Phase structure of a generalized Nambu Jona-Lasinio model with Wilson fermions in the mean field or large $N$-expansion

We analyze the vacuum structure of a generalized lattice Nambu--Jona-Lasinio model with two flavors of Wilson fermions, such that its continuum action is the most general four-fermion action with 'trivial' color interactions, and having a $SU(2)_V x SU(2)_A$ symmetry in the chiral limit. The phase structure of this model in the space of the two four-fermion couplings shows, in addition to the standard Aoki phases, new phases with $<\bar\psi \gamma_5\psi>!= 0$, in close analogy to similar results recently suggested by some of us for lattice QCD with two degenerate Wilson fermions. This result shows how the phase structure of an effective model for low energy QCD cannot be entirely understood from Wilson Chiral Perturbation Theory, based on the standard QCD chiral effective Lagrangian approach.


Introduction
Since the first numerical investigations of four-dimensional non-abelian gauge theories with dynamical Wilson fermions were performed in the early 80's [1,2], the understanding of the phase and vacuum structure of lattice QCD with Wilson fermions at non-zero lattice spacing, and of the way in which chiral symmetry is recovered in the continuum limit, has been a goal of lattice field theorists. The complexity of the phase structure of this model has been known for a long time. The existence of a phase with parity and flavor symmetry breaking was conjectured for this model by Aoki in the middle 80's [3,4], and confirmed later on .
The standard wisdom on lattice QCD with Wilson fermions is that even if chiral symmetry is explicitly broken at finite lattice spacing a by the Wilson regularization, this symmetry will be recovered and spontaneously broken in the continuum limit. However, it is difficult to understand why there exists a critical line at finite lattice spacing along which the three pions are massless. Indeed, the pions cannot be the three Goldstone bosons associated with the spontaneous breaking of the SU (2) chiral symmetry since, as previously stated, the Wilson regularization breaks explicitly this symmetry.
One of the main features of Aoki's picture was to clarify this point. In the Aoki phase, the charged pions are massless because they are the two Goldstone bosons associated with the spontaneous breaking of the SU (2) flavor symmetry down to U (1), with a non vanishing vacuum expectation value of the iψγ 5 τ 3 ψ condensate. The neutral pion, which is massive in the Aoki phase, becomes massless on the critical line because flavour symmetry is continuously recovered on this line which separates the broken (Aoki) phase from the unbroken (physical) phase. The other relevant feature of the Aoki scenario is that it provides a counterexample to the Vafa-Witten theorem on the impossibility to spontaneously break parity in a vector-like theory with positive definite integration measure [28][29][30][31][32].
Aoki's conjecture has been supported not only by numerical simulations of lattice QCD, but also by theoretical studies based on the Nambu-Jona-Lasinio model [5,33], on the linear sigma model [6], and on applying Wilson chiral perturbation theory (W χP T ) to the continuum effective Lagrangian [11]. The latter analysis predicts, near the continuum limit, two possible scenarios, depending on the sign of an unknown low-energy coefficient. In the first scenario, flavor and parity are spontaneously broken, and there is an Aoki phase with a nonzero value only for the iψγ 5 τ 3 ψ condensate, whereas iψγ 5 ψ = 0. In the other one (the "first-order" scenario) there is no spontaneous symmetry breaking.
This standard picture for the Aoki phase was questioned by three of us in [21], where we conjectured on the appearance of new vacua in the Aoki phase, which can be characterized by a non-vanishing vac-uum expectation value of the flavor-singlet pseudoscalar condensate iψγ 5 ψ, and which cannot be connected to the Aoki vacua by parityflavor symmetry transformations. More recently, we have obtained results from numerical simulations of lattice QCD with two degenerate flavors of Wilson fermions, suggesting that our conjecture could be realized [34]. Since these results seem to question the validity of the W χP T analysis [11], an approach which has been successfully applied in many contexts, it is worthwhile to analyze the possible origins of this discrepancy.
First, one should notice that the chiral effective Lagrangian approach is based on the continuum effective Lagrangian written as a series of contributions proportional to powers of the lattice spacing a, plus the construction of the corresponding chiral effective Lagrangian, keeping only the terms up to order a 2 [11]. This means that predictions based on this chiral effective Lagrangian approach should work close enough to the continuum limit, where keeping terms only up to order a 2 can be justified. However, the data reported in [34] were obtained at β = 2.0, and a very rough estimate gives a lattice spacing of order 3.0 GeV −1 at this β. Hence, a possible explanation for the discrepancies that we found relies on the necessity of including higher-order terms in the chiral effective Lagrangian.
We want to recall here that the Aoki effective potential was obtained from a strong coupling expansion combined with a 1/N expansion [4], i.e., far away from the continuum limit. Furthermore, Aoki's solution shows degenerate vacua with iψγ 5 τ 3 ψ = 0, iψγ 5 ψ = 0 and iψγ 5 τ 3 ψ = 0, iψγ 5 ψ = 0 respectively in the strong coupling limit. The inclusion of higher order contributions to the strong coupling-1/N expansions breaks the vacuum degeneracy by selecting the standard vacuum with iψγ 5 τ 3 ψ = 0, but the normalized difference of vacuum energy densities of the two vacua is of order 10 −14 at β = 2.0 and N = 3, showing the extremely high instability of the Aoki solution.
The second possible origin of the discrepancies between the nonstandard scenario of [21,34] and the W χP T analysis of [11], the analysis of which will be the main subject of this paper, lies in the following point. The chiral effective Lagrangian approach is based, as it is well known, on the assumption that the relevant low-energy degrees of freedom in QCD are the three pions. This assumption can be reliable in the physical phase, up to the critical line, and also in the Aoki phase, near the critical line, but it could break down as we go deep in the Aoki phase, where the neutral pion is massive. Indeed, QCD with two degenerate flavors of Wilson fermions of bare mass m 0 = −4.0 in lattice units should also show degenerate vacua with iψγ 5 τ 3 ψ = 0, iψγ 5 ψ = 0 and iψγ 5 τ 3 ψ = 0, iψγ 5 ψ = 0 respectively, as discussed in [34].
With the purpose of establishing the range of applicability of the standard QCD chiral effective Lagrangian approach, we will analyze in this paper the vacuum structure of a generalized Nambu-Jona-Lasinio model (N JL) with Wilson fermions in the mean field or leading order 1/N -expansion. The model has been chosen to possess the more general SU (2) V × SU (2) A symmetry in the continuum, in analogy to QCD. The election of the N JL model for our analysis was motivated by the fact that four-dimensional models without gauge fields, and with four fermion interactions, are considered as effective models to describe the low energy physics of QCD 1 [35,36].
The outline of the paper is as follows. In section 2 we describe the model in the continuum and its lattice regularized version with Wilson fermions, as well as the way in which the model can be analytically solved in the mean field-1/N -expansion with the help of eight auxiliary scalar and pseudoscalar fields. The gap equations and the phase diagram of the mean-field model in the various physically relevant cases are analyzed in section 3. In section 4 we show how the mean field equations of our generalized N JL model can be obtained in the leading order of the 1/N -expansion of a four-fermion model with non trivial color and flavor interactions, but where the action is local and free from the sign problem. Section 5 summarizes our conclusions.

The model
The most general four-fermion continuum Lagrangian in Euclidean space with SU (2) V × SU (2) A symmetry in the chiral limit and with trivial color dependence can be written as follows, (1) where ψ is a fermion field with four Dirac and two flavor components, and τ a are the Pauli matrices acting in flavor space. It is customary, in order to avoid the sign problem and/or to perform a 1/N -expansion, to add another ("color") degree of freedom to the spinors, and to straightforwardly generalize the interaction by replac-ingψBψ → N i=1ψ i Bψ i , where B is any of the matrices appearing in Eq. (1). Although the interaction is not diagonal in color space, it will become so after a Hubbard-Stratonovich transformation, and moreover it will be the same for every color: for this reason we will call it diagonal and trivial in color, with a small abuse of terminology. 1 For a review on the NJL model, see [37] and references therein.
In the following, we will refer to this straightforward generalization as the N -color model. The N JL model given by action (1) enjoys the same SU (2) V × SU (2) A symmetry of QCD and it is an effective model to describe the low energy physics of QCD [35]. This model, regularized on a hypercubic four-dimensional lattice with Wilson fermions, was analyzed in the G 2 = 0 limit and in the mean field or first order 1/N -expansion by Aoki et al. [5], who found a phase, for large values of G 1 , in which both flavor symmetry and parity are spontaneously broken, in close analogy to lattice QCD with Wilson fermions. The qualitative results of Aoki et al. were also corroborated by Bitar and Vranas in [33], where they found, using numerical simulations, the existence of this parity-flavor broken phase in the two-color model.
The lattice action of the N -color model in the Wilson regularization can be written as S = S 0 + S I , with the free part of the action being where now ψ is a fermion field with four Dirac, two flavor and N color components, and where the Dirac-Wilson operator ∆ is given by with r the Wilson parameter and m 0 the bare fermion mass. The interaction part is were we have conveniently redefined the coupling constants. As it is well known, the Wilson term breaks explicitly the full chiral symmetry, and so only parity and vector symmetries are kept in the lattice regularization. The four-fermion action can be bilinearized by performing a Hubbard-Stratonovich transformation, which implies the introduction of eight scalar and pseudoscalar auxiliary fields as follows, where the fermion matrix M is and moreover β i = 1/(4G i ). Here we are considering the case G 1 , G 2 ≥ 0.
In the G 2 = 0 case analyzed in [5,33] it is easy to see that the fermion determinant is real 2 and therefore the theory, with an even number of colors, is free from the sign problem. This allows to consistently perform the 1/N -expansion [5] and the numerical simulations in the two-color model [33]. Unfortunately, in the general case (G 1 = 0, G 2 = 0) the fermion determinant is complex, and even if the leading order of the 1/N -expansion is free from the sign problem for an even number of colors, the very consistency of this expansion is, at least, doubtful. This is the reason why we decided to study the infinite range model, or mean field approximation, where again one can easily show that the sign problem is absent for an even number of colors. However, in section 4 we will show how the gap equations of the infinite range model are just the same obtained at leading order in the 1/N -expansion of a four-fermion model with local interactions, the same symmetries, and free from the sign problem.

The phase diagram of the mean-field model
where V is the number of lattice sites. Performing again a Hubbard-Stratonovich transformation we get for the bilinearized action where the auxiliary fields are now constant fields, the fermion matrix is and again β i = 1/(4G i ), with G 1 , G 2 ≥ 0. The integral over the fermion fields can again be done analytically and in the limit of large volume V the model can be solved by writing down and solving the saddle-point equations.
Integrating out the fermionic degrees of freedom, the partition function of the mean-field model reads with V eff the effective potential per flavor and color. As we show in the appendix, the fermionic determinant is real in this case; since we are taking an even number of colors, Det M is also positive, so that there is no sign problem, and we can write In order to compute the determinant it is convenient to go over to momentum space. Starting from a finite lattice with periodic boundary conditions and then taking the limit of infinite volume, one obtains where and where B is the first Brillouin zone p µ ∈ [0, 2π], µ = 1, . . . , 4 (or equivalently p µ ∈ [−π, π] due to periodicity), and tr stands for the trace over Dirac, flavor and color indices. A straightforward calculation shows that cos p µ   +m 0 +σ+iγ 5 τ · π+iγ 5 η+ τ · ρ .
(13) The effective potential V eff can be computed explicitly, and reads with where we have set Notice that Q ≥ 0.
In the large volume limit, the partition function will be dominated by the contribution coming from the minimum of the effective potential, and so it can be computed through the saddle-point technique. In order to look for the minimum of the effective potential, it is convenient to reorder the terms in Eq. (15). A little algebra allows to rewrite it as Setting also we have Q = Q 0 − Q 1 , and the effective potential can be rewritten as (20) It is convenient for our purposes to group the various terms in two different ways. The first way is which as we will see is appropriate for the case β 1 < β 2 , and the second way is which is appropriate for the case β 1 > β 2 . The key observation is that Q 0 depends only on σ and on the combination z 2 ≡ Π 2 +η 2 +ρ 2 , so that the first line in both equations depends only on σ and z. Therefore, the minimization of the effective potential at fixed σ and z involves only the second line of Eqs. (21) and (22). To make things more transparent, let us introduce the new set of variables z, ω, ϕ, in terms of which one writes Π = z cos ω , η = z sin ω cos ϕ , The range of these variables is z ≥ 0, ω ∈ [0, π 2 ], ϕ ∈ [0, π], that corresponds to the range Π ≥ 0, ρ ≥ 0, η ∈ R of the original variables. In terms of the new variables, Eq. (21) reads (25) As we have already noted, the first line in Eqs. (24) and (25) depends only on σ and z. As a consequence, in order to minimize the effective potential with respect to ω and ϕ we have to focus on the second line only. Moreover, since Q ≥ 0, we have that Q 1 ≤ Q 0 , and so the last term in Eqs. (24) and (25) is positive or zero, Therefore, the second line in Eq. (24) is positive or zero if β 2 ≥ β 1 , and the second line in Eq. (25) is positive or zero if β 1 ≥ β 2 ; in particular, both terms are positive or zero. If we can find values of ω and ϕ such that these lower bounds are saturated, then we have automatically minimized the effective potential with respect to ω and ϕ. In order to do so, we need to make both terms vanish, and in particular we need that Q 1 vanishes identically as a function of the momentum. terms of our new variables, Q 1 reads One sees immediately that Q 1 vanishes identically if z = 0, or if sin ω = 0, i.e., ω = 0. If z = 0, ω = 0, then both terms in braces must be zero, and the second one must be so independently of p: this can happen only if sin ϕ = 0, i.e., ϕ = 0, π, which in turn requires that cos ω = 0, i.e., ω = π 2 . Summarizing, Q 1 vanishes identically only if 4 independently of the values of σ and z. Stated differently, in terms of the original variables, Q 1 vanishes identically only if It is immediate to check that in case 1 the second line of Eq. (24) vanishes, while in case 2 the second line of Eq. (25) vanishes, independently of ∆β. Let us now discuss the various cases separately.
Case β 1 < β 2 In this case, the minimum of the effective potential lies on the curve ω = 0, and is obtained by minimizing the functional with respect to σ and z. Since ω = 0 corresponds to η = ρ = 0, in this case z = Π. Clearly, the minimum will be independent of β 2 . The gap equations read therefore There are two solutions to these equations. The first one has Π = 0 and σ determined by the solution to the equation which always exists. 5 A second solution with nonzero Π is obtained by solving This solution is clearly degenerate since only the modulus of π is determined, while the direction is arbitrary on the sphere S 2 . As it has been shown in Ref. [5], a solution to Eq. (33) exists only in a closed and bounded region in the (β 1 , m 0 ) plane (see Fig. 1); on the other hand, when it exists it also minimizes the effective potential. Therefore, inside the region enclosed in the solid line in Fig. 1, we have a phase with Π = 0, so that ψ iγ 5 τ ψ = −2N β 1 π = 0, and thus parity and flavor are broken. On the other hand ψ iγ 5 ψ = −2N β 2 η = 0, so that the vacuum is of the standard Aoki type. 5 To see this it is enough to show that the right-hand side is always finite, and that it vanishes as 1/σ for large |σ|. The second point is trivial, while to prove the first one it is enough to bound the right-hand side as follows: < ∞ . Here we set r = 1.
Case β 1 > β 2 In this case, the minimum of the effective potential lies on the curves ω = π 2 , ϕ = 0, π, and is obtained by minimizing the functional with respect to σ and z. Since ω = π 2 , ϕ = 0, π, corresponds to Π = ρ = 0, in this case z = |η|. Notice that ϕ = 0 and ϕ = π give the same V > . The gap equations read therefore There are two solutions to these equations, one with |η| = 0 and σ determined by the solution to Eq. (32), which always exists, and a second solution with nonzero |η|, obtained by solving that exists only in a certain region of the parameter space. This solution is (at least) twofold degenerate, since only the absolute value of η is determined, while the sign is not. Numerical investigations show that the solution to Eq. (36), when it exists, is unique (up to the sign of η) and yields the absolute minimum of the effective potential, see Fig. 2. In particular, for any choice of ∆β = β 1 − β 2 > 0, there is a region in the (β 2 , m 0 ) plane where the solution exists and minimizes the potential, see Fig. 3. This region is symmetric with respect to m 0 = −4r, and does not extend beyond β = β c ≃ 1.43057. This region corresponds to a phase where the condensate ψ iγ 5 ψ = −2N β 2 η = 0, so that parity is broken; on the other hand, since Π = ρ = 0, flavor is not broken. The vacuum is therefore not of the standard Aoki type. Notice that η = 0 on the phase boundary.
Case β 1 = β 2 ≡ β In this case there are two degenerate sets of minima, one with ω = 0, and one with ω = π 2 , ϕ = 0, π, both obtained by minimizing the functional with respect to σ and z. The minimum at ω = 0 corresponds to η = ρ = 0 and z = Π, while the minima at ω = π 2 , ϕ = 0, π correspond to Π = ρ = 0 and z = |η|. The gap equations read therefore There are two solutions to these equations, one with z = 0 and σ determined by solving Eq. (32), which can always be done, and a second set of solutions with nonzero z, obtained by solving which is possible in a certain region of the (β, m 0 ) plane. Setting η = ρ = 0 and z = Π, there is a S 2 degeneracy, while setting Π = ρ = 0 and z = |η| there is a twofold degeneracy. Clearly, since this equation is identical to Eq. (33), up to the substitutions Π → z and β 1 → β, a region exists in the (β, m 0 ) plane where the absolute minimum lies at nonzero z, which coincides with the one found by Aoki and collaborators, see Fig. 1. In this case, however, there are two degenerate (sets of) vacua with broken parity, one with broken and one with unbroken flavor symmetry, that are not connected by a symmetry transformation. In this case we have the coexistence of standard Aoki and non-standard Aoki vacua.

A model with local interactions
In this section we describe a N -color model with local interactions, free from the sign problem, which in the limit of large N has the same phase diagram of the mean-field theory, discussed in the previous section. The action of the model is S (loc) = S 0 + S (loc) I , with the free part of the action being given in Eqs. (2) and (3), and the interaction part being where we are taking an even number of colors N , and where the N ×N diagonal matrix λ acting in color space is −1, 1, −1, . . . , 1, −1) .
After a Hubbard-Stratonovich transformation, the partition function takes the form where now the fermion matrix is It is easy to show that there is no sign problem in this case. Indeed, in color space the fermion matrix has the form and one easily sees that M ′ xy = C † M * xy C, with C = γ 1 γ 3 τ 2 . Therefore, which is clearly real and positive. We can therefore write for the effective action where Tr is here the trace over spacetime, Dirac and flavor indices.
In the limit of large N , we can parameterize the auxiliary fields as φ i x = φ i + δφ i x √ N , with space-independent values φ i , to be determined by the saddle-point equations δV eff δφx = 0, where we have denoted collectively {φ i x } = (σ x , π x , η x , ρ x ). Expanding the effective action with respect to δφ i x we find Going over to momentum space, and taking the limit of large volume, we have where we have made explicit the dependence on N . The effective potential at the saddle point reads with Q given in Eq. (15), i.e., the same result Eq. (14) obtained in the mean-field case. Obviously, the phase diagram at large N of the local model discussed here is therefore exactly the same found for the infinite-range model. The first correction is expressed through the matrix where i, j = σ, π, η, ρ, and moreover n σ = n π = 1, n η = n ρ = 2, the matrices M i are and the matrixM (p) is given in Eq. (13). The function g(p, z) is given by with z = 0 in the unbroken phase, z = | π| in the standard Aoki phase, and z = η in the non standard Aoki phase. The nonzero elements of the inverse propagator G −1 ij (q) in the various phases are given in the appendix. It is interesting to check the values in the broken phases of the masses of the excitations corresponding to π and η, defined through the small-q behavior of the corresponding diagonal terms of the inverse propagator. 6 Choosing for convenience π a = Πδ a3 , we set where q 2 = 4 µ=1 q 2 µ . Using the results reported in the appendix, we find that 7 in the standard Aoki phase (Π = 0, η = 0), and in the non standard Aoki phase (Π = 0, η = 0). The quantities I 0,2 (q, σ, z) = I 0,2 (0, σ, z) − q 2 I 0,2 (σ, z) + O(q 4 ) are defined in the appendix. Let us make a few remarks.
• In the "physical" phase, where parity and flavor symmetries are realized in the vacuum, the three pions and the η are degenerate. The pion degeneracy follows from flavor symmetry, but the fact that the η is degenerate with the three pions is probably an artifact of the large N -mean field approximation. The three pions and the η on the other hand become massless on the phase boundary.
• In the standard Aoki phase (G 1 > G 2 ) the "charged" pions are massless and the neutral pion and the η are massive and degenerate, except at the phase boundary where all of them become massless.
• In the non standard Aoki phase (G 1 < G 2 ) the three pions and the η are massive and degenerate, and all of them become massless on the phase boundary.
• In the special case G 1 = G 2 , the η and the neutral pion are massive and degenerate in any of the two vacua, and again become massless at the phase boundary.

Conclusions
With the aim of establishing the range of applicability of the standard QCD chiral effective Lagrangian approach, we have analyzed in this paper the vacuum structure of a generalized Nambu-Jona-Lasinio model (N JL) with Wilson fermions in the mean field or leading order 1/N -expansion. The model has been chosen to possess the more general SU (2) V × SU (2) A symmetry in the continuum. The election of the N JL model for our analysis was motivated by the fact that four-dimensional models without gauge fields, and with four fermion interactions, are considered as effective models to describe the low energy physics of QCD.
This generalized N JL model shows a rich phase structure in the three-parameter space (two four fermion couplings plus the bare fermion mass). Indeed, in addition to the standard "physical" phases in which both, parity and flavor symmetries are realized in the vacuum, new phases where flavor and/or parity are spontaneously broken are found. These phases are, for G 1 > G 2 , of the standard Aoki type with iψγ 5 τ 3 ψ = 0, and iψγ 5 ψ = 0, in agreement also with predictions of the chiral effective Lagrangian approach. However when G 1 < G 2 the new phase is characterized by iψγ 5 τ 3 ψ = 0 and iψγ 5 ψ = 0. In the special case G 1 = G 2 , where the continuum N JL action has, in the chiral limit, a U (2) V × U (2) A symmetry, as QCD, two degenerate vacua, not connected by parity-flavor symmetry transformations, with broken parity coexist, one with broken and another one with unbroken flavour symmetry.
These results show how the phase structure of an effective model for low energy QCD cannot be entirely understood from Wilson Chiral Perturbation Theory, based on the standard QCD chiral effective Lagrangian approach. Nonetheless, and as explained in [21,[23][24][25]34], our conclusions leave no practical consequences for lattice practitioners: Wilson Chiral Perturbation Theory is still applicable outside the Aoki phase, and the properties of the "physical" phase of lattice QCD with Wilson fermions remain unchanged.
We have also shown in section 4 that the mean field equations of our generalized N JL model can be obtained in the leading order of the 1/N -expansion of a four-fermion model with non trivial color and flavor interactions, and the same N JL-symmetries, but where the action is local and free from the sign problem, thus allowing to compute corrections to the 1/N -expansion and to perform numerical simulations at finite N in order to test the stability of the mean-field results reported here.
Finally, since M is diagonal and trivial in color, M =M 1 C , one has that for an even number of colors det M = (detM ) N = (detM detM † ) N 2 ≥ 0.
(59) The non-vanishing entries of the inverse propagator G −1 ij (q), Eq. (50), are listed below for the standard-Aoki and non standard-Aoki cases. In the unbroken phase the inverse propagator is diagonal, and can be recovered by simply setting Π = 0 or η = 0 in the equations below. (62)