Properties of magnetized neutral pions at zero and finite temperature in nonlocal chiral quark models

The behavior of $\pi^0$ meson properties in the presence of a uniform external magnetic field is studied in the context of a nonlocal extension of the Polyakov-Nambu-Jona-Lasinio model. The analysis includes the $\pi^0$ mass, the effective $\pi^0$-quark coupling and the pion-to-vacuum hadronic form factors, both at zero and finite temperature. Numerical results are compared with previous calculations carried out within the local NJL model, when available. The validity of chiral relations and the features of deconfinement and chiral symmetry restoration transitions are discussed.


I. INTRODUCTION
The behavior of strongly interacting matter under the influence of intense magnetic fields has become an issue of increasing interest in the last few years [1][2][3]. This is mostly motivated by the realization that the presence of strong magnetic fields should be taken into account in the analysis of some relevant physical systems, e.g. in the description of high energy noncentral heavy ion collisions [4], or the study of magnetars [5]. From the theoretical point of view, addressing this subject requires to deal with quantum chromodynamics (QCD) in nonperturbative regimes. Therefore, present analyses are based either in the predictions of effective models or in the results obtained through lattice QCD (LQCD) calculations. In this work we focus on the effect of an intense external magnetic field on various π 0 meson properties at zero and finite temperature. This issue has been studied in the last years following various theoretical approaches for low-energy QCD, such as Nambu-Jona-Lasinio (NJL)-like models [6][7][8][9][10][11][12], chiral perturbation theory (ChPT) [13,14] and path integral Hamiltonians (PIH) [15,16]. In addition, results for the light meson spectrum under background magnetic fields at zero temperature have been obtained from LQCD calculations [17,18].
In Ref. [19] we have studied the behavior of the π 0 meson mass and one of its axial decay form factors in the presence of a uniform static magnetic field at zero temperature, within a relativistic chiral quark model in which quarks interact through a nonlocal fourfermion coupling [20]. This so-called "nonlocal NJL (nlNJL) model" can be viewed as a sort of extension of the NJL model that intends to provide a more realistic effective approach to QCD. Actually, nonlocality arises naturally in the context of successful descriptions of low-energy quark dynamics [21,22], and it has been shown [23] that nonlocal models can lead to a momentum dependence in quark propagators that is consistent with LQCD results.
Moreover, in this framework it is possible to obtain an adequate description of the properties of light mesons in the absence of an external electromagnetic field at both zero and finite temperature [23][24][25][26][27][28][29][30][31][32][33][34]. Interestingly, as shown in Refs. [35,36], nlNJL models naturally allow to reproduce the so-called inverse magnetic catalysis (IMC) effect, previously observed from LQCD results. According to these calculations, the chiral restoration critical temperature turns out to be a decreasing function of the magnetic field B. In fact, the observation of IMC in LQCD calculations [37,38] represents a challenge from the point of view of theoretical models, since most naive effective approaches to low energy QCD (NJL model, ChPT, MIT bag model, quark-meson models) predict that the chiral transition temperature should grow when the magnetic field is increased [1][2][3]. In addition, nlNJL models lead to a B dependence of the π 0 mass that is found to be in good agreement with LQCD results [19].
The aim of the present article is to extend the work in Ref. [19], considering some additional properties of the magnetized π 0 mesons. As shown in Ref. [39], in the presence of a constant magnetic field B the pion-to-vacuum vector and axial vector amplitudes can be in general parametrized in terms of three "decay" form factors. One of them, f (A||) π 0 , corresponds to the pion decay constant usually denoted by f π . The behavior of this form factor under the magnetic field has already been analyzed in Ref. [19], together with those of the masses m π 0 and m σ , and the quark-meson coupling g π 0 qq . The other two decay form factors are a second axial decay constant, f (A⊥) π 0 , associated to momentum components that are perpendicular to the magnetic field, and a vector decay constant f (V ) π 0 . As shown in Ref. [6], another relevant feature induced by the presence of the external magnetic field is the fact that the π 0 dispersion relation turns out to be anisotropic, implying that the movement along the direction perpendicular to the magnetic field is characterized by a diffraction index u π 0 which is in general different from one. In this way, to complement the analysis carried out in Ref. [19], in this work we study the magnetic field dependence of f and u π 0 . In addition, we extend the analysis to a system at finite temperature T , considering the thermal behavior of these quantities and also of the masses m π 0 and m σ , the coupling g π 0 qq and the decay constant f (A||) π 0 , which have been studied in Ref. [19] only for T = 0. This article is organized as follows. In Sec. II we show how to obtain the analytical equations required to determine the relevant π 0 properties at zero temperature in the presence of the magnetic field. Our calculations are based on the formalism developed in Refs. [19,35,36], which make use of Ritus eigenfunctions [40]. In Sec. III we show how to extend the analysis in Sec. II to a system at finite temperature, taking also into account the coupling of fermions to a background color field (the so-called "Polyakov loop nlNJL model"). In Sect. IV we quote and discuss our numerical results, while in Sec. V we present our conclusions. Finally, in Appendices A and B we outline the derivation of some of the expressions quoted in the main text.

II. THEORETICAL FORMALISM
The Euclidean action for the nonlocal NJL-like two-flavor quark model we are considering reads Here m c is the current quark mass, equal for u and d quarks, while the currents j a (x) are given by where Γ a = (1 1, iγ 5 τ ). The function G(z) is a nonlocal form factor that characterizes the effective interaction. The action can be "gauged" to incorporate couplings to electromagnetic, vector and axial vector gauge fields A µ , W V,a µ (x) and W A,a µ (x), respectively. This is done by replacing whereQ = diag(q u , q d ), with q u = 2e/3, q d = −e/3, C = V, A, a = 1, 2, 3, Γ V = 1 and Γ A = γ 5 . For this nonlocal model, gauge symmetry also requires the replacements [23,29,34] where ℓ runs over an arbitrary path connecting x with y. As it is usually done, we take it to be a straight line path.
As stated, we assume the presence of an external uniform magnetic field B. Therefore, using the Landau gauge, and choosing the x 3 axis in the direction of B, we take A µ to be a static field given by A µ (x) = Bx 1 δ µ2 .
Since we are interested in studying light meson properties, we carry out a bosonization of the fermionic theory, introducing scalar and pseudoscalar fields σ(x) and π(x) and integrating out the fermion fields. The bosonized action can be written as [19,23,34] S bos = − log det D + 1 2G where withx = (x + x ′ )/2. We expand now the meson fields around their mean field values.
Since the external magnetic field is uniform, one can assume that the field σ(x) has a nontrivial translational invariant mean field valueσ, while the vacuum expectation values of pseudoscalar fields are zero. We separate the mean field piece of the first term of the action in Eq. (6), writing where the traces run over color, flavor, Dirac and coordinate spaces. The form of the mean field operator D 0 in the presence of the external magnetic field has been studied in detail in previous works, see e.g. Ref. [36]. It can be written as where is the so-called Schwinger phase, and a direct product to an identity matrix in color space is understood. The mean field quark propagators −1 can be obtained following the Ritus eigenfunction method [40].
As shown in Ref. [36] (see also the analysis carried out within the Schwinger-Dyson formalism in Refs. [41,42]), it is possible to write the propagators in terms of the Schwinger phase and a translational invariant function, namely where p ⊥ = (p 1 , p 2 ) and p = (p 3 , p 4 ). The expression ofS f (p ⊥ , p ) in the nlNJL model under consideration is found to be [36] S f (p ⊥ , p ) = 2 exp(−p 2 where the following definitions have been used. The perpendicular and parallel gamma matrices are collected in vectors γ ⊥ = (γ 1 , γ 2 ) and γ = (γ 3 , γ 4 ), while the matrices ∆ λ are defined as ∆ + = diag(1, 0, 1, 0) and ∆ − = diag(0, 1, 0, 1). The integers k λ are given by The functionsX ±,f k,p , with X = A, B, C, D, are defined asÂ where The functions M λ,f k,p play the role of effective (momentum-dependent) dynamical quark masses in the presence of the magnetic field. They are given by where M(p 2 ) = m c +σ g(p 2 ) , g(p 2 ) being the Fourier transform of the nonlocal form factor G(x). In Eqs. (12) and (18), L k (x) and L 1 k (x) stand for generalized Laguerre polynomials, with the convention L −1 (x) = L 1 −1 (x) = 0. The relation in Eq. (18) can be understood as a Laguerre-Fourier transform of the function M(p 2 ). It is also convenient to introduce the Laguerre-Fourier transform of the form factor g(p 2 ), thus one has The transform in Eq. (20) can be inverted to get To study the mass and decay form factors of the neutral pion, we expand the operator δD(x, x ′ ) in powers of the meson fluctuations and the external vector and axial vector fields, keeping up to linear terms in δπ 3 , W V,3 µ and W A,3 µ . We obtain where Here we have used the definitionsx = ( Given a definite model parametrization, the value ofσ can be found by minimization of the effective action at the mean field level. The corresponding "gap equation" reads [35,36] A. Pion field redefinition and quark-meson coupling constants The calculation of the π 0 mass in this model has been previously carried out in Ref. [19].
As shown in that paper, the piece of the bosonized action that is quadratic in the neutral pion fields can be written as where for integration in two-component momentum spaces we use the notation p q r...
Choosing the frame in which the π 0 meson is at rest, its mass can be obtained as the solution of the equation To normalize the pion field we can expand the action in Eq. (30) around the pion pole (t ⊥ = 0, t 2 = −m 2 π 0 ) up to first order in momentum squared. We define and renormalize the pion field according to π 3 (q) = g π 0 qqπ3 (q), where g π 0 qq = Z 1/2 is the meson-quark effective coupling constant. Thus, one has where From the above expressions of the quark propagators and δD π , after some straightforward calculation we find where the trace is taken over Dirac space. We have defined q ± = q ±t /2, while the function φ in the exponential is given by As stated in Ref. [19], the trace in Eq. (36) is given by with For simplicity we use the notation B f = |q f B|.
To work out the integrals in Eq. (36) it is convenient to use the Laguerre-Fourier transforms introduced above. In this way, the integrals over perpendicular momenta can be performed analytically. As found in Ref. [19], the expression for The normalization constant Z −1 can be obtained by derivation on the r.h.s. with respect to To obtain an expression for Z −1 ⊥ one has to expand F (t 2 ⊥ , t 2 ) up to first order in t 2 ⊥ . The calculation of the corresponding integrals over perpendicular momenta is sketched in Appendix A. One finally gets B. π 0 decay form factors The π 0 -to-vacuum amplitudes for vector and axial vector quark currents are given by As discussed in Ref. [39], in the presence of an external magnetic field these currents can be written in terms of three form factors. Following the notation in Ref. [43], in Euclidean If we write the corresponding piece of the bosonic action as it is easily seen that The functions F C µ (t) can be separated into three pieces F C,(i) µ (t) with i = I, II, III, coming from the various contributions to the effective action, namely The explicit calculation of F C,(i) µ (t) leads to and As in the case of the calculation of the π 0 mass and wave function renormalization, the integrals over transverse momenta can be performed analytically after Laguerre-Fourier transforming the nonlocal form factor functions. The steps to be followed in each case are outlined In what follows we just quote the results of this rather lengthy calculation.
The form factors are evaluated at the pion pole, i.e.
has been previously performed in Ref. [19], where the contribu- (t) are quoted. Summing all three contributions one has [19] t · F A (t) Taking account this result, and making use of Eq. (32) and the gap equation (29), one arrives at [19] f In the case of f (t) vanish identically, and the contribution from Finally, for f (A ) π 0 the calculations sketched in Appendix B lead to where g ′ λ,f k,q , g ′′ λ,f k,q indicate derivations with respect to q 2 , and we have defined Summing these three contributions, and using the relation (which arises from the properties of Laguerre polynomials), we arrive to a final expression , given by (64)

C. Chiral relations
It is interesting to study the relations involving form factors and renormalization constants in the chiral limit, m c → 0. Firstly, taking into account the expression in Eq. (40), the gap equation (29) and the relation it is seen that Thus, in the limit m c → 0, the second term on the r.h.s. vanishes and from Eq. (32) one obtains m π 0 = 0, as expected.
The validity of the Goldberger-Treiman relation in the presence of the external magnetic field have been shown in Ref. [19]. In these equations, subindices 0 indicate that the quantities have to be evaluated in the chiral limit. Now let us take into account the expression for f (A⊥) π 0 in Eq. (64). For m c = 0, m π 0 = 0, it is seen that the first term into curly brackets is zero owing to Eq. (65), while the last two terms also vanish since F λ,f (XY ) kk ′ ,q q is symmetric under the exchange between k and k ′ . Moreover, it is easy to see that from which the piece proportional to H λ,f kk,q q also vanishes. One gets in this way On the other hand, from Eqs. (41) and (65) it is seen that in the chiral limit one has After some algebra, it can be shown that the factor in square brackets is equal to Thus, taking into account Eq. (67), one has This result has been also found in the framework of the local NJL model in Ref. [43] and (using a different notation) in Ref. [6], where it is obtained from a modified PCAC relation.

III. FINITE TEMPERATURE
In this section we extend the previous analysis to a system at finite temperature using the standard Matsubara formalism. To describe the confinement/deconfinement transitions we include a coupling between the fermions and the Polyakov loop (PL), assuming that the quarks move on a uniform background color field. This type of interactions have been previously considered in nonlocal models [30][31][32][33]36], as well as in the local Polyakov-Nambu-Jona-Lasinio (PNJL) model [44][45][46][47] and in Polyakov-quark-meson models [48,49]. The background field is given by φ = igδ µ0 G µ a λ a /2, where G µ a are the SU(3) color gauge fields. Working in the so-called Polyakov gauge the matrix φ is given a diagonal representation φ = φ 3 λ 3 + φ 8 λ 8 , and the traced Polyakov loop Φ = 1 3 Tr exp(iφ/T ) can be taken as an order parameter of the confinement/deconfinement transitions. Since at mean field Φ is expected to be real owing to charge conjugation symmetry, one has φ 8 = 0 and Φ = [1 + 2 cos(φ 3 /T )]/3 [47]. In addition, we include a Polyakov-loop potential U(Φ, T ) that accounts for effective gauge field self-interactions. The mean field grand canonical thermodynamic potential of the system per unit volume under the external magnetic field is given by [36] where ∆ f k,p nc is the function in Eq. (17), and we have defined p nc = (p 3 , p 4 nc ), with p 4 nc = (2n + 1)πT + φ c . The sum over color indices runs over c = r, g, b, and color background Since Ω MFA B,T is divergent, it has to be properly regularized. We take the prescription followed in Ref. [36], in which one subtracts to Ω MFA B,T the thermodynamic potential of a free fermion gas, and then adds it in a regularized form. The regularized potential is given by In fact, the "free" piece keeps the interaction with the magnetic field and the PL. The explicit expression of Ω free,reg B,T , for which the Matsubara sums can be performed analytically, can be found in Ref. [36].
The form of the PL potential is an additional input of the model. In this work we take a widely used polynomial form based on a Ginzburg-Landau ansatz, namely [46,50] U(Φ, T ) where The numerical values for the parameters a i and b i , which can be obtained by a fit to pure gauge lattice QCD results, can be found in Ref. [46]. This potential leads to a first-order phase transition at T 0 , which becomes a further parameter of the model. In the absence of dynamical quarks, form lattice QCD one would expect a deconfimenent temperature of about T 0 = 270 MeV. However, it has been argued that in the presence of light dynamical quarks this parameter should be reduced. For definiteness we will take T 0 ≃ 200 MeV, as suggested in Refs. [48,49].
The values ofσ and φ 3 at the mean field level can be found by minimizing the regularized thermodynamic potential, i.e. by solving the coupled equations Finite temperature meson masses and decay constants can be then calculated from Eqs. (32), (40), (57), (58) and (64), following the prescription and taking the external momentum t = (i m π 0 (T ), 0). Notice that these mass values correspond to spatial "screening masses" for the zeroth bosonic Matsubara mode (t 4 = 0).
The reciprocals m π 0 (T ) −1 can be understood as the persistence lengths of this mode, in equilibrium with the heat bath.

IV. NUMERICAL RESULTS
To obtain numerical predictions for the behavior of the quantities defined in the previous sections, it is necessary to specify the model parameters and the shape of the nonlocal form factor g(p 2 ). We consider here the often-used Gaussian function In general, the form factor introduces an energy scale Λ that represents an effective momentum cut-off. This constant has to be taken as a free parameter of the model, together with the current quark mass m c and the coupling constant G in the effective Lagrangian. In the particular case of the Gaussian form factor one has the advantage that the integral in Eq. (20) can be performed analytically, which leads to a dramatic reduction of the computer time needed for numerical calculations of meson masses and form factors.
As in Refs. [35,36], we fix the free parameters by requiring the model to reproduce the empirical values (for vanishing external field) of the pion mass and decay constant, as well as some phenomenologically adequate values of the quark condensate f f (f = u, d).
Some parameter sets, corresponding to different values of the condensate, can be found in Ref. [36]. Here we take m c = 6.5 MeV, Λ = 678 MeV and GΛ 2 = 23.66, which lead to f f = (−230 MeV) 3 . This will be called Set I. As shown in Ref. [36], for this parametrization the behavior of quark condensates with the magnetic field, at zero temperature, are found to be in very good agreement with lattice QCD results. These parameters have also been used in the previous calculation of m π 0 and f (A ) π 0 for nonzero B carried out in Ref. [19]. In order to test the sensitivity of our results to the parameters we also consider two alternative sets, which correspond to quark condensates f f = (−220 MeV) 3 and f f = (−240 MeV) 3 in vacuum. The latter are denoted as Sets II and III, respectively.
In Fig. 1  form factor have been previously obtained in Ref. [19].
From the graphs in Fig. 1 it can be said that in general our results do not show a large dependence with the model parametrization. As shown in the upper left panel in Fig. 1, the dependence of the π 0 mass with the external field for the nonlocal effective model is π 0 as a function of eB. This form factor is zero at vanishing external field and shows a monotonic growth with eB, with little dependence on the parametrization.
In this case the growth is shown to be somewhat steeper for the local NJL model. Finally, in the lower right panel we quote the curves for directional refraction index u π 0 , which is found to get reduced for increasing external field. The results for the above discussed quantities, together with those obtained for qq (for the analysis of condensates and related quantities, see Refs. [35,36] with the magnetic field can be understood from Eq. (73), taking into account that u π 0 becomes significantly reduced for increasing B. In the NJL, it has also been shown that the relation π 0 ) holds in the chiral limit [43]. We have checked this relation numerically in the nlNJL model, finding that it remains only approximately valid (that is, within a 15% accuracy) for the chosen range of eB.
We turn now to our results for a system al finite temperature. As expected, at some critical temperature T c (B) the system undergoes a crossover transition in which chiral symmetry is partially restored. Moreover, as shown in Refs. [35,36], this model leads to inverse magnetic catalysis, in the sense that T c is found to be a decreasing function of B. This is in agreement with lattice QCD results [38]. It has been also shown that there is a very small splitting between chiral restoration and deconfinement transition temperatures, the latter being defined according to the behavior of the Polyakov loop Φ (see e.g. Ref. [36] for details).
Regarding the quantities we are interested in here, in Fig. 2  π 0 , and occurs at a slightly lower temperature. We recall that, at any temperature, f (V ) π 0 is zero for vanishing external field.   Fig. 3, the behavior of the directional refraction index u π 0 is found to be basically independent of the temperature.

V. SUMMARY AND CONCLUSIONS
We have studied the behavior of neutral meson properties in the presence of a uniform static external magnetic field B, in the context of the so-called nlNJL model. That is, a nonlocal effective approach based on the Nambu-Jona-Lasinio chiral quark model. Our analysis is a sort of extension of the work carried out in Ref. [19], where the pion mass and the decay form factor f (A ) π 0 , at zero temperature, were studied in this same framework. In the nlNJL model the effective couplings between quark-antiquark currents include nonlocal form factors that regularize ultraviolet divergences in quark loop integrals, and lead to a momentum-dependent effective mass in quark propagators. In order to obtain closed analytical expressions for meson polarization functions and pion decay constants in the presence of the external magnetic field, we have worked out a formalism that involves Ritus transforms of the Dirac fields.
We have firstly concentrated in the analysis at zero temperature of the form factors associated with pion-to-vacuum matrix elements of the vector and axial vector hadronic currents. In agreement with the model independent analysis in Ref. [39], it is seen that for nonzero B three nonvanishing independent form factors can be defined. Two of them, f π 0 , to the vector piece. We have also calculated the neutral pion directional refraction index, u π 0 , which in general is different from one. In addition, chiral relations are shown to be valid in the presence of the external field.
For the numerical calculations we have considered the case of Gaussian nonlocal form factors, choosing sets of model parameters that were previously found to reproduce the empirical values of the pion mass and decay constant at B = 0, and lead to values of quark condensates at nonzero B that are compatible with lattice QCD calculations. Taking into account external fields in the range from zero to eB = 1 GeV 2 , from our numerical results it is noticed that all studied quantities show a strong dependence with B. As discussed in Sec. IV, in most cases this dependence is found to be significantly larger than that observed in the local NJL model [43]. On the other hand, it is seen that in general the results are rather stable under changes in the model parameters. It has been also verified that chiral relations remain approximately valid for the chosen parameter sets.
We have extended the calculations to finite temperature systems, including the couplings of fermions to the Polyakov loop and a PL polynomial potential that accounts for effective color gauge field self-interactions. As expected, it is seen that the system undergoes a phase transition corresponding to the restoration of SU(2) chiral symmetry. The model predicts the existence of inverse magnetic catalysis, leading to a decrease of the critical temperature T c with the magnetic field. Concerning the behavior of meson masses, it is seen that the π 0 mass remains approximately constant up to T c (B), while the σ meson mass begins to drop earlier. Beyond the transition both masses become degenerate, as expected, and show a thermal growth for large T . Regarding the thermal behavior of the form factors, we observe that they remain approximately constant until temperatures close to T c (B) are reached, and then they show sudden drops. Finally, the directional refraction index u π 0 is found to be basically independent of the temperature. To provide an alternative view, we have also included some graphs showing the behavior of the studied quantities as functions of the magnetic field, for some selected values of the temperature. dw 2 e −w 2 F m (x, w) ± F m−2 (x, w) F n (y, w) ± F n−2 (y, w) = δ mn ± δ mn−2 F m (x, y) + δ mn ± δ mn+2 F m−2 (x, y) , In the case of calculations of t ⊥ · F A,(II) ⊥ (t), and f (V ) π 0 , some relations between the Bessel functions J ν (x) and the Laguerre polynomials are also required (see appendices in Ref. [19]).
In this way, after a lengthy calculation, one arrives at the expression for f (A⊥) π 0 in Eq. (64).