in Figure 2.1 obtained from ST (Vint($ + <S>e)(Xl) Vmt($ + $e)(*2) Vint($ + $ e ) ( x 3 ) Vint($ + $ e ) ( x 4 ) ) (2.5.12) where Vint{$) = e J fyipdx. (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 5G = S(qi + q2) % i -ki + kA)6{k5 + k3- kA)S(k2 - k3 - k5)6{k! + q2 - k2). (2.5.15) Let pi = &4 be the loop momentum for the larger loop and p2 = k5 be the loop momentum for the smaller loop. Integrating out the delta functions (except S(qi + q2)), we have h = qi+Pu k2=pi, k3=p2-pi. (2.5.16) Chapter 2. Preliminaries 28 Figure 2.2: G with loop momenta pi and p2 Using the loop momentum labeling of G given in Figure 2.2, we have d4qi Val(G) = J ^ + &) JWft.fc) ^ f a W f c ) (2-5-17) where j d4pi tr .j M I . -7 [Pi+m1 p'l+<£1 + ml ^ + m ^ + m Here we suppress the spinor indices of the 7 matrices and sum over repeated indices. V2 . (2.5.18) -r 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 Cs has length scale M~S,M > 1 being a fixed scale parameter. In effect, this decomposition resolves the U V and IR singularities of C. By successively integrating out the fields $ s (from high to low), one obtains a tree expansion for duren. The tree expansion 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~s (momentum scale M 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 = 0 for h ^ s. That $ | has length scale M s means the sliced propagator <??.(*, 2/) = / ^)dvQ satisfies l b a ^ C * ( p ) | | o o < c MV'-^HPl)' where P" Pi1 • nad Pd > 30 =d01 ••• dpd P wpi wPd \a\ = a>i + h ad, c is a positive constant and de is the dimension of the line : dt = (3.1.2) (3.1.3a) (3.1.3b) (3.1.3c) (3.1.3d) (3.1.4) (3.1.5) 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 Ct(x,y). Roughly speaking, a covariance satisfying (3.1.3) behaves like Cs ~ ciMsdte~C2M°\x~y\ In particular as \x — y\ —> 0, Ce(x,y) ~\x- y\~dt ~ ]T (M8)dte-cM'^-v\. (3.1.6) Chapter 3. The GN Tree Expansion and Renormalization 31 3.2 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 u (3.2.1) V e u m = \og(l[£s s=0 where $ e is in Schwartz space. With the removal of the local singularities in Ce (x,y), we see that V e c / ($ e ) has a well denned fps in A. For future convenience, we introduce the following notation: s=0 U £>k — n £s s=k+\ s=0 etc... Given a cutoff U we define a flow of effective potentials Vku, k = U, U — 1, following recursive relations (suppressing the superscript U): Vn = V (3.2.2) •1 by the (3.2.3) (3.2.4) Note that the flow represents successive Gaussian integration from k=U to k=0, and that Vl! = Ve. For each Vk-\, w e make the following cumulant expansion: V f c _ 2 = log(£ f c(e v*)) OO I = £ ^ > g ( 4 ( e ^ ) ) U ] 0 (3-2.5) -TP-OO 1 = Y-P=i y-£Z( vk,---,vk ) * — » — ' , p arguments J 0 Chapter 3. The GN Tree Expansion and Renormalization 32 For the expectation -\ £\T(Xi, • • •, Xp) , we represent it by the following fork: 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 [£(k^£j(Xi, • • • ,XP)] , we represent it by the following sub-tree: = k) X | X p function of input field VP s * fork = connected expectation £ ^ k •* highest scale of output field The succession of simple expectations £(k,s) after £j(Xi, • • •, Xp) in the above means setting = 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 . J-(T) is partial ordered such that for fu f2 G ^F(T), fi > h if fi lies above f2. 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 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), Hr(r) = { s \ r < s f < U ; s f l > s h iff fx > f2} we define 77 to be the subtree of r such that the root of Tf is 7 r ( / ) . For a given r and s, the value V(r,s) is defined inductively: V(e, s) (3.2.7) v(x f i.r) Chapter 3. The GN Tree Expansion and Renormalization 34 where we consider £ as a null tree consisting of only one vertex and /i, • • •, / p are forks or end points immediately above f. Note that V(r/,s) depends only on the scales Sf, f > f. The operations £ > S f in the above formula ensure that the highest scale of the fields in 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 E V(T,a) (3.2.8) T S 6 - 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/}/ e^-( T) . Each associated labeled graph Gs(r) consists of a set of vertices V(G), a set of labeled lines CS(G) and a set of external legs (half-lines) A(G). Each 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 $<r(x\) 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 Gft are being formed by contraction of legs. We view 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 Gs,u and Gs-U = j K^u(x)Ue{xe) dx (3.2.9a) where K°>u(x) = ( - 1 ) ^ cG A* n C { / ( l ( x , , % ) (3.2.9b) £e£(G) is the kernel of Gs'u, x = (xi, • • •, xv), and IIe(xe) = YlxeA(G) ®e{xx)- 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 xe and ye are the coordinates of the endpoints of £. Expressing (3.2.9a-b) in momentum space using (2.5.7), as in (2.5.11), after integrating out the delta functions, A e A ( G ) A e A ( G ) A G A ( G ) (3.2.10a) where q is the set of external momenta and the kernel K is an integral over loop momenta P/'s, K(q) = cG A" / J] C'wiPe) II 7 ^ , iec(G) iecP(G) where each Pe is a linear combination of loop and external momenta. Example 3.2.1: (3.2.10b) Figure 3.3: A tree r with forks / i , f2 and / 3 Consider the following graph G of QED 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 Ds(p) = PSB(P)D(P), Ss(p) = PSF(P)S(P) (3.2.11) where S(p), D(p) are given in (2.4.4), (2.4.7) respectively, and the px, X e {B,F}, are the slicing functions for restricting the support of the covariance in momentum space. The corresponding Val(Gs) (in momentum space) is d4Qi where Val(G) = J 7 & % i + ft) AfM)K^MiA2)Af2{q2) (3.2.12) *W2(gi>92) = c G A 4 f d4p2 [L^M - ( l - r / ) ^ ^ ) ] (3.2.13) J dipl tr P2 Sf2 Sf3 I ~J i 1 i ' -J , / _/ -J , " [pi+m' p/x + ix + m ' p!l+ml ^-pj^ + m Finally, we express V(r , s) in terms the values of the associated graphs of r: V ^ = ^T) E (3.2.14) where n(r) = fl/e^Cr) / V a n 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 Ve in which each topological graph that contributes to Ve has been decomposed into a sum over labeled 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 s , the subgraphs GSf are partially ordered 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 con-tributing 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 ) c i x (3.3.1) ? 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 e(x) and the internal vertices occurring only in Ks'u. When working in coordinate space (x-space), we shall estimate GS,U using the "pinned L 1 -norm" of its kernel 11^111= / K*'u(x) dxl---dxv_l. (3.3.2) J xv=0 The reason why we pin down the last coordinate of Ks in defining the norm is because, by translational invariance of Ks, we can write GS>U = j dx1---dxv_1K°'u(x1,---,xv-1,0) j dy ^{xx + y, • • •, xv-X + y, y) where \&(x) = llAeA(G') ®e(x'\)- When working in momentum space (p-space), GU = Y [ % i + " - + 9n) / K*>u(q, p)dp n e (q) dq (3.3.3) where q = (qi,---,q n) are the external momenta, and p are the loop momenta, and the corresponding norm is the sup-norm on Ks'u(qil p) without the delta function S(qi + \-qn), 11^1100 = sup / ^ ( q , p ) dp. (3.3.4) q 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 39 We can.bound | | / ^ s | | i 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) in the integral by || • H ^ ' s and then bound each of the leftover lines in A4(G) by || • | | i 's . Thus by (3.1.3c-d), \\K%~ < c n i i ^ i i o o n iic'iii £eC(G)/M(G) ecM(G) < c n MdtSl n M~dst. (3.3.5) eec(G) . i€M(G) Dually, in bounding H-K^Hoo, 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 i i^ i ioo n i i ^ i i i teM(G) ieC(G)/M{G) < c n MdlSl JJ M~dst. (3.3.6) £eC(G) £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(Gf)= de-d(v(Gf)-l) (3.3.7) tec(Gf) where dt is given by (3.1.5) and v(Gf) = |V(G/) | . Chapter 3. The GN Tree Expansion and Renormalization 40 Theorem 3.1 (Dyson-Weinberg power counting theorem) \\K*\\ < c n MD^sf (3.3.8) = c JJ MD{G^sf~s^\ (3.3.9) 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 \\KK\\ < c M ^ e a o ^ ' - E / e r o o r f M P / - 1 ) (3.3.10) By regrouping the sum in the above exponent, we get E d e h m - dhf(pf-l) = hf E de-d(pf-l) ieC(G) f€T(r) . / G ^ ( r ) l:f(t)=f = £ hfD(gf). (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 (af - a h ) b f = E («/ - Mf))Bf (3-3-12) / > / i f>h 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. The GN Tree Expansion and Renormalization 41 3.4 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 renormaliza-tion 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 SVU to the original potential Vu($-U). Because 5VU is required to be of the same form as the local monomials of Vu(^-u^), there will be extra graphs coming from the renormalized tree expansion in which modified potentials Vu + 5Vu,s are fed in at the top end points instead of Vu. 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, • • • ,.xv), and n / ( x ) = a x a l i $f(x 1 ) ••• dz$Zk{xv), k = s 7 r ( /) (3.4.1) Chapter 3. The GN Tree Expansion and Renormalization 42 where o% = Yli=i ( ) > \a\ = £i=i ai-> a n 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) dimQj d-2 2 (3.4.3) where if $j is a Bose field if $; is a Fermi field. (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 dimen-sionless so that 5(Gf) = Dd(Gf) - Nf (see (page 215) of [Ros90]). 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), - • • ,Xv(tf)), Xj(tf) =Xf+tf(Xj - Xf) (3.4.4) with 0 < tf < 1. The local parts of II/ and Gf are 0 £11/(x) = 7^n/(*('/)) tt-Q if 6{Gf) < 0 if S(Gf) > 0 (3.4.5) and LGf = J Kf(x) (LII/(x)) dx. (3.4.6) The renormalization of a Gf is then defined as RGf = (1 - L)G{. (3.4.7) Chapter 3. The GN Tree Expansion and Renormalization 43 By Taylor's theorem Gf if 6{Gf) < 0 RGj 7^? Jo dtf(l - tfYf~l J Kf(x)(A • d)^f Il/((x(t/))rfx if S(Gf)>0 (3.4.8) where fif = 5(Gf) + 1 and A • d = Y,j{xj — xf) • d x- such that the S's do not act on the 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 . (3.4.9) Sf — [if = —1 if R is applied at f From the Taylor subtraction, we also obtain the factor A^f. 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~^isf. 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^fSiT^. Thus we get a net improvement of M - ^ ^ - ^ ^ ) ^ in the power counting to 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 Kf(q) fi/(q) <5q dq where <5q = S(qi + h qv). Since when taking Fourier transform, the multiplication by Xi becomes —idp? and dx? becomes multiplication by ipf, it is not too hard to see that 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): for 8(Gf) > 0, LGf = j (LKf(q)) Ilf(q) £q dq (3.4.10) where LKf(q) = Kf(0) + £ ft • 3 f ttf,(0) + (3.4.11) l<i<t; E 6 ( G f ) ) ((9 1 1-^)---(C-^;)K / ( O ) ) ) | Q I | + - + 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 QED with momentum labeling given in Figure 3.6. Ignoring scales and combinatoric factors, in coordinates-space, LG = LJ 7p(Xl) (K(xux2)) ip{x2) dxxdx2 (3.4.12) = jdxi if)(xi) ^dx2K(0,x2)S) ip{xi) + jdxi4>{xL) (^jdx2x%K (0,x2)>j dx^{x1) = j K(0,x2) dx2 j tp(xi)tp(xi) dxi + j x2iK(0,x2) dx2 J V>(xi) dx»ip(xi) dxi, where K(xi,x2) = D(xi,x2)S(xi,x2) is the product of a Bose line and a Fermi line. In p-space, using the momentum labeling in Figure 3.6 LG = J ~^{qi) (LK(qi)) xP(q2) S(qi + q2) dQldq2 (3.4.13) Chapter 3. The GN Tree Expansion and Renormalization 45 = J ~^{q) K(0) <4>(-q) dq + J$(q) {q"dql>)k(0) 4>(-q) dq = K(0) J $(q)j>{-q) dq + dq,K(0) j i>(q)q^(-q) dq, 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) 4 factors), we have Jdx2K{0,x2) Ji>(xx)tP(x1) dxx (3.4.14) = J elipi+p^X2D(Pl)S(p2) dPldp2 J el^+q^x^{qx)^{q2) dqxdq2dxx = J D(px)S(-px) dpx J kliM-li) dq, = K(0) J $(q)i>(-q) dq, and Jx^K(0,x2) dx2 j $(xx)dx^{xx) dxi . (3.4.15) = j x^+^D(Pl)S(p2)dPldP2 j (dx,e^Xi) e*™${qx)]>(q2) dqxdq2dxx = H ) j D(pi) (-d_p^S(p2)) S(Pl + p2)dPldp2 (-i) j $(qi)(-q$)$(q2)6{qi + q2)dqxdq2 = j D(Px) dp,S(-px) dpx j i(qM(-qi) dqx = dq»K(0) j ^(q)q^(-q) dq. 3.5 Renormalized tree expansion We briefly describe how to implement the Taylor subtractions given by the L and R operations given by (3.4.5)-(3.4.8) into the tree expansion to cancel the divergences so that each graph G obtained from the expansion has the corresponding Gu uniformly bounded in U . (For more details, see §2 of [FHRW88]). A renormalized tree r is a tree defined as in the unrenormalized expansion except that it has an extra label o~f = R or C at each fork f, and the set of scale assignments is modified as: V.r{s, B) = {s | s7r(/) < Sf <U if Tf = R ; 0 < Sf < s^f) if Of = C} (3.5.1) Chapter 3. The GN Tree Expansion and Renormalization 46 where, here, S^F) = r for the lowest fork F. For a given a, the value V(r,a,s) is determined by the following rules. If Xx, • • •, Xp are monomials arising from V , or from an R-fork or C-fork, then = X(s > k) (1 - L) £J ($<*), • • -,Xp(&>j) X(8<k)(-L)£],(x1(^')1--.1XP(^')) *J=0; s<j<k By carrying the above definition inductively down the trees, then the counterterms are &VU = E E E Vu(T,C7,S) n.t.T ff:aF=C seric(r,S) (3.5.2) 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 SVU is a fps in the interaction coupling A whose coefficients are local polynomials in the fields . We define the renormalized effective potential VrU*e)=[loge<u{ev>+sv»)]0. (3.5.3) and as in (3.2.8) we have K e n = Vu($e) + E E E VU(T,0,S) (3.5.4) n.t.r a:aF=R seH(r,a) 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 Gren 6 Q(T, a) according to the choice of m / in (3.4.5) and to how the derivatives Stf act in (3.4.5) and (3.4.8). We write VU{T,3,3) = £ G% (3.5.5) where the value Gsr^n is similar to that of Gs,u 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 Gsren be a graph contributing to Vu(T,O,S) in (3.5.5) with r = -1. Then < c JX M* ' ( * ' -M/ ) ) (3.5.6) / where for the bottom fork sn(F) = 0. 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 < Sf < d and sf - s < 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, Cs ren Chapter 3. The GN Tree Expansion and Renormalization 48 when summing over the scales from 0 to sF: we get a factor of sF, the sum of the bounds 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 SF M < W =CK\{r + 2)K M5FT . (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: Coro l l a ry 3.1 (UV-Renormalizability) Consider an EQFT with dimensionless inter-action and positive masses. Let GSREN be a graph contributing to V(T,O,S) in the renor-malized tree expansion (3.5.4)- Then E \Gren\ < c0[G) K\(r + 2)K M5FT (3.5.10) se'Hr{T,cj) where c 0 is a constant independent of U, G, and r, K is the number of marginal C-forks in T, and SF — S(GREN) satisfies (3.5.7) or (3.5.8). The G N tree expansion that we have described can also be applied with some modifica-tions 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. The GN Tree Expansion and Renormalization 49 3.6 Running coupling constant tree expansion From the previous section, we have that V r f n ( $ e , A ) = [ \og£(ev+sv^~u+ipe'^) ] 0 exists 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 K e n , * ' = [ log£ f c + 1 ( e v - " ' f e + 1 ) ]o = L [ l o g £ f c + 1 ( e W + 1 ) ] 0 + R[\0g£k+1(ev— *+*)]<> (3-6.1) = - A f e - V f c + R[log£k+l(ev-> «+i)]o (3.6.2) where R , L are the operations defined in (3.4.5) and (3.4.8) and V = -\-V = -52Xi jdxd V? = J dx Ci($^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 f c . We call A f e the coupling at scale k, \ u the bare coupling, and A - 1 the physical cou-pling. Ken,*: is a function of $- f c and the coupling constants A f c , • • • , Xu. Thus the ( U V ) renormalizability of a theory (in perturbation theory) is equivalent to this: for a fixed physical interaction coupling constant A , there exists Ken a s a fP s m $ e a n d A, and each A f as a fps in A , i.e. oo VrUen(*e^U) = (3-6-4) ra=l ce Chapter 3. The GN Tree Expansion and Renormalization 50 Af = IX»*B> (3-6.5) n ( in particular, \ u = A + afA 2 + • • • ) such that l i m ^ o o K% exists. We remark that as U —> oo the coefficients af n diverge. On the other hand the actual RCC's Af may 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 Vren,k = -A* • V* + £ £ V(T^ (3-6-6) where J2n.t. r 'means sum over nontrivial trees and = {$ I sAf) < sf < u ; k = sn{F)} . (3.6.7) V(T, S) is defined inductively, if k = sn(£) then V(e, s) = -Xk • VK , (3.6.8) otherwise V(rf, s) = ^[R£jf (£>SfV(rfl,s), • • •, £>SfV(rfp, 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(Gfenr) is d4qi Val(G) = j A%(gi) [RK,lll2(qi,q2)} A^(q2)6(g1 - q2) U (3.6.10) where K^Aqi,q2) = <* (\[h)2W)2 f dtp* tr PF -J S / 3 ./xi PF [p2 + m p!2 + 4 i + m l/2 + m r , _ P l _ [ R K s h { p i ) ] jj2 + m (px) = . ( d4Pl P-\ \Lvv{px) ~ oTw{jti\ Yj—77-J Pi —V-, From (3.6.1)-(3.6.2), we have - \ k - V k = L[\og£k+1(ev^k+i)}0 (3.6.11) (3.6.12) = - A f c + 1 • Vk + This gives a system of discrete flow equations : \ k = A f c + 1 + fps involving in \ k + 1 , \ k + 2 - - - in terms of deg > 2, (3.6.13) = A f e + 1 + f3k(\k+1, Xk+2, • • •) (3.6.14) The flow equation given by (3.6.14) is the analog of the R G flow equation of (2.3.2) and (3k is the analog of the /3-function, i.e. ^ = /3(A) —» A f c + 1 -Xk = f3k . Chapter 3. The GN Tree Expansion and Renormalization 52 Here, (3h is only defined perturbatively. As in the continuous R G equation, A = 0 is a fixed point. The theory is U V A F if Xu ->• 0 as U -» oo for an initial fixed A-1. The theory is IRAF, if Xu diverges as U —> oo; roughly speaking, this corresponds to Xs —> 0 as s —¥ —oo. Chapter 4 Decomposition Of Field Via 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 nth slice 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 ap-plied 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 co-variance. 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 Sa defined on a hyper-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 = ad Eckv^)) 2 + \m^(x) + A P(<f>(x)) (4.1.1) xeA 1 1 53 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 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 n = (aLnZ)d. We partition A n into hyper-cubic blocks of lattice such that each block has volume Ld x (aLn)d and the center of each block belongs to A n . S = J dx lid^ix))2 + l-mcj>2{X) + A P^x)) (4.1.3) Example: ? I ? ? T Figure 4.7: A 3x3 block of sub-lattice of (aLnZ)2 Consider A n + 1 as a sub-lattice of A" , let T71 = {</>" | <j)n : A n ->IR}. (4.1:4) The block-spin average transformation (4.1.5) Chapter 4. Decomposition Of Field Via Block-Spin Method 55 is denned by (Qn4n)(x) = c £ 4>n(y) = 4>n+1(x) (4.1.6) \y-x\<!f for all x G A n + 1 where c is a positive constant to be chosen below (in (4.1.15)). It is convenient to work on the fixed unit lattice Zd. For now, we set a = 1 (we will reinstate a at later stage) and introduce the rescaling transformation S : F1*1 ->• Tn (4.1.7) defined by (S4>)(x) = Ll-^(Lx) (4.1.8) for all x G A" . The canonical factor L^1 is chosen so that / ( c ^ ) 2 dx is invariant under the rescaling operation. Intuitively, we want the </>n to decompose as a sum of its block-spin average 4>n+1 and a local fluctuation pn, i.e., 4>n = 4>n+1 + p. (4.1.9) Next we define 4>n = Sn(j)n. Applying S~n to the above equation yields cf)n = (S -V n + 1 ) (x) + pn{x) (4.1.10) where pn = S~npn. At a second glance, we see that (4.1.9) (as well as (4.1.10)) is mathematically unsound because </»" and S~1(j)n+1 do not belong to the same space, i.e., if x A 1 then S~lcj)n+l 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'iQCQY1, (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 (4.1.13) where <,>d^c ^s t n e m n e r product with respect to the Gaussian measure du.C- Notice 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>n. Notice that for 0 6 JF 0, Q:T°-* T°. (4.1.14) Explicitly, {Q<f>){x) = L-i-1 4>{Lx + y). (4.1.15) Furthermore, we have the following relations ^n+l = Sn+l^n+l = Sn+l Q S'n ((f>n). (4.1.16) It is easy to see that SQnS~1 = Qn and Qn~lS = SQn. (4.1.17) Hence <j)n+1 = SQ<f>n = Q4>n. (4.1.18) Iterating the decomposition (4.1.12a-b), we get the following independent decomposi-tion if = AnQcj)n + Qn (4.1.19a) Chapter 4. Decomposition Of Field Via Block-Spin Method 57 An = GnQ'iQGnQ')-1 = 0^0-1, (4.1.19b) gn = (I-AnQ)4>n (4.1.19c) where Gn =< <j>n, <j>n >d,c=< Qn<l>, Qn<t> >d»c = QnC(QT (4.1.19d) is the covariance for </>n. As a result, we have the decomposition, <f>=T.Xn + * N + 1 , (4.1.20a) n=0 where = X n = A0A1---An.l6n, $N = A0A1--.An^n- (4.1.20b) By taking N —> oo, we have the desired decomposition oo 0=Ex" (4.1.21) n=0 where the sliced fields are independent of each other as we now show. Let us first define the covariance Cn =< >d»c = C(Qt)nG~1QnC. (4.1.22) From the facts that QAn = I, (4.1.23a) AoA^-.An., = CiQ^G'1, (4.1.23b) and (4.1.19d), we have <§n+\xn>d»c = <<S>n+\$n>d,c-<<S>n+\$n+1>d»c = CiQ^G-^Q < 0", p >d,c G-lQnC - C n + l = Cn+\ — Cn+\ = 0. Chapter 4. Decomposition Of Field Via Block-Spin Method 58 Thus, we may decompose C according to the decomposition (4.1.20a): f n = Cn — Cn+i = covariance of x (4.1.24a) N C = J2fn + CN+1. (4.1.24b) n=0 4.2 Fn's bound in scalar theory From the block-spin decomposition of a lattice field, we obtain a corresponding decom-position of the covariance C given by (4.1.24b). A slice covariance r n has the form f B = C(Q*) B [G-1 - Q'G-^Q] QnC, (4.2.1) where Gn = QnS{Qt)n (4.2.2a) Q${x) = L ~ ( » ) / 2 $(Lx + y), x£A° (4.2.2b) yeB° An = (aLnZ)d (4.2.2c) B° = {y e A 0 I < L/2} . (4.2.2d) We would like to inquire about the exponential decay bound of Tn. In the case of scalar bosonic fields, [GK86] has shown that the r n has the following bound. Theorem 4.1 Let tn be a slice covariance given by (4-2.1) with C{x,y) = (~)d [ dpw{p)-leiv[-x-y) (4.2.3a) \2irJ JMa = 2 ^ l - c o s ( o p ^ ) + m 2 » a Ma = 7T 7T a ' o (4.2.3c) Chapter 4. Decomposition Of Field Via Block-Spin Method 59 Then \Tn(x,y)\ < C l a\Lnf-d e ~ c ^ , (4.2.4) where c l 5 c2 are constants independent of a and n. Here, I give only an outline of the proof. The reader can refer to [GK86] for details. Their strategy is first to rescale r„ as T n = kn fn (&')" = &*C(Q*) n [G-1 - Q'G^Q] Q n C ( & T > (4.2.5a) where &$(x) = L - ( d i m * ) / 2 $ ( L x ) , x e A " 1 . (4.2.5b) Then Tn is factored into r„ = AnX{Any, (4.2.6a) where An = & n C ( Q * ) n G ^ 1 (4.2.6b) * = G n - G n Q ^ - ^ Q G , , . (4.2.6c) They then establish exponential decay for An and X separately with \An(x,y)\<c3e-£^ \X{x,y)\< c 4 e ~ £ ^ (4.2.7) where c 3, c 4 are positive constants independent of a and n. The desired bound (4.2.4) is 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 p-space 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>(Lx + vY~iVV (4-2-8) Chapter 4. Decomposition Of Field Via Block-Spin Method 60 = L - ( « ) / 2 52 52(l)(Lx + y ) e i i { L x + y ) ^ = L-^)i252m^sL{P), where " = 1 l i / ^ K f e L -1 Therefore, n — z ^ - j — 1 = n —m—-EE = n - d t - (4-2.9) (Q>)(p) = L " ( " ) / 2 S L (p) 0(|). (4.2.10) From (4.2.10), / dpf(p)Qg(p) = L " ( " ) / 2 / dpf(p)sL(p)g(p/L) (4.2.11) JM„ JMa = L ^ d i m ^ 1 2 f dk f{Lk) sL(Lk) g(k) J LMa = L - ( " ) / 2 52 [ dk f{V) 8L{k>) g{k), jeJ1 where J" = { j e Z d , | b ' „ | < L " / 2 } , (4.2.12a) if = * , v e A - . (4.2.12b) Therefore, (Q7)>) = L - ( " ) / 2 E / ( ^ ' K M - (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) Gn has a pole at p = 0. It is then easy to see that both terms in X = Gn - GnQtG~]rlQGn (4.2.14) Chapter 4. Decomposition Of Field Via Block-Spin Method 61 have a pole at p = 0. Furthermore, X also has poles at p^ = (±27r/aL, • • •, ±2n/aL), coming from Q* operating on QGn in p-space. But these poles are actually removable. At p = 0, there is cancellation of poles between the Gn and GnQtG~\lQGn. And at P(i) = (±2ir/aL, • • •, ±2ir/aL), the zero of GnQtG~llQGn coming from the G~lx would remove the corresponding pole coming from the Gn. This allows one to display uniform bounds for An, X in Ma x i[0,e/a}. The exponential decay can then be obtained by the Contour Shift Lemma. Lemma 4.1 (Contour Shift Lemma) Suppose F(p,x) is analytic in each argument P f l in an open set containing \RePll\ < | , \ImPlJi\ < e/a. Furthermore, \F{p,x)\<c, (4.2.15) uniformly 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„, • • •)• (4.2.16) CL CL Then F { X ) = { ^ K ) 1 L P(P>x)eW'XdP (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) Implx = e - - < R e P l l < - . (4.2.19) a a Using the analyticity and periodicity of F(x), we have 1^ )1 = \F(p + ieelt,x)\\e*-*-e*>\dp Alt JMa < ce'^. Q.E.D. (4.2.20) Chapter 4. Decomposition Of Field Via Block-Spin Method 62 4.3 T n of lattice Fermion In this section, we would like to carry over a similar analysis for finding an exponential decay bound for Tn as done in the scalar bosonic case. There are many variations of the fermonic propagator S(x,y). We take S(x,y) to be the following : = - [ilvsm(apu) + 2sin 2(ap J //2)| + m (4.3.1) a, v where S(x,y) = {£f jM dp e**-v\S(p) (4.3.2) {1(1,1^} = 25^, and Ma is defined in (4.2.3c). The term 2sin2(ap^/2) is added to 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 <c5a {Ln)l-de-c^ (4.3.3) uniformly in n where c 5, c 6 are positive constants independent of n,a,x,y. 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 (Texpp)(Z X)=f dH^P e^-W'X-W p(4>,# (4.3.4) Using the Renormalization Group Transformation Texp, by assuming that the exponen-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 expo-nential decay (4.3.3) independent of n but dependent on the weight parameter a. In their analysis, r n is composed of terms involving factors of (S 1 " ) - 1 ^) where Sn = a'1! - QnS{Qt)n (4.3.5) Chapter 4. Decomposition Of Field Via Block-Spin Method 63 (see (3.11) of [B0S91]) is the nth transformed covariance (scaled by &) and otn — ot\_^_n (see Lemma II. l of [B0S91]). Since l im^oo QnS{Qt)n{±ir/a, ±ix/a) = 0, the bound (4.3.3) is no longer uniform in n in the limit a —>• oo. For finite an, what prevents Sn(p) from having zeros (so that Tn has the bound (4.3.3)) is the a~lI term which vanishes in the limit a —» oo. Because x , "0, Xi 4> a r e independent variables, (x~Q'4}iX~Q'lP) is n ° t definite in sign. Hence, it is in question that l i m ^ o o Texp = T since it is not clear why the exponential weight ea(*~Q^'x~Q^ would provide the decay to force x = Q4>, a n d X = Qi3- This 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 Tn(x,y) = jM dp eip^Tn(p,x,y), (4.3.6) where tn(p,x,y) = Hn(p,x)G-\p)Hi(p,y)-\Hn(p,x)L-^ E e-W^SLfr + 27rJ/aL) #(p + £/L, y) leJ1 , (4.3.7) where Hn(p,x) = (Lnyd-1 J2"jx,a s^tf) Stf), (4.3.8a) Gn{p) = ( L n ) - 2 d _ 1 52 s i V ) Sip1) (4.3.8b) d sm(Lnap„/2) SL*(P) = 11 —7-, T ^ - , 4.3.8c P(n) = Lnp — 2irj/a, for some j € J"™ so that |p(n)| < ir/a, (4.3.8d) and Jn, pP are defined in (4.2.12a-b). In applying the contour shift lemma to T n , peri-odicity for Tn(p, x, y) is obvious, we only need to verify the boundedness of Tn uniformly 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 aP{i) = 0 as in the bosonic case, our analysis also fails to yield the desired exponential decay bounds uniformly in n. The source of the problem is the fact that the lattice Dirac operator G{p) = -Yl^Hw») (4-3.9) a ^ in momentum space vanishes not just at ap = 0 but also at ap = ( ± 7 r , • • •, ± 7 r ) . By rationalizing S^ip) — G(p) + H(p), where H{p) = - £ s i n 2 ( a p M / 2 ) + m, (4.3.10) a fi we have b { P ) ~ Dip) + Dip) ( 4 - 3 - 1 1 } where * denotes complex conjugate and Dip) = S-\S-lYip). (4.3.12) We now describe briefly the difficulty which we shall investigate in the remain-der 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) is Q(L~n + aLnm), this suggests that Gn = iQ^SQ71 has zeros near ap = ( ± 7 r , • • •, ± 7 r ) . As a result, Tn can not be bounded uniformly in n since we have yet found a way to show that there are cancellations of poles coming from the zeros of Gn. Thus we cannot perform the Contour Shift Lemma to Tn to obtain the desire exponential decay bound on the sliced covariances. In Section 4.4, we provide the expressions for the components of Tn. 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 Gn(p). Chapter 4. Decomposition Of Field Via Block-Spin Method 65 4.4 Components of Tn In analyzing the boundedness of Tn, it is convenient to write Hn and Gn of (4.3.6) as a sum of components. First, by rationalizing S{jp>) using (4.3.1), we get stf) = { [ s rW 1 }" 1 [STV) = a2 ^ Xl^ (4A1) where [S*}-\p) = - Y [-i-YvSm(ap„) + 2 sin2(a^/2)] + m (4.4.2a) Dn{p) = 5 > i n 2 ( a p „ ) + [am + 2 £ s i n 2 ( a p „ / 2 ) ] 2 (4.4.2b) Substituting (4.4.1) back into (4.3.8a) and (4.3.8a), we write Hn(p,x) = a Y.-ilvKl,>n(p,x)+ Tn(p,x), (4.4.3a) V Gn(p) = o ^ - i ^ W + W ^ ) , (4.4.3b) where Kv,n(p,x) = (Ln)~d~1 Y e^*'*sLntf)*^Q , (4.4.4a) Tn(p,x) = (Ln)~d~l Y e ^ 5 L . ( r y ) 2 E - S i n 2 ( a f . ( 2 ) + a m , (4.4.4b) jejn Dn{p>) WJp) = £ ^ (Y^^'f^'"" , (4.4.4d) jeJn L>n{PJ) and p7' is defined in (4.3.8d). In analyzing the boundedness of | f n | in 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 = {p£Se/a | RePl/>o}. (4.4.6) Furthermore, Wlog, we choose the <it/l-component of p for the continuation into the complex plane, i.e., we take p = d in S£/a. We would like to introduce a couple of lemmas for showing that \(Ln)~dSLn(p)\ is bounded uniformly in S'e/a and for finding bounds for the components of TZUtU, Tn, Vu>n, Wn. Moreover, from the lemmas, we can easily see that TZ^n, Tn, V^n, W„ can only be singular at p = 0 where the source of singularity comes from the term 1/Dn(p°). Lemma 4.2 Let z = x + iy, x 6 [0, TT], \y\ < e, and e is sufficiently small. Then (1-) C12 \z\ < | sin ^| < c i 3 \z\ if x < n/2 (4.4.7a) C12 \n — z\ < | sin z\ < c 1 3 \n — z\ if x > n/2 (4.4.7b) Cu x < Re sin z < C i 5 x if x < n/2 (4.4.7c) cu (7r — x) < Re sinz < c i 5 (n — x) if x > n/2 (4.4.7d) (2.) c i e k - z\/2 < I cos(z/2)| < cn \n - z\/2 (4.4.8a) c i 8 (n -x)/2< Re cos(z/2) < c 1 9 (TT - x)/2 (4.4.8b) (3.) for j an integer and aq — z, C20 1 + -n sin(aq/2) sin(aqi/2) < C21 (4.4.9) where the Ci's are positive constants. 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)\2 = sin2(:r) + sinh2(y) cos(x + iy) = cos(x) cosh(y) — i sin(x) sinh(y) | cos(x + iy)\2 = cos2(x) + cosh2(y), and for 0 < x < n/2, — x < sinx < x n \ 2 (n — — X ] < cos x < — X 2 J ~ ~ n \2 As for (4.4.9), from the above, we have 1 s i n ( a g / 2 ) | 2 < ^ 2 + y 2) and Thus sin(a 9V2)| 2 > c 2 2 r _ n sin(ag) x±2jn\2 ( y 2Ln J \2Ln \aq\ n 1 \aq/n + 2j\ ~ + sin(aqi) The upper bound follows from a similar argument as well. Q.E.D. (4.4.10) (4.4.11) (4.4.12) Chapter 4. Decomposition Of Field Via Block-Spin Method 68 L e m m a 4.3 For j e J n , j ^ 0 and 0 < Re apd < ir, define k(j) e Jn by jv if Re apl" < TT/2 N-j„ if Re apl" > TT/2 then for pd real, and for complex pd, LZnDn(pi)>c2MJ)\2, LlnDn{p>)\ < c 2 5 Ykl(j) + (L-n\j\2 + aLnmy (4.4.13) (4.4.14) (4.4.15a) (4.4.15b) \L2nDn(tf)\ > c26\k(j)\2, 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) is analytic in \z — z0\ < R, sup|z_2o|<# f(z) < M and f(z0) = m, then l/f{z) is analytic in \z - z0\ < r = {\m\/M)RjA and \l/f{z)\ < 2/\m\ for \z - z0\ < r. For each fixed j ^ 0, p„, v ^ d, we apply this result to f{pd) = Dn{p>) with R = u/2a center at a pd with pd real. For pd real, applying (4.4.14) m = f(pd) = L2nD(p>)>c24 \k(j)\2. Q.E.D. Chapter 4. Decomposition Of Field Via Block-Spin Method 69 4.5 Cancel la t ion of poles We demonstrate the cancellation of poles arises from the singularities of Hn(p,x) and Gn(p,x). From the components of Hn(p,x), Gn(p,x) in (4.4.4a-d) and Lemma 4.3, we see that the only possible singularities of Hn(p,x) and Gn(p,x) would come from the zeros of Dn(p°). To find these zeros, we write Dntf) = G ^ [ s i n 2 ( a ^ / 2 ) + (Qn,2)2], (4.5.1) where 4(1 + am + 29n) 9n = £ s i n 2 ( a p > / 2 ) . Let aPd/2 = q + ik, then sm2(apd/2Ln) + (e£,2)2 = [sm(apd/2Ln) + i&nfi] [sin(apd/2Ln) - i®nft] = [sm{q/Ln) cosh(k/Ln) + i(e£>2 + cos(q/Ln) smh(k/Ln)} [sm(q/Ln) cosh(k/Ln) + i ( 9 ^ 2 - cos(q/Ln) smh(k/Ln)}. (4.5.3) Thus Dn(p°) = 0 when q = 0, sinh(k/Ln) = ± e n t 2 (4.5.4) Consequently, both Hn(p, x), Gn(p) have first order poles at the point p satisfying (4.5.4). At a closer look on Fn(p,x,y), these singularities cancel each other out. Let G*n(p) = E ' 7 , V „ , n ( p ) 4 - W n ( p ) , (4.5.5a) fn(p) = G*n(P)Gn(p)=Gn(p)Gn(p), (4.5.5b) J? = { k e j n | MO} . (4.5.5c) Chapter 4. Decomposition Of Field Via Block-Spin Method 70 Rationalizing tn(p,x,y) of (4.3.7), we get = Hn(p,x)y^G:(p)Hn(p,y) (4.5.6) Hn(p,x)L'^-lsL(p) /n+l(P(l)) where p(i) is defined in (4.3.8d). Taking out the / = 0 term from the sum Y^iej1 a n d combining it with the first term, we have Fn(p,x,y) lrin(p,x) (4.5.7a) [UP) L[P)fn+i(Pw) Hn(p,y) Hn(p,x)L-™-hdp)^HH E e - W a L ) ^ . ( P + ^ ) ^ ( p + ^ , 2 / ) /n+l(P(l)) leJi 2rd aL = Ql,n(p) ~ Q2,n(p) where Q\,n{P) Hn(p, x)G*n{p) ZieJl sl(P + gf)<?n(p + S)) Gn+Mi))Hi(p, V) fn(p)fn+l(P(l)) (4.5.7b) 2d-i„ / ^ G n + i b ( i ) ) v - ^ _ . v o , » n M . , , 2nL^, 2nl (4.5.7c) We now show that we can removed the poles of Hn(p,x), Gn(p,x) in Qi ,n and Q2,n-We separate out the j = 0 term from the j-sums in Hn(p,x) and Gn(p,x), so that we can extract the terms responsible for the singularities. Let us write Gn(p) = gn(p°) + E 0»(PO (4-5-8A) Hn(p,x) = MP'O + E W ) , (4.5.8b) Chapter 4. Decomposition Of Field Via Block-Spin Method 71 where gntf) = (Ln)~2d~1s2Ln(pj)S(pj), (4.5.9a) hn{pP) = ( L N ) ~ D _ V 2 7 R - 7 3 : / ' A sLn(pj)S(pj). (4.5.9b) Then UP) = gnWniP0) + E 9ntf)9n&) + g n t f K t f ) + E 9ntf) E 9ntf) (4.5.10) = I L - ^ S K P 0 ) + £ ^ ( P V L O ^ ) [[5,]-1(P0)5*(p'') + ^ J f f l - 1 ^ 0 ) ] + 7Jn(p°)[ £ L - ^ P O W ] [ E ^ v ^ ' W ) ] j T2nr]( o, S V ) = « • (4.5,1) where Similarly, + E ei2^/U- 3 d n S i(^>i(p0)5(?y)[5]-1(p0) + [ £ eL2^/FLL-^SL(^)5(^)] [ £ L - M n s 2 (p»')5V")] Since the sums in Q i > n 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°n(p) = L2nDn(p°). (4.5.13) to both the numerator and the denominator of Q i , n and Q2,n- Thus we write F(p,x,y) = AnAAn,2 - Ai,3, (4.5.14) Chapter 4. Decomposition Of Field Via Block-Spin Method 72 where -4^ i,l A n,3 D°(p)Hn(p,x)Gn(p) D°n(p)fn(p) (4.5.15a) L ' ^ E slip + ^ ^ ) ) g " ( P + g)^i ( P d))^n ( P , v) - 2 d - l E e - * ( 2 T ' / a i ) l / S L ( p + 2 7 r// a L) 5 i (p ) (4.5.15b) (4.5.15c) £>S+1(P(i))gn(p,g)C?^.1(P(i))^(p+2g,y) •P£+l(P(l))/n+l(P(l)) Note that in (4.5.15b), for Lp + ^ = if Z 7^ 0 then we get cancellation between D° + 1 (p( i ) ) and the singularity of G n (p + 2^f)G'*+ 1(p(i)), and if Z = 0 then we get can-cellation between D°+l{p^) and G*+ 1(p(i))i7*(p,y). Similar cancellations are done in (4.5.15c). Now the next source of possible singularities of Tn(p,x,y) is the zeros of fn(p). Only if we can show that either / n(p) has no zero or there are cancellations of zeros, then we can be sure that Tn{p,x,y) is analytic in S£/a. 4.6 Existence of zeros of fn(p) From Section 4.5, we found that the poles of Gn and Hn do not actually pose any singularities for f n . But there is another source of singularities which we believe that they can not be removed as the poles of Gn and Hn. These non-removable singularities come from the zeros of (4.6.1) given by the following proposition. Chapter 4. Decomposition Of Field Via Block-Spin Method 73 Propos i t i on 4.1 For a < L n < 1, aLn sufficiently small, £ independent of n,a, there exist n's, p G S'E/a with pd = 1 ~ such that fn(p) = 0. Before proving the above proposition, we state a couple of observations which suggest that fn{p) should have zeros near ap = ( ± 7 r , • • •, ± 7 r ) . 1. Since V„,n has odd symmetry under pv —> —pu, (4.6.2a) K ^ I / , VK )„ has even symmetry under pv —pv, (4.6.2b) W n has even symmetry under pv —> —pv (4.6.2c) and each of the above is periodic with period 27r/a in Re pv. Thus VuM\v^ = 0 (4.6.2d) and Vi,tn as a function of pv has an odd symmetry about apv = IT, i.e., V „ ) n ( " 7 T — pv) = V„,n(7T +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~n), thus w e s e e t n a t W„(P) - ( L T ^ E ( L T M 4 V ) ^ s i n t f / 2 ) = 0(^"n + aL n m). (4.6.3) Let us investigate more thoroughly on the behavior of V„,n and W n . Away from the poles of 1/D°(p), Wn and V„,n are analytic functions at each pv e S'E/a. Since the pole of Chapter 4. Decomposition Of Field Via Block-Spin Method 74 1/D^(p) is near p = 0, we divide S'e/a into two regions where one contains the pole and the other contains the zeros of Vv>n. Let S ; / a = 5 < V 2 u 5 < 1 T / 2 l (4.6.4) where S<„,2 = {pe S'e/a | \Re apv\ < TT /2} (4.6.5) and 5,<7r/2 ^ s the complement of S<n/2 relative to S'£/a. When considering upper bounds on \V^n\ and |W„| for p e S<n/2, one should consider bounds on |£)£(p)V„ ) n(p)| and |D°(p)Wn| instead . From the observations, let us write V„,n = (TT - ap„) A v > n (p) , (4.6.6a) Wn = L~n Bn(p) + aLnmCn(p). (4.6.6b) Propos i t i on 4.2 For e- sufficiently small: 1. forpeS'e/a, c27 <ReAUtn(p), ReBn{p), Re Cn{p) (4.6.7) 2. / o rp e S<*./2, c 2 9<|D°(p)A,, n(p)|, | /J°(p)5„(p)| , |D°(p)C„(p)| < c 3 0 (4.6.8) 3. for p e 5^/2, c 3 i < |4,,„(p)| , |B„(p)|, \Cn(p)\ < c 3 2 (4.6.9) where the Ci's are positive constants. Chapter 4. Decomposition Of Field Via Block-Spin Method 75 Because the proof of the bounds is laborious, we only demonstrate a sketch for Vu>n over S<nj2. The other bounds follow by similar techniques. By viewing Vv,n{p) as a function of pu with other components being fixed, V„)Tl is analytic over the given domain. By analyticity over a compact domain and choice of smallness of e, it is suffice to show that, for p G S<n/2 and p real, c 3 3 (7r - apu) < Vv>n(p) < c 3 4 (7r - ap„) , (4.6.10) where the Q'S are positive constants. Let us rewrite V„)Tl as V„>B(p) = ( L " ) - 2 d + 2 Pto)vu,ntf), (4.6.11a) -N<jK<N;kjiu where N = (Ln — l)/2. We further simplify vUtn(p>) into N vu,n{p>) = (2N + I)""3 sin 2(ap„/2) 2 £ b(p>) (4.6.12) iu=-N where V ) = ^ f f f • (4.6,3) Next we show that YYJ„=-N HP*) c a n ^ e written as a sum of non-negative terms where each term has apv = TT as its only zero. Let us rearrange J^^JI-NHP*) s o that the first 2N terms is written as a difference of two positive quantities. Using _ ap, - ( j „ + l)27r _ ( 2 7 r - a p „ ) + j „ 2 7 T ~ 27V + 1 ~ 27V + 1 1 ' = -a((2yr/a) - pv)iv, Chapter 4. Decomposition Of Field Via Block-Spin Method 76 the first N terms E KPO = -EV')> ju=-N j „ = 0 where for K ^ v, qK = pK, and qv = (2ir/a) — pv. Thus Y b(pi) = b(pN) + NYb(pi)-b(qi). 3u=-N >=0 We would like to extract a factor pv — n/a from b{p>) — b(q3) and b(pN). Since 7T — apv (4.6.15) (4.6.16) N ap = IT 27V + 1' we have Let and write N tan((7r-o P < / ) / (2 iV + l)) l P } Dn(p») </> = ap>/2,, d = aqi"/2. (4.6.17) (4.6.18) (4.6.19) = cot(fl-cot(fl) + COt(</>) - cot(0) 1 + cot(0) Z)n(p?") Dn{qi) Dn{qj) ~ Dnjp3)' [ Dn{pi)Dn{qi) (4.6.20) Dn(pi) We extract a factor pv — IT/a from cot((/>) — cot(0) and Dn(qj) — Dn(jpi). From the difference apt" (IT - apv) d-(j) = cot((/>) - cot(0) 2 2 y> ' sin(0) cos(</>) - sin(</>) cos(0) sin(c/>) sin(0) sin(0 — (/>) sin((7r — apu)/Ln) (4.6.21) sin(</>) sin(0) sin(</>) sin(0) Dntf) - Dn(p>) = [sin 2(2^)-sin 2(20)]+4[sin 4(^)-sin 4(0)] +2 [sin2(0) - sin2 (</>)] | am + 2 sin2(apj."/2) (4.6.22) Chapter 4. Decomposition Of Field Via Block-Spin Method 77 Further simplifying the first two terms of the above yields [sin2 (29) - sin 2 (20)] + 4[sin4(0) - sin4(0)] = 4[sin2(0) cos2(0) - sin2(</>) cos2(0)] + 4[sin4(0) - sin 4 (</>)] = 4[sin2(0) - sin2(0)]. (4.6.23) Using an interpolation integral, sin2(#) - sin2(</>) = 2(9 - <f>) sin(i# + (1 - t)<j>) cos(t9 + (1 - t)<j>) dt Jo = ^ ~LnP^ [ s i n ( 2 t g + 2 ( 1 ~ * M d t (4.6.24) Note that since , for 0 < jv < N — 1, 0 < <\> < 9 < TT/2, (4.6.25) and (j) — 9 only when apv — ix. Thus since for 0 < x < IT/2, cotx is strictly decreasing and sin 2 x is strictly increasing and both are positive, it is easy to see that (4.6.18) and (4.6.20) are strictly positive except for apv = IT where V„tn(p) = 0. From (4.6.16), we break Vv<n into three parts corresponding to the two terms in (4.6.20) and b(pN). Since each part is non-negative, we can bound Vv<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 corresponding to the cot(^) — cot(#) term is 2 V L-™^rUv)Y ( L ~ 2 " S I N W 2 ) )(L-nsm((ir-aplJ)/L»)\ -N<&N-*#, u{P\^0\sm(^msm(qir/2)))\ Dn(pi) J (4.6.26) 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 3 6 (TT - ap,,), where k(j) is defined in (4.4.13). From (4.4.2b), L~2nDn(p0) = L~2n sin2 (ap./L") + [am + 2 £ s i n 2 ( a p , / 2 L n ) ] 2 ^ (4.6.28) > C 3 7 £ ( a p i y ) 2 > c 3 7 7T2/4 V since one of the p„'s is > 7r/2. Thus it follows that the j = 0 term is also bounded above by c 3 8 (7T - ap„). Q.E.D. From (4.6.1) and (4.6.6a-b) /n(p) = E < „ ( P ) ( T - ap,) 2 + (L-nBn(p) + aLnmCn(p))2. (4.6.29) V From Proposition 4.2, we expect to find a zero of fn for p near (TT, • • •, TT) and L ~ " , aLnm sufficiently small. Let us prove Proposition 4.1 now. Proof of Proposition 4.1: Let Qk = (TT,-• • ,TT,TT + ik) (4.6.30a) a„> n(fc) = Al>n(qk), bn(k) = Bn(qk), cn(k) = Cn(qk). (4.6.30b) 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 Av>n, Bn, Cn has even symmetry at ap„ = TT. By the even symmetries, it is easy to see that av^n(k), Chapter 4. Decomposition Of Field Via Block-Spin Method 79 bn{k), cn(k) are of real values. Evaluating fn(Qk), since V„>n(gfc) = 0 for v ^ d, we get /«(?*) = -ad,n(k)2 k2 + (L~nbn(k) + aLnmcn{k))2. (4.6.31) Given an e > 0, by choosing n sufficiently large and aLn sufficiently small, using the bounds in Proposition 4.2, we see that fn(qe) < 0. Since fn(qo) > 0, By the Intermediate Value theorem, there exist an k such that fn(Qk) = 0. Q.E.D. We remark that a-d,n{k) and k is of the order 0(L~n + aLnm). Also, there are more zeros at (TT — Si, • • •, TT — Sd-i,TV + ik) (4.6.33) for Si are sufficiently small. For these points, ,,, ^J(L-nbn(k) + afrmcnik))* + E^daljk)5i fc = : 77^ : 4.6.34 and k is of the order 0(L n + aLnm + S) where S = ^Y^^dM-Not long after our failed attempt in showing the sliced Fermi covariance of the block-spin 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 Gn. This allows Gn to be bounded away from zero and establishes the desired bound by using the Contour Shift Lemma. Chapter 4. Decomposition Of Field Via Block-Spin Method 80 4.7 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 po-tential 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 nonpertur-bative 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 non-perturbative 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" 5 (*' (4.7.1) where ip is a field belonging to a coarser lattice A'1) and Q is the block-spin averaging (see [GK86] for the definition). Notice T effectively integrates out degrees of freedom of the lattice action with momenta between a - 1 , the lattice spacing, and p = a _ 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~Sl in (4.7.1), 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 - L^-dyiL(2-d)(N(x,y)-i) ( 4 ? > 2 ) where N(x, y) is the smallest positive integer N s.t. [L~Nx] = [L~Ny] and the interaction is Vm = - £ v($(x)) = - £ \p2 : $ 2 : + A : & : . (4.7.3) X X Z ^ , 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 Vn is local, the recursion (4.7.1) reduces to a recursion for a function of one variable: 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 vn = tnv given by (reftvphi) can be approximated by perturbation theory is that, using analyticity of tv in a strip, one can extract the <&2 and <&4 terms of tv from a Taylor expansion at $ = 0. The coefficient of the $ 2 and $ 4 terms can than be shown to have good approximation by the second order perturbation theory of the model. Take d = 4. If | $ / L + f | < 0 (A~ 1 / 4 ) and u2 is of order A 2 , it follows that \v\ < 1. Thus we make the following partition: <3> is in the small-field region if |$| < B\~llA where B is some large constant. Let x be a partition function such that tv($) = - log jdpr{0 e x p ( - L ^ ( L ( 2 - ^ 2 $ + 0)- (4.7.4) x(0 = x ( l ^ l < O ( 5 - 4 A - 1 / 4 ) ) (4.7.5) and express tv as tvx + tvx± (4.7.6) where tn x (4.7.7) toxx = Jdp(0 ( 1 -x ) exp( -L 4 u($ /L + 0 . (4.7.8) With the cutoff x, using the positivity of v, for |$| < . B A - 1 / 4 , we can show that tvxx is a small nonperturbative contribution, since tvx± = Q(exp(-cA- 1 / 2 )) < 0(A n ) , Vn , (4.7.9) Chapter 4. Decomposition Of Field Via Block-Spin Method 83 due to the small probability of |£| > 0(A1/4) i n the Gaussian d/j(£). Using the analyticity of tv in a strip | i ra$ | < A ~ 1 / 4 , we Taylor-expand tv at $ = 0 : where (d/d&)v'>6(0) = 0,i < 6. By Cauchy's theorem, for |$ | < £?A~ 1 / 4 , (4.7.10) - ($) < 0 ( e x p ( - c A - 1 / 2 ) ) , i = 2,4,6. (4.7.11) Thus the contribution of (tv)x± to the coefficients in the Taylor expansion is nonpertur-batively small. Furthermore, by Taylor expanding tvx (denoting u = L 4 u ( $ / L + £)), 1 1 rl tvx($) = const.+ < u >x - - < u2 >x +- J dt(l - t)2 < u3 > £ t , (4.7.12) where < w >x= jf J wxdp, < w >x,t= -A; j 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) tvx can be well approximated by our standard perturbation theory. From a second order 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: A' v' « > 6 L2p2 + 0(\2,p2X), A - & A 2 + O(AVA), 0(A2), O(A) • (4.7.15) (4.7.16) (4.7.17) (4.7.18) 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 In C [—CnA2, CnA2] of values for the bare mass p? in (4.7.3), such that (an.) exp(—vn(<$>)) is analytic on \Im$\ < B(Xn)"1^4 and satisfies \exp(-vn{§))\ < exp(-A 1 / 2|$| 2 + A„(/m<I>)4 + L>) , (4.7.19) where D = 0(1) constant. (bn.) For |$ | < SA - 1 / 4 , « " ( * ) = ^ 2 : $ 2 : + ^ : $ 4 : +r? n : $ 6 : + ^ 6 ( $ ) , (4.7.20) with (d/d&)v$6(0) = 0, i < 6, and 1/41 < C o A 2 , (4 < (4 ^>6 1 < C 2 A „ , (4 and 1 , 1 1 , - + c_n < A„ < - + c+n . (4.7.24) A A A n immediate corollary of the above result is that for A < A0, there exists a pCrit{X) in (4.7.3), such that Jim exp(-t nu($))-4 1 (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 anal-ysis 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 averag-ing (e.g. a Wilson average on gauge group elements which are defined on bonds instead of sites of the lattice [BJ86]) . Chapter 5 Loop Regularized Running Covariance 5.1 Running covariance From their multiplicative renormalization scheme, the physicists derived a R G flow equa-tion for determining the U V and IR behavior of the perturbative Green functions. In the QED model, the physicists' flow equation seems to depend only on the photon self-energy diagrams because of the uZi = Z 2 " condition. Although we have some reservations on 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 di-agrams 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 Hs to be described in the below. In the next section, we will determine the explicit form of the running covariances in a RC 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 IQED 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 RC 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 con-sists 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 (renor-malized 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 = / ^t(p)Ke(p)^(p) dp of the effective potential into the sliced covariance by means of the following lemma. Here (•)* denotes the transpose of (•). Lemma 5.1 Suppose (HKeY = KeH, then 1 ^ ^ logfevW-l*tK'*dnH(Z) = --BtReB + logc + E n = l L n! log Je^^d^iO , (5.1.1) A=0 where $ = B + £, H = (H^ + Ke)-1 = [l + KeH)-lH = H{l + HKe)-1 (5.1.2a) B = (l + HKe)-1B = (l-HKe)B (5.1.2b) Ke = {l + KeH)Ke. (5.1.2c) Furthermore, the covariance of B is J=(l + HKe)-1J(l + KeH)-1. (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 71 = 1 L pm r ^ log J e x v ^ e - ^ K ^ t K ° B + B t K ^ d i i H ( 0 (5.1.4) 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, {HKey = KeH, we have (HKey = [HKey{\ + (KHy)-1 = K£H(I + HKeyl = K£H. Hence C{H~l + Ke)£, + CKeB + BlK£ = (£ + HKeBy H (f + HKeB) - Bl KeHKe B. (5.1.5) Thus by making the change of variable £ —> £ — HKeB, | F ( 0 e-i«'^+f t i f « f l +^««d/i H (0 = c c i B t / f - ^ B J F(E- HKeB) d/z*(0. where c = [det(l + i f K e ) ] _ 1 is the normalization factor. Hence (5.1.4) becomes (B'K^l-HKe] B)+logc+Y 71=1 ^ log Jexv{^-kK^dpk{C) (5.1.6) A=0 The quadratic term Bl Ke[l - HKe] B = B* (1 + [HKey)Ke B = Bl (1 + # e # K e B = Bl KeB (5.1.7) yielding (5.1.1). Q.E.D. Chapter 5. Loop Regularized Running Covariance 90 Let us demonstrate the first step of the iteration of combining (5.1.1) and renom-ralization 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-u = p-uC, where C is the original full covariance, into J = C < u = p<uC and H = Hu = puC, with Ke = Ku being the original quadratic counterterms. In the QED model, the corresponding C and Ku have respective kernels, C(P) = D{p) 0 0 0 0 S{p) 0 -5*(p) 0 \ Ku(p) = V K%ip) 0 0 0 0 o -Kgipy KUAP) o \ ) (5.1.8a) where D(p), Sip) are defined in (2.4.7), (2.4.4) respectively and K%ip) = iX^-l)p2 + M2Ui\^-M-2U KuFip)=mi\li-l) + iXu2-l)pJ. (5.1.8b) Here we don't shift the id-A)2 term (see the discussion after (5.4.11) for an explanation). From Cip) and Kuip), it is straightforward to see that iHuKuY = KUHU. 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 n=l L | log Je^^dp^) A=0 1 ~ - °° 1 -B*6KUB + 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 expecta-tion w.r.t. dpfji^). We add the quadratic local terms ^B^SK11 B to further modify Ke. Now setting K"-1 = Ke-5KU = KU + KUHUKU-5KU, C < u = J = il + HUKU)-1 p<uC il+ KUHU)~\ (5.1.10a) (5.1.10b) Chapter 5. Loop Regularized Running Covariance 91 we then sp l i t C<u i n to c<(u-i) H u - i [1 + HuKu)~l P<U-VC (1 + KUHU)~\ ; i + HuKuyl pu~l C (1 + KuHuYl (5.1.11) for the sh i f t ing i n the next scale s = U — 1. C o n t i n u i n g the above process leads to the f o l l o w i n g i t e r a t i on : K*-1 = Ks-5Kl + KlHsKt H s-l 1 + HaK8e)-1pa-iC (1 + K'H3)-1 (5.1.12a) (5.1.12b) a n d Hs = (1 + KseHs)~lH\ C<s = (1 + HsKse)~l p<sC (1 + KIH3)-1 (5.1.13a) (5.1.13b) M o r e o v e r , by a n i n d u c t i o n a rgument , i t is easy to see tha t the r u n n i n g te rms have the f o l l o w i n g m a t r i x forms HS(P) 0 H*(p) = H%(p) 0 0 0 HsF{p) { 0 -(H'FY(p) 0 7 o N 0 0 -(HF(p)Y \ 0 H'M o , [ KB(P) o Kip) 0 o o -(KiF(p)y \ 0 KlF(p) 0 1 H%{p) 0 f C£s(p) 0 0 ^ c<s(P) = V 0 0 Cf{p) 0 - ( C < s ) J ( p ) 0 ] where each set K^x, H x , C x s } , X G {B, F}, also satisfy the equat ions (5.1.12a-b), (5.1.13a-b). Chapter 5. Loop Regularized Running Covariance 92 5.2 R C w i t h neighbourly sl icing 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. We let R = Cx and drop the subscript X in {Hx, Ks x , Hx, Cxs}. Let Ks be the kernel of the shifted local (photon or Fermi) quadratic terms in the shifting scheme which has the form Ks = Y XajKj (5.2.1a) 3 and A f 1 = A* - 8] (or Ks~l = Ks - 8KS), (5.2.1b) where Ks 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 Hs = ps R (l + ps+1 RKs+l)~2 (5.2.2a) u Kse = Ks + R52 P\jtf (5.2.2b) t=s+l and ps R H " = 1 + if R IC + [(1 + R A T - + 1 ) 2 - 1]' ( 5 2 - 3 a ) 6 < - = (if^RK-r ( 5 ' 2 ' 3 b ) where an d pu+l = K u + 1 = 0. 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 neigh-bourly, p8K8 = p8 [Ks + p8+lR(K8+l)2] . (5.2.5) Hence HsKse =ps R ( l + ps+l R K s + i y 2 [Ks + ps+1 R (Ks+1)2] (5.2.6a) and KseHsKse = psR{Js)2. (5.2.6b) From (5.2.6a), (5.1.12b) becomes H8-1 = ps~1 R (l + ps RK8)'2 . From (5.2.6b) and (5.2.4), (5.1.12a) becomes u K8-1 = Ks + R52 PV) 2 -SK" + psR{J8)2 (5.2.7) t=s+l U = K8-1 + RY,Pt(Jt)2-t=s From (5.1.13b), (5.2.6a) and the fact the slicing is neighbourly, we have (5.2.3b). Substi-tuting (5.2.2a-b) and (5.2.6a) into (5.1.13a), and using the fact that (ps + ps+1) = 1 on the overlapping support of p8 and p8+1, p8R(H8)-1 = (l + H8K8) (l + p s + 1 RKs+l)2 = ( l + ps+1 R K8+lf + p8R [K8 + p8+lR(Ks+lf\ = l + psRK8 + 2ps+lRK8+1 + (p8-rps+1)ps+1(RKs+l)2 Chapter 5. Loop Regularized Running Covariance 94 1 + ps R Ks + 2ps+1 R Ks+1 + ps+l (R Ks+l)2 1 + ps R Ks + ps+l [(1 + R Ks+lf - 1]. Q.E.D We shall call the term VS(X) ~ D>(\y (5.2.8a) where Ds{\) = (1 - ps - ps+1) + ps(l + RKS) + ps+l (1 + R Ks+1)2, (5.2.8b) the running slicing for the sliced covariance Hs. The running slicing serves as the mod-ified slicing which has a dependence on the running couplings Xs. Under a R C scheme, the coefficients of the R C C flow are computed using Feynman diagrams with lines corre-sponding to these modified covariances. 5.3 Loop Regularization We would like to incorporate the loop regularization of [FHRW88] in the running covari-ance 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 mod-ified by inserting additional terms as monomials in the fictitious spinor fields ^j,^fj, j = 1, 2, 3, where ^ is a Fermi field and \&2, ^3 are Bose fields with propagator M 2 ( A ) = m2 + 2A 2 , M 2 2 (A) = m 2 + A 2 , M 3 2 (A) = m2 + A 2 . (5.3.2) < >= 5(M,-) (5.3.1) where Chapter 5. Loop Regularized Running Covariance 95 For convenience, we define M 2(A) = m2, (5.3.3) and tf = ( t f 0 , - - - , t f 3 ) , tf = (tfo,---,tf 3), (5.3.4a) $ = (A , t f 0 , t f o , - - - , t f 3 , t f 3 ) , (5.3.4b) where tf0 = ip and tf0 = tp. The modified Lagrangian (cf. (1.2.5a-b)) depends on an UV cutoff U on the photon field and a loop regularization parameter A: C = C0 + d (5.3.5) where Co = \[A-(-A)A + V(l-V)-\d-A)2 + A2}+J2^J(Mj-ip)^j, (5.3.6) 3=0 Ci = ]Ttf, [ z A f ' A 4 - ^ A 2 / ' A - l ¥ + ( A 4 / ' A - l ) M J ] tf,- (5.3.7) i=o +\ \(^'A ~ I)A • {-A)A + M2U\fKA2 + A6U'A(<9 • A)2 + A^'M 4 ] . Here, the Af'A have a dependence on U and A. From the expanded Lagrangian, the corresponding free covariance C\ of $ has the following non-zero components : ( C A ) I I = D (CA)23 = -(CA)l = S(M0) ( C A ) 4 5 = - ( C A ) £ = SfMx) (5.3.8) ( C A ) 6 7 = (CA)l = (CA)89 = (CA)l = S(M2), where S(Mj) 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 GA 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 Uth slice of the ^-covariances corresponds to scales [U, oo). More explicitly, for h <U, Sh{Mj) = ph S{Mj), Dh = ph D . (5.3.9) and s u = p[u,oo) S(Mj^ Du = pu D (5.3.10) where ph is the neighbourly slicing functions defined in (B.2a) of Appendix B. 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 LA described in (3.31) of [FHRW88]. For graphs G with no fictitious legs or with LG = 0, LKG = LG. For a graph Gj with fictitious legs of 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: (LAGJ)(A^],^J) = j:{Jf) (LSG)(A^^3), (5.3.11) where L$G consists of the monomials of degree 8. Chapter 5. Loop Regularized Running Covariance 97 5.4 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 Kse in (5.2.2b) and Vs'(A) in (5.2.8a) are well defined and uniformly bounded in U and A. First, Kse 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 t + 1 R Kt+l in (5.2.4) stays away from zero. As for Vs(X), we need to examine its explicit form as a function of the running couplings Af. We make these imposed conditions explicit by imposing that the slicing ps at scale s, 0 < 5 < U, has the following support, (1 - e)M2s < p2 < M2s+2. (5.4.1) 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 IQED 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 CSA using the neighborly slicing ps. We then apply the R C scheme of Section 5.1 to modify the slicing ps of the components of CA into Vs(X) using (5.2.8a-b). 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 = (A- - 1) + M™Xl (5.4.5) and the corresponding Ks and R in Lemma 5.2 are Ks = [(A* - 1) p2L + M2s Xs5] , (5.4.6) Chapter 5. Loop Regularized Running Covariance 98 p 2 + 1 ^ (1 - r/) + p 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 are given by LV3 < G >fr=5^ A(p2L)A, LVs < G >„= A2. (5.4.7) Note the reason for introducing the factor M2s in the coefficient of the A2 counterterm is that by counting the naive degree of divergence, the coefficient 5\ of an A2-\oca\ term is Q)(M2s). By explicitly having the factor M2s, the corresponding running coupling Xs5 becomes dimensionless. From (C.1.5a-b) of Appendix C, the running photon covariance is HSB = HSL + H^ (5.4.8a) i + p i where V**W = i + ^ . + p.+i[(^+i + i ) 2 _ 1 ] ( 5 - 4 9 a ) V h> A X ) = 1 + P s k + P p s + ^ Y (5.4.9b) and Y = K + Xl, A - = ^ _ ^ ) + p 2 5 (5.4.9c) ^ = w - *) (if-*) > ^ = ^ r$- ( 5 A 9 d ) Chapter 5. Loop Regularized Running Covariance 99 In the Feynman gauge where r\ = 0, p2 + 1 and in the Landau gauge, where rj = 1, ~ Vj>A\) L + VSBT(X) T H'B = B M ^ , »'T{ 1 , (5.4.10) VSBAX)L H% = -ffj,-. (5.4.11) Note that because we prefer to use the Landau gauge in L R C for IQED, in the corre-sponding tree expansion, a (<9 • A)2 local part of a subgraph at scale t contracting at a lower scale s < t produces a zero kernel since the resulting lines from the contraction con-tain 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)2 counterterm in the running potential for making the 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 VBL(X), VSBT(X) have the following bounds. n . ( A ) < ^ < l + , ^ ; y - l > . ( 5 „ 3 a ) Vs (X) < 1 + (1 - V)M~2*-2 where JS = Xs3 + Xl (5.4.14) 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 = [Xs2-1)TI+(\\-1)M3 (5.4.16) R3 = feZ+M,)-1. The local parts L < G of selected Feynman diagrams G (e.g. the electron self-energy diagram) that are being shifted into the "external" quadratic term ip KF tp are given by LV2 < G >k= 52l- < G >H-= <J4i * M * . (5.4.17) where 82 and 84 are ^-independent. From (C.2.1)-(C.2.8) of Appendix C, we have HFJ = VpjRj = p'iDfi-iRj (5.4.18) where for s = U, Df = l + puA"; (5.4.19a) D] = 1 + psA] + ps+1B°+l- (5.4.19b) A] = RlKjRj, Bsj=R4j(ksJ)2 + 2R2jKsj, (5.4.19c) K] = A^Mj + A ^ + ^ - A ^ M ^ , (5.4.19d) R) = (p2 + M2)-\ A? = A ? - 1 . (5.4.19e) Also we impose the conditions, A^ > 1-e, (5.4.20a) A^ > 1, (5.4.20b) A > m a x ( M c / + 1 , 4 M c / + 1 sup \~XS2\), (5.4.20c) 0<s<U for s < U, 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 - > \us\ rs (5.4.20d) 4 1 1 S- > \us\rs+ 2\us+1Xs2+1\rs+1+ (us+1)2 (rs+1)2, (5.4.20e) where us = \\ — \\ and (rs)~2 = p2 + m2. 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 | P F , O I I O O < T 7 . (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 regular-ization 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 regular-ization 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. The first level removes the 102 Chapter 6. Two-Slice 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 renormaliza-tion 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']^ w n e r e pWM = Yli=aPli l s 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^°'u^ (Note that U is not the U V cutoff parameter 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 Mu. 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\ > Mu+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 IQED 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 A-singularities of the non-renormalized graphs. The third section describes how to remove these A-singularities by renormalizing at the top scale using only the F2 counterterm corresponding to the F2 local part of the scale U+1 vacuum polarization (VP) diagram. 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 Mu uniformly in U'. We denote p^+1 — p[u+1>u'} and puB = p^°'u^ respectively as the top photon slicing and the bottom photon slicing. As for the spinor fields, the top slicing is p f + 1 = plu+1'°°) and the bottom slicing is pF = p^°'u\ We apply the L R C to IQED with only two slices corresponding to the above slicing of the fields. The starting effective potential at the top scale has the form, E A 7 + 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 A4 counterterms by setting A f + 1 = A f + 1 = 0 here because the (d • A)2 and AA local parts of the graphs from the expansion do not contain 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 In the two-slice model A- = Af for 0 < s < U. We will see later in Section 6.3 that by choosing A ^ + 1 as the coefficient (which is of O(A); s e e Proposition 6.1) of the F2 local part of the VPU+1 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). O ( lnA) , (6.1.2) and for Af ^ A3 with 0 < s < U + 1 (6.1.3) 1. O(i ) (6.1.4) O(l ) if 3 = 0 (6.1.5) 2. Further suppose the condition in (6.1.2). t O ( anA)" 1 ) ifs = U + l , O ( l ) if s < U (6.1.6) Chapter 6. Two-Slice LRC 106 \nB oo 0(( lnA)- x ) 2 / 5 = [7 + 1 { O ( l ) ifs<U (6.1.7) 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 s = r+^Ag, and using (6.1.2) and (6.1.3), the bound in (5.4.13a) simplifies to V%<{ r i I 7 T r = 0(( lnA)- 1 ) ifs = £/ + l £r-77 = 0 ( l ) i f s<[7 Q.E.D. (6.1.8) Let us denote L%j1 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 Ln, n must be an even number (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 00 2. For n > 6, l l - ^ n j ^ l o o — I 0(1) if 3 = 0 0(A" 2 ) if j ? 0 O ( l ) if 3 = 0 0(A" 2 ) if3 7^0 (6.1.9) (6.1.10) Chapter 6. Two-Slice LRC 1 0 7 Proof: see Appendix D. Next we consider the spinor loops L 2 + l and L%+1. 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 n A ) P a r t of the F2 local part of VPU+1. Proposition 6.1 1. 3 VPU+1 = A[Y sign(j) L%fl ) A = /hF* + fo(d-A)2 + A I £ sign{j)RLu2^ ) A, \j=o / \j=0 ( 6 . 1 . 1 1 ) where (\U+1\2 \A2 ) fo = O ( i ) 0 ( 0 ( A _ 1 ) if 3*0 i O ( l ) if3=0 l ^ l l o o = { • ( 6 .1 -13 ) 2. ^ | 1 | | o o — O ( l ) if 3 = 0 { 0 (A~ 1 ) if3 + 0 ( 6 . 1 . 1 4 ) 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 108 6.2 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. Further-more, 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 Af at the next scale are constants inde-pendent 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 VPU+1 graph and by choos-ing a counterterm which corresponds to the F2 local part of VPu+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'A = j dQdPKs(P,Q)H^(p). (6.2.1) i 2. | | G A | | = || / dQ\Ka(P,Q)\ |U = sup J dQ\Ks(P,Q)\. (6.2.2) 3. Halloo = M o o (6.2.3) where Hi is the covariance corresponding to the line L Chapter 6. Two-Slice LRC 109 4. L N is a spinor loop containing n spinor lines. 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. CBT+1{G) is the set of scale U + l photon lines of G. 9. GL{G) is the subset of spinor loops of G excluding the loop L 2 + L loops; a is the index labeling the loops in QL. 10. QLU+L(G) is the set of L 2 + 1 spinor loops'of G; a2 is the index for QLu+i(G). 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 The following proposition allows one to narrow down the search of A-divergent graphs. P ropos i t i on 6.2 Suppose GSA contains more scale U+l photon lines than L2+l's or of G. it has a fictitious line not belonging to an L2+1, then 0 (6.2.4) Proof: A l l lines are estimated by | | ^ | | o o 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 GSA contains more scale U+1 photon lines than L 2 + 1 , s then \\G%\\ < c ( l n A ) ' ^ + l h | £ r i Vol(Q) \ \ J] I I ^ I U < 0 ( ( l n A ) - x ) . (6.2.5) 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 - VPu+l = J dp A(p) K^ip) A(-p). (6.2.6) where - K™ = L^H^jr) • • • L ^ H ^ L ^ (6.2.7) *• ' n—l copies 2. For n > 0, let u+i u + 1 u+i U+1 U+1 u + 1 u + ) = I dpdr^p-r)i{r)Hl+l(r)Kvn^p{r)^{-~p) (6.2.8) Note that VQ_VPU+I is just an ordinary vertex. Chapter 6. Two-Slice LRC 111 Lemma 6.3 If GA+1 is not a generalized vertex Vn_vpu+i or a graph consisting only of the Vn_vpu+i's connected by Fermi lines then lim \\GuA+l\\ = 0. (6.2.9) A—>oo Proof: Note that each Vn_vpu+i has the same number of scale U+1 photon lines as the number of L2+l's and each internal n — VP subgraph has one more scale U+1 photon lines than its L2+l,s. Hence if the reduced graph GA+l/Vn-vpu+i, obtained by replacing each Vn-vpu+i with an ordinary vertex v in GA+l, contains fictitious lines or scale U+1 photon lines then it satisfies the hypothesis of Proposition 6.2 and thus lim | |GJ [ + 1 | | =0 . (6.2.10) A—>oo Q . E . D . Corollary 6.1 The only (nonrenormalized) GuA+l that can have l i m ^ o o | | G A + 1 | | ^ 0 are the graphs: n — VPU+1; L^+1{Ni, • • •, Nn) with n VNi^vpu+i vertices and n > 4; n-chained V^-ypU+i, 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 — VPu+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 GeA containing an n — VPu+1&s 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 VPU+1 and we have the following result. Chapter 6. Two-Slice LRC 112 Figure 6.8: n - VPu+l Chapter 6. Two-Slice LRC 113 U+l • • • U+1 U+l Figure 6.10: VN._vpu+i, 1 < i < n From (6.1.12), let us write h{VP +) = bvp = b\nA + C3 + o(l) (6.2.11) where (Af + 1 ) 2 6 = " C ( A l # ' ( 6 ' 2 ' 1 2 ) C is a positive the constant, and C 3 is a constant independent of A but with a possible, dependence on U. Proposition 6.3 By choosing A f + 1 = Klt \ u 2 + l = K2 — K4, = b\nA + C3 + K3, = K5 (6.2.13) (6.2.14) where the Ki are independent of A, then lim Af = Ki A->oo (6.2.15) Chapter 6. Two-Slice LRC 114 6.3 Renormal ized 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 IQED 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 GA obtained at the root scale containing n — VPu,s or Vn_vpu's then lim \\G%\\ = 0. (6.3.1) A— » o o In other words, by Corollary 6.3 and the above, in the limit A —¥ oo, a non-vanishing Ge is not allowed to contain any unrenormalized L^ 's , U scale photon lines, or fictitious lines. Actually (6.3.1) might not be true in some cases where an n — VPU subgraph is external. 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 Ve C {p | p2 < (1 - e)M2U} (6.3.2) where e is the constant defined in the slicing functions ps. 2. For a region V | | / (p) | | P = sup|/(p)|. (6.3.3) pEV 3. Let (G) denotes the graph G with all its legs amputated. In particular, u + 1 U+l U+l U+l (n-VPu+1) = U+l U+l u + 1 u + 1 = Lu2+\p){HuB+\p) Lu2+\p)T-' (6.3.4) Chapter 6. Two-Slice LRC 115 and u+i u+i (V, n—vp' 0 U+1 U+1 U+1 U+1 {HuB+\p)Lu2+\p)) n (6.3.5) 4. For a given G, we group the non-renormalized L 2 ' s of G into the following two types of subgraphs with legs amputated. (a) A maximal k-chained n — VPu+l of a graph G is a sequence of n — VPu+1,s chained together by scale U photon lines. The sequence is obtained by am-putating all the legs of a subgraph Gmax-n-vp which is a chain consisting of n — VPu+1,s and two Vn-vpu+i's with the following arrangement. Each Vn-vpu+l is not a part of a (non-renormalized)!/^1, the n — VPu+1,s and the Vn_vpu+i's are connected by scale U photon lines with the two Vn^vpu+i's are situated at both ends of the chain. We denote Max(n — VPu+l)nu...^nk as a maximal k-chained n - VPu+l. Graphically, a Max(n - VPu+1)nu...tTlk is the following diagram. (V n. v pu+,)vAy-(n 2-VP u + 1) • • • vA^(n k -yp u + 1 ) v / V X Vn.VP-) 1 k = (Vni_vpU+i)H^(n2 - VPu+l) • • • HuB{nk^ - VPu+1)H^(Vnk_vpv+1), where k > 2; nx, nk > 0; for 1 < i < k, rii > 2; £*=o n; > 0(note that V1_vpu+i = VPu+l are all renormalized at scale U+1). The reason for calling it maximal is that, by the definition of a Max(n — VPu+1)nu...>nk where the two U U U (6.3.6) Chapter 6. Two-Slice LRC 116 Vn_vpu+i's are not part of a (non-renormalized)^4"1), it cannot be contained in a longer Max(n - VPu+1)nu...tnk where k > k. (b) In a similarly manner, we define a maximal k-chained Vn_vpu+i (an external subgraph with legs amputated) as the sequence obtained from the a chain like a Gmax-n-vp except that there is only one Vn_vpu+i situated at one end of the chain. We denote Max(Vn_VPu+i) n\,—,nk a s a maximal k-chained Vn_vpu+i. Graphically, a Max(Vn_VPu+i)nit...;nk is the following diagram. U U (y n. v p U + 1)vyv^(n2-vpu + 1) • • • v/v-\n k -vp u + 1 ) = iVni_vpv+i)HuB(n2 - VPu+l) • ••HuB{nk - VPU+1), (6.3.7) where k > 1; nx > 0, ; for 1 < i < k, rii > 2; YH=O ^ > 0. To get control on bounds of graphs G\ containing Vn_vpu+i and n - VPU+1 contracted at lower scales, it suffices to consider bounds on maximal k-chained n — VPu+l and maximal k-chained Vni_vpu+i. We consider the following two lemmas on bounds of n — VPU+1 subgraphs. Lemma 6.4 Let nx > 1 and n 2 > 2 be positive integers. 1. Fort<U -1, (i) (n 2 - VPu+l)HB = 0 (6.3.8) (") ( V n i _ ^ + 1 ) ^ = 0. (6.3.9) 2. (i) Alim ||(V n i_ t ) p t/ +i)|| 0 0 = l (6.3.10) (ii) lim | | ( V n i _ , p C / + 1 ) ^ | | o o = 0. (6.3.11) A—>CO Chapter 6. Two-Slice LRC 117 3. i) lim \\{n2-VPu+l)HuB\\o0< 1, • (6.3.12) A—>-o (ii) lim 11 (n 2 - VPu+l)H%\\x = 0. (6.3.13) A—>o Proof: See Appendix D L e m m a 6.5 1. (a) For (n^n^) / (0,0), 2. Urn || Max(n - V P u + 1 ) n i ! . . . ! r i f e IU = 0, (6.3.14) (b) For (m,nfc) = (0,0), Um || Max(n - IU < • (6-3.15) Um || Max(n - VPu+1)nu...tTlk \\x = 0 (6.3.16) || Max{Vn_vpu+i)nu...,nk \\Ve = 0 (6.3.17) ^. Suppose that the external momentum feeding into Max(n — VPu+1)nu...ink is re-stricted to the domain Vere = {P\P = Y1 aiPi ; \Pi\<re; <k = 0 or ± 1}, (6.3.18) where re is a positive constant. If m re < (1 — e)l/2Mu+1 then 11 Max{n - VPu+1)nu...>nk \\^ = 0. (6.3.19) Chapter 6. Two-Slice LRC 118 Proof: Part (3-4) follows from the fact the support of pu+l is disjointed from the Ve,s. For Part 1(a), Wlog, we suppose n\ ^ 0. Using the fact that fc—I || Max(n-VPu+1)nu...tnk < {{(V^u^H"^ JJ \\(ni-VPu+1)Hu\\00 11(V„fe_w^ +i)11 (6.3.20) from (6.3.11), (6.3.12), (6.3.10) of Lemma (6.4), we have (6.3.14). If (n 1 ; n f c ) = (0,0), then the first term of (6.3.20) IKKj-opU+O-H^ Hoo = H a l l o o < ^jj M 2 U + 2 " (6.3.21) Since the kernel of an ordinary vertex is 1, the last term ||(Vo_upi/+i)||oo = 1- Thus by (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 f c) = (0,0), || Max{n-VPu+1)ni,.,nk ||a < H ^ I U n Wi^-VP^1)^^ | | ( n f c _ 1 - V P 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 limA_>oo | | C A | | = 0 if it contains an n— VPU+1. The strategy of the proof is that we factor the kernel of G\ into sets of Maximal chains, L f + 1 ' s , and the remaining lines. We can first select a subset of internal chains corresponding to the loop momenta (excluding L2+l) 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 ^ + 1 ' s . Chapter 6. Two-Slice LRC 119 In proving the proposition, we adopt the notations defined in Section 6.22 and intro-duce more in the following. Let us drop the subscripts nu-,nk from the Maximal chains. Given a G, 1. Nlmax(G) is the set of internal Max{n - VPU+1) of G; 2. N^ax(G) is the set of external Max{n - VPU+1) of G; 3. Vmax(G) i s the set of (external) Max(Vn_vpu+i) of G; 4. Q\(G) is the set of loop momenta (excluding L 2 + l loops)of G that run through the Maxin - VPu+lYs; 5. we arrange the loop momenta so that for each q e Qi(G), we select a Max(n — VPu+l)(q) where q is the momentum feeding into Max(n — VPU+1); NLax(G) i s the set of these selected internal Max(n - l /P c / + 1 ) (g) ' s ; 6. Q2(G) = Q(G)/Q1(G); 7. C^+l(G) is the set of scale U photon lines of G not belonging to a Max(n — VPu+1) or a Max(Vn-vpu+i); 8. C(G) is the set of lines of G not belonging to a spinor loop, a Max(n — VPu+l) or a Max(Vn_vpu+i). Proposition 6.4 lim IIGlll = 0, (6.3.23) A-K30 if GeA satisfies one of the following: 1. it contains a fictitious line; 2. it contains an internal Max(n — VPU+1); Chapter 6. Two-Slice LRC 120 3. it contains a (external) Max(Vn_vpu+i); 4- it contains a scale U+l photon line not contained in a Max(n — VPU+1) or a Max(Vn_vpu+i); 5. it contains an external Max(n — VPu+l) and m external sources where the support of each of the source is a subset of the domain Ve = {p\p2 < ( l /m) 2 ( l - e)M2U+2}. (6.3.24) 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 \\Max(Vn_vpU+i)\\De II \\Max{n-VPu+l)\\be " m a x i y m a x voi(Q2) (A3J+I)-I^+II n nwioo n imioo aegL (.ecicuB+l II WMaxin-VP^1)]^ ]J \\Max(n - 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 2 ,-,n f c_ 1 ,o in the limit A —> oo since the number of external 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 2 ,- ,n f c_i,o by an 1-norm. Chapter 6. Two-Slice LRC 121 Theorem 6.1 Suppose each external source <$>\ has bound Halloo < c e . (6.3.26) where ce is a positive constant. If G\ contains a fictitious line or a scale U photon line then H m | G A | = 0. (6.3.27) where \G%\ is the "absolute value" of the graph with external sources included. Proof: Let {(j>t}i<i<m be the set of external source for G\. We consider two cases. 1. Suppose that N^ax is empty, then m \GeA\ < \\Gl\\ n i l t f l l o o (deY 1=1 < l | G A | | (ce)m ( r e ) 4 m - 4 - (6.3.28) which, by Part 1-4 of Lemma 6.4, goes to zero in the limit A -» co. 2. Suppose N^ax 0. Let Max(h — VP) be an element of N!^ax and N^ax = N*max/{Max{h-VP)}. m \GeA\ < \\GA/N*max\\ II H a l l o o (^ e) 4 m - 8 n \\Max{n-VPu^)\U i=l Me max [ dPi\Max(h-VP)(Pi + P ) | , (6.3.29) where G A / A ^ a x * s the reduced graph obtained by deleting all the external Max(n — VPU+1) in GeA, Pi is the external momentum of <j>\ and P is a linear 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%\ < \\GejNemax\\ (ce)m(de)4m-8 JJ \\Max{n-VPu+l)\\oa \\Max(n - VP)\\x. (6.3.30) fife J max From the bound in (6.3.25) and Lemma (6.5), it is easy to see lim \\GejN*max\\ (6.3.31) 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 Mu+1. In the less restrictive models, one must do a multi-scale expansion on the bottom slice corresponding to the slicing p ' 0 ' ^ . In the remaining chapters, we will present a L R C model with "full" renormalization where the top slice is A-renomalized by only a F2-counterterm and the bottom slice is renormalized by the R C 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 limA_><x> coupling flow, we may discard any GA+l that contains a fictitious line or a scale U+1 photon line. The following corollary describes what kind of GuK+l can survive the limA_>oo-Corollary 6.2 If G% is non-vanishing in the limit A —> oo then any G A + 1 subgraph of G\ must be a graph consisted of only ordinary vertices connected by Fermi 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 A f + 1 = Ki, i^3 (6.3.32) Af 4 " 1 = blnA + K3, 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 GrA be a graph ofV0r. lim lim \GrA\ < KG(U). (6.3.33) where KG is a finite constant depending only on U and the perturbation order of G. The above corollary implies that only the top scale renormalization is required for ob-tained 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^°'u^ is further sliced up into [7 + 1 slices, and after integrating out the scale U + l shifted slice as described in the two slice model, the RC 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 Vs($-S) and a flow of running 124 Chapter 7. Multi-Slice LRC 125 external quadratic terms Q-SK8$-S are generated from iterating the procedure described 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 VS(Q-S) and &-sKse$-s. Recall that in the running 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 VS(<S>-S) as depicted in (3.6.12). In L R C , a flow equation is obtained in a similar way except that the graphs of the running potential V s ( $ - S ) contain no local quadratic 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 3 3 vi = £ ^ ¥ % ^2 = 5 ; * ^ , v4 = E W i - (7.i.i) j=0 j=0 j=0 V3 = F2, V5 = A2, V6 = (d-A)2, V7 = A\ 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 -\SV° = - A T 1 ! ? + £ LVsG (7.1.2) where Qs is the set of graphs obtained from the scale s potential y s($< s + 1). The corre-sponding flow equation is then A J + 1 - A J = § (7.1.3) Chapter 7. Multi-Slice LRC 126 where St = E Pi(G) (7-1-4) Gegs and Pi{G) are the coefficients in the localizations LViG = P!(G)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 A f + 1 = Ki(U), (7.2.1) X3+1 = b\nA + K3(U), 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 V3 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 GV, 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 Qs as the set of these graphs obtained from the trees The corresponding flow equation is then A * + 1 - A? = St (7.2.3) where st = E Pt(G) Gegs (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 V4, V5 local parts, the photon mass coupling Xs5 and electron mass coupling A 4 should be dimensionful couplings which respectively should carry a factor M2s and Ms. 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 ESE diagram in Appendix E). Thus we single out Af as the only dimensionful coupling. In anticipating the M2s factor in the A 5 equation, we rescale the equation at scale s + 1 by M2s and get A * + 1 - A * = — ( M 2 — l ) A g + 1 + M~2s8sb+l. (7.2.5) 7.3 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 p[U,oo)_ 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, Af = K$(U) where Chapter 7. Multi-Slice LRC 129 K3(U) is independent of A and Af = A f + 1 (7.3.1) 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 RC model, and that the scale U + l and U Fermi lines can be combined as scale U Fermi lines of a non-loop-regularized RC 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 + ^B,R where the first part is V% restricted to the left overlapping 5ff and the sharp region S^h of pu and the second part is V% restricted to the right overlapping region SR of pu. Let psL, respectively psR, denotes ps restricted to the left overlapping region S[, respectively the right overlapping region. From (5.4.9a) and the fact that p^ + pu+l = 1, 1 - o u + 1 Vu — H L (7 I 0) B'R (i-pD(f+i)+r(f+i+i)2 { ] where js is defined in (5.4.9c). Thus it is easy to see that, since j u + 1 = 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 £ H ( p ) | | i = 0. (7.3.3) Thus' after taking the limit A -+ oo, we can disregard the VBR P a r t °f 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 7. Multi-Slice LRC 130 line of the subgraphs represents the sum H^+1 + HUF = + VUF)R (7.3.4) where Vp+1 and Vp7 are the running Fermi slicing functions, and R(q) = (m + We show that Vp+l + Vp has the same form as VF+1 except the slicing function pu of Vp is replaced by pu + p[ c / + 1 '°°] = pluM. We illustrate our point by explicitly adding the two running slicing functions VF+1, Vp together. From (7.3.1) Af = Af + 1, and so for simplicity, we set Af = A f + 1 = A and we write the Fermi slicing functions (see (C.2.31a)) as •pu+i _ P 1 + ^ + 1 . 0 0 ) ^ - 1 ) ' i + pu(x-i) + Piu+^)(\2-iy 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 VVF. By (E.l.24) of Lemma E.2, we see that on Sg, VuF+l + VVp = ^f-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 conve-nience 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(sa), a>0, AXs = o(Xs), 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(-^~a s) 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~as) irrelevant. 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-B = 3m (7.3.7) where VSB 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 s is defined in (5.4.9c). In calculating the j5 function of low order diagrams, we would like to approximate 1 + 7 s by 7 s = \ s 3 + Af for extracting leading terms. From the fact that Xs Xs 1 + 7 s 7' s \ M 2 S ) A'+Y^" 1' Xs (1 + 7 S ) 7 S = 0 ( ( M 2 - 1 ) ( A » ; ) r ' ( A ! " 1 ) ) . (7-3.8) where xs is the characteristic function of the support of p s, in the calculation of the (5 functions, we may replace the 1 + Y by js in V% defined in (5.4.13a) with the caution that there is an error of Q((M2 - l)\sb(*ys)~2) fr°m e a c h photon line. Next we consider the Fermi running covariance which has the form HSF = VSFR (7.3.9) Chapter 7. Multi-Slice LRC 132 where R(p) = (tf+m) 1 and Vp is the running Fermi slicing function defined in (5.4.18). By assuming the order in (7.3.5), from (C.2.31a) of Appendix C, ns ns Es n = * - j B V + & > (7"3-10) where Bs = (1 - p 5 - p s + 1 ) + p s A* + p s + 1 ( A ^ 1 ) 2 , (7.3.11) - W § v &) ' (7-3-12) Es(p) = us (ps R) + 2us+1 Xp-1 ( p s + 1 R) + {us+1)2 (ps+1 R2), and us = m(X\ — Xs,). We show that Es is irrelevant. Observe that, for a graph G, each insertion of a term psR in a Fermi line of G lowers the degree of divergence D{G) by 1. Thus two insertions of psR in a 2P diagram (defined in Figure 7.11) reduces the corresponding D(G) = 0. This means, in a zero order localization of the resulting diagram, there can only be a V5 local term with coefficient of o(M2s). But since the A 5 equation is scaled by M~2s, this local term is irrelevant. From the above observation, we see that only the 2P and 2F diagrams (defined in Figure 7.11) with an insertion of one psR can produce relevant terms. Thus by assuming the orders in (7.3.5), effectively in our calculation, we may use p>s _ P_ P O I o\ 5 s (1 - ps - p°+l) + p° Xs2 + ( A ^ 1 ) 2 ' ^ ' provided that we include the following extra diagrams obtained from replacing the run-ning slicing function of some Fermi lines of the relevant diagrams by the relevant part of the second term psEs(Bs)~2 of (7.3.10). More explicitly, let HSEF = -mV^'R2 (7.3.14) where ,E,S __ SPS (A^ - A*) + 2 p*+! A $ + 1 ( A S + 1 - A £ + 1 ) F ~ P ( B O * : ' (7.3.15) Chapter 7. Multi-Slice LRC 133 Given a diagram < G >B where each Fermi line corresponds to a H , let < G >E B denotes a diagram where one of the Fermi lines corresponds to a HEF and each of the remaining Fermi lines corresponds to a HSF with Vp defined in (7.3.13). To account for the contributions corresponding to the term Es in (7.3.10) , for each 2P or 2F diagram G, we include the relevant local terms of all < G >E>H: in the set up of the coupling flow. Actually, of the extra diagrams < 2P >E B and < 2F >E jj, only the latter has relevant local terms. We first demonstrate in a simple case of a one loop < 2P >E ^ using a sharp slicing where the slicing functions ps are characteristic functions. The sharp slicing versions of Vp and VE'S are obtained by setting ps+l = 0 and Bs = A | in VSF of (7.3.13) and in VE'S of (7.3.15). Consider the following one loop <2P >E B and its local parts from a zero and a first order localizations. ( A ^ - A l O l < 2^ >U ( A § ) W 2 J dpA<°(p)K^(p)A<s(-p), Ks^{p,m) = m J dqps(q) ps{p + q)tr[f1 R{p + q) Y R(q)2}- (7.3.16) Let us write m q1 + m2 q2 + m2 (7.3.17) In the zero order localization, K'mo = mj dq ps(q) tr [f 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 dqps(q) tr Y R(q) Y R(q> m = 0 (1) . q* + Now since the A 5 equation is scaled by M _ 2 s , Ks^u0V5s is an irrelevant term. The kernel of the first order localization is KUM = P°M J d ( 1 PS^) T R [T" R(qhaR(q) iv R(q)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/{q2 + m2), is zero. For a general < 2P >E fi, we use the same argument as the one loop case to show that there are no relevant terms. Note that, firstly, the degree of divergence D(< 2P >E fi) of < 2P >E fi is 1 and so there can only be relevant terms from zero or first order local-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 o(M2s). Since the A 5 equation is scaled by 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 PidPi 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 >Efi remain irrelevant. To account for the contributions from the < 2F >E ^'s, it is better to combine the contributions with LVi < 2F >fi in the following way. Since D(< 2F >Ejfi) = 0, there can only be relevant terms from zero order localizations contributing to the (5 function Chapter 7. Multi-Slice LRC 135 o~4 of A 4 . Thus, we consider LV4 < 2F > with each running Fermi covariance replaced by Ha+tF = HSF + HSEF = VSFR - mVpsR2 (7.3.18) where VSF and VF'S are defined in (7.3.13) and (7.3.15) respectively. Since 2 q 2 - m 2 + 2m<JL m-<$ ~ R W = / 2 , 2^ 2 , R(q) = 2 , 2 , ( g 2 + m 2 ) 2 g 2 + m 2 we have H V t e ) - m(T"C + ^ + ^ - n^H- (7.3.19) g 2 + m 2 ( g 2 + m 2 ) 2 g 2 + m2 Next we show that the relevant part of L y 4 < 2F > can be computed effectively with using only part of Hs+F. As in the argument of the irrelevancy of < 2P >E it suffices to consider an 2F that has no subgraph which requires renormalization. Since LVi < 2F > is obtained from the zero order localization, it is easy to see that the kernel 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+ F(q) by m(VsF + Vps) q2 + m2 and the rest of the other F(g)'s by (7.3.20) (7-3-21) qz + m 2 Since Vp + VF'S is linear in the A4's, the above implies that LVi < 2F > has coefficients linear in the A4's. Subsequently, since the effective Vp defined in (7.3.13) involves only A 2 and no X\, except for the A 4 equation, all other equations of the coupling flow have no A 4 appears in them. Chapter 7. Multi-Slice 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 Ae(p) are used in the generating functional, the Feynman diagrams that contain lines carrying only external momenta p with \p\ bounded do not require renormalization because ps(p) = 0 for sufficiently high s. For example, let us consider < V , VP > = R S X W • (7.3.22) From the calculation of the V5 local part of < VP > in Appendix E, it is easy to see that the relevant zero order local part of (7.3.22) is b 5 X\ M2s < v,A2 >H-3 where b 5 is defined in (E.2.76a). Since u £ M 2 S < «, A2 >ks | (7.3.23) s=r = dp dq i)e{p) 52 i i ^ s 2^ ^ (let n q2 7° 9 the restricted support of ps(q) and Ae(q) being compactly supported allow (7.3.23) to be 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 V6. Since V6 = A p2T 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 4 and A 6 in the fi functions in the following proposition. Proposition 7.1 The fi function 5\ is linear in A 4 and contains no Xe- For the fi functions 5s with i ^ 4, Sf contain no A 4 nor Xe-Chapter 7. Multi-Slice LRC 137 7.4 Summary of multi-sl ice LRC for IQED We summarize the multi-slice LRC 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 + Xu2+lM2UV2 + - 1)V3 + ( A ^ 1 - 1)V4 +\%+1M2UV5 + \V+1V6 + \V+lV7} where 3 3 3 3=0 3=0 3=0 V3 = -A p2L A, V5 = -A2, V6 = -Ap2TA, V7 = -A\ M 2 ( A ) = m 2 , M 2 ( A ) = m 2 + 2 A 2 , Ml {A) = m 2 + A 2 , M 2 ( A ) = m 2 + A 2 . T - A _ P^P" T - P t i P u (7 A 0\ L ^ ~ V p 2 ' V - ~p^T- [7 A.2) 2. Running terms Running fields: - scale s transformed "external" fields u+i <|><s = JJ ^ (7.4.3) i=s+l where and Ti is the scale i shifting transformation defined in Lemma 5.1; Chapter 7. Multi-Slice LRC 138 scale s decomposition of the field: = $<«-i + £s (7.4.4) where £ s is the scale s shifted slice field. Running external quadratic term $ - s Kse <3>-5: U+l K t , t+l r> (]St+l\2 Kse = KS + RJ2 p'iJ1?, Jt = [ —-• t^+i 1 + Pt+l R Kt+l The nontrivial components of Ks and F t (w.r.t. the 9 components of $- s ) are : K'n = KB = (\l-l)p2L + M2° Xs5, RU = RB, K'j+2J+3 = .signU) (K°+W)T = K'j = (Xs2 - l)p!+ (X\ - l)Mjt Rj+2,j+s = sign(j) (Rj+3J+2)T = Rj, R b = TTp^> R3 = tp!+M3)-1, (7.4.5) where 0 < j < 3, sign(0) = sign(l) = -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+p2 1 - (ps + ps+1) + (p s7 s + 1) + /9S+1(7*+1 + l ) 2 .2 \ M2s where L is defined in (7.4.2). Chapter 7. Multi-Slice LRC 139 • Effective Fermi covariance: Hp = Vp Ro V s = t F (1 - P s -ps+1) +ps \S2 +PS+1 (\S2+l)2 where RQ is defined in (7.4.5). • Running potentials VS: — top scale potential vu+i(Q<u+i) = [Af+1y1+A2/+1M2t/Vr2 + ( A 3 7 + 1 - l ) V 3 + (A^ + 1-l)y 4 +\V+1M2UV5 + Af + 1y 6 + \U7+LV7] ( $ ^ + 1 ) where the V^($) are defined in (7.4.5); - top scale shifted potential vu+i^<u+^u+i) = x"+lv^u+^+1)+\%+lv6(^uHU+1); (7.4.6) — scale s running potential OO -I n=l n -= <PS Kse <p5 + T > ( < | < s ) where is the scale s + 1 shifted slice field, £ ^ s + 1 ( - ) is the connected expectation w.r.t. dp^3+1, and VS(Q-S) 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 VR contain no top scale photon lines and fictitious lines. Chapter 7. Multi-Slice LRC 140 3. Imposed conditions on parameters • Forms of the top scales couplings: A f + 1 = Ki{U) i ^ 3 , A f + 1 = Mn A + K3(U), (7.4.7) b - c^+l)2 b - 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: Xs = 0(sa), a>0, AXs = o(Xs), l<t<7 (7.4.8) A* > \XS5\, Xs2>l-e, XSA>1, where A A | = A | + 1 - Xs. 4. A —>• oo coupling flow equations Af = A f + 1 = UU) z#3, . XU3=KS(U). Let QGV = the set of relevant generalized vertex diagrams obtained at scale s Q2P = the set of relevant 2 photon legs diagrams obtained at scale s Q2F = the set of relevant 2 Fermi legs diagrams obtained at scale s Q\p = the set of relevant 4 photon legs diagrams obtained at scale s. (7.4.9) Chapter 7. Multi-Slice LRC 141 For 0 < s < U - 1, AA; = 6'+1 = £ Pi+l(G) (7.4.10) AA5 = <$|+1 = £ ^ S + 1 ( G ) G e S 2 F + i A A 3 = = £ $ + i ( G ) A A 4 = S'A+1= £ Gee,^1 A A 5 = -5M A f 1 + M~2s5s5+1 = -5M A f 1 + M~2s Y Gee**1 A A * = S'6+1 = Y Pt\G) A A ? = 5s+l= Y, ^7+1(G) G e ^ 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 /34, the Fermi lines are replaced by the effective Fermi covariance; • pi with i 7^ 4 are independent of A 4 and A6 , and /?4 is linear in A 4 and inde-pendent of A 6 . We will make further analysis of the A —>• co coupling flow to determine the depen-dence 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 LRC 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) rs = (^i)2 f« i n - ( A ^ A i + Af)- ^ • } In perturbation theory, it is desirable to keep £ s small through out the flow of the running 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 s ^ . 0 0 £ s = 0. In Section 8.2, we analyze a one variable O D E in £ s obtained from a primitive flow of IQED with sharp slicing. We discover that, by choosing the value £ r of £ at the root scale r sufficiently small, the flow of £ s of IQED is U V free. From the solution of £ s , one then can recover the O D E estimates of the Af's. In Section 8.3, we apply the running 142 Chapter 8. LRC coupling flow with sharp slicing 143 covariance technique to the negative charge 4>4 perturbative model and demonstrate that 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 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 QED for following two reasons. Since, for Q E D with real A f + 1 , the coefficient of the V 3 local part of the scale U+1 VP diagram is negative (see Proposition 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 pu+l, i.e., the denominator otherwise (8.1.2) with the property plp> = <% p\ (8.1.3) (8.1.4) Chapter 8. LRC coupling How with sharp slicing 144 of VB+l has a zero for pu+1 sufficiently small. 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 QED coupling flow. Since the IQED flow can be obtained from the QED flow by switching A x to iXi, we do not distinguish the 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 = — ^ (8.1.5a) H8F = p'iD")-1^ (8.1.5b) where R~1(p) = m L is the projection defined in (7.4.2), and 7* = W - U ^ + A ^ (8.1.6a) D , = 1 + p , ( A S - i ) m 2 + ( ^ - i y + ( A | - A ; w f i pz + mz The corresponding effective sharp Fermi slicing functions are obtained by setting ps+l 0 and Bs = Xs, in Vp of (7.3.13) and Vp + Vp's of (7.3.20); n = £ (8-i-7) w , -n*,. PL , PS ( A 4 ~ A 2 ) = P ^ A | X°, (A|) 2 (A | ) 2 ' Vp + Vp Chapter 8. LRC coupling flow with sharp slicing 145 8.2 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 GV, 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, VP and ESE respectively. For relevant diagrams with n loops, GV^n\ 2P<~n\ 2F^n\ 4PW denote the n loop G V , 2P, 2F, 4P diagrams. The relevant (3 functions of the primitive flow are the following: 61 = Pi(GVW), 52 = (32{2F^), (8.2.1) h = fc(VP), S4 = (34(ESE), S5 = fo(VP), 5, = MVP), 57 = /M4P(1)). We put in the two loop diagrams in the Ai and A 2 equations because in the Landau gauge, LVl < V > and Ly2 < ESE > are irrelevant (see Section E.2 of Appendix E). 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: 5M 51 5M fis 5M °4 5M 51 5M 5M 5\ 5M b\2 &22 h (Af) s\5 (-M)4(r)2 (Af) 4 (Ai) 3 (r) 2 (Af) 2 Ag (Af) 2 (AI) 2 3(A|) 2 (Af)2 (1 + * U A ' ) ) (1 + E 6 2 2 (A S ) ) (8.2.2) A AT A l 7 s (Af)2 5 (A!) 2 (Af) 2 6 (AI) 2 (Af) 4 (AIT where b3, 6 4, & i 2 , 6 2 2 a r e constants independent of Xs and /?|, Af are described below. If A? = A,-+ o(l), l < j < 2 where A i and A 2 are constants, then each of As, A|> A 7 has the following form (8.2.3) A i - Af A 2 - A Af A2 (8.2.4) Ai / \ 2^ where is a constant. In the calculation, there is a factor 5M = M2 — 1 coming from the size of the support of the slicing function pl. We divide each 5- by this factor in anticipating making a switch to its O D E analogue. Each of Ebl2(Xs), Eb22(Xs) is of 0(5M A |/(7 s ) 2 ) and comes from replacing 1 + 7 s by 7 s in the photon running slicings (see (7.3.8)). Chapter 8. LRC coupling flow with sharp slicing 147 8.2.2 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/8M where 8M = M2 — 1, and the discrete index 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 ds d\2 ds dXz ds dA 4 ds dX5 ds dX6 ds dX7 ds 022 1 (Ai) 5 7 2 (A 2) 4' 1 (Ai) 4 7 2 (A 2) 3' A? A? \2 q\2 A6(Al,A 2), A2 OA2 b4 X 1 (Ai) 2 7(A 2) 2 -A 5 + /35(Ai,A2) Ae(Ai,A2), /?7(Ai,A2), where 7 = A 3 + A 5 , 63, 64 are constants defined in (E.2.75), and "A? A?" A5(Ai,A2) = b5 Ae(Ai,A2) = b, /?r(Ai,A2) = b7 X2 A 2 A ? _ A | X22 A 2 . X\ A\ A 4 A 4 (8.2.5a) (8.2.5b) (8.2.5c) (8.2.5d) (8.2.5e) (8.2.5f) (8.2.5g) (8.2.6) where b5, b6 and b7 are constants defined in (E.2.75). Note that the terms corresponding to the F 's of the discrete flow become zero because they are 0((8M)2) as 8M —> 0. Since the subsystem (A 1 ; A 2 , A 3 , A 5) has no dependence on other couplings, it is con-venient to study the subsystem first. In analyzing the subsystem, we first drop the term Chapter 8. LRC coupling flow with sharp slicing 148 A 2 /3e(Ai, A 2) in the A 3 equation and justify its dismissal after finding estimates for the solution of the resulting subsystem. In analyzing the subsystem, we follow the wisdom of the physicists by introducing the variables a EE A 2 , WEE A 2 . (8.2.7) We study the flow of the a, UJ, and 7 instead of Ai , A 2 , and A 3 . From (8.2.5a)-(8.2.5c) and (8.2.5e), the corresponding O D E in the new variables is dot , cn2 . n n . — = 2 a bi2 - 5 — ^ 8.2.8a ds UJ2 x ^ = 2ujb22-^, (8.2.8b) ds UJ2 j2 dy a — = 6 3 7 ds a; 7 A 5 - 05 -UJ. (8.2.8c) ^ = - A 5 + /? 5 (A 1 ,A 2 ) . (8.2.8d) The U V asymptotic behavior of the Aj's can easily be obtained once we find the asymp-totic behavior of the new variables. Observe that from (8.2.8d) A ! ~ / ? 5 ( A i , A 2 ) and ~ 0 ( 8- 2- 9) and the rest of the system, asymptotically, is independent of the A 5 equation. Using (8.2.9), we rewrite (8.2.8a)-(8.2.8c) as I £ = 2 612 (8.2.10a) a ds \u> 7/ I £ = 2b22 ( - " - ) ' , (8.2.10b) UJ ds \u> 7/ LtL „ ft3^. (8.2.10c) 7 as UJ 7 Chapter 8. LRC coupling How with sharp slicing 149 From (8.2.10a-c), one sees that the system naturally yields a flow of the variable C = — . (8.2.11) UJ 7 We would like to interpret £ as the product ( Z i / Z 2 ) ( l / Z 3 ) in the physicists' multiplicative renormalization scheme. The physicists derive the flow of the running charge square a from a flow equation of the renormalization multiplicative factor (l/Z3) of a. In their set up, Zi = Z2 and there is no A 5 contribution in 1/Z3 because of the Ward Identities; £ corresponds to 1/Z3 and only the F2 local part of the 2P diagrams contributes to l/Z3. It turns out that the flow of £ determines the flow of all the running couplings. Differentiating £ by s yields 1 dC = ]_da _ l_duj_ _ 1 dj (8 2 12) £ ds a ds co ds 7 ds 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 QED: ^ = & £ 2 + & £ 3 , (8.2.13a) ds 02 = -h, & = 2 (61 2 - b22). (8.2.13b) As for IQED, since Ai —> iXi, we have the change £ —> — £ in the £ equation: ^ = -A>£2 + &£ 3. (8.2.14) ds Since the RHS 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 b3 (see (E.2.75) of Appendix E), we have b3 < 0; thus /?2 > 0. From the sign of the term j32 C 2 [ l + {P3/P2) (], we sketch 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 r < |A>/A|, C ~ , r ( C v (8-2.15) C r c 3 (s - r) + 1 where c 3 = —b3 > 0, 7" is the root scale, and — 1 < r < s. From (8.2.15), we find the asymptotic behavior of the running couplings of IQED. Let 4 - x W . x ( S ; c 3 - , 7 ^ 1 + C 3 ( s _ 1 r ) ( e ; / r ) . (8-2.16) Then C = e2r/Y, C ~ (er/Y) s a n d s o 1 da din OJ ct ds ds 2 b12 C 2 - 2 x(s)2, (8.2.17) 1 da; din a; „ , ~ , / e 2 N 2 = 2 6 2 2 C - 2 6 2 2 :x(s)2, a; ds ds Y7' 1 d7 din 7 e2 7 d 7 = ~ d s ~ = c 3 C - c 3 - , ( s ) . This implies that a \ 2b12 el Hence \u)rJ c3 Y J' a ~ a r exp ^ [z(s) - i f ) , (8.2.19) / 2 622 e2 \ ^ ~ w exp I —— — [a;(s) - 1] J , 7 ~ 7 r x(s) = c 3 (s - r) e2 + Y • Chapter 8. LRC coupling flow with sharp slicing 152 This suggests that both a, to approach a non-trivial fixed point: oo _ l- / \ r „ „ „ (o 1^2 er 2 a- = }}ma(s) = ar exp \2 ^ ^ j , (8.2.20) o;°° = lim u(s) = OJT exp (2 — ^ ] . From (8.2.19), (8.2.7) and (8.2.5d), for C UVAF, we obtain the following U V asymp-totic 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 SM.) For IQED, A l „ A ^ x p ^ ^ (8.2.21) A 2 ~ X2 exp ^ ^ [x(s) - 1] j , A 3 ~ c 3 (s - r) e2r + Xr3, 64 / C 3 4 ~ ^4 AA ~ A R By setting Al = A ' e x p f t l ) ' A2 = A ' e x p f t r ) ' (8'2'22) It is easy to see that, for i = 1, 2, 2 £ ^ = j » W + O M , ) 2 ) . (8.2.23) i=l,2 A * C3 7 This justifies the dropping of the /?6(Ai, A2) term in the A 3 equation. From (8.2.5f-8.2.5g), (8.2.9), (8.2.6), A 5 ~ -2 h (bu -b22) et x(s) ( g 2 2 4 ) A 6 ~ h(bx2 - b22) lnx(s), A 7 ~ 67(^ 12 - b22) lnx(s). We summarize the U V limit of the couplings of IQED in the following table. Chapter 8. LRC coupling how with sharp slicing 153 UV fixed points of (UVAF) IQED zero fixed point nonzero fixed points divergent couplings C°°, A5°° \ oo \ oo \s \s \s \s A3> A4> A6> A7 Remarks 1) It is desirable to pick the boundary conditions of the couplings so that one can de-termine for what set of values (or the asymptotic values) of Xu, the resulting coupling 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 £ s stays close to origin. 2) Even though A 5 is a zero fixed point, the actual coupling of the photon mass coun-terterm is of Q(M2U/U) and hence not a zero fixed point. 3) If we consider the primitive flow of QED, from the phase diagrams, we cannot de-termine 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°r 7 > 0), V v ^ 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 ex-ponent 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 slicing 154 8.2.3 The "Zi = Z 2 " condition of the coupling flow Because of an U V cutoff U introduced in the scale decomposition for regularizing the divergence in the tree expansion, the Ward Identity of QED 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 Z1 — Z2. We do not wish to take such assumptions on faith without a rigorous justification. Nonetheless, we would like to implement the physicists' idea of "Zi = Z2 in QED. In the coupling flow of the tree expansion scheme, we interpret the "Zx = Z 2 " condition in IQED as e\s2 = X[(l + o(l)) where Xu lim -rjj = e. (8.2.25) (7-foo A f ; To study the "Zx = Z2 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°° = e2. Let m be the order of the perturbation (in () used in the primitive flow and b\n and b2n 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 _ 1 da 1 du e ds a ds u ds = 1^2 - 622) C2 + (6 1 3 - M C 3 + • • • + (61m - b2M)CM] = K(C).(8.2.26b) Let K(s) - K(r) = Jrs K(C(*)) dt so that e(s) = e{r)eK^s)-K{r). (8.2.26c) Chapter 8. LRC coupling how with sharp slicing 155 Since ((s) = 0(l/s), it is easy to see that e°° = e(r)e-K(r\ Thus e(r) = e°°eK^ in order to establish UZ\ = Z2 for the primitive flow. Remarks 1. Note that if the initial condition e(r) = e2, then it is easy to see that we must have K(C) = 0. Thus b\n = b2n for all n < m; otherwise, e°° would shift away from e2. In this instance, the "Zi = Z2 condition of IQED in perturbation theory is equivalent to b\n = b2n for all n. The condition that (8.2.26b)=0 has stronger implication than Ai = e A 2 ( l + o(l)): it actually implies that Ai = e X2 at every step of the flow. 2. So far, our analysis of the the uZi — Z2 condition has been done in the Landau gauge. To establish criteria for the "Zi = Z2 condition in other gauges, one might try considering the equation A 2 d(\xj\2) = 1 d\1 1 d\2 Ai ds Ai ds A 2 ds using ^ = Pi(X) and ^ = P2(X) computed with the gauge parameter n ^ 1. Chapter 8. LRC coupling flow with sharp slicing 156 8.3 R C scheme for other models 8.3.1 (f>4 model We would like to apply the RC scheme to </>4 and analyze the primitive O D E analogue of 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 RC scheme for c/>4: 1. The running potential is vs = w + (M2SXS2 - m)v2s + ( A 3 - i)v3; 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 (p4 is fl=-TT—a- (8-3-2) 4. The external term for generating the running covariance is K\ = M2s\s2 - m + (A* - l)p2. (8.3.3) 5. The couplings Af satisfying the flow equations Af"1 = Af + 6? (8.3.4) where = LVl <V>&.= b 1 6 M ^ V 1 (8.3.5) M~2sSs2V2s = M~2s LV2 <SE>fi3=b26M{-^V2, SS3V3S = LV3 <SE>B.= b 9 6 M ^ V 3 is = K, + A 3 . From (5.2.3a) of Lemma 5.2, the iterated running covariance is Vs(X) os Hs = VS(X) = (8.3.6) p2 + m2 v ; 7 s v ' where as usual, with the assumptions 7* = 0(sa), a > 0 and 7 s > C , we denote = as equality up to 0(M~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 slicing 158 Setting g = Xi/'j2 and assuming that A 2 - b2 g2 7 = 0(g3), we have 1 dAi Ax ds dg ds = h g, = 9 1 dj 7 ds 1 dAi 2d7 = h g2 + 0(g3), = h g2 + 0(g3)-(8.3.8) Ai ds 7 ds We interpret g as the Z\jZ\ 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 9 Ai 7 A 2 A 3 1 Ai -g' 61 (s + 1) + 1 1 -gr h [s + 1) + 1 (8.3.9) 7 exp I -h (s + 1) + 1 &2 (Pr)2 7 r {-gr bX (S + 1) + 1)2 exp &! L - 5 r 61 (S + 1) + 1 7 exp 61 bx (s + 1) + 1 - 1 A,. In a > 0; thus, for g to stabilize around the origin, we need g < 0 which means Ai < 0. For g UVAF, we have Ai = 0 ( l / s ) a n d A 2 = 0 ( V 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 3 in order to determine that the system would stabilize. On the other hand, under the RC 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 3 flows to a non-trivial fixed point (8.3.10) \ 00 A 3 y e — 6 3 / 6 1 for the U V A F case. Chapter 8. LRC coupling Row with sharp slicing 159 8.3.2 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\ • • • Ar. In applying the R C scheme, each dimensionful coupling Aj of a dimensionful local countert-erm 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 XfMdiSVi. For example, in Q E D , the photon mass counterterm is expressed as Xs5M2sA2. We would like to determine the cor-responding "g" equation since it supposedly governs the flow of all the couplings involved in the model. Observe that the g's of QED and (/>4 correspond to the renormalized charge 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 QED, we did not want to have square-root factor in the equation, thus we were considering C = Af/[(A2) 2(A 3 + A5)] instead of 9 = A i / [A2\ /A3 + A5].) Differentiating g by the scale s yields dg = ds 1 dX -^v 1 dji (8.3.12) X ds i 2 7J ds 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 pertur-bation. 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 gn. Thus (8.3.13) is just gn with the 1/^/77 s associated to the uncontracted legs removed, i.e., = 9n II STi- (8-3-14) nUeC{G)li fceA(G) where & denotes a leg of G. For the case that G is a generalized vertex diagram, H v ^ = ^ ; (8-3.15) &eA(o y hence, n X \ = A ^ - 1 . (8.3.16) 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 A2, the coupling term is of the form gn 7,. Let 7$ be of the form where { } is the set of couplings for the counterterms Ki = J2j ^ ijVij u s e d m the shifting process for modifying the covariance of Ai. Since dXij/ds 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 Md^sXsn = Mdi^s+^XSitl + Ssi+l Chapter 8. LRC coupling how with sharp slicing 161 where is the naive degree of divergence of It is easy to see that the above dimensionful flow can easily be converted into the following dimensionless one. A Xs- MDIJs8sJrl Mdij - 1 13 Md>j - 1 Thus the corresponding O D E analogue is of the form dXn ds -Xij + cjign + ---. (8.3.17) Hence A -^ ~ c 7, gn, and they have no effect on the g equation since dX^/ds ~ 0. Hence we see that (l/ji)(dji/ds) is also a polynomial in g. Chapter 9 L R C coupling flow w i t h smooth sl icing 9.1 Pre l iminar ies 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 Hs (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 LRC 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 £ 5 = A ^ A ^ A f + Af)) stabilizes and 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 slicing 163 by using a rigorous contraction argument. 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 Es = Xs — A s where As is the primitive solution As described in Section 8.2. We would like to give more on the preliminary setting for the (A —>• oo) coupling flow with smooth slicing. The smooth slicing {ps} used here is the one defined in Appendix 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), (9.1.1) K = O 0 0 , K = 0(sa), a>o, xs5 = o(l), Xs6 = 0 ( M , A* = 0(ms), AAf = o(Af) A 2 > 1 - e, 7 5 = A 3 + Af > C 7 , A* > 1, where both e and C1 are some positive constants with e < 1. Later, we will see that for 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 ps (see Appendix F). In expanding ps, the lowest order ESE diagram has a nontriyial relevant contribution to the A 2 equation. As seen in the lowest order flow under sharp slicing, it is the absence of the relevant contribution from the vertex and ESE diagrams and the VP 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 £ s than the leading contribution coming from 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 s _ 1 , ps, ps+1 and their intersections. Thus it is convenient to separate ps into the following three parts. From (B.3a), we write ps = psL + pssh + psR with support Ss = ((1 — e)M2s, M2s+2), where the corresponding supports are SSL = ((1 - e)M2s, M2s+2), Sssh = [ M 2 s , (1 - e)M2s+2}, SSR = ((1 -e)M2s+2,M2s+2). In computing these integrals, it is often convenient to rescale Ss to S1 by making a change of variable q —> Msq in the integration momenta. When scaling back to the scale s=l, we drop the superscript 1 in the above notations. Correspondingly, we also separate the running slicing Vs into three parts, VSL, Vssh, VR. 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 ps. Because of the presence of the derivative, one no longer can extract a factor 8M 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 FMr), X„(l - Mr)), A ) ™ ' ^ P * W ) ' A ) + ) u dr F(,»(r),l - P . W . A l ^ ' ^ M f " ( r ) ' 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 f1 dx F(x,0, X ) d V { x : ° ' X ) + F ( l - x,x, A) Jo dx dV(l - x,z, A) dz dV(y,x, A) dy y=l-x It is not obvious that the integral above is 0(5M) unless the integrand explicitly contains a factor 6M-Chapter 9. LRC coupling flow with smooth slicing 166 9.2 The p r imi t ive flow From the O D E analysis in Chapter 8, the relevant terms of (3 functions are estimated by orders in powers of ( 8 = (A;) 2/(A|)7 S where 7 s = A 3 + Xs5. From the asymptotic estimates (8.2.21), (8.2.24) of A 5 , we see that it is convenient to estimate the relevant terms of the (3 functions by order in powers of 1/7S. In this section, we set up and solve a primitive flow of (7.4.10) which is obtained from using only the 0 ( V 7 S ) a n c l OiK/Y) terms in the one loop (3 functions. The solution of the one loop flow is then expressed as the solution A.s of the primitive flow plus an error term which can be shown to be o(A 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, VP, ESE, 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 con-tributions from Appendix F are 0 (V (Y ) 2 ) a n d they contribute only to /32. From the 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 s by A* for the convenience in indexing and use the notations os = Af/A 2 , and E = A i / A 2 introduced in Appendix E. P'(VP) = 5M{b,[(as)2-E2]+Ja}, {3s(4PU) = 5M{b7[(as)4-ZA]+J°}, where SM = M2 — 1, the coefficients 6, are defined in (E.2.75) of Appendix E , and the Pl(V) = 0, PS2(ESE) = SM J*2, PSs(VP) = SM j [h (as)2 + J>] - bj [(a*)2 - S 2] j , (9.2.1) 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 Ts — r - — J i ~ 8 M 1 M E A A ' ^ . + tf, .7=1,2,7 E AAJ<% + # i=l,2,7,4 for i = 3, 5, 6, 7 1 8M where o , ( 7 s ) 2 , E AAJG: .7=1,2 m o for i — 3, 5, 6 and £ = 1,2 A? Gij = O s \ 2 ' = O U; rAj and for i = 4,7, O (*') 7 sAf = o V ) (9.2.2) (9.2.3) (9.2.4) (9.2.5) (9.2.6) (9.2.7) Since from the sharp analysis, we expect 7 s = 0(s)> the terms involving AAf in (9.2.2-9.2.4) may be regarded as terms of 0 ( V s 2 ) - We set up a primitive flow by us-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 s of the primitive 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 solu-tion with deviation of o(A s) from A 5 . Now by dropping the terms (9.2.2-9.2.4) in (9.2.1), the (3 functions of the primitive flow become Chapter 9. LRC coupling how with smooth slicing 168 At = A4S o, $ = o 6. 3 <TS)2 (9.2.8) SMb3 (cr5)2 - | [(a*)2 - E" 64 <5M A 4 AI M 2 5 " 2 Ae <W M2 b5 [(a s) 2 - E 2 ] 8M h [(crs)2 - S 2 ] SM b7 [(as)4 - E 4 Al = Because /?i and A2 are trivial, we set Af = A x and A?, = A 2 where A x , A 2 are constants. Subsequently, since E = A 2 / A 2 , , AI = A6S = A? = 0. (9.2.9) By denoting e2 = E 2 , the non-trivial A functions become A3 = h 5M e2 (9.2.10) A Xs bi 8M e2 — /yo In the IQED flow, we make the switch A x —> zAi in the flow equations (7.4.10), so the corresponding squared charge e2 ->• - e 2 . For convenience, we adopt the convention that the coefficients e3- are positive, so that each negative coefficient is written explicitly as a minus sign multiplying a positive coefficient £j. Thus we let c 3 = - 63 > 0, c 4 = —64 > 0, £3 = 8M c 3 e2 > 0, e 4 = 8M c 4 e2 > 0. (9.2.11) (9.2.12) Using the above notation and (9.2.10), the primitive flow for IQED reduces to Chapter 9. LRC coupling how with smooth slicing 169 A A 3 = e3 (9.2.13a) AAf = (9.2.13b) AAf = -SMXs5+l. (9.2.13c) Since the A 3 and A 5 equations are independent of A 4 , let us leave the solution of A 4 until after we have solved for A 3 and A 5 . From the fact that $ = 0 for i = 1,2,6,7, (9.2.13a) and (9.2.13c), it is clear that by choosing a boundary condition for A 3 and A 5 at the root scale r, and for Ai where i = 1,2,6,7, the constants Ai independent of s, then \{ = Ax, A* = A 2 , xs3=e3(s-r) + Xl, (9.2.14) Af = M - 2 ( s - r > A £ , Af = A 6 , A * = A 7 , where A 5 , A^ are the boundary values at r. Here we do not consider the solution (M~(s~r^\r5, A 6 , A 7 ) as the primitive solution for (Af, Af, A7) since they are not the domi-nant part of the flow when higher order terms are included in the j3 functions. In the above, the solution of A 3 and A 5 are determined in terms of the boundary condition chosen at the root scale r. If we had chosen to fixed the boundary condition of A 3 and A 5 at scale U, then the corresponding solution of A 3 and A5 is \i[ = \V + s3(U-s), XS5 = M^U-Shl 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 A 5 , 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 3 and A5 at the root scale r. 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 ) A 5 in the denominator, *=*"K,(.+i%+j- <9-2-w» The solution of (9.2.16) is, for s > r ^ ^ - M ^ r <9'2-i7) where A 4 is the boundary value at s = r. Extracting the leading term, we get A* = A^exp ( - In f l £ * = * (K3 + £ K ^ ~ r ) ) " + O ((*. + e3(s- r))^) (9.2.18) t=r+i \ KA + ez(t-r)t 'Kz + ez_(s-r)\% , ^(lTjr , _ ( where K3 = Xr3. From (E.2.75) of Appendix E , e4 _ 64 SM e2 £3 h 5M e2 , n ( » - 3 m f c n I n M 3 (2TT)4(M2 - 1) I ( 2 7 R ) 4 M 2 _ 1 _2kn\nM = (i + O ^ ) ) 9 ? -Chapter 9. LRC coupling how with smooth slicing 171 9.3 Higher order flow 9.3.1 The order N (3 functions We wish to show that the primitive solution (Af, A£, A | , A 4 ) in (9.2.14), (9.2.18) of the subsystem (Af, A|, A|, Af) stabilizes, that is, any higher order flow of the subsystem ob-tained as a perturbation of the primitive system with additional higher order diagrams has a solution with the same asymptotic behavior as (Af, Ks2, A | , Af). Since the O D E analysis in Chapter 8 suggest the effective coupling C = J ^ i l ! _ f 9 3 i) C (AI) 2 r [ 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 (3s from each relevant diagram has order given by a power of C3. From the primitive flow, we saw that 1/7* = 0 ( l / 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 (As5 = 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 contribut-ing 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, AP diagrams have relevant local parts. Since for i / 4, (3f do not contain A 4 , we first 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(FGTNI(\')) (9.3.2) where FGv,niW = C ( n i ~ 1 ) / 2 A ! , m > 3 F 2 F,n 1 (A) = C n i / 2 A 2 , n x > 2 ^ ( A ) = C l / 2 7 , n x > 2 i ? 4 p,n 1 (A) = C n i / 2 7 2 , nx > 6 (9.3.3) where £ is defined in (9.3.1). The reason for having the restriction on n x for the various 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 x = 4, FIP(\) is actually 0 ( C 3 7 2 ) because of vestigial gauge invariance (see Section E.2 of Appendix E). After 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 A7. 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 £ LlG(h)= Y Pi(G,s,h)Vt (9.3.4) where Li is localization operator and h is the scale assignment of the lines of the graph, U8{T) = {h\U>hh>hfl>hF = s for h > h > F ; fuf2 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) = J dPAG(X,h,P)KG{h,P), (9.3.6) where KG(h, 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 G(A, h, P) = A^VG, A, h) V(CG, A, h) (9.3.7) where A ^ V G A M = n AI"> ( 9 - 3 - 8 ) vevo v(cG,x,h) = n 'PF1 n vi*, VSF, V% are the covariance respectively denned in (E.2.5), (5.4.9a), and VG is the set of Vi-vertices of G, £G<F (respectively £G,B) is the set of Fermi (photon) lines of G, and hv, 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 AG(X, h, P) being removed. 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 \KG(h,P)\ < cG JJ M5{-Gi){hf-h<»\ (9.3.9) where h^^p) = 0 and cG is a positive constant independent of scales. We are now ready to show that Pf(G) = Q(FGtni(Xs)). Let us denote E £ e W by E^> s and 12n>aPi(G,s,h) i^5 { M-2sZfi>sPi(G,s,h) i = 5 (9.3.10) Proposition 9.1 Assuming (9.1.1-d) and that G contains no V-j-vertices, then \fr(G,s,h)\ < C0FGtni(Xs)cGM(G,h) \PSi(G)\ < CG C0FGtni(Xs) (9.3.11a) (9.3.11b) where FQjTIi is defined in (9.3.3) and C0 is a positive constant depending on the conditions in (9.1.1-d), ~ (9.3.12) M{G,h)= JJ Ms{Gf^hf-h^\ and CG>{ cGZn>sM(G,h) i^5 ( cGM-2sZn>sM(G,h) i = 5 is a positive constant independent of s. (9.3.13) 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 slicing 175 term AG(X,h,P). Let where AG(X,h,P) AG(X,h,P) AG(A, h) ' and yh = A 3 1 + A£. From the definition of js in (8.1.6a), XSh 7" 1 + 7 min I inf - , mi •2h -1 < 2 (9.3.14) (9.3.15) (9.3.16) Kp2esh 1 +p2 P^esh 1 +p2/ where e is the overlap parameter and Xsh is the characteristic function of the support of the slicing function ph. By the fact that A ^ < 1 , (1 + M < 1 , (9.3.17) where x5/> is the characteristic function of the support of ph, it is easy to see that AG(X,h,P) < 2 | £ g - b | (9.3.18) where | £ G , B | is the number of photon lines of G. Thus from the conditions in (9.1.1-d), we have AG(X,h,P) < 2^AG(X,h) < 2' £ g > b' A G(A, h) < C0AG(X,s), (9.3.19) where A G ( A ) S ) = n K n ^ n ^ From counting identities (9.3.20) nx = \VG\ = \CG>F\ + l^J- = 2|AG)B| + |AGiB (9.3.21) Chapter 9. LRC coupling how with smooth slicing 176 where \AG>F\ and |AG,B| respectively are the number of Fermi legs and photon legs of G , we have AG(s, h) = F G > N I ( A S ) and AG(\h,P) <C0FG>ni(Xs). From the bound on A G(A, h, P) and Theorem 9.1, \Pi{G,s,h)\ < J dPAG(\,h,P)\KG(h,P)\ (9.3.22) < C0FGtni(Xs) J dP \KG(h,P)\ < cGC0FG,ni(\s) M(G,h). 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 ( ^ ) ) ) N { G V ) > x mm = o ( ( C » ) ^ ) ) > N(2F) > 1 = Q((Cs) i V ( 2 P )), N(2P) > 1 fl(4P) (7 5^2 (9.3.23) 0((CS)JV(4P)) A^(4P) > 3 0(C3) 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 4 , each relevant term of A|(G) is obtained by one of the replacements described in the paragraph after (7.3.18) where the VF and VF'k 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 VSF + VF,S with no constant term. From (E.2.28) VSF + Vps = \\{VF)2 + VF +1 \ s+ l F BR + n ( i - P i ) - P l + 1 ( A I + 1 ) 2 ( 9 3 2 4 ) R Chapter 9. LRC coupling how with smooth slicing 177 where Bs = 1 - ps - ps+l + psXs2 + p s + 1 ( A 2 + 1 ) 2 , and terms subscripted by R, respectively 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), vsF + vps = x ,s A 4 (W [i + (xi + xsR) O ( i ) ] , (9.3.25) where xs i s the characteristic function of the support of ps. From the above, it is easy to see that a Pl(2F) has the form I dPAe,G(\,h,P)KG(h,P) (9.3.26) where AG(X, h, P) is of the form defined in (9.3.7-9.3.8) and each AeiG(\, h, P) is obtained from A G (A, h, P) by replacing a VhFl with VFe+VF'ht. Since we assume A£ = 0(sa) with a > 0, we need to borrow a little bit of exponential decay from M(G, h) (9.3.12) to make cG J2 ^M(G, h) < CGA, h>s and obtain the bound \ v i ( m \ < E h>s G(X,h,P)\ KG(h,P) < C 0 C G - f A G ( A , S ) A 2 where A G (A, s) is defined in (9.3.20). From above, it is easy to see that Ai ='0«cr( 2 F ) ) . (9.3.27) (9.3.28) If there were no V7 counterterm vertices, then we would define an order N flow with N > 1 as a flow constructed from using graphs GV, 2F, 2P, AP with the following restrictions on their ni(G)'s: ni(GV) < 2N + 1, m(2P)<2N, rn(2F)<2N, 2N N>3 0 N < 3 ni(4P) = { (9.3.29) Chapter 9. LRC coupling flow with smooth slicing Example 9.3.1: 178 VX2 yy "^\^ ^^xr^^ ^ " ^^^^ X , Xl 1 / X 2 F V = X] I (X\y) = F Y P = X*/ X22= C Y *ESE = ^ / <»-2Y) = C \ Figure 9.13: The diagrams of a order 1 flow The order 1 flow uses only the V diagram (m(V)=3), the ESE diagram (ni(ESE) = 2), and the V P diagram (rii(VP) = 2). Example 9.3.2: \ t / s\ / s G = 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 L x G ( M ) = ( A ; ) 3 ( A i ) 2 / (^(T))2 ( P M A 2 ) ) 2 (74(A 2 )) 2 L i i ^ ( Q ) V , (9.3.30) where LiKs,t(Q) is the localized kernel. 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 3 ( A i ) 2 / ( n ( 7 ) ) 2 ( n ( A 2 ) ) 2 {VtF{\2)fLlK^(Q) dQ. (9.3.31) t>s By assuming (9.1.1-d), ^ i ( g ' s ) i - c g ^ i ^ i M " ( t " s ) - C G ( c s ) 2 A -where c G and CG are positive constants independent of s. To extend the definition of order N flow including graphs with V7-vertices and Pl(G), we first extend the expression A G ( A , h, P) in (9.3.7) to include A 7 and Vp + Vp,s factors. Let us introduce some notations for convenience. Given a graph G containing n 7 number 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 V7-vertex of G, there is a factor A7* and at 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,U7 and the set of the remaining photon lines not in CG,B,U7 is denoted by CG,B,dB- Let us write the extended version of (9.3.7) in the following form. A G ( A , h, P) = A i ( V G , n i , A, h) A 7 ( V G , n 7 , A, h) V(CG, A, h) (9.3.32) where A ^ V ^ A K ) = n Ai"> A 7 (V G ,„ 7 ,A,K) = Ar"> (9-3-33) v(cG,x,h) = n VB n PB X < Y.eeca,r CP"/ + Vf**) YleecG,Fw Vh^ i = 4 Chapter 9. LRC coupling Bow with smooth slicing 180 By including the V7-vertices, the counting identity (9.3.21) becomes nx + 4n 7 = 4n 7 + \CG,F\ + = 2 | £ G , B | + |AG,B| (9.3.34) where n\ is the number of ordinary Vi-vertices of G, and n7 is the number V7-vertices of 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)\ < CG C0 FGinun7(Xa) (9.3.35) where CQ is a positive constant depending on the conditions in (9.1.1-d) and [ (T%Y7 FGnA\) i ^ 4 [ (^) FG,ni(\)% z = 4 where FGjTll is defined in (9.3.3). From the O D E estimate, we expect A 7 to behavior like 0((Af/A^)2 Ins). Hence, from Proposition 9.2, and the fact that for the order one V and ESE diagrams, p\(V) = 0 and p2(ESE) = 0 (see the calculation of px(V) = 0 and P2(ESE) = 0 in Section E.2 of Appendix E), we expect fi(GV) Af ff(2F) A? ff(2P) ryS ff(4P) (r)2 = < O ( ( C ) ( n i _ 1 ) / 2 + 2 n 7 (Ins)"7), O ( ( C T l / 2 + 2 " 7 ( M n 7 ) , i = 2,4 0((C s ) n i / 2 + 2 n 7 ( lns) n y ) , i = 3,5,6 ' 0 ( (CT l / 2 + 2 n 7 (ms)" 7 ) ni > 6 0(C 3 + 2 n 7(lns)" 7) . ni = 4 • O(C2"70ns)n7) ni = 0 (9.3.37) 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 V7 vertex as a 4P^ diagram constructed Chapter 9. LRC coupling how with smooth slicing 181 from 4 ordinary vertices in which an extra factor 0( m ( s ) ) is attached. An (ny, n7) graph is defined as a graph G consisting of ny Vi-vertices and n7 Vy-vertices. We define the order N A -> co coupling flow equation for IQED as the system of equations in (7.4.10) with the P^s computed using (ny,n7) graphs satisfying the following conditions: for n7 = 0, < 2N + 1 G is of type GV ny I ; (9.3.38) <2N G is of type 2P, 2F, 4P for n7 > 1, < 2N + 1 G is of type GV ny + 4(n7 - 1W . (9.3.39) < 2N G is of type 2P, 2F, 4P The reason for having n7 — 1 instead of n7 is because of the O 0 n ( s ) n 7 ) factors. An order N flow {A"}5,.!^ is a solution satisfying the order N flow equation with a 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 2 and at the root scale for the remaining couplings. Thus we only consider order N flows with boundary conditions of the form: (9.3.40) (9.3.41) where A 1 ; A 2 , and Ki with 3 < i < 7 are constants independent of the U V cutoff 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 2 , and Ki so that the resulting flow has the same asymptotic behavior as the primitive flow under the same boundary condition. = A i , Af = A 2 A 3 = K3, A^ = KA A 5 = K5, A£ = K6, Ar7 = K7 Chapter 9. LRC coupling how with smooth slicing 182 9.4 The flow of E* For finding a solution A s of the order N flow equation where Xs is a perturbation of the primitive solution A* described in Section 8.2 and satisfies (9.1.1), it is convenient to study the flow of the difference Es = Xs — As. In this section, we describe the set up of a subsystem of the flow of Es. From Proposition 7.1, the /? functions of the subsystem Af, i = 1,2, 3, 5, 7 do not con-tain 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 A5 = (Af, A|, A|, A|, A7) and its corresponding primitive solution as A s = (Af, A£, A | , Af, A 7 ) . We would like to write the flow of Xs as a perturbation of the primitive flow A s . For convenience, we take the components of As to be the solutions obtained in (9.2.14) with the modification that the exponential decay term Xr5 M ~ 2 ( s _ r ) is dropped so that As5 = 0, i.e., As = (A l 5 A 2 , e 3 (s + 1) + Xr3,0,0). (9.4.1) More precisely, for a choice of (A 1 ; A 2 , A 3 , Af, A7) satisfying (9.1.1), and A : ^ 0, A 2 / 0, A 3 + A 5 > 1, and for a given order N, we would like to show there exists a flow A s satisfying the order N flow equation of (7.4.10) such that Af = Ai + E[, E[ = o{l) (9.4.2) Xs2 = A 2 + £ | , Es2=o{l), XI = £ 3 ( s + l) + A 3 + £ 3 s , El =0(8), Xt = El, Es5=o(l), X7 = Es7, Es7 = 0(\ns), where ( A 1 ; A 2 ) is the asymptotic value of (Af,A 2) as s —> 0 0 , (A 3 ,A5 ,A 7 ) are the initial values of (A3, Xs5, A7) at the root scale r = - 1 , El = 0, El = Xr5, and E7 = A 7 . We shall 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 Xs = As + Es where Es = (ES,ES,ES3,E§,ES) is the error to the full solution at a 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, X7)}. The equations of a higher order flow have the form AXS = fis+1{A + E). (9.4.3) For our convenience, we shall introduce the notion of 1/s-order for describing a decom-position 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), fii(K) = 0(l/s2), i = l,2 (9.4.4a) fii(K) = O(IA), i = 3,7, fi5(K) = 0(\n s/s). (9.4.4b) The OQ n 5 / 5 ) °f 05 comes from the term (9.4.9) below obtained from a self-contraction of a V7 vertex. 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 Bs+l{As) of fis+1(A + E): fis+l{A + E) = BS+1(AS) + 6EBS+1(AS, Es) + BS+1{A, E) + AS+1(A + E) (9.4.5) where <5 E ^ + 1 (A S , Es) = Bs+l(As + Es) - BS+1(AS), (9.4.6) Chapter 9. LRC coupling Row with smooth slicing 184 Bf+1, i = 1,2,3,5 are the lowest 1/s-order (3 functions given by (9.2.8), Bs7+l = 0, Bf+l are the higher 1/s-order (3 functions coming from the order one diagrams given by (9.2.2-9.2.4), and AS+1(A + E) are the (3 functions coming from higher order diagrams. By the fact that A A* = B - + 1 ( A S ) , (9.4.7) applying the difference operator A to A s = As + Es, the flow equation for the Es is AE* = 5EB°+1(AS, Es) + Bt+X(A, E) + AS+1(A + E). (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 (3S+1(E). Now 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 * + 1 by a f3t+2{E) for i = 1,2,3, and A A f 1 by -5M{h M2 [(a8)2 - e2} + £ | + 2 ) + (3S5+2(E) where as = (Af/A|) 2 , so that the corresponding terms have an order of a higher power in I/7 5. In these replacements, the / 3 - + 2 ( £ ) are given by (9.4.5) in which A A f 2 = A f 3 - A f 2 (we do not want to recursively replace A A - 1 by for h > s + 1, beyond h = s + 1). Let us rewrite the higher order flow using the notation in (9.4.5)-(9.4.6) and replacing A A | + 1 by the appropriate f3f+2(E) in (3s+l(E). Also we would like to write down some 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,(E7) be the f32 contribution of the graph G obtained from a self-contraction of a V7 vertex as displayed in Figure 9.15. P5(E7)=c5Es7 [ drVs(j) (9.4.9) where V^y) is the running photon slicing using l+js instead of 7 s = A 3 + A 5 (see (5.4.9c) for the definition of 7). Chapter 9. LRC coupling flow with smooth slicing 185 Figure 9.15: A self-contraction of a V7 vertex Next we consider terms of the form (a 5) 2 - e2 or (a8)2 - e4 where o8 = (X{/Xs2)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{ U8 ^2 (9.4.10) A i ' ~* A 2 For example, let us consider the term 5M M2 b5 [(o8)2 — e2] from the /?f of (9.2.1). Using the variables in (9.4.10), '(M + Ei)2 5M M2 65 s\2 = 6MM2b5 5M M2 b5 e L(A2 + £ ! ) 2 (1 + Ut)2 — e (9.4.11) _ ( l + [/|)2 Let us introduce the following notations for expressing terms like (9.4.11) more concisely. Let X8(E) = YS{E) = (1 + ut)2 (1 + Ul)2 (1 + ^f) 4 (1 + ui)4 - 1 = o(ut) + o(u8) - 1 = Oiut) + 0(ui) (9.4.12) (9.4.13) and £3 = c3 5M e2, e5 EE 65 M2 e2, e7 = b7 SM e4, (9.4.14) Chapter 9. LRC coupling how with smooth slicing 186 where c 3 = —b3, and b3, b5, be, b7 are the constants defined in (E.2.75-c) of Appendix E . Using (9.4.12), we can write (9.4.11) as 6M M2 b5 e2 Xs {E). Now the higher order flow equation (9.4.8) can be expressed as AE* = ~p[+\E)= A[+l{E) (9.4.15) AES2 = P^(E)=Y: GS+\E) P°+2(E) i=l,2 +GS2+\E) [-6M(E*+2 + e5Xs+2(E)) + P*+2(E) + PS3+2{E)\ +Hs2+l{E) + As2+l(E) AE* = ps+'(E)=e3Xs+l(E)+Y: Gs3f (E) p*+2 (E) + As3+1 (E) i=l,2 AEl = Pt+1(E) = -5M[Es5+l+e5Xs+l(E)} + £ Gtf(E) P?+2(E) + It'iE) + Ap-\E) i=l,2 AES7 = ~ps7+l{E) = e7 YS+1(E) + £ Ga7f{E) Pl+2{E) + AS7+1(E). t=l,2 From (9.3.37) and (9.1.1), the P\ can estimated by order of powers of 1/Y where 7 s = Af + Af. Since we would like to express A* as a perturbation of As (9.4.1) as described in (9.4.2), we would like to estimate the Pl in terms of the "primitive" part Vs of 7 s . From (9.4.1), A 3 = A 3 + e3(s + 1) and A 5 = 0. Let T s = A 3 + Af = A 3 + e3(s + 1) (9.4.16) where K3 = A 3 > 1 and e3 is defined in (9.2.11). Since A 3 = Vs + E3 and X2 = Eg, it is easy to see that Y = TS + Es3+ Es5. (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 x O(l) , A* = A 2 0 ( 1 ) , 7 s = r s O(l)- (9.4.18) Hence we expect that (9.4.19) fc{E) + 8MEt = o ( ^ r ) , M(E) = o ( ± y ®{E) = O ( ^ ) -Note that for the order one V and ESE diagrams, Pi{V) = 0 and 02{ESE) = 0 and that the dominant term of /3|(F) + 5ME^ is not Xs(E) but I§(E) since it contains the iactorVsB(E) Es7 = 0(\nFs/Ts). 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 s = As + Es from the primitive solution A3 (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 E3 for j ^ i; hence the solution Ei{£) of each scalar non-homogeneous linear equation obtained from the replacement has a simple explicit expression analogous to the quadrature of a linear ODE. In this way, for a given boundary condition EB of E, we rewrite the original equation as E = JC(£, EB) which we treat as a 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, EB)\\ 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(£,EB) in terms of the Lipschitz bound for the /3 functions Bi(£). Similarly, bounds for ||/Cj(0,EB)\\ can also 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 G (A, h, P) of (9.3.32). These primitive parts are expressed in terms of the primitive parts A l 5 A 2 , Vs, Vs respectively of Ai, A 2 , 7 5 = A?, + Af, 1 + 7 where 7 is defined in (8.1.6a). We would like to describe Ts, Vs in more detail. Recall at the end of Section 9.4, we introduced the primitive part Vs of 7 s where Vs is defined in (9.4.16). To express 7 s as a perturbation of Vs, we introduce the variable W° = ^ , (10.1.1) where E° = Esz + Eg. From (9.4.17), Ei + E? 7 Vs ( 1 + 3 + 5) = rs(i + ws). (10.1.2) The running photon slicing V% of (5.4.9a) is actually expressed in terms of 1 + 7 s = 1 + Af + Af instead of 7 s = Af + Af, where by' rescaling p ->• Msp H = / 3 V ) (AS - 1), Af = fl{p2) Af, (10.1.3) and A V ) = / 5 V ) = j p ^ , (10.1.4) where 1 — e < p2 < M2 and e is the overlap parameter. Thus as for 7 s , we define W* = | ; (10.1.5) where f s = mP2) T s - K(P2) {Kz + e3(s + 1)) (10.1.6) E; = fs(P2) + mP2)ES3 + f°(p2)Eg MP2) = I-/I(P2) = F ^ 2 -Chapter 10. The Fixed Point Solution 190 and K3 == A 3 > 1. Expressing 1 + Y in terms of Ws, 1 + 7 S = £S + E° = fs(l + Ws). (10.1.7) Note that for the overlap parameter e sufficiently small and the slicing parameter M > 1 sufficiently close to 1, say e < 1/3 and M2 < 3/2, then / aV ) < M~2s, 1/2 < fl(p2) < 1, 1/2 < fl{p2) < 3/2. (10.1.8) From (9.3.37) and (8.2.15) with C r = e2r/y = 0(e2), for i = 1,2 and an order N graph 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 K = ui+un, ( f f * » 2 - S ) 2 = e 2 ( r r ® 2 - ( 1 0 - " ° ) where e2 = k\lk\. We set up a Banach space B for £ in the following way. We remind the reader that we are not including the EA and E6 at this stage. Let us denote E° = {UIUIEIEIE>7) (10.1.11) and the whole sequence {Es} by E. Correspondingly, let S = ( £ u £ 2 , £ 3 , 8 b , £ 7 ) , (10.1.12) where each £j is a sequence { £ • } • From the expected orders (9.4.19), we expect U[ = O ( ^ ) , ^ 2 = 0 ( ^ 7 ) . EI = 0(lnTs) (10.1.13) /In r 5 \ E S 5 = O — , ^ = o ( i n r - ) . Chapter 10. The Fixed Point Solution 191 Note that because of the J | term (defined in (9.4.9)) of fil+l{E) in (9.4.15), the expected order of F | is O ( I n P / P ) instead of O 0-/Ts). We define a norm for each component of £ according to the expected estimates (10.1.13) by the following. For F a sequence {Fs}U=r where each Fs is a function of £ and p2 with 1 — e < p2 < M2, (recall the root scale r = -1), let \\F\\n,m= sup sup (r s) n(l+ l n T 5 ) m | F S | . (10.1.14) l-e<p2<M2 r<s<U Note that since K3 = Xr3 > 1, 1 + ln Ts > 1 for s > - 1 . Let ll^lll(l) = |N|l,0 11^211(2) = W&Wlfi (10.1.15) ll^3||(3) = ^3 H ^ l l o . - l 11^ 511(5) = h e~2 H l ^ l l l - l ||^7||(7) =• k7 e~2 ||^7||o,-i where k3, k5, k7 are small positive constants to be determined later. The reason for having the factor e~2 in the definition of || • ||(7) is that we think of £1 ~ 5(r s) _ 1 and £2 ~ 5(r s) _ 1, from the "primitive" part of (37 (see the (37 of (9.2.8)) , we expect ^ - e ^ O ^ + O ^ ) - ^ 2 I n F . t=s A similar reason applies for having the e~2 factor in the definition of || • ||(5). We will see 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 e2, £\, £2, £5, £7/Vs. The purpose for having k3, k5 and k7 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 Also, we denote 11^ 111(1) > II^IL) , ||^3||(3) > IN|(5) , IN|(7) • (10.1.16) Bs = {£ | \\£\\ < 5}, (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 equa-tions by replacing E by £ in the /3 functions except for the linear term in (9.4.15). AE* = B}+1(£), i = 1,2,3,7 (10.1.18) AE* = -5ME°+1 + B°5+1(£), where B [ + \ £ ) = ( 1 ( U i l 9 ) B2+\£) = T { E G$\£) A 2 l i = i , 2 + G'£(£) [ ~ S M ( £ t 2 + e5Xs+*(£)) + p>+*{£) + {%»{£)] + H°+1(£) + As+l(£)} Bl+\£) = e 3 X ° + 1 ( £ ) + Y : Glf{£)(5t+\£) + A?\£) i=l,2 Bt+1{£) = n + l { £ ) - s b 5 M X s + \ £ ) + Y G s 5 y ( £ ) f i r 2 ( n + As5+l(£) 1=1,2 Bs+\£) = e 7 Y s + \ £ ) + Y : Gs7f{£)^\£) + A7+\£). i=l,2 Consult with the discussion starting at (9.4.5) for the definitions of the various terms in the above. (Do not confuse the above B's with the B's in Section 9.4.) As mentioned in the analysis of the primitive flow in Section 9.2, because we wish the coupling flow to respect the conditions (9.1.1), unless one has very precise estimates on the (3 functions, it is more convenient to specify the boundary conditions at the root scale r for divergent variables (as s —> oo) and at the scale U for o(l) variables. We prescribe the following boundary conditions for E(£) satisfying the equations (10.1.18). For Ei with i=3,5,7, we prescribe W == {El, El, Er7) = (0, XI, Xr7), (10.1.20) Chapter 10. The Fixed Point Solution 193 and for U\ and U2, we prescribe f/f = U2 = 0. From the prescribed conditions, the solution to (10.1.18) is, for r < s < U u U[ = £ B{{£) (10.1.21a) j=s+l Ui = £ B{(£) (10.1.21b) j=s+l El = Y,Bl{£) (10.1.21c) 3=0 Eg = J2M-2^S-^BJ5(£) +M~2^Erb (10.1.21d) 3=0 s Es7 = Y.Bi{£) + E7- (10.1.21e) 3=0 Note that from the prescribed boundary conditions, (10.1.21a-10.1.21b) are sums running from s + 1 to U while (10.1.21c-10.1.21e) are sums running from 0 to s. Thus for each Er, we obtain a map defined by (10.1.21a-e) K : (£, Er) —»• (JC1} /C 2 , AC3, KA, JCS) = (Uu U2, E3, E5, E7). (10.1.22) For a function / on the ball B$ 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 \\£-£\\, (10.1.24) where 5{ are small constants. 2. K{BS) C B5. Chapter 10. The Fixed Point Solution 194 For the second condition, since \\JC(£,Er)\\ < \\K{£,Er)- JC(0,Er)\\ + \\)C(0,Er)\\, (10.1.25) using the contraction property of IC, it suffices to show that ||/C(0, Er)\\ 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, £ ' r ) | | can be expressed in terms of Lipschitz bounds for 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, " lnM ( K3 d M~STS = -M~s [ln M Vs + e3] = -M'se3 (10.1.26) ds where 6M — M2 — 1, and Vs and e3 are defined respectively in (9.4.16) and (9.2.11). Since K3 > 1 and c 3 > 0, (10.1.26) is negative for sufficiently large s. Hence it is easy to see that there exists a constant CM depending only on M, such that for s > j > 0 p ( P ' ) - 1 < CM. (10.1.27) For example, let M — 1.1. Since ^ + (s 5M)) - 1 > s l n M - 1, (10.1.28) OM Vc 3e 2 / d M ~ s s l " < 0 for s > 11. Thus, for 11 < j < s, M " T S < M~jTj. For the remaining scales 0 < j < s < 10, M-l*-fl rs(T3)-1 < K z + }}ez < 12. (10.1.29) - K3 + £3 K3 + e3 -Hence, here we may chose CM = 12. Chapter 10. The Fixed Point Solution 195 Let (mi.nx) = (m 2 ,n 2 ) = (2,0), (m3,n3) = (m7,n7) = (1,0), and (m 5 ,n 5 ) = (1,-1). Lemma 10.1 Suppose Then WVBiisMU.m <MM \ \ s - i \ \ \\Bi(0)\\mi,ni<8?6M. VKi{£,£) < CBi 5t \\£-£\\, ll^(0)| | ( i ) < CBi6° + Cl, where CBi CB3 ci Cl c 3 e2 ;, 1 = 1,2 G B5 c 3 0, i = 1,2,3, k 5 c M K 3 \E;\ 2 he, CM Cr7 = C B7 k7 c 3 e4 k7 \EJ}\ (l + \nK3) e 2' (l + l n K 3 ) e2 ' and CM is defined in (10.1.27). Proof: From (10.1.21a-b) and the hypothesis (10.1.30a), for i=l,2 < c3 ez \£-£\\. Let d3 = k3 and d7 = k7e~2. From (10.1.21c) and (10.1.21e), for i=3,7 ^ ( l + l n r 8 ) - 1 \VK\{£,£)\ < ^ ( l + l n r 8 ) " 1 ^ \\£ - £\\ £ (10.1.30a) (10.1.30b) (10.1.31a) (10.1.31b) (10.1.32a) (10.1.32b) j=r+l 5M < dj c3 e' ;Si\\S-£\\. Chapter 10. The Fixed Point Solution 196 From (10.1.21d), (10.1.27), and the hypothesis (10.1.30a), ^ s \ - i i n cw / s wc c11 ^ sM M 2 ( s ^ m + inr-? pa+inrr 1 !^^^)! < 6 s \ \ £ - £ \ \ g , ^ T J i + l n r s < 65 CM \\£-£\\ ± M-l-n^L j=r+l M < 2S5CM \\£-£\\. 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 Er7 of (10.1.21d-10.1.21e) satisfy the bounds \\M-2^Er5\\{5) < Q , 11^ 711(7) < Cr. It is easy to check that for i = 5, 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~2, 52e~2, k353e~2, k555CMe~2, and k757e~4 sufficiently small. Chapter 10. The Fixed Point Solution 197 10.2 Lipschitz continuity of the 0 functions 10.2.1 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 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 Kijn has a Lipschitz bound similar to (10.1.30a) with constant 5i>n which can be made arbitrary small by choosing e2 = A 2 / A 2 and S, the 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 N (10.2.1) n=l (£) = Bt+l{£) + B- (£) + At (10.2.2) where Bs+1{£) Bs3+1(£) Bs7+\£) £3XS+1(£) 0 Bs5+1(£) = -e5 SM Xs+l{£) + I 5 S + 1 ( £ ) (10.2.3) s7 Ys+\£); Chapter 10. The Fixed Point Solution 198 BSCG\£) = 0 (10.2.4) Bi$V) = E G$\e)K+\E) j =3,5,7 i=l,2 B${£) = T"(E Gff(£) ft+*(£) A 2 [ i=l ,2 [-M*l + 2 + e5Xs+2(£)) + #+2(£) + $+2(£)]} ; 5 ^ ( 5 ) = i = l,2, (10.2.5) = Ai+1(£) i = 3,5,7. The i?*^ 1^) are the higher order terms coming from higher order diagrams, and B-+1(£), Btfi~{£) are terms from the order one diagrams. Bf^(£) are terms coming from 0 functions of order one diagrams in which A A S + 1 has been replaced by the 0f+2(E)'s (see (9.4.15) and (9.2.2-9.2.4)). Because 0t+2{E) can be split into first order and higher order terms, let us write # + 2 ( £ ) = B°+2{£) + A\+2{£) (10.2.6) where As+2(£) are the higher order terms and Bf+2(£) denotes the terms from (9.2.1) and J | defined in (9.4.9). Correspondingly, we further divide the terms of Bf^l(£) into two classes. Let Bi£1{£) = B$1(£) + B&1.A(£), (10.2.7) where B[%\£) = 0, £ G$\£) B!+2(£) j =3,5,7 (10.2.8) i=l,2 = E Gff(£)Bt+i(£) A 2 l i= i ,2 +Gff(£) [-6M(£t2 + s5Xs+2(£)) + B'6+2{£) + Bs3+2(£)}} ; Chapter 10. The Fixed Point Solution 199 B'+G\A(£) = 0, B°%\{£)=Y G$\e)A?*{£) j = 3,5,7 (10.2.9) i=l,2 B&AV) = i [ E G$2(£) A?\£) + Gft{£) JA^(£) + As^(£)} j . 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(£) and BiGl(£)- 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{£ - £), 0 < * < 1, (10.2.10a) and that V f {£,£)= [l ±f(£(t))dt= ^ def(£(t))-(£-£)dt (10.2.10b) Jo at Jo where dsf(£) denotes the sequence {dsrf(£)} of partial derivatives and d£f (£)•(£-£) = Y,d£ff (£)(£?- £?)• i,s' For a function F of £ , we denote the sup of the norm of F over the range of the interpo-lation £(t) as \\F(£,£)\\m,n= sup | | F ( £ ( t ) ) | | 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 || • | | m , n . Let K(£) be some sequence of differentiable functions of £ such that for each index s, Ks{£) only depends on £s', s < s' < U. Here differentiability means the partial derivatives dsKs(£) exist. Chapter 10. The Fixed Point Solution 200 Lemma 10.2 Suppose that for E II [dSf,K](€,£) ILi,m2 < Oi (10.2.12a) s'>0 where a, is some positive constant. Then for p\,p2 > 0 or —p2 = p\ > 0, \T> K(£,£)||mi+Pl!m2+p2 < £ ai \\£i — £i IPl)P2• (10.2.12b) Proof: By (10.2.10b), ( r s ) m i + p i (1 + In r s ) r o 2 + P 2\V Ks(£, £) I < ( r s ) m i ( i + i n r s r 2 f1 dt £ £ | c v Ks{£{t)) ( r s ) p i ( i + i n r s ) P 2 \ef-£f\ < £ s'>s / P _ \ P 1 / l + l n r ' V 2 ' m i , m 2 VT S' / I 1 + ln Vs' J \\£i £i\\pi,. ,P2 < E ^ - i. i \ lpi,P2-Note that in the case — p2 = 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 + £5 fz(p2) + h(p2)£z + h(p2)£5 (10.2.13) where / | , / 3 S , / 5 S , and fs are defined in (10.1.4) and (10.1.6). Because the photon running slicing VSB(E) of (5.4.9a) and its approximation VB(E)\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 VSB{E) and VB(E)\l+^7 in (10.2.10b), we would like to use W and VV as the "error" variables instead of £ 3 and £ 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: e2 = A 2 / A 2 , K3 = A 3 , k3, k5, k7, 5, M, e, where the fcj's are the parameters defined in definition of 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 e2 < min((10 b5 M 2 ) - 1 , k5), K3 > 1, e < 1/2 (10.2.14) h < 1/2, k5 < 1/2, k 7 < l , 5 < k3/2 < 1/4 (10.2.15) 1 < M 2 < 3/2, e < 1/3. (10.2.16) where b5 is denned in (E.2.75-c) of Appendix E. Lemma 10.3 l l W - V V l l ! , - ! ^ - 1 ^ - ^ ! . (10.2.17) l i v v - ^ l l i , - ^ ^ - 1 ! ^ - ^ ! ! . (10.2.18) Proof: From (10.2.13), rs (1 + l n r s ) _ 1 | W s - Ws (10.2.19) < (1 + l n P ) - 1 l ^ - ^ l + l ^ - ^ l k3 1 + lnT 5 L ' 3 1 ' k5Ts e2 (l + lnT 8 ) So So I ~f~ Chapter 10. The Fixed Point Solution 202 From (10.2.14-10.2.15), K3 > 1, e2/k5 < 1, and k3 < 1/2. Thus from (10.2.19) < 1 11^ -5311(3) + - ^ ||55_55||(5) K3 /C5 A 3 (10.2.20) From (10.2.13) and the definition o f f in (10.1.6), T s (1 + I nT s)- 1 1 W s - V V < (1 + l n T 5 ) - 1 < (1 + l n T 5 ) - 1 '/aT / 5 T S 3 " 1 cs cs 1 J t> cs cs\ c3 °3 °5 c5\ fi 1^3 - ^ 1 + ^ 5 - ^ 1 J3 Also, from (10.1.4) and (10.2.16), it is easy to check that / | / / | < 2. Hence, | | W - V V | | l , - l < ^ ||53 -f 3||(3) + T ^ l i f t -55||(5) K3 K5 A 3 (10.2.21) (10.2.22) 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, £ 2 ' s , ft's, and 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 s = Xi, + Af . Let xs denotes a A?, or 7 s , or 1 + 7 s (10.2.23) Let us drop the superscript s from ps and write a running slicing function as V{xs,xs+1) D(xs,xs+1) (10.2.24) Chapter 10. The Fixed Point Solution 203 where D(xs, xs+1) = 1 + p(xs -1) + XR PR{{XS+1Y ~ 1) +1\2 (10.2.25) PR and XR denote respectively p restricted to the right overlapping region SR and the characteristic function of SR, and pR = (1 - pR). Let D(xs, xs+1) be denoted by DS, and D restricted to the support of pR be denoted by DR. The third and fourth types are respectively of the forms Zx EE {Ax*)-1 [V(xs,xs) -V(x3,xa+l)\ PR PR (Xs + xs+l) DR(xs,xs+1)DR(xs,xs) PR PR X + PR PR X" DR(XS, XS) DR(XS, XS+^) DR(XS, X S + X ) DR(XS, XS) PR p R X +vR(x>y P R xs(pR + pRxs) DR(xs,xs+1) "X 'PR + PRXS (the last line is obtained by the fact that DR(xs,xs) = Xs(PR + pRxs)) and Z2 EE (Arc?) s \ - l PR PR [l + pR(xs+l-l) l + pR(x'-l) (PR)2 [l + pR(x°-l)) (l + pR(x<+i-l)) (PR)2 (10.2.26) (10.2.27) (pR + PR xs) (pR + pR xs+l)' They come from expressing the G - j ( £ ) / ? - + 1 ( £ ) ' s (see (E.l.7) and (E.l.29) of Appendix E) using differences instead of interpolation integrals. The fifth type is the relevant part of Z3 EE VV(js,js+l) = V(l + f , l + Y+1)-V(ys,js+1) p (pVY + pR(ys+l + 1 + 7 S + 1)737 S + 1) ~ 7J(1 + 7 S ,1 + 7 S + 1 ) / J ( 7 S , 7 S + 1 ) Chapter 10. The Fixed Point Solution 204 where V is defined in (10.2.24), and p2 e S1 = (1 — e < p2 < M2). ZZ comes from the term # | ( £ ) arised from replacing 1 + 7 s by Y in the photon covariance (5.4.9a). Discarding irrelevant terms, Zs = K fZ3(p)V(Y,Y+1)V(l + f,l + Y+1) (10.2.28) + A f 1 / l f b ) ^ ( 7 s , 7 s + 1 ) -DR{l + YA + i s + 1 ) + A S + 1 / i r ( p ) ^ ( i + r . i + y + 1 ) P f l y > r + 1 ) where / | 3 (p) = (p2 - l ) / p 2 and p 2 e S i . Note that since p2 e S i , e/(l - e) < \fz3\ < 5 M / M 2 . Even though / | 3 = Q(SM), we cannot extract a factor 52M from H2 because the relevant part of H2 contains the term V'F which makes it O(^M) (See Appendix F for the relevant terms of E S E from expanding the slicing function and the later part of Section 9.1). Let us first consider the derivatives of the second type factors which are of the form p/D where D is defined in (10.2.25). Let a product FG of two sequences F, G be defined by (FG)S = FSGS and an inverse { ( F s ) - 1 } of F be denoted by F~l. Since dxp/D = — p/D (dxD/D), we would like to consider bounds on \dxD/D\. From the definition (10.2.23) of xs, and the decompositions A 2 = A 2 ( l + U2), 7 s = r s ( l + W 5 ) , f = fs(l + Ws), let us correspondingly write x = z(l + y) where (VV, T) or (VV, f ) for the Bose case (y,z) = Also, let both D(xs,xs+1) and 1 + ys be denoted by Ds. (10.2.29) (£ 2 ,A 2 ) for the Fermi case Chapter 10. The Fixed Point Solution 205 Lemma 10.4 Let £, £ G Bg where 8 satisfies the inequality in (10.2.15). I dvtD D -(y,y) < 4. (10.2.30) Proof: Note that Ds only depends on £ s and £ 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 — - — L L J - > 1 — 25 > -zs ~ K3 ~ - 2 (10.2.31) x Bose case : — > 1 zs ~ (A) M W i , - i 1 + lnT > 1 - 5 A;,"1 > (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 8 dxsDs Z Ds P zs p Xs Ds Xs dyS + \DS zs+l Ds zs+l dxs+\Ds Ds 2pR x s + l Ds 2 z s+l rS+l PR (XS+1)2 Ds with \pxs/Ds\ < 1 and \pR (xs+1)2/Ds\ < 1, we have the desired bounds (10.2.30). Q.E.D Next we consider terms from the third and fifth type factors (10.2.26-10.2.28). Let PR Fsz{y) Gsz(y) Hsz(y) PR + PR zs(l + ys) PR zs+l{l + ys+1) DR(zs{l + ys),zs^(l + ys+1)) 1 PR zs{l + ys) PR + PR zs(l + ys)' (10.2.32) Chapter 10. The Fixed Point Solution 206 Note that the factors (10.2.26-10.2.28) are composed of the F z ' s , Gz% and Hz's. We 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). Let fz be one of the FZ,GSZ,HZ defined in (10.2.32). \\zfz\\Q<2 (10.2.33) I k dytfz(y,y) | | 0 < 16 where z is defined in (10.2.29). Proof: From (10.2.29) and (10.2.31), 1 + ys = xs/zs > 1/2. Hence 1*7,'I < l + 2/s < 2. and we have (10.2.33). As for (10.2.34), we consider the following bounds. (PR)2 * S zs \dysFsz (pR + pRzs(l + y*))2 PR XS 1 [i + y°y PR + PR XS < 4. It is easy to check that (10.2.34) Hence it follows that Z 3y3 G^ < 1. -1 — pR pR XSZS+1 (DR)2 zs zs \dys+iGsz(y)\ = z zs+1(l + ys)(l + ys+1) 2 (PR)2 XS+1ZS+1 PR x" D R 2 zs ^+1(1 + ^ + 1 ) 3 PR (XS+1)2 D R PR (*S+1)2 DR < 4, < 16. Chapter 10. The Fixed Point Solution Lastly, zs\dySHsz(y)\ < -< + ys l PR PRZ (PR + PR zs(l + ys)y < < (1 + Vs)2 1 (i + ys)2 2 PR PRX (PR + PR XS)2 PR + 1 (i + ys)2 PR PR PR + PR zs(l + ys) PR + PR XS PR + PR XS s PR X PR + PR XS + 1 ( l + y S ) 2 < 8. Q.E .D. Chapter 10. The Fixed Point Solution 208 10.2.2 Lipschitz bounds of B{£) Let us consider Lipschitz bounds on functions that are of the form K{£) = P(i,2,5)(£) p(7)(£) Q(£)- (10.2.35) where - P ( i , 2 , 5 ) ( £ ) , P(7)(£) and Q(£) are defined as the following. - P ( i , 2 , 5 ) ( £ ) is a product of £is, £ 2 ' s , and £ 5 ' s : P(1,2,5)(<?)S = ^ , 2 ) ^ ) ^ ( 5 ) ^ ) (10-2.36) rip ph){£) = II s < Sj < U ii = l,ov 2 .7=1 pk(£) = w n \ s < S l < u . P(7)(£) is a product of {£7/(l + lnr ) ) ' s : p ^ £ ) s = n r n ^ 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, • • •, qr) where the g^ 's are loop momenta of the integral; 2. n 7 (1 _ | _ l n p s ^ SJ nB-n7 S b nF S f Q - ( £ , Q ) = U ( £ )P n huh' (ia2'39) j=l B 6=1 B f=l U F where Dp and DSB are defined in (10.2.25) respectively with xs = A| , and xs = 7 s or 1 + 7 s , and s < s&, Sf, Sj < U; Chapter 10. The Fixed Point Solution 209 3. T°(£,Q) = i f fZfa) i f I I G'Z*(W) (10.2.40) <2=1 t=l u=l where s < sq, s f, s u, sv < U, and each fsz is a F | , G ^ , Hz defined in (10.2.32); 4. / dQ\G(Q)\<5MCG, (10.2.41) where S^JJp)(Q) denotes the support of the product of the slicing functions from QSTS(£, Q) and 5M = M2 - 1. Using Lemma 10.2 - Lemma 10.5, we wish to establish Lipschitz bounds for the components of (10.2.35). Let us denote || • | | 0 as || • 11o,o where || • | | m ) J l is the norm defined in (10.1.14). We first consider the case of (10.2.35) with Q{£) = 1 and P( 7)(£) = 1. Recall that es = A f / 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). Lemma 10.6 Let P(i,2,s), P(i,2) , and P^ be the products defined in (10.2.36). i v p ^ i s ^ ) ^ < \\e-e\\. (io.2.42) 2. n5 e 2 n 5 8 n & - 1 \VP(6)(£,£)\\i,-i < \\£-£\\. (10.2.43) e 2 n 5 ^ n 5 + n p - l |2>^(i,2^)(f,f)||i,-i < ^ - i , B B [nP + n5] \\£-£\\. (10.2.44) Chapter 10. The Fixed Point Solution 210 Proof: S i n c e | H | i , _ 1 < | H | 1 > 0 < l / l C and (pi,p2) - (1,0), ), by Lemma 10.2 with (mi, m2) = (np — 1,0) \\VP(l,2)(£,£)\\ 1,-1 < 1 T \\VP{l,2)(£,£)\\npfi (10.2.45) A 3 j \£-£\ np-1,0 It is easy to see that where n is the number of repeated £•] in P(S1)2)> a n d f ° r £ £ #<5> ft ,. ps _ „ C 1 ' 2 ) (10.2.46) < n n^ .111,0^ -^1. (10.2.47) np-1,0 j=l,j& By counting the number of terms in P*lt2)' a n ( 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{5)(£,£)||i,_i with (mi,m 2 ) = (0,0) and [pup2) = (1,-1). Since 1 + l n P e2 | | ft | | ( 5 ) 1 + l n P e2 | t s | S - P fe d ' (10.2.48) and (1 4- mz)/z < 1 for z > 1, we have II - i fe2 \ n 5 _ 1 (10.2.49) From (10.2.49) and Lemma 10.2, using the fact that \£5 - ft||l-1 = T~Hft - ft 11 (5) fe it is easy to see that we have (10.2.43). 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 < l|P(5)(£)||0 ||©P(l,a)(f^ )||l,-l + | | P ( l ,2 ) ( 5 ) | | o | | 2 > P (5 ) ( f , f ) | | l > -e 2 n 5 Sn5 rn p < x RHS of (10.2.42) + — ^ x RHS of (10.2.43) < RHS of (10.2.44). 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 \VP{7)(£,£)\\0<n7 l - l ^ l l f - S H . Proof: We apply a similar argument as in proof of 1. of Lemma 10.6. vpfa(e,i)\ < Y, dP, (l + m r < ) ^ % , £ ) d. < - E BP (l + \nr<)?-W(£,£) \£^ -£7'\ (1 + lnT s ' ) l l ^ - ^ l l -Since m £7° (1 + \nTSt)d£nPU£) = n TT — i ^ (7)1 ; . = ^ i + inr-i where n is the number of repeated £ 7 l in Pfo(£), and 1 + ln I> < iN|(7) < e2 5 A ) 7 A ; 7 ' by counting the number of terms in the product Pfo {£) P, E (l + l n r " ) ^ ( £ , 5 ) /e 2<5\n 7 _ 1 Thus from (10.2.54) and (10.2.51), we have (10.2.50). Q.E.D. (10.2.50) (10.2.51) (10.2.52) (10.2.53) (10.2.54) Chapter 10. The Fixed Point Solution 212 Next we consider the case of (10.2.35) with - P ( i , 2 , 5 ) ( £ ) = 1 and Pcj)(£) = 1. Let us denote a Q(£) (defined in (10.2.38)) with a fZF, fZB or, Gz factor removed from T(Q,£) (defined in (10.2.40)) by Qfx(£). Let NF = nF + nz,F, NB = nB+ nZtB + hB, NQ = 32 (NF + k^Ng) where the n's are defined in (10.2.39-10.2.40). (10.2.55) (10.2.56) Lemma 10.8 Suppose there exists a constant CQ satisfying the following bounds: for all £ E Bs, | | Q ( £ ) I L , n 2 < $M CQ, \z~1Qfz(£)\\n1,n2 < 0~M CQ, (10.2.57) where z is defined in (10.2.29). 1. "•1 + 1,712-1 2. For Q(£) contains only Fermi running slicings, i.e., NB = 0, \\VQ(£,£)\\ni+hn2 < 32 SM CQ NF \\£ - £\\. Note that the norm in the second case is different from the first. Proof: Since D s ( £ ) only depends on £ s and £ s + l , by Lemma 10.4 and Lemma 10.5, (10.2.58) (10.2.59) £ 1 1 ^ W , £ ) | | , S f + r t f=l t r=0,l d s f + r D F f D + £ l l ( ^ ) - 1 Q ^ ( f t ^ ) l k n 2 £ \\^ d£s+rfz(£,£)\ q=l r=0,l < 5M CQ [8 nF + 32 nz>F}. (10.2.60) Chapter 10. The Fixed Point Solution 213 Similarly, by Lemma 10.4 and Lemma 10.5, we have E 1 1 ^ 0 ( ^ ^ ) 1 1 m,n2 w Kw + Kyy < 5M CQ [8 nB + 32 (nz>B + nB) Using the above and applying Lemma 10.2 and Lemma 10.3, \VQ(£,£)\\ni+1>, 712 — 1 8(nF + 4nZtF) \\£2 - £2\l^i + Kw | | W - W | | i , _ i + \\W -nF + 4 n 2 , F - \\£2 - f2||(2) + Skeins + 4 ( n z , B + hB)\ \\£ - £\ < 5M CQ - S m C q r l + l n K 3 < 5M CQ 32 [NF + k^NB] \\£-£ < SMCQNQ \\£-£\\. The proof for Case 2 is similar to the above except that there is no (1 + \nK3) 1 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 2 ) = {NB,—n7) in (10.2.57) where NB and n7 are de-fined respectively in (10.2.55) and (10.2.39), then for NB > n7, we can choose CQ = CG 2Nb+Nf/A%F where CG is defined in (10.2.41). Lemma 10.9 Suppose NB > n7. \\Q(£)\\NB,-ni < $MCQ, \z~lQfA£)\\NB,-n7 < SM cQ, (10.2.61) where 0 CQ 2Nb+Nf ~~ATF (10.2.62) Chapter 10. The Fixed Point Solution 214 Proof: Recall the definitions oixs, zs, and ys in (10.2.23) and (10.2.29). From (10.2.25) and (10.2.31), each running slicing pt/Dt oiQs{£,Q) in (10.2.39) is bounded by 2/zK From (10.2.33), each j\, Qz factor of TS{£,Q) in (10.2.40) is also bounded by 2/zt. Thus it is easy to check that, for NB > n7, TS(£,Q)QS(£,Q) (TS)NB (1 + ln T s ) " 7 where Sj,Si > s. From (10.2.38), (10.2.41), and (10.2.63), since Sj,Si > s, we have l l<5(^)ll iv B , -n 7 < (10.2.64) The bound for \\z~lQfz(£)\\NB,-n7 can be shown by a similar argument. Q.E.D. We now consider more specific Lipschitz bounds on K(£) defined in (10.2.35). Let ( m i , m 2 ) = (1,0), (0,1), or (0,0), (10.2.65) dB = NB - n7 where NB and n 7 are defined respectively in (10.2.55) and (10.2.39). We would like to consider the following norms of V K{£, £): \\VK(£,£)\\mi+lt_m2. (10.2.66) with following specification dB>0, CQ = e2NKCQ (10.2.67) where dB is defined in (10.2.65), nK > 0, CQ is the constant given in (10.2.62) and CQ is a constant independent of the parameters e2, K3, ki, S. The parameters NB,n7,np,mi,m2 ONB+NF m p s / i I ] „ r s , \ NB-n7 ™ Chapter 10. The Fixed Point Solution 215 satisfy one of the following conditions: 1. ) NB = 0, np > 1, m x = 0 (10.2.68) 2. ) dB > 2 3.) dB = l, mi = 0 4.) n7 > 2, mi = 0 5.) n7 = 1, m x = 0, m 2 = 1. 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\m1 + l-NB,n7-m2 < ^ B - f m i + l ) (1 + l n ^ 2 ' 2. For n7 = 1 and mi = 0 and m2 = 1; or /or n 7 > 2 and mi = 0, (10.2.69) 2 I l i l U i - J V B . m z + r i T < (10.2.70) w/iere (mi,m 2) = (1 + mi, — m2), (mi, — m2), or (mi, 1 — m 2). Proof: From the fact that dB = NB — n7 and I I , , 1 + lnz l - i , i < sup < 1, z>l Z the cases: dB > 2, d B > 1 and mi = 0 n7 = 1, mi = 0 and m 2 = 1 , n7 > 2, m x = 0 and m 2 = 1 Chapter 10. The Fixed Point Solution 216 are clear. For the case n7 > 2, m x = 0 and m 2 = 0, using the fact that | | l | | _ i , 2 = sup^ >-< 3/2, 2>1 Z we have ^•\\mx-NB,m-m2 5: 11111 —djs — (?i7—2),ri7—2 | | l | < 3 - 1 , 2 2 K 3 B 1 -2,2 < Q - 1 , 2 < 6 2 A f Q . E . D . We consider the \\V K { £ , S ) \ \ \ + M U - M 2 with CQ, - / V b , n 7 , n p , m i , m 2 satisfying the specification (10.2.67)-(10.2.68). Lemma 10.11 \\VK(£,£)\\1+mu_m2<5VK \ \ S - £ \ \ , (10.2.71) where <W < — 2 rHTT^ [n*> + n s + 7 1 7 + 6 N<d (10.2.72) and NQ, CQ are defined respectively in (10.2.56), (10.2.67). Proof : Using differencing by part, VK(£,£) = P W ) Q { £ ) VP{7)(£,£) + P{7){£) VPW)Q{£,£) = P(i,2,5)Q(£) VP{7){£,£) + P{7){£) \Q{£) V P { W ) ( £ , £ ) + P{1,2,5)(£)VQ(£,£)] Suppose NB = n7 = 0, np > 1 and m i = 0. Since P(j)(£) = 1, we only have to deal with two terms. \\VK(£,£)\\1+MU_M2 < | | Q ( f ) | |o | | 2 ? P ( i , 2 , 5 ) ( f t f ) | | i 1 - r o a + | |P ( i > 2 l 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,2I5)(f,f)||l,-m2 < SMCQ \VP{l>2t5)(£,£)\\h0 < (1 + l n K 3 ) m 2 $M CQ (nP + n 5 ) e2^+n^ 5np+n5~l k?K?-1 (l + lnK 3 )™ 2 \e-e\\. (10.2.73) From (10.2.59) of Lemma 10.8, the second term |-P(l,2,5)(£)||o |P(l,2,5)(£)||o,-7n2 VQ(£,£) < 32 6M CQ NF e2(^+nK)Snp+n5 < 32 5MCQNp \\£-£\\ £-£\ K?k?(l-rlnK3)m' (10.2.74) Collecting the two bounds (10.2.73) and (10.2.74) and using the fact that K3 > 1, we have (10.2.71). Suppose the parameters satisfy one of the following cases: ds > 2, dB > 1 and mi = 0 n 7 = 1 and mi = 0 and m2 = 1, n7 > 2 and mi = 0. The corresponding norms can be bounded by a sum of three terms. \\VK(£,£)\\i+mu_m2 < \\P{i,2,5)Q(£)\\ l+mi , -m2 \\VP{7)(£,£)\\o (10.2.75) +\\P(7)(£)\\o Q(£) VPih2,5)(£,£)\\i,_i Imi,—m2+l + ||P(7)(£)||o ||P(1,2,5)^)||0 \\VQ(£,£) 11 l+mi ,—m 2 We consider each of the three terms in the RHS of (10.2.75) separately. From Lemma 10.7, l|P(l,2,5)Q(£)|| l+mi,—7Ti2 \\VP{7)(£,£)\\Q < ||P(l,2,5)(£)||o | | l | | l + m i - J V B , 7 i 7 - m 2 NB,-nr I |2>P(7) (5,5)110 Chapter 10. The Fixed Point Solution 218 < g 2 n 6 £ n p + n 5 \l+mi-NB,m-m2 SMCQn7 5n7~le2n7 /07 \\£-£\ < llll l + T O l — NB,m— 7712 8M CQ HJ ( jnp+^+nr - l^CriB+nT+nx:) P L.nS 1,717 From Lemma 10.6, • 1 1 ^ 0 1 1 . 11^>, ,„,„-„« < HP(7)(f) | |0 Hl|L,-„.,„ -"•3 ft5 ft7 | P P ( 1 > 2 ) 5 ) ( £ , £ ) | | W | | £ - £ | | . (10.2.76) + 1— T712 0 ( f ) NB,-m \VPM)(£,£)\\ < < 2 \ " 7 5 M C q (n p + n 5 ) e 2 n 5 ^ + " 5 - 1 &7 I l+mi-N B ,m-m,2 v5 -"-3 I l + m i -NB ,m-rri2 SM CQ (np + n5)e2^+n7+n^5n"+n5+n7-1 TfTip-l,n5 in7 (10.2.77) From Lemma 10.8, l^( 7)(f)||o | | P ( i , 2 > 5 ) ( f ) | |o PQ(5,5) l + m i , - m 2 < 11^(7)^)110 11^(1,2,5)^)11 0 1111 |mi — NB,m+l —m,2 1+NB,-717-1 C g J V 0 | | 5 - 5 | | < £7 fe £ M C Q A^Q e 2 ( " « + " 7 + n j r ) g n p + n B + n 7 ^ l l J - l | m i + l - J V B , n 7 - m 2 r /"p i .n 5 i .n 7 IP ^ I (10.2.78) For d B > 2, or for d B > 1 and mx = 0, from (10.2.76)-(10.2.78) and the bounds of Case 1 of Lemma 10.10, SM CQS""^-1 e^+m+nK) ^ + n ^ + K3l{n7 + 5 NQ) k77 K$lp~1)+dB~il+mi) (! + l n K^m2 | |PA(£,5)|| 1 + M I,_ 7712 — < 6-VK-Thus 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 Solution 219 10.2.3 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.(£, Er) defined by (10.1.21a-e) with the B(£)'s restricted only to the first order terms B i , 1 . t ( £ ) = B'i(S) + B'itG(£), (10.2.79) where B?(£) and B?tG(S) are defined in (10.2.3) and (10.2.8) (the JC(£,Er) here corre-sponds to a first order flow). We show the existence of the fixed point by showing that IC(£, Er) is a contraction with respect to the norm defined in (10.1.16) over the domain 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(£,Er) 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 fori=l,2 (10.2.80) k3 53 e~2 < c3 (10.2.81) k5 55 e~2 < (2 CM)'1 (10.2.82) k7 57 e~A < c3 (10.2.83) where the fcj's are defined in the norms || • in (10.1.15), e2 = A^/A 2 , , c3 = —b3, b3 is defined in (E.2.75). If we take the radius 8 of the Bg to be Q{e2), then Lemma 10.11 says that for a K^t(£) satisfying the specification (10.2.67)-(10.2.68) in which for i = 1,2,3,5, n 5 + n7 + nv + nK > 2 (10.2.84) for i = 7, n7 + np + nj< > 3 or n7 > 2, then the corresponding Lipschitz constant 5VK is 0 ( e 4 ) for z ^ 7 and 0( e 6 ) f ° r i = 7. Hence for sufficiently small e2, we can make 5VK small so that (10.2.80)-(10.2.83) can be satisfied. Chapter 10. The Fixed Point Solution 220 Since not all the terms of the first order (5 functions Bitist[£) satisfy (10.2.84). Let the first order terms in Bitxst{£) that do not satisfy (10.2.84) be denoted by K(£). By choosing appropriate value for the kiS defined in the || • norms, for K(£), we can again make the corresponding Lipschitz constants 6X>K from Lemma 10.11 sufficiently small so that the map JC(£,Er) is a contraction. We isolate K(£) from other first order terms by extracting leading terms from X s and Ys defined in (9.4.12). From (9.4.12), let us write Xs(£)=Xsa(£)+Xsb(£), Y°(£) = Y:(£) + Yb°(£), (10.2.85) where X°(£) = 2 ^ ~ ^ 2 , Xsb(£) = Xs(£)-Xsa(£), (10.2.86) (1 + U$) Y*'W = 4 n \ " 3 ' Ybs(£) = Y°(£)-Y:(£). The terms that do not satisfy (10.2.84) are: Kl{£) = e3Xsa(£) of B»{£) (10.2.87) Kl{£) = Is5(£)-SMe5Xsa(£)ofBI(£) K°{£) = e7Y:(£)o{B°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 2); (n5,n7,n7,np,nK, NB, NF); and CQ. For each of Ki(£), let us specify the cor-responding parameters and the Lipschitz constant 5VK of Lemma 10.11. K3{£): (mum2) = (0,0); (n5,n7,np,nK,NB,NF) = (0,0,1,1,0,2), CQ = 4|c3 - 66/3|; 8VK < 8 | c 3 - 6 6 / 3 | e2 5M [1 + 64 5]. (10.2.88) Chapter 10. The Fixed Point Solution 221 K5(£): (mi,m 2) = (0,1); i) I (n5,n7,np,nK,NB,NF) = (0,1,1,0,1,0), CQ = |c 5| where C5 is the coefficient from J | (see (9.4.9)); ii) 5Me5Xa(£) (n5,n7,np,nK,NB,NF) = (0,0,1,1,0,2), CQ = b5 where 65 is defined in (E.2.75) of Appendix E; SVK < 2 SM e2 i p + 4 o 5 (1 + 64 5) k7 K7(£): (mi,m 2) = (0,1); (n 5, n 7 , np, nK, NB, NF) = (0, 0,1, 2, 0, 4), CQ = 4 |67| where 67 is defined in (E.2.75); < 8 |6 7 | e4 5 M [1 + 128 S\. (10.2.89) (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 fc3, k5, k7, and 5. Let = min k7 = min C3 1 16\c3-b6/3\'2j' k5 = min ,8C M ( ( | c 5 | /A; 7 )+ 4 6 5 ) ' 2 r C3 16|67 5 < min ^C,? e2, ^ (10.2.91) where C 5 is a positive constant to be determined later from the bounds of ||-Bj(0)||m i ) n i 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 b3, b5, be, b7 from the first order diagrams (see Appendix E for the derivation of these coefficients). Recall the solution E{£) in (10.1.21a)-(10.1.21e) for defining the map K,(£,ER). Let EKB(£) denotes the part of E{£) corresponding to the contribution KB of B(£). Proposition 10.1 For the terms Ki defined in (10.2.87), by choosing S and ki as in (10.2.91), for£, £ 6 B5, l ^ ^ l l w ^ Q + C i e 2 ) | | 5 - £ | | , where Ci is a positive constant independent of e2. Proof: From (10.2.88) and (10.2.91), and applying Lemma 10.1, we have VEK%{£,£) < 8 e 2 | C 3 - & 6 / 3 | [ l + 6 4 C 5 e 2 ] ^ | | £ - £ | | w C3 < (I + C 3 e 2 ) | |£-£| | , where C 3 = 512 |c3 — be/3\ C$/c3. 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 0(e2). • The Xb and Yb defined in (10.2.86) has np = 2 We also observe that the i 7 | ( £ ) of B2(£) has n 5 = 1 and nK = 1. Next we consider the factors of BsiG{£) defined in (10.2.7-10.2.8). These terms are of the form Gsitjl3°+\ where p?+h is a Bsi+1 defined in (10.2.6) or -5M{£1+2 + e5Xs+2{£)) if % = 7 (see the discussion after (10.2.1)); and the Gitj's are defined in (9.2.1-9.2.7). Chapter 10. The Fixed Point Solution 223 From (9.2.6-9.2.7), it is easy to see that 0(e 2) for iy£ 7 Q(e 4 ) fori = 7 G 1,3 (10.2.92) • For 2 = 1,2, 3, 5, each P?~[Ji is Q(e2) or has n 7 = 1. 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( e") °f t 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 K{£) : BT m • B2 e3Xl : B3 E$b~M XI : B$ £7 Xsb : B7 • Q(e 4 ) 0(e 4) 0(e 4) 0(e 6 ) K(£) : BITG G\jP$t : BUG i ± 7 GhPjfi1 : Br<G 0(e 4) 0(e 6) Next we consider the bounds for the terms from B i ) Pt(0). Observe that by setting £ = 0, terms involving explicit ft factors are zero. Thus it is easy to see that -B;(0) = 0. As for the terms GjjPj^}t(0), we consider the following estimates: • a Pj\}t{0) contributes a factor of Q)(e2); • for j = 1, 2, a Pj*h(0) contributes at least a factor of Q(l/F); • for j = 7, the interpolation form of G -^(O) shows that it contributes a factor of Q ( l / ( r ) 2 ) (see (9.2.5-9.2.7). Chapter 10. The Fixed Point Solution 224 From the above estimates, for B^p^O), we give a list of estimates for the norms ||i?i,p*(0)||mij with (mi,ni) as defined in Lemma 10.1 in Table 10.2. Table 10.2: Bounds on 13^(0) Bi(0) 11^ (0)112,0 = 0 ||£2,i-(0)||2,o = 0(e4) ||53,i-(0)|ko = 0(e4) ti^5,i-(0)lli,-i = 0(e4) l | £ 7 , i - ( 0 ) | | i , 0 = O(e 6 ) From the estimates in Table 10.2 and (10.1.31b) of Lemma 10.1, it is easy to see that I ( 0 ) | | ( i ) < CBotl.t e2 + Cl (10.2.93) where Eitlst{£) denotes Ei{£) defined (10.1.21a-10.1.21e) with £ ; ( £ ) = Bitl.t, Cl are defined in (10.1.32b) and CB°,\ is a positive constant independent of e2. To find bounds for the nontrivial C\ and Cr so that fZ(Bs) C B$, we need to set bounds for the boundary conditions El and E7 of (10.1.18). Let us chose the following bounds for Cs, \El\, and \Er7\: C$ > 8 CBO,!**, \Er5\ < Cs (l + \nK3)e4 8 k5 K3 CM ' From (10.1.32b) and (10.2.94), we have \Err\ < Cs (l + lnK3) e4 8 k7 ci< Cse2 8 Cr7 < Cse2 8 (10.2.94) (10.2.95) We conclude this section by stating the map JC(£, Er) defined in (10.1.22) with using only the first order contributions, i.e., setting the Bi(£) in (10.1.21a-10.1.21e) to Biti*t(£). Chapter 10. The Fixed Point Solution 225 Proposition 10.2 Suppose fC(£,Er) is the map defined by (10.1.21a-10.1.21e) with E>i(S) = Bi^st(£). For sufficiently small e2, by choosing Cs, E\, and E7 satisfying (10.2.94), for £,£ eBs with 8 = Cs e2, then 1. \\VIC(£,£)\\ < \\£-£ \\)C(0,Er)\\ < where Cx>B,iat is a constant depending only on Cs; 2. K{BS) C B6. Proof: Let £,£ e Bs. From (10.1.31a) of Lemma 10.1, Table 10.1, and Proposition 10.1, \\V }d(£,£)\\{i) < (I + C W , i " e2) l | 5 - 5 | | (10.2.98) By choosing e2 satisfying C T ? B , I " E * < \ (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 e2 in (10.2.99), \\K.(£,W)\\ < | | / C ( 5 , ^ ) - / C ( 0 , ^ ) | | + i m o , ^ ) | | < (\ + CVB,I>* e 2) CS e2 + ±CS e2 < C5e2. Thus \\JC{£, Er)\\ < 8 and it follows that K{BS) C Bs. Q.E.D. (10.2.96) (10.2.97) Chapter 10. The Fixed Point Solution 226 10.3 Higher order terms In this section, we extend result of Proposition 10.2 for the map fC(£, Er) defined in (10.1.22) with the higher order terms BfA{£) and BfG.A{£) (defined in (10.2.5) and (10.2.9) of a fixed order included. We apply the same strategy as in the proof of Proposi-tion 10.2 by showing that the corresponding Lipschitz bounds and bounds of the higher order terms are also of Q(e4) for i ^ 7 and are of 0( e 6) f ° r i = 7. 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 x = | V G > T I I | VG,m = the set of V7 vertices of G, n7 = | V G J T 1 7 | CQ = the set of lines of G CG,B = the set of photon lines of G, n B = | £ G , B | CG,F = the set of Fermi lines of G, nF = \CG,F\ £-G,B,m = the set of selected photon lines attaching to the V7 vertices (see (9.3.33) £G,B,dB = CG,B/CG,B,m Note that \CG\ = nB + nF, \CG,B,n7 \ = n7, and \CG,B,dB \ = nB - n7 > 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) AG(X,h,P) A i ( V G , n i , A,h) A7{VG,n7, A, h) V(CG, A,h) (10.3.3) Chapter 10. The Fixed Point Solution 227 A i ( V ( ? , n i , A, h), A7(VG,n7, A, h), V(CG,X,h) are defined in (9.3.33) and E^> s denotes We would like to bound (3-(G) by a similar argument as done in the proof of Propo-sition 9.1. Here we do not assume conditions (9.1.1) since the running couplings A = (A 1 ; A 2 , A 3 , A 5 , A7) are replaced by (A + £) = (Ai(l + £{), A 2 ( l + ft), T + <f3, ft, ft) where £ £ Bs and T is defined in (9.4.16). Also, we do not estimate 1 + 7 (defined in (5.4.9c)) by 7 = A 3 + A 5 in the photon running slicing functions VB defined in (5.4.9a) and we keep better account of the factor C0 of the bounding constant in (9.3.35). 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 B5. We set the radius S of Bs as Cs e2 where Cs is a constant to be determined later. We would like to further introduce the following notations for the account of the C0 corresponding to Pf{G). T(£,h) CG nK where cG, M(G,h), 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){£) = E4G,ni(£) KlG{£) (10.3.5) h>s = II ^ II r \ (10.3.4a) i'ecG,B,n7 ft iecG,B,dB = cG Y,M{G,h) (10.3.4b) h>s = 2 n B + n F ^ C K G = 2 ^ B + n F C G ^ C S _ y 7 ^ (10.3.4C) / Cx \ n r =. 2n'+2nB+nFCG[j-j , C°PG = 2nBCG, (10.3.4d) ^ if nx is odd — if ni is even 2 (10.3.4e) nK + 2n7, (10.3.4f) Chapter 10. The Fixed Point Solution 228 where KlG(S) = j dP AG((A + £)XP) AG((A +£),h,P) KG(h,P). = A?A7(VG,n7, (A + £), h) V(CG, (A + £), h) (10.3.6) (10.3.7) where A7(VGtTl7, X, h), V(CG, X, h) are defined in (9.3.33). The reason for writing (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 A7(VGtTlT, (A + £), h) V(£G, (A + £ ) , h) of K^G(£) which is similar to the proof of Lemma 10.11 . Let KIG{£) = T.KIG{S). (10.3.8) h>s Recall the p functions Bi(£) for the map }C(£,Er), for i=l,2, the P functions are scaled 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 < 2ni> \ztK?G(£)\ < C A g e ^ ° G M { ° J H ) h G K n ~ |r(5,/i)| ( P ) no e 2 n K (10.3.9a) (10.3.9b) (10.3.9c) (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), Th/Th < 2. From the definition of Vs in (9.4.16), for h > s > - 1 , l + lnT' 1 l + lnT s 1 1 p / i — ps ' p / i — ps ' Hence, using the fact that \S7 \ < (Cs/k7) e4 (1 + \nTh) and nB > n7, 1 / T T 1 T T I 7 I T T < TT — TT TT < J f t T T T " (TS)nB eecGBrh< < { e ^ r B ( 1 0 3 1 0 ) From (10.3.9b), the definition of CG, CKa in (10.3.4b-10.3.4c), and the bound (10.3.10), we have \*iKla(£)\ ^ Y.\^KIG{£)\ h>s < £ C A G ^ G M { G : K ) < CAG + C g E M { Q t K ) ^ ' h>s - p-j^ • (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(£)\ < \AG(£,h)\ J dP\KG(h,P)\ < \AG(£,h)\cG M(G,h), (10.3.12) Chapter 10. The Fixed Point Solution 230 where AG(s,h) = A - n n A A " 1 eecG\„7 ' eeCo,B ^ A 2 ( l + ^ ) 1 + W*< J r(5,/i) n n (10.3.13) For £ e Bs, from the bounds (10.2.31) and the definition of C A g in (10.3.4c), n * IT _ J _ < Ar ni = c, A G Ar ' (10.3.14) From the identity ri\ = \£G,F\ + \AG,F\/2, we have nF = ny for i = 3,5,7, and nF = r i i — 1 for i = 1, 2. Hence from the definition of n y in (10.3.4f), it is easy to see that Ar (10.3.15) From (10.3.12-10.3.15), we have \zt K?JG(£)\ < CAG cG M(G, h) en>< (10.3.16) \r(£,h)\ (10.3.9d) can be obtained by setting £ = 0 in Pi(G, s,h)(£) and applying the a similar argument as the above. Q . E . D . Chapter 10. The Fixed Point Solution 231 We now turn to finding Lipschitz bounds for the Zi (3-(G)(8). Let us consider the | | • | | m i , - m 2 norms of V (3f(G) where the range of values of (m 1 ,m 2 ) are described in (10.2.66). Proposition 10.4 Suppose 8,8 G B5 with 5 = Cse2, and (3-(G)(8) as defined in (10.3.5). 1. where \zi D 0i(G)(8, 8)||l+mi,-m2 < eVPG 11^ — f £vpG — C, (10.3.17a) (10.3.17b) 2 Cs and NQ is defined in (10.3.29) and C$G, nv are defined in (10.3.4d), (10.3.41) respectively, 2. M|(G)(0) | | 1 + m i , _ m 2 <e%G (10.3.18a) where £P°c = C k e 2 n K > (10.3.18b) UK are defined in (10.3.4e), and C°^G is the constant independent of Cs defined in (10.3.4d). Proof: From (10.3.5), V pst(G)(8,8) = £ [V 4GNI(8,8) KlG(8) + 4GJ8) V K*G(£,£)] . (10.3.19) Chapter 10. The Fixed Point Solution 232 Hence, from (10.3.9a) 1+7711,-7712 < II* E VEl>ni(£,£)KlG(S)\\l+mu.m2 (10.3.20) h>s +2"1 \\ZiV J2 Kf>G(£,£)\\1+mi>_m2. (10.3.21) 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 B6, since for hv > s, [X —t £iv\ < 2 and l / r h " < i / r s , VEiGni(£,£)\ < n i 2 n i " 1 r f g | 1 . (10.3.22) Hence, -7712 / l > S < m 2"1"1 | | ^ A ^ 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, dB = dB-n7 = nl + 3n7 - \AG,B\ > 2 (10.3.24) where | A G , 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) CKg e2nv zi Ki,G(£)\\mi,-m2 < K d B - m i ^ + l n K ^ m 2 < C K G e 2 n v . (10.3.25) Chapter 10. The Fixed Point Solution 233 Collecting the bounds in (10.3.23) and (10.3.25), since CpG = 2niCK(,, the first term (10.3.20) has Lipschitz bound 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 AlG(£) = 2^ZiAnC A7(VG,„7, (A + £), h) V(CG, (A + £), h). A\G(£) has a form similar to the K{£) of (10.2.35) with P{£) = 1, T(£,P) = 1, and (n5, rip, NB, NF) = (0, 0, nB, nF). From the proof of the bound (10.3.9b), it is easy to check that we can pick the corre-sponding CQ of Lemma 10.11 as m CPG e2"v \\£-£ (10.3.26) 2 \VAlG{£,£)\\l+m^m2 < (n7 + SNQ) \\£ - £\\ (10.3.28) where NQ = 8(nF + k3lnB) (10.3.29) and kz is defined in (10.2.91). From (10.3.28), ( r s ) 1 + m i ( i + i n r s ) m 2 2"1 \ziY,VK£G(£,£)\ h>s < h>s < h>s £ / dP\\VAlG(£,£)\\1+mu.m2 \KG(h,P)\ < (10.3.30) Chapter 10. The Fixed Point Solution 234 Since 8 = C^e2, from the definitions of CpG, nv in (10.3.4d), (10.3.4f) respectively, (10.3.30) can be simplified to CPG e 2 ^ - D n7 _ + e2 NQ \\£-£\\. (10.3.31) C s 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 BiiA(£), nv > 2 for i ^ 7 (10.3.32) > 3 i=7 from (10.3.17a) of Proposition 10.4, the corresponding Lipschitz bounds for Zi(3-(G)(£) are C)(E4) for z 7^ 7 and 0 ( e 6 ) for i = 7. Similarly, since setting 5 = 0 implies a nonzero ft-(G)(0) has its corresponding n v = nK, it follows from (10.3.32) and (10.3.18a) of Proposition 10.4 that Zi(3?(G)(0) are 0(e 4) for i ^ 7 and 0(e 6) for i = 7. From the bounds on the higher order terms of BfA(£), it follows that the terms from BfG.A(£) have similar bounds (see (10.2.7-10.2.9) for the description of B-G.A(£)). We summarize our results on the Lipschitz bounds on the higher order terms by the following corollaries. Let KBi(£) be a term of BsiA(£) or BsiG.A(£) with £ £ Bs-Corollary 10.1 For £,£ G Bs with 8 = Cse2, and e2 sufficiently small, \\VKBi(£,£)\\m^ni < CKe2ai \\£-£\\ (10.3.33) | | A ^ ( 0 ) | U , n i < C°Ke2a<, (10.3.34) where = 2 for i ^ 7, 07 = 4, CK, C°K are positive constant independent of e2, and C^ is also independent of Cs-Chapter 10. The Fixed Point Solution 235 For a given order N flow defined in (9.3.39), Let EA,N{8) denotes the higher order contributions to E{8) defined in (10.1.21a-10.1.21e) corresponding to terms of Bi<A{8) and BsiG.A{8) that are of order up to N. From Lemma 10.1 and Corollary 10.1, we get the following bounds for EA,N{8). Corollary 10.2 For 8,8 e Bs with 8 = Cse2 and e2 sufficiently small, VEA>N(8,8)\\ < CVA,N e2 \\8 - 8\\, (10.3.35) l l ^ ( 0 ) | | < CAo,Ne2 (10.3.36) where CVA,N and CA°,N a-re positive constants depending on N with CT>A,N further de-pending on Cs-From the results of Corollary 10.2, we wrap up the fixed point argument for showing existence of a solution to the subsystem (Ai, A 2 , A 3 , A 5 , A7) satisfying the boundary con-dition (Af = A 1 } Xu2 = A 2 , Ar3 = KZ>\, XI = El, A 7 = W7) (10.3.37) for a given order N flow, where Erb and E7 are the boundary values of £ f (£) and E7{8) of the map JC(8, ET) defined in (10.1.22) and the ranges of ET5 and E7 are to be chosen below. From the bounds of ||£P*(0)|| in (10.2.93) and the bounds of | | £ A i i V ( 0 ) | | in (10.3.36), let us choose Cs>8 (CBo,i + CAo,N) • (10.3.38) where Cs is the constant for defining the the radius 8 of the ball Bs, i.e 8 = Cs e2. Let the boundary values of El and Er7 satisfy the following bounds: C5 (l + ln iv 3 )e 4 C , ( l + lnif 3 ) e4 ^ 5 l - 8k5K3CM ' 8 ^ • - ( m 3 - 3 9 ) where k5 and k7 are defined in (10.2.91), CM is defined in (10.1.27), and Cs satisfies (10.3.38). 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 e2 where e2 = ( A 2 / A 2 ) . Then for For e2 sufficiently small, K3 > 1, E\ and ET7 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 Bs-From (10.3.35) and (10.2.98), \\VJCs+1(£,£)\\{i) < + (CVB,I« + CVA>N) e 2) \\£ - £\ \ . (10.3.40) By choosing e2 satisfying (CDB,I«* + CVAtN) e2 < ^, (10.3.41) then \\VK(e,i)\\ < ^\\£-£\\ (10.3.42) 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) \\JCt(0,Er)\\{l)<^-. (10-3.43) From (10.3.42) and (10.3.43), \\K,(£,Er)\\ < \\JC(£,Er) - K,(0,Er)\\ + ||/C(0, £ r ) | | < C6 e2. Hence K{BS) C Bs. Q.E.D. Chapter 10. The Fixed Point Solution 237 10.4 The solution for the running couplings 10.4.1 The solution for A| and Af To make the solution of the full coupling flow complete, we determine XSA and Af for given boundary conditions Af and A 4 chosen at the root scale since A^ and Af diverges 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 /?|(A4,A*) = A ^ ( A * ) + ^ + 1 ( A 4 , A * ) , Bi(A.) = 6M e2 b4 A A f = (3^ • -b~Mbe,e2 '1 + E{ (10.4.1) (10.4.2) 1 + ElJ where A* is the primitive solution of A*, and ^ ( A 4 , A*), A\(A*) are the higher order terms. From the discussion in Section 9.1, A\ is linear in A 4 . Furthermore, since A 4 (1 ,A») and AQ(\*) are 0 ( C 2 ) where ( = (Ai /A 2 ) 2 /7 , from the solution A*, it is easy to see that A*(1,A*) and Af (A*) are of 0(l/(Ts)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 A5 = AS + 5 ]#(A, ) . (10.4.3) Since the leading term of /?|(A*) is 5Mbee2 'l + E( t=o - 1 sM e4 o ( v n and the remaining terms are of Q(l/(TS)2), it is easy to see that IAS — A5| < t=0 < CRr e 2 ln (P) ln(r°) (10.4.4) Chapter 10. The Fixed Point Solution 238 where Cpe is a positive constant depending on N. 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°+lBl+1(A,). (10.4.5) From (9.2.18), A\ has leading behavior tf4(-J , (10.4.6) where K4 is a positive constant, a 4 = 6 4 / 6 3 ~ 9m/2 and m is the "electron mass" of the 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 + 0(1). (10.4.7) Let us consider the difference equation of r 4 . From (10.4.1), X ^ + ^ + M - ^ r ^ A ) (10.4.8) Art = K+'A(A^)-1 + ( A ^ ^ A A l _ _|_ ( .A^- 1 AS+lt \-4 1v4 1V4 From (10.4.5), the first two terms of (10.4.8) add to zero. Setting r 4 = 1 + E4, we have the following difference equation for E±. AEl = (A^) - 1 ^ + 1 (A 4 (1 + E 4 ) ,A , ) (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(\*) is sum of terms where each is of the form £ JdPAG(\4,\*,h,p)KG(h,P) (10.4.10) h>s+l Chapter 10. The Fixed Point Solution 239 where if \X\\ are only of order in powers of s then the exponential decay from KG(h, P) allows the following bound. T,jdP AG(X4,X*,h,p)KG(h,P) h>s <CG\AG{X4,X*,h ,1)|. (10.4.11) where h denotes setting all components of h to s. Hence by assuming that E\ = o(l), from the the leading behavior of A 4 in (10.4.6), | ( A 4 ) _ 1 ^ 4 + 1 ( A 4 ,A*)| < C p * S M e * (10.4.12) where C@4 is constant depended on the order N. From this fact, we see that ii El = o(l), then the RHS of (10.4.9) is of e4 O ((H - 2)- Hence we expect Es4 = e2 O ((rs)-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 = {£± | | | £ 4 | | i , o < Q e 2 } , (10.4.13) l- J2 ( A 4 - 1 ) - 1 4 ( A 4 , A , ) t=s+l 1,0 where || • | | 1 ) 0 is the norm defined in (10.1.14) and Cs > 2 We consider the map E4 : Bs —> Bs defined by E4 = 0 and EKSt) = E ( A 4 - 1 ) - 1 A 4 ( A 4 ( 1 + <?4),X). i=s+l Since A\(A4 (1 + £4), A*) is linear in £ 4 , from (10.4.12), it is easy to see that (10.4.14) (10.4.15) \VES4(£4,£4)\ < E ( A n - 1 ^ A ) t=s+l \£± — £4111,1 < C V E i e £4 — (10.4.16) Chapter 10. The Fixed Point Solution 240 where CVE4 is a positive constant depending on N. From above, one can easily establish the Lipschitz bound \\VEs4(£4,£4)\\lfi < l- H ^ - ^ I k o - (10.4.17) From the (10.4.14), | | £ 4 ( 0 ) | k o < C8e2/2. Thus from (10.4.17), it is easy to see that the map E±(£±) 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 = K4 (1 + e2F(K4)) with F(K4) = 0(1) as e2 —> 0. Since F(KA) is continuous in K4, for KA e S4 = (1/2, K^) with KI a fixed constant, for e2 sufficiently small, a subset of S4 can be chosen so that XKK4) satisfies the condition 1 < A 4 < k\ where k\ is a constant. Chapter 10. The Fixed Point Solution 241 10.4.2 The solution for the full system Let us write down the solution of the coupling flow for the full system. Let QSN 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: ^ = £ /?••"«?) ( 1 0 . 4 .18) G^G+V\N G ^ N A ^ = £ PT\G) A K = E Pt+1(G) G ^ N A A* = -SM Xs5+1 + M~2s £ PS5+1(G) A A e = E P ? \ G ) AX* = PT+1(G)-G ^ N with boundary conditions: A f = Ax, Ao/ = A 2 , (10.4.19) Ar3 > 1, K4>XrA>l, XI = K5 Af = A 6 , Xr7 = K7, where A 4 , K5, Ke, K7 are finite constants. Recall that T s = £ 3 (s + 1) + A 3 , e3 = c3 5 M e2, e4 = 64 5 M e2. Chapter 10. The Fixed Point Solution 242 Theorem 10.2 For e2 = (Ai/A2)2 sufficiently small, there exits a positive constant Cg depending on N such that for | * . | < | * | < + (1o.4.20) 8k5K3CM 8k7 where k5, k7, and CM are respectively defined in (10.2.91) and (10.1.27), then there is a unique solution Xs satisfying (10.4-18), (10.4-19), and (10-4-20) with the following asymptotic behavior. For r < s < U, X{ = M0. + E1), \ E S \ < ^ , (10.4.21) A* = A 2 ( I + E | ) , m < C s e 2 XI = As3(l + Es3), \Es3\<Cse XI = Al{l + Efi, \ES4\< 2 (1 + l n P ) k3T' ' Cse2 CdeA (1 + lnP 5) k5T° \K\ < \Xl\ + Cse2 ln(P) Cse4 (1 + l n P ) k7 where E\ = 0, A 3 = rs A4 = ^ n ( i - ^ ) _ 1 t=o v 1 7 and K4 is a constant depending on Xr4. Note that A f + 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 Ward 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. where M is the 4 x 4 diagonal matrix with diagonal entries M3- defined in (5.3.2)-1. S (11.1.1) (5.3.3). 2. * = ( * 0 , * 1 , * 2 , * 3 ) , $ = ( A * , * ) -3. 3=0 243 Chapter 11. Ward Identities 244 4. = Af+Vx + ( A ^ 1 - 1)V2 + ( A ™ - 1)V3 + ( A f + 1 - l)V4. 5. V £ + 1 = M 2 C / A 5 U + 1 V 5 + A ? + 1 V 6 + A f + 1 V 7 . 6. VJJ+1 — vgi + * V 7. V r U + 1 ( * r ) = l o g - A - / % + i ] ( $ ) e ^ ( $ + $ , ) . where = J dP[0>u+1](<S>)evuUZ^ (11.1.2) 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 p[o,u+i] = £ pi ( 1 L 1 3 ) i=0 where for 0 < i < U, pl = pl , pu+l = plu+1>°°\ and the pl are the neighbourly slicing functions defined in (B.2a). It is easy to see that S satisfies the identity e - ^ S " 1 eiex = S - 1 + e pX- (11.1.4) Chapter 11. Ward Identities as in (2.4.5). Under the change of variables A' = A, AT' = Ar, it is easy to check using (11.1.4) that we have the following. 1. For i ^ 2, Vr<($' + $r') = Vi($ + $r) (11.1 and V2($' + = _(^r + g-ie X(_-^ye* + ^ = V2($ + $r) + e Vxidx, tf + tfr,tf + #r) = V 2($ + $r) + e Vx{dx, + * r ' , + *r')-2. V rj /+ 1 1($ + $ r ) = y^($ ' + $r') + e(A2 a + 1-l)Vl(^x,*' + * r \ * ' + *r')-(11.1 3. dPu+1($) = cdPu+1(&)eeVl(W>*'l (11.1 where c is a constant. Let us consider the root potential Vru+1 under the change of variables Chapter 11. Ward Identities 246 From (11.1.6) and (11.1.7), Vru+l(V) = log-^— [ d P u ^ e M V ^ 1 + (11.1.8) x exp (e [Vi(0x, *') + ( A ^ 1 - l)Vi(0x, + * r ' , + ^ r')]) • The exponent of the second exponential term in (11.1.8) can be simplified into eAf+Vx^x, + * r ' , + * r ') - e [Vi(<9x, + * r' , + * r') - Vi(<9x, *')] = eAf + V 1 ( ax ,* ' + * r\*' + ^rVeV'i(^X>^,>*'^T'',*r') (11.1.9) where = vi (dx, * r \ $') + vi(a x , * r ') + vi (a x, * r # , f ' ) -From (11.1.9), since V 3 ($ + dx) = V 3($) where $ + 9% = (f, , A + dx), Vu£(® + $r') + e A f + ^ a x , tf' + tfr', tf' + tfr') = v^ +H '^ + ^ ' + ax) (n.i.io) +^ c / + 1 (9x, $r') + ( e A ^ 1 - A f + 1 ) VI(9x, + ^ r ' , tf' + tfr'), where w^+1(<9x, $ r) = vguv+l($ + $r) - vguv+1($ + $ r + ax). (n . i . i i ) Thus the RHS of (11.1.8) becomes log ^ / dPu+l(&) exp ( V f f i 1 ^ ' + <T' + 9x) - cVi(9 X , * r ')) x exp (Wu+l(dx, $r') + ( e A ^ 1 - A f + 1 ) Vi(<9x, tf' + tfr', tf' + tf7"')) . (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 2 " condition Xu+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 + 1 / A 2 7 + 1 and not allowing gauge variant coun-terterms in Vff+l in the LRC set up, then the root potential Vru+l satisfies the Ward Identities Vru+1($r) = l o g - i - /dPu + 1(<S>) exp ( V ^ C * + <F') - Vi(dx, W, % , (11.1.14) where W' = (Ar',$>r',qr') = {Ar + dX, eiexW, e~iexW). (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 Xgv — (A5, A 6 , A7) to be selected at the root scale r. Moreover, the fixed point argument does not indicate by choosing A ^ + 1 = (0,0,0), the corresponding X would have a finite limit for its components as U —> 00. 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 248 11.2 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 / 1 = (Af + 1 , A f + 1 , A f + 1 ) set to (0,0,0), the resulting 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:^>a. (11.2.1) where % are the components of tfe = $ r 2 , $>r3). 2. LVo,r = E A ' V J (11.2.2) \xpl=ipV-0,i>0 where each \ \ denotes a formal sum of contributions from all graphs at root scale r and VI = Vj($r). 3. H(x) is the dimension zero local part of ipeiexe(Sfix + $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^1 has gauge in-variant form such that A f + 1 ; A f + 1 , A f + 1 are all zero. Let e = A f + 1 / A f + 1 . Then for any smooth polynomially bounded function x{x)> Vru+l(Ar,Vr,$>r) = Vru+1(Ar+ dx,(l + SX)eiexW,qre-iex{l + XS)) + j Vre-iex(X + XSX)eiexmr (11.2.3) where X = e$x-Chapter 11. Ward Identities 249 Lemma 11.2 (analogue of Lemma 4.3 of [FHRW88]) Suppose has gauge in-variant form. Then for any smooth polynomially bounded function x{x), LVoAAr^r^r) = LV0tr{Ar + dx,etex^r^re-iex) + e j ^ 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 Bs = l-HSKS (11.2.5) where Ks and Hs are defined in (5.2.2b) and (5.2.3a). Thus at the end of the flow, the external field $ r becomes u l r = I ]B s * r - (11.2.6) s=Q Lemma 11.3 LMiV) = Vi($r). (11.2.7) Proof: From the formulae of Hs and Ks, it is easy to see that we can write u l[Bs = l + B, (11.2.8) s=0 where B is a sum of terms in which each term contains the ps's as factors. Let K($ r ) be 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) $ r(p). (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(Ar + dX)2 + K j[d • (Ar + dX)}2 + Xr71[Ar + dxf (11.2.10) Comparing the formal sum of both sides of (11.2.4), since x is arbitrary, it follows that Ag, Ag, and Xr7 are formally zero. Let us state our observation in the following Theorem. Theorem 11.1 Suppose the top scale couplings of a LRC expansion for IQED has the form A f + 1 = A l 5 A ^ 1 = A 2 , (11.2.11) X\>+1 = b\nA + K3(U), XV+1 = K4(U), A f + 1 = 0 z = 5,6,7, (11.2.12) where bin A is the coefficient of the V3 local part of the VP diagram with lines as the scale U+1 shifted spinor covariances and 6 " C ( A 2 » « ) 2 C, Ai and A 2 are constants independent of scales and the loop regularization parameter A, and K$(JJ), K4(U) are functions of the scale U but independent of A. Then the resulting root scale couplings Xr5, Ag, X7 are formally zero. Chapter 11. Ward Identities 251 The above theorem suggests we must choose the boundary condition in Theorem 10.2 as Af = Ai , Af = A 2 , AS > 1, K4>XrA>l, \ r 5 = 0, XI = 0, A 7 = 0, 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*"1 of the fixed point solution of the order N flow is zero since the theorem only provides bounds on the components of A ^ + 1 . We can not apply the uniqueness of the solution to argue that A ^ + 1 is zero since we have not shown that Xrgv is zero in an order N perturbation. Chapter 12 Conclusions and Outlook 12.1 Summary of the LRC scheme for IQED 12.1.1 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 Iden-tities (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 Vs, the local quadratic terms of Vs 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 ps of the slice covariances are replaced by the running slicing functions Vs 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 Xs = {Af}J=1 in the Landau gauge for determining the asymptotic form of As as s —> oo. The resulting renormalized connected Green's functions of the expansion 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 Xs is a fixed point argument in a suitable Banach space whose norm captures the asymptotic form of Xs. We chose the Landau gauge in our set up because the corresponding relevant V\ local part of the vertex diagram and the relevant V2 local part of the electron self-energy diagram vanish (see the calculation of these local part in Section E.2 of Appendix E) and the kernel of the divergence term Ve is orthogonal to the corresponding photon free 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 254 12.1.2 Two-slice LRC In the L R C scheme for IQED, there are two types of U V divergence in the limit of remov-ing 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. By 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"3 local part of the top scale V P diagram, and lower level 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 s}^L 0 as follows. For the photon field, pB+1 = p[u+1'u>] is the top photon slicing and PB = P^°'U^ I S t n e bottom photon slicing, where U < U'. As for the spinor fields, the top slicing is puF+l = plu+1'°°} and the bottom slicing is puF = p' 0 , 1 7 ' . 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 A f + 1 = Ki z ^ 3 , (12.1.1) Chapter 12. Conclusions and Outlook 255 A 3 = MnA + 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 < M2U, 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 Ks (see (5.2.2b) for the definition of Ks) which flow down the scale via the shifting transformation. From the form (5.2.2b) of Ks, 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 The mult i-sl ice 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 AA? = <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 re-garding 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 4 . The second observation is that, by choosing the Landau gauge, the kernel of the free photon propagator is propor-tional to Lfj,v (defined in (1.2.8)). Since the kernel of V6 is proportional to the projection T^y (defined in (1.2.8)) which is orthogonal to L^v, contracting a V 6 vertex in a graph gives zero. Hence, for having a cleaner form of the running photon slicing function, we chose not to shift V6 in the shifting procedure and left it in the running potentials. From 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. O D E 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. An O D E analogue is then obtained by dividing the discrete flow equations with M 2 — 1 and taking the limit M —> 1 + , where M is the slicing parameter of the smooth slicing. From considering Chapter 12. Conclusions and Outlook 257 the subsystem Xsub = (Ai, A 2 , A 3 , A 5 , A7) of the ODE, we discovered that the O D E flow is governed by an O D E flow of the generalized coupling where the solution of the couplings can be recovered from solution of ( s . For the flow of C satisfying a sufficiently small initial condition, as s —> oo. The O D E analysis indicates that it is the flow of £ that is ultra-violet asymp-totically 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 ( s. In Section 9.3, we defined a perturbation order for the A -> oo coupling flow in terms of the orders of the /3/(G)'s in powers of Cs (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 ( s stays around the origin. As in the O D E case, we 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 s corresponding to the (3 functions in (9.2.1) by explicitly solving for the solution. For the boundary condition (12.1.2) (A|) 2 (A| +Ag) C = 0(1/8) (12.1.3) A 2 (12.1.4) A- = Ki 3 < i < 7 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 2 , Af = £ 3 ( s - r ) + K3, (12.1.5) Al = K4 IT fl ^ \ —) , V^A e3(t-r) + K3) Af = AT 2< s- r> tf5, K = Ke, A* = A 7 . where e3 and e4 are defined in (9.2.11) and (9.2.12). In the subsequent sections, we then analyzed the higher order flows by expressing an order N flow Xs as a perturbation of the primitive flow A s where XS = AS + ES. (12.1.6) An order N flow equations (9.4.15) of Es is obtained by substituting (12.1.6) in the order N flow equations for Xs. For convenience, we set A 6 = K7 = 0, and for the corresponding order N flow of Es, we first considered the subsystem with boundary condition EB: U? = 0, U? = 0, Er3=0 (12.1.7) El = XI z = 5,7. In Sections 10.1-10.3, we showed that for sufficiently small e2 = ( A i / A 2 ) 2 , Af, and Ay, there exists a unique stable solutions of Es satisfying EB. Hence there exists a unique solution of the subsystem Xsub = (A l 5 A 2 , A 3 , A 5 , A7) satisfying the corresponding boundary condition Xfub. The existence of a unique solution of E satisfying EB is obtained by showing that, under the conditions on EB 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 K(8) where the terms K(8)s of the sequence satisfy the the boundary condition EB and the fixed point equation obtained by replacing the non-linear terms Bni(E) with Bni(8) in the flow equation of E (see (10.1.22)). In Section 10.4.1, from the solution of Xsub and choosing boundary condition of A 4 and A 6 at the root scale, we obtained the solution of Ag and X\. Finding a solution for Xs4 required a fixed point argument similar to the fixed point argument used in finding the solution Xsub. Collecting all the results on the components of the order N flows, for 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{ = A1(l + Esl), \ E S \ < ^ , (12.1.8) A* = A 2 (1 + £ 2 S ) , | £ 2 ' | < ^ £ r n -i- in r s N i A* = T"(l + E'3), \Es3\<Cse' k3Ts ' £ 4 \ ~ l M , ™N , ™, ^ C5 e2 < C , e 4 ( l + l n P ) k5T" \K\ < l ^ l + C . e 2 I n Q Cge* (1 + l n P ) | A t I " hr ' where Ts = K3 + e25M(s + 1), e2 = ( A x / A 2 ) 2 , 8M = M 2 - 1, Erz = 0; Cg is a constant depending on TY; k5, k7, and CM are respectively defined in (10.2.91) and (10.1.27). 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 A f + 1 = e A f + 1 , A f + 1 =0, 5 < i < 7. (12.1.9) where e = A i / A 2 . 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 , Af = A 2 (12.2.1) A?3 > 1, \\>l, A- = 0, 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 ( m s ) > a n d | | is bounded by O 0 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 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 3 J + 1 = b\nA + 0(\n(e25MU)), Af = Q(A?) = Q((e25MU)9m/2) (12.2.2) where e2 = (Ax/A 2 ) 2 , 5M = M2 - 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 V3 local part of the V P diagram should be bin A + O ( l ) a n d not 6 In A + A 3 + c3e25M(U + 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 Xug+l = (0,0,0) where A5„ EE (A 5 ,A 6 ,A 7 ) , then Xrgv = (0,0,0). But because we were not able to show that Xrgv = (0,0,0) in an order-by-order fashion as done 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 Xgv 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 A f + 1 = A A ? + 1 = A 2 5 < i < 7. 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 LRC 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. 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D (3), 2:1438-1472, 1970. [Wil71] Kenneth G. Wilson. Renormalization group and strong interactions. Phys. Rev. D (3), 3:1818-1846, 1971. [Wil72] K. G. Wilson. Phys. Rev. D (6), pages 419-426, 1972. A p p e n d i x A Vest ig ia l Gauge Invariance Because of the gauge invariance of the Lagrangian of the model, in a perturbative ex-pansion of an external source generating functional for a gauge model of a QFT, 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 Xgv of type A4, A2, or (d • A)2. However, because of the presence of a cutoff for regularizing the divergence of Feynman diagrams, the WI 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 Xgv is marginal, i.e. its degree of divergence is zero, then "vestigial gauge invariance" still renders Xgv 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 4 - local part of the 4-photon legs diagram and the (d • A) 2-local part of the vacuum polarization diagram are bounded uniformly in U . 265 Appendix A. Vestigial Gauge Invariance 266 Lemma A . l Suppose that LiGu is a marginal local term of G which is formally zero because of WI but which is nonzero because of an UV cutoff ((M~up) € <S(IRd). Then LiGu is bounded, uniformly in U. Proof: The coefficient of LiGu has the form (3(LiGu) = JR{p)C,{M-up) 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(LiGu) = J Q(p)M-uC(M-up) 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~up, f3(UGu) = J M^uQ(Muq)C'(q) dq. (A.3) Now lim M^uQ(Muq) = S(q), (AA) U—>oo where S is a homogeneous rational function of deg 1 — d. Hence lim P(LiGu) = f S(q)C(q)dq, (A.5) (7->oo 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(LiGu) is uniformly bounded. Q.E.D. Appendix A. Vestigial Gauge Invariance Example A . l : 267 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 4 1 X M ™ M 4 = / d ± p t r j { - Yj P-U{PY (A.7) j = \ V "f" m where p-u(p) is an UV-cutoff. There are many choices for the UV-cutoff, for examples, P~U(p) = e ~ M 2 U p 2 for an exponential decay slicing and p-u(p) = 1 — h(M~2U~2p2) for a neighborly slicing where h(p) is defined in (6.1.1). Wlog, here we use the exponential decay cutoff p^u(p) = e-M~2Ur2. Let C{M~up) = (p^u{p))\ and < • >= J dp tr(-). A neat way to compute X is to use the identity -PadPaR(p) = R{p)plR(p). (A.8) and to integrate by parts. Thus (A.7) becomes 1 1 1 ~ r/+m°p^ pf+m1 p!+m1 ; > — _J£"M2£»lM3 / i 4 _ J£-/J2 /J3/J1/J4 _|_ y 7 * l M 2 W M ^ where Appendix A. Vestigial Gauge Invariance 268 = < fi v 3 y^d c > P + m M2 p+m M1 (A.10) where ^ / + m ^ ^ + m ^ ^ , 2 ^ > jrY-i. ' ^ 3 » 2 ?Vl s (A.11) By (A.10) and (A.l l) / Y C I M W W A A A A — ^ I M-M ft-Hi ^ 3 ^ 4 — l y M i / i 2 / i 3 / i 4 f A A A A y ^ 1/il / 1Al 2 / 1A»3 / 1 ^4 1 v/Jiwww f A A A A J SI in.-ft-HI ^ V 3 ^ V t -(A.12) Now 7 V4 trpi^ _ 4 P / i 4 where D = p2 + m2. Thus integrating by parts on Z / i l / i 2 / i 3 ' t 4 , (A.13) = ±jdp^dp^dPndp^{M-up) = A j dp = 4 y.dp - 5 M3M4 2PWP/*4 £ > 2 9Pli2dPnC{M~up) _ 2(5M3/Mp / i2 2(5n2lj,3Pfj,A + 5^2/i4p^3) L>2 D3 D2 dPfiiC(M-up). Let (A.14) M - c / c ' ( M - f / P ) = a P f l i c(M-^), Appendix A. Vestigial Gauge Invariance 269 then 'i^jmnPiii + °"/ i2/ i3P/i4 + ^tJLiinPiiz) _j_ 47J/x2P/i3P/i4 r>2 M-UC(M-Up). D3 It is easy to see that Q ( P ) = has deg(Q) = —3 and ( ^ / X 3 M 4 ^ 2 + ^ 2 / 4 3 ^ 4 + ^ A M P / ^ ) , ^P^jP^Pjlj D2 D3 (A.15) (A.16) 5(g) = lim M3UQ(Muq) = -f7—>oo 3 ( ° ~ / t 3 / M ° / J 2 + ^ 2 / ^ 3 ^ 4 + ^/J2^4?w) , ^?M2?M3^M4 For p-u(p) = e M 2 U p 2 , £ is Euclidean invariant and (A.17) (A.18) and X " 1 " 2 " 3 " 4 becomes | dp 2 4 M - 4 C V 4 M - 2 V ^^3tljPlJ-lP^2 + °"^2M3P/tiP/i4 + ° " /U2/ i4^ / t lP / i3 4p f t lP / t 2P j U 3P / L i 4 8 3 2 - 2 M 2 r y (A.19) £>2 £ ) 3 From (A.19), by the identities (E.l.l) in Section E . l and the change of variable g = 2M~up, we have X ™ M 4 I ^ 4 ^ ^ = p(L7Gu)A'i, (A.20) where 3g2 4g4 l(q2 + (2M-um)2)2 (q2 + (2M-Um)2)3 (A.21) (g2 + (2M" c / m) 2 ) 3 (9g 1 2g 2 2-g 1) + 3g 2 (2M- c / m) 2 ] . Appendix A. Vestigial Gauge Invariance 270 Changing to polar coordinates: (<?i,<72) ->• (r cos 9, r sin 9) (93 ,^4) (£ cos 0, £ sin <£) and letting mrj = 2M~um, 8 ^ /-oo ^oo e - ( r 2 + t 2 ) r 4 " ( i ' G » = 3 2 ^ + + ( A ' 2 2 ' where A = f2n d9[9cos29sm29-cosi9}=9-- — = — Jo 1 J 4 4 2 / d9 cos2 6=^-. (A.23) «/ 0 ^ B = 3 # 2 0 = Changing to polar coordinates: (t,r) —» (s cos cr, s sin a), and following with another change of variable: s2 —> s, yields roo e~s2 / ' 7 r/2 /3(L7GU) = 8n2[l + ml] ds s ^ 5 - ^ - / da cos a sin 5 a Jo {sz + mfjY Jo 2 /T 2 r°° e~s s3 = T ^ ^ l dsJ7T^Y' ( A 2 4 ) which is positive for all U and has limit 2TT r°° 2TT lim (3(L7G ) = — / ds e's = —. (A.25) U->oo 6 JO O Hence, f3(L7Gu) is positive and bounded uniformly in U. Example A.2: Consider the (d • A)2 term in QEDA. The vacuum polarization diagram in Figure A . 17 has value = j^y\I dpA^U[p) * £ ( P M F ( - P ) . ( A - 2 6 A ) where the kernel KM = j d q p~u^+^ p~u^)tr [-f R(p+1) ^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)2 part from the second order Taylor expansion R(p + q) = R(q) - R(q)p!R{q) + R(q)j/R(q)j/R(q) - (A.27) This yields j ~ j dp [| dqC:{M-uq)tr >f Rjf Rjf R-f R^ A,{p)Au{p) ~ J^y J dP K^AMMP) (A.28) where R = R(q) and C = (p-u)2. We project out the local-(<9 • A)2 part by the following. Since C is Euclidean invariant, then the kernel K in (A.28) can be written as K = Ap2T + Bp2L (A.27) where L and T are defined in (1.2.8), and pKp A = p* B = K^,2(P) A 3 p 2 (A.28) (A.29) The coefficient of the local-(<9 • A)2 part is just A. Appendix A. Vestigial Gauge Invariance 272 We perform an integration by parts using the identity (A.8) to obtain A = ±r[ dqpv(dq£) tr yR? - 2-1dq C tr (r/i?)4. Hence (A.30) (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 AT -u K dqu' (A.32) D(q)2 D(q)3 It is clear from (A.32) that we need Euclidean invariance, i.e., £ = ((M~2Uq2), in order that a be a constant, in which case A = 8 3p4 8 3 8 h h - 3 p2p-q 4(p-q)4 + M-2UC(M-2Uq2) D(q)2 D{qf M-2UC(M-2Uq2) ~H + Ml D(q)2 D{qf + 4q\ _(q2 + M-2Um2)2 (q2 + M-2Um2)3 For p^u(M'up) = e - P 2 / M 2 U , we have CV)- (A.33) -2(7„2\ _ „-2M-2UP2 Q'(M~2Uq2) = -2e~2M~2Up2 C(M-2Uq2) = e By an almost identical calculation as in (A.21)-(A.24) with mu = V2M~um, 2ir ds e ss3 (s + ml) 2 ^3 ' (A.34) Hence A is positive and bounded uniformly in U . A p p e n d i x B Neighbour ly sl icing 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 ( 1 for x > 1 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 h(x) = 0 for x < 1 - e, 0 < e <• 1 - M~2 (B.l) ps(x) = h{x/M2s) - h(x/M2s+2), (B.2a) and p°{x) = 1 - h(x/M2). (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 1/2 Figure B.18: Example of h(x). s s+1 P(x) P (x) j A J (1-e) M (1-e) M Figure B.19: Only adjacent slices have overlapping supports. Su{pl) n Su{p>) ± 0 i f f | i - j | < l , (B.3b) Su(ps) n Su(ps+l) = ((l-e)M2s+2, M2s+2) = Is,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 C l Photon field In the following, using Lemma 5.2, we derive an explicit form of the running covariance Hs for the L R C scheme on IQED described in Chapter 5. From the derived form, we impose conditions on the running couplings so that Vs (A) in (5.2.8a) and KI in (5.2.2b) are well defined and uniformly bounded in U and A. We make these conditions explicit by imposing that the slice function ps at scale s, 0 < s < U, has the following support, (1 - e)M2s < p2 < M2s+2, (C . l . l ) 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 Xs, = (XI-1)f3(p), Xs5 = Xs5h(p). (C.1.2a) where v2 - M2s AW = ftiP) = ^ ( d . 2 b ) By the fact that L2 = L, T2 = T, LT = 0, we have (A* - 1) p2L + M2sXs5) 275 RKS = L + [l-ri) p2 + l (l-r))+p-Appendix C. LRC running covariance 276 (**•+')' = (^ + ^ )'^ +(((11"_y^)'r, (C.1.3b) where K s and R are defined in (5.4.6). Let * = + ^ { f ^ - ^ Using the fact that [a L + b T]'1 = a'1 L + b~l T, and substituting (C.1.3a-b) into (5.2.3a), the running photon covariance Hs = H[ + H^, where H = K W j ^ (C.l.Sa) % = W > 7 ^ ^ . (C.1.5b) and = 1 - {ps + p s + 1 ) + ps{js + 1) + p-+i(7«+i + 1)2 (C1.6a) w> = 1 + ^ . / ; + 1 ( ) t f l ) 2 . (c.i.6b) Also, the denominator 1 + p s + 1 R Ks+1 in (5.2.4) is 1 + ps+1 (jsL + A*T) . (C.1.6c) 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 ps(p2) defined in ( C . l . l ) , M2sp2 ~ 1 and ^^w-i|)(i-rr^) * 1^1 < IAS -11 (1 - jTjr^s)^-!-^) - + M - 2 « - K l ~ 1 + ' M - 2 . ' ( C ' 1 ' 7 b ) ( 1-" ) 1 A |' < Kl<„ ( C 1 . T C ) (1 - ri)M-2s + M2 ~ 1 5 1 - (1 - r))M-2s + 1' Appendix C. LRC running covariance 277 T h u s the cond i t i ons XI > 0, Af > 0 , (C.1 .8) are sufficient to guarantee 1 + psjs > 0. In the case of choos ing the L a n d a u gauge, we d o n ' t need to require A5 to be pos i t ive . T h e c o n d i t i o n XI > \XS5\, (C.1 .9) is sufficient. L e t 7 s = Af 4- Xs5. F r o m the above requi red cond i t ions , i t is easy to see t ha t n w < ' < i + T--('+y> L K 1 - 1 + 7 s - 7 s 7 5 (1 + Y) < i t i ^ ^ f - ' l ( d . l O a ) 1 1 + f l _ n)M~2s'2 Appendix C. LRC running covariance C .2 Spinor fields For each component Hj = VjRj of the spinor covariance, let us write Vj=pa{DS)-1, where {OTS = U, Df = l + puAf; for s<U, D) = 1 + psA] + ps+lB°+l; A] = R, Kj, Bs+l = (1 + Rj K°+l)2 - 1. and Kj and Rj are defined in (5.4.16). Let us introduce more notations here. 1. For k = a + bp1, let us denote k* = a — bp", RE(k) = ^ = a, A- — k* AH(k) = = H k2 = kk* = a2 + b2p2. 2. A - EE (/<-')*/<-; where Af = Af — 1. Appendix C. LRC running covariance 279 Let us rewrite Aj and B*+1 in terms of the above notations. For convenience, we suppress the indices j and s except at places where they are necessary for avoiding confusion. K = A 4 M 2 + A 2 p 2 + (A2 - A 4 ) M j / (C.2.6) A = RR* [(iT 1)* K] = R2 K, (C.2.7) B = {RK + l ) 2 - 1 = {RK)2 + 2RK = R4 K2 + 2R2 K. (C.2.8) We wish to impose conditions on A- so that \\D/ps\\ and ||(1 + psA)/ps\\ stay away from zero and | | p D _ 1 | | is bounded uniformly in U and A (here || • || is the Hilbert Schmidt norm). We start by first considering lower bounds for RE{D) and 1 + psRE{A). From (C.2.7) and (C.2.8), 1 + psRE(A) = 1 + psR2 RE{KS) RE{D) = l+(ps RE{A) + ps+1 RE{B)) = 1 + [psR2 RE{KS) + ps+1 [R4 RE{[KS+1}2) + 2R2 RE{KS+1)) = {1-ps+l - ps+l) + ps{l + R2 RE{KS)) +ps+1 [l + R4 RE{{Ks+l}2) + 2R2 RE{Ks+l)] . (C.2.9) In trying to make the quantities in consideration non-negative, we first impose that A^ > 1, (C.2.10a) A > Mu+1. (C.2.10b) 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), = ^ R4RE{[K}2) = R4 ( [A 4 M 2 + \2p2}2 - (A2 - A 4 ) 2 M V ) M2{M2-p2)~2 M2p2 -2 " (p2 + M 2 ) 2 A 4 (p2 + M 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 ^ + 2R2RE(K) > i ^ s f (C.2.14) \ + &RF(ft\ > ( A 2 - | W ) (Af + 1 )M 2 l + i2J*£?(lO > 2 ( p 2 + M 2 ) + ( p 2 + M 2 ) • (C.2.15) Since A —>• oo before [/ —> oo, we may impose that A 2 > 1 - e , (C.2.16) A > max(Mu+\4Mu+1 sup |A||), (C.2.17) 0<s<U where £ is a constant satisfying 0 < e < 1/2. From (C.2.15), and the above imposed conditions, ps{l + RE(A)) > pl (AS + 1)M 2 M? + M2s > Psj- (C2.18) From (C.2.11) and the imposed conditions, (C.2.13) and(C.2.14) are non-negative. Thus by dropping the p s + 1 ( l + RE(B)) term in RE{Df) and using (C.2.18), RE(DSA > ps(l + RE{A)) > ps ^ (C.2.19) l + psRE{A) > ps{l + RE(A))> ps ^ (C.2.20) (C.2.19) implies that Hal loo < | -< 2. (C.2.21) A 4 (C.2.20) together with using external sources with compact support implies that Kse is well defined and \\Kse\\ is bounded uniformly in U and A. Appendix C. LRC running covariance 281 As for the case j = 0, we use the fact that psRl ~ M 2 s to control bounds. Let r(p) = Jl% = llyji? + m\ rs = psr, u8 = (X\ - Xs2) m. By rewriting ps RKS = p8 Xs2 + us psR, (C.2.22) we have \\p8 (l + psRK8) s\-l 1 + PSXS2 1 + =- psR l + PsXs2 , (C.2.23) Here in addition to the imposed conditions in (C.2.10a) and (C.2.16), we require \us\ rs < -. - 4 From the imposed conditions (C.2.16) and (C.2.24), since e/(l — e) < 1, | | / / (l + p s i2 X s ) - 1 || < l - e l - ( e / 4 ( l - e ) ) 3(1 - e)' Similarly, by (C.2.22), VS(X) = p'iB'y^l + iB^E8}-1, Bs = [1 - ps - ps+1) + ps {Xs2 + l) + ps+l (A* + 1 + l ) 2 , Es = us (p s R) + 2us+l (X2+1 + 1) (Ps+l R) + (us+l? (ps+l R2). If we require that (C.2.24) (C.2.25) (C.2.26) \us\rs + 2\us+lXs2+1\ rs+1 + (us+1)2 ( r s + 1 ) 2 < - , 2 (C.2.27) then | |P S (A) | | < tf\B'\-x ( 1 - d l ^ H / I B ' l ) ) - 1 14° < ps IB8]'1 \ l -1 1 _2_ - X82 l - ( e / 2 ( l - e ) ) " A l < ^ - < 4 . 1 — £ r8 + 2\u8+1Xs2+1\ rs+1 + (us+1)2 ( r s + 1 ) 2 - l \BS (C.2.28) (C.2.29) Appendix C. LRC running covariance 282 We would like to estimate Vs (\) by a simpler form in the case that K = Q(sa), K = 0 ( 0 , a > o. (C.2.30) Assuming 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 rs = Q(M~S), it is easy to see from (C.2.26) and (C.2.30), that \\Es(p)\ \ = Q(M-Ss2a) and so where Es = Pi 1 psEs ~ l + Es(Bs)~l ps(Es)2 g f-E8^1 (B°Y u v B° and \\Es{p)\\ = 0(M-2ss4a). Since the part Es is irrelevant, we can approximate Vs (X) by (f_ psEs B~s ~ (Bs)2' (C.2.31a) (C.2.31b) (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 " + 1 = 0( lnA), (D . l . l) and for Af ^ A ^ 1 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, Vol(q) be the volume of the domain of q, and i be an index for the lines of L ^ u . It is easy to see that \\L'nf\\ao< 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 < flWVpjWoo [ dqf[ + i = l 7 i = l ~ n < J dqH \Rj(q + Pi)\ (D.1.4) 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 - 2 f dq |1 + 4 + M - V i l - 1 (D.1.5) J i=l 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~m) where m is the number of fictitious spinor lines in the integral. Recall that the support of the spinor slicing p f + 1 ( p 2 ) is [(1 — e)M2U+2, oo). This implies that integrand with 1 — p f + 1 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 + 1 by unity and attach an error of Q(A~m). We substantiate the above claim by the following lemma. For convenience, we drop the subscript F in p f + 1 . Let us write down a more explicit expression of the scale U+1 spinor covariance H^+l. From (5.4.18), (5.4.19a-e) we have Hf+1 = pu+1(l.+ pu+1RjK%1)-1Rj = pu+lRu3+\ (D.1.6) where K^f1 and Rj are defined in (5.4.16), and R?+1 = [(1 + pWll+^Mj + (1 + pu+l\u2+1)p\-\ (D.1.7) Appendix D. Bounds on Spinor Loops 285 where \f+l = A f + 1 - 1, Thus Hf+1 is like Hf+1 with the M,- and ^ being modified respectively by 1 + pu+lX^+1 and 1 + pu+1\%+1. Let [Fj(p)](P) = JdQ[K3(p)](P,Q) (D.1.8) be a function of a set of external momenta P = {pi, • • • ,pm} where each pi is restricted to a compact domain and Q = {<?i, • • •, qn}- The kernel Kj is a product of lines which depends on a slicing function p. By suppressing possible 7 matrices placed in between the lines, we write n rik [Ki(p)](p,Q) = n n & ( p ) ] ( ? * + p i ) (D.1.9) k=l 1=1 where ej(p)=pRj(p), (D.1.10) A,(p) is defined in (D.1.7) with pu+1 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 j*0 where d = Y^k nk-( D . l . l l ) Proof: Let us introduce an interpolation between Fj(p) and ^ (1 ) . For 0 < t < 1, Now ^0 dt dF[ dt = ^ dt j dQ (P) 5p (D.l.12) (PU+1 ~ 1) ( W - 1 3 ) Appendix D. Bounds on Spinor Loops 286 To consider the derivation SKj(p)/Sp, we first consider the derivation on a line £(p). Let S£3(p) 6p Sp ' Using the facts that and 5p ~ ,c5R-\P) ~ , -RAP) § p Rj(p) m ( p ) ) - 1 = Ku+i 5p e'j (D.l.14) (D.l.15) (D.l.16) where Kftl is defined in (5.4.16), we have lj{p) = R ^ - p R ^ K l f 1 R^p). Since Kj(p) is a product of lines, by the product rule, we have (D.l.17) k,l (k,l)^(kj) (D.l.18) where (-)(fe'') denotes a term with momentum qk + pt and Vol(Q) is the volume of the domain of Q restricted by the respective (1 — pu+1Yk'l^s. From the definition of the lines £j and £j, it is easy to check that both ||^-||oo and | |^||oo have the following estimates O(i) if i = o [ 0(A" 1 ) , if j 7^ 0 (D.l.19) Thus from (D.l. 18), it follows that Q.E.D. O(l) if J" = 0 Q(A-d) if j 7^ 0 (D.1.20) Appendix D. Bounds on Spinor Loops 287 D.2 Proof of Proposition 6.1 of Section 6.1 We compute the coefficients of the 1/j-local parts of VPu+l. The formal expression of the VPU+1 is (\U+l\2 1 3 < V P > ^ + 1 = ^ f l - / dPAf[p) Y,M9n(j)K?£(p)AZu(-p), where sign(0) — sign(l) = 1, sign(2) = sign{2>) = —1, and K?£(P) = J dotr [f H^\p + q) Y H?+\q) } . (D.2.1) The kernel of the projection of the a type-j VPu+l diagram onto F2 and (<9 • A)2 has the form {LK^HP) = J dq (p(q)U+1)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. Note that since pu+l has no U V cutoff, (LKj)2^l, is not well defined without summing the integrands of {LKj)2illv 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 IR4 (N —>• oo) after making the 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)2^(p) = p^A^ + B M , (D.2.3) Appendix D. Bounds on Spinor Loops 288 where and LM„ are defined in (1.2.6). As in Appendix A, (summing over p and v) p„ (LK)2^u,j Pv A3 = (LK)2tlilid(p) _ Aj_ 3p2 3 (D.2.4) (D.2.5) We would like to show that Aj = Q(A m) where m is the number of fictitious lines in the loop, and that £ Bi = 0(lnA). . (D.2.6) 3 We now consider the coefficient Aj. We apply Lemma D . l and set pu+1 = 1 in Aj. Subsequently, by the Ward Identities, the corresponding integrand of Aj becomes an exact derivative. Thus, accounting for the Q)(A~m) error where m is the number of fictitious line on the loop, we have ' O(l) if J = 0 0(A"2) i f " 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, A3 = (D.2.7) KumM = j dqtr -1» \{p<rPadqadq„R) tr + 0(A" m) r(pA,R) 1" (PAM + 0(A-m) (D.2.8) where here since R = ( A 4 M + A 2 (D.2.9) (D.2.10) Appendix D. Bounds on Spinor Loops 289 To evaluate the trace in (D.2.8), we first consider the term padqaR. Using the fact that we have Rp!=p!R* + 2X2{p-q) R\ -padqaR = X2RpjR = X2R2 [VJ+2X2(p-q){RrlyR?] = X2R2 [pJ+2X2(p-q)(X4M - X2i)R2 = A 2 JR 4 [(2A4A2(p • G/) M) + (R-y- 2X2(p • q = X2R4{a+k), where (D.2.11) (D.2.12) a = 2XAX2{p-q)M k = R-2p-2X22(p-q)q. In terms of a and k, the trace term in (D.2.8) is X2R8 tr [7" (a + y ) 7 / i (a+ff)} = X2R8 (a2tr[YY]+tr[Yf/Yffi) = X2R8 (-16a 2 + tr [7" (7"A;2 - 2kilk) ]) = X2R8 (-16a2 + k2tr[YY + 2}) = -8X22R8 (2a2 + k2) = -8X2R8 [8X2X2{p-q)2M2 + R-y-AX2R-2{p-q)2 + AXi{p-q)2q = -8X22R8 [R-Y + 4(p-q)2X22 (2X2M2 + X2q2-R-2)} (D.2.13) (D.2.14) -8A 2p 2 32A2A4(p • q)2M2 ( A 2 M 2 + A 2c- 2) 2 ( A 2 M 2 + A 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)2f(q2) = ^ J' dq q2f(q2), (D.2.16) we have [LK)2^{p) 4 A 2 f dq 3p2 3 J [\lM2 + \2q2)2 AX2XA2M2 r dqq2 + ^ 1 f + 0 ( A -m ) (D2 17) 3 J (\2M2 + \ 2 q 2 y + U [ 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)/ - 2, { L K ] 2 - / P ) = ^ r + I* + O ( A - ) , (D.2.18) 3p2 3A| 3A2, where 00 dx x ( M 2 + x)2 J b = M2 [°° r^X\* (D.2.19) Jo (M2 + x)4 v ' and is the angular contribution of the integral (D.2.17) in spherical coordinates. Note that Ia is a divergent integral without summing over j. Let us compute the integrals Ia and Ib. We now reinstate the index j back in our discussion. roo ~\ roo ~\ Ji = / d x 772 Mi / d x 7T77 3 Jo M2 + X 3 Jo (M2 + X (M2 + x)2 [M2 + N\ = lim ln 3^0 - 1. D.2.20 Upon summing 7" over j , we have Appendix D. Bounds on Spinor Loops 291 . / [m -= In 2 1_ yY2)2 (m 2 + 2A 2 )m 2 J ' (m 2 + A 2 ) 2 l n ( A ) + 1 ^ ( m 2 + 2 A 2 ) m 2 A 2 ; 2 ln(A) - ln (2m2) + o(l). (D.2.21) As for Ij, integrating by part, we get L M2 r, D X X \ . = M 2 ( - X 2 X o (M2 + x)A V 3 ( M 2 + x ) 3 3 ( M 2 + x) 2 3(M2+x) = \ - (D.2.22) Upon summing 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 ^ 4 C n ( A f + 1 ) 2 1 , A S ln(2m2) , s , n n n A s E g j = " , U h ( A ) " V + o(1)' (d-2-24) In writing down /?3, we remind the reader that there is extra minus sign needed because 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)Kn(q, P) be the kernel of a Ln(P) where ({q,P) is a product of the pu+l's from the lines, and (by suppressing the Ai's, combinatoric constants, and the gamma matrices multiplied to the lines) Kn{q, P) is a product of lines with loop momentum q. We write Kn(q,P) = f[£(q + Pi) (D.2.25) i=i 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)Kn(q, P). Rather, as in the VPU+1 case, the kernel of the local part is the Taylor terms taking from just Kn(q,P) alone. More explicitly, LLn(P) = Jdq C(q, 0) L(Kn(q, P)) , (D.2.26) where L(Kn(q,P)) = ±dkKj^tP) (D.2.27) k=0 0 , 1 t=0 and 5 is the degree of divergence of L%+1 (here 8{L±) = 0 and 8(L2) = 2). Since each t-derivative dijq + tPi) ~ ^d{R-\q + tpj) -f — = -R{q + tpi) — R{q + tpi) = -(l + p\2)e(q + tPi) £ e(q + tPi), (D.2.28) L{Kn(q,P)) has the form E pk I I 4/, (D.2.29) fc=0 i=l where kl indexes the lines of LLn and P^ are products of linear combinations of the PiS. By add and subtract the same term / dq ((q, P) L(Kn(q, P)) to RLn(P) = (1 — L)Ln(P), we write RLn(P) = IX(P) + 72(P) (D.2.30) where h(P) = jdqaq,P)R(Kn(q,P)), (D.2.31) J 2 (P) = Jdq(aq,P)-<:(q,0))L(Kn(q,P)). (D.2.32) The term h(P) is easy to bound. First, we write aq,P)-aq,0) = jo1dtd^^. (D.2.33) Appendix D. Bounds on Spinor Loops 293 Since the derivative dpu+1(x)/dx, where here pu+1(x) is viewed as a function of a real variable x, is bounded, we have '\dC(q,tP)\ sup t dt = C( (D.2.34) where is a positive finite constant which has a dependence on U. Let Vpu+i be the support of the derivative dpu+l(x)/dx. Since Vpu+\ is contained in the open interval ((1 — e)Mw, M2U), it is easy to see that the volume Vol(q) of the domain of q restricted by the support of dC(q,tP)/dt is finite. By bounding the lines £ki with sup-norms and using Lemma 6.1, we have \h,3{P)\ < fdq\(((q,P)-C(q,0))\ \LKnd(q,P)\ = CcVol(q) J2 W I I H a l l o o A;=0 (=1 = < O ( l ) i fj = 0 0(A" n ) if J 7^ 0 As for the Ii(P) term, we first have By applying Lemma D . l , we get ' 0(1) if J = 0 0 ( A ~ ( n + m ) ) if3*0 where (D.2.35) (D.2.36) Iij(P) = fdq RknJ(q,P) + (D.2.37) (D.2.38) Kn(q,P) = Kn{q,P)\pU+i=1 and that each of the £'s in Kn(q,tp) is now the R in (D.2.9). After taking the t-derivatives, RKn(q,P) can be written as a sum of terms as the following. RKn(q,P) = J2(RK){n>m)(q,P) (D.2.39) Appendix D. Bounds on Spinor Loops 294 where each (RK)^m)(q,P) has the form p\ n+l+5 P(n,m) / dt(l-t)S KQ + tPi) J 0 • i (D.2.40) i=i and P(n,m) 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 , m ) | dt dq TT |A 4 + A 2 ^ + t M 7 - V i ) r 1 • 7 o ^ i=l = 0 ( M r x ) . (D.2.41) Thus O( i) ifi = o (D.2.42) Q.E.D. 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 Vni_vpu+, = ( L ^ + 1 H u + 1 r . (D.3.1) Subsequently, it suffices to show that each term of the power has its sup-norm equals to l + 0 ( ( m A ) - 1 ) . Let us write down the detailed expressions of the various terms involved in the bounds. L 2 + l = Kpp2L +RL"+1 (D.3.2) 1 l + p 2 Hs = - P S T ~ ^ L (D-3.3) 0u+i Vu+1 = Sr-: (D.3.4) l + P ^ W 1 - ! ) ^ u P U + l v = i + - i ) ^ r + P u + i m + 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 RL2+1 in (6.1.13) and the value of bvp in (6.1.12), we have I A 3 % ^ + l V u + l n p +O(0nA) - 1 ) (D.3.6) where From (D.3.4), we have nr = J ^ . (D.3.7) a " + i " k + i - = ! + S • - i i ) ^ , = 1 + ° ( < i n A r , ) - ( a 3 - 8 ) Appendix D. Bounds on Spinor Loops 296 Thus since we have Jvp M n A + C 3 + 0(( lnA)- 1 ) A 3 7 4 - 1 61nA + C 3 + A r 3 l + OianA)- 1), L " + l H u + l = l + 0((lnA)" 1). For Part 2 (ii), we write 0^ni-vPu+l)Hu = (V(ni-i)-vPu+l) H u + 1 L , 2 + l H u . From (D.3.10) and (D.3.2-44), we have (D.3.9) (D.3.10) (D.3.11) l l ( V n i _ ^ + 1 ) ^ | | o o < \\L™H™\\^ \\HU+1L™H < (X^)2 {\v+1)2VuVu p2 (I+P2¥ + 0((lnA)" 1). (D.3.12) Inserting (D.3.5) and (D.3.4) into (X^+1)2VUVU p2/(l+p2)2, we have (X^+1)2VUVU p2 {1+p2)2 (1 + (A^1 - l)pu^np) p2 1 + pU{\}> - l)np + pP+HK^1 ~ IK + I]2 - 1}' (D.3.13) Since the above involves the product pupu+l, we have pu+l = pf+1 and pu = 1 — pf+1 (where pf+1 is pu+1 restricted to the intersection of the supports of pu+1 and pu), and (X^+1)2VUVU p2 (1+P2)2 1 - P I + 1 PUL+\^+lfn2p (1 + (Af+1 - I K + 1n p)P 2 1 + (1 - PUL+1)M ~ I K + P L + 1 { [ ( A 3 7 + 1 - I K + I]2 - 1} (D.3.14) (D.3.15) < M-2U-2 (1-e)-1+ 0{(\nA)~l). From (D.3.12) and (D.3.15), we have M-2U~2 b, \\(Vni-vpu+i)H |oo _ vp (1 - e) (Af"1"1)2 + 0(( lnA)- 1 ) = 0(( lnA)- 1 ) . (D.3.16) Appendix D. Bounds on Spinor Loops 297 For Part 3, by the fact that (n2 - VPu+l)Hu = (Lu2+1Hu+l)n*-2 Lu2+lHu+1Lu2+lHu (D.3.17) and (D.3.10), it suffices to prove it for the case n2 = 2. For part3 (i), again we expand L 2 + 1 and apply the estimate (6.1.13) of R L 2 + l . W L ^ H ^ L ^ H ^ < \\bvp p2L Hu+1 bvp p2L + 2||6,p p2L Hu+lHu RLU2+1\ + \\HU+1 Hu (RLu2+l)2\U < + 2-V ( A 3 / + 1 ) 2 ( A f + ^ M 2 ^ 1 + 0( ( lnA)- 1 ) -< ( l + 0(( lnA)' 1 )) | | ( A ^ + 1 ) V ^ + 1 ^ l | o o + Q ( ( lnA)- 1 )p.3.18) We further simplifying the in (D.3.18). \\{\\J+1)2p*Hu+lHu\\00 < {\u3+lyvu+lvu P 4 (1 + P 2 ) 2 From (D.3.14), { \ u z + l Y V u ^ y v p4 (1+p 2 ) 2 1 - Pi+1 PUL+1M+I)2n2p 1 + W+1 - l)pt+lnp 1 + (1 - puL){Xuz - l)n p + P L {[(A3 - l)np + I]2 - 1} (D.3.19) < l + CXQnA)-1). Thus I I L ^ ^ + ^ H o o < l + O(0nA)- 1 ). (D.3.20) For part 3 (ii), from the proof of part 3 (i), it is easy to see that 2 ' \ { \ I i + 1 ) 2 v u + l v u p 4 (1+p 2 ) 2 + 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 (\%+1)2Vu+1Vu pA\ (1+p2)2 < (D.3.22) To estimate the integral (D.3.22), we make the change of variable r = p2. The domain of integration of the variable r is [r,,,^] where r0 = (1 — e)M2U+2, r\ = M2U+2. Note that pu+1(r0) = 0 and pu+l(n) = 1. Now (D.3.22) < ci r -JT0 1 < 2 c i r i f Jr, dr r (1 - pu+l) + (A 3™ - l ) ^ + 1 ( i ^ ) (D.3.23) r a 2 + (X^-l)puL^-where c\ is a positive constant. To show the above r-integral is zero in the limit A —> oo, we choose a point r 0 < e(A) < r\ satisfying p c / + 1(e(A)) = ( lnA)- 1 / 2 , (D.3.24) and spilt the integral into two parts according to e(A). Dropping constant factors, (D.3.23) becomes dr fe dr r 1 dr ra 2 + - l)pU+ By the continuity of p u + l , + •„ 2 + (Af"1"1 - l)pu+l Je 2 + ( A 3 7 + 1 -(ri - c) < (e-r0) + 1 + (lnA) 1 / 2 ' lim e — r 0 = 0. A->oo Hence Q.E.D. lim A—too ( l+p 2 ) ' = 0. (D.3.25) (D.3.26) (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 p2 and A is a four-vector then J dpgp^ = ^ „ j dpgp2, (E . l . l ) J dpg{A.pY = (A2)2 J dpgp\. 2. trYYF^ = - 4 F W , (E.1.2) L l I I I I 1 £il/i2A»3M4 ^ I1 / i l / i lA»3M3 J /il/^2MlM2 ~ 2 Ml£*3M3/i l J ) where repeated indices are summed over. 3. / = V , (E.1.3) 299 Appendix E. (3 functions of the first order diagrams 300 4. Let R{p) = m)'1, then -PadPaR(p) = R(P)PIR(P)- (E.1.4) We would like to establish a couple of handy lemmas for computing marginal dia-grams. For a first order marginal diagram, the coefficient of its zero order localization is usually of the form cG J Ts(q)Ks(q) (E.1.5) where Ts(q) is a function of the the running slicing functions {/°'}t>s and {X\}t>s, and Ks(q) is a rational function of degree —4 in q. After changing to spherical coordinates and letting r = q2, the relevant part of the integral breaks up into a sum of terms each of the form IK = cGkuIps Ips = [ -T(Xs,Xs+l,ps,ps+l). (E.1.6) Jsps r where Sp* is (the support of the neighbourly slicing ps defined in Appendix B, and &n is the value of an integral involving only the angular spherical coordinates. We would like to extract dominant terms up to o(A|) from (E.1.6) in the case that AAf = Xi+1 - Af is o(A|). Let us write T(AS, A s + 1 , ps, Ps+1) = u(Xs, Ps,ps+1) + ps(Xs, Xs+\ps, ps+1) (E.1.7) where u(Xs,ps,ps+1) = T (A s ,A s ,p* ,p s + 1 ) (E.1.8) and ps is the corresponding error. ps can be expressed as the following interpolation integral. ps(Xs, Xs+\ps,ps+1) = ^ ( A r 1 - AJ) PKA S , Xs+\ps,Ps+1) (E.1.9) Appendix E. (3 functions of the first order diagrams 301 where pt(Xs,Xs+\ps,ps+1) = [ dtj^T(Xs,Xs + tAXs,ps,p°+1). (E.1.10) Thus, correspondingly, Jp» can be split into two parts Ipa=[ ± u { x s , p \ p s + l ) + Y:&Xl f -pl{X\Xs+\p\ps+l). (E.1.11) JspS r ^ Jsps r 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 ) 2 -I)" ( E ' L 1 2 ) where {xs} is a sequence of numbers. We further subscripts x by i to denote a member {xf} of a set of sequences indexed by i. Let XSR be the characteristic function with support as the right overlapping region of ps. The following lemma estimates the resulting error of T from replacing f(xs,xs+1) by f(xs,xs). Lemma E . l 1. Suppose that xf > 0 and A x f = o(xf). Then for 0 < t < 1 — f a p — - X E ° I W F J ' ( E 1 1 3 ) 2. Suppose that Es = Hi/(xf, xf + t A x f ) . Then ^ r = £o(lliV (E.1.14) faf x> - I V X , Proof: It is clear that Part 2 follows from Part 1. As for Part 1, dropping the subscript i from Xi, 5f 2psps+1xs+l 5xs+l [1 + ps{xs - 1) + ps+l{{xs+l)2 - !)]< (E.l.15) Appendix E. (3 functions of the hrst order diagrams 302 Since ps < 1, the [• • •] term in the denominator of (E.1.15) can be bounded below by psxs or ps+1{xs+1)2. Thus, since Axs = o(xs) 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 IpS. Next we split the integral I pa into three parts respectively corresponding to the left overlapping region SSL = ((1 - e)M2s, M 2 s ) , the sharp region S*h = [M2s, (1 - e)M2s+2] where ps = 1, and the right overlapping region SSR = ((1 — e)M2s+2, M2s+2) where ps _|_ ps+i _ -p L e t u g denote ps restricted to SI, S*h and SSR by respectively psL pssh, and psR. Splitting Ip* as described above, we get dr, IpS -- [ -r(xs,xs,pi,o) JssL r + [ —T(XS, Xs, 1,0)+ [ -T(XS, Xs,psR, (1 - psR)). (E.1.16) Js\ r Js% r sh K By rescaling r —>• M~2r in the first integral, then since / -f{Pi)=f -m-M, (E.1.17) JssL r JssR r (E.1.16) = / —T(XS, Xs, 1,0)+ / - (T(XS, Xs, psR, (1 - psR)) + T(A S , Xs, (1 - psR), 0)). Js°sh r Js'R r (E.l.18) Equivalently, by rescaling r -> M 2 r in the third integral, then psR(M2r) becomes (1 — psL{r)) and (£.1 .16)= / -T(XS,XS,1,0)+ [ -(T(Xs,Xs,psL,0) + T(Xs,Xs,(l-pl),psL)). Js°h r JS'L r (E.l.19) Appendix E. f3 functions of the hist order diagrams 303 The term T(Xs, Xs, psR, (1 - psR)) + T(A S ,A S , (1 - pR),0) can actually be simplified further. Since in (E.l.19), the argument A s + 1 of T is replaced by Xs, in studying I p s , we may further suppose that the running slicing functions Vs are of the form V ^ x ) = l + ^ ( x . ^ - l ) + p ^ ( ( ^ ) 2 - l ) ' ( E ' L 2 0 ) where {xs} is a sequence of positive numbers. This form allows further simplifications when restricted to each of the three regions Si, S*h and SR. Let us denote Vs restricted to SI, S*h and SR by respectively V[ Vssh, and VR. Clearly, Vgh = pssh/xS- When restricting to Si, and when restricting to SR, n = 1 + pR(x° - 1) + (1 - psR)((x°)2 - 1) = x>{p>R + (I- psR)x°) • (E.l.22) For / i (r) a function with support 5£ , let fi(r) = ./^(Af ~ 2r) where the support of / L (r) is SSR. Similarly for /#(r) with support SSR, let /i?(r) fR(M2r) with support ££ . Let be a set of sequences indexed by i, V\ EE Vs(xi), and = support of ps, XSL = X(SSL), (E.1.23) X ^ = x (^ ) , 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. (E.l.24) (E.l.25) X ^ + X | A ^ i Q / M _ ( R 1 2 6 ) X R x\ + HR x i x\ vt + V\f = VU + Appendix E. (3 functions of the hrst order diagrams 304 2. dr Li=l i=l n -> A Ts / n i i=l x« j x i \i=l x i . (E.l.27) where the' index j in Yj Ax? runs over the set of distinct Xi in 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 s , and xf. Let us denote p = PL and pL = (1 — p). From (E.l.21) and (E.l.22), when restricted to SR 1 4- p(x - 1) x(px + p x ) ± 1 p1- + px x(px + p1-) X As for (E.l.26), from the form of V[ in (E.l.21), it is easy to check that (E.1.28) (l + p i + 1 ( x a + 1 - l ) ) ( l + P i + 1 ( x 8 - l ) ) ^ s W W ' v ' ' Thus from (E.1.28) and (E.l.29), V+V?1 = VsL + Vssh + VR + VsL+l XSR Axs ^ ( l L s f t X s X s Vx' P £ + + O (A ) • (E.1.30) x x \x Part 2: From (E.l.26) of Part 1, (E.1.29), and Vs(x) = ps O ( l /x s ) , n ^ + p f f i - n ^ 1 ) i=l i=l Appendix E. (3 functions of the first order diagrams 305 i=l n xk xf n A rs / n -j npe'j+xis-^ofni i = l j x j s LL i = l 2 = 1 n s A r s / n i N n ^ + x t E ^ o f n i j ^0 \i=l x i / i = l Xi (E.1.31) Now by scaling back, r J(i-e) r 7 7** < - M 2 rtr — = 21nM, 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 i = i (E.1.32) The second integral is identical to the first when scaling back by r —>• rM2. Thus the difference of the two is zero. Q.E.D. Suppose now A*>0, 7 s >0, AA*=o(A*), A 7 s = o( 7 s ). (E.1.33) Let us apply Lemma E.2 to marginal terms of the form IG = cGJ dqrs(q2)KG(q2) = cGj T s ( r ) ^ where KG(q2) is a rotational function in q of degree —4, V% = Vs (j), Vp Vs{x) = f{xs,xs+1) where / is defined in (E.l.12). (E.1.34) (E.1.35) Vs(A2), and Appendix E. functions of the hrst order diagrams 306 Lemma E.3 IG - cG2 ln M — 1 \ ™s / 1 \ n p where EjG = 0(1), £ is the overlap parameter, and = denotes equality up to irrelevant terms (Q(M~as) terms with a > 0). „ [AY AX 1 + eElG ( — 4 - + (E.l.36) 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 AXS2\ s where (E.1.37) Y = (VSB + VsB+l)nB (fsF + 7 3 ^ ) n F - {V'B+J;)nB(ni1)"', (E.1.38) V8B(i) and VSF(X2) are of the forms in (E.l.20), and Applying Part 2 of Lemma E.2 to / dr us(r)y, we have (E.l.36). Q.E.D. Appendix E. j3 functions of the first order diagrams 307 E.2 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 . T r . Lv. < V > , T r LV2 < ESE > _ n „ . b i V l = -^-z , b2V2 = -^—= , (E.2.1a) OM OM . LV3 <VP> . LV4<ESE> . . b 3 V3 = — ? — , b 4 Vi = —— , (E.2.1b) OM OM b 5 V5 = Vj , b 6 T/6 = , (E.2.1c) 0~M UM LVr < 4P > b 7 V 7 = , (E.2.1d) OM where 8M = -A^2 — 1 and = denotes equality up to irrelevant terms. In our calculation, we discard contributions from the gauge variant local parts that are zero when summed over scales. Here Vp(p) and VsB{p) respectively denote the slicing functions for the Fermi and pho-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(q2) = JdtoJ r3drf(r2) = y / r drf(r) (E.2.2) where kn = 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, AA? = o(A|) (E.2.4) and A| = O ( s 0 ) 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 B1 = 1 - p> - p°+i + p°\s2 + pS+1(XS2+1)2 ^ E ' 2 ' 5 ' ) with an relevant error —mVp'sR, where R is the Fermi propagator (m + #0_1, Vfs = (BT2 [(K - (Psf + A ^ + 1 ( A f 1 - A^ + 1) ps p 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 V4 local terms, this error contributes only irrelevant terms. Also, for the case of V4 local part of the one loop ESE, the relevant part is obtained by replacing the running Fermi line by H'+iF(q) = VFR(q) + VpsR2(q) (E.2.7) where R(q) = (m + 4)~l, Vp and Vp's are defined as in the above. We also estimate V% by P 2 s") 1 - ps - ps+1 + p A Y + p S + L ( Y + 1 ) 2 ' ' with an error of 0 ( S M X S 5 / ( Y ) 2 ) ( see (C.1.10a) of Appendix C ). Appendix E. (3 functions of the hrst order diagrams 309 E.2.1 Vertex diagram Figure E .20: The vertex diagram <V>=J^hy JdPd^<S(p)K(P,r)A<s(r)^<s(-p-r) (E.2.9) where K'0(p,r) = J dqTs(p,q,r) ^ M-fR(p - q)YR(p - q + r)Y, (E.2.10) and Ts(p,q,r) = (X[)3PaF{p-q)VaF{p-q + r)VB(q) (E.2.11) + E(A*)2^ [PsF(p -q)VF(p-q + r) VB{q) t>s + (VF(P - <?) VFiP -Q + r) V%{q) + VsF(p - q) VF(p - q + r ) ) VsB(q) + E(M)3 [PF(P-q)VtF(p-q + r)VtM t>s + (VF(p -q)VF(p-q + r) VB{q) + VF{p - q) VF(p - q + r)) VsB(q)] . Zero order localization We show the relevant contribution of LVl < V > is zero. T f % , 0 , 0 ) A ^ ( 0 , 0 ) = Jdq-± + q2 (YR(-qh°R(-qhn iR{-q)rR{-q)i (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 and iR{-q)YR{-q)4 (q2 + m2)2 7 M [4l°4 + m[j, 7 a ] + m2ja) 7^ (q2 + m 2 ) 2 Y (q2ja - 2qa4 - 2mqa + m 27 C T) 7^ (q2 + m2)2 2(q2 + m2)Y -Aqa{4-2m) (q2 + m2)2 + m)ja(4 + m) (q2 + m2)2q2 ,2 U<7 f-flf -mq2[r,4} + ™ 4Y4 (q2 + m2)2q2 _ 7<J(<72 + m2)q2 + 2mq2qa — 2m2qa4 (q2 + m?)2q2 From above, by scaling and dropping irrelevant terms, we have "27CT AqJ 7 0 7^(0,0) = Jdqrs(Msq,0,0)^ = J d q r \ M % o , o ) - ^ 7 (q2)2 q2 4q„4 (E.2.13) (E.2.14) (E.2.15) From (E.l . l) , it easy to see that Ksa{0,0) = 0. Thus b i = 0. Appendix E. j5 functions of the hrst order diagrams 311 E.2.2 Electron self energy diagram Figure E.21: The electron self energy diagram < ESE >= I dp 4><s(p) Ks(p) ^<s(-p) (E.2.16) where K'ESE(P) = jdq Y ^ l " [fS(Q,P - Q) R(P ~ O) ~ mTE(q,p- q) R2(p - q)] Y (E.2.17) and T'{q,p-q) = (AJ ) J n(p - « ) n ( « ) + E W ) 2 fa(P " Q) VB{q) + VF(p - q) VsB{q)) t>s (E.2.18) T%(q,p-q) = .T'( g,p-g)|^ ?, (E.2.19) where VSB, Vp and Vfs 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). KESE{0) = K'{0) + K'E(0) (E.2.20) where >(q,-q)R(-c YTs(q,-q)R(-q)r-q1 (E.2.21) Appendix E. (3 functions of the first order diagrams 312 —m f dq —-J 1 + YTsE{q,-q)R\-q)Y-^E(q,-q)R2(~ (E.2.22) Let us compute the terms Ks(0) and KE(0) separately dq Ts(q,-q) Ks(0) = J • J (1 + q2)(q2 + m2) dq Ts(q, -q) 3(4 - m). (E.2.23) (l + q2)(q2 + m2) By using the fact that an integral with an odd integrand in a rotational invariant domain vanishes, Ts(q,-q) (£ .2 .23) = -3m k2Y —3m kn f dr (E.2.24) where Ts(r) = Ts(q, — q)\q2=r and kn is defined in (E.2.3). We now compute KE(0). . « r , —m J (0) = j dq = —m j dq j to -m TsE(q,-q) 7 + q2 T%(q,-q) = — m = —m ( l + o 2)(o 2 + m 2 ) 2 r%(q,-q) (1 + q2)(q2 + m2)2 rsE(q,-q) (1 + q2)(q2 + m2)2 3mkn f dr Y(4 + M) J 2 ^ + m)24' Y(-Q2 - 2m4 + m2)Y 3(q2 - 2m4 — m2), -q2 — 2m4 + m2)4 (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 Ts(r) + TsE(r). From (E.2.18) and (E.2.19), Ts(r) + T%(r) is linear in VSF + VF'S. From (E.2.5) and (E.2.6), it is easy to see that VSF + VF'S is linear in A 4 . Let us extract leading terms by making the substitution Xs+1 —> Xs in X s ( r ) + TsE(r). For a term K(XS, Xs+1) which is a function of A and Xs+1, let us denote K(XS, Xs) by K^. By denoting V + T F = V'F + V$'A, we have ?U = (T s (r) + T ^ ( r ) ) | A 5 + 1 ^ A 5 = ( A D 2 [(n,F,->+n+AL^){vsB^+n+i^) - n+,h^n+i^\ • (E.2.27) To determine Tt>, let us first simplify Vs+F 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 + ^ W f l : ^ + « W W + 1 - M " ) | — Vs -4- Vs (E.2.28) where V L F = K(vsFf + K+1 vsF 2 p s + 1 A ^ + 1 Vs — Vs R0,F — r F Bs - psXs2-2 ps+1{Xs2+1)2 B1 l-{ps + ps+1)- ps+l{Xs2+1)2 Bs (E.2.29) (E.2.30) Let us determine the expressions of VQF_> and V{ F_+. Using the fact psR — (1 — psL+1), p£(i-p£)-pr1(i-pi+1)(Ar1)2 0,F P £ ( I - P I ) P i + 1 ( i - p r ) 3+U [(1 - pi) + pm)2 [(1 _ p f 1 ) - ^ + PsL+1xs2+Y (E.2.31) Appendix E. (3 functions of the hrst order diagrams 3 1 4 Hence Vs M),F,- A'L - AL+\ (E.2.32) where A", p£(i-pi) KI - P D + p i A i r (E.2.33) For the term V%4 F , (V s \2 F) Vi 2 p s + 1 \ s 2 + l Ws + AA* s+l \ s+l v% 2 P L+ i \ s 2 Bs (E.2.34) Hence, Vs 9s \ PSL+1^Y Bs + Pl PL a 2 B» + EJA _|_ PR + 1-PR (l-p£) + pLA| A§ A § [ ^ + ( 1 - ^ ) A 5 ] K + ( l - p ) \ ) A | ] , 2 / c+1 \ 2 5 s , Pi _j_ Xs/t _|_ X i? ; i - P i ) + p LAi A§ A§; VPfl + ^ - ^ i Xsh+R (A!)2 + P L P l + 1 . ( I - P D + P L A S ; V ( 1 - P L ) + P L + 1 A | We show that the term ASL — AsL+l contributes zero in the localization. From (E.2.27), since Y_> is linear in V+!F,^, let us consider the expression T . = (Ai ) 2 \ {A ' L - A L + 1 + AS^)(VB^ + VsB+l^) - AsL+1VsB+l i s + i - T j s + i = " (A s i ) 2 ASLVSB^ AsL+lVsB% (E.2.36) For a marginal integrand, the above term is irrelevant, i.e., (E.2.37) Appendix E. j3 functions of the hrst order diagrams 315 Hence, from (E.2.27), (E.2.32), and (E.2.35) dr ss r A? (E.2.38) After extracting the leading term, we write the contributions from the remaining terms as e £1=1,2,4,7 0(AAj/Aj) where A7 = 7 and e is the overlap parameter. Hence it is easy to work out that s\2 (A?) 1 + e AA* AA? AA2 A 7 S + r~~~ + + —-Af \ S \ S n/" A2 A4 7 ' (\ss 1 + 5 M O - 7 0(i)| (E.2.39) where the 1 + 5M O ((7s) *) factor comes from replacing 1 + js by 7 s in VB. First order localization K[(p) = j d q ^ f ^ - L,„rP«(dv„R(p-q)\p=»)r T S M ) 'Po 0. (E.2.40) 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 b 4 = 0 s\2 -3mkn\nM\s4 (X[) (2TT)4 8M (Ai)2 AA? AA? A A i A 7 S 1 H r~ A~ —T~~ ~\~ —T~~ ~\~ Af ryS O(i) i + sMo (E.2.41) Appendix E. (3 functions of the first order diagrams 316 E.2.3 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 = - ^ ~ J dpA<s(p)K^(p)A?(-P), KM = J dqTs(q,p)tr[rR(p + q)YR(q)}- (E.2.42) where Ts(q,p) = (X{)2 PMP + 9)n(g) + E W ) 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 RC 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 1 ; A 2 ) . Appendix E. fd functions of the first order diagram's 317 Recall that VPU+1 has an A2 local part whose coefficient is O ( l ) with respect to the loop regularization parameter A —> oo. However, it is Q>{M2U) with respect to U —> oo since it equals the contribution from the remaining slices. Let us determine the exact relevant contribution of this leftover term. Let * ™ = i - ^ C ^ ( A f ) ( K 2 - 4 5 ) WW) = ' U 1 - pu - pu+l + pP(\%) + p^+HAf) 2 ' where {pu, pu+l} is a two neighbourly slices decomposition of unity (recall that \f+1 Af for i / 3). Proposition E.l where LV5LU2^ = ( - p f + 0 ( A - 1 ) ) V5 (E.2.46) /?f = J tu(r)dr (E.2.47) fu(r) = (Af) 2 [(VFT)2 + 2VUFVUF+1] (E.2.48) Proof: Recall that we argued that the A2 local part of L f + 1 ( p c / + 1 ) is O ( l ) by using the fact that L2+1{pu+1 = 1) = 0 from Ward Identities and LU2+\pU+l) - Lu2+\l) = 0(1). (E.2.49) Since the terms involving the j ^ 0 spinor lines are 0 ( A _ 1 ) , it suffices to consider the 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 = {vuFy + 2vuFvuF+l + (vuF+l)2 = (PuF+vuF+xy. (Af) 2 (E.2.50) Since {pu, pu+l} is a two neighbourly slices decomposition of unity, we only need to verify the above on the overlapping region. In the overlapping region, let us denote p = p c / + 1 , p = pu, and A 2 = Af. From (E.2.45), since p + p = 1, VUF + VuF+l + P A 2 (p + pA2) p + pA2 A 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) (q2 + m2)2 Ts(g,0) t r (-4 + m) f (-4 + m) } -45, (q2 + m2)2 Ts(g,0) Zq^qu - 5^v(q2 + m2) (q2 + m2)2 -5^ 2kn JdrTs(r), 9 ^ 9 9 q - -q + ml where Ts = {\{)2(VF)2 + 2{\S1+1)2(VSFVF+L1). (E.2.51) (E.2.52) Let us also denote (E.2.52) by T S (A S , Xs+1) so that Ts(x,y) denotes (E.2.52) with A s and A s + 1 respectively being replaced by x and y. Appendix E. /3 functions of the hrst order diagrams 319 Assuming (E.2.44), let us extract the leading terms of TS(XS, Xs+l). Let Ys(p) = T s ( ( A i , A 2 ) , (Ai, A 2)). Let us write TS(XS,XS+1) = Ts(p)+Elr + EsbJ E. TS(XS,XS+1) - T S ( A S , A S ) = T S ( A * , A S ) - T s ( p ) = x s xl(Af) 2 (Af)2 A? A A | + A A | Af A 2 O(i). (E.2.53) O ( l ) (E.2.54) L(AI)2 A5. Let us show by setting (Af, Af) = (Ai ,A 2 ) , the leading term A| (A) = / drTs(p) of A | (A S ) = /drT s(r) sums to (E.2.47). Using similar notation for denoting replacement of arguments in T s , let us denote Vp(A2, A2) by Vs. Also, as from (E.1.21)-(E.1.22), let Vssh = PSsh/^2. Pl Vi = Vs = rR — ( l - p i ) + p ! A 2 ' PR (E.2.55) A2(p%A2 + (1 - pR) A2y Let us sum the T s(p)'s from s = 0 to s = U. Proposition E.2 By setting (Af, Af) = (Ai ,A 2 ) , f ; r » = A 2 [{VuFf + 2VF!VpT+1 s=0 where Vp andVF+1 are defined in (E.2.45). (E.2.56) Proof: E T » = (Ai) 2 E T O 2 + TO2 + (n ) 2 + 2 ( n n + 1 ) s=0 s=0 Since V\ — 0, rearranging the sum, we get u-i J2r°(P) = ( A 0 2 x : ( n ) 2 + ( n + ^ + 1 ) s s=0 s=0 +(Ai) 2 [Vl? + [VuRf + 2 (VURVUL+') (E.2.57) (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 l " V (E.2.59) Hence by setting Af = Aj with i = 1, 2, u 77-1 53 Xs/i+R + X s / l s=0 A? [ ( ^ ) 2 + 2 ( = Aj [(VvFf + 2VFTVFT+1' . Q.E.D. Thus from Proposition E . l and Proposition E.2, the leading contribution Z)f = 0 A f (A) of £!L0 A f (As) exactly cancels out the "leftover" relevant part (E.2.47) of the coefficient of the V 5 local of VPU+1. By discarding the leading term A f (A), from (E.2.53), it is easy to see that (At) 2 (Al ) 5 b 5 = o 'Af - A x ' , Ax + o 'Af, - A 2 A? s\2 AAJ AA* (AH ( A l ) 2 L A f ' A i O ( l ) . (E.2.60) Second order localization The kernel of the projection of the V P diagram onto V3 and (d • A)2 has the form K%M = j ^ T S M ) tr where T s is computed in (E.2.52). Let 7" \{PaPadqadqaR{q)) 7" R(q). (E.2.61) K^(P)=AA'TI», + B'LIU,) (E.2.62) where and are defined in (1.2.8). Using the trace of Ks 2(p) m (E.2.62), we have pKp As = Bs = p* K^,2(p) As (E.2.63) (E.2.64) 3p2 3 Because (d • A)2 is inert to other graphs, we will not calculate As in detail. However, because (d • A)2 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 b5, we may discard the leading term of As obtained by setting Af to Aj. It is easy to see the remaining relevant part is s\2 As = 5M (Af) 0 ( ^ ) + 0 ' A 2 - A g ' + e AA? AA? 1 + 2 Af AS O ( l ) (E.2.65) Since we will show that Bs = 0(1)> (E.2.65) implies that the sign of Bs does not depend on A8. We now compute the coefficient Bs. From (E.2.61), 1 3p2 3p2 J dq Ts(q, 0) tr [7" -(p„padqadqaR(q)) 7" /%)] • (E.2.66) To compute the trace in (E.2.66), we first compute 7 M \{PaPadqadqaR{q)) 7" \paPadqadqa('fR{q)'f) 1 a n ,-24-Am. 1 q1 + m2 (E.2.67) 2p2 Hence tr [• • •] tr 4]/ (p • q) (q2 + m2)2 (q2 + m 2 ) 2 W + 2m) (g2 + m 2 ) 3 (g2 + m 2 ) 2 J ' 2p2 8(p-q)2 (g2 + m 2 ) 3 (g2 + m 2 ) 2 y '8(p-(z)2 4(p • g)2 - (g2 + 2m2) ( - 2p2 (g2 + m 2 ) 3 Using the identities (E.l.l) and above, we have q2 + m2 ~4 + m\ q2 + m2 J (E.2.68) 3p2 4 f . T s(g,0) (<7 2 + m 2\3 , 2m2(q2 + 2m2) g 2 + v ' (g2 + m 2) (E.2.69) Applying Lemma E.3, up to an error of the form (E.2.54), (E.2.69) = i f c n l n M J ^ . (E.2.70) Appendix E. (3 functions of the first order diagrams 322 Hence (recall that we need an extra minus sign), b 5 = b 3 = kn (Af) s\2 2(2TT)4 (A!)2 -Akn InM (Af)2 6SM (2TT)4 (A§)2 where the E's are of the form £ b 5 + o ' A x - A Af O ' A 2 - A ^ A 2 - A 2 AA? A A 2 s H s Af A 2 O ( i ) . (E.2.71a) (E.2.71b) (E.2.72) E.2.4 Four photon legs diagram Figure E.23: The four photon legs diagram To compute the coefficient of the A4 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)4 (A!)4 O Al ~ + o (— Af AS (E.2.73) Appendix E. /3 functions of the hrst order diagrams 323 E.2.5 Summary of coefficients Smooth slicing We summarize our calculations. Let cr = h = h = Xs £ = A 2 2 J i2 -2kn I n M 3 (2TT) 4 (M 2 - 1)' 2(2TT)4 -3m kn ln M (2TT)4 M 2 - 1 (E.2.74) (E.2.75) M 2 _ 1 y Xah+fl-RVp.Vfe, &7 = ^ 2 ~ 7 J T Xah+RK. AP,V7, where KVp,v6 and KAPtVl respectively are the relevant part of the V% and V7 localized kernels (excluding the running slicing functions) from the V P and the 4P diagrams. bi = 0, b 3 = M ^ ) 2 ( i + £ b 3 ) - f [ ( < 7 s ) 2 - £ 2 ] Xt (a*)2 Ao ^ ,-, ——Eb ryS be 3 b 4 = 64 b 5 = h [(as)2 - E2] + (as)2Eb5 b 6 = h[(as)2-Z2}+(as)2Eh6 b7 = b7\(as)A + [o-sYEh (E.2.76a) (E.2.76b) (E.2.76c) (E.2.76d) (E.2.76e) (E.2.76f) where, for the coefficients obtained from the Fermi loops, the errors E are of the form O ( l ) , . (E.2.77) AAf AA? 1 + 2 Af and for the coefficients b 2 and b 4 Eb2 = £ AAf AAf, A 7 S Af Xs, AS O W + ^ O S U ^ O ( ^ ) (E-2.78) Appendix E. ft functions of the first order diagrams 324 6 \S Xs \s O/S . Al A2 Ai 7 AA? , A A 2 , A A 4 , A 7 ' (E.2.79) Sharp sl icing 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). 0, b 2 = 0 (E.2.80a) (E.2.80b) (E.2.80d) (E.2.80c) (E.2.80e) b 7 b7 (as)4 - S 4 . (E.2.80f) A p p e n d i x F The contr ibut ion from expanding the sl icing functions In this appendix, by assuming the conditions (9.1.1-d) on the couplings, we show that there is no 0(1/7 S ) contribution to the (3 coefficients from expanding the slicing functions 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 ps(p — q) = p1(M~s(p — q)) can only leads to irrelevant 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 ESE 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 S ) 2 ) contribution from the E S E diagram. In the renormalization procedure of the tree expansion scheme, counterterms are ob-tained 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 gen-eral, the counter terms corresponding to these local vertices are irrelevant. We illustrate this statement with the example of the ESE 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 psF(p - q) pB(q) ^ ^ ^ M (F.l) where CQ is the usual combinatoric factor and psF and pB 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 2 flow equations) have kernels K(0, s,t) and dq /4(tf)y + g 2 7 (F.2) —Q + m + dP^psF{-q)--q + m y -$ + m_ 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 cG Pa J dq p f ^ ^ r d^p?{-q)-± <i + m (F.3) Appendix F. The contribution from expanding the slicing functions 327 Since the term dqapFu(—q) restricts the support of the integrand to be in the scale U 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 V2 local part of the ESE diagrams in the L R C scheme. The running slicing Vp and VB are the one defined in (E.2.5) and (E.2.8) respectively. From (E.2.17) of Appendix E , the kernel of the < ESE >jjs diagram is KESE = [dq ^ M^s{q,p- q) R(p ~ q)Y (F.4) J l + <r where Ts(q,p-q) = X{X[VF{p-q)VB(q) + 'Z(>\)2 (VsF{p - q) VB{q) + VF{p - q) VsB(q)) t>s (F.5) From (F.4), by expanding VF(p - q) at p = 0, extracting the first derivative term, we obtain the following relevant local term (we replace denominators m + q2, 1 + q2 by q2 and the resulting error is only irrelevant) K>, = - J d q f ° ( q 2 ) 2 p . q Y ^ Y ^ = - J dqfs(q2) where 2p-q q2 q2 [q2f (F.6) fs(q2) = (X{)2 (VF)'(q2) VsB(q2) + (AJ+ 1 ) 2 ((VF)'(q2) VsB+1(q2) + (VF+1)'(q2) VB(q2)) (F.7) Appendix F. The contribution from expanding the slicing functions 328 and f'(q2) denotes the derivative df(q2)/dq2. Applying the 7 algebra, and using Euclidean symmetry, we have Kl' = - ^ / ^ 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; , = -C2P1 f dr Ts(r) (F.9) where Ss is the support of ps(r), c2 = k^ci and is defined in (E.2.3). We show that / s . dr f s = 0(1/(7 S) 2)- Since (p°ah)' = 0, from (F.7) t s = (K)2 {(VsFJVBtL + (VFJVB:R) +(K+1)2 {(nj n:i + (n%l)"pB,R) • (F.IO) We would like to write Ts as a sum of two terms y4-p(As) and Bfs(Xs, AXS) where the first term is obtained by having all the A 5 + 1 being replaced by A s in T s and the second term is the error as a result of the replacement. It is easy to see that the first term Af. {Xs) = (x[)2 {[(vsFtR + r F j ( v B t R + vB,L)} + [(vsFjvsB,L - ( n , L ) ' n , J } • (F.l l) where Vs is defined as in Section E . l of Appendix E. For convenience, let us denote A 7 = 7 S = A* + A*. (F.12) The second term Bfs(Xs, AA S ) can be written as the interpolation integral E ^r-EhP, (F.13) i=l,2,7 i where Ei Appendix F. The contribution from expanding the slicing functions 329 From (E.l.24) of Lemma E.2 in Appendix E , (ntR+nj = (h' = o- (F.i5) Thus (F.l l) = (X{)2 [{VFtL)'VBiL - (PFtLyPBtL\ . (F.16) By scaling the second term back to the SSL, it is easy to see that Jsm dr (A?)2 [(VFtL)'VBtL - (PFJVB,L] = 0. (F.17) We now consider Bfs(Xs, AA S ) . Since for i = 1,2,7 f ^ o ( i ) , we are left to show that (F.14) = Q)(l/js). Let us consider the partials derivatives in (F.14). r)T s x{iw = 2 a ; a ; + 1 ^ + ^i 1)'^ .*) ( R 1 9 ) A ^ = ( A D 2 ^ ( ^ ^ ) ' n ^ + ( A r 1 ) 2 ^ ( ^ n , f i ) , n - 2 + ( A f + 1 ) 2 A^^rni 1 ) '^ (F.20) +( A*i + 1) 2 ^ ^^(ni 1 ) ' (F.21) We apply the following estimates and formula of derivatives to show (F.14) = 0 (V7 S ) -VFMR) = O (^ ) . * W ) = O (£) • (F.22) Appendix F. The contribution from expanding the slicing functions 330 2. Let Vi = VB and V2 = VF. Af d Af dy+i rITR <9Af+1 i l L 2p^Pl + 1 A|AJ + 1 < ( ^ A J + ^ + 1 ( A f + 1 ) 2 ) 2 A * ' P £ + 1 P £ + 1 A? , Af ( l + ^ + 1 ( A f + 1 - l ) ) 2 " (Af + AAf) (F.23) dVp;l _ l + rf,+1(AS+1-l)-^+1(AS+1-l) <9p£+1 (1 + p f W 1 (F.24) (1 + P l + 1 ( A ^ + 1 - 1 ) ) 2 ' P^A| + (1 - ^ ( A f 1 ) 2 - pR [AS - (Af 1 ) 2 ] (p^AI + a - p ^ X A ^ 1 ) 2 ) 2 ( A f 1 ) 2 ( p ^ + ( l -p | J ) (A^ 1 ) 2 ) 2 ' 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 € Ss, from (F.23) 1 1 } 7 djs+1 1 f ' r ) < (Af 5 s r s+l\2 1 7s M 2 ( l - e ) ( A f 1 ) 2 1 Y M2(l-e) ( A f 1 ) 2 1 YXs2 M 2 ( l - e ) A! UW) rM2 JM2(l 2 ( l - e ) PR=0 dr(VsFJ Vs r F,R PR = 1 i + O i + O /Ay V 7s 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 bpiV<i be the contribution from expanding the slicing function in < ESE >, s\2 P A | 7 S A A f , AA^_ , A 7 Af + + H(XS,XS+1) where H(XS,XS+1) = Q(1). (F.26) (F.27) Next, we examine the contribution of the one loop VP diagrams from expanding the slicing functions. s s s+1 s 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 con-tribution from expanding the slicing function. Proof: We first look at the V P diagram with all s lines. The kernel has the form KVP(s,s) = cG f Ts(q,p + q) t r ^ ^ p + q)^uR(q)\ dq. Appendix F. The contribution from expanding the slicing functions 332 where rs(q,p + q) = (Xt)2 VF(p + q)VsF(q) + i:(^)2 VsF(p + q) VF{q)+PF(p + q) VF{q) (F.28) t>s Let t V ) = (K)2n(q)2 + (K+1)2n+\q) vF(q). (F.29) In the expanding the slicing factors, we get the relevant contribution cG P2 J [fs(q2)}' tr[%R(q)^R(q)} dq = cG p2 j [Ts(q2)]' (F.30) Making the usual change of coordinates, (F.30) = cG M2sp2 f dr4- [ts(r)l2 = 0, (F.31) Jss dr 1 J since VsF(r) is zero at both ends of 5S. Q.E.D From the above analysis, we conclude that for the one loop diagrams, there are no 0(1/7 S ) relevant contributions to the (3 functions from expanding the slicing functions.
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Imaginary charge quantum electrodynamics : a running coupling analysis Ho, Andy C.T. 1998
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Title | Imaginary charge quantum electrodynamics : a running coupling analysis |
Creator |
Ho, Andy C.T. |
Date Issued | 1998 |
Description | 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 QED model in perturbation theory. By integrating out a sequence of slice fields, the LRC scheme generates a flow of effective potentials V⁸. Here we do not demand that the flow of V⁸ be gauge invariant but only that the Ward Identities hold at the end of the flow. From the flow of V⁸, we obtain a flow of the couplings ʎ⁸ of the local parts of Vs. Using a fixed point analysis in a suitable Banach space whose norm captures the asymptotic form of ʎ⁸, we determine the asymptotic behavior of ʎ⁸ satisfying boundary conditions partially fixed by the Ward Identities. At each step of the flow, the slice covariance is transformed by shifting the local quadratic terms of V⁸ to the Gaussian measure. In this way, the corresponding ʎ⁸ is governed by a flow of an effective coupling ζ⁸which is ultra-violet asymptotically free around the origin. The UV asymptotic freedom of ζ⁸ provides the stability of ʎ⁸ so that the asymptotic form of ʎ⁸ can be obtained from a primitive flow corresponding to only a few low order diagrams of the LRC expansion. |
Extent | 11434987 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-06-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080040 |
URI | http://hdl.handle.net/2429/9483 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1998-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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