Axial U_A(1) Anomaly: a New Mechanism to Generate Massless Bosons
aa r X i v : . [ h e p - l a t ] J a n Axial U A (1) Anomaly: a New Mechanism to GenerateMassless Bosons
Vicente AzcoitiDepartamento de F´ısica Te´orica, Facultad de Ciencias, andCentro de Astropart´ıculas y F´ısica de Altas Energ´ıas (CAPA),Universidad de Zaragoza, Pedro Cerbuna 9, 50009 Zaragoza, Spain
Abstract
Prior to the establishment of
QCD as the correct theory describing hadronic physics,it was realized that the essential ingredients of the hadronic world at low energies are chi-ral symmetry and its spontaneous breaking. Spontaneous symmetry breaking is a non-perturbative phenomenon and thanks to massive
QCD simulations on the lattice we haveat present a good understanding on the vacuum realization of the non-abelian chiral sym-metry as a function of the physical temperature. As far as the U A (1) anomaly is concerned,and especially in the high temperature phase, the current situation is however far fromsatisfactory. The first part of this article is devoted to review the present status of latticecalculations, in the high temperature phase of QCD , of quantities directly related to the U A (1) axial anomaly. In the second part I will analyze some interesting physical implicationsof the U A (1) anomaly, recently suggested, in systems where the non-abelian axial symmetryis fulfilled in the vacuum. More precisely I will argue that, if the U A (1) symmetry remainseffectively broken, the topological properties of the theory can be the basis of a mechanism,other than Goldstone’s theorem, to generate a rich spectrum of massless bosons at the chirallimit. Keywords:
Chiral Transition, Lattice
QCD , U (1) Anomaly, Topology, Massless Bosons Introduction
Nowadays we know that symmetries play an important role in determining the Lagrangianof a quantum field theory. There are essentially two types of symmetry, local ones or gaugesymmetries, and global ones. The gauge symmetries are characterized by transformations whichdepend on the space-time coordinates while in global symmetries the transformations are space-time independent. Also gauge symmetries serve to fix the couplings of the Lagrangian, andglobal symmetries allow us to to assign quantum numbers to the particles and to predict theexistence of massless bosons when a continuous global symmetry is spontaneously broken.In what concerns
QCD , the theory of the strong interaction, and prior to the establishmentof this theory as the correct theory describing hadronic physics, it was realized that the essen-tial ingredients of the hadronic world at low energies are chiral symmetry and its spontaneousbreaking. Indeed these two properties of the strong interaction have important phenomenologi-cal implications, and allow us to understand some puzzling phenomena as why pions have muchsmaller masses than the the proton mass, and why we do not see degenerate masses for chiralpartners in the boson sector, and parity partners in the baryon sector.Chiral symmetry breaking by the vacuum state of QCD is a nonperturbative phenomenon,that results from the interaction of many microscopic degrees of freedom, and which can beinvestigated mainly through lattice QCD simulations. As a matter of fact, lattice
QCD is themost powerful technique for investigating non-perturbative effects from first principles. How-ever, putting chiral symmetry onto the lattice turned out to be a difficult task. The underlyingreason is that a naive lattice regularization suffers from the doubling problem. The addition ofthe Wilson term to the naive action solves the doubling problem but breaks chiral symmetryexplicitly, even for massless quarks. This is usually not considered to be a fundamental problembecause we expect that the symmetry is restored in the continuum limit. However, at finitelattice spacing, chiral symmetry may still be rather strongly violated by lattice effects.On the other hand, staggered fermions cope to the doubling problem reducing the numberof species from sixteen to four, and to reduce the number of fermion species from four to one,a rooting procedure has been used. Even if controversial, the rooting procedure has allowed toobtain very accurate results in lattice
QCD simulations with two and three flavors.The doubling problem cannot be simply overcome because there is a fundamental theoremby Nielsen and Ninomiya which states that, on the lattice, one cannot implement chiral sym-metry as in the continuum formulation, and at the same time have a theory free of doublers.However, despite this difficulty, the problem of chiral symmetry on the lattice was solved atthe end of the past century with a generalization of chiral symmetry, through the so-calledGinsparg–Wilson equation for the lattice Dirac operator, which replaces the standard anticom-mutation relation of the continuum formulation Dγ + γ D = 0 by Dγ + γ D = aDγ D . Withthis new concept a clean implementation of chiral symmetry on the lattice has been achieved.The axial transformations reduce to the continuum transformations in the naive continuumlimit, but at finite lattice spacing, a , an axial transformation involves also the gauge fields, andthis is how Ginsparg-Wilson’s formulation evades Nielsen-Ninomiya theorem.All these features are well established in the lattice community, and the interested readercan find in [1], for instance, a very good guide.Returning to the topic of QCD phenomenology, there is also another puzzling phenomenonwhich is known as the U (1) problem. The QCD
Lagrangian for massless quarks is invariantunder the chiral group U V ( N f ) × U A ( N f ) = SU V ( N f ) × SU A ( N f ) × U V (1) × U A (1), with V and A denoting vector and axial vector transformations respectively. Below 1 GeV the flavorindex f runs from 1 to 3 (up, down and strange quarks), and the chiral symmetry group is U V (3) × U A (3). The lightweight pseudoscalars found in Nature suggest, as stated before, that2he U A (3) axial symmetry is spontaneously broken in the chiral limit, but in such a case wewould have nine Goldstone bosons. The pions, K -meson, and η -meson are eight of them butthe candidate for the ninth Goldstone boson, the η ′ -meson, has too great a mass to be a quasi-Goldstone boson. This is the axial U (1) problem that ’t Hooft solved by realizing that the U A (1)axial symmetry is anomalous at the quantum level. ’t Hooft’s resolution of the U(1) problemsuggests in a natural way the introduction of a CP violating term in the QCD
Lagrangian, the θ -term, thus generating another long standing problem, the strong CP problem.Thanks to massive QCD simulations on the lattice, we have at present a good qualitativeand quantitative understanding on the vacuum realization of the non-abelian SU A ( N f ) chiralsymmetry, as a function of the physical temperature, but as far as U A (1) anomaly and itsassociated θ parameter are concerned, and especially in the high temperature phase, the currentsituation is far from satisfactory, and this makes understanding the role of the θ parameter inQCD, and its connection with the strong CP problem, one of the biggest challenges for highenergy theorists [2].The aim to elucidate the existence of new low-mass weakly interacting particles from atheoretical, phenomenological, and experimental point of view is intimately related to thisissue. The light particle that has gathered the most attention has been the axion, predicted byWeinberg [3] and Wilczek [4], in the Peccei and Quinn mechanism [5], to explain the absence ofparity and temporal invariance violations induced by the QCD vacuum. The axion is one of themore interesting candidates to make the dark matter of the universe, and the axion potential,that determines the dynamics of the axion field, plays a fundamental role in this context.The calculation of the topological susceptibility in QCD is already a challenge, but calculat-ing the complete potential requires a strategy to deal with the so called sign problem, that is,the presence of a highly oscillating term in the path integral. Indeed, Euclidean lattice gaugetheory has not been able to help us much because of the imaginary contribution to the action,coming from the θ -term, that prevents the applicability of the importance sampling method [6].The QCD axion model relates the topological susceptibility χ T at θ = 0 with the axionmass m a and decay constant f a through the relation χ T = m a f a . The axion mass is, on theother hand, an essential ingredient in the calculation of the axion abundance in the Universe.Therefore a precise computation of the temperature dependence of the topological susceptibilityin QCD becomes of primordial interest in this context.In this article I will focus on the current status of the lattice calculations, in the hightemperature chirally symmetric phase of
QCD , of quantities directly related to the U A (1) axialanomaly, as the topological and axial U A (1) susceptibilities, and screening masses, and will alsodiscuss on some interesting physical implications of the U A (1) axial anomaly in systems wherethe non-abelian axial symmetry is fulfilled in the vacuum. I will briefly review in section 2 sometheoretical prejudices about the effects of the axial anomaly in the high temperature phase of QCD , and will analyze what the results of the numerical simulations on the lattice suggest onthe effectiveness of the axial anomaly in this phase. In section 3 I will argue that the topologicalproperties of a quantum field theory, with U A (1) anomaly and exact non-abelian axial symmetry,as for instance QCD in the high temperature phase, can be the basis of a mechanism, otherthan Goldstone’s theorem, to generate a rich spectrum of massless bosons at the chiral limit.The two-flavor Schwinger model, which was analyzed by Coleman [7] many years ago, is anexcellent test bed for verifying the predictions of section 3, and section 4 contains the results ofthis test. The last section contains a discussion of the results reported in this article.3
Theoretical biases versus numerical results
The large mass of the η ′ meson should come from the effects of the U A (1) axial anomaly andits related gauge field topology, both present in QCD . Despite the difficulty of computing thecontribution of disconnected diagrams to the η ′ correlator in lattice simulations, these obstacleshave been overcome and lattice calculations [8], [9], [10] give a mass for the η ′ meson compatiblewith its experimental value, and this can be seen as an indirect confirmation that the effects ofthe anomaly are present in the low temperature phase of QCD.Conversely, the current situation regarding the fate of the axial anomaly in the high tem-perature phase of QCD , where the non-abelian axial symmetry is not spontaneously broken,is unclear, and this is quite unsatisfactory. The nature of the chiral phase transition in two-flavor
QCD , for instance, is affected by the way in which the effects of the U A (1) axial anomalymanifest themselves around the critical temperature [11]. Indeed, if the U A (1) axial symmetryremains effectively broken, we expect a continuous chiral transition belonging to the three-dimensional O (4) vector universality class, which shows a critical exponent δ = 4 . U A (1) is effectively restored, the chiral transition is first order, or second order withcritical exponents belonging to the U V (2) × U A (2) → U V (2) universality class ( δ = 4 . U A (1) axial anomaly in the chiral symmetryrestored phase of QCD started a long time ago. The idea that the chiral symmetry restoredphase of two-flavor QCD could be symmetric under U V (2) × U A (2) rather than SU V (2) × SU A (2)was raised by Shuryak in 1994 [14], based on an instanton liquid-model study. In 1996 Cohen [15]showed, using the continuum formulation of two flavor QCD, and assuming the absence of thezero mode’s contribution, that all the disconnected contributions to the two-point correlationfunctions in the SU A (2) symmetric phase at high temperature vanish in the chiral limit. Themain conclusion of this work was that the eight scalar and pseudoscalar mesons should have thesame mass in the chiral limit, the typical effects of the U A (1) axial anomaly being absent in thisphase. Also Cohen argued in [16] that the analyticity of the free energy density in the quarkmass m , around m = 0, in the high temperature phase, imposes constraints on the spectraldensity of the Dirac operator around the origin which are enough to guarantee the previousresults.Later on, Aoki et al. [17] got constraints on the Dirac spectrum of overlap fermions, strongenough for all of the U (1) A breaking effects among correlation functions of scalar and pseu-doscalar operators to vanish, and they concluded that there is no remnant of the U (1) A anomalyabove the critical temperature in two flavor QCD , at least in these correlation functions. Theirresults were obtained under the assumptions that m -independent observables are analytic func-tions of the square quark-mass m , at m = 0, and that the Dirac spectral density can beexpanded in Taylor series near the origin, with a non-vanishing radius of convergence.The range of applicability of the assumptions made in [17] is however unclear. As stated bythe authors, their result strongly relies on their assumption that the vacuum expectation valuesof quark-mass independent observables, as the topological susceptibility, are analytic functionsof the square quark-mass, m , if the non-abelian chiral symmetry is restored. The two-flavorSchwinger model has a non spontaneously broken SU A (2) chiral symmetry and U A (1) axialanomaly, and Coleman’s result for the topological susceptibility in this model [7] χ T ∝ m e shows explicitly a non-analytic quark-mass dependence, and thus casts doubt on the generalvalidity of the assumptions made in [17].In Ref. [18] a Ginsparg-Wilson fermion lattice regularization was used, and it was arguedthat if the vacuum energy density is an analytical function of the quark mass in the high4emperature phase of two-flavor QCD , all effects of the axial anomaly should disappear. Themain conclusion of [18] was that either the typical effects of the axial U A (1) anomaly disappear inthe symmetric high temperature phase, or the vacuum energy density shows a singular behaviorin the quark mass at the chiral limit.On the other hand, an analysis of chiral and U A (1) symmetry restoration based on Wardidentities and U (3) chiral perturbation theory has been carried out in [19], [20]. The authorsshow in their work that in the limit of exact O (4) restoration, understood in terms of δ − η partnerdegeneration, the Ward identities analyzed yield also O (4) × U A (1) restoration in terms of π − η degeneration, and the pseudo-critical temperatures for restoration of O (4) and O (4) × U A (1)tend to coincide in the chiral limit.The first lattice simulations to investigate the fate of the U A (1) axial anomaly [21], [22]also started in the 90s. In Ref. [21] the authors report results of a numerical simulation ofthe two-flavor model with staggered quarks. They compute two order parameters, χ π − χ σ forthe SU A (2) chiral symmetry, and χ π − χ δ for the U A (1) axial symmetry, where χ π , χ σ and χ δ are the pion, σ and δ -meson susceptibilities, and they show evidence for a restoration ofthe SU V (2) × SU A (2) chiral symmetry, just above the crossover, but not of the axial U A (1)symmetry. Ref. [22] contains the results of a similar calculation in two-flavor QCD usingalso a staggered fermion lattice regularization. As stated by the authors, the relatively coarselattice spacing in their simulations, a ∼ Fermi, does not allow for conclusive results on theeffectiveness of the U (1) A anomaly.After these pioneering works, this issue has been extensively investigated using numericalsimulations on the lattice, and Refs. [23]-[43] are representative of that. We will focus fromnow on the most recently obtained results.In Ref. [29] (2 + 1)-flavor QCD is simulated, using chiral domain wall fermions, for tem-peratures between 139 and 196 MeV. The light-quark mass is chosen so that the pion mass isheld fixed at a heavier-than-physical 200 MeV value, while the strange quark mass is set toits physical value. The authors report results for the chiral condensates, connected and discon-nected susceptibilities and the Dirac eigenvalue spectrum, and find a pseudocritical temperature T c ∼
165 MeV and clear evidence for U A (1) symmetry breaking above T c .Ref. [31] contains also a study of QCD with (2 + 1)-flavors of highly improved staggeredquarks. The authors investigate the temperature dependence of the anomalous U A (1) symmetrybreaking in the high temperature phase, and to this end they employ the overlap Dirac operator,exploiting its property of preserving the index theorem even at non-vanishing lattice spacing.The pion mass is fixed to 160 MeV, and by quantifying the contribution of the near-zero eigen-modes to χ π − χ δ , the authors conclude that the anomalous breaking of the axial symmetry in QCD is still visible in the range T c T . T c .The thermal transition of QCD with two degenerate light flavors is analyzed in [34] bylattice simulations, using O ( a )-improved Wilson quarks and the unimproved Wilson plaquetteaction. In this work the authors investigate the strength of the anomalous breaking of the U A (1)symmetry in the chiral limit by computing the symmetry restoration pattern of screening massesin various isovector channels, and to quantify the strength of the U A (1)-anomaly, they use thedifference between scalar and pseudoscalar screening masses. They conclude that their resultssuggest that the U A (1)-breaking is strongly reduced at the transition temperature, and thatthis disfavors a chiral transition in the O(4) universality class.Results for mesonic screening masses in the temperature range 140 MeV T QCD , using the highly improved staggered quark action, are also reported bythe
HotQCD
Collaboration in [41], with a physical value for the strange quark mass, and twovalues of the light quark mass corresponding to pion masses of 160 and 140 MeV. Comparingscreening masses for chiral partners, related through the chiral SU L (2) × SU R (2) and the axial5 A (1) transformations, respectively, the authors find, in the case of light-light mesons, evidencefor the degeneracy of screening masses related through the chiral SU L (2) × SU R (2) at or veryclose to the pseudocritical temperature, T pc , while screening masses related through an axial U A (1) transformation start becoming degenerate only at about 1 . T pc .A recent calculation in (2 + 1)-flavor QCD [42], using also the highly improved staggeredquark action, shows, after continuum and chiral extrapolations, that the axial anomaly remainsmanifested in 2-point correlation functions of scalar and pseudo-scalar mesons in the chiral limit,at a temperature of about 1.6 times the chiral phase transition temperature. The analysisis based on novel relations between the nth-order light quark mass derivatives of the Diraceigenvalue spectrum, ρ ( λ, m l ), and the ( n + 1)-point correlations among the eigenvalues ofthe massless Dirac operator, and the calculations were carried out at the physical value of thestrange quark mass, three lattice spacings, and light quark masses corresponding to pion massesin the range 55 −
160 MeV.Ref. [43] contains the latest results of the JLQCD collaboration. In this work the authorsinvestigate the fate of the U A (1) axial anomaly in two-flavor QCD at temperatures 190–330MeV using domain wall fermions, reweighted to overlap fermions, at a lattice spacing of 0 . U A (1) susceptibility, χ π − χ δ , and examine the degeneracy of U A (1) partners in meson and baryon correlators. Their conclusion is that all the data abovethe critical temperature indicate that the axial U A (1) violation is consistent with zero withinstatistical errors.All the results discussed so far mainly refer to the temperature dependence of the axialsusceptibility U A (1), screening masses, and related quantities. The topological susceptibility, χ T , is another observable that can be useful in investigating the fate of the axial anomaly in thehigh-temperature phase of QCD , and its dependence on temperature has also been extensivelyinvestigated, [35], [36], [37], [39], [43].The authors of Ref. [35] explore N f = 2 + 1 QCD in a range of temperatures, from T c toaround 4 T c , and their results for the topological susceptibility differ strongly, both in the sizeand in the temperature dependence, from the dilute instanton gas prediction, giving rise to ashift of the axion dark-matter window of almost one order of magnitude with respect to theinstanton computation.The authors of Ref. [36], however, observe in the same model very distinct temperaturedependences of the topological susceptibility in the ranges above and below 250 MeV; thoughfor temperatures above 250 MeV, the dependence is found to be consistent with the diluteinstanton gas approximation, at lower temperatures the falloff of topological susceptibility ismilder.On the other hand, a novel approach is proposed in Ref. [37], i.e., the fixed Q integration,based on the computation of the mean value of the gauge action and chiral condensate at fixedtopological charge Q ; they find a topological susceptibility many orders of magnitude smallerthan that of Ref. [35] in the cosmologically relevant temperature region.A more recent lattice calculation [39] of the topological properties of N f = 2 + 1 QCDwith physical quark masses and temperatures around 500 MeV gives as a result a small butnon-vanishing topological susceptibility, although with large error bars in the continuum limitextrapolations, pointing that the effects of the U A (1) axial anomaly still persist at these tem-peratures.The JLQCD collaboration [43] reports also results for the topological susceptibility in two-flavor QCD , in the temperature range 195 −
330 MeV, for several quark masses, and their datashow a suppression of χ T ( m ) near the chiral limit. The authors claim that their results are notaccurate enough to determine whether χ T ( m ) vanishes at a finite quark mass.In short we see how, despite the great effort devoted to investigating the fate of the axial6nomaly in the chirally symmetric phase of QCD , the current situation on this issue is far fromsatisfactory. U A (1) anomaly in models with exact SU A ( N f ) chiral symmetry We will devote the rest of this article mainly to analyze the physical effects of the U A (1) anomalyin a fermion-gauge theory with two or more flavors, which exhibits an exact SU A ( N f ) chiralsymmetry in the chiral limit. However we will also give a quick look to the one flavor model, andto the multi-flavor model with spontaneous non-abelian chiral symmetry breaking. Althoughmany of the results presented here can be found in Refs. [18], [44] and [45], we will make therest of this article self-contained for ease of reading.We will show in this section that a gauge-fermion quantum field theory, with U A (1) axialanomaly, and in which the scalar condensate vanishes in the chiral limit because of an exactnon-Abelian SU A (2) chiral symmetry, should exhibit a singular quark-mass dependence of thevacuum energy density and a divergent correlation length in the correlation function of thescalar condensate, if the U A (1) symmetry is effectively broken. On the contrary, if we assumethat all correlation lengths are finite, and hence the vacuum energy density is an analyticalfunction of the quark mass, we will see that the vacuum energy density becomes, at least upto second order in the quark masses, θ -independent. In the former case, the non-anomalousWard-Takahashi (W-T) identities will tell us that several pseudoscalar correlation functions,those of the SU A (2) chiral partners of the flavor singlet scalar meson, should exhibit a divergentcorrelation length too. We will also argue that this result can be generalized for any number offlavors N f > To begin, let us write the continuum Euclidean action for a vector-like gauge theory with global U A (1) anomaly in the presence of a θ -vacuum term S = Z d d x N f X f ¯ ψ f ( x ) ( γ µ D µ ( x ) + m f ) ψ f ( x ) + 14 F aµν ( x ) F aµν ( x ) + iθQ ( x ) (1)where d is the space-time dimensionality, D µ ( x ) the covariant derivative, N f the number offlavors, and Q ( x ) the density of topological charge of the gauge configuration. The topologicalcharge Q is the integral of the density of topological charge Q ( x ) over the space-time volume,and it is an integer number which in the case of QCD reads as follows Q = g π Z d xǫ µνρσ F aµν ( x ) F aρσ ( x ) . (2)To keep mathematical rigor we will avoid ultraviolet divergences with the help of a latticeregularization, and will use Ginsparg-Wilson (G-W) fermions [46], the overlap fermions [47],[48] being an explicit realization of them. The motivation to use G-W fermions is that theyshare with the continuum formulation all essential ingredients. Indeed G-W fermions show anexplicit U A (1) anomalous symmetry [49], good chiral properties, a quantized topological charge,and allow us to establish and exact index theorem on the lattice [50].The lattice fermionic action for a massless G-W fermion can be written in a compact formas 7 F = a d ¯ ψDψ = a d X v,w ¯ ψ ( v ) D ( v, w ) ψ ( w ) (3)where v and w contain site, Dirac and color indices, and D , the Dirac-Ginsparg-Wilson operator,obeys the essential anticommutation equation Dγ + γ D = aDγ D (4) a being the lattice spacing.Action (3) is invariant under the following lattice U A (1) chiral rotation ψ → e iαγ ( I − aD ) ψ, ¯ ψ → ¯ ψe iα ( I − aD ) γ (5)which for a → e − i α a tr ( γ D ) (6)where a tr ( γ D ) = n − − n + = Q (7)is an integer number, the difference between left-handed and right-handed zero modes, whichcan be identified with the topological charge Q of the gauge configuration. Equations (6) and(7) show us how Ginsparg-Wilson fermions reproduce the U A (1) axial anomaly.We can also add a symmetry breaking mass term, m ¯ ψ (cid:0) − a D (cid:1) ψ to action (3), so G–Wfermions with mass are described by the fermion action S F = a d ¯ ψDψ + a d m ¯ ψ (cid:16) − a D (cid:17) ψ (8)and it can also be shown that the scalar and pseudoscalar condensates S = ¯ ψ (cid:16) − a D (cid:17) ψ P = i ¯ ψγ (cid:16) − a D (cid:17) ψ (9)transform, under the chiral U A (1) rotations (5), as a vector, just in the same way as ¯ ψψ and i ¯ ψγ ψ do in the continuum formulation.In what follows we will use dimensionless fermion fields and a dimensionless Dirac-Ginsparg-Wilson operator. In such a case the fermion action for the N f -flavor model is S F = N f X f (cid:26) ¯ ψ f Dψ f + m f ¯ ψ (cid:18) − D (cid:19) ψ f (cid:27) (10)where m f is the mass of flavor f in lattice units. The partition function of this model, in thepresence of a θ -vacuum term, can be written as the sum over all topological sectors, Q , of thepartition function in each topological sector times a θ -phase factor, Z = X Q Z Q e iθQ (11)where Q , which takes integer values, is bounded at finite volume by the number of degrees offreedom. At large lattice volume the partition function should behave as8 ( β, m f , θ ) = e − V E ( β,m f ,θ ) (12)where E ( β, m f , θ ) is the vacuum energy density, β the inverse gauge coupling, m f the f -flavormass, and V = V s × L t the lattice volume in units of the lattice spacing. Q = 0 topological sector. The one-flavor model and the multi-flavormodel with spontaneous chiral symmetry breaking In our analysis of the physical phenomena induced by the topological properties of the theory,the Q = 0 topological sector will play an essential role, and because of that we devote thissubsection to review some results concerning the relation between vacuum expectation values oflocal and non-local operators computed in the Q = 0 sector, with their corresponding values inthe full theory, which takes into account the contribution of all topological sectors. In particularwe will show that the vacuum energy density, and the vacuum expectation value of any finiteoperator, as for instance local or intensive operators, computed in the Q = 0 topological sector,is equal, in the infinite volume limit, to its corresponding value in the full theory. We willalso show that this property is in general not true for non-local operators, the flavor-singletpseudoscalar susceptibility being a paradigmatic example of this. However there are non-localoperators, as for instance the second order fermion-mass derivatives of the vacuum energydensity, the value of which in the Q = 0 sector match their corresponding values in the fulltheory, in the infinite lattice volume limit.We will also analyze in this subsection the one-flavor case, as well as the multi-flavor casewith spontaneous chiral symmetry breaking, and will show how, although the aforementionedproperties will imply that the U A (1) symmetry is spontaneously broken in the Q = 0 topolog-ical sector, the Goldstone theorem is not realized because the divergence of the flavor-singletpseudoscalar susceptibility, in this sector, does not originate from a divergent correlation length[18].The partition function, and the mean value of any operator O , as for instance the scalarand pseudoscalar condensates, or any correlation function, in the Q = 0 topological sector, canbe computed respectively as Z Q =0 = 12 π Z d θZ ( β, m f , θ ) (13) h O i Q =0 = R d θ h O i θ Z ( β, m f , θ ) R d θZ ( β, m f , θ ) (14)where h O i θ , which is the mean value of O computed with the lattice regularized integrationmeasure (1), is a function of the inverse gauge coupling β , flavor masses m f , and θ , and we willrestrict ourselves to the case in which it takes a finite value in the infinite lattice volume limit.Since the vacuum energy density (12), as a function of θ , has its absolute minimum at θ = 0for non-vanishing fermion masses, the following relations hold in the infinite volume limit E Q =0 ( β, m f ) = E ( β, m f , θ ) θ =0 (15) h O i Q =0 = h O i θ =0 (16)where E Q =0 ( β, m f ) is the vacuum energy density of the Q = 0 topological sector.Taking in mind these results, let us start with the analysis of the one-flavor model at zerotemperature. The results that follow apply, for instance, to one-flavor QCD in four dimensionsor to the one-flavor Schwinger model. 9n the one flavor model the only axial symmetry is an anomalous U A (1) symmetry. Thestandard wisdom on the vacuum structure of this model in the chiral limit is that it is uniqueat each given value of θ , the θ -vacuum. Indeed, the only plausible reason to have a degeneratevacuum in the chiral limit would be the spontaneous breakdown of chiral symmetry, but sinceit is anomalous, actually there is no symmetry. Furthermore, due to the chiral anomaly, themodel shows a mass gap in the chiral limit, and therefore all correlation lengths are finite inphysical units. Since the model is free from infrared divergences, the vacuum energy density canbe expanded in powers of the fermion mass m u , treating the quark mass term as a perturbation[51]. This expansion will be then an ordinary Taylor series E ( β, m u , θ ) = E ( β ) − Σ ( β ) m u cos θ + O ( m u ) , (17)giving rise to the following expansions for the scalar and pseudoscalar condensates h S u i = − Σ ( β ) cos θ + O ( m u ) (18) h P u i = − Σ ( β ) sin θ + O ( m u ) (19)where S u and P u are the scalar and pseudoscalar condensates (9) normalized by the latticevolume S u = 1 V ¯ ψ (cid:18) − D (cid:19) ψ P u = iV ¯ ψγ (cid:18) − D (cid:19) ψ (20)The topological susceptibility χ T is given, on the other hand, by the following expansion χ T = Σ ( β ) m u cos θ + O ( m u ) (21)The resolution of the U A (1) problem is obvious if we set down the W-T identity whichrelates the pseudoscalar susceptibility χ η = P x h P u ( x ) P u (0) i , the scalar condensate h S u i , andthe topological susceptibility χ T χ η = − h S u i m u − χ T m u . (22)Indeed the divergence in the chiral limit of the first term in the right-hand side of (22) iscanceled by the divergence of the second term in this equation, giving rise to a finite pseudoscalarsusceptibility, and a finite non-vanishing mass for the pseudoscalar η boson.In what concerns the Q = 0 topological sector, we want to notice two relevant features:1. The global U A (1) axial symmetry is not anomalous in the Q = 0 topological sector.2. If we apply equation (16) to the computation of the vacuum expectation value of thescalar condensate, we get that the U A (1) symmetry is spontaneously broken in the Q = 0sector because the chiral limit of the infinite volume limit of the scalar condensate, thelimits taken in this order, does not vanish.Equation (14) allow us to write for the infinite volume limit of the two-point pseudoscalarcorrelation function, h P u ( x ) P u (0) i , the following relation h P u ( x ) P u (0) i Q =0 = h P u ( x ) P u (0) i θ =0 . (23)This equation implies that the mass of the pseudoscalar boson, m η , which can be extracted fromthe long distance behavior of the two-point correlation function, computed in the Q = 0 sector,10s equal to the value we should get in the full theory, taking into account the contribution of alltopological sectors. On the other hand the topological susceptibility, χ T , vanishes in the Q = 0sector, and hence the W-T identity (22 ) in this sector reads as follows χ Q =0 η = − h S u i Q =0 m u . (24)This identity tell us that, due to the spontaneous breaking of the U A (1) symmetry in the Q = 0sector, the pseudoscalar susceptibility diverges in the chiral limit, m u →
0, in this topologicalsector. This is a very surprising result because it suggests that the pseudoscalar boson wouldbe a Goldstone boson, and therefore its mass, m η , would vanish in the m u → h P u ( x ) P u (0) i Q =0 correlationfunction at any finite space-time volume V verifies the following equation h P u ( x ) P u (0) i Q =0 = R d θ h P u ( x ) P u (0) i c,θ e − V E ( β,m,θ ) R d θe − V E ( β,m,θ ) + R d θ h P u (0) i θ e − V E ( β,m,θ ) R d θe − V E ( β,m,θ ) (25)where h P u ( x ) P u (0) i c,θ is the connected pseudoscalar correlation function at a given θ . Thefirst term in the right-hand side of equation (25) converges in the infinite lattice volume limitto h P u ( x ) P u (0) i θ =0 , the pseudoscalar correlation function at θ = 0. In order to compute thelarge lattice volume behavior of the second term in the right-hand side of (25) we can expand h P u (0) i θ , and the vacuum energy density in powers of the θ angle as follows h P u (0) i θ = ( m u χ η + h S u i ) θ + O ( θ ) . (26) E ( β, m u , θ ) = E ( β, m u ) − χ T ( β, m u ) θ + O ( θ ) (27)and making an expansion around the saddle point solution we get, for the dominant contributionto the second term of the right hand side of (25) in the large lattice volume limit, R d θ h P u (0) i θ e − V E ( β,m,θ ) R d θe − V E ( β,m,θ ) = 1 V ( m u χ η + h S u i ) χ T . (28)Since the topological susceptibility χ T is linear in m u , for small fermion mass (21), and thescalar condensate h S u i is finite in the chiral limit, this contribution is singular at m u = 0.Equations (25) and (28) show that indeed the pseudoscalar correlation function in the zero-charge topological sector converges, in the infinite volume limit, to the pseudoscalar correlationfunction in the full theory at θ = 0. These equations also show what we can call a clusterviolation at finite volume for the pseudoscalar correlation function, in the Q = 0 topologicalsector, which disappears in the infinite volume limit. This cluster violation at finite volumeis therefore irrelevant in what concerns the pseudoscalar correlation function but, conversely,it plays a fundamental role when computing the pseudoscalar susceptibility in the Q = 0topological sector. In fact, if we sum up in equation (25) over all lattice points, and take the11nfinite volume limit, just in this order, we get for the pseudoscalar susceptibility in the Q = 0topological sector χ Q =0 η = χ η + ( m u χ η + h S u i ) χ T . (29)This equation shows that the pseudoscalar susceptibility, in the Q = 0 sector, diverges in thechiral limit due to the finite contribution of (28) to this susceptibility. Hence we have shownthat, although the Q = 0 topological sector breaks spontaneously the U A (1) axial symmetry togive account of the anomaly, the Goldstone theorem is not fulfilled because the divergence ofthe pseudoscalar susceptibility in this sector does not come from a divergent correlation length.The multi-flavor model with spontaneous non-abelian chiral symmetry breaking, as for in-stance QCD in the low temperature phase, shows some important differences with respect tothe one-flavor case. The model also suffers from the chiral anomaly, and has a spontaneouslybroken SU A ( N f ) chiral symmetry. Because of the Goldstone theorem, there are N f − m u = m d = m . The anomalousW-T identity (22) for the flavor-singlet pseudoscalar susceptibility reads now χ η = − h S i m − χ T m (30)while the non-anomalous identity for the pion susceptibility is χ π = − h S i m (31)where S = S u + S d . The Q = 0 sector breaks spontaneously the U A (2) symmetry, and the W-Tidentities for this sector are χ Q =0 η = χ Q =0 π = − h S i Q =0 m − (32)The analysis done in this subsection allows to conclude that, although χ π is a non-localoperator, it takes, in the infinite lattice volume limit, the same value in the Q = 0 sector asin the full theory. Conversely, that is not true for the flavor-singlet pseudoscalar susceptibility,which diverges in the chiral limit in the Q = 0 sector, while remaining finite in the full theory.A straightforward analysis, as the one done for the one-flavor case, shows that, again, thedivergence of χ Q =0 η does not come from a divergent correlation length.The case in which the SU ( N f ) chiral symmetry is fulfilled in the vacuum will be discussedin detail in the next subsections. SU A (2) chiral symmetry There are several relevant physical theories, as for instance the two-flavor Schwinger model or
QCD in the high temperature phase, that suffer from the U A (1) axial anomaly, and in whichthe non-abelian chiral symmetry is fulfilled in the vacuum. I will discuss in what follows whatare the physical expectations in these theories. I will argue that a theory which verifies theaforementioned properties should show, in the chiral limit, a divergent correlation length, anda rich spectrum of massless chiral bosons. To this end we will start with the assumption that12ll correlation lengths are finite and will see that, in such a case, the axial U A (1) symmetry iseffectively restored.We consider a fermion-gauge model with two flavors, up and down, with masses m u and m d ,exact SU A (2) chiral symmetry, and global U A (1) axial anomaly. The Euclidean fermion-gaugeaction (10) is S F = m u ¯ ψ u (cid:18) − D (cid:19) ψ u + m d ¯ ψ d (cid:18) − D (cid:19) ψ d + ¯ ψ u Dψ u + ¯ ψ d Dψ d (33)where D is the Dirac-Ginsparg-Wilson operator.If we assume, as in the one-flavor model, that all correlation lengths are finite, and the modelshows a mass gap in the chiral limit, the vacuum energy density can also be expanded, as inthe one-flavor case, in powers of the fermion masses m u , m d , as an ordinary Taylor series E ( β, m u , m d ) = E ( β, , − m u χ s u,u ( β ) − m d χ s d,d ( β ) − m u m d χ s u,d ( β ) + . . . (34)The linear terms in (34) vanish because the SU A (2) symmetry is fulfilled in the vacuum, and χ s u,u , χ s d,d and χ s u,d are the scalar up, down and up-down susceptibilities respectively χ s u,u ( β ) = V (cid:10) S u (cid:11) m u = m d =0 χ s d,d ( β ) = V (cid:10) S d (cid:11) m u = m d =0 χ s u,d ( β ) = V h S u S d i m u = m d =0 (35)where S u and S d are the scalar up and down condensates (20), normalized by the lattice vol-ume. The disconnected contributions are absent in (35) because the SU A (2) chiral symmetryconstrains h S u i m u = m d =0 and h S d i m u = m d =0 to vanish, and χ s u,u ( β ) = χ s d,d ( β ) because of flavorsymmetry. Moreover we know that the vacuum energy density of the Q = 0 topological sector,in the infinite volume limit, will also be given by (34). In the presence of a θ -vacuum term, expansion (34) becomes E ( β, m u , m d ) = E ( β, , − m u χ s u,u ( β ) − m d χ s d,d ( β ) − m u m d cos θχ s u,d ( β ) + . . . (36)The scalar up and down susceptibilities for massless fermions get all their contribution fromthe Q = 0 topological sector, and therefore we can write χ s u,u ( β ) = χ s d,d ( β ) = χ Q =0 s u,u ( β ) = χ Q =0 s d,d ( β )Since the SU A (2) chiral symmetry is fulfilled in the vacuum, the vacuum expectation valueof any local order parameter for this symmetry vanishes in the chiral limit. We have also seenthat any local operator takes, in the thermodynamic limit, the same vacuum expectation valuein the Q = 0 topological sector than in the full theory. Therefore the SU (2) A chiral symmetryof the Q = 0 sector should also be fulfilled in the vacuum of this sector.The scalar up and down susceptibilities, in the Q = 0 sector, for non vanishing quark masses,also agree with their corresponding values in the full theory, in the infinite volume limit In Ref. [45] it was implicitly assumed that the vacuum energy density of the Q = 0 sector is also a C function of m u and m d . We will show here that this assumption is justified. The simplest way to see that is true is to take into account that these susceptibilities can be obtained assecond-order mass derivatives of the free energy density, and the free energy density of the Q = 0 sector and ofthe full theory agree if both quark masses are of the same sign. Q =0 s u,u ( β, m u , m d ) = χ s u,u ( β, m u , m d ) χ Q =0 s d,d ( β, m u , m d ) = χ s d,d ( β, m u , m d ) . Therefore these quantities can be obtained from (34) χ Q =0 s u,u ( β, m u , m d ) = χ s u,u ( β ) + . . .χ Q =0 s d,d ( β, m u , m d ) = χ s d,d ( β ) + . . . where the dots indicate terms that vanish in the chiral limit.The pseudoscalar up and down susceptibilities, in the Q = 0 sector, χ Q =0 p u,u ( β, m u , m d ) = V (cid:10) P u (cid:11) Q =0 , χ Q =0 p d,d ( β, m u , m d ) = V (cid:10) P d (cid:11) Q =0 , can be obtained from the W-T identities in thissector (24) beside (34) χ Q =0 p u,u ( β, m u , m d ) = χ s u,u ( β ) + | m d || m u | χ s u,d ( β ) + . . .χ Q =0 p d,d ( β, m u , m d ) = χ s d,d ( β ) + | m u || m d | χ s u,d ( β ) + . . . (37)where the absolute value of the quark masses is due to the fact that these susceptibilities areeven functions of the quark masses, and again the dots indicate terms that vanish in the chirallimit.The difference of the scalar and pseudoscalar susceptibilities for the up or down quarks, χ s u,u − χ p u,u , χ s d,d − χ p d,d , is an order parameter for both, the U A (1) axial symmetry, and the SU A (2) chiral symmetry. We can compute this quantity, in the full theory, making use of (34),the W-T identities (22), and the topological susceptibility χ T ( β, m u , m d ) = m u m d χ s u,d ( β ) + . . . the last obtained from (36), and we get χ p u,u ( β, m u , m d ) = − h S u i m u − χ T m u = χ s u,u ( β ) + . . .χ p d,d ( β, m u , m d ) = − h S d i m d − χ T m d = χ s d,d ( β ) + . . . (38)where, also in this case, the dots indicate terms that vanish in the chiral limit. We see fromequation (38) that indeed, and in spite of the U A (1) anomaly, χ s u,u − χ p u,u and χ s d,d − χ p d,d ,which are also order parameters for the SU A (2) chiral symmetry, vanish in the chiral limit, asit should be. Conversely if we compute this order parameter in the Q = 0 topological sector we get χ Q =0 s u,u ( β, m u , m d ) − χ Q =0 p u,u ( β, m u , m d ) = − | m d || m u | χ s u,d ( β ) + . . .χ Q =0 s d,d ( β, m u , m d ) − χ Q =0 p d,d ( β, m u , m d ) = − | m u || m d | χ s u,d ( β ) + . . . and therefore this order parameter for the non-abelian chiral symmetry only vanishes in thechiral limit if χ s u,d ( β ) = 0. Thus we see that, under the assumption that all correlation lengths A non-local order parameter for a given symmetry, which is fulfilled in the vacuum, can diverge if thecorrelation length diverges, as for instance the non-linear susceptibility χ nl ( h ) = ∂ m ( h ) ∂h in the Ising model atthe critical temperature. However we are assuming here that all correlation lengths are finite, and hence thenon-local order parameter should vanish. SU A (2) chiral symmetry in the Q = 0 sector requires a θ -independentvacuum energy density (36), which implies, among other things, that the axial susceptibility χ π − χ δ , an order parameter that has been used to test the effectiveness of the U A (1) anomaly,vanishes in the chiral limit.Note, on the other hand, that a non-vanishing value of χ s u,d ( β ) not only implies that the SU A (2) chiral symmetry of the Q = 0 sector is spontaneously broken, but also the SU V (2)flavor symmetry, as follows from (37). Even more, a simple calculation of the sum of the flavorsinglet scalar χ σ and pseudoscalar χ η susceptibilities for massless quarks give us χ Q =0 σ mu = md =0 + χ Q =0 η mu = md =0 = 2 χ s u,u ( β ) + 2 χ s d,d ( β ) = 4 χ s u,u ( β ) (39)while if, according to standard Statistical Mechanics, we decompose our degenerate vacuum, orGibbs state, into the sum of pure states [52] , and calculate χ σ + χ η in each one of these purestates, we get χ Q =0 σ mu = md =0 + χ Q =0 η mu = md =0 = 4 χ s u,u ( β ) + 1 + λ λ χ s u,d ( β ) (40)with λ = | m d || m u | . We see that the consistency between equations (39) and (40) requires again that χ s u,d ( β ) = 0.Therefore, even if one accepts that the Q = 0 sector spontaneously breaks the SU A (2) axialand SU V (2) flavor symmetries, even though all local order parameters for these symmetriesvanish, we have found that the consistency of the vacuum structure with the theoretical preju-dices about the Gibbs state of a statistical system requires, once more, that χ s u,u ( β ) = 0, andhence a θ -independent vacuum energy density in the full theory.In the one-flavor model we have not found inconsistencies between the assumption thatthe correlation length is finite, and the physics of the Q = 0 topological sector. The chiralcondensate takes a non vanishing value in the chiral limit, and hence the U A (1) axial symmetryis spontaneously broken in the Q = 0 sector, giving account in this way of the U A (1) axialanomaly of the full theory. In the two-flavor model, and under the same assumption of finitenessof the correlation length, we would need a non-vanishing value of χ s u,d ( β ) to have an effective U A (1) axial symmetry breaking which, also in this case, would imply the spontaneous breakingof the global U A (1) symmetry in the Q = 0 sector. However, in such a case, we find stronginconsistencies that lead us to conclude that, either χ s u,d ( β ) = 0, and hence the U A (1) symmetryis effectively restored, or a divergent correlation length is imperative if the U A (1) symmetry isnot effectively restored. We have argued that the two-flavor theory with exact SU A (2) chiral symmetry and axial U A (1)symmetry violation should exhibit a divergent correlation length in the scalar sector, in thechiral limit. In this subsection we will give a qualitative but powerful argument which stronglysupports this result. To this end we will explore the expected phase diagram of the model inthe Q = 0 topological sector [44], and will apply the Landau theory of phase transitions to it.Since the SU A (2) chiral symmetry is assumed to be fulfilled in the vacuum, and the flavorsinglet scalar condensate is an order parameter for this symmetry, its vacuum expectation value h S i = h S u i + h S d i = 0 vanishes in the limit in which the fermion mass m →
0. However, if weconsider two non-degenerate fermion flavors, up and down, with masses m u and m d respectively,and take the limit m u → m d = 0 fixed, the up condensate S u will reach a non-vanishingvalue 15im m u → h S u i = s u ( m d ) = 0 (41)because the U (1) u axial symmetry, which exhibits our model when m u = 0, is anomalous, andthe SU A (2) chiral symmetry, which would enforce the up condensate to be zero, is explicitlybroken if m d = 0.Obviously the same argument applies if we interchange m u and m d , and we can thereforewrite lim m d → h S d i = s d ( m u ) = 0 (42)and since the SU A (2) chiral symmetry is recovered and fulfilled in the vacuum when m u , m d → m d → s u ( m d ) = lim m u → s d ( m u ) = 0 (43)Let us consider now our model, with two non degenerate fermion flavors, restricted to the Q = 0 topological sector. As previously discussed, the mean value of any local or intensiveoperator in the Q = 0 topological sector will be equal, if we restrict ourselves to the region inwhich both m u >
0, and m d >
0, to its mean value in the full theory, in the infinite volumelimit. We can hence apply this result to h S u i and h S d i and write the following equationslim m u → h S u i Q =0 = s u ( m d ) = 0lim m d → h S d i Q =0 = s d ( m u ) = 0 (44)The global U (1) u axial symmetry of our model at m u = 0, and the U (1) d symmetry at m d = 0,are not anomalous in the Q = 0 sector, and equation (44) tells us that both, the U (1) u symmetryat m u = 0 , m d = 0 and the U (1) d symmetry at m u = 0 , m d = 0 are spontaneously broken inthis sector. This is not surprising at all since the present situation is similar to what happensin the one flavor model previously discussed.Fig. 1 is a schematic representation of the phase diagram of the two-flavor model, in the Q = 0 topological sector, and in the ( m u , m d ) plane, which emerges from the previous discussion.The two coordinate axis show first order phase transition lines. If we cross perpendicularly the m d = 0 axis, the mean value of the down condensate jumps from s d ( m u ) to − s d ( m u ), and thesame is true if we interchange up and down. All first order transition lines end however at acommon point, the origin of coordinates m u = m d = 0, where all condensates vanish because atthis point we recover the SU A (2) chiral symmetry, which is assumed to be also a symmetry ofthe vacuum. Notice that if the SU A (2) chiral symmetry is spontaneously broken, as it happensfor instance in the low temperature phase of QCD , the phase diagram in the ( m u , m d ) planewould be the same as that of Fig. 1 with the only exception that the origin of coordinates isnot an end point.Landau’s theory of phase transitions predicts that the end point placed at the origin ofcoordinates in the ( m u , m d ) plane is a critical point, the scalar condensate should show a nonanalytic dependence on the fermion masses m u and m d as we approach the critical point, andhence the scalar susceptibility should diverge. But since the vacuum energy density in the Q = 0 topological sector, and its fermion mass derivatives, matches the vacuum energy densityand fermion mass derivatives in the full theory, and the same is true for the critical equation Since the two flavor model with m u < m d < θ = 0 is equivalent to the same model with m u > m d >
0, this result is also true if both m u < m d < ritical point m d m u Figure 1: Phase diagram of the two-flavor model in the Q = 0 topological sector. The coordinate axisin the ( m u , m d ) plane are first order phase transition lines. The origin of coordinates is the end pointof all first order transition lines. The vacuum energy density, its derivatives, and expectation values oflocal operators of the two-flavor model at θ = 0 only agree with those of the Q = 0 sector in the first( m u > , m d >
0) and third ( m u < , m d <
0) quadrants (the darkened areas). of state, Landau’s theory of phase transitions predicts a non-analytic dependence of the flavorsinglet scalar condensate on the fermion mass, and a divergent correlation length in the chirallimit of our full theory, in which we take into account the contribution of all topological sectors.More precisely, we can apply the Landau approach to analyze the critical behavior aroundthe two first order transition lines in Fig.1 near the end point, or critical point. In the analysisof the m d = 0 transition line we consider m d as an external ”magnetic field”and m u as the”temperature”, and vice versa for the analysis of the m u = 0 line. Then the standard Landauapproach tell us that the up and down condensates verify the two following equations of state − m u h S u i − = − C m d h S u i − + 4 C − m d h S d i − = − C m u h S d i − + 4 C (45)where C and C are two positive constants. If we fix the ratio of the up and down masses m u m d = λ , the equations of state (45) allow us to write the following expansions for de up anddown condensates h S u i = − m u (cid:18) C (cid:19) + C (cid:0) C (cid:1) λ m u + . . . h S d i = − m d (cid:18) C (cid:19) + C λ (cid:0) C (cid:1) m d + . . . (46)17quation (46) shows explicitly the non analytical behavior of the up and down condensates. Inthe degenerate flavor case, m u = m d = m , the scalar condensate and the flavor-singlet scalarsusceptibility near the critical point scale as h S i = h S u i + h S d i = − (cid:18) C (cid:19) m + . . .χ σ ( m ) = 13 (cid:18) C (cid:19) m − + . . . (47)showing up explicitly the divergence of the flavor singlet scalar susceptibility in the chiral limit.We see that the critical behavior of the chiral condensate in the Landau approach (47) isdescribed by the mean field critical exponent δ = 3. Mean field critical exponents are expectedto be correct in high dimensions, while in low dimensions, the effect of fluctuations can changetheir mean field values. This means that, in the latter case, the Landau approach give us a goodqualitative description of the phase diagram, but fails in its quantitative predictions of criticalexponents.To finish the Landau approach analysis we want to point out that all these results can begeneralized in a straightforward way to a number of flavors N f > Beyond the Landau approach, we can parameterize the critical behavior of the flavor singletscalar condensate and of the mass-dependent contribution to the vacuum energy density, in thetwo degenerate flavor model, with a critical exponent δ > h S i m → ≃ − Cm δ . (48) E ( β, m ) − E ( β, ≃ − Cδδ + 1 m δ +1 δ . (49)where C is a dimensionless positive constant that depends on the inverse gauge coupling β .Equation (48) gives us a divergent scalar susceptibility, χ σ ( m ) ∼ Cδ m − δδ , and hence a masslessscalar boson as m → SU A (2) non-anomalous chiral symmetry χ ¯ π ( m ) = − h S i m , (50)we get that also χ ¯ π ( m ) diverges when m → Cm − δδ , and a rich spectrum of masslessbosons ( σ, ¯ π ) emerges in the chiral limit. The susceptibility of the flavor singlet pseudoscalarcondensate fulfills the anomalous W-T identity (30), and because of the U A (1) axial anomaly,the η -boson mass is expected to remain finite (non-vanishing) in the chiral limit.The hyperscaling hypothesis, which arises as a natural consequence of the block-spin renor-malization group approach, says that the only relevant length near the critical point of a mag-netic system, in what concerns the singular part E s ( β, m ) of the free or vacuum energy density,is the correlation length ξ . Since equation (49) contains only the singular contribution to thevacuum energy density, we can write E s ( β, m ) ≃ − Cδδ + 1 m δ +1 δ ∼ ξ − d (51)18nd the following relationship between the correlation length and the fermion mass ξ ∼ m − δ +1 dδ (52)which implies that the pion and sigma-meson masses scale with the fermion mass as follows m ¯ π , m σ ∼ m δ +1 dδ (53)In the presence of an isospin breaking term, the fermion action can be written in a compactform as S F = (cid:18) m u + m d (cid:19) ¯ ψ (cid:18) − D (cid:19) ψ − (cid:18) m d − m u (cid:19) ¯ ψ (cid:18) − D (cid:19) τ ψ + ¯ ψDψ (54)where ψ is a Grassmann field carrying site, Dirac, color and flavor indices, and τ is the thirdPauli matrix acting in flavor space.If we include also a θ -vacuum term in the action, this θ -term can be removed through achiral U A (1) transformation, which leaves the ¯ ψDψ interaction term invariant, and if next wealso perform a suitable non-anomalous chiral transformation, we get the effective fermion actionthat follows S F = M ( m u , m d , θ ) ¯ ψ (cid:18) − D (cid:19) ψ + A ( m u , m d , θ ) i ¯ ψγ (cid:18) − D (cid:19) ψ + B ( m u , m d , θ ) ¯ ψ (cid:18) − D (cid:19) τ ψ + ¯ ψDψ (55)where M ( m u , m d , θ ), A ( m u , m d , θ ) and B ( m u , m d , θ ) are given by M ( m u , m d , θ ) = 12 (cid:0) m u + m d + 2 m u m d cos θ (cid:1) (56) A ( m u , m d , θ ) = 2 m u m d sin θ ( m u + m d ) (cid:16) m u + m d − m u m d m u + m d +2 m u m d tan θ (cid:17) (57) B ( m u , m d , θ ) = − m d − m u θ (cid:16) m u + m d − m u m d m u + m d +2 m u m d tan θ (cid:17) (58)Since we do not expect singularities at non-vanishing fermion masses, the vacuum energydensity E ( β, M, A, B ) can be expanded in powers of A and B as an ordinary Taylor series,and taking into account the symmetries of the effective action (55), we can write the followingequation for this expansion up to second order E ( β, m u , m d , θ ) ≡ E ( β, M, A, B ) = E ( β, M, ,
0) + 12 A χ η ( β, M ) + 12 B χ δ ( β, M ) + . . . (59)where χ η ( β, M ) and χ δ ( β, M ) are the flavor singlet pseudoscalar susceptibility and the δ -mesonsusceptibility in the theory with two degenerate flavors of mass M ( m u , m d , θ ), respectively. Notethat this expansion should have a good convergence if θ and m d − m u are small.The vacuum energy density, to the lowest order of the expansion (59), is that of the modelwith two degenerate flavors of mass M ( m u , m d , θ ), in the absence of a θ -vacuum term. We havepreviously shown that this model should show a critical behavior (48), (49) around the chirallimit, and hence we get, to the lowest order of this expansion,19 ( β, m u , m d , θ ) − E ( β, , ,
0) = − C δ +1 δ δδ + 1 (cid:0) m u + m d + 2 m u m d cos θ (cid:1) δ +12 δ + . . . (60)The free energy density depends on m u , m d and θ through (cid:0) m u + m d + 2 m u m d cos θ (cid:1) , and itsdominant contribution in the chiral limit is given by the power-law behavior of equation (60). The flavor-singlet pseudoscalar susceptibility, χ η ( β, M ), fulfills the anomalous W-T identity(30), and hence it is expected to remain finite in the chiral limit. Since the SU A (2) chiralsymmetry is exact in this limit, the same holds true for χ δ ( β, M ). In such conditions, therelevance of the second order correction to the zero-order contribution to the vacuum energydensity (59), for two degenerate flavors, turns out to be A ( m, θ ) E ( β, M, − E ( β, , ∼ m − δ sin θ (cid:0) cos θ (cid:1) δ (61)while in the isospin breaking case, and for small θ values, we have A ( m u , m d , θ ) E ( β, M, , − E ( β, , , ∼ m u m d θ ( m u + m d ) δ B ( m u , m d , θ ) E ( β, M, , − E ( β, , , ∼ ( m d − m u ) ( m u + m d ) δ (62)Since δ > δ = 3 in the mean field model), we see that the critical behavior of themodel, which describes the low energy theory, is fully controlled in both cases by the zero-order contribution to the vacuum energy density (63), and the second order contribution canbe neglected in what concerns the chiral limit of the theory.Let us now look at some interesting physical consequences that can be obtained from equa-tion (60). In the degenerate flavor case, m u = m d = m , equations (60), (52) and (53) become E ( β, m, θ ) − E ( β, ,
0) = − Cδδ + 1 (cid:18) m cos θ (cid:19) δ +1 δ + . . . (63) ξ ∼ (cid:18) m cos θ (cid:19) − δ +1 dδ (64) m ¯ π , m σ ∼ (cid:18) m cos θ (cid:19) δ +1 dδ (65)For non-degenerate flavors, the vacuum energy density (60) at θ = 0 is a function of m u + m d ,hence the vacuum expectation values of the up and down condensates are equal, and the sameholds true for their susceptibilities: h S u i = h S u i = − C δ +1 δ ( m u + m d ) δ X x ( h S u ( x ) S u (0) i − h S u ( x ) i h S u (0) i ) = X x ( h S d ( x ) S d (0) i − h S d ( x ) i h S d (0) i ) = Note that if we apply this expansion of the vacuum energy density to two-flavor
QCD at T = 0, where chiralsymmetry is spontaneously broken, and hence δ = ∞ in (48) and (49), we get the vacuum energy density of thelow energy chiral effective Lagrangian model [51]. x ( h S u ( x ) S d (0) i − h S u ( x ) i h S d (0) i ) = C δ +1 δ δ ( m u + m d ) − δδ . (66)We don’t see any dependency on m d − m u , and isospin breaking effects are therefore absentin these quantities, which on the other hand show a singular behavior in the chiral limit. Thenormalized flavor singlet scalar susceptibility, χ σ , χ σ = C δ δ ( m u + m d ) − δδ . (67)diverges in the chiral limit, while the δ -meson susceptibility, χ δ , vanishes in the zero-orderapproximation to the vacuum energy density, this indicating that it is a good approximationwhen the ratio of the σ and δ meson masses is small, mσm δ ≪ χ T = C δ +1 δ ( m u + m d ) − δδ m u m d (68)showing that this quantity is sensitive to the isospin breakdown.The W-T identity for the charged pions, π ± , reads χ π ± = − h S u i + h S d i m u + m d (69)and hence we get χ π ± = C δ ( m u + m d ) − δδ . (70)Like the σ -susceptibility, the charged pions susceptibility diverges in the chiral limit.To calculate the susceptibility of the neutral pion, we use the following W − T identities X x h P u ( x ) P u (0) i = − h S u i m u − χ T m u X x h P d ( x ) P d (0) i = − h S d i m d − χ T m d X x h P u ( x ) P d (0) i = − χ T m u m d (71)which give us X x h P u ( x ) P u (0) i = X x h P d ( x ) P d (0) i = − X x h P u ( x ) P d (0) i = C δ +1 δ ( m u + m d ) − δδ (72)and for the normalized neutral pion susceptibility we get χ π = C δ ( m u + m d ) − δδ . (73)Equations (70) and (73) show that the π ± and π susceptibilities are equal and independentof m d − m u . Again isospin breaking effects are absent in these quantities, and even though m d − m u = 0, the three pions have the same mass. In what concerns the flavor-singlet pseudoscalarsusceptibility, χ η , equation (72) shows that it vanishes.21ast but not least, if for simplicity we consider two degenerate flavors, equations (53) and(68) imply that the pion mass m ¯ π (or the σ -meson mass) and the topological susceptibility χ T verify the following relation m ¯ π ( χ T ) d = k ( β, L t ) (74)where k is a dimensionless quantity that depends on the inverse gauge coupling β , and eventually,at finite temperature T , on the lattice temporal extent L t , but that is independent of the fermionmass m .In summary, we have seen that, in the zero order approximation to the vacuum energydensity, that accounts for the chiral critical behavior of the theory, isospin breaking effects onlymanifest in the topological susceptibility. The three pions have the same mass, the ratio of thepion (73) and σ -meson (67) susceptibilities is equal to the critical exponent δ , and the pion (or σ -meson) mass is related with the topological susceptibility, as seen in equation (74). Quantum Electrodynamics in (1+1)-dimensions, is a good laboratory to test the results reportedin the previous section. The model is confining [53], exactly solvable at zero fermion mass, hasnon-trivial topology, and shows explicitly the U A (1) axial anomaly [54]. Besides that, theSchwinger model does not require infinite renormalization, and this means that, if we use alattice regularization, the bare parameters remain finite in the continuum limit.On the other hand, the SU A ( N f ) non-anomalous axial symmetry in the chiral limit of themulti-flavor Schwinger model is fulfilled in the vacuum, and this property makes this model aperfect candidate to check our predictions on the existence of quasi-massless scalar and pseu-doscalar bosons in the spectrum of the model, the mass of which vanishes in the chiral limit.The Euclidean continuum action for the two-flavor theory is S = Z d x { ¯ ψ u ( x ) γ µ ( ∂ µ + iA µ ( x )) ψ u ( x ) + ¯ ψ d ( x ) γ µ ( ∂ µ + iA µ ( x )) ψ d ( x ) } + Z d x { m u ¯ ψ u ( x ) ψ u ( x ) + m d ¯ ψ d ( x ) ψ d ( x ) + 14 e F µν ( x ) + iθQ ( x ) } (75)where m u , m d are the fermion masses and e is the electric charge or gauge coupling, which hasmass dimensions. F µν ( x ) = ∂ µ A ν ( x ) − ∂ ν A µ ( x ), and γ µ are 2 × { γ µ , γ ν } = 2 g µν (76)At the classic level this theory has an internal SU V (2) × SU A (2) × U V (1) × U A (1) symmetryin the chiral limit. However, the U A (1)-axial symmetry is broken at the quantum level becauseof the axial anomaly. The divergence of the axial current is ∂ µ J Aµ ( x ) = 12 π ǫ µν F µν ( x ) , (77)where ǫ µν is the antisymmetric tensor, and hence does not vanish. The axial anomaly inducesthe topological θ -term iθQ = iθ R d xQ ( x ) in the action, where Q ( x ) = 14 π ǫ µν F µν ( x ) (78)is the density of topological charge, the topological charge Q being an integer number.22he Schwinger model was analyzed years ago by Coleman [7], computing some quantitativeproperties of the theory in the continuum for both, weak coupling em ≪
1, and strong couplingor chiral limit em ≫ θ = π and some intermediate fermion mass m separating a weak coupling phase ( em ≪ Z symmetry of the model at θ = π is spontaneously broken, from a strong coupling phase( em ≫ Z symmetry is fulfilled in the vacuum. This qualitative result has beenrecently confirmed by numerical simulations of the Euclidean-lattice version of the model [55].What is however more interesting for the content of this article is the Coleman analysis ofthe two-flavor model. As previously stated, the theory (75) has an internal SU V (2) × SU A (2) × U V (1) × U A (1) symmetry in the chiral limit, and the U A (1) axial symmetry is anomalous.Since continuous internal symmetries can not be spontaneously broken in a local field theoryin two dimensions [56], the SU A (2) symmetry has to be fulfilled in the vacuum, and the scalarcondensate, which is an order parameter for this symmetry, vanishes in the chiral limit. Hencethe two-flavor Schwinger model verifies all the conditions we assumed in section 3.We summarize here the main Coleman’s findings for the two-flavor model with degeneratemasses m u = m d = m :1. For weak coupling, em ≪
1, the results on the particle spectrum are almost the same asfor the massive Schwinger model.2. For strong coupling, em ≫
1, the low-energy effective theory depends only on one massparameter, m e cos θ , the vacuum energy density is then proportional to E ( m, e, θ ) ∝ e (cid:18) m cos θ (cid:19) , (79)and the chiral condensate, at θ = 0, is therefore h ¯ ψψ i ∝ m e (80)3. The lightest particle in the theory is an isotriplet, and the next lightest is an isosinglet. Theisosinglet/isotriplet mass ratio is √
3. If there are other stable particles in the model, theymust be O (cid:16)(cid:2) em (cid:3) (cid:17) times heavier than these. The light boson mass, M , has a fractionalpower dependence on the fermion mass m : M ∝ e (cid:18) m cos θ (cid:19) (81)Many of these results have been corroborated by several authors both, in the continuum[57], [58], [59], [60], [61], and using the lattice approach [62], [63]. Coleman concluded his paper[7] by asking some questions concerning things he didn’t understand, and we cite here two ofthem:1. Why are the lightest particles in the theory a degenerate isotriplet, even if one quark is10 times heavier than the other?2. Why does the next-lightest particle has I P G = 0 ++ , rather than 0 −− ?23he results of section 3 allow us to qualitatively understand the main Coleman’s findingsfor the two-flavor model with degenerate masses in the strong coupling limit, as well as to givea reliable answer to the previous questions.In section 3.5 we predicted, from the interplay between the U A (1) anomaly and the exact SU A (2) chiral symmetry, a singular behavior of the vacuum energy density (49), (63) in thechiral limit limit as E ∼ C (cid:0) m cos θ (cid:1) δ +1 δ . In the Schwinger model, a simple dimensional analysistell us that C must be proportional to e δ − δ . Therefore our result matches perfectly Coleman’sresult (79) if we choose δ = 3.In what concerns the masses of the light bosons, our prediction (65), m ¯ π , m σ ∼ (cid:0) m cos θ (cid:1) δ +1 dδ matches, for δ = 3, Colemans’s result (81) too.In section 3.5 we also predicted that the flavor-singlet scalar susceptibility (67), and the”pion” susceptibility (70), (73), should diverge in the chiral limit as Kδ m − δδ e δ − δ and Km − δδ e δ − δ respectively, and for δ = 3 we have χ σm → = K m − e ∼ | h | ˆ O σ | σ i | m σ χ π m → = Km − e ∼ | h | ˆ O π | π i | m π (82)where K is a dimensionless constant.We have also seen that the σ and ¯ π meson masses, in the strong-coupling limit, scale withthe quark mass as m ¯ π , m σ ∼ m e . (83)Taking into account that the SU A (2) symmetry is exact in the chiral limit, equations (82) and(83) imply that lim m → | h | ˆ O σ | σ i | = lim m → | h | ˆ O π | π i | ∼ e (84)and therefore we have lim m → χ π ( m, e ) χ σ ( m, e ) = lim m → m σ ( m, e ) m π ( m, e ) = 3 (85)These results show that indeed the lightest particle in the theory is an isotriplet, and thenext lightest is an isosinglet I P G = 0 ++ . However our result for the ratio m σ m ¯ π = 3 [45] is indisagreement with Coleman’s result m σ m ¯ π = √ m u + m d , and we have shown that in such a case only the topologicalsusceptibility is sensitive to isospin breaking effects. The three pion susceptibilities (70), (73)and masses are equal, and to see isospin breaking effects we should go to the second ordercontribution. The relevance of the second order correction to the zero-order contribution tothe vacuum energy density has also been estimated (62), and for θ = 0 turns out to be of theorder of ( m d − m u ) ( m u + m d ) e , a result that justifies the validity of the zero-order approximation in thestrong-coupling ( em u,d ≫
1) limit. The factor e δ − δ comes again from dimensional analysis in the Schwinger model. Georgi has recently argued [64] that isospin breaking effects are exponentially suppressed in the two-flavorSchwinger model as a consequence of conformal coalescence. U A (1) anomalous symmetry. What is a two-dimensional peculiarity is the fact that in thechiral limit, when all fermion masses vanish, these quasi-massless bosons become unstable, andthe low-energy spectrum of the model reduces to a massless non-interacting boson, in accordancewith Coleman’s theorem [56] which forbids the existence of massless interacting bosons in twodimensions. Thanks to massive
QCD simulations on the lattice, we have at present a good qualitativeand quantitative understanding on the vacuum realization of the non-abelian SU A ( N f ) chiralsymmetry, as a function of the physical temperature. As far as the U A (1) anomaly, and its as-sociated θ parameter are concerned, and especially in the high temperature phase, the currentsituation is however far from satisfactory. With the aim of clarifying the current status concern-ing this issue, we have devoted the first part of this article to analyze the present status of theinvestigations on the effectiveness of the U A (1) axial anomaly in QCD , at temperatures aroundand above the non-abelian chiral transition critical temperature. We have seen that theoreticalpredictions require assumptions whose validity is not always proven, and lattice simulationsusing different discretization schemes lead to apparently contradictory conclusions in severalcases. Hence, despite the great effort devoted to investigating the fate of the axial anomaly inthe chirally symmetric phase of
QCD , we still don’t have a clear answer to this question.In the second part of the article we have analyzed some interesting physical implications ofthe U A (1) anomaly, recently suggested [45], in systems where the non-abelian axial symmetry isfulfilled in the vacuum. The standard wisdom on the origin of massless bosons in the spectrumof a Quantum Field Theory, describing the interaction of gauge fields coupled to matter fields,is based on two well known features: gauge symmetry, and spontaneous symmetry breaking ofcontinuous symmetries. We have shown that the topological properties of the theory can bethe basis of an alternative mechanism, other than Goldstone’s theorem, to generate masslessbosons in the chiral limit, if the U A (1) symmetry remains effectively broken, and the non-abelian SU A ( N f ) chiral symmetry is fulfilled in the vacuum.The two-flavor Schwinger model, or Quantum Electrodynamics in two space-time dimen-sions, is a good test-bed for our predictions. Indeed the Schwinger model shows a non-trivialtopology, which induces the U A (1) axial anomaly. Moreover, in the two-flavor case, the non-abelian SU A (2) chiral symmetry is fulfilled in the vacuum, as required by Coleman’s theorem[56] on the impossibility to break spontaneously continuous symmetries in two dimensions. Thismodel was analyzed by Coleman long ago in [7], where he computed some quantitative proper-ties of the theory in the continuum for both weak coupling, em ≪
1, and strong coupling em ≫ m , in the chiral limit, and the flavor singlet scalarsusceptibility diverges when m →
0. In addition, our results provide a reliable answer to somequestions that Coleman asked himself.It is worth wondering if the reason for the rich spectrum of light chiral bosons near the chirallimit, found in the Schwinger [7] and U ( N ) [65] models, lies in some uninteresting peculiarities oftwo-dimensional models, or if there is a deeper and general explanation for this phenomenon. Wewant to remark, concerning this, that the analysis done in section 4 strongly suggests that the25xistence of quasi-massless chiral bosons in the spectrum of the two-flavor Schwinger model, nearthe chiral limit, does not originates in some uninteresting peculiarities of two-dimensional modelsbut it should be a consequence of the interplay between exact non-abelian chiral symmetry, andan effectively broken U A (1) anomalous symmetry. What is a two-dimensional peculiarity is thefact that, in the chiral limit, when all fermion masses vanish, these quasi-massless bosons becomeunstable, and the low-energy spectrum of the model reduces to a massless non-interacting boson[66], [67], in accordance with Coleman’s theorem [56] which forbids the existence of masslessinteracting bosons in two dimensions.In what concerns QCD , the analysis of the effects of the U A (1) axial anomaly in its hightemperature phase, in which the non-abelian chiral symmetry is restored in the ground state,has aroused much interest in recent time because of its relevance in axion phenomenology.Moreover, the way in which the U A (1) anomaly manifests itself in the chiral symmetry restoredphase of QCD at high temperature could be tested when probing the
QCD phase transition inrelativistic heavy ion collisions.We have argued in section 3 that a quantum field theory, with an exact non-Abelian SU A (2)symmetry, and in which the U A (1) axial symmetry is effectively broken, should exhibit a singularquark-mass dependence in the vacuum energy density, and a divergent correlation length inthe correlation function of the scalar condensate, in the chiral limit. On the contrary, if allcorrelation lengths are finite, and hence the vacuum energy density is an analytical function ofthe quark mass, we have shown that the vacuum energy density becomes, at least up to secondorder in the quark masses, θ -independent. The topological susceptibility either vanish or is atleast of fourth order in the quark masses and, in such a case, all typical effects of the U A (1)anomaly are lost. QCD in the chirally symmetric phase, T ' T c , shows an exact non-abelianaxial symmetry and hence, either the vacuum energy density is an analytical function of thequark masses, and QCD becomes θ -independent, or the screening mass spectrum of the modelshows several quasi-massless chiral bosons, whose masses vanish in the chiral limit. Which ofthe two aforementioned possibilities actually happens in the high temperature phase of QCD is a difficult question, as follows from the current status of lattice simulations reported in thisarticle.A recent lattice calculation [39] of the topological properties of three-flavor
QCD with physi-cal quark masses, and temperatures around 500
M eV , gives as a result a small but non-vanishingtopological susceptibility, although with large error bars in the continuum limit extrapolations,suggesting that the effects of the U A (1) axial anomaly still persist at these temperatures. If weassume this to be true, and hence that there is a temperature interval in the high temperaturephase where the U A (1) anomalous symmetry remains effectively broken, we can apply to thistemperature interval the main conclusions of section 3.Taking into account lattice determination of the light quark masses [68] ( m u ≃ M eV , m d ≃ M eV , m s ≃ M eV ), we can consider
QCD with two quasi-massless quarks as a goodapproach. The results of section 3 predict then a spectrum of light σ and ¯ π mesons at T ' T c .The presence of these light scalar and pseudoscalar mesons in the chirally symmetric hightemperature phase of QCD could, on the other hand, significantly influence the dilepton andphoton production observed in the particle spectrum [69] at heavy-ion collision experiments.Lattice calculations of mesonic screening masses in two [34] and three [41] flavor
QCD ,around and above the critical temperature, give results that are unfortunately not enough toallow a good check of our spectrum prediction. However, the results of Ref. [41] show a smallchange of the pion screening-mass when crossing the critical temperature, and a decreasingscreening mass, at T ' T c , when going from the ¯ us to the ¯ ud channel.26 Acknowledgments
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