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On bound states and non-trivial fixed points in quantum field theories Gat, Gil 1992

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O N B O U N D S T A T E S A N D N O N - T R I V I A L F I X E D P O I N T S IN Q U A N T U M FIELD THEORIES By G i l Gat B. Sc. (Mathematics/Physics) Hebrew University of Jerusalem M . Sc. (Physics) Tel-Aviv University  A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR  OF PHILOSOPHY  in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH C O L U M B I A  October 1991 © G i l Gat, 1992  In  presenting  this  degree at the  thesis  in  partial  University of  fulfilment  of  of  department  this or  publication of  thesis for by  his  scholarly purposes may be  or  her  permission.  (c s  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  representatives.  ;5. V. 3 2.  for  an advanced  Library shall make  it  agree that permission for extensive  It  this thesis for financial gain shall not  Department of  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  Abstract  The calculation of the binding energies in Quantum Field Theories (QFT's) is a hard and long standing problem. Even for weakly coupled field theories like Q E D the extraction of information on bound states from the perturbative expansion is considered an "art". In the first part of this thesis we define those bound states that can be recovered from the perturbative expansion (threshold bound states) and calculate their mass in various models. It is shown that the method of Pade approximation and the Bethe-Salpeter equation (supplemented by a proof of absence of on mass shell singularities) provide a systematic way of calculating threshold bound states masses from the perturbative expansion of the S-matrix. We check these methods on 1+1 and 2+1 dimensional models where there exist a good expansion for the S-Matrix i.e. either weak coupling or 1/N. In the second part of this thesis a less rigorous approach is taken. This part concentrates on the B=2 sector of the SU (3) Skyrme model. We show that one can generate classical configurations (skyrmions) corresponding to bound states of two particles of an effective field theory (the Skyrme model), starting from classical solutions of the euclidean SU(3) Yang-Mills theory. The parity doubling of the ground state in this sector is also investigated. The third and last part deals with non trivial fixed points in QFT's. It is shown that the infra-red fixed point of the chiral phase transition in d=3 is the critical GrossNeveu model. This is yet another proof to the nonperturbative renormalizability of fourFermi interactions in 2+1 dimensions. The critical exponents of this phase transition are calculated within the 1/N expansion. The renormalizability of the G N model is also demonstrated explicitly to next to leading order in l/N by calculating /N function,  eifective potential and ultraviolet dimension of various operators. Finally the addition of operators of the type (îJip)^ is considered.  Table of Contents  Abstract  ii  List of Tables  vil  List of Figures  viii  Acknowledgement 0.1  x  Overview of this thesis  xi  I  Threshold Bound States  xiii  1  Introduction  1  1.1  9  2  3  The Method of Fade Approximants  15  2.1  Quantum Mechanics  17  2.2  1+1 dimensional Thirring Model  19  2.3  1+1 dimensional Gross-Neveu model  23  Binding Energies in Weakly Coupled 2+1 Dimensional Q F T ' s  26  3.1  27  3.2  II  General concepts and methods  Scalar theories Fermionic theories  Exotic Bound States Of Skyrmyons  31  37  4  A short introduction to the Skyrme model  38  4.1  SU{3) Skyrmions from instantons  45  4.2  Parity Invariant Solution In The B=2 Sector Of The Skyrme Model  III 5  6  7  8  Non Trivial Fixed Points in Q F T ' s  . .  50  56  Introduction  57  5.1  Flows, Universality and Scaling  58  5.2  Relevant, Irrelevant and Marginal Operators  59  5.3  The flow near the critical surface  61  5.4  Conformai invariance  62  Chiral Conformai Point  65  6.1  Chiral IR fixed point  68  6.1.1  73  U V dimensions of local operators near x C P  Renormalizable four Fermi interaction  77  7.0.2  Beyond leading order  81  7.0.3  R G improved effective potential  85  7.0.4  Critical exponents of the G N model  88  Six Fermi Interaction  91  8.1  Leading order  91  8.1.1  Triviality of (rp^jf near x C P  93  8.1.2  Townsend - Pisarski unstable fixed points  95  8.2  Discussion and Conclusions  Appendices  98 101  A Some properties of Fade approximants  101  B  105  Proof of absence of mass-shell singularities  C The physical meaning of the auxiliary  field  D Non Linear o model E  113  Calculation of U V dimensions in the framework of the four - Fermi  models F  108  118  Equivalence of (V'V')^ and  G Table of Integrals G.l  2+1 bosonic integrals  G.2 2+1 fermionic integrals Bibliography  models  120 122 122 123 125  List of Tables  1.1  examples of Q M bound states  6.2  Ultraviolet dimensions (in units of energy) of various operators in the Yukawa model  10  74  List of Figures  1.1  Diagramatic representation of the Bethe-Salpeter equation  2.2  The physical sheet in terms of rapidity variables  20  3.3  The lowest order diagram contributing to  36  3.4  The lowest order diagrams contributing to  in the scalar theory eq.(l) . in the massive Thirring  model. The dashed line represents a fictitious photon with propagator 1. 3.5  The lowest order diagrams contributing to  53  The radial energy density e (solid curve) for the parity-invariant solution. The dashed curve shows 20pb for comparison  6.9  52  The radial baryon density pi, (solid curve) for the parity-invariant solution. The dashed curve shows the radial baryon density ior U2  4.8  36  The function Xp (solid curve). For comparison the functions x (dashed curve) and ip (dot-dashed curve) in the solution [1] U2 are also plotted. .  4.7  36  in 0(2N) symmetric Gross-  Neveu model. The wiggly line represents the a propagator 4.6  5  Phase diagram of the Z2 invariant Yukawa model  54 70  6.10 Diagrams contributing to the 'ipip propagator to order 0(N). Solid lines denote fermion propagators and wavy lines - scalar propagators  73  6.11 Diagrams contributing to the correlator < ^)ip{Q)'4){x)ip{y) > to order 0(N). 75 7.12 Feynman rules for the 1/N expansion in the Gross - Neveu model  80  7.13 The effective potential for the field o. The solid line is the leading order result VQ{O). The dashed line is the next to leading order correction V\{CF). Vi(a) is not shown for small cr, when it becomes complex 8.14 R G flow of the couplings C and j] as functions of t = log(m)  88 97  8.15 continuous lines represent deformations of the Yukawa type c/cr^/'V'- dashed arrows represent deformations of the form (•0AjjV')^ while dotted arrows represent deformations of the form m'^^  99  C. 16 Diagrams contributing to the 0(1/N) corrections to the a propagator . .  109  D. 17 Feynman rules for the 1/N expansion in the Non-Linear a model  116  Acknowledgement  I would like to thank B. Rosenstein, A . Kovner, N . Weiss, and A . Schakel for helpful and illuminating discussions. This thesis is dedicated to my parents Ariela and Eliahu Gat.  Don't get mad get even !  0.1  Overview of this thesis  This thesis is divided into three parts. The first part which includes chapter 1 to 3 deals with the calculation of bound state masses using the perturbative expansion of the Smatrix (i.e. weak coupling or expansion in some other small parameter like 1/N). One cannot expect to recover all the bound states from this expansion, only a certain kind of bound states, threshold bound states, can be captured. In chapter 11 define and classify these states. I give examples of Quantum Mechanical threshold bound states and their Q F T analogs. Problems with direct perturbation theory for the binding energy are discussed. In chapter 2 the method of Padé approximants to the S- matrix is applied to problems where threshold bound states exist. First I show that it works in Quantum Mechanics and then apply it to 1+1 dimensional solvable models [1]. The [1/1] Padé approximant is shown to reproduce exactly the first and second terms in the expansion of the lowest bound state energy in the Thirring and Gross-Neveu models. The aim of chapter 3 is the calculation of the binding energy of the lowest bound state in the newly discovered nonperturbatively renormalizable 2+1 dimensional Gross-Neveu model [2]. Here I use the well known method of Bethe-Salpeter equation. However unlike the general case I can show that for a large class of theories in 1+1 and 2+1 dimensions the perturbative expansion of the BS kernel leads to a systematic (and simple, in leading order) expansion. The second part of this thesis deals with exotic bound states of skyrmions (chapter 4). In particular I concentrate on the B = 2 sector of SU(3)F  Skyrme model. After a brief  introduction to the Skyrme model I show that it is possible to construct approximate analytic configurations with energies very close to those of the numerical solution by integrating over the phase factor of Yang-Mills instantons [3]. In chapter 5 it is shown  that for the SU{S)F  skyrmion, if one imposes parity invariance the resulting configuration  is already classically unstable. It is however claimed that this configuration plays a crucial role in the tunneling process between the two degenerate classical vacua [4 . The third and last part deals with non trivial fixed points in QFT's. The models considered are four-Fermi interactions in 2-1-1 dimensions [5],[7]. These models were recently shown to be renormalizable within the 1 / N expansion . After introducing the basic ideas of renormalization group in chapter 5 I show in section 6.1 that the infra-red fixed point of the chiral phase transition in d=3 is the critical G N model. I calculate ultraviolet dimension of various operators in the vicinity of this point and examine various deformations. The renormizability of the four Fermi interaction is shown to next to leading order in chapter 7. The P function and effective potential are calculated to next to leading order. In section 7.3 I calculate critical exponents of the G N model and show that they obey scaling. Finally in chapter 8 "marginal" operators of the {'tpip)^ type are considered. We show that they are irrelevant near the chiral fixed point. However the extension of the parameter space leads to the appearance of new non trivial U V fixed points. Unfortunately in all the cases considered these points are located outside the stability region of the theory. In appendix C we comment on the calculation of two particle binding energies using the 1 / N expansion. Appendix D contains a calculation of the next to leading order /? function in the non-linear a model and a comparison to lattice simulation.  Part I  Threshold Bound States  Chapter 1  Introduction  Formation of bound states is one of the most interesting problems in Quantum Field Theory ( Q F T ) . The physical states of Q F T are often very different from the "elementary particles" that comprise it. For example, in Quantum ChromoDynamics (QCD) where the elementary particles are quarks and gluons one finds only mesons (bound states of quark anti-quark), baryons (bound states of three quarks arranged in a color singlet) and (possibly) glueballs which are color singlet bound states of gluons. This is also true for theories where the interaction is weak (in the physical region) for example Quantum ElectroDynamics (QED), the elementary particles are electrons and protons but the low energy spectrum consists of the hydrogen atom. In general, given a theory we usually know very little about its spectrum. Some qualitative features of the spectrum can be deduced even without calculating it explicitly. For example if the theory is invariant under some non-trivial global symmetry group G , then this will be reflected in the spectrum. The bound state spectrum can be classified in multiplets of G. Even if the symmetry is broken it may still have an imprint on the spectrum. For example if a global continuous symmetry is broken spontaneously (in d > 2), it can be shown that massless particles (Goldstone bosons) appear. However an exact solution of Q F T ' s which gives the whole spectrum of a theory, is available only for a handful of 1-f 1 dimensional theories. Before we continue let us clarify what is meant by elementary particle. A n elementary or composite particle is not a fundamental concept. While sometimes at high energies  one can see the composite nature of bound states in other cases particles which we are accustomed to call bound states are so tightly bound that they remain essentially pointlike even at very high energies. A n example of this is the isoscalar particle arising near the chiral phase transition in 3+1 dimensions [11 In nonrelativistic Quantum Mechanics we do have a clear picture of what a bound state is. We interpret a state as composed of N elementary particles (if the wave function is normalizable) held together by an attractive force as bound. The classic examples are atoms and molecules which are composed of the nucleus and electrons, bound together by the attractive Coulomb force. Furthermore we have a well defined way of computing the energies of all the bound states (at least in principle). A l l we have to do is to solve the Schrodinger equation with the appropriate potential in the N particle sector and make sure that our solution is normalizable (i.e. square integrable). This picture does not change very much when we pass to non-relativistic field theories. The field operator in such theories: V i ( x , t ) = / e'"P-«(p,t)a(p) jff contains only annihilation operator. A n interaction term like for example ip^ipip^i/) will have zero matrix element between states belonging to different sectors of the Hilbert space. This is why, for example, for two particle bound state we can reduce the problem to a quantum mechanical one. The amplitude < 0|Tip{x)tp{y)\B  > can be shown [9] to  satisfy Schrodingers equation in the center of mass frame. In relativistic Q F T this picture is no longer valid. This is because of the following reasons.  First, generally the binding force is mediated by particle exchange produc-  ing retardation. Secondly, the elementary particles are "dressed" by a cloud of virtual particles  Because the covariant (say, scalar) field operator is now <t>{x,t) = / ^ - ^ an interaction term like  [e-*^a(fc) + e'*^-^at(A;)  will contain terms like aaaa or a^a^a^a and will necessarily  mix different sectors of the Hilbert space. The ability of Q F T to create and destroy particles, makes the whole concept of a bound state very hazy. The concepts of potential and wave function which had well defined meaning in quantum mechanics do not have well defined counterpart in Q F T except in limiting cases. Consider a relativistic field theory involving a set of Heisenberg field operators {rpi{x), '4>2{x)... ipnÇx)}. By the usual assumption of asymptotic completeness there are " i n " and "out" fields that create asymptotic states of elementary particles. A bound state i n field theory is defined as a state of discrete mass and definite spin and possibly quantum numbers of internal symmetries, orthogonal to all asymptotic "elementary", " i n " and "out" states. A complete set of states in the Hilbert space of the field theory must include all the bound states. Although there is some ambiguity in choosing the bound state operator, it was shown [10] that one can generally choose a local operator to describe it. Matrix elements of this operator can then be shown to obey integro-differential equations like the Bethe-Salpeter equation. These equations are Poincaré invariant and there is no general way of defining their non relativistic limit (even worse, in some cases it doesn't exist). For these reasons we shall try to avoid dealing with the bound state amplitude (the analog of the bound state wave function in quantum mechanics). Instead we shall be interested in the location of poles in the full relativistic S-matrix. These will directly give information on binding energies and (for Breit-Wigner resonances) widths. There exist many approximate methods to calculate binding energies. The crudest method, potential model, is based on the following idea:  i . compute the scattering amplitude T/j defined in terms of the S-matrix element Sfi Sfi =< /,out|i,in >= ôfi + i{2iry-H''-\Pf  - Pi)Tfi  in lowest non-trivial order in perturbation theory. i i . find the non-relativistic limit iii. obtain the potential V"(r) as the Fourier transform  and solve the Schrodinger equation for the bound states of this potential. There have been many attempts to incorporate higher relativistic corrections (recoil effects) into this calculation but in general there is no systematic way to do this. Potential models necessarily stay within a given sector of Fock space. The number of particles and anti-particles is fixed once and for all and therefore no further degrees of freedom can be accommodated. For example pair creation effects cannot be described by this method. It is therefore only a qualitative tool in analyzing the spectrum. The next and most important development in this field was the introduction of the Bethe-Salpeter equation. In 1951 Bethe and Salpeter wrote down an equation which is formally an exact equation for the relativistic bound state problem. For simplicity we give here a sketch of the basic idea applied to a two particle bound state (a more detailed discussion is found in chapter 4). We will therefore be interested in the 2 —>• 2 scattering amplitude or equivalently (up to phase space factors), the on mass- shell four point function. The Greens function G{xi,X2,xs,x^)  =< 0|r(f>{xi)<j){x2)(l>*{Xi)(l)*{x^)\0  > .  where T is the time ordering operator, can be written as a Feynman path integral. Bethe and Salpeter (BS) showed that one can rearrange the expansion of the four point  P/8*p  PI2*%  ^ s _ A ^  W»»P  P«.,  "»-p  ^ ^ i s ^ y  P/a.p- V - ' ^ ^  P/a.p  p/a.*  Figure 1.1: Diagramatic representation of the Bethe-Salpeter equation function to obtain an integral equation for G. Diagramatiealy this is shown in fig.(1.1) where K is a kernel and the first term on the left hand side describes the motion of two noninteracting particles. Usually we only know the expansion of the kernel K to some finite order (assuming a small expansion parameter can be defined). However, even if a reliable perturbative expansion for the kernel exists, one should baxe in mind that the BS equation is not an eigenvalue equation. The highly nonlinear nature of this equation makes its solution and in particular, its perturbative solution very difl&cult. A n alternate approach which is commonly used is to perturb around an exact solution of a simplified model. Assuming that the full solution of the model is not much different from it one then treats the difference terms as perturbations. A typical example is the calculation of binding energy of the hydrogen and positronium in  QED4.  The unper-  turbed model assumes instantaneous Coulomb interaction, that is, one takes as kernel of the BS equation  This is equivalent to summing all n photon exchange diagrams where the photon lines do not cross each other and is known as the "ladder approximation". In this case it is possible to reduce the problem to a Schrodinger equation that can be solved exactly. One then tries to take into account the neglected terms due to retardation, radiative corrections and recoil effects as perturbations. The testing ground for all these methods is the calculation of the hydrogen and positronium (muonium) binding energy. The experimental data on the hydrogen and  positronium spectrum has been and still is more accurate than the theoretical one. The most recent theoretical result for the hydrogen hyperfine splitting between the singlet and triplet ground states is u = 1420.45195(14) M H Z and corresponds to calculation of all terms up to  log(Q) in the "expansion" of the  binding energy. This agrees very well with the most accurate experimental data u = 1420.425 751 766 7(9) M H Z , the relative difference being 32.56(10) ppm. However, generally even for weakly coupled theories like Q E D the BS equation doesn't give the whole spectrum (actually as we shall show in chapter 2. one shouldn't expect it to) and in many cases generates many copies of the same solution or even spurious ones. Various improvements like renormalization methods and normalization conditions on the BS amplitude were added to the BS equation but the basic idea is the same until today. The development of Q F T during the 70s brought new problems and puzzles with it. The possibility of confinement and a complete failure of the perturbative expansion was considered. Although this problem is still far from being solved the search for non perturbative methods which was initiated by it turned out to be fruitful. New methods like the semi-classical, variational gaussian approximation and the 1/N expansion were found. Towards the end of the 70's even an exact solution of some nontrivial 1-1-1 dimensional relativistic Q F T ' s was found [19 . The availability of an exact analytic expression for the spectrum of some lower dimensional Q F T ' s enabled an easy check of methods for calculating binding energies. Recently the list of renormalizable theories was extended even more with the discovery of a large class of 2-1-1 dimensional theories that are renormalizable within the framework of the  1/N expansion. Despite all this continuous effort a systematic (i.e an expansion of the binding energy in some small parameter) method for calculating binding energy of some, if not all, of the bound states in a theory where they exist, is still not available. One may wonder why do we bother using all these complicated methods even in weakly coupled field theories. Why can't we simply calculate binding energies using perturbation theory ? A simple quantum mechanical example in one dimension will show the problem. Consider the scattering of a particle off an attractive —aS{x) potential. The full S matrix in the even parity channel is S = in this channel reads S=l  while the perturbative expansion of S  — ij + ^ + -- -. The exact binding energy can be read  from the location of the pole in the S matrix Ef, =  If instead we try to use the  perturbative expansion i n a we find that to any finite order the only divergencies are at fc = 0 i.e on the threshold and in general they are not pole singularities. A similar situation prevails in Q F T ' s ; there one finds only threshold singularities (i.e. square root or log type singularities) at  = (nM)^ in on-mass-shell Greens functions at any finite  order in perturbation theory. The previous example shows the need for a resummation technique for obtaining poles from the perturbative expansion. But even if we succeed in resumming some or all of the perturbative expansion, will we get all the bound states ? Again, a simple one dimensional example shows that in general the answer is no. Consider the spectrum of a one dimensional square well (1.1) The number of bound states depends on the depth of the well, as one decreases the depth most bound states disappear long before it reaches zero. In fact all excitations will disappear one by one for some finite non-zero values of a. However the even parity  ground state remains for arbitrary small depth of the well and disappears only when the well depth is zero. Since perturbation theory is, by definition, analytic in the coupling, it is clear that, if any, we may expect to recover only the ground state binding energy from the perturbative expansion.  1.1  General concepts and methods  I will be using throughout this work examples and analogies from quantum mechanical problems. This is because there are only a few nontrivial Q F T ' s which are exactly solvable and even there we don't always have all the information we would like to have (for example the exact expressions for Greens functions are not available). The alternative of using usual perturbation theory does not work for the bound state problem since the effective expansion parameter (e.g. | in Q M or ^2,4^(^2 in QFT's) actually becomes large near the two particle threshold. In contrast, in Q . M . there are many problems which exhibit the same features and for which an exact analytic solution is known so it is much easier to check new methods on them. Since the basic tool in Q F T is perturbation theory I will be particularly interested in the dependence of the binding energy on the expansion parameter. Table 1.1 lists a few examples of potentials for which an exact expression for the ground state is known. The hydrogen spectrum in d = 3, the (only) bound state of the one dimensional 6 potential, and the ground state of the square well and Poschel- Teller potentials ( cosh^cj)^  )  in one and two dimensions have the common property that no matter how small the coupling {a, the coefficient of the potential) is they remain bound. We will call states that have this property threshold bound states because as the coupling decreases they become closer and closer to the threshold for elastic scattering (E = 0). The importance of these states is the fact that they are the only bound states that can (in principle) be reliably studied using perturbation theory. Non threshold bound states hke those of 3D potential are clearly out of reach of perturbation theory. Among threshold bound states one can essentially distinguish two kinds. Analytic threshold bound states, that is states whose binding energy is analytic in the coupling.  Potential  -£^0  d=2  d= 1 -a 0  -a(5(x) if|x|<a otherwise a cosh(x)2 a IX  9.  -  2maW + 0 ( ^ 3 )  ^^exp(|-27-^)  ^  + 0{a^)  m  -exp(-^) ma''  ~ mo!^ + cma^ log(o!^)  2  a X ax^ Table 1.1: examples of Q M bound states  For example all the bound states in the hydrogen spectrum or the ground state of the square-well potential i n one dimension are analytic. The rest of the threshold bound states have binding energy that depends non-analytically on the coupling. A typical example is the e x p ( - ^ ) behavior of the binding energy of the square-well and PoschelTeller potentials in two dimensions. This classification will become important because we will have to somehow change variables in order to get a reliable perturbation expansion for the non-analytic threshold states. It may seem that threshold bound states are rare, however using bounds ( from the theory of trace class operators) on the operator |wp''^(-^ + aj'^v^^"^ where v^/^ means sign{v)\v\^^'^, Simon [24] proved that for one dimensional potentials that obey f dx\v{x)\{l + x^) < oo (bounded), and / dxe^v{x)  < oo for some R> 0 (short range)  and / dxv{x) < 0 (attractive), there exists exactly one threshold bound state (the parity even one). Moreover, it is not a coincidence that its energy is analytic in the coupling but rather a general feature. The ground state energy to third order in a is [24]: Eo =-^a^  (^Jdx v{x)^ -a^ JJdxdyv(x)\x-y\v{y)  Jdzv{z)+0(a^).  (1.2)  In [24] it was also proved that in two dimensions potentials that have the following general  behavior / cPx  < oo, / d'^x \v{x)\{l + x^y+^ < oo for some € > 0 and / d^x v{x) < 0  support at least one threshold bound state. However its binding energy is never analytic in the coupling. Nevertheless one can still define the following expansion for the ground state energy. oo  -  Yl  ClmnA'B  (1.3)  m,n=0,/=2  A  =  B  =  d  d e « d d —e " a  =  where C200 =  4 exp  -2  ( y d'^x K x ) )  j d^xv{^) log(|x - y|)Ky) + 2C  and C is Euler constant. In three dimensions it is well known that short range potentials do not have a threshold bound state. The ground state energy is non analytic in the coupling ~ (a — a^)^ so for small enough a there is no bound state. Long range potentials like coulomb,  (for  an infinite discrete set of values of a) and -a (j - j r f ^ ^ ) , where CQ and Ci are arbitrary, have a spectrum which is analytic in the coupling and therefore have many threshold bound states. It is interesting to note that the spectrum of confining potentials (e.g. a  log(3;)  or  ax^) is also "threshold" in the sense that it is continuous in the coupling. However, here it is difficult to define the threshold since the potential increases with x and the ground state energy is — oo. One can still talk about différences between energy levels since they are finite. A l l these diflîerences go to zero continuously as the coupling is reduced to zero although they are usually not analytic in the coupling. For example for power law potentials t> ~ r^T^, AE = a".  It is easy to find the analogs of threshold bound states in massive QFT's. The threshold here is just the n-particle threshold, or in the simplest case, the two particle threshold  = 4 M ^ where M is the mass of an elementary particle. Unfortunately there  are only a few examples of non trivial field theories where the full spectrum or even part of it is known exactly. In 1+1 dimensions we have: 1. the 0{2N) symmetric Gross-Neveu model  is the simplest four-Fermi interaction model, where [19] M„  =  M^?S) sm(|:)  forn  =  l...A^-l  (1.4)  and M is the physical mass of the elementary fermion. 2. the Thirring model jC = ï^ii^ - m)iP + ^{i^^.i^f is a theory of self interacting fermions. It was shown by Coleman [28] that this model is equivalent to the sine-Gordon theory of self interacting boson fields described by the lagrangian C=^d^(f>d'^<l>+~{cos{Pcf>)-l) where a has dimension mass and is related to the Thirring model fermion mass and ^ = ^ (|j — l ) - It was also shown using large N expansion [18] that this theory is equivalent to a U{1) gauge theory. The mass spectrum was calculated from the exact S-matrix [19] to be 71TT M„  =2Msin(2^^-^)  where re = 1,2...[1 +  (1.5)  and M is the physical mass of the elementary fermion. 3. and V[4>]2 theories  ^  where  ^  n=3  > 0. These models are superrenormalizable in 1+1 dimensions and for  A > 0 have no bound state. For A < 0 there exists one threshold bound state whose mass is infinitely differentiable in A. The leading term in this expansion is given by [31] ^2  = 2 M - ^ ^ + -..  .  (1.6)  where M is the physical boson mass. The expression for the binding energy of the lowest state in the hydrogen spectrum in three dimensions is known only approximately (i.e. from the perturbative solution of the BS equation [29]) Ebinding = - y  + aa^ + ba^ log(a) + ca^ log(a)2 + Oia')  (1.7)  where a, b, c are constants of order 1. Before we continue to deal with the field theory problems it seems instructive to check what happens in quantum mechanics when we try to imitate the field theory problem accurately. To do that one has to use Rayleigh-Schrodinger perturbation theory where the free Hamiltonian would be just Hq =  and the whole potential is treated as a  perturbation. Since Hq has only continuous spectrum one needs to define some infra-red regulator in order to be able to use perturbation theory. Wigner [3] looked at this problem for the three dimensional hydrogen atom. He tried regularizing by enclosing the system in a large sphere of radius R and taking the limit i? ^ oo at the end of the calculation. He found that the first order perturbation vanishes (as it should) while the second order one gives a finite but wrong result E2 = —0.1093»^ instead of — ^ . Even worse, if we go  to higher orders the result diverges in the limit R —> oo. Wigners explanation for this discrepancy is quite simple, the exact result has the form Eb — —^ + ce~°'^^. If we first expand in a we get in lowest order Eb = —{^ — c)a^ independent of R but higher orders diverge in the limit i? —> oo. To get the right result we have to take the limit in R first, but then we are back to continuous spectrum. Trees [25] found that for the coulomb potential there are at least two ways to avoid these e~"^ terms. One is to replace Hohy HQ- f as the unperturbed Hamiltonian and take the limit /? —»• 0 at the end of the calculation. The other method takes advantage of the hidden SO (4) symmetry of this potential and embeds the problem in a hypersphere of radius R. The Schrodinger equation then becomes 1 d 2 [R^sm\x) dx V""^  Ze2 dx j  R'^s\n\x) '\  R  cot(x)* = E *  (1.8)  where x is the angle of the stereographic projection from S^ to R^ and we replaced the coulomb potential by _ a ^ L £ (which becomes - - ^ in the limit r —^ 00). Binding energies are found by taking the limit i? -> 00 at the end of the calculation. In both cases one gets the exact result from the second order perturbation while all higher terms vanish. It is evidently seen that more sophisticated methods of resummation are needed for the Q F T case.  Chapter 2  The Method of Padé Approximants  It is generally very diiScult if not impossible to obtain information about bound states from the perturbative expansion. In contrast to the usual perturbation theory for scattering of "elementary" particles the calculation of the bound states characteristics "requires some artistic gifts" [8] even in Q E D . Available analytic methods include (for different theories): the nonrelativistic limit, truncations of the Bethe - Salpeter equation [27], variational [12], semiclassical [13] and adiabatic (Born - Oppenheimer) approximations 14] and the method of Padé approximants to the perturbative scattering amplitude [15]. In the The first three methods are not systematically improvable in the sense that they are not part of an approximation scheme which converges to the correct result. While the semiclassical or adiabatic approximations are in principle improvable, the expansion parameter (h or ^) is frequently not small and even leading order is extremely diflBcult to evaluate. The Padé approximants method based on a reliable perturbation theory is in principle an improvable and simple one and has the special advantage of explicit Lorentz (and gauge) invariance. It was used successfully some time ago in quantum mechanics and some quantum field theories [15, 16]. For threshold bound states we will show that it provides a systematic (in the sense of an expansion) method for calculating binding energies. The basic idea is to consider an expansion of the scattering amplitude in terms of a small parameter g A{s) = J2A,{s)gn=0  (2.9)  The [i/j] Padé approximant is a rational function ^''/•''(s) , N = i + j, of g A " « ( . ) = g|  (2.10)  where P*(s) = Y^]=QC^{s)g' , Q^(s) = Y,i=QD'{s)g^, such that its expansion in g to order N coincides with eq.(2.9). As i , j (or both) become large the results, converge to the correct one (the proof of convergence is valid under physically acceptable conditions, see discussion in [15],[17]). For example, for Coulomb, delta, square well and other quantum mechanical potentials the spectrum is very quickly reproduced [16]. The diagonal Padé approximants A'^-^'I have an additional merit of making the S - matrix unitary [15 . In relativistic quantum field theory the only models studied so far (in connection with phenomenological models of mesons and nucléons scattering) are 0^ and the Yukawa theory [15]. It was however very difl&cult to estimate the validity of the approximation for the first few Padé approximants in these models because one could not compare them with solid results obtained by other means. Meanwhile exact S - matrices of certain 1+1 quantum field theories like the sine-Gordon Thirring model (TM) , various cr-models and the Gross - Neveu (GN) model were found [19]. In the T M and the G N model a rich spectrum of bound states as well as a reliable expansion ( 1 / N in 0(2N) symmetric G N model and weak coupling in T M ) exists [17, 23]. It became possible to test the Padé approximants method on bound states of such a model. In what follows we have two goals. First, we show that already the [1/1] Padé approximant works well in the one dimensional quantum mechanical models with weak binding. Then we calculate the meson binding energies in 1+1 dimensional T M and G N models using weak coupling and 1 / N expansion respectively and compare them with the exact ones. We also speculate on the applicability of the method to more realistic theories.  2.1  Quantum Mechanics  We first show that the [1/1] Padé approximant gives precisely the expression eq.(1.2). consider the Hamiltonian (we take m = l ) 1 2dx^  (2.11)  — 0iv{x) .  The starting point for a calculation of the binding energy using Padé approximants is the perturbative (Born) expansion of the S-matrix. <k'|S'|k>  =  6kk  < k'|r|k >  =  Q; < k>|k >  (2.12)  r^6(A; - A;') < k'|T|k > < k'\v{x)G{x - y)v{y)\k >  < k'\v(x)G{x - z)v(z)G{z - y)v{y)\k > + • where  çik\x-y\ 2  '  2  2k  is the free Green's function. In order to define the appropriate channel we note that the asymptotic scattering states invariant. ip+ >=  >=  and |V'2 >= ^^=- are not parity (reflection)  The parity eigenstates can be chosen as linear combinations of these two  -^ilfpi  >  +|V'2 >) and IV'- >= :^(|'0i > -|'02 >)• Since the lowest bound state  has even parity we should expand (2.13) where Sr{k)  =<  " ^ i l ^ j ^ i > and St{k)  =< ip2\S\ip2 > are the reflection and transmission  amplitudes. To first two orders in a one gets; Sevenik) = 1 + aSi(k)  +  S2{k)  +  (2.14)  where Si{k)  =  s^ik)  =  -'-jdx{l  (2.15)  + e'''^'')v{x)  j jdxdy{é^^^+y^ + é^^y-'''^)v{x)é^\''-y\v{y)  The [1/1] approximant is given by  and the binding energy is calculated by solving perturbatively for the zero of the denominator. Expanding the exponentials in Si{k) and S2{k) in eq.(2.16) we get the following equation for the location of this zero _ ia { —[/dx v{x)\^ — ik\JJdxdy v{x)\x — y\v{y) + 2Jdx v{x)x J dyv{y)] + • • •} / dx v{x) + 2ik J dx v{x)x H (2.17) Substituting k = i{aa + ba^ -\  The energy  ) in eq.(2.17) and expanding in a we find  a =-  jdx v{x) ,  (2.18)  6 =—  J j dxdy v{x)\x — y\v{y) .  = Y ~ -\o^oP' — aha^ coincides with eq.(1.2).  The situation in two dimensions is a little bit different. The free Green's function  i.e. it depends non analytically on the energy E . However, the Padé approximants method still works. We write the first few terms in the Born series for the scattering matrix in the / = 0 channel S{k)  =  l + 2TraJ d^xe'^''v{x) +27ra'J Jd2^d2^e'k-^;(x)i^[log(l|l(x  (2.20) - y)^) + 2C]v{y) + 0{a^)  again since we anticipate the binding energy to be (exponentially) small we can expand the exponentials in 2.21 to get S{k) 5*1 (fc) S^ik)  =  (2.21)  i + aSi{k) + a^S2{k) + ---  =  ira J (fxv{x)  =  jdVyv{x)[log{^-^iK-yf)+2C]viy)  +  0{a').  Constructing the [1 /1] Fade approximant we find that the location of the pole is determined by 1 +  «  27r  / /  d^xd^yv{x)[\og{f) + 2\og |x - y| + 2C]v{y) ^ ^ J d'^xv{x)  ^2.22)  solving for E we get exactly the first term in the expansion 1.4. To conclude, in quantum mechanics of short range potentials the leading term in the binding energy is determined by the [1/1] Fade approximant. The [0/1] non unitary Fade approximant misses the first term in the expansion by a factor of 2, while higher, diagonal approximants increase the accuracy. Although the arguments leading to the expansion eq.(1.2) are purely non-relativistic [24] the Fade approximation has a natural Q F T analog.  2.2  l - h l dimensional Thirring Model  It is remarkable that in Q F T for which the exact spectrum is known (like T M and GN) binding energies happen to be analytic in small coupling. The Thirring model is defined by C = ^(z^ - m)V + | ( ^ 7 M ^ ) '  (2.23)  For sufficiently small coupling there exists only one bound state of fermion-antifermion. The mass [13] of this bound state and the mass of an "elementary fermion" are Mb =  2Mfsin( i  (2.24)  , ^ ^2(1 +  22  Ree  -3Jtl  -2n:  2B:  3JI!  Im e  Figure 2.2: The physical sheet in terms of rapidity variables Mf  =  —(1 + —) TT TT  .  The binding energy is therefore 4p3 AE = 2Mf - Mfe = M / ( / - ^ + 0 ( / ) ) TT  (2.25)  .  Motivated by the previous result we calculate it by the Padé method. In order to form the 1/1] Padé approximant we need the perturbative expansion of the fermion-anti-fermion scattering amplitude. Let us consider i n detail fermion-(anti) fermion scattering in 1+1 dimensions. Energymomentum conservation P i + P2 = P i + p2 and the mass-shell condition  = p| =  =  Pi — P2 iniply tliat only forward (pi = P i , P2 = P2) and (for distinguishable particles) backward (pi = p2, P2 = Pi) scattering is possible. The S-matrix acting on a 2-particle state by <Pi,7;P2,<5|«5|pi,a;p2,/? =  y  s-4m^  ^^^^^ ~  Pi^'^^P^ -  (27r)H(pi + P2 - Pi - P2)',sAs)ai3 P2) -  <5(Pi -  P2)<^(P2 -  (2.26)  Pi)]^s As)a/3  where a, (3, 7 and 8 are sets of quantum numbers describing the incoming (outgoing) particles and A{s) (where s =pi +P2) is the invariant amplitude'^. For simplifying further * n o t e t h a t i n 1+1 t h e r e is o n l y one i n d e p e n d e n t M a n d e l s t a m v a r i a b l e  discussion we introduce the complex rapidity variable 6 defined by cosh(^) =  .  (2.27)  The conformai mapping eq.(2.27) is one to one between the infinitely many sheeted complex s-plane and the complex 9 plain. The physical sheet is required to be mapped into the strip 0 < Im^ < TT (see fig.(2.2)). A l l other sheets are mapped into similar strips above and below the physical strip. For future convenience we shall assume that the S-matrix is symmetric under 0{N) (for T M N=2) and the fermions belong to an N-plet of that symmetry. We choose to work with real fermions i.e. N-even. The invariant amplitude can be written as  where (J2,  are the transmission and reflection amplitudes respectively and oi describes  annihilation type processes. These amplitudes, however, do not have definite transformation properties under 0{N). We can construct such amplitudes from linear combinations of the cTj by using the following projection operators: Visa =  ^^apà-y6  (2.29)  The resulting amplitudes are a(d)uo = ^iO)a.s =  Na^ie) + a2{e) + asie)  (2.30)  O2{0)-az{e)  symm In order to describe fermion-antifermion scattering we should pass to complex representation of the fermion fields. For N = 2 (the T M ) this means defining tp{x) =  'ipi{x) + irp2(x)  and  = tpiÇx) — ixp2{x). Rewriting the S-matrix in terms of these fields  we find the following relations between the old and new amplitudes S{e) = a2{e) + aaie)  Srie) = a,{e) + a^{e)  St{e) = ai{e) + a^iO) .  (2.31)  where St{d) and Sr{9) are transition and reflection amplitudes for the fermion antifermion scattering and S{6) is the scattering amplitude for identical fermions.  These  amplitudes are not invariant under charge conjugation (C-parity). The odd C-parity amplitude is S{9)odd = k{St{d) — ST(9)).  The transmission and reflection scattering  amplitudes 5^(0) , St{e) are [20] + 0(9') _  . sin(7r(l + 2g/ir))  (2.32)  Near the threshold S{6)odd has the following behavior: S{e)odd = 1 + 9  J  6  +  (2.33)  The unitary [1/1] Padé approximant to S{6)odd can now be formed 6 Vz n r) ^ 6 ^ 6 The location of the pole i n the denominator is given by  (2.34)  Aie = -8g{l + —) (2.35) 7f substituting 9 = i{ag + bg^) and solving perturbatively for a and b we get 9 = —i{2g — ^)  + 0{g^). Plugging this expression for 9 into eq.(2.27) and expanding in powers of g  we get that the pole is at s = Mf{4 - a^g^ - 2abg^) + 0 ( / ) = 4 M 2 ( 1 - / + 1^3) ^ ^ ( ^ 4 )  (2.36)  which leads to an expression for the binding energy that is identical to eq.(2.25) to order  2.3  1+1 dimensional Gross-Neveu model  The asymptotically free G N model 2 C = ^i^^ + ^{Uf  (2.37)  was introduced i n [23] and its exact S-matrix was calculated in [19]. Bound states appear in the isoscalar and anti-symmetric tensor channels and have degenerate mass spectrum: M„ = where n = 1,2 • • • <  (2.38)  and M is the dynamically generated mass of the fermion. This  spectrum is analytic in 1/A'' so for example M2  = M  '  7r2  47r2  1  (2.39)  It is convenient to rewrite the lagrangian eq.(2.37) in terms of 0,^/» and an auxiliary field a ~  in order to facilitate the l/N expansion. £ = Vii^V + <^î>'^ - ^cP-  (2.40)  The isoscalar mesonic bound state is interpolated by the field a. In leading order the o propagator [39] = ~  (2.41)  does not contain a pole. A l l we see is just a branch cut at p"^ = 4M^. This cannot be taken as evidence for the existence of a pole as was claimed in the original paper [23]. It is often said, however, that within the 1/N expansion the leading order already captures the isoscalar bound state because the leading order propagator or the four point Greens function contains a pole that coincides with the two particle threshold. It is assumed that higher order corrections will separate this pole from the cut ^. ^ A n a t t e m p t t o e x p l i c i t l y c a l c u l a t e h i g h e r o r d e r corrections by S c h o n f e l d [21] d i d n ' t clarify this p o i n t . See d i s c u s s i o n i n a p p e n d i x C  Quite generally, one cannot find singularities other than cuts (on the physical sheet) in Feynman diagrams therefore we will consider Padé approximants of the first few orders of the 1/N expansion. There is a close relation between the invariant amplitude in the isoscalar and antisymmetric-tensor channels ^. Sisoscaiar{9) = —-. -^Santisym{e) ITT — a  (2.42)  SO we concentrate on the antisymmetric one. The S-matrix element Santisym{9) is SantisymiO) "  1+ ^ 27r2 ^2  (sinh(^) ^ 27r2  +  (1  1 N^ 2  \  i9. TT  47r2  \  sinh(^) /  9  TT^  sinh(^)2 ^ cosh(|)2  1  9 sinh(^)  ^À9\ TT J  ',Tr-i9. TT  (2.43) where $ is the digamma function ^. The invariant amplitude expanded around the two body threshold ^ = 0 to order 1 /A'^2 ig. SanUsyr.i9) = 1  1^(1 - ^ + • • •) _ ^ ( ^  + iô +  + . . .) + O ( ^ )  (2.44)  from which we can form the [1/1] Padé approximant. 1 [1/1]  /  ^antisymi^)  i_/-8+8tfl\ OTTATV  = T~T  ;  /  ^_8-8^^e^"  (2.45)  Solving perturbatively for the location of the pole in eq.(2.45) we get eq.(2.39). The same pole exists in the isoscalar amplitude as can be seen from eq.(2.42). Conclusions The unitary [1/1] Padé approximant gives correctly the binding energy of mesons in the G N and Thirring models to third order in the coupling. We don't have a rigorous ^ T h i s is consequence of t h e existence of conserved charges w i t h n o n l i n e a r m o m e n t u m dependence i n this model ^ T h e first t w o n o n t r i v i a l orders i n t h e e x p a n s i o n of t h i s a m p l i t u d e were c a l c u l a t e d [22] a n d agree w i t h t h e exact r e s u l t  proof that this is always the case but these examples indicate that it might be the case. We expect that higher diagonal approximants will increase the accuracy. One might think that the method works only with analytic binding energies and therefore is of limited importance. However as we have seen this class is surprisingly large. It contains probably all the loosely bound states in 1+1 dimensions produced by short range forces.  In higher dimensions the binding energy (or a simple function  of it) is analytic in the coupling [24, 26]. B. Simon showed this for two dimensional short range potentials in Q M while Wigner and Trees [25] calculated binding energies of long range coulomb type potential in 3 dimensions using perturbation theory. The Padé method is presumably applicable in 1+1 Q F T where the exact S-matrix is not known like scalar and Yukawa theories and in higher dimensions [15] although some modifictions may be required. The advantage of the method compared to the conventional truncated Bethe-Salpeter equation approach is that it is Lorenz invariant (and gauge invariant for gauge theories) and therefore much simpler. There is no need to divide the interaction to instantaneous and retarded parts.  Chapter 3  Binding Energies in Weakly Coupled 2+1 Dimensional Q F T ' s  The general framework for calculating bound state characteristics in relativistic Q F T is the Bethe-Salpeter (BS) equation [27]. It is impossible however to extract any information from this integral equation unless a systematic method for calculating its kernel is available. In some theories like Q E D , weak coupling perturbation theory is such a method. One hopes that the kernel calculated perturbatively will give a systematic expansion for bound state characteristics like binding energies. One doesn't get usually an expansion of say, binding energy since the BS equation is nonlinear and does not, in general, have an analytic solution as a function of the coupling. For example, the expression for the positronium hyperfine splitting [29] is AJE; =  + i - ^(iog(2)+ f)  +  i^g(^) + . .  Sometimes, however, one finds that certain quantities are i n fact expandable and therefore can be recovered from the above approximation. In many 1+1 dimensional Q F T ' s it turns out [30, 31] that the binding energy of the lowest bound state is expandable in the coupling. However the kinematics of scattering in 1+1 is very restrictive and (from the 3+1 dimensional point of view) unrealistic. One would like to find more physical examples, yet still simple, where bound state characteristics can be recovered from the weak coupling or some other expansion. In what follows we show that the perturbative expansion of the scattering amplitude provides a systematic method for calculation of binding energies for a large class  of theories in 2+1 dimensions. We demonstrate this by calculating the leading order contribution to the binding energy of threshold bound states for scalar ^ and fermionic theories i n 2+1 dimensions. We compare our results with previous calculations using variational approximation. 3.1  Scalar theories  To demonstrate the method we now calculate the lowest bound state (s-wave) mass i n the theory C = ^d^(t>d^<f> + i m V ' - A<^^ + (cfP .  (3.46)  The signs of the (f>^ and the cf)^ terms are arranged in such a way that there exists an attractive interaction that creats a bound state without ruining the boundedness from below of the potential. This theory was studied in 1+1 dimensions using BS equation [31], variational approximation [12] and potential model [32, 33] and was shown to contain one threshold bound state. We start with a brief review of the BS formalism. Let us define kernels  where  Do{xi,X2,X3,X4)  =  D{Xi,X2,X3,X4)  =  G2(x,y)  and  ^4(0;,  G2{XI,X3)G2(X2,X4)+  G2(XUX4)G2(X2,X3)  (3.47)  G4{xi,X2,X3,X4)-G2{Xi,X2)G2ix3,X4)  y, 2, s) are the full 2 and 4 point functions respectively. The  BS kernel K is defined (in matrix notation) by D = Do-  DQKD  (3.48)  and can be calculated perturbatively as the sum of all two particle irreducible (2PI) graphs. A graph is called 2PI if after removing up to two internal lines from it, each ^for s c a l a r theories i n 2 + 1 d i m e n s i o n s this question was a l r e a d y considered ( i n t h e framework of a x i o m a t i c field t h e o r y ) b y B r o s a n d l a g o l n i t z e r [30]. H e r e we c o m p l e m e n t t h e i r general t r e a t m e n t b y a n explicit calculation  connected component of the reminder of the graph must contain at least one " i n " leg and at least one "out" leg. It is convenient to use the translational invariance of the theory to introduce relative coordinates R = ^{xi + X2 — x^ — 2:4), Vi = xi — X2, Vf = X3 — X4 and Fourier transform all functions to momentum space: Do{P,qi,qf)  D{P.Mî) where DQ{P,qi,qj)  =  j^JdRdridrfe'^''^+^^^'+^f'-f^Do(R,ri,rf)  =  2Do{P,qi)6iqi + qf)  (3.49)  -^JdRdudrfe'^''^-''^-^^+^f^^^DiR,r,,rj)  =  — (52(|P + qi)G2{^P — qi). From now on we work in the frame of  reference where P = 0. functions of s = P^, Çio,  In this frame in 2+1 dimensions D{P,qi,qf),  K{P,qi,qf)  are  ; Qfo, Qf and cos(^), 9 being the angle between qj and qf ^ .  Since the rotation group is 0(2) partial wave decomposition is much simpler than that in 3+1 dimensions and is reduced to the Fourier series with respect to 9. Eq.(3.48) can be rewritten as a set of decoupled equations for each partial wave D'iP, Qi, Qf) = Do{P, qi, qf) - Jd^s dHDç,{P, qi, S)K\P, where  is defined by  S, t)D\P, t, qf)  (3.50)  = ^ J^^ de e'^^D.  Before we continue with the calculation we discuss here the relation between bound states and poles of Greens functions. It is easier to exemplify this relation in a quantum mechanics. Consider the correlator < TO{0)O{t) >= 9{t) < O{0)Oit) > +9{-t) < O{t)O{0) > of an arbitrary operator 0{t).  Using the time translation operator e'^* where H is the  hamiltonian of the system, and inserting a complete set |n > of eigenfunctions of the ^We have a l r e a d y t h e L o r e n t z a n d r o t a t i o n a l i n v a r i a n c e  connected component of the reminder of the graph must contain at least one " i n " leg and at least one "out" leg. It is convenient to use the translational invariance of the theory to introduce relative coordinates R = \(xi + X2 —  ~ X4), Vi = xi — X2, rf = X3 — X4  and Fourier transform all functions to momentum space: Do{P,qi,qf)  D{P,qi,qf) where Do{P,qi,qj)  =  j^JdRdridrfe'^''''-^^'^'^^f^f^Do{R,ri,rf)  =  2 D o ( P , Ç i ) % + 9/)  =  (3.49)  -^JdRdridrfe'^^''+^^^'+^f^f^DiR,ri,rf)  = ^2(5^ + qi)G2{^P — qi). From now on we work in the frame of  reference where P = 0. In this frame in 2+1 dimensions D{P,qi,qf),  K{P,qi,qf)  are  functions of s = P^, Çjo, qf, Ç/o? <lf and cos(^), 6 being the angle between qi and qf ^ . Since the rotation group is 0(2) partial wave decomposition is much simpler than that in 3+1 dimensions and is reduced to the Fourier series with respect to 6. Eq.(3.48) can be rewritten as a set of decoupled equations for each partial wave Jd''sdHDo{P,qi,s)K\P,s,t)D\P,t,qf)  D'{P,qi,qf}^D,{P,qi,qf)where  (3.50)  is defined by D ' = ^ J^"" d9 e''^D.  Before we continue with the calculation we discuss here the relation between bound states and poles of Greens functions. It is easier to exemplify this relation in a quantum mechanics. Consider the correlator  < ro{o)o{t) >= e{t) < o{o)o{t) > +e{-t) < c>(i)c(o) > of an arbitrary operator 0{t).  Using the time translation operator e*^* where H is the  hamiltonian of the system, and inserting a complete set |n > of eigenfunctions of the ^ W e have a l r e a d y t h e L o r e n t z a n d r o t a t i o n a l i n v a r i a n c e  hamiltonian we can rewrite this in the form:  <roioMt)>^ll^  ^  ,^,<0|(g(Q)|n ><n|O(0)|0>  L„  E-En  + ie  where the sum over n includes integrals over indices corresponding to continuous spectrum and we used the integral representation of the 6 function. Fourier transforming this correlator we obtain the spectral representation  / e - < O(0m)  >= E < ° l " ( ° ) l " > < " l f ) l ° > - (e - , - 0 . E -  Jt  En~ie  It is therefore seen that i f < n|C(0)|0 > is non zero, an energy eigenvalue of a discrete state will contribute to a pole in the correlator. This result can be generalised to field theory, for a good discussion of the subtleties involved see [31]. The four point function D{x,x,y,y)  is related to the propagator of the field D{x,x,y,y)  =< <t>\x)(t)\y) >-<  (t>\x) >< (l)\y) >  we thus expect that any bound states with quantum numbers of (fy^ contributes to a pole in D We now return to the calculation. From eq.(3.48) it is seen that the inverse of the full four point function D is proportional to the operator 1 + KDQ. This means that (after transforming all operators to p space) the bound state spectrum in the neighborhood of the threshold 2m is given by the values of PQ for which the operator K'DQ{P)  has  eigenvalue -1. In lowest order we take (fig.(3.3)) Do(P,quqf)  =  r/„ P^2 +, m ^ 22 1 (?. +, f)2 ] [^ ( g , - fP^TTZ:^ ) 2 + m2  K%P,q,,qf)  =  -6\n.  (3-51)  where XRÏS the renormalized quartic coupling, higher orders will be discussed later. The  eigenvalue equation is in this approximation: J-^3K'D,{P,q)x{q)  =  (3.52)  -x{p)  from which it is seen that the only eigenfunctions x are constants. The unique eigenvalue is determined by l = ? M î 2 ^ .  (3.53)  We solve this equation perturbatively by substituting P = 2m - AE into eq.(3.53). 2m-AE 2m  = tanh  7r(2m-AE)  ~l-2e  ir(2m-AE) 3AR  (3.54)  The inverse logarithm of the binding energy AE = 2m - M g can be found perturbatively from eq.(3.53). Exponentiating the result we get AE = 4me  + • • •.  (3.55)  The analytic behavior of the binding energy AE obtained here is in accordance with the general theorem of [30]. Note that the pre-exponential factor in AE is twice that obtained using gaussian variational approximation [12]. We do not know to what effect within this approximation to attribute this discrepancy. A similar gaussian calculation in 1+1 dimensions [12] shows no discrepancy with BS results found in [31 . In 1+1 dimensions it was shown in [32, 33] that the potential model one obtains in the nonrelativistic limit gives the correct leading behavior of the binding energy. If we try to take the same limit in the 2+1 dimensional theory we arrive at a two dimensional 6(x.) potential. This potential produces fall to the center in 2D (unless one uses a renormalization procedure similar to the one used in Q F T [34]). The potential model based on it therefore fails.  In our calculation we used  and DQ calculated to leading order. One would like to  know whether higher orders will alter this result. This could happen if higher order terms in K^DQ contained mass shell singularities. To illustrate this point suppose we have a diagram whose contribution to K^DQ is of the form ~  . ^"  If the binding energy is  the order of this contribution is g^~^. This characteristic mixture of orders needs  to be addressed in any kind of perturbative expansion of the BS equation. We show in the appendix that for massive scalar theories in 2+1 dimensions this does not happen.  3.2  Fermionic theories  Bound states frequently appear in theories containing fermions. The case of gauge theories, although interesting, cannot be treated reliably with perturbation theories due to confinement^. We would like to consider here weakly coupled theories. The natural candidates are 4-Fermi and Yukawa theories (Although 4-Fermi theory is not renormalizable in weak coupling perturbation theory it was recently shown to be renormalizable within the 1/N expansion [37]). Let us note first the general features peculiar to fermions. The BS kernel is as before eq.(3.48) but all operators are now matrices carrying flavour (color) indices in addition to the Lorentz indices. The operator DQ is now  where Sp is the full fermion propagator, while K°{P,qi,qf)ap^s is the sum of all 2PI connected graphs. However, the formal series defined by the eq.(3.48) diverges already in leading order because of the U V behavior of Z^o- A method to solve this problem was suggested in [19]. ^ A b e l i a n gauge theories like Q E D i n 2+1 are c o n f i n i n g  As a warmup to the treatment of the "matrix" BS equation we first calculate the lowest bound state energy in the massive 1-1-1 dimensional Thirring model in weak coupling perturbation theory ^  Sp and K are now matrices. To lowest order in g we take K%P,  qiAs)aMS = 25  [(7/.)a/3(7M)7«  - (7M)a6(7/.)/3T]  (see fig.(3.4)) and Sp is just the free propagator. The BS amplitude X{PA)  (3-56) is a 2 X 2  matrix, i n order to separate the different channels we expand it in terms of a basis of matrices that have simple transformation properties under the Lorentz group X =  $ 5 ! + $P75 +  ^vln-  (3-57)  For renormalizable theories the series eq(3.48) is not well defined. This is because eq.(3.48) is essentially a rearrangement of the graph expansion of the four point function , thus the bad large momentum behavior of the integrals cannot be avoided without modifications. The axiomatic (but practically impossible) way of remedying this is by modifying Do i.e. DQ ^  DQ{P,qi)x{\P  + qi)x{\P  change the analytic properties since  - Qi) where x{k) = e-''(^'-^'). This does not  e~P^''^~>^^^  is equal to one on the mass shell. A more  practical (at least in lowest orders) consistent way of modifying the kernel was suggested by Bros and Docoumet [36]. Their method uses the B.P.H.Z [35] procedure of expanding the integrand in a power series in P. It was shown in [36] that this method preserves the analytic properties of the four point function and in particular leads to a unitary scattering amplitude. ^In 1 + 1 d i m e n s i o n s this m o d e l is equivalent b y a F i e r z t r a n s f o r m a t i o n to the massive G r o s s - N e v e u model C — — m)ip + g['ipijj)'^  Plugging the expansion eq.(3.57) into eq.(3.56) and equating coefficients we get : d'^q  where /^^^ means subtraction at P=0. As can be seen from eq.(3.58) the only eigenfunctions are constant functions. Eq.(3.58) can be written as a set of three (in general coupled) equations $5  =  2g  yjAw?- —  P2arcsin(^)  m  (3.59)  TT  2pFarcsin(^) 7rV4m2 -  -f-  P2  4pmarcsin(^) P7rV4m2 -  P2  2ff(p2$^-P/-p.$y)arcsin(£)  V'4m2 - p 2 a r c s i n ( ^ )  m  P7rV4m2 - P2  TTP  TT  4gmavcsin(£^) 2m-' p^ffk P 7 r V 4 m 2 - P 2 ' 'eu:.P ''' $P where we have already rotated P back to Minkowski space.  The lowest bound state  appears in the P (V) channel. The location of the pole is given by the solution of the following algebraic equation: 1 =  2ffParcsin(^) 7rV'4m2 -  (3.60)  P2  Expanding p 2 = 4 m 2 — cg"^ and plugging this back into eq.(3.60) we get in leading order M l  =  P2 =  4m2 -  4^2  m  which agrees with the exact result [33, 19 . We now turn to the 0(2N) symmetric Gross-Neveu model (3.61) This model was recently shown to be a renormalizable field theory in 2-j-l dimensions within the 1 / N expansion [42]. In l - f - l dimensions the spectrum of this model is known  from Zamolodchikovs exact S-matrix [19].  For large enough N it contains only two  degenerate 2-particle bound states in the 0(2N) isoscalar and antisymmetric channels. In 2-t-l dimensions no such solution exists and there are no numerical simulations to compare to.  It is reasonable to assume that attraction in the antisymmetric tensor  channel results in formation of a threshold bound state. We start by calculating the kernel K{P,p^qflf^f  which is as before the sum of all 2PI graphs. In order to facilitate  the l / i V expansion we shall use the auxiliary field formalism. The 2PI in this language means 2PI with respect to the fermionic lines. In lowest order this means (see fig.(3.5)): K{P, qi. qft';\s = ^ [SapS.sô'^ô'^SBiP) + è^,èps8'H^'S{qi + q^) - ô^e^^S^'^^Se^qi - qf) (3.62) where Ssiq) = (4m2-g2)^tInh-'(^)  ^ meson propagator [42]. We now expand the t  and u channel vertices around ç = 0 to get an efi"ective four-Fermi interaction. In order to separate the different channels we use the projection operators eq.(2.30) and write Hq)% 27r  L  = X^^iq)Viso + xU{q)Pa.s + xlTiq)'Psym.  Plugging $ back into eq.(3.52) we get  "^^^^ + ^ ^ ) [ ^ ^ ^ * K x ( g ) i . o ) + Xiqho + x{q)lo]SF{q - \p) 27r N  l^/Fiq + lPMq)a.s  + x(q)L]SF{q-lp)  ~N isub ^^^^ 2'^^^^^^^'^"' ~ Xiq)Jym]SF{q -  =  KPUxiph  = X{P)a.sXip)a.. = KP)symX{p)sym (3.63)  where A(P)s are 2 x 2 matrices and  is the transposed matrix. In lowest order the  contribution to K from the symmetric channel vanishes while the contribution of the isoscalar channel mixes orders in 1/N. For simplicity we calculate leading term in the binding energy of the lowest bound state in the antisymmetric channel. The eigenfunctions of eq.(3.63) are again in lowest order constant functions, so the location of the pole can be easily determined. The eigenvalue equation in this channel is similar to eq.(3.53)  when we replace g by 2'K/N: 1 =  N  47rP  (3.64)  from which we get A£^ = 4m e~^^ H discussion In this chapter we have used the perturbative solution of the BS equation to find the leading term in the expansion of the binding energy of the lowest bound state in the nonperturbatively renormalizable 0{2N)  symmetric Gross-Neveu model in 2+1 dimen-  sions We have shown that for scalar and fermionic renormalizable massive theories i n 2+1 dimensions this method is systematic so that our leading order result is the exact leading term. Our solution for —\<j)^ + C4>^ differs from the variational one [12] by a factor of 2. It should be noted that one may hope to capture only threshold bound states i.e. those states that persist in the spectrum for arbitrarily small value of the coupling (a famous example being the hydrogen and positronium spectrum in 3-1-1 dimensional Q E D ) . This was demonstrated for weakly coupled theories {(ffi) and theories where the expansion parameter is external like 1 /N. Nonthreshold bound states (like all the bound states i n 3+1 dimensional Yukawa model) disappear from the spectrum when the coupling reaches a finite critical value and therefore cannot in principle be recovered from the expansion.  Figure 3.3: The lowest order diagram contributing to A'°.in the scalar theory eq.(l)  Figure 3.4: The lowest order diagrams contributing to in the massive Thirring model. The dashed line represents a fictitious photon with propagator 1.  Figure 3.5: The lowest order diagrams contributing to K° in 0(2N) symmetric Gross-Neveu model. The wiggly line represents the a propagator.  Part I  Exotic Bound States Of Skyrmyons  Chapter 4  A short introduction to the Skyrme model  In the previous chapters an attempt was made to find systematic and rigorous methods for calculating binding energies. These methods relied on a perturbative expansion in some small parameter, although such expansions exist also for Q C D the computational difficulty in calculating even leading order prohibits their use (so-far). In the mean time it is still possible to extract information on the low energy behavior of Q C D by using as much as possible the experimental information and the constraints imposed by the symmetries of the model. The Skyrme model is an attempt to do just that. The Skyrme model was suggested (1960) as a non linear field theory that describes strongly interacting particles. Within this model mesons are treated as the elementary fields 7r(r) and baryons emerge as solitons whose winding number is identified with baryon number. It took many years to partly justify this approach. The most convincing justification comes from the large Nc limit of Q.C.D (A^^ is the number of colors). It was first pointed out by t'Hooft [53] that 1/Nc is a good (although not so obvious) expansion parameter. In the limit Nc —>• oo, with g'^Nc fixed, Q.C.D simplifies considerably. A l l mesons constructed of qq pairs become stable non interacting particles (meson-meson scattering cross sections behave as 1/iV?), meson masses behave as N^, Zweig rule, which states that processes like cj) are supressed compared to (f)  ir'^  TT" involving three gluon exchange  K~ involving one gluon exchange, becomes exact  and there are no exotics (like qqqq). Witten [55] showed that baryon masses behave in this limit as N^ that is l/{l/Nc) where l/Nc is the expansion parameter. He also showed  that Nc scales out of all equations which determine baryon size and shape, baryon-baryon or baryon-meson scattering. These features are shared also by special solutions, called solitons, in nonlinear field theories such as the Skyrme model. Thus Witten's results suggest that baryons may be regarded as solitons in an effective meson theory. Since there is (so-far) no derivation of the large Nc limit of Q.C.D [69], we are left to construct the effective field theory on the basis of symmetry principles. A s a result the effective theory will contain coupling constants that should in principle be calculated from large Nc Q.C.D. The physical degrees of freedom in the Skyrme model are the Goldstone bosons (pions). In order to construct an effective theory for the Goldstone bosons one considers them as mappings from 4 dimensional space time to the energy degeneracy manifold. For Np flavours the target space is ^ ( ^ ^ f f l ^ V ^ ( ^ ) - The pion field is described by an Np x A^^ matrix U{x,t) of determinant 1. Following Skyrme and including also the Wess-Zumino term we will assume that its dynamics is governed by the action ^ : S(U)  =  Ssk{U)  =  S^.iU)  =  S,km  + S^.{U)  (4.65)  ld'x^Tr{R,Rn-~Tr{[R„R^]') -^rj;^  / d^xe'''''"'^Tr{R^'R''R''R''R'^)  where R^ = Wd^U, F^r ~ 186 Mev, e=5.45 and D is a five dimensional disk D whose boundary is the four dimensional compactified space time. The Wess-Zumino term does not depend on the metric and therefore does not contribute to the energy momentum tensor and especially to the energy functional (although it will change the equation of ^In general there are three t e r m s t h a t are c h i r a l l y s y m m e t r i c a n d less t h a n order four i n t i m e d e r i v a tives. F o r S U ( 2 ) one c a n show t h a t t h e y a l l reduce t o the S k y r m e t e r m , however for S U ( 3 ) there is a n o t h e r t e r m [70] Ua ~ tx{Rf,R^)tx{WW) - \.x{Rf,W)ii{RyBy). T h i s w i l l not affect t h e s p h e r i c a l l y s y m m e t r i c c o n f i g u r a t i o n w i t h m i n i m a l energy because the c o n t r i b u t i o n of t h i s t e r m to the energy of s u c h c o n f i g u r a t i o n s vanishes  motion) E(U)  =  -j  éx[j^Tr{Ili)  +  (4.66)  j^Tr(lRo,R,?)  Finiteness of the energy functional requires that U approaches a constant Uoo when r —> oo.  Uoo can be reduced by chiral transformations to 1, so that U{x,t) —> 1 as  r —> oo (for fixed t). In other words, requiring finite energy implies that a point in spatial infinity maps into a fixed element in the field manifold Uoo- This means that has been compactified to i?^ U oo which is topologically S^. Such mappings are classified by the third homotopy group U.3{SU{NF))  —  Z  {Np > 1)  or equivalently by integer valued winding number:  ^^^)=2^'"V which is a topological invariant.  (4.67)  '^''^TriRiR^Rk)  Witten [68] showed that this winding number can  be identified with the baryon number that is defined in Q.C.D. The winding number refiects the boundary conditions in the problem. A configuration with a given winding number cannot be continuously deformed into one with different winding number. The homotopy class of a given configuration cannot therefore be changed by its hamiltonian time development; B(U) is a constant of the motion. The fact that the configuration space is split into sectors makes the problem of finding the ground state more complicated since we have to vary our solution within a given B sector to make sure it is a minimum. For skyrmions a lower bound on the energy exists. It is derived from eq.(4.66) by completing it to a square E{U)  = =  -^Jd'xTr RiR' + 3Tr^B(U)F^/e .  (-^€ijkRj,Rky  er Tf  >^Jd^x  \€i,kRiRjRk\ (4.68)  This is very similar to the Bogomolny [59] bound one gets for the Yang-Mills instantons action. However for skyrmions this bound cannot be satisfied unless B = 0. This is because the right current i?^ trivially satisfies the Maurer-Cartan equation: d^R^ - d^R^ + [R^,,R,] = 0  (4.69)  while the saturation condition is (4.70)  Ri = -^[Ri,Rj]  It is easy to show, by diagonalizing Ri, that the only solution satisfying both equations is U = const. The condition needed for saturating the bound actually means that U{x) the Skyrme field is an isometry between space-time and the group manifold. In particular for SU(2) the map that saturates the bound is the identity map. This approach was pursued by Manton et al [71] who showed that one can get approximate solutions using the identity map.  In the next section we will use a different approach starting from  instantons to derive approximate analytic solutions for the equation of motion of the skyrmions. For a given baryon number a static solution UQ, when it exists, is expected to have maximal symmetry [57]. A static classical configuration at rest (with its center of mass at the origin) may be acted upon by the internal symmetry group SU(N)i  x  SU{N)ii,  producing another solution. Static solutions are not changed by time translations while space translations change the center of mass location. Lorentz boosts set the solution in motion thereby changing the energy. Only the spatial rotation group 50(3) may produce a solution at rest at the origin. The group SU(N)L  X  SU(N)H  acts on the static solution UQ by UQ  AUQB^  produc-  ing a new solution. Since UQ has finite energy and we chose (7o(oo) = 1, f/o(oo) is invariant only under the diagonal subgroup SU{N)  i.e. the pairs {A, A) where A  G  SU(N).  We  are left with the group SU{N)F  X SU{N)J.  A n element {A, R) of the group operates on  a solution t/o(x) as follows C/o(x) —> AUo{R'x)A^.  However this is not the ground state  symmetry group since this is not the largest subgroup of SU{N)F  X  SU{N)J  that (for  B ^ 0) leaves the ground state unchanged. Using the mathematical concept of equivariance we can build a higher symmetry (slowly varying) solution by demanding that AUQ(X)A^  — UQ{R{A)X).  subgroup of SU{N))  In other words UQ{X) should be equivariant under A  that acts on space through rotations x  R{A)x)  {=SU{2)  and simulta-  neously, on the SU{2) flavour subgroup manifold by conjugation U —> AUA^. In the language of infinitesimal transformations this reads: - i{xi X U{x) + [9i, U{x)\ = 0 where 6i generate any SU{2)  or SO{Z)  subgroup of SU(N)F-  symmetry group is SU{2)K the diagonal subgroup of SU(2)F  (4.71) Thus the ground state X S0{3)J which is often  refered to as the generalized spherical symmetry. Solving the truncated equation of motion for configurations having this symmetry ensures only that the minimal configuration is a local minimum with respect to variations in the SU{2)K direction and an extremum with respect to any other variations. For NF = 2 the static properties of baryons were first calculated by Adkins Witten and Nappi using Skyrme's ansatz I7(r) = cos(/(r)) + «sin(/(r))cr • r [54] and were found to be within 30 % in agreement with the experimental data. In particular MB=I = 3 6 . 5 ^ . The extension to NF = 3 can be done in various ways: 1. Choosing 9i = l/2(7j (i=1..3) where a are Pauli matrices U{r) =  ' e'/(^)<^'- 0 ^ 0  l)  (4.72)  This ansatz leads to the lowest state in the B = 1 sector. After quantization this leads to a reasonable description of the baryon properties. For B > 1 the classical energy of  these solutions increases rapidly {E2 =  S.lEi).  By discretizing the exact equation of motion for the field matrix U and solving it on a three dimensional lattice ^, Braaten et al [67] have shown that the lowest energy configuration with B = 2 is not in fact spherically symmetric. It is actually axially symmetric and the baryon density has the form of a torus. The energy of this configuration isEB=2 = 1.92 EB^I. 2.  Choosing the 50(3) subgroup Oi = A7, 62 = —A5, O3 = A2 the classical ansatz  satisfying eq.(4.71) is: [/(r) = e'-^('-)'-''+'K'-^^'-tl«('-).  (4.73)  It turns out that this ansatz does not allow states with odd B number. Instead one gets two families of even B states: one with P = • • • — 2,2,6,10 • • • and one with B = •••-4,0,4,8,12---. The energy of the lowest non trivial baryon number (B=2) configuration is £^5=2  =  1.92 X SQ.dF^/e"^. Although we do not have a proof that this is the global minimum, it was shown to be a local minimum and an axially symmetric variational ansatz similar to that of SU(2) turns out to converge to the spherical one for minimum energy. The general form of a static solution UQ that fulfills eq.(4.71) will not be invariant under the flavour group SU{N)F  signifying the presence of collective coordinates (zero modes)  corresponding to this symmetry of the action. These collective coordinates are introduced by: U(r,t) where A 6 SU{N)F.  = A{t)Uo{r)A^{t)  (4.74)  Substitution of eq.(4.74) into the action eq.(4.66) allows us to  rewrite it in the form where the effective action for A{t) is given by; SiU)  = ~J  dtMiUo) + J dt(Csk{A^À) + CWZ{AU)).  m a t t e r of a few tens of c.p.u h o u r s o n a C r a y - 2 s u p e r c o m p u t e r  (4.75)  For the SO{3) soliton Swz(A.^À.) vanishes since Cwz becomes proportional to  Tr(A^dA)  which is zero since A^dA is valued in the SU{S) Lie algebra. The problem is akin to the problem of a free particle constrained to move on the energy degeneracy manifold 5^(3) X 50(3) SO(3)  One expands A^dA = Y^a^ada^, a = 1- •• Np — 1, and using the canonical method replaces the classical observables by quantum operators bearing in mind the constraint that A^A = 1. From here on the problem is to find the irreducible representations of the appropriate algebra of observables.  Quantization of this action is then a first approx-  imation to the quantization of the classical action. Quantization of the configuration eq.(4.73) yields the following mass formula: 1 1 J{J+1) M(,„). = M , = 2 + [ ^ - ^ J - ^  , 2{p^ + q'^ + 3{p + q) + pq)  + ^  ^  for a spin J state corresponding to a {p,q) representation of SU(3), 243^r^, l3{Uo) « 365jr^.  (4.76)  where a{Uo) w  For the singlet and the two spin values of the octet and  decouplet it becomes: M(o,o)o =  (4.77)  MB=2  ,  .  1  5  «([/o)  M,,,2  =  ^ - 2  , M(3,0)3  =  .  + ^  m )  +  6  MB=2 + -fiT\  ^  6 +  SU(3) Skyrmions from instantons  4.1  Recently Atiyah and Manton [56] proposed a method to construct multiskyrmion configurations in three dimensional space {R^) using SU(2) instanton configurations defined on four dimensional Euclidean space-time (P^). Their main observation is that the phase factor of SU(2) Yang-Mills fields along time lines reduces naturally to static SU(2) skyrmions with topological numbers identified. They have suggested to use t'Hooft i n stantons to investigate the dynamics of the one and two skyrmion system. Their proposal is based on purely geometrical considerations and does not follow from dynamical ones. The holonomy (phase factor) offers just a bridge between exact solutions of an exact gauge theory (Yang-Mills) and approximate solutions of an approximate chiral theory (the Skyrme model). In particular it was shown that applying this procedure to an SU(2) gauge field with topological charge 1 leads to a simple and good (accurate to within 2 % ) approximation to the B = l Skyrmion. It was also claimed that a good approximation to the more complicated two skyrmion manifold can be obtained, using similar methods, from the two instanton configuration. In this section we extend the analysis of Atiyah and Manton to the case of SU(3). We shall construct SU(3) Skyrme field configurations (depending on x ) from SU(3) Yang-Mills instantons (depending on x,a;o). For SU(3) it was shown by M a and X u [58] that there are no spherically symmetric (i.e. equivariant under  S0{4)j  x SU{3)F  ) instantons. Since we are interested in max-  imal symmetry [57] we can next try configurations with "cylindrical" symmetry, that is configurations which are equivariant under the diagonal group SU{2)j  X SU{2>)F.  SU{2)j  X SU{2)F  S0{3)j  x  SU{3)p  or  The latter case is trivial in the sense that it can be reduced to the case.  The S0(3) instanton however has new structure. The general form of such instanton is: Ao  =  -Mao-Mao  (4.78)  A  =  - M m i + rxK^-ti^-Ky+(oi-^ài,</.o-^<^o,<^i^<^i)  (4.79)  where M  =  ^{A  • f +  Qabfafb)  M  =  ^{A  • f -  Qabrah)  Ka  =  {8ab -  Ka  =  {6ab - fafb){^A  fafb)i^A  • f +  Qabrah)  • f -  QabfJb)  are generators of SU(3) written in an 0(3) symmetric form [61] with A = (A7, - A 5 , A 2 ) and  Qab^af'b  = \{{A • r)^ — |) .  9^1) ^O) <^i) OQ, a i , OQ, ài are functions of r and t that  are yet to be determined. Following Atiyah and Manton we define the associated SU(3) Skyrme field in R ^ , U{r), to be the holonomy of A^(a;) along time lines r,t where r is fixed as: U{T) = Vexp{- r  J—00  (4.80)  At{T,t)dt)  If Af^{x) decreases fast enough at infinity U{oo) reduces to the vacuum value 1. This means that U{x) is a mapping fron 6*^ = i?^ U 00 —> SU{3) and therefore characterized by a non-trivial winding number Q. Modulu magnetic charges Q matches exactly the topological charge k of the instanton. To see that one uses the fact that A; is a total divergence in four dimensional Euclidean space. In unitary gauges this means fc = g + j - ^ jd^xV  -{UVU  xA)  ,  (4.81)  and if the gauge field falls fast enough ~ ^ at infinity there are no magnetic charges so that the second therm on the right hand side vanishes. Under this definition t/(r) is  gauge invariant up to constant gauge transformations at infinity since U transforms as QooUg^ under a gauge transformation g^. In order to find f/(r) one would, in general, need to solve the first order differential equation for the auxiliary field U(x) dtÙÛ-^ = -At{x)  (4.82)  but in all the cases considered here At{T, t) along a time line is proportional to a constant element of the Lie algebra so the integration in equation (4.80) can be done analytically. The analysis of Witten [60] and Bais and Weldon [61] shows that there are two classes of solutions to the self duality equations: T)jpe A Solutions These are solutions which correspond to embedding of SU(2) in SU(3) via the canonical homomorphism from SU(2) to S0(3). ha^h-^ = aiRij  h G SU{2) R G 80(3)  (4.83)  For topological charge k they depend on | parameters. They are generated by a function ^ W = n^  (4.84)  where z = r + it and c,- are arbitrary complex numbers with Re{ci) > 0. The functions a^(r) and à^(r) where / i = 0,1 in eq(2) are given by  = ^tiudi,'^  (4.85)  and  ^ = ^ = i-g(r^) The topological charge is A; = 4/ - 4 so the first non trivial configuration has k = A with / = 2. From (3) we now get a Skyrme field U{T) = e'^'^^'')^-'' where  with a, b arbitrary real numbers > 0. Plugging this configuration into the energy functional of reference [62] and minimizing with respect to ab we get that the minimum occurs for ab = 8 . 5 ^ ^ and  E^^^  =  4.033EB=I  where  = 36.478P^/e is the energy^  EB=I  of a B = l SU(2) skyrmion .This should be compared with the numerical solution for a skyrmion having the same symmetry -E^f^ericoZ = 4EB=I [62 . T\/pe B Solutions These are irreducible solutions that is they can not be transformed by a gauge transformation into an element of the Lie subalgebra. Again the field A^(a;) is determined by the function G(z) eq.(4.84) but ,  1, 2  3r2(2 + GG*)  /  ''\{l-GG*Y{l 1,  (  \  + 2GG*Y /  Zr\l + 2GG*)  \  the possible values of the Pontryagin index are fc = 6Z — 4 so the first non trivial one is k=2 (1=1). To simplify the calculation we take the Q in eq.(4.84) to be real. We then get a Skyrme field of the form  where /(,)  =  ( \ ~  J 3 ^ " + r rrl i 2V^(3a2-|-2ar-F3r2)l 2v^(3o2 - 2ar-|-3r2)l J 7r\/3 f 3r - Q 3r-h a 1 2  I ( 3 a 2 - 2 a r + 3r2)ï  (3a2 + 2ar-F 3r2)l J  (4.88)  ^'  '  One can now check that these functions have the correct boundary conditions /(O)  =  ^(0) = T T , /(oo) = ^(00) = 0 mod 2%. Plugging this into the energy functional of [62] and minimizing with respect to a we find the minimum at a = 3 . 1 ^ and the corresponding '^F„ a n d e are p a r a m e t e r s i n the S k y r m e l a g r a n g i a n  energy El^, = 1M16EB=I  to be compared with El,^,^,,i  = 1.9191£;B=I  Since by construction this Skyrmion has the same symmetry as the instanton it will also be parity degenerate (i.e U(T) and W(—r) have the same energy but belong to two disconnected patches of the energy degeneracy manifold [4]). In conclusion we have shown that one can get good approximate SU(3) skyrmions by using Atiyah and Manton's method of computing the phase factor of Yang-Mills instantons along time lines. This method involves simple integration and in our case led to an analytic function for the Skyrmion profile while the exact solution requires the solution of two point boundary value problem for coupled non linear equations.  Parity Invariant Solution In The B=2  4.2  The 50(3) equivariant soliton in the  SU{3)F  Sector Of The Skyrme Model  Skyrme model admits only even baryon  number solutions [1]. The lowest (nontrivial) of these possesses baryon number 2 and energy  £(1/2)  =  7 0 . 3 ^ and could presumably be identified with the H dibaryon 7^ = 0+,  y = O S ' = —2of the M I T bag model [64]. The quark model predictions of this state mass is M H = 2150 Mev which is 80 Mev below the A A threshold. It has already been mentioned by Balachandran, Lizzi, Rodgers and Stern [63] that the classical ground state in this sector of the Skyrme model is not parity invariant. Parity acts on any solution U{r, t) by U(r, t) -> W{x, t). This operation maps a classical ground state C/2(r) into another classical static ground state in a disconnected part of the energy degeneracy manifold ; i.e., there exists no SU{S) matrix A such that U2{T) =  AU2(r)AK  In the following we shall show that there is a new parity invariant classical solution in the B = 2 sector and calculate some of its properties. The main motivation in constructing this configuration comes from the analogy to the much simpler one dimensional double well problem. There one also has a parity degenerate ground state although in the ful quantum theory the energy levels are split. One finds that the parameters that are needed to estimate the splitting (and in particular the splitting between the ground state and the first excited state) are crucially determined by the only parity invariant point of the potential V^(0). Our configuration is the field theory (infinite dimensional) analog of this point. The model We study the Skyrme model for the flavor group SU(3)F-  The lagrangian density £  is £ = ^Tr(R^R^)  + y ^ T r ( [ P ^ , P,]^)  (4.90)  where  A classical ground state of this theory is supposed to have maxi-  = —iWdyU.  mal symmetry. Its symmetry group is therefore a maximal subgroup of (Poincaré) xSU{Z)i,  X SU{Z)R  [(Poincaré) is the Poincaré group] that leaves the energy, center of  mass location and boundary conditions unchanged [i.e. SO{Z)j x 5*0(3)^] and furthermore allows nontrivial solutions. This group is S0{3)s: namely, the diagonal subgroup of SO{S)j x S0{3)F  where S0{3)F  is the subgroup of real orthogonal matrices in SU{3)F-  The general configuration satisfying these constraints [62] is given by the ansatz U(r) = é^^I + i sin(x)e-'^/2^ • r + (co5(x)e-''^/2 _  • r)^  (4.91)  where x and ip are functions of the variable r = |r( and r denotes the unit vector in the radial direction. boundary conditions We are looking for a parity-invariant configuration Up. Since U{v) and AU{v)A'^ are degenerate i n energy for any A G SU{3) we look for a solution U(r) which satisfies [/t(r) = AU{T)A^  S O that C/(r) and its parity transform are in the same connected  component of the energy degeneracy manifold. Since U G SU{3) it satisfies à.ei{U)=l so that we get two conditions on Up : namely Tr[Up{v)] = Tr[Up{vf\  =  Tr[Ul{-v)]  (4.92)  TT[Ul{-vf]  A n additional restriction comes from the demand that the solution belongs to the B=2 sector. W i t h this restriction we get two solutions ^ = ±2%. The corresponding expressions for U are Up{r)  =  e2'>^/ + i s i n ( x ) e - ' ' ^ A - f + (co5(x)e-'^-e2'>^)(A-r)2  U^{r)  =  e-2'-x/ + i sm{x)e'^A • f + {cos{x)e-'^ - e-^'^)(A •  (4.93)  Figure 4.6: The function Xp (solid curve). For comparison the functions x (dashed curve) and ip (dot-dashed curve) in the solution [1] U2 are also plotted. Both solutions Up and U^ belong to the same connected part of the energy degeneracy manifold. The demand that Up{oo) = I leads to the boundary condition •0(oo) = 0, while the demand that Up{0) be well well defined and belong to the B=2 sector leads to X(0)  = T.  equations of motion To find the explicit form of x(r) one has to minimize the energy functiona poo  E[Up] = 2TT— /  Jo e JO  dr {r'f  + C) + (IOC + QB)f  +  (4.94)  where C = 1 - c o s ( x ) cos(3x), B = sin(x) sin(3x)) and the radial coordinate r is measured in units of  The Euler-Lagrange equation for the function x is  (^r2 + sin(x)' + Asin{2xf)x  + \rx + \{sni{2x) + &sin{Ax))x^ -  \sin{2x) - ^sin(4x) - ^ ^ ^ ( 4 s i n ( 2 x ) 2 - 3sin(x)') =: 0  (4.95)  Figure 4.7: The radial baryon density (solid curve) for the parity-invariant solution. The dashed curve shows the radial baryon density for U2. The asymptotic behavior of the solutions follows an irrational power low. Thus near the origin  %('') =  TT —  cor^~^ while for large values of r we have 5 = ^"^  x{f) =  Coo/^^  with  « 2.7912  (4.96)  The constants CQ and Coo were determined numerically to be CQ = 20, CQO = 73.1. classical properties Solving eq.(4.96) with the boundary conditions and inserting the solution Xp (fig-(4.6)) into the energy functional eq.(4.94) we get M[Up]  —  ^.\ZMB=\-  Inserting the solution  into the expression for the radial baryon density one gets Pb = 47rr2&o = - - [3sin(x)sin(3x) + 1 - cos(x) cos(3x)] X  (4.97)  TT  The graph of this function is shown in fig. (4.7). It has two maxima near the points where x{r) — |7r and x('^) = f^f and one minimum near the point where xiy) — \''^ • The thin  70 C2 54  - /) -/  r-T-  » r  T r- T r-T I- r »• ! •  -  4e - 1 96  » 1  30 22  C)  -  1  14 ~ , // 1 ' 6 1 1 ~T II ' -2  -  1 * *  /  ^^^^^^^^  /  \ /  . 1 . 1 i 2  11 I ' l l 3 4 5  1 • 1 > 1 11 6 7 8 »  10  Figure 4.8: The radial energy density e (solid curve) for the parity-invariant solution. The dashed curve shows 2Qph for comparison. spherical shell of negative baryon density contributes about -0.1 to the baryon number integral. Using the baryon density one can calculate the rms radius of this state Tl/2  Jo  J  (4.98)  This turns out to be 5.2/eF-^ which is to be compared with the value 3.57/eF^ for U2. The radial energy density e(r) is plotted in fig.(4.8). Some of its features are clearly correlated with those of the baryon density. conclusions We have shown the existence of a new parity-invariant, classical, static solution in the B=2 sector of the SU{3)F  Skyrme model . This state cannot correspond to a stable  quantum state since its energy is l54FT^/e or 4 . 2 1 M B = I above the energy E[U2] of the classical degenerate states. It is known [3] that an instanton connecting two classical  ground states belonging to disconnected components of the energy degeneracy manifold cannot have spherical symmetry. However, such an instanton could still pass through the new solution.  Part II  Non Trivial Fixed Points in Q F T ' s  Chapter 5  Introduction  In the next chapters an extensive use of the renormalization group (RG) will be made. We therefore give here a summary of the basic ideas and methods that will be used. The main theme behind the name R G is the idea that starting with a theory with a large (actually infinite) number of degrees of freedom one can successively map the theory into a series of theories, each with fewer degrees of freedom than its predecessor and each characterized by a new set of coupling constants. This procedure is called the R G . It can be defined in many ways. As an illustration I consider here Kadanoff's block spin method applied to the two dimensional Ising model. This model was originally intended to simulate the order-disorder phase transition in magnetic systems. Here we have a lattice of "spin variables" <TJ taking values ± 1 , at each lattice site. For large enough J this system undergoes phase transition from a disordered (high temperature) phase J < Jc to an ordered (low temperature) one J > J^. In the original variables CTJ, the statistical weight  is given by (5.99)  with h small. Near Jc the correlation length ( becomes large. We therefore expect that fluctuations within a given block will be inessential as long as its size is less than C- The transformation Rb which will map the original lattice (with lattice spacing a) to the new lattice (with lattice spacing ba where b is an integer) is defined as follows: one groups  together blocks of  spins in clusters. To each block we associate a new variable (5.100)  a}, = Sign^Oi  (with an additional convention for the case that this sum vanishes). If the correlation length C, is large enough we can neglect fluctuations within the block. We thus obtain a new action J\6{abh  Sign^Ui) e^. ieb  (5.101)  This action is in general very complicated and is as hard to solve as the initial one, however we expect that the symmetries of the theory are preserved by this transformation and also (assuming b ban be continuous) that for 6  1 Tiab  Tia- Neglecting next to  nearest neighbor and higher interactions the new action is + ^6 E  Hi,^JbJ2 (6,6')  + ^0  (5.102)  6  where HQ is a symmetry breaking term. Thus up to an additive constant we just get the old action with new couplings and a correlation length C' which is C/6.  5,1  Flows, Universality and Scaling  The effect of iterating this transformation can be analyzed qualitatively. Starting with some general hamiltonian H{gi) where gi is a set of couplings, and applying a renormalization group transformation  to it we get Rbn{gi) = Hb{g'i).  (5.103)  Hb defines a theory with correlation length ^ and new couplings g\. If at the starting point the values of the couplings g^ are not on the critical surface (i.e. with finite correlation length Co) ! successive applications of this transformation will result in an effective theory  where the correlation length —> 1 and therefore physical scales like masses will be driven to the cutoff scale (A ~ 1/a where a is the lattice spacing). If however we start at a point on the critical surface i.e. where ( = oo then Rf, will not take the theory out of this surface (since ^ = y = oo). The only change will be in the couplings. The set of points in the parameter space of the theory corresponding to the successive application of R G transformations is called an R G trajectory. In principle a critical trajectory might eventually end up at infinity or it might wander around on the critical surface forever in some chaotic manner. However a more common possibility is that the critical trajectory ends up at a fixed point g* : RbHig*) = n{g*)  (5.104)  In general many different theories (points on the critical surface) flow to the same fixed point. Since critical theories are classified according to the IR, or long distance behavior of various correlators (i.e. by critical exponents) this means that many different theories will have the same critical behavior. This is the statement of universality which is seen very well in the experimental data on phase transitions. 5.2  Relevant, Irrelevant and Marginal Operators  In the vicinity of the fixed point we can linearize the R G transformation. In this region we can write 7i = Ti* + gQ where g is small and Q is some operator. Assuming an infinitesimal transformation n'=  RbH = n* + gRbQ + Oig"^)  (5.105)  RbQ is a linearized R G transformation operator and may be expected to have a spectrum of eigenvalues Ai(b) and associated eigenoperators Qi ( called scahng operators) such that  RbQi = Ai{b)Qi. Each of the eigenvalues should be expressed in the form Ai{b) = 6^'  (5.106)  where Aj is independent of b (this is a consequence of the semi group properties of the R G transformation Rbu2 — RbiRb2- acting with Rb on H then yields n' = n* + Yl9iAiQi i  + 0{9f,gigj)  (5.107)  and after / iterations n'C)  =  H* + Y:9?Qi  9?  =  9iA'i = b'''9i  + 0igf,gigj)  As / increases there are three possible courses for gf\  (5.108)  If A,- > 0 then A, > 1 and gf^  grows rapidly driving the system away from the fixed point. Operators Q, corresponding to such ^fiS are called relevant. If Aj < 0 then A, < 1 and gf-' shrinks steadily to zero. For sufiiciently large I it should be possible to ignore the operator Qi corresponding to such coupling. This kind of operators are called irrelevant. Finally it might happen that Aj = 0, A i = 1. In this case further analysis is needed, one has to go beyond the linear approximation. Eigenoperators corresponding to such Aj are called marginal. Relevance or irrelevance of operators is determined with respect to a given fixed point. If there is more than one fixed point it may happen that a relevant operator near one becomes irrelevant near the other or vice versa. Adding relevant operators does not change the short distance behavior of the theory. However as we look at longer and longer wavelengths these operators become more and more important. Addition of irrelevant operators on the other hand will not be felt at long distances but will be more and more important at short distances.  5.3  The flow near the critical surface  A theory defined on the critical surface contains massless particles. Therefore if one tries to calculate physical quantities one encounters IR divergencies. This is in particular problematic in dimensions d < A. If one tries to use perturbation theory, divergencies at m = 0 become more and more severe. For instance in scalar  theory the 4-point  function behaves as p~^'^~'^\ Hence its iteration n times will produce IR divergencies if n(4 — d) > d in diagrams that are U V convergent. On the other hand there are no IR problems in d = 4 so one can study the massless theory directly. If this limit (m —> 0 ) also corresponds to the ^ —> 0 limit then one can use a double expansion in g and e = 4 — rf to proceed to lower dimensions. A different approach which we shall be using in the next sections is the CallanSymanzick (CS) approach. Instead of dealing with the massless theory directly one keeps nonzero mass m to control IR divergencies. The critical theory appears in the domain 1 and ^ <C 1. From the renormalized theory point of view the scale A is absorbed by a redefinition of Tc by the use of the correlation length  as a fundamental parameter  and a rescaling of the fields (this automatically takes care of the second condition ^ -C 1). To approach the critical surface we consider a change of scale in the correlation functions. This can be done by varying the mass m at  fixed  and A . For simplicity we  consider here the 4>^ theory. For a bare vertex function we obtain d  m dm r"(p, mo, go, A)  drriQ  = m dm  r"'^(p,ç = 0,mo,go,A)  (5.109)  The bare and renormalized functions are related through {p, mo, go. A)  =  {p,q, mo, go, A) =  Zr"ri{p,m,g) Zr"Z,Tf{p,q,m,g)  (5.110)  Introducing the following quantities Pig)  =  m  dg dm  (5.111)  = 72  =  rr-l  Z2  m  9Z2  dm  and expressing the bare functions in terms of renormalized ones we get the following equation: , 0 1  d  Tlip,m,g)  =  rfip,0,g,m)  Z2m^'^'  dm  (5.112)  so,A  where /?, 71 and 72 do not depend on the momenta or n. Using the renormalization conditions rl{0,m,g)  = m^  Vf (0,0, g,m) = 1  (5.113)  z^m^  (5.114)  we find that (2 - 7 i ) m 2 ^  So the final form of the equation is m-  d  ^d  1 - - n 7 i nip, dm ' '' dg 2' dg  m, g) = (2-  dm  go,A  ji)m'r"/{p,  0, g, m)  (5.115)  This equation is of little use unless the right hand side vanishes. For large p, r^'^(p, 0) is of order p~^r^(p), up to powers of log(p), order by order in perturbation theory. Therefore if the log(p)s do not sum up to compensate for the p~'^ the right hand side of eq.(5.115) can be neglected. The solution of this equation will then give the dependence of the bare coupling on 1/m, the correlation length, and therefore the flow. 5.4  C o n f o r m a i invariance  In many cases critical models possess in addition to scale invariance also invariance under special transformations called conformai transformations. A conformai transformation is  a transformation which preserves angles but not necessarily distances. These transformations form a group which includes (in three dimensional space) the Euclidean group as a subgroup (translations, 3 parameters, rotations, 3 parameters) dilatations (one parameter) D:  x-^e"x  (5.116)  and special conformai transformations (3 parameters) defined by inversion, rotation and a second inversion X  — aa;2  In three dimensions these describe the connected part of 5'0(4,1) which has 10 parameters. The generator of infinitesimal special conformai transformations K°' obeys the following commutation relations with the other generators of the conformai group. i[P>',K'']  =  -2^^"£) + 2 M ^ "  i[D,K"]  =  -iT"  i[K'',Kl^]  =  0  (5.118)  Since the commutator of an infinitesimal conformai transformation and infinitesimal translation (P^) is a linear combination of infinitesimal Lorentz transformations (M'^") and infinitesimal scale transformations a Poincaré invariant theory which is also conformally invariant is scale invariant. The condition for the opposite to be true is [73] that one can define an energy momentum tensor 9'^'' such that the scale current is 5'^ = a:,^^^  (5.119)  Although this is not guaranteed for an arbitrary theory it is known that for all (perturbatively) renormalizable theories, eq.(5.119) can be satisfied. In particular this is true  for the Yukawa model which we shall use later.  Chapter 6  Chiral Conformai Point  A current paradigm in relativistic Q F T is that a renormalizable theory in four dimensions is either asymptotically free or trivial. Recently, however a great eflFort has been made to find an example of an interacting theory with a finite fixed point [76]-[75]. Reasonable candidates are the strongly coupled Q E D with possible addition of the four fermion coupling or some other perturbatively irrelevant operators.  A theory of this kind, if  exists, might provide an alternative to the scalar sector of the standard model [78 . The key idea is that some perturbatively irrelevant operators may acquire large ultraviolet (UV) anomalous dimension and become relevant at strong coupling. Whether the renormalizable field theory defined around the new U V fixed point is an interacting one is a separate important question. Since the main nonperturbative method used, Monte Carlo simulation on the lattice, is a numerical rather than analytic one, the physical interpretation of results in the critical region is rather complicated ^ One would like to understand the possible mechanism behind the appearance of such a strong coupling fixed point by examining simpler models using some analytical yet reliable nonperturbative method. Although in 3-f 1 dimensions no such model is known, it turns out that they are quite common in lower dimensions. In 2+1 dimensions this phenomenon has recently attracted attention. Renormalizable theories possessing nonperturbative U V fixed points like certain a models [79],[80] and four-Fermi interaction models [81],[82],[83] are known. ^In p a r t i c u l a r w h e t h e r t h e s c a l i n g b e h a v i o r is the m e a n field one is not yet a settled issue [74],[75].  In this work we continue the study of four Fermi theories in 2+1 dimensions from the point of view of their relation to critical phenomena. Some of the material overlaps with ref. [5] and [84],[6. There is a deep relation between existence of continuum limit (or renormizability of a Q F T ) and second order phase transitions [85]. Regularized Euclidean Q F T can be considered as a statistical model. Continuum limit is defined near a critical hypersurface in the parameter space of the theory. This critical surface generally contains special points at which the theory is not only critical (contains massless particles) but also conformally invariant. Dimensions of various local operators are generally different at different conformai points (CP). Therefore each C P has its own set of relevant operators. In the vicinity of any of these conformai points one can define continuum theory deforming the conformai field theory ( C F T ) with these relevant operators. Whenever perturbation theory is used one explores just the vicinity of the gaussian C F T . In particular since in d=4 no non-gaussian C F T is known, in particle physics one always studies the neighborhood of gaussian C F T . As is well known from the theory of critical phenomena the situation in d=3 is markedly different. The universal properties of abundant phase transitions in condensed matter systems are described by nongaussian IR fixed points. Examples are order-disorder phase transitions in magnetic systems generally described by (j)^ theories. A gaussian C F T in these cases is an ultraviolet fixed point of the R G flow. The universal critical behavior is described by a different, IR fixed point. The continuum theory describing for example the universality class of Heisenberg antiferromagnets, is the critical nonlinear 0(3) symmetric a model. The non -gaussian character of the IR fixed point (theoretically predicted and experimentally measured) follows already from the nonzero value of the critical exponent rj. Near this C F T there is only one relevant direction: the coupling /  of the a model [86] 2. There exists another important class of phase transitions: chiral phase transitions. Since there are massless fermions on the symmetric side of these transitions, they are expected to be qualitatively different from magnetic phase transitions. Even for the same symmetry breaking pattern G  H (the same order parameters) critical exponents  are expected to be different. This expectation is born out by recent calculations i n 1 / N expansion [87]. A n analog of Ginzburg - Landau description for a chiral phase transition is a Yukawa model. A spinorial representation of the Lorentz group 5*0(2,1) in three dimensions is provided by two component Majorana spinors with the corresponding representation of the 7 matrices 7° =  7^ =  7^ =  ^2  iai  However, there is no other 2 x 2 matrix that anticommutes with all the gamma matrices. Therefore one cannot define chiral symmetry in the usual sense for the two component theory. The massless theory has no more symmetries than the massive one. We can combine two Majorana spinors to form a complex representation of the Lorentz group. The 4 x 4 7 niatrices can be taken to be oz  0  0  \  0 0  -az  \  —iai  ia2  0  \ 0  -ia2  (6.120) j  there are two 4 x 4 matrices that anti-commute with all the 7 matrices 7^ - i  '0  1'  1 0I  7^ - i  '  0  1^  -1  0;  (6.121)  so the massless theory is invariant under the "chiral" transformations  ^In c o n t r a s t n e a r the g a u s s i a n fixed p o i n t i n ci* one o b v i o u s l y has two relevant d i r e c t i o n s (j)^ a n d <i>^.  In the following chapters we concentrate on a simple case where the chiral symmetry is a discrete subgroup of these transformations i.e. the Z2 chiral symmetry corresponding to V —> 75'0- We show that the IR fixed point of this universality class is the critical four-Fermi model. This C F T , the chiral conformai point (xCP) and its deformations by relevant operators are therefore non-perturbatively renormalizable. We find that t/ji/j and [tpi'y are relevant while (ïiipY is irrelevant with U V dimension close to 3 for large N .  6.1  Chiral IR fixed point  To study chiral phase transition in d=3 we start by considering the simplest example of Z2 chiral symmetry. The generalization to more complicated cases is straightforward and is briefly discussed in chapter 8. The theory which describes this phase transition is the (Euclidean) Yukawa model ^. + l(d,af  £ = -ri^r Here  a = 1,...,  + ^ c r ^ - •^<yrr  (6.122)  is a four component complex spinor. For any N the theory has  a discrete chiral symmetry: ip — 7 5 ^ , a  —a. A s will become apparent later, the  theories with different N belong to different universality classes. We show now that the critical G N model £ =  - ^ ( ^ ^ ) '  (6-123)  is the IR fixed point of Yukawa model eq.(6.122) and therefore determines the universality class. The value of the critical coupling Tc oc A (where A is an U V cutoff) is given later. Throughout, we use mostly the 1 / N expansion. The existence of the IR fixed point is also shown in the standard framework of R G improved perturbation theory [85]. The ^ N o t e t h a t this t h e o r y i n 2+1 d i m e n s i o n s is p e r t u r b a t i v e l y s u p e r r e n o r m a l i z a b l e a n d t h e o n l y relevant c h i r a l i n v a r i a n t t e r m , (T^ does n o t generate divergent r a d i a t i v e c o r r e c t i o n s .  same conclusion is arrived at using e expansion around d=4 [81],[90] * In the leading order in 1/N, one finds a phase in the Yukawa model with nonzero V E V of the order parameter a: < a >= v. Chiral symmetry is spontaneously broken and fermion's mass M is dynamically generated. The equations determining M and v are: mlv  =  M  =  4AOM//VJ  (6.124)  ^  where  The theory has a line of second order phase transition points defined hy M = v — Q (see fig.6.9): 2_ ml = — A ^ A  (6.126)  It is convenient to introduce a finite scale t that measures the distance from this critical line Ao/2A_ A In the broken phase it is equal to the dynamically generated fermion mass M. In the unbroken phase where fermions remain massless it determines the width of the unstable a resonance. The continuum massive theory contains particles whose mass is small on the cutofi" scale it is therefore defined near the critical line (in the scaling region) [85]. To find the IR fixed point we now calculate the Callan-Symanzik /^-function for the dimensionless coupling g. The renormalized dimensionless coupling of the Yukawa model ^ N o t e t h a t a p p l i c a b i l i t y of t h e e a n d t h e weak c o u p l i n g expansions is not r e s t r i c t e d t o large n u m b e r of fermions. O n the o t h e r h a n d , i n b o t h cases the expansion p a r a m e t e r s are n o t s m a l l . F o r the e e x p a n s i o n i n d = 3 , e = 1, a n d t h e f i x e d p o i n t value of dimensionless c o u p l i n g g* (see eq.(6.140)) is of order 1. T h i s is s i m i l a r to t h e c o r r e s p o n d i n g d i s c u s s i o n i n the m o r e f a m i l i a r case of " m a g n e t i c " phase t r a n s i t i o n s d e s c r i b e d b y the 0{N) invariant theories ( N = l for I s i n g , N = 2 for X Y , N = 3 for H e i s e n b e r g m o d e l s ) [85].  symmetric phase  Figure 6.9: Phase diagram of the Z2 invariant Yukawa model g is defined as the dimensionless (renormalized) vertex function at zero momentum V^g = r''\pi = o,p2 =  o,p3  =  0) = z y % r i ' 2 ( o , o, o)  (6.128)  where m is the renormalized mass of cr ^. We impose the following wave functions normalization conditions dp^  = 1  di)  =  1.  (6.129)  p2=0  To leading order in 1 / N there are no corrections to the fermion propagator and the vertex function. The bare inverse propagator for the boson is 1 ^1{P) =P^ + -^g^MPl +  + const  (6.130)  g{m,\o) 8  (6.131)  The wave function renormalizations are consequently  ^ S t r i c t l y s p e a k i n g , i n M i n k o w s k i space m defined as m = T'"^\O) has an i m a g i n a r y p a r t (a decays i n t o massless f e r m i o n s ) . I n t h e E u c l i d e a n f o r m a l i s m t h a t we use, m is real.  The Callan-Symanzik (CS) /5 function is defined as dg{m,Xo)  (6.132)  Prom a calculation of the vertex part we obtain (6.133) The I R fixed point g* is defined by the m —> 0 limit. In our case g* = 2\/2. The 7-functions which determine the IR asymptotics of the critical theories can now be calculated using eq.(6.133) dlogZ,  la = 5 log m  7/ = 0  8 '  (6.134)  The I R fixed asymptotics of the inverse propagators at g* are given by [85] (6.135)  The I R dimensions of the fields tp and a are [V»] = [cr] = 1. The connection between the bare and renormalized couplings is determined by the R G equation: ^  ogj m  (6.136)  = 0.  The solution is Ao(g) = x / m g e x p j - i ^ ^ As g  g*, the bare coupling AQ diverges, AQ  1  /?(a;)  2  X  dx\  (6.137)  oo. Along the critical line the ratio  between AQ and the bare mass ml is fixed by eq.(6.126). Therefore in the critical strong coupling limit one also has ÎTIQ  —y  oo. It follows then that the kinetic term of the  a -  field in eq.(6.122) is negligible compared to the mass term and can be dropped. The disappearance of a's kinetic term turns it into an auxiliary field. Integrating over a one arrives at the four Fermi theory eq.(6.123) with (6.138)  using eq.(6.126) one recognizes in eq.(6.138) the finite fixed point condition for the four Fermi theory[83' 2A 2  (6.139)  The use of the 1 / N expansion is by no means mandatory. The same result is obtained in the loop expansion. The one loop CS /^-function is (6.140) The numerical value of g* is changed for small N but all the qualitative features remain the same. We feel that this R G picture gives a most clear and convincing argument in favor of nonperturbative renormalizability of G N model in d=3. We now summarize the physical picture. The Yukawa theory eq.(6.122) has two physical scales, one is the mass of particles, M (i.e. fermions and their bound states). The second is the interaction scale p ~ \ / M A . This scale governs scattering of the various particles in the theory. At scales p »  /X the theory is essentially free (gaussian) while for p < / i it is interacting. The  existence of second order phase transition means that M can be taken to zero at fixed p. Therefore the two scales are decoupled. The x C P IR fixed point is approached when 0 0 , m —^ 0. In the limit / i  oo at fixed m, for large momenta p > M the theory  is essentially x C P . This theory is consequently well defined in the continuum limit and contains all the degrees of freedom important for phase transition. The x ^ P , which is the IR fixed point of the Yukawa model, becomes therefore the U V fixed point for this four Fermi theory. It is therefore seen that the U V asymptotics of the four Fermi theory is the same as the IR asymptotics in the massless Yukawa model. The basic difi'erence between the gaussian fixed point (/i  0, m —> 0) and the x C P (/i -> oo, m -> 0) is that near the  latter the interaction is nonvanishing.  Figure 6.10: Diagrams contributing to the tpip propagator to order 0(N). Solid lines denote fermion propagators and wavy lines - scalar propagators. It is clear from this discussion that Green's functions, not only of fermion field but also of the order parameter field <T, are multiplicatively renormalizable in the four Fermi theory. The (finite) renormalized Green's functions of G N model are obtained in the critical strong coupling limit of the Yukawa model. Green's functions of tp and a are renormalizable and well defined for arbitrary values of the physical couplings of the Yukawa model. In the critical strong coupling limit the physical couplings remain finite and the Green's functions do not diverge. The proof of this point, although physically obvious, requires tedious manipulations with overlapping divergencies in the formal framework of 1 / N expansion. The detailed renormalizability proof for Green's functions of ip is given in [37] [41].  6.1.1  U V dimensions of local operators near x ^ P *  Having established the existence of the x C P , it is interesting to calculate U V dimensions of various local operators near it. We start with the field a. To leading order in 1/N the bare scalar correlator (in the broken phase) is [84],[6  where  Operator  Weak coupling dim.  Strong coupling dim.  i;  1  1  ipip  2  1  3  3  4  2  6  3  1/2  1  1  2  3  4  2  4  2i  2  a  Table 6.2: Ultraviolet dimensions (in units of energy) of various operators in the Yukawa model ^  \  arctan(/g:)  27r3 ( P + M2)((fc + p)2 + M 2 )  47rp  ^  ^  At weak coupling it has a standard large momentum behavior: G{p) —^ ^ which defines the canonical scaling dimension of the field a: [a] = 1/2. On the other hand, in the (critical) strong coupling regime AQ  oo,  = const therefore for p large but p «C A  G{p) ~ ^ .  (6.143)  Consequently the dynamical U V dimension becomes [a] = 1. As a result the kinetic term of a becomes irrelevant: [(d^a)^] = 4 > 3. This is another way of saying that this term can be dropped from the Lagrangian. Note that at strong coupling the dimension of ip remains 1 but polynomials of  acquire large anomalous dimensions. The connected  (bare) correlator of the bilinear •ipip in the leading order in 1/N is given by the diagram  Figure 6.11: Diagrams contributing to the correlator < ipip{0)ip{x)ilj{y) > to order 0 ( N ) . fig.6.10  where J is defined in eq.6.125. At strong coupling 8 72  it behaves at large momenta like 1 /p (after subtraction of the contact term —4 J ) as opposed to the "canonical" weak coupling behaviour p which can be read from the AQ —> 0 limit of the propagator ^ G^^ip) ~ 4 J - 2(p2 + 4 M 2 ) / ( p ) .  (6.146)  The U V dimension of -ipip as read off eq.(6.145) is 1 (recall that G^^ is related to the four point function). Similarly the U V dimensions of ('ipipy becomes 2 instead of ®The c o r r e s p o n d i n g m u l t i p l i c a t i v e r e n o r m a l i z a t i o n f a c t o r Z c a n b e f o u n d f r o m t h e c o m p o s i t e t h r e e p o i n t G r e e n ' s f u n c t i o n fig.6.11  T h i s d e t e r m i n e s t h e c u t o f f d e p e n d e n c e of Z^^ = T h e r e n o r m a l i z e d c o r r e l a t o r is therefore  at s t r o n g c o u p l i n g (at weak c o u p l i n g Z^^ = 1).  4 and that of (iptpY becomes 3. This follows from the mean field factorization of the leading order in 1/N. The scaling dimensions of various operators are summarized i n Table 6.1.1^. As discussed in the previous section, the U V dimensions coincides with the IR dimensions in the critical Yukawa model eq.(6.134).  '^These vaJues contraxiict those obtained in [87] but coincide with those given in ref. appendix E .  [81].  see also  Chapter 7  RenormaUzable four Fermi interaction  Deformations of the C F T eq.(6.123) with relevant operators 'ipip and (ipipY should have no effect on the ultraviolet asymptotics. The renormizability properties of the theory therefore should not be spoiled. One expects that the Lagrangian C =  + m)iP -  - ^ ( « '  (7-147)  describes a finite continuum theories and has a x C P as its U V fixed point. Here A/? = — T~^, so that the four Fermi coupling is not necessarily fixed at its critical value. This expectation is substantiated by the results from three independent sources i. The direct 1/N expansion in G N model^[82],[83 . ii. The 2+e expansion starting from the asymptotically free G N model in d=2 [90 . iii. The direct lattice simulation [91 . We concentrate here on the first source, the 1/N expansion. The N - component Gross - Neveu [23] theory  £ = -UHi  - ^i^i^if  (7-148)  where z = 1 to iV, and ^ is a four-component Dirac fermion. The model has been analyzed in the framework of the 1/N expansion [42, 43]. At leading order in 1/iV the ^It is clear f r o m t h e discussion i n s u b s e c t i o n 2.1 t h a t G r e e n ' s functions n o t o n l y of f e r m i o n field b u t also of t h e order p a r a m e t e r field a are m u l t i p l i c a t i v e l y r e n o r m a l i z a b l e i n t h e four F e r m i theory. T h e p r o o f of t h i s p o i n t , a l t h o u g h p h y s i c a l l y obvious, requires t e d i o u s m a n i p u l a t i o n s w i t h o v e r l a p p i n g divergencies i n t h e f o r m a l f r a m e w o r k of 1 / N e x p a n s i o n . T h e d o u b t i n t h e r e n o r m a l i z a b i l i t y p r o o f of t h e G N m o d e l ref.[83] expressed i n t h e recent p r e p r i n t [87] is w i t h o u t basis.  theory exhibits two phases, distinguished by the chiral order parameter  In the  broken phase the fermion mass is dynamically generated like in the corresponding 1 -f 1 dimensional model [23 . In order to facilitate the 1 / N expansion we'll use an equivalent representation of the lagrangian eq.(7.148): NT C = —tpi^Tp + atpTp  —a  .  To see that this is indeed an equivalent representation all one has to do is to complete the quadratic form involving the a field to a square and perform the gaussian integration over (7. Classically a is just a lagrange multiplier and its equation of motion is a = •^•ipip. the discrete chiral symmetry is now: tp — 7 5 - 0  'ip —>• —•^75  a -> —a.  The trick of introducing the auxiliary field makes the action quadratic in the fermionic fields and will be useful for integrating over them in the path integral. Since we will be interested in the scaling behavior of various operators we work from now on in Euclidean space. We define the generating functional Z(r], rf) by coupling external sources to the fermionic fields and regularizing with momentum cutoff. Z{r],fi)  J DaD^PDiPe-^^+'>'^+''''>  =  (7.149)  Here Z^ and Za are wave function renormalization constants and gi is the bare coupling. The introduction of Z^r enables us to renormalize also Green's functions of the auxiliary field a which interpolates the bound state meson. Integrating over the fermionic fields we get 2(77,77)  =  J ^^^-'S,ff+JcPxZ^nG(x,y)rj  Scfj{a)  =  N jd?x-  trlog(-i^+ Z y V ) +  (7.150)  where G{x,y) =< x\{^ — (j)~^\y >• Eq.(7.151) shows that in the large N limit we are justified in evaluating the functional integral using saddle point approximation. We can find the saddle point perturbatively by demanding that  = 0 order by order in  perturbation theory. This leads to the gap equation ^^^-^^ 9b  =  ^J^7^''^^^^-'^^^l'"^)  =  G{x,x\a)  (7.151)  .  The meaning of this equation becomes clear if we notice that < ï>(x)iP{y) >  =  (7.152)  j DxpDÎ}e-^^î){x)i}{y) f Dae~^'ffG(x,y)  ^,  where a* is the saddle point value of a. Comparing eq.(7.152) and eq.(7.153) we find  This is just the quantum manifestation of the constraint one gets from the classical equation of motion for a. Fourier transforming to momentum space and expanding the trlog in 1 / N we get in leading order Zy^T <a>=  / f d^x tr {  ~  1  \ \.  (7.153)  This equation has a symmetric < a >= /j, = 0 solution for all values of the coupling and an asymmetric solution / i > 0 for values of T in the range T <Tc, where d^p 1 (27r)3 p2 • In this case the mass fj. can be non zero and is related to the coupling by y  f d'p  1  80  Chapter 7. Renormalizable four Fermi interaction fermion propagator  + m 2irp  a propagator  (4m2 + p2)arctan(^)  1  crtptp vertex  1  Figure 7.12: Feynman rules for the 1 / N expansion in the Gross - Neveu model. When both solution exist the phase of dynamically broken symmetry < a >^  0 is  preferred since it corresponds to a global minimum of the effective potential. The symmetric solution actually corresponds to a local maximum of this potential and therefore expanding around it will result i n tachyonic modes. In order to renormalize Greens functions of ip and a we have to define our renormalization conditions r'  =  0  (7.156)  =  0  (7.157)  where = means equal up to finite parts, we will show that these are the only counter terms we need (i.e. no Zg). We will assume that we can expand Z^,Za. T, < a > m powers of 1/N. In lowest order we get from eq.(7.157) two conditions, one on the derivative of the fermion propagator from which we deduce Z^ = 1 and one on the mass, from which we get < a >= p. Solving eq.(7.157) we get Z° = 1. Plugging this back into the gap equation we find ^  2A  The dimensionless baxe coupling constant defined by (7.158) approaches in the continuum (large A) limit the finite value Ac = ^ . The corresponding P - function in the neighbourhood of this ultraviolet fixed point is [42] /5(A) = Ac - A  (7.159)  The leading order result does not necessarily establish the existence of the U V fixed point even within the framework of the l/N expansion. Thus it could happen, in principle, that logarithmic corrections in higher orders force Ac to infinity, analogously to the perturbative running coupling in QBD3+1.  Alternatively Ac could be driven to zero, as in  QCD. Therefore it is interesting to see what happens to the finite U V fixed point in the Gross - Neveu and similar models beyond the leading order. 7.0.2  Beyond leading order  In order to calculate higher order corrections we'll have to take into account fluctuations around the saddle point. To this end we shift and rescale a by cr —>  (^/j, +  This  changes the effective action to  expanding the effective action in powers of 1 / N and minimizing with respect to < <7 > the gap equation becomes  r° < â >  +  <à>  -  =  0 .  (7.161)  We can use eq.(6.145) for the leading order propagator to rewrite this equation as  5  ^<^>^-T,+ D{0)  .  (7.162)  The condition on the fermion propagator now reads:  (1 +  +  + ^  )  -  =^  0  (7-163)  from which we find (using the tables in appendix D) p.Zl + y/N <â>-^log{^)  =  0  (7.164)  Solving for Z^j, and < â > we find  Note that at this order the expectation value of a becomes cutoff dependent^. Prom the condition on the vertex function  Z,Zl/^ + 1  ^  we get after collecting terms of order ^i =  - 1  ^ 0.  (7.167)  (the two loop diagram turns out to be finite)  ± I o g ( ^ ) - 2 4  = ii.og(;^) .  ^ T h i s also h a p p e n s i n t h e l o o p corrections i n other theories.  (7.168)  Finally plugging all this back into the gap equation eq.(7.162) and solving for ^ we get '2A  8^ , , A ' + ^log(-) N \7r^ ' 37r3  (7.169)  One must check that these renormalizations do indeed render all Green's functions finite, and this is so because the underlying chiral symmetry relates the divergences in the diagrams in eq.(7.161) and eq.(7.163) to those of the vertex function eq.(7.167). Physical quantities like the S-matrix do not depend on the scale p, that is they are invariant under the transformation / i , g{p), M{p)  p', g{p'), M{p').  Written as an  r^(p,,5,/.) = o  (7.170)  infinitesimal transformation this means rfr^(j?j,g,//)  '9  ,d  1  dfj,  '  with (7.171) =  lia) where T^{pi,gi„h.) = Z ^l'^V^ipi,g,p)  /^^iog(z)U,A is one particle irreducible N point function of ^  or a. We can now calculate the beta function for the dimensionless coupling  A  7r2  TTA A^V7r2"^37r3A^V7  (7.172)  It has the general form /?(A) = a(A. - A)  (7.173)  where (7.174)  The solution of the R G equations for the running coupling A is A(A) = A, - ( - ) - " A*  (7.175)  The finite U V fixed point still exists, albeit shifted downwards by a fraction l/N [37]. Notice also that the /3-function slope has been reduced to (1 — &/3'K'^N); it must be positive for consistency and this holds true even for A'' = 1. The deep euclidean behavior of the various Green's functions are just power laws. The powers can be obtained simply from knowing the renormalization constants in eq.(7.167, 7.168) and using the 7 functions =  16  2  7. =  For example, the fermion propagator behaves as B^ÈK.^  , , (7.176) The R G has exponentiated  the £np dependence of the "rainbow" diagram in 7.163, and this corresponds to the summation of "nested rainbows." We can codify this power by defining the ultra-violet (critical) dimension of the fermion field, M  = 1+ 3  ^  •  (7.177)  Similarly, the meson two-point function depends on Zi and Z2, and behaves asp~^^"'"3lv^\ Thus W = 1- 5 ^  .  (7.178)  The high energy behavior of the connected, truncated Green's function with n fermion legs and m cr-legs is then ~  where P = 3 - n[^] - m[a] .  (7.179)  We note however that the existence of the fixed point to all orders in 1/N cannot be inferred from this calculation by the R G argument. Ttiis is a general feature of the  1/N expansion as opposed to the weak coupling perturbation theory, in which such a statement can be made at least for asymptotically free theories [48]. The point is that in weak coupling perturbation theory the expansion parameter is precisely the quantity whose p - function is calculated. In an asymptotically free theory an expansion parameter is small exactly in the asymptotic region. Therefore the /3 - function calculated in the first nontrivial order becomes reliable at suflBciently high scale. Therefore the R G flow of the coupling constant near the U V fixed point can be firmly established using lowest nontrivial order. On the other hand, 1/N is not a running coupling. One calculates the /3 - function of the coupling A as series in 1 /N and therefore cannot use the previous reasoning. There is no guarantee a priori that the higher order corrections in the asymptotic region are less important than the leading order. In other words, there is no linkage between the large momentum scale limit and 1/N  0 limit.  It is interesting to note that in the G N model it is possible to renormalize the Green's functions of the field a. This is probably connected with the fact that a interpolates a bound state. This is unlike the 0{N)  a model [45], where the auxiliary field does not  correspond to physical particle and its Green's functions are not renormalizable. 7.0.3  R G improved effective potential  The effective potential is equal to the expectation value of the energy density in the state where < cr >= Cc. As such it is best suited for the study of the vacuum sector of the theory and in particular dynamical symmetry breaking. We define the effective action T(ac) as r(a,)  =  j d^xac{x)J{x)-W{J)  W(J)  =  -log^  DuD^DÏje-  (7.180)  where W ( J) is the generating functional for connected n-point a Greens functions. Translational invariance ddictates that  be a constant. We then use ^  = — J to eliminate J  from r in eq(7.181) r((Te) = J d^xV{ac) .  (7.181)  The effective potential can be written as (7.182)  Viac) = NVo + Vi . Working in momentum space we find in lowest order Fo(ae) = Z^Tal - trlog(;5 -f-  .  (7.183)  It is easier to calculate the derivative ^^gl^"^ = M  =  Z^Tac-^JZ^trJ-^^j-^,  (7.184)  Zy^a,  The leading order VQ is a cubic function of \a\ (solid line on Fig.(7.13)) y„ = - ^  + ±.f.  (7.185)  It means that at this order four and higher Green's functions at zero momentum vanish. The effective potential of the field o to the order 0(1). V{(j) = NZ.Tal  - NixXogii + M) + \lr\ogD-\p)  (7.186)  where D-\p)  _arctg(^)  = -p + M +  (7.187)  It is convenient to take the derivative with respect to a before performing the momentum integral, so Via,)  = Z^Ta, - iZ,N)'^Tr{^  + M)-'  +  pdparctgi^)D(p)  (7.188)  The integral is well-defined provided D{p, a) remains non-zero for all euclidean momenta. By looking at small p we see that this requires M > ^p, and this is just the region where the leading order potential has positive curvature. Performing the momentum integration we get  J/pdparctg(^Mp) =  _  +  5  l o g ( - ) + finite .  (7.189)  Substituting the renormalization constants eqs.(7.168),(7.169), in eq.(7.186) and using eq.(7.188) we get a finite effective potential. po"-  1 -  8  log  (7.191)  " 2 7  As we have seen the explicit expression for the renormalized eflfective potential involves a mass p.. However this mass is arbitrary thus in the absence of dimensionless couplings a small change in the mass can be compensated for by an appropriate small rescaling of the field. This can be expressed as follows: dV dp  dadV op, oa  dV p  dV /i  dV da  (7.192)  where we used the definition of 7 , eq.(7.172). It is convenient to work with dimensionless quantities, we therefore rewrite V = (T^f{x) where x = ^ . Plugging this expression into eq.(7.192) we find: ( l + 7 ) ^ + 37)/(:r) = 0 where t =  log(3;).  (7.193)  The general solution of this equation is 3T  f = Ax  i+T  (7.194)  ^ N o t e t h a t o u r choice of r e n o r m a U z a t i o n c o n d i t i o n s w a s s u c h t h a t one o n l y needs t o p e r f o r m a shift i n (7 t o w o r k w i t h the p h y s i c a l field.  finite  (7.190)  Chapter 7. Renormalizable four Fermi interaction 0.8  -3  -2  -1  1  Figure 7.13: The effective potential for the field a. The solid line is the leading order result Vo(cr). The dashed Une is the next to leading order correction Vi(cr). Vi(cr) is not shown for small a, when it becomes complex. The arbitrary constant A can be determined by demanding that in the limit N  oo  we recover the leading order effective potential. Substituting the explicit expression for 7 from eq.(7.176) we get lim Ax-^  (7.195)  = Vo  Thus A = Vo and the R G improved effective potential is y(cT) =  p(y^\  ^ 37r  —  27r  ,0"  —  __i6_  Nt2  (7.196)  One can now check that expanding this expression in 1 /iV we recover eq.(7.191). A similar analysis can be extended to the four-fermi models with continuous chiral symmetry [49 .  7.0.4  Critical exponents of the G N model  Near the critical point the relevant scale is the divergent correlation length and all physical quantities scale with respect to it. At this region correlations at large distance are  not sensitive to the details of microscopic interaction. Their behavior is dictated by the essential characteristics of the model like dimension of space, nature of the order parameter and the underlying symmetries. The universal behavior of critical theories can be characterized by a set critical exponents. These determine how quantities like correlation length, specific heat and susceptibility scale in the critical region. We would like to calculate this set for the d=3 G N model. For convenience (and ease of calculation) we work in the symmetric phase whenever it is possible. The temperature (distance from criticality) is defined by A T = |T - Tel  (7.197)  where Çcr was defined in eq.(7.154). The critical exponent v determines the dependence of the correlation length ( on the temperature in the critical region. We can easily compute u by considering the renormalized two point function D~^(p)  M ^ ( A T ) " . To next to  leading order in 1 / N this propagator in the symmetric phase is (7.198)  where we used | = A T . Therefore  The critical exponent rj is determined by J d^xe^''^ < 0{x)0{y) > ~ fc'^+f? •\Ye actually calculated the value of this exponent when we calculated the high energy behavior of the a correlator Da{p) ~ —-^^g— therefore p2p 3^r^  The dependence of the correlator of the order parameter at zero momentum on the temperature is determined, at the critical region, by the exponent 7 : Da-{0) ~ AT~'' In our case in the symmetric phase  Therefore 7 = 1+ ^  (7.202)  The dependence of the expectation value of the order parameter on the temperature is related to the exponent (3. We can calculate this dependence from eq.(7.161). Since there are no log terms in the next to leading order there is no correction to the leading order (mean field) value /? = 1 . We can now check that this set of exponents obeys hyperscaling relations 7 = (2-77)v to next to leading order in 1/N.  ^5=^(1+77)  (7.203)  Chapter 8  Six Fermi Interaction  8.1  Leading order  In chapter 6 we have seen that in the leading order in 1/N the operator {•ipip)^ is marginal at strong coupling. In this chapter we examine in more detail the effect of adding this term and more general ones. Our interest is thus in the theory A = î^i-i^ + mo)V - ^ ( ^ » ' + ^  W)' •  (8-204)  However the direct calculation in the 1/N expansion for this theory, although possible, is quite cumbersome. Instead one can work with the theory Nn  ATT  £2 = ^ ( - « ^ +  mo)ip + aî>i)  + —^2 +  -^a^  (8.205)  In appendix F we give an explicit demonstration that the two models C\ and £2 are equivalent to next to leading order in  The equivalence is in the sense that renor-  malized Greens functions are the same in the limit A  oo-^. Since we shall calculate  the /? function only to next to leading order this equivalence is sufficient to our purposes. Although lagrangian £2 is classically unbounded from below we shall see that to next to in the leading order \u 1/N \t is bounded. Since the Feynman rules for £2 are simpler we work with it. We introduce the standard wave function renormalization for both fields a and ip. £ = Z ^ ^ ( - i ^ -f mo)V' +  Z^Zl'''aî,ilj  -f-  ^Z^a''  + ^Zl'^a^  (8.206)  ^ T h i s is s i m i l a r t o t h e équivalence between t h e 0* a n d Y u k a w a m o d e l s i n 3 + 1 d i m e n s i o n s [89]  The finiteness of the following quantities is required: rJ(o)  =  P=0 = T'''\p = 0,q = 0) d  ^r'tip) P=o  -  Tfip = 0,q = 0)  fermion mass  M 1  fermion wave function  1  a fermion fermion vertex  1  a propagator  V  0-3 vertex  (8.207)  The leading order gap equation is Ta + 7]a^ - (mo + a)  2A  mo + (J  (8.208)  = 0.  TT  It has a nonzero solution: a = M. For mo 7^ 0 or 77 7^ 0 the Z2 chiral symmetry is broken explicitly. The bare couplings are determined to be  ^0  -  Z/T = 1  ^  2A  (8.209)  vM  \M  +  TT  mo  =  ^2 2m TT 2  M  m2  where m is a finite scale. The a propagator is given by: ( 4 M 2 + p2)arctan(2^)  2m2  rjM  - +' 7r2M +' 2  27rp  (8.210)  For I77I < |77c| = f it does not contain tachyonic pole. Consistently for these values of rj the eff"ective potential is bounded from below V(a) = N  ^1 2m  / m2  2"^ + j^M  7]M 4  M\ 27r^  a^ +  3  ^ Tja^ 3!  -HO(l)  (8.211)  The fermionic fluctuations at this order stabilize the classically unstable a^ term. However the problem of stability will haunt us at higher orders.  8.1.1  Triviality of {^tpf near x C P .  Naive power counting indicates that the standard renormalizabihty proof goes through in this model like in the four Fermi theory Indeed the U V asymptotics of propagators did not change compared to (ipipy. The new vertex, a^, causes the appearance of log A divergencies i n three point function of a which are absorbed in rj. This by itself does not guarantee the existence of a nontrivial theory in the continuum limit. Leading order of an expansion is frequently not indicative of an existence of a U V fixed point. For example the leading order of loop expansion (classical approximation) does not difi'erentiate between Q E D and Q C D . Since there are no U V divergencies in this approximation, the /?-function is zero in both cases. Taken at its face value this means a line of fixed points. However as is well known the one loop corrections change this behaviour drastically making Q E D trivial and producing a U V fixed point at zero for Q C D . In our case in the leading order in 1 / N one also has /3{ri) = 0. Our goal in this subsection will be to calculate /? function for the couplings g and rj in the next to leading order in 1 / N and map the U V fixed points of the R G flow. To obtain the renormalization constants to next to leading order we calculate divergent parts of the efi'ective potential to 0(1). The efi'ective potential to next to leading order is given by  V(a) = N  2T  +  trlog i^-i^ + mo + yjZ^aj  + ^trlogD;i  (8.212)  The propagator Da should be calculated at arbitrary shift a. Expanding the inverse a propagator in 1/p to order l / p ^ we obtain p  8|(7  3  Substituting this back into eq.(8.212) we find the eff"ective potential 2. The renormalization constants  and 77 are given by the cubic terms in the efi'ective potential  V(a) = N  +•  3!  (8.214)  +  37r  We do not give explicit expressions for the cutofi" dependence of mo and T since this is not essential for the discussion. Z„ 77  = =  1+  16 37r2Ar  log  (8.215)  r]{p) 1 + 4 ^ ( 1 - 2 7 7 V ) ) l o g ' ' ' ^ ^  The (3 and 7-functions are (8.216) 16 la  =  37r2Ar  The origin 77 = 0 is U V unstable fixed point. The coefiicient of the linear term in the /^-function determines the U V dimension of the operator cr^ near the x C P : k^] = 3 +  8  (8.217)  We find therefore, that this operator is irrelevant near x C P . ^Taken literally the terms i n eq.(8.213) w o u l d l e a d t o i n f r a r e d divergencies. I n r e a l i t y however, t h e I R cutoff is f u r n i s h e d b y n o n z e r o value o f a a n d we j u s t s u b s t i t u t e i t for t h e cutoff whenever i t is necessary. T h i s p r o c e d u r e i s v a l i d for l a r g e values o f t h e field a » M. ^ T h e t w o other r e n o r m a l i z a t i o n c o n s t a n t s a r e :  ml =  4  + 2r]M + 7)^M -  167/771*  7r4M2 -I-  47/m2  TT^M  47/2 M 2  1+  21og(A)  41og(A) 2  167/2^2  ^2 4771  TT^M  7r*M  -f-  47/M2  A 87/2m2  T/TT 4  7/2  ^2 87n 3?r2M  ^2 4m 2M  3 ^ ^ 7/M  37r  8Mr72  8.1.2  Townsend - Pisarski unstable fixed points.  The extension of the parameter space, in principle can lead to the appearance of a new U V stable fixed point. Indeed, we find that the points  = ± ^ are U V stable fixed  points. These values of 77* are however outside the leading order stability region. To check the stability in the next to leading order we calculate the R G improved efi'ective potential. The R G improved efi'ective potential obeys the equation dV _ ^ dfj,  d  d  d  da  dp  dr}^  subject to the boundary condition that for A?"  (8.218)  00, F ^ VQ the leading order potential.  The explicit solution with (3 and 7-functions given in eq.(8.217) is 16  V{a)  =  7^  N  16  16  _»2  (8.219)  +  The large field asymptotics is 16  n.) = ^(èw'+|.0 The theory is unbounded from below since \r)*\ =  (8.220) > |.  To summarize, the new U V fixed point found in this subsection does not define stable continuum theory. This is quite similar to the situation in (j)^ theory [88], where one also finds U V fixed point outside the stability region. This phenomenon is not unique and occurs in a large class of theories. We have checked several models in search of a stable U V fixed point. In all the cases the U V fixed points were found outside the stability region. Here is the partial list of theories considered. 1. The minimal theory of this type includes only one 0(N) multiplet of two component real Majorana fermions £ =  -a)iP  + ma + ^ - + '^  (8.221)  The stability region here is \r]\ <  while the U V stable fixed point is found at 7y = j ^ .  2. The 0(2) symmetric theory of 2N Majorana fermions C = ^V^(-^/9- a - a i r i - (TaTg)^ + ' " ^ + ^  + ^  + '?|r +  (^"^^S)  The stability region here is defined by the two conditions: |?7| < ^ and [77 + 3C| < 7. There is an unstable U V fixed point at (77, () = (0,0) and four U V stable fixed points at (0, ± ^ ) and (±.105, ±.402). A l l are again slightly outside the stability region. 3. The Z3 symmetric theory with 2N complex Weyl fermions £E = ^{-i^  + ar+ + ^ V _ ) ^ + A ^ T a V + ^{a*^  + a^)  (8.223)  o  where ipf^ is an SU(N) vector of two (a = 1,2) complex two component Weyl spinors, a is a complex scalar field and 7 matrices are two component. This theory is interesting since the symmetries forbid any mass term. At 77 = 0 the lagrangian is equivalent to a four-Fermi theory with U{1) chiral symmetry. This lagrangian preserves also parity X -> -X,  y  The addition of  y,  —>• 72^2'^, a  a* and chiral symmetry -0  e'^'^'ip, a  e'^a.  + a*^ interaction breaks chiral U{1) down to Z3 a  : 2irn  e'~a  xp  e'-^'i) .  (8.224)  while preserving the two discrete symmetries. The remaining (discrete) symmetries prevent the appearance of all possible mass terms. Other interaction terms are excluded as well. The symmetric theory therefore contains only two parameters 77 and g. The calculation of the P function to next^ to leading order gives (3(TJ) = ^rj.  The origin is  again unstable and there is no U V stable fixed point. 4. SU{2) broken to C/(l) type theory C =  ^ff.r)^p + ma^ + ' ^ + r^^+  (8.225)  Figure 8.14: R G flow of the couplings ( and 77 as functions of t = log(m) where i = 1,2. The stability region is more complicated than 3. The R G flow in the 77 — ^ plane is shown in fig.(8.14). The locations of the stable U V fixed points (C, rj) = (0, and (5,5)  are again outside the stability region.  We conclude this subsection with the following comment.  A lattice (regularized)  (•^V')3 theory i n principle can not be unbounded from below, since the fermionic Hilbert space at each point in space is finite. However in our analysis the terms of the form (^)" were dropped. If one keeps all the cutoff dependent terms, the effective potential will be stabilized and will have a global minimum at cr oc A. This is the characteristic behaviour of an irrelevant (nonrenormalizable) operator: if one insists that its effect be felt at small values of cr, the large values of the field are affected so strongly that the theory is driven far away from the criticality and continuum limit.  8.2  Discussion and Conclusions  We examined the deformations of chiral Z2 invariant conformai field theory in d=3. This theory was shown to be an IR fixed point of a chiral invariant Yukawa model. The x C P was identified with the critical G N model eq.(6.123). At the x C P the operators ijjip and {'ipipy are relevant while (tptp)^ turned out to be irrelevant. To show the relevance of m'tpip and A/?(V'V')^ we used leading order 1/N expansion while irrelevance of ('i;^'^)^ is seen only in the next to leading order. The operator (iptp)^ is similar to (j)^ around gaussian C P . Both are trivial near the corresponding fixed point. The addition of this irrelevant operator to the Lagrangian leads to appearance of a Townsend - Pisarski U V fixed point. This fixed point is however located outside the stability region of the theory and does not define a viable continuum limit. The same approach can be used in search for other nontrivial renormalizable continuum field theories. One starts from a gaussian fixed point of free fermions and scalars. Near this fixed point the operators of the following types are relevant: cr^, a^, a'*, ipip and a-ipip. The free massless Lagrangian perturbed by these operators generally has IR fixed points. These correspond to zeroes of the Callan-Symanzik /^-function. In this way one arrives at new three dimensional C F T s . Each C F T found in this way defines its set of relevant operators O, (O,- are generally not relevant near gaussian theory). The U V dimensions of these operators coincide with their IR dimension on the critical (massless) hypersurface of the original Lagrangian £ .  Deformations of the C F T with Oi define  nonperturbatively renormalizable massive Q F T . In this procedure symmetries are very helpfulT~yymmetnes ofX are preserved at t h F I R f i x e d point. The straightforward generalization of the chiral Z2 case, considered in detail in the previous chapters is a Yukawa model with a larger chiral group G. The IR fixed point will be a G- invariant critical four Fermi theory. The relevant deformations generally  Figure 8.15: continuous lines represent deformations of the Yulcawa type gatpip. dashed arrows represent deformations of the form ("tpXijip)"^ while dotted arrows represent deformations of the form mipip include fermionic masses and four Fermi couplings (at least as suggested by the large N limit). Different four Fermi models are connected by the R G flow of relevant couplings. In R G picture any local Q F T corresponds to a particular R G trajectory in the space of theories. A trajectory starts at U V fixed point and either ends at another C P (this line is sometimes called interpolating trajectory [94]), the IR fixed point or flows towards a massive theory. For simplicity we consider the case of 0{M)  x 0{N)  symmetric four Fermi theories (8.226)  C =l^f{~-i^^la-AT^i^^ where ^ is a Majorana fermion, a = 1 . . . M , « = 1 . . . A'',  are real symmetric matrices  and the coupling T"^ is at its critical value. Deformation of the theory eq.(8.226) with mass of one of the fermions 7ni/i°Vi (tiiis does not spoil the 0{M)  symmetry), makes  massive. This massive multiplet disappears at the IR fixed point. This is then the interpolating trajectory leading from 0(M)  x 0{N)  to 0{M)  x 0{N - 1) invariant x C P .  The deformation with four Fermi interaction term (JAA.(3AB(^B reduces chiral 0{M)  to  the symmetry of A/5. For example starting with 0(2) symmetric model JC =  _  - a2T2)iP + ^  (8.227)  the deformation with A/^trl tracks the interpolating trajectory towards the Z2 critical G N eq.(6.123). In this way one gets a grid of C F T s Fig.8.15. Here empty circles represent free theories containing M massless bosons and M N massless fermions. The full circles represent non gaussian fixed point. The first row contains "magnetic universality classes": Ising, X Y etc. The first column represents the Z2 symmetric x C P s that we have considered. The (M,N) full circle represents the 0 ( M ) x 0(iV) invariant xCP^. This picture is only a sketch of the general pattern. Other theories with symmetry breaking of the type G/H can be considered in a completely analogous fashion. Incorporating in C eq.6.122 scalar self interaction cr'* opens a possibility to find new fixed points, not necessarily of magnetic or four Fermi type. For example at M=N=1 one should find a supersymmetric conformai point. A n interesting question is whether this class of theories exhausts all possible C F T s constructed of scalar and spinor fields.  *It is i n t e r e s t i n g to n o t e , t h a t at least i n 1 / N e x p a n s i o n (if one takes i t seriously for i V = 1) i t is i m p o s s i b l e to i n t e r p o l a t e between t h e second a n d t h e first rows of the d i a g r a m . T h e a d d i t i o n of a f e r m i o n i c mass t e r m to t h e c o r r e s p o n d i n g four F e r m i m o d e l , either breaks the 0{M) s y m m e t r y or drives all the b o s o n s massive, p r e v e n t i n g the appearance of the n o n l i n e a r cr- m o d e l as a n I R fixed p o i n t .  Appendix A  Some properties of Padé approximants  In this appendix we give some general properties of Fade approximants and bounds on their convergence. We also define Padé approximants for analytic matrices and discuss the modification of these theorems and their relation to the scattering problem. 1. definition Given a formal power series oo  j=0  the [L/M] Padé approximant /[^/•^'(^) =  , is a rational function defined by:  h-  fl+l  det ft  fl+l  fl+M •  fl-M+l  fL-M+2  EUfj^^  (A.228)  fl+l  det fl  fl+l  fl+M 1  where  = 0 if j is negative. This definition may be useful only for the lower order  Padé approximants, for numerical calculations a recursion method due to Baker [17] is much faster.  2. Existence B y the previous definition it is not clear whether the Padé approximant for a function always exists. In fact it is easy to give a counter example e.g. f{z) = l+z"^. However the addition of the condition QM(0) = 1 takes care of such cases. 3. Uniqueness When it exists the [L/M] Padé approximant is unique. This result is easy to prove. Suppose ^(^) = # 7 7 i = ^  + 0(.^^+^+^)  then by multiplying both sides by QM{Z)QM{Z) QM{Z)PN{Z)  - PN{Z)QM{Z)  (A.229)  we get =  (A.230)  but since the left hand side of this equation is a polynomial of degree N + M this is possible only if QM{z)Pr,iz)  = 0 => ^ = ^ QM[Z) Quiz)  - P^{z)Q^iz)  (A.231)  q.e.d 4. Invariance properties Let g{z) =  and /(O) # 0 then /[^/^l(^) =  ^ i ^ .  This theorem is also easy to prove  '^'^  QM(£)  _  PLiz)g{z) - Quiz)  p,{z)  -  p;xz)  -  P,{z)fiz)  _ PL{Z) -  -^('  the theorem now follows by the uniqueness theorem 3. 5. Convergence  QMiz)f(z)  i[z)p;x^)  ^  (^-232)  The most recent and advanced result in this area, is due to Stahl [66], however since it depends heavily on modern potential theory we give here another less general theorem by Nutall [65 . Let f(z) be any meromorphic function, R an arbitrary positive number. Let e, 6 be two arbitrary numbers. Then it is possible to find an integer N such that îov n> N the [N/N] Padé approximants of f{z)  satisfy  |/(^) - /™(-^)|  (A.233)  < e  for all z < R except for a domain D„ of measure smaller than 6. This is often called convergence in the measure. Under more stringent conditions one can prove [15] uniform convergence in the measure. Padé approximants of analytic matrices For analytic matrices S{z) one has the expansion Siz) =  '£AnZ^  (A.234)  n=0  where A„ are matrices belonging in general to a non-commutative algebra. One can therefore define in general left hand and right hand Fade approximants depending on where we place the denominator. However, it is easy to prove that if S{z) has a left and right Fade approximants they are identical. This means that all left hand approximants are identical and all right hand approximants are identical (by the scalar uniqueness theorem) and therefore the approximation is unique. invariance properties 1. If S{z) is symmetric then 5*1^/^'(^) is symmetric. 2. If S{z) is hermitian then S^^^^^{z) is hermitian.  Appendix A. Some properties of Padé approximants  3. If S{z) is unitary then SW^\z)(S^^/%))^  = 1  therefore the [N/N] diagonal Padé approximants are unitary  Appendix B  Proof of absence of mass-shell singularities  In order to show that there is no mixing of orders in the expansion of KDQ due to mass-shell singularities we'll use Landau's analysis of mass-shell singularities of Feynman diagrams [40]. A n arbitrary bosonic Feynman diagram can be written as  As a function of the external momenta G may have two kinds of (real) singularities. 1. End point singularities-that is one of the singular points approaches one of the end points of the contour. 2. Pinch singularities- that is the contour is pinched by two or more singularities that approach it from above and from below^ . A necessary condition for such singularities is given by the following set of equations either or  ai  =  mf  =  0  (B.236)  for each i together with d dki  E«i(ç?-mf) = 0 ^  E9i«' = 0  (B.237)  for each loop momentum integration variable fcj. ^for e x a m p l e t h e i n t e g r a l  <^'^ (w-z)(w~a)  a n d 2 = 1 a n d a p i n c h s i n g u l a r i t y at ^ = a  =  {j?-a)ï)  ^""^ P"^'^* s i n g u l a r i t i e s at z = 0  This condition means that in the configuration that produces singularity, either all internal lines are on shell or the corresponding a,- is zero. In the latter case the momentum Qi is irrelevant except for overall conservation of momentum so the singularity of the diagram is exactly the same as the singularity of a the same diagram with that line absent (cut). The singularity of a given diagram corresponding to no a,- = 0 is called the leading singularity. Lower order singularities can be obtained by contracting lines (i.e setting one or more of the aj s to 0). Once we know where the singularity points are, the nature of the singularity can be obtained by expanding around these points. The nature of the real singularities of an arbitrary bosonic Feynman diagram is given by: if 7 > 0 or ( 7 ^ Z) < 0 JTITI log(^)  if 7 < 0, 7 e Z  where T is the solution of the Landau equations [40] which determine the surface of real singularities of G{pi, • • • Pn) and 7 = i [ / — c H- - l | L , here / is the number of internal lines in G, L is the number of loops, d the dimension of space-time and c is the number of cuts (for the leading singularity c = 0). For any diagram in the scalar theory eq.(8.208) 7 can be related to the number E of external legs and the numbers V4 of Ve of (j>^ vertices 7 = ^ 4 ( 1 - f ) + and d = 3 we get 7 = 1 -  vertices and  - rf) + ( f - D i d - 1) - f. Taking E = 4, c = 0 The only singular diagram is therefore the one with  y4 = 2 and H = 0 7 = 0 i.e. the "fish" diagram. This classification of mass shell singularities can be used as a bound on the singularities one may expect in a similar fermionic theory (i.e Gross-Neveu or Thirring model in 1-1-1 dimensions or G N model in 2-1-1 dimensions). Since any fermionic diagram can be written as a sum of bosonic diagrams with coeflBcients that are polynomials in p the degree of divergence of such diagram is not more than that of the same diagram where all the lines are bosonic. In the case of large N models there may be an additional complication due to the fact that the leading order composite field propagator may not have a  simple form. However if the full propagator describes an asymptotic state it must admit spectral representation:  Dcip")  =  -2  2 +  /  pi — rw^ Jim^  2  2-  p^ — m'^  This means that in fact we are back to the ordinary bosonic theory.  Appendix C  T h e p h y s i c a l m e a n i n g of the a u x i l i a r y field  The 1 / N expansion is powerful enough to resum the perturbative, small coupling , expansion and sometimes produce bound states. These bound states correspond to the auxiliary fields introduced "by hand" i n order to integrate out some of the field variables. Typically, in theories with more than one parameter a true bound state appears in the leading order (i.e. as a pole in auxiliary field propagator) for some range of parameters. However in the case of the Gross-Neveu model and other models, one finds in the leading order only threshold behavior. In this case the leading order singularity that one finds in the Greens function of the auxiliary field should not be taken as evidence for a bound state. Since the choice of auxiliary field is dictated by our calculational difficulties it may happen that the composite field we introduced is just a hox. For example in the exactly solvable 0{N)  NLcrM one finds that there are no bound states except the elementary  bosons of the theory. This is not so surprising since we can think of this model as the infinite coupling limit of the (p^ model C = ^^^n^^'n + g{n'^ - if  (C.238)  /Li  which has the wrong (repulsive) sign for the interaction. On the other hand the GrossNeveu model is known to contain a rich spectrum of bound states. The auxiliary field o has the same quantum numbers as the lowest isoscalar bound state. It is therefore interesting to see how will this be seen in the 1/N expansion. In order to check this more carefully we shall calculate the leading order propagator for the a field in the 1 / N expansion. The starting point for this calculation is the generating  Figure C.16: Diagrams contributing to the 0{1/N) corrections to the a propagator functional of a Greens functions. Z(J) = JD^J'i''^^^^iogm^^^'^^^<^)+k^.<^'+-^'^ we shift the a field hy a  p+ ^  and expand the trlog  + •'^'KZ)  trlog(^ + a) =  239)  +'"•'{^^2)^  (*^-240)  in leading order the a field becomes dynamical. The a two point function < a(x)a{y) > is found by taking the second functional derivative of Z, Dtr{x,y) = zfô)sjix)sjly)leading order in 1 / N we get  in 1+1 and 2+1 dimensions the integral in eq.(6) is divergent but substituting the solution of the gap equation (eq(3)) for ^ this divergence cancels. Rotating back to Minkowsky space we get: 1+ 1  D,{p)=  i^^^/p^ V 4 M 2 - p ^ a r c s i n (2M ^)  2+ 1  Da{p) =  i27rv^ (4M2-p2)arctanh(^)  Both propagators are singular at  = 4j\^2 ^ however the singularity is not a pole  singularity. In 1+1 dimensions it is just a square root type singularity while in 2+1 dimensions the pole coincides with a logarithmic singularity. It is useful to write the Kallen Lehman representation for these propagators. D.  ds'  S'  —  (C.242)  S  -ImD,(s)  p{s) =  TT  In the 2+1 dimensional case p{s) =  A'Ky/s ( 4 M 2 - 5 ) [ l o g ( | ^ | ) 2 + 7r2  while in the the 1+1 dimensional case it is easier to express p in terms of rapidity variables P{0) =  27rcoth(f)  7r2 + ^2  In both cases the spectral density function p diverges at 5 = 4 i.e. on the threshold and has no other peaks. Higher order corrections will, in general, shift the location of this divergence thus creating a pole separated from the cut. The diagrams contributing to the order 1 / N corrections to the a propagator are shown in fig.(C.16). In 1+1 dimensions one can use the Kallen-Toll cutting rule [72] to reduce all these diagrams to one loop bosonic diagrams with complicated coefficients. The result is : -2 g Bjp - q) f 27r2 q2 _ 4 ^ 2 B{q){q'^ - 2p • q) \ ' q^m^f 4g2 g2-4M2 \2M'^{q^-2p-q) q'-{p-qf 1 B{p + q) f^_2{p + qf\ 1 + q^ + 2p-q B{q) q^+p-qj q^-2p-q 2{p-qf\ q^-p-q)  +'  +i4:M^-p-q)'  2M\q^-p.qf >.Bip-q)\ Biq)  ip-qfBiq) \ B{p - q)  +  p-qm^-p-q)ip-qf 2M\q^ ~ P • qf ^  Bip - q) B{q)  (1-  2ip.q)iAM^-p-q) (C.243)  where fx'' - 4'  ,2_4\l/2  1/2  m  log  X^  + 1  (C.244)  This integral is naively equal to zero. To see that one has to translate q in the terms  2 M 2 ( ç 2 _ p . ç ) 2 \^  B{p-q)  /  to q + p (which is legal because this term is only logarithmicaly divergent) the whole integrand then becomes odd under q  -q.  Schonfeld [21] who first calculated these  diagrams used the following method for calculating the binding energy: for s — AM'' <C 4M2  ^  ,  lAA/r2 AM^-s_ c  bh  4M2  or M2 4M2  1 -  (C.245)  62  am  where b is the sum of all I P I diagrams contributing to the 0(1/N) a propagator evaluated at p2 = 4 M 2 . We therefore see that a 1/N correction to  becomes at most 1/N''  correction to the binding energy. Sconfeld concluded that since the integral eq.(C.243) vanishes AM  = 2Mf — M^ = 0 + 0{^).  expansion of the exact result eq.(12).  This is however in contradiction with the Haymaker and Cooper [21] who were already  aware of the exact result repeated the calculation and got the same result. They suggest that the special kinematics of 1+1 dimensions may allow other graphs to shift the a mass. There are two possible explanations to this apparent contradiction. One is the existence of mass-shell singularities in higher order diagrams, thus for example if an order 1 / N 2 diagram contained a term that goes like 1 /yjArn' — p' then neer the threshold this diagram is in fact of order 1/N (assuming the binding energy is proportional to 1/N''). Although Haymaker and Cooper do not mention this point, Schonfeld claims that there  are no such diagrams (i.e. all higher order contributions to the inverse propagator are regular at p' = 4M^). The calculation of the S-matrix elements for the 2  2 scattering  suggest that such diagrams could be the ones in fig.3 however an explicit calculation shows that this diagram has no infrared problems.  We therefore have to go back to  the 0{1/N) correction and check the integral more carefully. Expanding the integrand around ç = 0 shows that it is singular at this point. However, the structure of this singularity is very complicated and makes the numerical evaluation of this integral very complicated.  Appendix D  Non Linear a model  Similar analysis applies to the calculation of the /? - function in the 2+1 dimensional 0{N)  symmetric nonlinear a - model (NLo-M).In this case we can even compare the 1/A^  predictions with recent lattice data [52 . The lagrangian density for this model is (D.246)  JC = ^d^ndf'n 2g  where  fields  are 0{N)  vectors with the additional constraint that n^-n' = 1 .  (D.247)  The generating functional for n{x) Greens functions is ^  Z = JDn6{n^ - lyï'^^i^s^^^'^^-J-n)  g^g^  In order to implement the constraint in a convenient way we introduce an auxiliary field a: Z{J)  =  J £)njDûe-^^+J-"  Spin, a)  =  ^ fd^xd^nd''n 2 J  (D.249)  + a{n^-^)  .  g  where we rescaled the n fields as to comply with the conventional normalization of the kinetic term. We follow Arefeva's derivation of the 1/N expansion [44] and choose a coordinate system in isotopic space such that its axis coincides with the rii direction.  Integrating over the remaining N-1 n fields we get the following result (D.250)  SE{ni,a)  =  -NSi{nua)  + S2(ni,a)  Siinua)  =  ^trlog(a2 + a) + Jd^x ^d^n.d'^n, + |(n? -  S2{ni, a)  =  /  J  =  ^)  J i m + ^J{d^ + a)~'3 {J2,---JN)  We shall calculate Z{J) by expanding the action —NSi + 5*2 in the neighbourhood of the saddle point of -Si. The saddle point (nxcCKc) is defined by the following set of equations (in momentum space): d^p 1 f ^27^3 ^ p2 T -_|. Q,^+ \g2+ <  =  0  d'unie + OLcUic =  (D.251)  0  ^ The first equation contains a divergent term which can be renormalized by the following fine tuning:  gl  47r d^p 1  o'er  J 27r3p2  where / i is an arbitrary mass. There are two solutions to these equations  1.  «e = 0  2.  nic = 0  The first solution corresponds to g' > g^, 0(N)  symmetry is broken to 0{N  nL = i ^ = ^ gets non-zero expectation value and the  - 1). The n i field becomes massless as it should  according to Goldstone theorem. For  < gl the symmetry remains intact  ^We w i l l o n l y t r e a t here c o n s t a n t s o l u t i o n s of these equations.  = 0.  In order to renormalize the only counter terms that are needed ^ are  where a and b are renormalization constants and we assume that they have an expansion in 1 / N . Both these constants can be determined order by order by calculating the self energy corrections to the n i particle. In leading order the Feynman rules in fig.(7.12) give  ao(i'^+ < a >)  = 0  -  (D.253)  Prom which we get, using the table in appendix D, 2"° = 1 and p^ = ND{0) Choosing bo =  -  1 + 6o 2^2  1 / A V27r2  +2  p 47rJ  (D.254)  - ^ ) we get STT  In the next to leading order we have the following contributions to the rii propagator  0 a^iP'+p'')-  d_  +  = 0 (D.255)  * T h e n a i v e u l t r a v i o l e t d i m e n s i o n of t h e a a n d n fields are 2 a n d 1/2 r e s p e c t i v e l y so i n p r i n c i p l e we have t o a d d also n * , n^, an^ c o u n t e r t e r m s . However we do n o t t r y t o r e n o r m a l i z e G r e e n s functions of a p a r t i c l e t h a t is G r e e n f u n c t i o n s w i t h a legs w i l l r e m a i n divergent. T h e reason for t h i s u n c o n v e n t i o n a l r e n o r m a l i z a t i o n p r o c e d u r e is t h a t s u c h a n a t t e m p t is inconsistent w i t h t h e c o n s t r a i n t < > = 1 [51]  Appendix D. Non Linear a model  116  1  n propagator  a propagator  8irp  ^ W  a'^ctan(^)  1  1 VN  ann vertex  N_  n tadpole  charge renormalization counter terna  _  wave function counter term  ——6  ap2  Figure D.17: Feynman rules for the 1/N expansion in the Non-Linear a model. This gives two equations, one for the coefficient of  from which we get  and one for the mass part from which we deduce  In the last equation the contribution of the first two diagrams cancel. Rewriting the Lagrangian in terms of renormalized quantities Tib =  yfz'nu \ Zn = a a  gl so we get  =  g'^Zg  ;  Zg^ab  (D.258)  The next-to-leading order /^-function (in notations of [44] A = g^A) is ^(A) =  1-f  32 STT^N  (D.260)  ) ( i - ^ ) + o(iï)-  Comparison with experiment Manousakis and Salvador [50] used Monte-Carlo simulation on the lattice to investigate the phase structure of the 0(3) NLcrM. They found that in the limit T = 0 this model undergoes a phase transition in the coupling at  = 1.45(1) where a is the  lattice spacing and m.c. means momentum cutoff. We would like to compare this result with the results we got from the 1/N expansion. To this end we go back to eq.(4) and 2  note that gm.c. = lattice spacing a ~  If we used the naive relation between the momentum cutoff and the we would obtain  = ^ + OÇl/N"^) i.e. for N = 3 this means  = 2.1. Actually this is just a crude estimate, we can obtain a much more accurate comparison by calculating the analogue of eq.(7.154) on an infinite cubic lattice: (D.261) For N=3 this yields  = 1.32 which is within 8 % in agreement with the Monte-Carlo  result. Manousakis and Salvador also calculated the slope of the /? function at the critical point,  and found pi = 1.28 ± 0.05. In the large N limit gm.c. ~ a so the leading  order result is /5i = 1 . In the next to leading order we get from eq.(D.260) for iV = 3 Pi =  1.36  Appendix E  Calculation of U V dimensions in the framework of the four - Fermi models  In this Appendix we show that U V dimensions of various operators calculated within (strong coupling) Yukawa model can be obtained directly in the framework of the four Fermi model. The Lagrangian of the four - Fermi model is C{ï>, xP) = -ï,i^ï> - ^ ( # ) '  (E.262)  The generating functional can be rewritten introducing an auxiliary field a ^.  Z = yX>i/iPVÎ^ae-/^>(^'^''^)  (E.263)  where NT Cx (ip, ip, a) = -xpi^ip + aipxp + —-a^ Classically, fpip = -NTa.  (E.264)  This relation remains true quantum mechanically. More  precisely, all Green's functions of a coincide with those of the composite operator - ^-ipip up to contact terms. This is seen as follows. Green's functions of a are generated by introducing a source j Z[j]=  JvrpVxpVae-f^+^''  ^ N o t e t h a t t h e r e n o r m a l i z a t i o n of t h e t h e o r y c a n be p e r f o r m e d i n e i t h e r f o r m .  (E.265) T h e n u m b e r of  n o r m a l i z a t i o n c o n s t a n t s for L a g r a n g i a n is larger since t h e r e n o r m a l i z a t i o n of the a u x i l i a r y field is r e q u i r e d  For example, the propagator of a is 1 Z[0]  (52  6j{x)èj{y)  (E.266) j=0  Integrating over a in eq.(E.265) one obtains < (r{x)a{y) >  =  1  <52  Z[0] Sj(x)Sj{y) 1  I < il;tp{x)i;i){y) > +-77^S{x - y)  (E.267)  The same is true for any Green's function of a. Thus operators a and xpip are proportional after subtraction of contact terms. The proportionality factor is the renormalization constant Z^.^ of the composite operator ipip. The U V dimensions of the operators are equal since contact terms should be subtracted in order to define the renormalized correlators 6  The correlator of a, to leading order in 1 / N has the following U V asymptotics: Da{p) oc l/\p\. Therefore we obtain that the U V dimensions of [cr] = [ipip] = 1 i n agreement with the results of section 2. To this order i n 1/N more complicated Green's functions factorize and consequently we arrive at U V dimensions given in Table 1.  ®The p r o p o r t i o n a l i t y o f these o p e r a t o r s was n o t p r o p e r l y u n d e r s t o o d i n [87]. T h i s is t h e source of t h e i n c o r r e c t values for U V d i m e n s i o n s o f ipip, {îpipY a n d aipijj o b t a i n e d i n t h a t p a p e r .  Appendix F  Equivalence of {fpiif and  models  In this Appendix we show that two Lagrangians £i and £2 eq.(8.204) and eq.(8.205) are equivalent at least to next to leading order in 1/N expansion. Introducing by standard methods two auxilliary fields a and a the Lagrangian Ci can be rewritten as N Nn L = -ijji^ij) + moiVcr - — 0 - 2 4- —J-a'^ - a{Na - ipip)  (F.268)  The leading order effective potential is V{a, a) = N{moa - — + ^a"" - aa -  + 7r2  37r  ^}  (F.269)  The gap equations are -mo + -  a  \a\  + -  =  2A  a  (F.270)  — a  The bare coupling T expressed via the finite parameter t is (F.271) We introduce renormalized fields by a  —> Zua  a  —>  +  Zi2a  ^210" + Z22O!  (F.272)  The leading order effective potential is V{a,a)  =  + aZu)-<y'(Jj;Zl  N^mo{aZn  ZnZi2 + ^\aZ22  + ZnZ2i  + Z11Z22 + Z12Z21  + (TZ2I\^ + ^ {aZu  +  2A  + ^Zl^  Z21Z22J  (F.273)  + (F.274)  + «^12)^  The wave function renormalization that renders V finite is Zn =  2A  TT  Z22 —  ^12 — ~  Z2I  =  The cutoff dependence of the bare couplings mo and 77 is mo ~  -TT  77 ~ p-. Note that  the renormalized effective potential is independent of a. As a function of a it is identical to the potential eq.(8.211). The inverse propagator matrix (for arbitrary values of the fields) elements is -1  v-\p)  (F.276)  = [  where D^pY^  2  -i-,D.{pr'  A  is the leading order inverse a propagator defined in eq.(8.210). In order  to find the next to leading order effective potential we calculate the eigenvalues of this matrix 2  " V  4  4A2  The second eigenvalue A2 does not depend on the fields and therefore the effective potential is given by TrlogP-^ = T:ûogD-\p)  (F.278)  Therefore it is the same as the effective potential for a when one uses the lagrangian Ci.  Appendix G  Table of Integrals  G.l  2+1 bosonic integrals  A  m  27r2  47r  iirp  (G.279)  (G.280)  o  P.  " Sm^ + p^  327r(p2 + 4m2)2  F  (G.282)  F  0 —I  - , —^— \. o  =  47r(p2 + 4m2)2  ^^''^^'^^  8 , .A ^ l o g ( - ) +finite 7r2 m  (G.284)  0/'  -0  (G.285)  '4A2  0  0  27r2  +  8mA\ TT  +  finite  8  A •log(—) + finite 37r2 ""^m'  --t.  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