Axial $U_A(1)$ Anomaly: a New Mechanism to Generate Massless Bosons

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 chiral symmetry and its spontaneous breaking. Spontaneous symmetry breaking is a non-perturbative phenomenon and thanks to massive $QCD$ simulations on the lattice we have at present a good understanding on the vacuum realization of the non-abelian chiral symmetry 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 from satisfactory. The first part of this article is devoted to review the present status of lattice calculations, 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 implications of the $U_A(1)$ anomaly, recently suggested, in systems where the non-abelian axial symmetry is fulfilled in the vacuum. More precisely I will argue that, if the $U_A(1)$ symmetry remains effectively 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 chiral limit.


Introduction
Nowadays we know that symmetries play an important role in determining the Lagrangian of a quantum field theory. There are essentially two types of symmetry, local ones or gauge symmetries, and global ones. The gauge symmetries are characterized by transformations which depend on the space-time coordinates while in global symmetries the transformations are spacetime independent. Also gauge symmetries serve to fix the couplings of the Lagrangian, and global symmetries allow us to to assign quantum numbers to the particles and to predict the existence 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 establishment of this theory as the correct theory describing hadronic physics, it was realized that the essential ingredients of the hadronic world at low energies are chiral symmetry and its spontaneous breaking. Indeed these two properties of the strong interaction have important phenomenological implications, and allow us to understand some puzzling phenomena as why pions have much smaller masses than the the proton mass, and why we do not see degenerate masses for chiral partners 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 be investigated mainly through lattice QCD simulations. As a matter of fact, lattice QCD is the most powerful technique for investigating non-perturbative effects from first principles. However, putting chiral symmetry onto the lattice turned out to be a difficult task. The underlying reason is that a naive lattice regularization suffers from the doubling problem. The addition of the Wilson term to the naive action solves the doubling problem but breaks chiral symmetry explicitly, even for massless quarks. This is usually not considered to be a fundamental problem because we expect that the symmetry is restored in the continuum limit. However, at finite lattice 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 number of 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 to obtain very accurate results in lattice QCD simulations with two and three flavors.
The doubling problem cannot be simply overcome because there is a fundamental theorem by Nielsen and Ninomiya which states that, on the lattice, one cannot implement chiral symmetry 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 at the end of the past century with a generalization of chiral symmetry, through the so-called Ginsparg-Wilson equation for the lattice Dirac operator, which replaces the standard anticommutation relation of the continuum formulation Dγ 5 + γ 5 D = 0 by Dγ 5 + γ 5 D = aDγ 5 D. With this 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 continuum limit, but at finite lattice spacing, a, an axial transformation involves also the gauge fields, and this is how Ginsparg-Wilson's formulation evades Nielsen-Ninomiya theorem.
All these features are well established in the lattice community, and the interested reader can find in [1], for instance, a very good guide.
Returning to the topic of QCD phenomenology, there is also another puzzling phenomenon which is known as the U (1) problem. The QCD Lagrangian for massless quarks is invariant under the chiral group (1), with V and A denoting vector and axial vector transformations respectively. Below 1GeV the flavor index 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, that the U A (3) axial symmetry is spontaneously broken in the chiral limit, but in such a case we would have nine Goldstone bosons. The pions, K-meson, and η-meson are eight of them but the 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) problem suggests 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 qualitative and quantitative understanding on the vacuum realization of the non-abelian SU A (N f ) chiral symmetry, as a function of the physical temperature, but as far as U A (1) anomaly and its associated θ parameter are concerned, and especially in the high temperature phase, the current situation is far from satisfactory, and this makes understanding the role of the θ parameter in QCD, and its connection with the strong CP problem, one of the biggest challenges for high energy theorists [2].
The aim to elucidate the existence of new low-mass weakly interacting particles from a theoretical, phenomenological, and experimental point of view is intimately related to this issue. The light particle that has gathered the most attention has been the axion, predicted by Weinberg [3] and Wilczek [4], in the Peccei and Quinn mechanism [5], to explain the absence of parity and temporal invariance violations induced by the QCD vacuum. The axion is one of the more 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 calculating 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 gauge theory 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 axion mass m a and decay constant f a through the relation χ T = m 2 a f 2 a . The axion mass is, on the other 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 susceptibility in QCD becomes of primordial interest in this context.
In this article I will focus on the current status of the lattice calculations, in the high temperature chirally symmetric phase of QCD, of quantities directly related to the U A (1) axial anomaly, as the topological and axial U A (1) susceptibilities, and screening masses, and will also discuss on some interesting physical implications of the U A (1) axial anomaly in systems where the non-abelian axial symmetry is fulfilled in the vacuum. I will briefly review in section 2 some theoretical 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 on the effectiveness of the axial anomaly in this phase. In section 3 I will argue that the topological properties 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, other than 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 an excellent test bed for verifying the predictions of section 3, and section 4 contains the results of this test. The last section contains a discussion of the results reported in this article.

Theoretical biases versus numerical results
The large mass of the η ′ meson should come from the effects of the U A (1) axial anomaly and its related gauge field topology, both present in QCD. Despite the difficulty of computing the contribution of disconnected diagrams to the η ′ correlator in lattice simulations, these obstacles have been overcome and lattice calculations [8], [9], [10] give a mass for the η ′ meson compatible with its experimental value, and this can be seen as an indirect confirmation that the effects of the anomaly are present in the low temperature phase of QCD.
Conversely, the current situation regarding the fate of the axial anomaly in the high temperature 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 twoflavor QCD, for instance, is affected by the way in which the effects of the U A (1) axial anomaly manifest themselves around the critical temperature [11]. Indeed, if the U A (1) axial symmetry remains effectively broken, we expect a continuous chiral transition belonging to the threedimensional O(4) vector universality class, which shows a critical exponent δ = 4.789(6) [12], while if U A (1) is effectively restored, the chiral transition is first order, or second order with critical exponents belonging to the U V (2) × U A (2) → U V (2) universality class (δ = 4.3(1)) [13].
The first investigations on the fate of the U A (1) axial anomaly in the chiral symmetry restored phase of QCD started a long time ago. The idea that the chiral symmetry restored phase 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 the zero mode's contribution, that all the disconnected contributions to the two-point correlation functions in the SU A (2) symmetric phase at high temperature vanish in the chiral limit. The main conclusion of this work was that the eight scalar and pseudoscalar mesons should have the same mass in the chiral limit, the typical effects of the U A (1) axial anomaly being absent in this phase. Also Cohen argued in [16] that the analyticity of the free energy density in the quark mass m, around m = 0, in the high temperature phase, imposes constraints on the spectral density of the Dirac operator around the origin which are enough to guarantee the previous results.
Later on, Aoki et al. [17] got constraints on the Dirac spectrum of overlap fermions, strong enough for all of the U (1) A breaking effects among correlation functions of scalar and pseudoscalar operators to vanish, and they concluded that there is no remnant of the U (1) A anomaly above the critical temperature in two flavor QCD, at least in these correlation functions. Their results were obtained under the assumptions that m-independent observables are analytic functions of the square quark-mass m 2 , at m = 0, and that the Dirac spectral density can be expanded 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 by the authors, their result strongly relies on their assumption that the vacuum expectation values of quark-mass independent observables, as the topological susceptibility, are analytic functions of the square quark-mass, m 2 , if the non-abelian chiral symmetry is restored. The two-flavor Schwinger model has a non spontaneously broken SU A (2) chiral symmetry and U A (1) axial anomaly, and Coleman's result for the topological susceptibility in this model [7] χ T ∝ m shows explicitly a non-analytic quark-mass dependence, and thus casts doubt on the general validity of the assumptions made in [17].
In Ref. [18] a Ginsparg-Wilson fermion lattice regularization was used, and it was argued that if the vacuum energy density is an analytical function of the quark mass in the high temperature phase of two-flavor QCD, all effects of the axial anomaly should disappear. The main conclusion of [18] was that either the typical effects of the axial U A (1) anomaly disappear in the symmetric high temperature phase, or the vacuum energy density shows a singular behavior in the quark mass at the chiral limit.
On the other hand, an analysis of chiral and U A (1) symmetry restoration based on Ward identities and U (3) chiral perturbation theory has been carried out in [19], [20]. The authors show in their work that in the limit of exact O(4) restoration, understood in terms of δ−η partner degeneration, 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 of the two-flavor model with staggered quarks. They compute two order parameters, χ π − χ σ for the 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 of the 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 using also a staggered fermion lattice regularization. As stated by the authors, the relatively coarse lattice spacing in their simulations, a ∼ 1 3 Fermi, does not allow for conclusive results on the effectiveness of the U (1) A anomaly.
After these pioneering works, this issue has been extensively investigated using numerical simulations on the lattice, and Refs. [23]- [43] are representative of that. We will focus from now on the most recently obtained results.
In Ref. [29] (2 + 1)-flavor QCD is simulated, using chiral domain wall fermions, for temperatures between 139 and 196 MeV. The light-quark mass is chosen so that the pion mass is held fixed at a heavier-than-physical 200 MeV value, while the strange quark mass is set to its physical value. The authors report results for the chiral condensates, connected and disconnected 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 staggered quarks. The authors investigate the temperature dependence of the anomalous U A (1) symmetry breaking 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 eigenmodes 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 1.5T c .
The thermal transition of QCD with two degenerate light flavors is analyzed in [34] by lattice simulations, using O(a)-improved Wilson quarks and the unimproved Wilson plaquette action. 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 masses in various isovector channels, and to quantify the strength of the U A (1)-anomaly, they use the difference between scalar and pseudoscalar screening masses. They conclude that their results suggest that the U A (1)-breaking is strongly reduced at the transition temperature, and that this disfavors a chiral transition in the O(4) universality class.
Results for mesonic screening masses in the temperature range 140 MeV T 2500 MeV in (2 + 1)-flavor QCD, using the highly improved staggered quark action, are also reported by the HotQCD Collaboration in [41], with a physical value for the strange quark mass, and two values of the light quark mass corresponding to pion masses of 160 and 140 MeV. Comparing screening masses for chiral partners, related through the chiral SU L (2) × SU R (2) and the axial U A (1) transformations, respectively, the authors find, in the case of light-light mesons, evidence for the degeneracy of screening masses related through the chiral SU L (2) × SU R (2) at or very close to the pseudocritical temperature, T pc , while screening masses related through an axial U A (1) transformation start becoming degenerate only at about 1.3T pc .
A recent calculation in (2 + 1)-flavor QCD [42], using also the highly improved staggered quark action, shows, after continuum and chiral extrapolations, that the axial anomaly remains manifested 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 analysis is based on novel relations between the nth-order light quark mass derivatives of the Dirac eigenvalue spectrum, ρ(λ, m l ), and the (n + 1)-point correlations among the eigenvalues of the massless Dirac operator, and the calculations were carried out at the physical value of the strange quark mass, three lattice spacings, and light quark masses corresponding to pion masses in the range 55 − 160 MeV.
Ref. [43] contains the latest results of the JLQCD collaboration. In this work the authors investigate the fate of the U A (1) axial anomaly in two-flavor QCD at temperatures 190-330 MeV using domain wall fermions, reweighted to overlap fermions, at a lattice spacing of 0.07 fm. They measure the axial 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 above the critical temperature indicate that the axial U A (1) violation is consistent with zero within statistical errors.
All the results discussed so far mainly refer to the temperature dependence of the axial susceptibility 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 the high-temperature phase of QCD, and its dependence on temperature has also been extensively investigated, [35], [36], [37], [39], [43].
The authors of Ref. [35] explore N f = 2 + 1 QCD in a range of temperatures, from T c to around 4T c , and their results for the topological susceptibility differ strongly, both in the size and in the temperature dependence, from the dilute instanton gas prediction, giving rise to a shift of the axion dark-matter window of almost one order of magnitude with respect to the instanton computation.
The authors of Ref. [36], however, observe in the same model very distinct temperature dependences of the topological susceptibility in the ranges above and below 250 MeV; though for temperatures above 250 MeV, the dependence is found to be consistent with the dilute instanton gas approximation, at lower temperatures the falloff of topological susceptibility is milder.
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 fixed topological charge Q; they find a topological susceptibility many orders of magnitude smaller than 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 QCD with physical quark masses and temperatures around 500 MeV gives as a result a small but non-vanishing topological susceptibility, although with large error bars in the continuum limit extrapolations, pointing that the effects of the U A (1) axial anomaly still persist at these temperatures.
The JLQCD collaboration [43] reports also results for the topological susceptibility in twoflavor QCD, in the temperature range 195 − 330 MeV, for several quark masses, and their data show a suppression of χ T (m) near the chiral limit. The authors claim that their results are not accurate 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 axial anomaly in the chirally symmetric phase of QCD, the current situation on this issue is far from satisfactory.
3 Physical effects of the 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) anomaly in a fermion-gauge theory with two or more flavors, which exhibits an exact SU A (N f ) chiral symmetry in the chiral limit. However we will also give a quick look to the one flavor model, and to the multi-flavor model with spontaneous non-abelian chiral symmetry breaking. Although many of the results presented here can be found in Refs. [18], [44] and [45], we will make the rest 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) axial anomaly, and in which the scalar condensate vanishes in the chiral limit because of an exact non-Abelian SU A (2) chiral symmetry, should exhibit a singular quark-mass dependence of the vacuum energy density and a divergent correlation length in the correlation function of the scalar condensate, if the U A (1) symmetry is effectively broken. On the contrary, if we assume that all correlation lengths are finite, and hence the vacuum energy density is an analytical function of the quark mass, we will see that the vacuum energy density becomes, at least up to second order in the quark masses, θ-independent. In the former case, the non-anomalous Ward-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 divergent correlation length too. We will also argue that this result can be generalized for any number of flavors N f > 2.

Some background
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 where d is the space-time dimensionality, D µ (x) the covariant derivative, N f the number of flavors, and Q(x) the density of topological charge of the gauge configuration. The topological charge 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 To keep mathematical rigor we will avoid ultraviolet divergences with the help of a lattice regularization, 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 they share with the continuum formulation all essential ingredients. Indeed G-W fermions show an explicit 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 form as where v and w contain site, Dirac and color indices, and D, the Dirac-Ginsparg-Wilson operator, obeys the essential anticommutation equation a being the lattice spacing. Action (3) is invariant under the following lattice U A (1) chiral rotation which for a → 0 reduces to the standard continuum chiral transformation. However the integration measure of Grassmann variables is not invariant, and the change of variables (5) induces a Jacobian where is an integer number, the difference between left-handed and right-handed zero modes, which can 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, mψ 1 − a 2 D ψ to action (3), so G-W fermions with mass are described by the fermion action and it can also be shown that the scalar and pseudoscalar condensates transform, under the chiral U A (1) rotations (5), as a vector, just in the same way asψψ and iψγ 5 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 where m f is the mass of flavor f in lattice units. The partition function of this model, in the presence of a θ-vacuum term, can be written as the sum over all topological sectors, Q, of the partition function in each topological sector times a θ-phase factor, where Q, which takes integer values, is bounded at finite volume by the number of degrees of freedom. At large lattice volume the partition function should behave as where E (β, m f , θ) is the vacuum energy density, β the inverse gauge coupling, m f the f -flavor mass, and V = V s × L t the lattice volume in units of the lattice spacing.
3.2 Q = 0 topological sector. The one-flavor model and the multi-flavor model 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 this subsection to review some results concerning the relation between vacuum expectation values of local and non-local operators computed in the Q = 0 sector, with their corresponding values in the full theory, which takes into account the contribution of all topological sectors. In particular we will show that the vacuum energy density, and the vacuum expectation value of any finite operator, 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 will also show that this property is in general not true for non-local operators, the flavor-singlet pseudoscalar susceptibility being a paradigmatic example of this. However there are non-local operators, as for instance the second order fermion-mass derivatives of the vacuum energy density, the value of which in the Q = 0 sector match their corresponding values in the full theory, in the infinite lattice volume limit. We will also analyze in this subsection the one-flavor case, as well as the multi-flavor case with spontaneous chiral symmetry breaking, and will show how, although the aforementioned properties will imply that the U A (1) symmetry is spontaneously broken in the Q = 0 topological sector, the Goldstone theorem is not realized because the divergence of the flavor-singlet pseudoscalar 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 scalar and pseudoscalar condensates, or any correlation function, in the Q = 0 topological sector, can be computed respectively as where O θ , which is the mean value of O computed with the lattice regularized integration measure (1), is a function of the inverse gauge coupling β, flavor masses m f , and θ, and we will restrict 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 = 0 for non-vanishing fermion masses, the following relations hold in the infinite volume limit 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 zero temperature. The results that follow apply, for instance, to one-flavor QCD in four dimensions or to the one-flavor Schwinger model.
In the one flavor model the only axial symmetry is an anomalous U A (1) symmetry. The standard wisdom on the vacuum structure of this model in the chiral limit is that it is unique at each given value of θ, the θ-vacuum. Indeed, the only plausible reason to have a degenerate vacuum in the chiral limit would be the spontaneous breakdown of chiral symmetry, but since it is anomalous, actually there is no symmetry. Furthermore, due to the chiral anomaly, the model shows a mass gap in the chiral limit, and therefore all correlation lengths are finite in physical units. Since the model is free from infrared divergences, the vacuum energy density can be 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 giving rise to the following expansions for the scalar and pseudoscalar condensates where S u and P u are the scalar and pseudoscalar condensates (9) normalized by the lattice volume The topological susceptibility χ T is given, on the other hand, by the following expansion The resolution of the U A (1) problem is obvious if we set down the W-T identity which relates the pseudoscalar susceptibility χ η = x P u (x) P u (0) , the scalar condensate S u , and the topological susceptibility χ T Indeed the divergence in the chiral limit of the first term in the right-hand side of (22) is canceled by the divergence of the second term in this equation, giving rise to a finite pseudoscalar susceptibility, 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 the scalar condensate, we get that the U A (1) symmetry is spontaneously broken in the Q = 0 sector because the chiral limit of the infinite volume limit of the scalar condensate, the limits taken in this order, does not vanish.
Equation (14) allow us to write for the infinite volume limit of the two-point pseudoscalar correlation function, P u (x) P u (0) , the following relation This equation implies that the mass of the pseudoscalar boson, m η , which can be extracted from the long distance behavior of the two-point correlation function, computed in the Q = 0 sector, is equal to the value we should get in the full theory, taking into account the contribution of all topological sectors. On the other hand the topological susceptibility, χ T , vanishes in the Q = 0 sector, and hence the W-T identity (22 ) in this sector reads as follows This identity tell us that, due to the spontaneous breaking of the U A (1) symmetry in the Q = 0 sector, the pseudoscalar susceptibility diverges in the chiral limit, m u → 0, in this topological sector. This is a very surprising result because it suggests that the pseudoscalar boson would be a Goldstone boson, and therefore its mass, m η , would vanish in the m u → 0 limit. The loophole to this paradoxical result is that the divergence of the susceptibility does not necessarily implies a divergent correlation length. The susceptibility is the infinite volume limit of the integral of the correlation function over all distances, in this order, and the infinite volume limit and the space-time integral of the correlation function do not necessarily commute [18]. The infinite range interaction Ising model is a paradigmatic example of the non-commutativity of the two limits.
Let us see with some detail what actually happens. The P u (x) P u (0) Q=0 correlation function at any finite space-time volume V verifies the following equation where P u (x) P u (0) c,θ is the connected pseudoscalar correlation function at a given θ. The first term in the right-hand side of equation (25) converges in the infinite lattice volume limit to P u (x) P u (0) θ=0 , the pseudoscalar correlation function at θ = 0. In order to compute the large lattice volume behavior of the second term in the right-hand side of (25) we can expand P u (0) 2 θ , and the vacuum energy density in powers of the θ angle as follows and making an expansion around the saddle point solution we get, for the dominant contribution to the second term of the right hand side of (25) in the large lattice volume limit, Since the topological susceptibility χ T is linear in m u , for small fermion mass (21), and the scalar condensate S u 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 zerocharge topological sector converges, in the infinite volume limit, to the pseudoscalar correlation function in the full theory at θ = 0. These equations also show what we can call a cluster violation at finite volume for the pseudoscalar correlation function, in the Q = 0 topological sector, which disappears in the infinite volume limit. This cluster violation at finite volume is therefore irrelevant in what concerns the pseudoscalar correlation function but, conversely, it plays a fundamental role when computing the pseudoscalar susceptibility in the Q = 0 topological sector. In fact, if we sum up in equation (25) over all lattice points, and take the infinite volume limit, just in this order, we get for the pseudoscalar susceptibility in the Q = 0 topological sector This equation shows that the pseudoscalar susceptibility, in the Q = 0 sector, diverges in the chiral limit due to the finite contribution of (28) to this susceptibility. Hence we have shown that, although the Q = 0 topological sector breaks spontaneously the U A (1) axial symmetry to give account of the anomaly, the Goldstone theorem is not fulfilled because the divergence of the 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 instance QCD in the low temperature phase, shows some important differences with respect to the one-flavor case. The model also suffers from the chiral anomaly, and has a spontaneously broken SU A (N f ) chiral symmetry. Because of the Goldstone theorem, there are N 2 f − 1 massless pseudoscalar bosons in the chiral limit and, in contrast to the one-flavor case, the infinite-volume limit and the chiral limit do not commute. However, in what concerns the flavor-singlet pseudoscalar susceptibility, the essential features previously described for the one-flavor model still work in the several-flavors case.
Let us consider the simplest case of two degenerate flavors, m u = m d = m. The anomalous W-T identity (22) for the flavor-singlet pseudoscalar susceptibility reads now while the non-anomalous identity for the pion susceptibility is where S = S u + S d . The Q = 0 sector breaks spontaneously the U A (2) symmetry, and the W-T identities for this sector are The analysis done in this subsection allows to conclude that, although χ π is a non-local operator, it takes, in the infinite lattice volume limit, the same value in the Q = 0 sector as in 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, the divergence 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 discussed in detail in the next subsections.

Two flavors and exact 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 which the non-abelian chiral symmetry is fulfilled in the vacuum. I will discuss in what follows what are the physical expectations in these theories. I will argue that a theory which verifies the aforementioned properties should show, in the chiral limit, a divergent correlation length, and a rich spectrum of massless chiral bosons. To this end we will start with the assumption that all correlation lengths are finite and will see that, in such a case, the axial U A (1) symmetry is effectively 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-gauge action (10) is 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 model shows a mass gap in the chiral limit, the vacuum energy density can also be expanded, as in the one-flavor case, in powers of the fermion masses m u , m d , as an ordinary Taylor series The linear terms in (34) vanish because the SU A (2) symmetry is fulfilled in the vacuum, and χ su,u , χ s d,d and χ s u,d are the scalar up, down and up-down susceptibilities respectively where S u and S d are the scalar up and down condensates (20), normalized by the lattice volume. The disconnected contributions are absent in (35) because the SU A (2) chiral symmetry constrains S u mu=m d =0 and S d mu=m d =0 to vanish, and χ su,u (β) = χ s d,d (β) because of flavor symmetry. 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). 1 In the presence of a θ-vacuum term, expansion (34) becomes The scalar up and down susceptibilities for massless fermions get all their contribution from the Q = 0 topological sector, and therefore we can write su,u (β) = χ Q=0 s d,d (β) Since the SU A (2) chiral symmetry is fulfilled in the vacuum, the vacuum expectation value of any local order parameter for this symmetry vanishes in the chiral limit. We have also seen that any local operator takes, in the thermodynamic limit, the same vacuum expectation value in the Q = 0 topological sector than in the full theory. Therefore the SU (2) A chiral symmetry of 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 2 . Therefore these quantities can be obtained from (34) χ Q=0 su,u (β, m u , m d ) = χ su,u (β) + . . .
where the dots indicate terms that vanish in the chiral limit.
The pseudoscalar up and down susceptibilities, in the Q = 0 sector, , can be obtained from the W-T identities in this sector (24) beside (34) where the absolute value of the quark masses is due to the fact that these susceptibilities are even functions of the quark masses, and again the dots indicate terms that vanish in the chiral limit.
The difference of the scalar and pseudoscalar susceptibilities for the up or down quarks, χ su,u − χ pu,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 the last obtained from (36), and we get where, also in this case, the dots indicate terms that vanish in the chiral limit. We see from equation (38) that indeed, and in spite of the U A (1) anomaly, χ su,u − χ pu,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, as it should be. 3 Conversely if we compute this order parameter in the Q = 0 topological sector we get and therefore this order parameter for the non-abelian chiral symmetry only vanishes in the chiral limit if χ s u,d (β) = 0. Thus we see that, under the assumption that all correlation lengths are finite, an exact SU A (2) chiral symmetry in the Q = 0 sector requires a θ-independent vacuum 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 flavor singlet scalar χ σ and pseudoscalar χ η susceptibilities for massless quarks give us while if, according to standard Statistical Mechanics, we decompose our degenerate vacuum, or Gibbs state, into the sum of pure states [52] , and calculate χ σ + χ η in each one of these pure states, we get with λ = |m d | |mu| . 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) axial and SU V (2) flavor symmetries, even though all local order parameters for these symmetries vanish, we have found that the consistency of the vacuum structure with the theoretical prejudices about the Gibbs state of a statistical system requires, once more, that χ su,u (β) = 0, and hence a θ-independent vacuum energy density in the full theory.
In the one-flavor model we have not found inconsistencies between the assumption that the correlation length is finite, and the physics of the Q = 0 topological sector. The chiral condensate takes a non vanishing value in the chiral limit, and hence the U A (1) axial symmetry is spontaneously broken in the Q = 0 sector, giving account in this way of the U A (1) axial anomaly of the full theory. In the two-flavor model, and under the same assumption of finiteness of 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 breaking of the global U A (1) symmetry in the Q = 0 sector. However, in such a case, we find strong inconsistencies that lead us to conclude that, either χ s u,d (β) = 0, and hence the U A (1) symmetry is effectively restored, or a divergent correlation length is imperative if the U A (1) symmetry is not effectively restored.

Landau approach
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 the chiral limit. In this subsection we will give a qualitative but powerful argument which strongly supports this result. To this end we will explore the expected phase diagram of the model in the 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 flavor singlet scalar condensate is an order parameter for this symmetry, its vacuum expectation value S = S u + S d = 0 vanishes in the limit in which the fermion mass m → 0. However, if we consider two non-degenerate fermion flavors, up and down, with masses m u and m d respectively, and take the limit m u → 0 keeping m d = 0 fixed, the up condensate S u will reach a non-vanishing value lim mu→0 S u = s u (m d ) = 0 (41) because the U (1) u axial symmetry, which exhibits our model when m u = 0, is anomalous, and the SU A (2) chiral symmetry, which would enforce the up condensate to be zero, is explicitly broken if m d = 0.
Obviously the same argument applies if we interchange m u and m d , and we can therefore write (42) and since the SU A (2) chiral symmetry is recovered and fulfilled in the vacuum when m u , m d → 0, we get 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 intensive operator in the Q = 0 topological sector will be equal, if we restrict ourselves to the region in which both m u > 0, and m d > 0, to its mean value in the full theory, in the infinite volume limit. 4 We can hence apply this result to S u and S d and write the following equations 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 symmetry at m u = 0, m d = 0 and the U (1) d symmetry at m u = 0, m d = 0 are spontaneously broken in this sector. This is not surprising at all since the present situation is similar to what happens in 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 the same is true if we interchange up and down. All first order transition lines end however at a common point, the origin of coordinates m u = m d = 0, where all condensates vanish because at this point we recover the SU A (2) chiral symmetry, which is assumed to be also a symmetry of the vacuum. Notice that if the SU A (2) chiral symmetry is spontaneously broken, as it happens for instance in the low temperature phase of QCD, the phase diagram in the (m u , m d ) plane would be the same as that of Fig. 1 with the only exception that the origin of coordinates is not an end point.
Landau's theory of phase transitions predicts that the end point placed at the origin of coordinates in the (m u , m d ) plane is a critical point, the scalar condensate should show a non analytic dependence on the fermion masses m u and m d as we approach the critical point, and hence 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 density and fermion mass derivatives in the full theory, and the same is true for the critical equation of state, Landau's theory of phase transitions predicts a non-analytic dependence of the flavor singlet scalar condensate on the fermion mass, and a divergent correlation length in the chiral limit 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 around the two first order transition lines in Fig.1 near the end point, or critical point. In the analysis of 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 Landau approach tell us that the up and down condensates verify the two following equations of state where C 1 and C 2 are two positive constants. If we fix the ratio of the up and down masses mu m d = λ, the equations of state (45) allow us to write the following expansions for de up and down condensates Equation (46) shows explicitly the non analytical behavior of the up and down condensates. In the degenerate flavor case, m u = m d = m, the scalar condensate and the flavor-singlet scalar susceptibility near the critical point scale as 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) is described by the mean field critical exponent δ = 3. Mean field critical exponents are expected to be correct in high dimensions, while in low dimensions, the effect of fluctuations can change their mean field values. This means that, in the latter case, the Landau approach give us a good qualitative description of the phase diagram, but fails in its quantitative predictions of critical exponents.
To finish the Landau approach analysis we want to point out that all these results can be generalized in a straightforward way to a number of flavors N f > 2.

Critical behavior of the two flavor model with an isospin breaking term
Beyond the Landau approach, we can parameterize the critical behavior of the flavor singlet scalar condensate and of the mass-dependent contribution to the vacuum energy density, in the two degenerate flavor model, with a critical exponent δ > 1 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) ∼ C δ m 1−δ δ , and hence a massless scalar boson as m → 0.
If on the other hand we write the W-T identity for the isotriplet of "pions" which follows from the SU A (2) non-anomalous chiral symmetry we get that also χπ (m) diverges when m → 0 as Cm 1−δ δ , and a rich spectrum of massless bosons (σ,π) emerges in the chiral limit. The susceptibility of the flavor singlet pseudoscalar condensate 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 renormalization group approach, says that the only relevant length near the critical point of a magnetic 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 the vacuum energy density, we can write and the following relationship between the correlation length and the fermion mass (52) which implies that the pion and sigma-meson masses scale with the fermion mass as follows In the presence of an isospin breaking term, the fermion action can be written in a compact form as where ψ is a Grassmann field carrying site, Dirac, color and flavor indices, and τ 3 is the third Pauli matrix acting in flavor space.
If we include also a θ-vacuum term in the action, this θ-term can be removed through a chiral U A (1) transformation, which leaves theψDψ interaction term invariant, and if next we also perform a suitable non-anomalous chiral transformation, we get the effective fermion action that follows where M (m u , m d , θ), A (m u , m d , θ) and B (m u , m d , θ) are given by Since we do not expect singularities at non-vanishing fermion masses, the vacuum energy density 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 following equation for this expansion up to second order where χ η (β, M ) and χ δ (β, M ) are the flavor singlet pseudoscalar susceptibility and the δ-meson susceptibility in the theory with two degenerate flavors of mass M (m u , m d , θ), respectively. Note that 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 model with two degenerate flavors of mass M (m u , m d , θ), in the absence of a θ-vacuum term. We have previously shown that this model should show a critical behavior (48), (49) around the chiral limit, and hence we get, to the lowest order of this expansion, The free energy density depends on m u , m d and θ through m 2 u + m 2 d + 2m u m d cos θ 1 2 , and its dominant contribution in the chiral limit is given by the power-law behavior of equation (60). 5 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) chiral symmetry is exact in this limit, the same holds true for χ δ (β, M ). In such conditions, the relevance of the second order correction to the zero-order contribution to the vacuum energy density (59), for two degenerate flavors, turns out to be while in the isospin breaking case, and for small θ values, we have Since δ > 1 (δ = 3 in the mean field model), we see that the critical behavior of the model, which describes the low energy theory, is fully controlled in both cases by the zeroorder contribution to the vacuum energy density (63), and the second order contribution can be neglected in what concerns the chiral limit of the theory.
Let us now look at some interesting physical consequences that can be obtained from equation (60). In the degenerate flavor case, m u = m d = m, equations (60), (52) and (53) become 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 same holds true for their susceptibilities: We don't see any dependency on m d − m u , and isospin breaking effects are therefore absent in these quantities, which on the other hand show a singular behavior in the chiral limit. The normalized flavor singlet scalar susceptibility, χ σ , diverges in the chiral limit, while the δ-meson susceptibility, χ δ , vanishes in the zero-order approximation to the vacuum energy density, this indicating that it is a good approximation when the ratio of the σ and δ meson masses is small, mσ m δ ≪ 1. The topological susceptibility is given by showing that this quantity is sensitive to the isospin breakdown. The W-T identity for the charged pions, π ± , reads and hence we get 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 which give us and for the normalized neutral pion susceptibility we get Equations (70) and (73) show that the π ± and π 0 susceptibilities are equal and independent of 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 pseudoscalar susceptibility, χ η , equation (72) shows that it vanishes.
Last but not least, if for simplicity we consider two degenerate flavors, equations (53) and (68) imply that the pion mass mπ (or the σ-meson mass) and the topological susceptibility χ T verify the following relation 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 fermion mass m.
In summary, we have seen that, in the zero order approximation to the vacuum energy density, that accounts for the chiral critical behavior of the theory, isospin breaking effects only manifest in the topological susceptibility. The three pions have the same mass, the ratio of the pion (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).

Two-flavor Schwinger model as a test bed
Quantum Electrodynamics in (1+1)-dimensions, is a good laboratory to test the results reported in the previous section. The model is confining [53], exactly solvable at zero fermion mass, has non-trivial topology, and shows explicitly the U A (1) axial anomaly [54]. Besides that, the Schwinger model does not require infinite renormalization, and this means that, if we use a lattice 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 the multi-flavor Schwinger model is fulfilled in the vacuum, and this property makes this model a perfect candidate to check our predictions on the existence of quasi-massless scalar and pseudoscalar 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 where m u , m d are the fermion masses and e is the electric charge or gauge coupling, which has mass dimensions. F µν (x) = ∂ µ A ν (x) − ∂ ν A µ (x), and γ µ are 2 × 2 matrices satisfying the algebra At the classic level this theory has an internal SU V (2) × SU A (2) × U V (1) × U A (1) symmetry in the chiral limit. However, the U A (1)-axial symmetry is broken at the quantum level because of the axial anomaly. The divergence of the axial current is where ǫ µν is the antisymmetric tensor, and hence does not vanish. The axial anomaly induces the topological θ-term iθQ = iθ d 2 xQ(x) in the action, where is the density of topological charge, the topological charge Q being an integer number.
The Schwinger model was analyzed years ago by Coleman [7], computing some quantitative properties of the theory in the continuum for both, weak coupling e m ≪ 1, and strong coupling or chiral limit e m ≫ 1. For the one-flavor case, Coleman computed the particle spectrum of the model, which shows a mass gap in the chiral limit, and conjectured the existence of a phase transition at θ = π and some intermediate fermion mass m separating a weak coupling phase ( e m ≪ 1), where the Z 2 symmetry of the model at θ = π is spontaneously broken, from a strong coupling phase ( e m ≫ 1), in which the Z 2 symmetry is fulfilled in the vacuum. This qualitative result has been recently 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 of the 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 theory in two dimensions [56], the SU A (2) symmetry has to be fulfilled in the vacuum, and the scalar condensate, which is an order parameter for this symmetry, vanishes in the chiral limit. Hence the 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 degenerate masses m u = m d = m: 1. For weak coupling, e m ≪ 1, the results on the particle spectrum are almost the same as for the massive Schwinger model.
2. For strong coupling, e m ≫ 1, the low-energy effective theory depends only on one mass parameter, m The results of section 3 allow us to qualitatively understand the main Coleman's findings for the two-flavor model with degenerate masses in the strong coupling limit, as well as to give a reliable answer to the previous questions.
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 where K is a dimensionless constant.
We have also seen that the σ andπ meson masses, in the strong-coupling limit, scale with the quark mass as Taking into account that the SU A (2) symmetry is exact in the chiral limit, equations (82) and (83) imply that and therefore we have lim m→0 χ π 0 (m, e) χ σ (m, e) = lim m→0 m σ (m, e) m π 0 (m, e) = 3 These results show that indeed the lightest particle in the theory is an isotriplet, and the next lightest is an isosinglet I P G = 0 ++ . However our result for the ratio mσ mπ = 3 [45] is in disagreement with Coleman's result mσ mπ = √ 3 [7]. In what concerns the first Coleman's question, we have argued in section 3.5 that the strong coupling limit performed by Coleman corresponds to the zero-order contribution to the vacuum energy density expansion (59). This zero-order contribution depends on the quark masses only through the combination m u + m d , and we have shown that in such a case only the topological susceptibility 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 order contribution. The relevance of the second order correction to the zero-order contribution to the vacuum energy density has also been estimated (62), and for θ = 0 turns out to be of the order of (m d −mu) 2 (mu+m d ) , a result that justifies the validity of the zero-order approximation in the strong-coupling ( e m u,d ≫ 1) limit. 7 6 The factor e δ−1 δ comes again from dimensional analysis in the Schwinger model. 7 Georgi has recently argued [64] that isospin breaking effects are exponentially suppressed in the two-flavor Schwinger model as a consequence of conformal coalescence.
The analysis done in this section strongly suggests that the existence of quasi-massless chiral bosons in the spectrum of the two-flavor Schwinger model, near the chiral limit, does not originates in some uninteresting peculiarities of two-dimensional models, but it should be a consequence of the interplay between exact non-abelian chiral symmetry, and an effectively broken U A (1) anomalous symmetry. What is a two-dimensional peculiarity is the fact that in the chiral limit, when all fermion masses vanish, these quasi-massless bosons become unstable, and the low-energy spectrum of the model reduces to a massless non-interacting boson, in accordance with Coleman's theorem [56] which forbids the existence of massless interacting bosons in two dimensions.

Conclusions and discussion
Thanks to massive QCD simulations on the lattice, we have at present a good qualitative and quantitative understanding on the vacuum realization of the non-abelian SU A (N f ) chiral symmetry, as a function of the physical temperature. As far as the U A (1) anomaly, and its associated θ parameter are concerned, and especially in the high temperature phase, the current situation is however far from satisfactory. With the aim of clarifying the current status concerning this issue, we have devoted the first part of this article to analyze the present status of the investigations on the effectiveness of the U A (1) axial anomaly in QCD, at temperatures around and above the non-abelian chiral transition critical temperature. We have seen that theoretical predictions require assumptions whose validity is not always proven, and lattice simulations using different discretization schemes lead to apparently contradictory conclusions in several cases. Hence, despite the great effort devoted to investigating the fate of the axial anomaly in the 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 of the U A (1) anomaly, recently suggested [45], in systems where the non-abelian axial symmetry is fulfilled in the vacuum. The standard wisdom on the origin of massless bosons in the spectrum of 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 of continuous symmetries. We have shown that the topological properties of the theory can be the basis of an alternative mechanism, other than Goldstone's theorem, to generate massless bosons 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 dimensions, is a good test-bed for our predictions. Indeed the Schwinger model shows a non-trivial topology, which induces the U A (1) axial anomaly. Moreover, in the two-flavor case, the nonabelian 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. This model was analyzed by Coleman long ago in [7], where he computed some quantitative properties of the theory in the continuum for both weak coupling, e m ≪ 1, and strong coupling e m ≫ 1. In what concerns the strong-coupling results, the main Coleman's findings are qualitatively in agreement with our predictions. The vacuum energy density, and the chiral condensate show a singular dependence on the fermion mass, m, in the chiral limit, and the flavor singlet scalar susceptibility diverges when m → 0. In addition, our results provide a reliable answer to some questions that Coleman asked himself.
It is worth wondering if the reason for the rich spectrum of light chiral bosons near the chiral limit, found in the Schwinger [7] and U (N ) [65] models, lies in some uninteresting peculiarities of two-dimensional models, or if there is a deeper and general explanation for this phenomenon. We want to remark, concerning this, that the analysis done in section 4 strongly suggests that the existence of quasi-massless chiral bosons in the spectrum of the two-flavor Schwinger model, near the chiral limit, does not originates in some uninteresting peculiarities of two-dimensional models but it should be a consequence of the interplay between exact non-abelian chiral symmetry, and an effectively broken U A (1) anomalous symmetry. What is a two-dimensional peculiarity is the fact that, in the chiral limit, when all fermion masses vanish, these quasi-massless bosons become unstable, 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 massless interacting bosons in two dimensions.
In what concerns QCD, the analysis of the effects of the U A (1) axial anomaly in its high temperature 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 restored phase of QCD at high temperature could be tested when probing the QCD phase transition in relativistic 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 singular quark-mass dependence in the vacuum energy density, and a divergent correlation length in the correlation function of the scalar condensate, in the chiral limit. On the contrary, if all correlation lengths are finite, and hence the vacuum energy density is an analytical function of the quark mass, we have shown that the vacuum energy density becomes, at least up to second order in the quark masses, θ-independent. The topological susceptibility either vanish or is at least 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-abelian axial symmetry and hence, either the vacuum energy density is an analytical function of the quark masses, and QCD becomes θ-independent, or the screening mass spectrum of the model shows several quasi-massless chiral bosons, whose masses vanish in the chiral limit. Which of the 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 this article.
A recent lattice calculation [39] of the topological properties of three-flavor QCD with physical quark masses, and temperatures around 500M eV , gives as a result a small but non-vanishing topological 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 we assume this to be true, and hence that there is a temperature interval in the high temperature phase where the U A (1) anomalous symmetry remains effectively broken, we can apply to this temperature interval the main conclusions of section 3.
Taking into account lattice determination of the light quark masses [68] (m u ≃ 2M eV , m d ≃ 5M eV , m s ≃ 94M eV ), we can consider QCD with two quasi-massless quarks as a good approach. The results of section 3 predict then a spectrum of light σ andπ mesons at T T c . The presence of these light scalar and pseudoscalar mesons in the chirally symmetric high temperature phase of QCD could, on the other hand, significantly influence the dilepton and photon 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 to allow a good check of our spectrum prediction. However, the results of Ref. [41] show a small change of the pion screening-mass when crossing the critical temperature, and a decreasing screening mass, at T T c , when going from theūs to theūd channel.