The Quartic Interaction in the Large-N Limit Of
THE QUARTIC INTERACTION IN THE LARGE-N LIMIT OF
QUANTUM FIELD THEORY ON A NONCOMMUTATIVE SPACE
By
Philip Albert DeBoer
B. Sc. University of Prince Edward Island, 1999
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MASTER OF SCIENCE
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DEPARTMENT OF PHYSICS AND ASTRONOMY
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THE UNIVERSITY OF BRITISH COLUMBIA
August 2001
© Philip Albert DeBoer, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of Physics and Astronomy
The University of British Columbia
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Date: Abstract
With a view to understanding generic properties of quantum field theories defined on spaces with noncommuting spatial coordinates (termed noncommutative quantum field theories), two simple models are considered. The first model is a theory of bosonic vector fields having an 0(iV)-symmetric quartic interaction. The second model is the fermionic counterpart of the bosonic theory, and is known as the Gross-Neveu model. In both cases the study is conducted in the simplifying large-TV limit.
Unlike in the commutative case, the noncommutative theory gives rise to two inequiv-
alent quartic interactions of the form (2)2 and ( interaction. ii Table of Contents Abstract ii Table of Contents iii Acknowledgments v 1 Introduction 1 1.1 Motivation 1 1.2 Quantum Field Theory 4 1.2.1 Saddle-Point Approximation 5 1.2.2 Effective Action 7 1.2.3 Large-iV 8 1.2.4 Renormalization 10 1.3 Moyal Product 10 1.3.1 Construction 11 1.3.2 Properties 14 1.4 Noncommutative Quantum Field Theories 14 2 Noncommutative O(N) 04 Theory 16 2.1 Introduction 16 2.2 Local Auxiliary Field 18 2.3 Two-point Propagator 19 2.4 Self-Energy 20 iii 2.5 Symmetric Vertex 22 2.5.1 Four Dimensions 26 2.6 Beyond the Classical Case 27 3 Noncommutative Gross-Neveu Theory 32 3.1 Introduction 32 3.2 Local Auxiliary Field 33 3.3 Perturbation About the Classical Solution 33 3.4 Two Dimensions 35 3.4.1 The Commutative Theory 35 3.4.2 The Noncommutative Theory 36 3.4.3 A Double-Scaling Limit 38 3.5 Three Dimensions 38 3.5.1 The Commutative Theory 39 3.5.2 Noncommutative Theory 40 4 Conclusions 44 Bibliography 46 iv Acknowledgments Thanks are due to my supervisor Dr. G. Semenoff for his guidance and insightful com• ments during the course of this work. I would also like to thank Emil for his willingness to answer all my little questions. Thanks also to Mark and Ben for their encouragement. Finally, I would like to thank Drs. J. McKenna and especially K. Schleich for their support during the final stages of this work. Great is our Lord and mighty in power; his understanding has no limit. Psalm 147:5 NIV Chapter 1 Introduction 1.1 Motivation Recently there has been much interest in noncommutative quantum field theories. The low-energy limit of string theories with background antisymmetric tensor fields [1, 2, 3, 4, 5, 6] gives rise to certain noncommutative quantum field theories. They retain some of the interesting features of string theory such as nonlocality, which can be studied in the more familiar context of quantum field theory. Since the string theories are consistent quantum mechanical theories, the noncommutative field theories which are their consistent zero slope limits should also be internally consistent, that is, unitary and renormalizable. In fact, for some theories, unitarity has been demonstrated explicitly at one-loop order [7]. Because the theory generated by the string theory limit is difficult to study, simpler quantum field theories are being studied in the noncommutative regime. The field theories which arise from string theory are the Yang-Mills theories; these quantum fields have a nonabelian gauge symmetry. These theories are technically diffi• cult to study; one of the problems is a lack of local gauge-invariant observables in the noncommutative case. As a result, in order to better understand generic features of these theories, simpler examples have been chosen for study. In particular, the theories studied here are theories of N fields with an O(N) symmetry and having quartic interactions. Both bosonic and fermionic models will be examined. These models have the advantage of being solvable in the large-N limit, but since these theories do not arise as string Chapter 1. Introduction 2 theory limits there is no reason a priori to assume internal consistency. The concept of field theory on noncommutative geometry is not entirely new, however. Because quantum field theories are plagued by divergences arising from short-distance behaviour, noncommutativity has been suggested as a way to tame these divergences. The argument is that the uncertainty relation can be used to establish a short-distance cutoff so that distances shorter than the cutoff can be interpreted as long distances. It was shown by Filk [15] that this need not be the case. For the theories considered here it will be seen that although the noncommutativity partially reduces the short-distance ultraviolet divergences, new divergences appear in the long-distance infrared regime. The remainder of this chapter introduces the important techniques required for study• ing noncommutative quantum field theories. In Chapter 2 the noncommutative O(N) c/>4 theory is studied in the large-N limit. In the critical dimension the coupling is dimensionless and the theory is marginally renormalizable by power counting. Here this occurs in four dimensions, and this is where the theory is studied in this paper. Although two possible interactions exist in the noncommutative case, only the sym• metric vertex will be examined in detail. This allows the sum of diagrams contributing to the self-energy of the field to be written as a geometric series, greatly simplifying the analysis. A momentum-independent solution to the self-energy is assumed. In the commutative case this solution is stable but nonrenormalizable. It is found that the noncommutative theory behaves similarly, but the nonrenormalizability is less severe. In Chapter 3 the fermionic Gross-Neveu theory is explored in the noncommutative regime. This theory is the fermionic analogue of the 04 vector model in that it has a quartic interaction. Again there are two possible interactions but only the symmetric one is studied. The critical dimension is two. Chapter 1. Introduction 3 In two dimensions the Gross-Neveu model is marginally renormalizable by power counting. The commutative version, however, is perturbatively renormalizable in the coupling constant; this is also true in the noncommutative case. Here, though, the model is studied in a 1/N expansion which is nonperturbative in the coupling since it sums contributions to all orders in the coupling. Unlike in the commutative case, the theory is found to be nonrenormalizable in the sense that the dependence on the coupling cannot be removed. A running coupling is introduced, and this effective coupling runs to zero as the cutoff is taken to infinity, leaving behind the trivial free field theory. This is a very interesting result, since the noncommutativity has destroyed the renormalizability despite early expectations that noncommutativity could actually improve it! Since the study is conducted in two dimensions, space and time are noncommutative. In this case the action contains infinite numbers of time derivatives, which ruins the ordinary Hamiltonian interpretation. But these are physical constraints and do not affect the mathematical analysis, so the study is justified on the basis that generic properties of the theories may still be exhibited. In Chapter 3 the Gross-Neveu model is also studied in three dimensions, where the coupling constant has the dimension of inverse mass. The commutative model is non• renormalizable in the coupling constant expansion, but is renormalizable in the non• perturbative large-iV expansion at the second-order phase transition. As in the two- dimensional case, the noncommutativity destroys the renormalizability of the large-./V expansion even near the critical point. Finally, in Chapter 4, the conclusions are summarized and discussed in the context of string theory. Chapter 1. Introduction 4 1.2 Quantum Field Theory Quantum field theory is described by an action S expressed as a functional of a Lagrangian C through (1.1) where D is the dimension of the space-time manifold. In this thesis the manifold will always be RD with Euclidean metric. Here d> represents the field configurations and d^d) their space-time derivatives. The field configurations (1.2) Analogous to the partition function of statistical mechanics, quantum field theory has a generating functional (1.3) This integral is over all possible field configurations. A source term J has been included explicitly. This allows correlation functions to be obtained from Z by taking functional derivatives with respect to J. The integral (1.3) can be completed explicitly, for arbitrary space-time dimension, when the integrand is a Gaussian. A Gaussian integrand corresponds to a free-field Lagrangian, which is quadratic in fields. In general, though, the path integral is not well defined and except in a few special cases must be treated through an approximation scheme. Chapter 1. Introduction 5 1.2.1 Saddle-Point Approximation In some cases this integral can be completed using the saddle-point approximation [8]. In mathematical contexts this is typically called the method of steepest descent, although in complex analysis it is also referred to as the stationary-phase method. In field theory it is sometimes referred to as a mean-field approximation. This technique requires that the action have a sharp minimum in field space. The minimum is found by solving the Euler-Lagrange equation of motion (1.2). At the mini• mum, corresponding to the classical field configuration that the dominant contribution to the integral comes from this point. It is possible that the action has several minima; in this case the contributions from these minima should be summed. The validity of the saddle-point technique for the action uS is governed by the scale factor v. For suppose the action has a single shallow minimum. Then when v is large the integrand of the partition function is very large around the minimum of the action, and falls off quickly. In this case the saddle-point approximation is accurate even to low order. On the other hand when v is small the corrections will be much more important. The action evaluated at quantum corrections proportional to 1/v. In fact v is always proportional to 1/h so that as h goes to zero classical field theory is obtained. In this document h will be set to one for convenience, so this limit will not be obvious. The terms in the perturbation expansions can be represented graphically by Feyn- man diagrams. These diagrams contain no new information but often provide a useful visualization tool. This tool has led to the interpretation of higher-order corrections as corresponding to multiple self-interactions. In this representation the nth-order term in the expansion corresponds to graphs with n loops, so for example first-order corrections Chapter 1. Introduction 6 are often referred to as one-loop corrections. It is often possible to rewrite an action so that it has an explicit coefficient v. As an example, consider the action of a scalar field with a polynomial interaction S = Here D is an appropriate differential operator. For an interaction of this sort one scales the fields by a power of the coupling constant A to make u scale as an inverse power of the coupling. So here rescale the fields as S = X^ (if;Dtp + ipn) = X^S'. (1.5) -2 Now for this theory the expansion parameter would be v = X 2-». The limitation to this approach is that interesting features far from the small-coupling limit cannot be explored. At the classical point d>0, ^ = Jand m>0 (L6) so that here — S[ about the mean field d)Q. The linear term will not appear because of (1.6), so oo uS[4\ = uSfa] + ^2)[o] - th Here S^[(f)o] refers the the n derivative of the action evaluated at 0. In practical applications the sum over n is often finite as higher derivatives of the action vanish. Now the field d> can be written as a classical field plus corrections, Then the expression for the generating functional is given by s s(a) a Z[J] = j ^e-" We-5 W* e-5:r=3 T^W*". (1.8) Chapter 1. Introduction 7 As the first exponential is constant, it can be removed from the integrand. The last exponential can be Taylor-expanded in its argument as „ l-n/2 - OO -I / co ,.l-n/2 \ m ( e-E-a^ "W» = 1+E J_ (-Z'—S^S") • (1-9) m U m=l - V n=3 - ) Then to first order the integrand is Gaussian and the integral can be represented as the determinant of an operator, (2) Z[J] = e~s^ det "5 (S M) - • • • (1-10) where the omitted terms correspond to the perturbative corrections of order i/m(1-"/2). The perturbation expansion expresses interactions in terms of free-particle correlation functions. To evaluate the determinant it can be rewritten as detA = elndetA = eTrlnA. (1.11) The trace is now readily evaluated when In A can be diagonalized. 1.2.2 Effective Action The dependence of the generating functional (1.10) on the source term is hidden in the definition of the classical field configuration because of (1.6). In this way the source generates a given classical field [9]. Given a fixed source, the free energy can be defined as (2) F[J] = - In Z[J] « S[4>0] + ^Trln (S [o]) • (1.12) But this is still a functional of the external source, and it would be more natural to have the energy as a functional of the classical field itself. To do this, the free energy should be Legendre transformed with respect to the source. The result is the effective action D 5 r[0o] = F- jd xJ £. (1.13) Chapter 1. Introduction 8 This effective action is extremized by the classical solution when the source is set to zero. The effective action is an extensive quantity; it can be written as an integral of an effective Lagrangian over space-time volume. Since the interesting classical fields to be studied are often independent of space-time it is convenient to define the intensive effective potential Ve/f [fa] = Info] (1.14) where v is a factor of space-time volume. 1.2.3 Large-iV In theories where the fields satisfy an O(N) symmetry, an expansion in N is possible and is particularly suitable when N is large [10, 11]. This approach can reveal information about the theory that is nonperturbative in the coupling. For example, a theory which is nonrenormalizable order-by-order in the coupling constant may in fact be renormalizable at a critical point. But since the critical point may not be at small coupling a study perturbative in the coupling may not apply. The existence of the O(N) symmetry allows the path integral over the fields l to % % be rewritten in terms of d> 4> . This is analogous to a spherically symmetric integral, for which the only nontrivial integral is the integral over the length coordinate. In this case the saddle-point approximation can be used with v — N. The typical model for this type of analysis is the bosonic O(N) dfi theory, whose action is S\<\r\ = j d4x (1.15) where d>2 is The large-AT limit is taken holding A fixed. Consider for example the propagator to one loop. If the vector index in the loop contracts with itself, the factor of N from the Chapter 1. Introduction 9 loop contraction will cancel the 1/TV of the vertex to give a contribution of zeroth order in TV. On the other hand if the loop index is contracted with the external propagator the graph will go as 1/TV. At two loop order there is again a contribution of first order, as well as contributions like 1/TV and 1/N2. In contrast to the small-coupling expansion, which considers all graphs order-by- order in the number of loops, to lowest order in N the large-TV expansion considers contributions from all loop orders and thus all powers of the coupling. However at each order in the coupling only some of the graphs are considered. The terms that are included in the expansion to first order are called bubble diagrams or daisy diagrams; they must have equal numbers of vertices and loops. Thus although this model is solvable in this approximation, it is nonperturbative in the coupling. This is the essence of the large-TV limit whenever it is used. To solve this model an auxiliary field cr is introduced through (1.16) Since the model has an O(N) symmetry the 0(TV)-direction of the fluctuations about saddle-point will not affect the variation of the action. It is useful to choose the fluctua• tions about the saddle-point to lie in the 4>N direction. The remaining (j)1, i = 1... TV — 1 are renamed ir\ i = 1... TV — 1. After the introduction of o the integral over the TV — 1 n1 is Gaussian and can be completed to give TV — 1 determinants. The partition function is then Z = J VNVae-s>>W (1.17) where here S'[o, 0"] = / dPx ^ (-d2 + o) r - ^a2 + + \(N - l)Tr fn(-d2 + o). (1.18) Chapter 1. Introduction 10 For space-time independent fields, the saddle-point equations are 5S a 1.2.4 Renormalization The integrals in these approximations always diverge. Fortunately these divergences can often be removed with a systematic physically sensible method. Theories where this is possible are said to be renormalizable, while all others are nonrenormalizable. In particle physics a major goal of the study of quantum field theory is to distinguish between the two classes of theories. The divergences are called UV divergences as they arise from integrands which diverge at large momenta. One standard approach to handling these divergences is to reduce the upper limit on the momentum of integration from infinity to a finite UV cutoff A. This new parameter is unphysical in particle theory, so experimentally available parameters must be independent of it. The removal of the dependence on A involves the use of so-called running parameters in place of the original bare parameters. When these parameters are forcibly introduced in a nonrenormalizable theory, the running coupling, which replaces the bare coupling and can be measured experimentally, flows to zero as the cutoff is taken to infinity. The resulting theory is simply a free field theory. 1.3 Moyal Product Noncommutative geometries have a long history [12]. Their most recent appearance in particle physics is due to an idea from string theory that perhaps the space we live in is Chapter 1. Introduction 11 itself noncommutative. In the simplest noncommutative generalization of Euclidean space the position oper• ators have a constant nonzero commutator given by [x",xv] = -i6'M/. (1.21) When the noncommutativity tensor 9^" vanishes, becomes an element of ordinary RD. The effect of a non-zero invertible 9liV on quantum field theories will be explored in this thesis. When quantum fields are written as expansions in these noncommuting coordinates the fields themselves become operator-valued. This makes traditional methods very awk• ward to deal with. Fortunately, the algebra of this new class of theories is equivalent to the algebra of functions of real numbers when products between functions are replaced by the Moyal star product given by f(x)*g(x) = limexp [~\^f(x)g(y). (1.22) 1.3.1 Construction This correspondence can be constructed as follows [13]. Let d>(x) be an ordinary field in RD, and let m = / (J^^y**; (i-23) this is just the analogous operational inverse Fourier transform of 4>{x) = [ dDxcj)(x)A(x,x) (1.24) Chapter 1. Introduction 12 where the map A(x,x) establishing the Weyl-Moyal correspondence is given by {X,X) \dete\J (27r)^e 6 6 = A(a;,£)t. (1.25) Note that when 9^v vanishes the correspondence operator reduces to A (a;, x) = 5(x — x) (1.26) as it should. A trace operator for the ip fields can now be defined by (1.27) Tr0 = j dDx(j)(x) which satisfies the normalization condition (1.28) Trh(x,x) = 1. With this definition the correspondence operators are now orthonormal so that (1.29) Tr[A(x,x)A(y,x)) =8{x-y). In addition the correspondence is now one-to-one since (1.30) 4>(x) = Tr (J>(x)A(x,x)). Derivatives of the fields <\> living on the noncommutative space should also be defined. The important commutators [14] are iS (1.31) and 4,#J - -«'(0 (1.32) Chapter 1. Introduction 13 Note how (1.21) has forced an analogous deformation of the derivative algebra. Then 4, A(x, £)] = -0,,A(x, z) (1.33) and [4,0] = y dDxdn4>(x)A(x, x) (1.34) follow. The translation generators are now given by the unitary operators ewd for v a c-number, so that eiv-°A{x, x)e~iv-B = A{x + v,x). (1.35) Finally, consider the products between fields. Using the definition (1.24) above, D D fa(x) = 4>i(x)fa(x) = J d yd z But, using the trace operator, 4>3{x) = Tr^^Afx.z)) D D D = j d xd yd z(j)1(y)(j)2(z)A(y,x)A(z,x)A(x,x) (1.37) = (2^)^/^^ (L38) = faWe-^Vfaix) (1.39) = fa + fo (1.40) 1 2 where d acts only on 1.3.2 Properties v Ufi Although it is clear from (1.39) that the star product is not commutative since 9^ ^ 9 1 it is associative. This follows directly from the associativity of ordinary multiplication: dCfcl fc3 ttl ( Another very useful property is that under an integral one star product can always be dropped. Using (1.38) this becomes a straightforward calculation. Dropping explicit reference to the vector indices, 1 1 10 L42 /d°xh*h = (2ir)»\det9\ I^^^iCy)^^)^^" ^" ^*^" - ( ) The first term in the exponential immediately vanishes by the antisymmetry of 9. Then completing the integral over x leaves 5(9'1(y — z)). Since this implies y = z, the final term in the exponential vanishes and completing the integral over y cancels the normalizing prefactors to leave an ordinary product between the fields. A corollary of this is that products under integrals are invariant under cyclic permutations of the fields. These properties will be used to simplify Lagrangians, which always appear as integrands. 1.4 Noncommutative Quantum Field Theories Using the Moyal star-product and its properties the methods of ordinary quantum field theory can be used to study the noncommutative theories. Because of the Weyl corre• spondence there is no need to deal with operator-valued functions. Instead, to construct a noncommutative quantum field theory, use the Weyl correspondence to replaces the products between ordinary commutative fields by the Moyal star-product. But by the discussion above the star-products have no effect on the terms in the Lagrangian that are quadratic in fields. However, this does not mean that the noncom- mutativity has no effect on free field theories since the star-product will still appear in Chapter 1. Introduction 15 operator product expansions for example. The noncommutativity really does do more than just change the form of interactions. Since the nonlocality of the Moyal product distinguishes between noncyclic permu• tations of momenta about vertices, an important new distinction between planar and nonplanar diagrams in momentum space arises. Diagrams which are planar in momenta are modified from ordinary Feynman diagrams only by constant phase factors, that is, phase factors that depend only on external momenta and are independent of the loop momenta. On the other hand, phase factors appearing in nonplanar diagrams depend on the momenta of integration and thus affect the behaviour of the integrals [15]. Although the phase factors in the nonplanar diagrams can improve convergence of integrals over loop momenta, this need not be the case [16, 17]. But even when the noncommutativity softens the ultraviolet (UV) divergences, which can still be removed through a usual renormalization prescription, new infrared (IR) divergences crop up which can destroy the renormalizability. This is because after completing the nonplanar integrals with the cutoff A, the result depends on A through a new effective cutoff Aeff = 1 /2 (A-1 + p99p)~ ' . For nonzero external momenta the effective cutoff approaches a finite limit as A is taken to infinity. Thus these nonplanar diagrams are no longer UV-divergent. However, once A has been removed the effective coupling is IR-divergent. This the UV/IR mixing of [18, 19]. Chapter 2 Noncommutative O(N) 04 Theory 2.1 Introduction One simple theory to study is the O(N) scalar (f)4 theory in the large-N limit. In the commutative case a local auxiliary field can be introduced which allows the resulting path integral to be calculated by a saddle-point or mean-field approximation. The expansion parameter is 1/N. This solution is exact in the sense that it sums contributions from all orders in the coupling. The commutative 0(iV)-symrnetric c/>4 theory is described by the Euclidean Lagrangian a a 2 £ = I^(-a + m )^ + A(0y) . (2.i) Here i is a group index running from 1 to iV and Einstein's summation notation applies. The (f>% are N bosonic fields invariant under 0(N) transformations. The bare mass and coupling are given by m2 and A respectively. In four dimensions A is dimensionless and the theory is marginally renormalizable. In the large-TV limit, for A > 0, this is a free theory unless a UV cutoff is imposed [20]. Then the ground state of the theory depends on the values of the cutoff and bare mass and coupling. In the large-TV limit it is determined by the sign of m2 A2 T + 32^ <2"2> where A is the UV cutoff. The ground state exhibits spontaneous symmetry breaking to an 0(N — 1) state if (2.2) is negative, but otherwise maintains the full O(N) symmetry. 16 Chapter 2. Noncommutative 0(N) tf Theory 17 Of course this analysis is valid only for momenta much less than A. When A < 0 the theory is inherently unstable. The noncommutative version of this theory has two interactions which are distin• guished by the cyclic order of the momenta about the vertex; C = \tf (-d2 + m2) tf + ^ (tf * tf) (tf * tf) + ~tf * tf *tf* tf- (2.3) When 9 goes to zero the star-products become ordinary products and the coupling is given by A = | (Ai +A2 ). It is useful to rewrite the interaction portion of the Lagrangian in momentum space, where it becomes Ant = / (ft |^) e^^M^D^D) {pi+p2+p3+pi) X tf iPi) tf (ft) tf (ft) tf (P4) W + ^fS***} • (2-4) Although less obvious in momentum space, the Ai vertex is symmetric in the fields, whereas the A2 vertex is not. This will be established concretely in the next section. In Section 2.3 the propagator is considered for large-AT and small coupling. This allows the full theory, with A2 nonvanishing, to be studied. However summing contributions to all orders in the coupling is not possible. In an attempt to find other nonperturbative solutions to the theory with A2 present, the Schwinger-Dyson equation of the theory is introduced in Section 2.4. A recursive definition of the self-energy is found, but nonper• turbative solutions dependent on A2 are not discovered. The theory is then studied with only Ai present. The theory then becomes very similar to the commutative one. The traditional method of solving the model will work, so in Section 2.5 the saddle-point approximation is introduced explicitly. This approximation has the effect of summing contributions to the propagator from all orders in the coupling Ai . To lowest order the theory is identical to the commutative model, so the same phase structure is found. Quantum fluctuations, which do depend on the noncommutativity, are Chapter 2. Noncommutative 0(N) studied in Section 2.6. Compared to the commutative fluctuations, the noncommutative corrections are found to reduce the stability and improve renormalizability. 2.2 Local Auxiliary Field The usual method of solving the O(N) model in the large-N limit involves the introduc• tion of a local auxiliary field o ~ 4>l(j)1. When this field is introduced the Lagrangian is quadratic in both the 4>l and o. Completing the path integral over u results in the origi• nal Lagrangian, while completing the path integral over the In the present case, to determine whether o can be successfully introduced only the exponential term in (2.4) must be checked. This is because the Kronecker delta functions can clearly not be written in any other way. Of course the momenta of integration can 1 easily be relabeled if necessary. Denoting p^O^p ^ by pm x pn, the exponential is X gf [Pl (P2+P3+P4)+P2X(p3+P4)+P3Xp4] ^2 5) This can be rewritten using conservation of momentum and the antisymmetry of the product as g£(PlXP2+P3Xp4) (2.6) or as e^(PlXp3+P2Xp4)gip2Xp3 (2 7) from which it is clear that the Ai term factors symmetrically while the A2 term does not. Thus no local momentum-conserving auxiliary field exists. This is an unfortunate result, but it also reveals the deep impact of the noncommutativity. Chapter 2. Noncommutative 0(N) 4 Theory 19 2.3 Two-point Propagator The propagator, or two-point correlation function, gives the amplitude for a particle to propagate from one point in space to another. For an interacting theory, such as the one studied here, this amplitude is modified from the free-particle amplitude by self- interactions. In the large-iV limit, ordinary perturbation theory in the coupling can still be used. In momentum space the propagator has a single one-loop correction in the large-iV com• mutative model. Here, since there are (|) cyclically distinct ways to order the momenta about the single vertex, there are six one-loop corrections to the propagator. To find the diagrams that are not suppressed by 1/N only the O(N) index structure needs to be considered. Two important contributions are found; the correction to Ai is planar in momentum space while the other is a nonplanar correction associated with X2 . In light of this it is not surprising that it is the A2 vertex which prohibits the existence of a local auxiliary field. Using the result above it is now straightforward to calculate the inverse two-point propagator r = S^1 to one loop order when the couplings are small and N is large. The planar correction is 4 2 2 cv Ai f d k 1 Ax Tr (k2 2l /A \ x _ \ ,no. 1 CP _ _L / — = A -m In — + constant terms (2.8) 1 8 J (2TT)4 k2 + m2 8(2TT)4 \ \m2 J ) and the nonplanar correction is 2 2 TC.«P = *L /• ****** = (A - m In (%U constant terms) , (2.9) 2 8 J (2?r)4 k2 + m2 8(2?r)4 \ eff \m2 ) ) where 1 . . Chapter 2. Noncommutative 0(N) tf Theory 20 and A is a UV cutoff. Thus the inverse propagator to one loop, , is given by [18] as ^='+™2+(A2 - ™2 >° (£))+^ K - ™2in (^)) • ^ Note that as p only occurs in the combination p9 in the one-loop corrections, this theory will look planar in the small-momentum limit, at least naively. However, this limit needs to be taken carefully, and is discussed in [18]. One can certainly add multi-loop corrections to this result; this corresponds to higher- order corrections in ordinary perturbation theory. Ideally one could add contributions from all orders in the coupling immediately. However as described above this will not be possible when A2 is nonvanishing. When A2 does vanish the resulting expansion in Ai is a geometric series, whose sum is of course known. In fact the series, which is defined as the self-energy, corresponds precisely to the auxiliary field which will be introduced in Section 2.5. 2.4 Self-Energy The self-energy E of a particle is the sum of loop corrections to the bare propagator defined through s = VTs <2'12> so that m? + E is the effective mass of the particle. Here S is the full propagator and SQ1 — (p2 + m2) is the inverse bare propagator. In the large-Af limit of commutative tf theory the only diagrams which contribute to E are the bubble or daisy diagrams, which have equal numbers of loops and vertices. These diagrams survive the large-N limit since each loop provides a factor of N and each vertex goes as 1/N. This sum of bubble diagrams can then be factorized into a geometric series and the sum can be computed. Chapter 2. Noncommutative 0(N) (j)4 Theory 21 It has been shown in Sec. (2.2) that a local auxiliary field cannot in general be introduced into the noncommutative version of this theory. Equivalently, the bubble diagrams can no longer be factorized into a geometric series. The problem with the A2 vertex is that the phases entangle adjacent loops so that a two-loop contribution is not the square of the one-loop contribution. Thus, the self-energy must be studied directly through its definition (2.12). The full propagator is S (q) 8mn n = S0(q)6™ + ( mn = S0(q)6 + I ^^e^<^H^)DiW (Pi +P3 +P4) (^M" + |^*) mi kl n X 46 S0(q)6(q - Pl) {8 V S(-P3)8 (-p3 - p4) 6 (~P2 - q) l kn + V 5 S(-P2)5 (-P2 - Pi) 8 (-p3 - q) jk ln + 8 8 S(-P2)8 (-P2 - p3) 8 (-P4 - q)} S(q) (2.13) which to first order in 1 /N after summing over the group indices leaves S{q) = S0(q) + / T^p ^S0(q)S(p)S(q) + ±-J>* In the large-iV limit this gives the self-energy £ = S'-1 — SQ1 as - _[^P_ 1 (^_ + ^eiPxq) (215) J (2TT)d + m? + E(p) V 2 2 )' 1 ] If the general solution to this equation could be found, it would allow a very detailed nonperturbative study of the theory. But since it is not clear how to solve this equation in general, solutions must be found perturbatively or in special limits of the theory. Chapter 2. Noncommutative 0(N) 4>A Theory 22 The nonplanarity of this theory is associated with A2 , so to simplify (2.15) without losing interesting behaviour it seems best to set Ai to zero. Interestingly, unlike the commutative case, there is no momentum-independent solution. However, perturbative solutions can be found. Gubser and Sondhi [21] argue that this leads to a striped phase due to the oscillatory phase associated with A2 . The contribution of the A2 term at large loop momentum diverges as the external momentum goes to zero. This prevents the condensation of the small momentum modes of the fields. Therefore the ordered state to which the fields condense must break translation invariance. This will not be studied here. On the other hand, setting A2 to zero allows only the momentum-independent solution of the more familiar commutative theory. Noncommutative corrections will still be found when perturbations about the solution are considered. This will be studied below. 2.5 Symmetric Vertex Keeping only Ai in the theory allows progress to be made. The terms in the bubble series now factor and the local auxiliary field can be introduced. This allows the running of the coupling and the phase structure of the theory to be explored. At the classical level all dependence on 9 will disappear, but quantum corrections will depend on the noncom• mutativity. The noncommutative quantum corrections will mildly affect the stability of the phases and the renormalizability. The Euclidean Lagrangian is given by { 2 CW} = l The interaction now factors, so a local auxiliary field can be introduced. Note that the Chapter 2. Noncommutative 0(N) 4 Theory 23 auxiliary field o does not have a kinetic term. The resulting Lagrangian is TV (2.17) 2Ai Here and below, the notation -kio-k indicates that the •-product is important in the relevant operator expansion. Although this Lagrangian has an extra albeit nondynamical field, it describes the same physics as the original Lagrangian. To see this, one must consider the path integral of the exponentiated action. This is given by (2.18) The original Lagrangian is obtained by performing the path integral over cr. Alternatively, one may integrate over the (j)1 to leave a Lagrangian which depends only on cr. The resulting description will be solvable in the large-TV limit. Before integrating over the Z[f] = / VtfVp2 exp (2.19) Since p2 does not correspond directly to an observable field it is not given a source term. If it were given a source term, then the TV + 1 degrees of freedom in the sources would not be independent. It is convenient to introduce an analogue of the Helmholtz free energy as . \nZ[J% (2.20) Chapter 2. Noncommutative 0(N) tf Theory 24 since then jjr = -M,s-*- <221> l where i—nl — U>=o — 0. (2.23) Hi \J=0 Before proceeding further, the integral over (jf in Z[Jl] will be completed. First, shift 1 (2.24) -d2 + *[x2* and note that Vtf = Vtf. Now 2 D 2 2 2 Z[f} = JVtfVn exp{- j d x \tf (-d +^)tf-^-^ -m y -Iji L 2 -d2+V*' 2 2 N jP/i exp|-yTrln (-d + V*) + JdDx 2A7 (2.25) 2 -o2 + *p2* This latter integral can only be evaluated when iV is large, since then the saddle-point approximation, or method of steepest descents, is valid. This is the reason that the earlier analysis focused on the large-iV limit. The argument of the exponential in (2.25) must be minimized as a functional of u2 for fixed sources. This is how u2 acquires a classical interpretation. Introducing the classical //Q as the minimum of the functional in this way is equivalent to including a source term H for p2 and subsequently solving the constraint placed on the N + l source terms J1, H. Since the classical solution is really the weighted average of all possible quantum fields, Chapter 2. Noncommutative 0(N) 04 Theory 25 pi is also known as the mean field, and in this context the saddle-point approximation is often called the mean-field approximation. In mathematical discussions this technique is also referred to as the method of steepest descents. To lowest order this approximation yields N 1 F[f} = -yTrln (-<32 + *//2*) + J dDx (pi - m2) J1 2A7 2 -d2 + *$* (2.26) Finally, taking the Legendre transform N 2 2 2 1 IM*)] = j TV In (-d + */.2*) -fdDx (pi - m ) - \j J 2 -d2 + *tj%* (2.27) l Using the definition of the classical solutions 2 2 D 2 2 2 2 r[p (x), ft(a)] = jTr In (-d + d x (p 0 - m ) - |fl (-d + 4 (2.28) For constant pi the star-products can be dropped and the Trln can be evaluated, leading to IK, <&{x)] = jJdD*lj05^{p2 + ti) ~ 2^" {ti - ™f + ^ {~d2 + ti) 4 (2.29) It is also convenient to introduce the effective potential 2 2 2 2 2 Veff = lr[p 0] = ^Tr In (-d + p ) - ~ (/xg - m ) + (2-30) which is minimized by the constant solutions p\ and (f>0. Here v is an infinite factor of space-time volume. Since constant classical solutions have been chosen, the •-product no longer affects the expansion to first order. Chapter 2. Noncommutative 0(N) 2.5.1 Four Dimensions Now dV ^'lf -rf« ..2,i = 0 (2.31) l so that either (j) 0 or u-l vanish. This means that the theory has two phases, and along 2 l the critical line separating the phases both 4>\ and /i , vanish. When (f) 0 is nonvanishing, so that these fields have acquired a vacuum expectation value, the O(N) symmetry of the original Lagrangian is spontaneously broken to an 0(N — 1) symmetry. In the other % phase the 4> 0 vanish and the theory maintains the full symmetry. In the unbroken symmetry phase where pi is non-zero, the mass-gap equation is dVeff _ N f d*p 1 N 2 2 J (27T)* 2 ^2 {*> rn ) dp P + Ai 2 2 N a2 N 2l A N 2 Nm n ,nnn. A ln 0 2 32 r - ^rr^l 2 0 72 - T-fi + -T- = ( - ) 32TT2 327r ^ /Z§ AI ™ ' AX in which a UV cutoff A has been imposed and it is assumed that pi <§C A2. Thus in this phase pi must satisfy 2 2 2 2 / 1 1 , A \ m A + ^{^ ^7J = xr+^- (2-33) So here the running coupling and mass should be defined as 1 1 1 , A2 . 2 + ^ln2 —2 (2.34) Ar(M ) Ax 32TT M and 2 2 2 2 mr (M ) _ m A 2 + {Z 6b) Ar(M ) ~~ Ax 32TT2- - 2 where M is the renormalization scale. Note that since Ai is positive, Ar —>• 0 as the cut-off is taken to infinity; this is the Landau pole. Using these definitions the mass-gap equation becomes simply 2 1*1 = m r (2.36) Chapter 2. Noncommutative 0(N) so that pi is identified with the renormalized mass of the (f>1 and acts as an order parameter of the theory. Note though that Bardeen and Moshe [20] choose pl/Xr as their order parameter. In the other phase the symmetry is spontaneously broken, pi = 0, and the mass-gap equation becomes 2 2 dp 0 32TT Ai so that ml « = —r~- (2-38) Although in general all couplings up to the dimension of Ax and compatible with the symmetry of the Ai vertex are needed to remove UV divergences, it was consistent in the large-iV limit to consider the case where A2 vanished. This is because to first order in 1/N the UV divergences arise solely in planar diagrams where the ordering of the momenta about the vertices is unimportant. But since either coupling can be associated with planar diagrams, the divergences can be removed by counterterms of either form. The nonplanar sector is UV-finite in the large-iV limit since the terms become convergent due to the phases. However, the nonplanar sector is plagued by IR divergences which can destroy renormalizability even when all UV divergences have been appropriately removed from the theory. This will be studied below. 2.6 Beyond the Classical Case In the above first-order phase structure analysis, 6 plays no role, so that the theory has reduced to the classical case. To see the effects of nonzero 6, perturbations about pi, the minimum of T (2.28), must be considered. This corresponds to including the first-order corrections to the saddle-point approximation in (2.26). It will be useful to reserve the % l symbol pi approaches zero. To properly define the expansion of Trln(—d2 + pi +*i8o*), where So is a small perturbation about the classical solution pi, it is best to return to (2.25). Thus recall that 2 2 exp j-yTrln {-d + p 0 + *i6o*J j = j Vft exp j-^ j d4xft * (-a2 + a2 + iSo) * ft J = yp^expj-^l d4xft*(-d2 + p2)*ft}f^^[-^J d4x5oft*4>^)n(2.39) Dividing out the infinite constant Z[Jl — 0], the nth Feynman diagram in this expan• sion is i\n 1 d Xl (_ 2~) So(x )r (x ..., x ) (2.40) I" n! / * ''' ^"M^OM^a) • • • n ly n with T [x\, . . . , Xn) j j \x-i) • ftixt) * • • • * 4> (xn) • 4> (xn))ji=Q 4 2 2 / IWOn) • • • • ^'(*n) * ^n)exp{-| / d xft (-d + p ) ft) .(2.41) / Vfi exp {-§ / d4a;<^ (-d2 + ^i) Note that in the exponentials both ^-products could be dropped since pi is assumed to be constant, and the expression occurs in an integrand. Now define rc to be the connected diagrams defined by r. Then the sum above can be rewritten as an exponential so that taking logarithms gives TV" , / iSo \ -Trln 1 + 2 V ~d2 + ti m NTV g°° (—i)(-0 r -— V / d4x • • • d4x 8o(x ) • • • 8o(x )Tc(x , ...,x ) m=l x m 1 m 1 m 2 ~~, m J N m=l .(2.42) 4 4 4 c i J d xiSo(x1)T°(xi) + ^ j d xid x25o(xi)Sa(x2)T (xi, x2) + ... Chapter 2. Noncommutative 0(N) tf Theory 29 With p2 — pi + i8u the effective action becomes 2 iSo lV,0i = fTrln(-a + ^) + |Ttln(l+ _<2 - N j dA x ((pi - m2) + iSo) 2AX 1 tf (-d2 + pl) p-^p + iS**? (2.43) 2N Since 8 a is an auxiliary field, its equation of motion is simply = 0. To first order in 8a, _8T_ : c 2 2 N (x) + -J d*x18a(x1)T (x, xx) --^-(p 0- m ) + —So + ~tf *tf = 0. 8(8a) (2.44) By conservation of momentum one may write Tc(X, XI) = T%(—id)8(x — xi). Thus •^tf-ktf + l-rc(x) m2> M*) ( \l2 + 2 2 (p 0 - m ) . (2.45) Because pi still satisfies (2.32), the last two terms cancel and the expression reduces to i8a(x) - •l " 1 2N -(j)* (2.47) *eff ^1 * So from (2.43), and using (2.32) again, 2 2 4 2 2 2 iWo] = jTrln (-d + p )-£-Jd x(p 0-m ) 4 2 (2.48) + J d x \\tf0 (-d + pi) + (ti * ti)' Note that the part dependent on has the same form as the original Lagrangian (2.16), with the bare coupling Xi replaced by a new effective coupling Ae//; this confirms that the choice of Xeff is appropriate. Chapter 2. Noncommutative 0(N) (p4 Theory 30 But Xeff is important for another reason. For consider the second-order term in 5o in (2.43); this is given by 1 [ d4x5o^—5o. (2.49) 2 J Aeff So Xeff is the propagator of the o field. If Xefj becomes negative the propagator is tachyonic, meaning that the ti condensate, and the 0(iV)-symmetric phase, is unstable. The critical line (f)0pl = 0 then corresponds at least to a local minimum. Whether this is a global minimum is not examined here. To study the behaviour of Xeff, then, the four-point coupling must be found ex• plicitly. In Fourier space, 4 iqxp 1 d d1 q 1 + e T2 f \ f W [Q2 + ti] [{q - P? + ti l xp (l i)+M2o s (2 so) 2TT^/o2 ^^^ {-24^ - ( ^)} - 2 1 11^in ^+A inif)'P2<<^ (2.51) ln^ + ln^),p2»^, with 1 -r^+p92p (2.52) 2 2 A eff ~ A as before. Thus here 1 1 / N = 7- + ^ (p) Ac//(p) 2 2 1 I 1 lnA2 1 1 lnAg2// Ax 647T2 p2 64-K2 p2 jr+^ln^-ehla(1+•p2» * (2-53) Chapter 2. Noncommutative 0(N) By taking a derivative of the exact expression (2.50) for Xeff it can be seen that the effective coupling is a monotonically increasing function of momentum. It is always pos• itive, which establishes the stability of the constant solution as at least a local minimum. The interesting difference between the above \eff and the usual effective coupling of the commutative theory is the additional logarithmic term in (2.53) which reduces the inverse effective coupling as the cutoff is taken to infinity. Thus the new effective coupling decreases more slowly than the commutative one for large cutoff. It also means that the solution is slightly less stable. To summarize, the introduction of noncommutativity in the Ai vertex has reduced but not destroyed the stability of the constant vacuum, and it has softened but not cured the nonrenormalizability. Gubser and Sondhi [21] suggest that the A2 vertex has further- reaching consequences in that it completely destroys the stability of the constant vacuum and leads instead to a striped phase. They find a first-order transition from the ((f)) = 0 phase to the striped broken-symmetry ordered phase. Chapter 3 Noncommutative Gross-Neveu Theory 3.1 Introduction This fermionic counterpart [22, 23] to the O(N) tf theory is more closely related to the Yang-Mills theory in that it is asymptotically free. The full Euclidean Lagrangian is C[ft = -ft 0ft - ± (ft* ftf - ± (ft * ft *ft* ft) . (3.1) Here the spinors tpl are taken to be Majorana fermions so that nonplanar diagrams appear in the leading order in the 1/N expansion. Being Majorana, they are symmetric under simultaneous charge and complex conjugation. In two dimensions they must have two components, while in three dimensions they can have two or four components. They will always be treated as two-component spinors here. As the arguments which prevented the introduction of a local auxiliary field in the presence of the X2 coupling apply here, in this analysis of the Gross-Neveu model A will be set to zero immediately. Because of chirality, in the free fermionic theory the two components of the spinors have an O(N) symmetry separately. The Gross-Neveu A interaction breaks this O(N) x O(N) symmetry to leave only an O(N) x Z2 symmetry. The remaining O(N) symmetry is a diagonal subgroup of the original symmetry, and the Z2 symmetry is the chiral symmetry. A mass term was not included in the above Lagrangian since the chiral Z2 symmetry forbids this. The fermions can acquire a mass only if the chiral symmetry is spontaneously broken. The order parameter for this symmetry breaking is (^ftft^. 32 Chapter 3. Noncommutative Gross-Neveu Theory 33 3.2 Local Auxiliary Field As found for the bosonic model after A2 was set to zero, a local auxiliary field can be introduced. The resulting Lagrangian is 2N C[ft,a} = -ft(0 + *cr*)ft + —a2. (3.2) A Integrating out ft in the saddle-point approximation, as done for the bosonic model, gives the generating functional 2 Z[H] = J Daexp {yTrln (0 + *o*) - j dPx —a -ft0(d + ft + Ha } , (3-3) and then the effective action is T[ft, a] = -yTrln (0 + + J dDx ^cr2 + ft (0 + ft . (3.4) Taking the fields to be constant, the effective potential becomes Veff = ^Tr\n{0 + o) + ™o2 + ftfto. (3.5) As in the bosonic model this theory has two phases corresponding to o = 0 and ft0 = 0 = ft0, and the mass-gap equation is D dV Nd r d p 1 AN -• ;i ,n D. Here d = 2 is the dimension of the gamma matrices and v is again the volume of space- time. 3.3 Perturbation About the Classical Solution Since the effective potential depends on constant fields, the effects of noncommutativity can only be probed by considering perturbations about these solutions. Chapter 3. Noncommutative Gross-Neveu Theory 34 Let o — M2 + So where M2 satisfies the conditions above, and So is a perturbation about this classical solution. Now, using the formalism developed in the previous chapter, the effective action is given by r[VM = -yTrln(> + M2) D c D D C -y |y d xiSo(xX)T (xi) - ^ d xid X2SO(XI)SO(X2)T (X1, X2) + ... + / dPx \^M4 + ^-M2So + ^5o2 + $ U + M2 + So) ^1 • (3.7) J L A A A ^ J In this chapter the r functions will always be the fermionic counterparts to the r functions described in (2.41). The formal definition is easy to obtain from (2.41) simply by replacing the boson fields 4>l with the spinor fields ipl and replacing the Klein-Gordon propagator d2 + *pl* with the Dirac propagator ^ + *<7*. To calculate the commutative r functions the parameter 6 can be set to zero immediately. Using the mass-gap equation for nonzero M2, the equation of motion for So is ST N AN. D c J d x1r (x, x1)So(xl) + ^-So + i>i^l = 0 (3.8) 5(So) ~2 Thus (3.9) with T— = T + U(-id). (3.10) Aeff A O Substituting this back into the effective action gives a new effective action for the classical fields: Wo) = j dDx hll fJ.- (3.11) $(0 + *)^ SN This analysis is completely analogous to that performed in Sections (2.5) and (2.6). To complete the analysis the dimension D must be fixed so that the integrals can be completed explicitly. Chapter 3. Noncommutative Gross-Neveu Theory 35 3.4 Two Dimensions In two dimensions the effective potential is The classical value of M minimizes this potential and corresponds to the dynamically generated fermion mass. Since it is a function of M2, the sign of M is undetermined; this is due to the chirality of the model. The gap equation is solved when \ = b»w- <313> After the running coupling is calculated, it will be possible to replace the dependence on A and A with a dependence on M. This technique is called dimensional transmutation. Note that if a renormalized coupling is defined at the renormalization scale \x by 2 Ar A 47T /J, then if A is taken to infinity along fixed Ar, A must go to zero. This is the asymptotic freedom of the Gross-Neveu model. The unphysical IR Landau pole is also present here; the renormalized coupling goes to infinity as /J? is taken to zero along fixed A and A. 3.4.1 The Commutative Theory To study the four-point coupling A the four-point correlator must be found. In the commutative case this is given by 1 C T0 = -- (3.15) 2TT M ^ \V 4M2 2M Euler's constant is denoted by 7. The regularization used here to tame the UV divergences is the same as that used in [18, 19]. Using the gap equation to eliminate A, the effective Chapter 3. Noncommutative Gross-Neveu Theory 36 coupling is K„ = ~j= —^- . (3.16) ^m(yrrg:+i)+7-i 2M For p2 3> M2, the coupling Xeff « (3.17) 111 M becomes small, as expected. As the momentum is lowered to small momenta the coupling increases, but it stops increasing when p reaches the scale M, at which point the coupling freezes at Xeff « r. (3.18) 7 — 1 3.4.2 The Noncommutative Theory As in the bosonic model, Tj receives contributions from both planar and nonplanar graphs through r| = + T%NV when 9 is nonzero. The planar contribution T^9 is again half of the commutative four-point function. On the other hand, the nonplanar contribution to the connected four-point function is q292 _1_ ~4~ + A* + where K0 denotes the modified Bessel function. Note that this term has a finite limit as A is taken to infinity, unlike the planar contribution. If A is removed from the resulting expression for Xeff the dependence on A is not removed so that for any p > the effective coupling vanishes as A is taken to infinity. 2 2 The UV behaviour of Ae// can be studied in the limit where both p M2 < A2. Chapter 3. Noncommutative Gross-Neveu Theory 37 Then when p2 3> AM2 and p2 » the planar contribution is r^-^ln^ (3-20) z 8-7T p while Tc,np _ e-M8p (3 21) so that Ae// * *-2ln£ - dig' (3-22) The bare coupling A has been removed using the gap equation. But when p2 < p2 < AM2, and p2 > ^ then ^-^•4 <3-23> and c np 2 2 2 r2 ' « In (e p M ) ; (3.24) thus, using the gap equation again, Here the effective coupling still goes to zero as the cutoff is removed for finite nonzero external momentum. Therefore in this noncommutative theory renormalization does not remove the cutoff dependence. The interaction goes to zero as the cutoff is taken to infinity, leaving behind the trivial free field theory. The arguments above depend heavily on the satisfaction of the gap equation. It is certainly possible to choose the bare coupling so that it does not satisfy the gap equation but rather removes the UV cutoff dependence. This approach destroys the stability of the model. Chapter 3. Noncommutative Gross-Neveu Theory 38 3.4.3 A Double-Scaling Limit A double-scaling limit can be taken by sending A —> oo and 9 —>• 0 along fixed 9A = jfc for some fixed parameter C. This corresponds to a regularization of the commutative model since the theory is noncommutative only at energy scales beyond the UV cutoff A. Even though the noncommutativity lives beyond the UV cutoff, it still affects the theory at all scales through the UV/IR mixing. In this case the expression for the effective coupling is 47T Kff = 7=^ • (3.26) | In (1 + CV/M*) + In + & + 5fr) Both terms in the denominator have cuts in the complex-p plane. The square root cut starting at p = 2iM corresponds to the pair production of fermions; this is familiar from the commutative model. However the logarithmic cut in the first term, present only when C is nonvanishing, is new to this double-scaling limit. This cut could correspond to the pair creation of some nonlocal solitons present in the noncommutative model which survive this double-scaling limit. Unfortunately no construction of these solitons has been found. 3.5 Three Dimensions In three dimensions the coupling has the dimensions of inverse mass, so it is not renor- malizable in the small-coupling expansion. But in the lavge-N expansion the theory is renormalizable, at least near the second-order phase transition. Near this phase transi• tion the coupling constant acquires a large anomalous dimension, rendering it effectively dimensionless and marginally renormalizable. Chapter 3. Noncommutative Gross-Neveu Theory 39 The gap equation is now M(\- T^e"^ ) = 0 (3.27) .A 16TT so that either M = 0 or M satisfies A 2M . -e~~ (3.28) A 16TT analogous to the solutions in two dimensions. But while in two dimensions the M = 0 solutions is never stable, here its stability depends on the coupling. For A < f (3.29) vanishing M is the stable solution, the fermions remain massless, and the chiral symmetry is unbroken. Otherwise the fermions acquire a mass according to (3.28) and the chiral symmetry is spontaneously broken. These two phases are separated by the second-order phase transition discussed above. 3.5.1 The Commutative Theory Massless, Symmetric Phase When A < I671-/A the four-point function is c r2 = -— + H (3.30) 2 16TT 16 V 1 The effective coupling is then given by Xeff = I _ JL + H' (3'31) A 16TT 16 If A is tuned to be close to but still less than the critical value, 1 A u2 ,„ . — = h —, (3.32) A 16TT 16' v ; Chapter 3. Noncommutative Gross-Neveu Theory 40 then the theory is renormalized and the effective coupling becomes e = ^ ff 2 i i i • (3.33) p2 + \p\ At the critical point, when p2 = 0, this becomes a nontrivial conformal field theory whose scaling properties can be found in the 1/TV expansion. The condensate {^%ip%^j has conformal dimension 1, instead of the classical value of 2, at this point. Note also from (3.32) that as the cutoff is removed along fixed p,2, the bare coupling flows to zero. This is what happens in asymptotically free theories. This is important because it is a major qualitative feature which the large-iV expansion of the Gross-Neveu model shares with the Yang-Mills theory which arises directly from string theory. Massive, Broken-Symmetry Phase In this phase A > 167r/A and the correlation function is 2 2 A 2M AM +p T bl ,n nl. To = —e~~ + , i tan"1 (3.34) 2 16TT 4ATT|P| 2M V ' and, using the gap equation (3.28) to remove the cutoff and bare coupling, XEFF = 3 35 4M2+P2 . ! bp ( - ) So here the coupling vanishes like 16/|p| for large momenta, and it has a finite constant limit 4n/M for small momenta. 3.5.2 Noncommutative Theory Massless, Symmetric Phase As now expected, there are two contributions to r|; one is half of the commutative result and comes from a planar diagram, while the other is nontrivially modified by Chapter 3. Noncommutative Gross-Neveu Theory 41 nonplanarity. In the massless phase the latter contribution reduces to i 12 ~ 32h + h) + ^) - (-WIWV=(^) • (3-36) The effective coupling is now given by 11 A , | Xeff X 327T 16 327T \ 4 (3.37) where the cutoff dependence has not been removed. The cutoff dependence in the fourth term is useful as an infrared regulator, but it is the second term which prevents the A —>• oo limit from being taken. In an attempt to renormalize this coupling, it is appropriate to introduce £ = (3-38) since this will cancel the cutoff dependence at zero momentum. This is the smallest that A can be taken as while keeping Xeff positive since q = 0 is a minimum of the expression above. This renormalized coupling must remain positive to satisfy the phase conditions, since taking j- to zero takes the theory to the critical point. With this definition 1 \"5 1 A- nJ 1 +£- f , ** w(-\q9\\q\Ja(l-aj). Xeff Xr 16 327T v V 4 A' ' Vs (3.39) From this expression it is clear that there is no Landau pole since Ae// is positive for all momenta. However, the cutoff dependence has not been removed, and for large cutoff 327T Kff ~ (3.40) If the cutoff dependence were removed at finite momentum through Chapter 3. Noncommutative Gross-Neveu Theory 42 then the theory becomes unstable, with tachyonic modes appearing for momenta near \q9\ « Ar/167r. Hence this theory is not renormalizable, becoming trivial in the large-A limit. The Double-Scaling Limit Another double-scaling limit can be defined in the three-dimensional case, if 9°l vanishes and 9*j = elj9 for spatial i, j. Here eJJ represents the Levi-Civita tensor with e01 = 1. Now take the infinite-cutoff limit while keeping A3#2 = C constant. Note that C has units of inverse momentum, as did C/M defined in the two-dimensional case. So 1 1 q C'q* = T + - + ——. (3.42) v ; Xeff X 8 2567T This means that the particles mediating the force between the fermions has a nonrela- tivistic dispersion relation given by W = I^V? + ^' (3'43) This is a long-range force, but it is not scale invariant. Even though the noncommutativ- ity occurs at distances scales beyond that of the UV cutoff, it still influences the model because of the UV/IR mixing. This was seen in the two-dimensional case as well. This double-scaling limit has important implications for the uncertainty relations. For consider the original noncommutativity relation Ax Ay = 9 = (C'/A3)1/2. If Aa; and so Ay are finite then Heisenberg's Uncertainty Principle AxAPx « 1 gives Ay « 3 1 2 Apx • (C'/A ) / . This is shorter than the usual short-distance cutoff 1/A. Thus the noncommutative uncertainty relation should be invisible in the cutoff theory. Following the same analysis in the two-dimensional theory shows that in that case the commutators would be of the same order and therefore marginally observable. Chapter 3. Noncommutative Gross-Neveu Theory 43 Massive, Broken-Symmetry Phase In this phase the gap equation (3.28) must be satisfied to get a stable solution. The cutoff dependence of the bare coupling A is also determined by the gap equation, but the cutoff dependence will not completely cancel in Ae// since this theory is nonrenormalizable. From the earlier calculations of the contributions to r| it is clear that for large momenta, above 1/A, the dependence on momentum in Xeff is small compared to the term of order A. The effective coupling is then given approximately by 1 (3.44) Therefore in three dimensions the effective coupling goes to zero linearly as the cutoff is taken to infinity, leaving behind the trivial theory. This is even more severe than the two dimensional case, which ran to zero as an inverse logarithm of the coupling. Chapter 4 Conclusions Bosonic and fermionic O(N) models with quartic interactions were studied in the large-A7' limit. In the noncommutative case there are two possible cyclically inequivalent orderings of the the fields in the quartic interaction, but it is argued that the theories are consistent even when only one interaction is considered. Progress was only made for the theories containing just the factorable interaction. This allowed the introduction of a nondynam- ical auxiliary field which facilitated a study of the phase structure and running couplings of the theories. A momentum-independent solution was assumed for the four-dimensional bosonic model. It was found to lead to a stable but nonrenormalizable theory. In the commutative case this theory is also sensible only with a fixed cutoff, but in the noncommutative case the nonrenormalizability is less severe. In two and three dimensions the fermionic Gross-Neveu model was found to be non• renormalizable, unlike its behaviour in the commutative case. In commutative theories, even massless ones, a mass scale is generated which cuts off the coupling as it runs to large values in the infrared. However, the noncommutativity has prevented mass from solving the IR divergence. Since the noncommutativity blurs space, it could naively be proposed as a way to solve the UV divergences and nonrenormalizability of quantum field theory. 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