Light pseudo-scalar meson masses under strong magnetic fields within the SU(3) Nambu-Jona-Lasinio model

We calculate the pole masses of pseudoscalar mesons in a strongly magnetized medium within the framework of the SU(3) Nambu-Jona--Lasinio model, using a magnetic field-independent regularization scheme. We employ both a constant and a magnetic field-dependent coupling $G(B)$, the latter being fitted to reproduce lattice QCD results for the pseudocritical chiral transition temperature. Numerical results for the pole masses are obtained for definite parametrizations of the model. For neutral mesons, the use of $G(B)$ provides closer agreement with lattice QCD results, which reveal a decrease of the mass with the external field. On the contrary, charged mesons masses are enhanced by $B$, showing no sign of the non-monotonous behavior found in recent lattice QCD simulations.


I. INTRODUCTION
The behavior of strongly interacting matter under the influence of intense magnetic fields has been attracting a lot of interest in recent years. This interest is partly motivated by the fact that strong magnetic fields have been achieved or hypothesized in several physical situations. For example, in non-central relativistic heavy-ion collisions, magnetic fields of magnitude as large as B ∼ 10 19 − 10 20 G are generated due to the motion of charged spectator particles, essentially at the earliest times of the collision [1]. In addition, strong magnetic fields may also play an important role in astrophysics scenarios, such as matter formation in the early Universe [2] or in the dynamics of magnetars, where the inner core can possibly harbor magnetic fields strengths as large as B ∼ 10 19 G [3]. One expects that new and detectable effects in the phase diagram and properties of strongly interacting matter will emerge due to these extreme magnetic fields, causing numerous phenomenological consequences. For example, the Chiral Magnetic Effect [4][5][6], Chiral Separation Effect [7], Chiral Magnetic Wave [8][9][10][11][12] and related phenomena are supposed to be experimental signals of the influence of strong magnetic fields in the QCD matter [13,14].
From the theoretical point of view, first principle analytical QCD calculations are very difficult to perform in the non-perturbative regime given the complexity of the theory. Therefore, one has to make use of alternative procedures to tackle the problem. In this regard, great progress has been made in recent years on the investigation of the QCD phase diagram by using either lattice QCD (LQCD) simulations or effective models, which can work together in a complementary manner in the face of lack of experimental evidence in some observables. In fact, in many of these models available experimental or LQCD results are used to fix their phenomenological parameters, allowing for improved results. In particular, results from LQCD calculations at zero temperature and physical pion masses show that an external constant magnetic field enforces the quark condensate favoring the breakdown of chiral symmetry, an effect known as magnetic catalysis (MC) [15]. This result is in agreement with most effective model calculations [16,17]. At finite temperature, the majority of these models predict the increase of the pseudo-critical transition temperature T pc with the magnetic field [16,17]. Nevertheless, accurate LCQD results [18][19][20][21] have shown the opposite pattern; T pc decreases with the magnetic field, a phenomenon dubbed as inverse magnetic catalysis (IMC). The explanation for IMC at finite temperature is still controversial and under study [see Ref. [22] for a mini-review on the IMC effect]. It is not clear that the nonmonotic behavior of the quark condensate is in fact the mechanism behind the IMC effect [23][24][25]. In the context of effective models, several possibilities have been explored in the recent literature to incorporate the IMC effect phenomenologically. Within the Nambu-Jona-Lasinio (NJL) model for example, which we will use in this work, these improvements include going beyond mean-field calculations [26] or taking into consideration the anomalous magnetic moment of quarks [27][28][29][30].
Motivated by the running of the QCD coupling, one of the simplest modifications available consists of introducing a coupling constant that depends on the magnetic field (and in some cases also on the temperature) and can be fixed by fitting some LQCD results, such as the quark condensate or the chiral pseudocritical temperature. This strategy has shown that the NJL model can satisfactorily reproduce LQCD results in a broad range of temperature and magnetic fields [31][32][33][34][35][36]. In this regard, an interesting possibility was recently proposed in Ref. [37]. There, the magnetic field dependent four fermion coupling is fitted to reproduce constituent quark masses, which are obtained from the LQCD calculation of baryon masses by assuming in a simplified way that the baryon mass can be obtained by merely summing the masses of their constituents. Lastly, calculations using the non-local NJL model have shown that IMC is obtained naturally [38,39].
The presence of strong magnetic fields also has a meaningful impact on hadron properties. In this work we will focus on its consequences over the masses of the light pseudoscalar meson nonet, which has drawn a lot of attention in recent years. Most calculations in the literature have been performed for the lightest mesons. For pions at zero temperature, LQCD simulations show an overall decrease of the neutral pion mass with the magnetic field (both in quenched QCD and using staggered fermions), while charged pions exhibit the opposite behavior within the quenched approximation [40,41]. A recent simulation using highly improved staggered fermions with a slightly heavier-than-physical pion mass of 220 MeV was performed in Ref. [42], where the masses of many pseudoscalar mesons are computed. There, the decreasing trend of the neutral pion (and kaon) mass is confirmed. Moreover, charged pions (and kaons) reveal an initial increase with the magnetic field up to values eB ∼ 0.6 GeV 2 , in accordance with previous results from Ref. [18] where eB < 0.5 GeV 2 values are considered for the charged pion mass using stout smeared staggered fermions. However, in stark contrast with previous quenched results from Refs. [40,41], for stronger magnetic fields this increasing tendency is found to be reversed, resulting in a nonmonotonous behavior.
In order to study the behavior of the masses of the light pseudoscalar meson nonet in the presence of an external constant magnetic field, we will use the SU(3) NJL model. We remind that the NJL model is a non-renormalizable model and a regularization procedure has to be adopted, which may be considered as part of the definition of the effective model. In fact, the choice of an appropriate regularization scheme is a crucial issue for the description of physical systems. It has been shown that the use of an inappropriate regularization scheme causes strong oscillations in meson masses and tachyonic or discontinuous behavior of masses. When working with quark matter immersed in a magnetized medium, performing an exact separation of magnetic from non-magnetic contributions for all physical observables is a key point for their correct description, a strategy known as the "magnetic field independent regularization" (MFIR) scheme. The importance of the regularization procedure has been reviewed in Ref. [70], where it is shown that the MFIR scheme is free of these unphysical behaviors, which are due to an improper regularization. An improvement within the MFIR scheme was recently suggested in Ref. [36] for the calculation of many mean-field observables.
However, this modification is not relevant for the quantities we will study in this work, namely quark condensates and meson masses, so we can safely omit it.
Regarding the determination of meson masses within the NJL model, one important point is the proper calculation of charged mesons. In this case, polarization functions have to be carefully handled in order to be diagonalized, since Schwinger phases arising from quarks propagators do not cancel, leading to a breakdown of translational invariance. As shown in Ref. [58], an appropriate treatment involves the use of the Ritus basis [71]. Unfortunately, this issue has not been properly addressed in several calculations. Of course, for neutral mesons the usual momentum basis can be used since the Schwinger phase factor cancels out in that case. In this regard, the NJL model shows an enhancement of the neutral pion mass for sufficiently strong magnetic fields, in contradiction with lattice results. One possible approach to overcome this issue, which we will adopt in this work, is to introduce a magnetic field dependent coupling constant, determined by fitting LQCD results for the quark condensate, which, as mentioned earlier, also allows the model to incorporate the IMC effect at finite temperatures. Results obtained following this strategy agree very well with LQCD simulations [34,57]. An alternative approach comprises the use of the non-local version of the SU(2) NJL model, where the neutral pions mass has been shown to naturally decrease with the magnetic field in agreement with LQCD results [65,66].
As mentioned earlier, there are very few calculations of meson properties incorporating the strange quark. The aim of the present paper is to study the behavior of the pseudoscalar meson nonet masses as functions of the magnetic field. To that end we use the SU(3) NJL model including the 't Hooft-Maekawa interaction which breaks the U A (1) symmetry. We work within the MFIR scheme and consider both the case of a fixed and of a B-dependent four-fermion coupling constant. At the mean-field level we calculate quark condensates, which are compared with LCQD results to find qualitative agreement. For the calculation of the light pseudoscalar meson nonet we adopt the RPA formulation, where special care has to be taken to the fact that constituent quark masses are different for each flavor. For charged mesons the Ritus basis is used to diagonalize the polarizers, resulting in monotonically increasing masses for both constant and magnetic couplings. For neutral mesons, the polarizers calculation is simplified since Schwinger phases cancels out. Note that the 't Hooft-Maekawa interaction together with the uniform magnetic field B induce a mix of neutral mesons states with equal flavors, i.e. π 0 , η, η ′ [72]. We see that, except for η ′ , neutral pseudoscalar mesons display a non-monotonous behavior when using a constant coupling, which shifts to a monotonous decrease in concordance with LCQD results when a magnetic coupling is introduced.
As already known from the usual SU(3) NJL model at B = 0, the η ′ meson comes out in the model as a resonance or unstable particle. In this case, the propagator becomes a complex number and from the analysis of the complex pole, the mass of the resonance is obtained. This situation gets intricate when the magnetic field is present, thus, we have developed a new formalism to treat this case.
We organize this work as follows. In Sec. II we introduce the theoretical formalism used to obtain neutral and charged pseudoscalar meson masses. Then, in Sec. III we present and discuss our numerical results, while in Sec. IV we provide a summary of our work, together with our main conclusions. We also include Appendices A and B to quote some technical details of our calculations.

A. Effective Lagrangian and mean field properties
We consider the Euclidean action of the SU(3) NJL model which includes a scalar-pseudoscalar interaction and the 't Hooft six-fermion interaction in the presence of an external magnetic field.
It is written as where G and K are coupling constants, ψ = (ψ u , ψ d , ψ s ) T represents a quark field with three flavors, d ± = det ψ (1 ± γ 5 ) ψ andm = diag (m u , m d , m s ) is the corresponding current quark mass matrix. In addition, λ 0 = 2/3 I, where I is the unit matrix in the three flavor space, and λ a with a = 1, ..., 8 denote the Gell-Mann matrices. The coupling of quarks to the electromagnetic field represents the quark electric charge matrix with Q u /2 = −Q d = −Q s = e/3, e being the proton electric charge. In the present work we consider a static and constant magnetic field in the 3direction. Using the Landau gauge we have A µ = δ µ2 x 1 B.
In order to study meson properties, we proceed by bosonizing the action in terms of scalar σ a (x) and pseudoscalar π a (x) fields and the corresponding auxiliary s a (x) and p a (x) fields. Following the standard procedure, we start with the partition function By introducing functional delta functions, the scalar (ψλ a ψ) and pseudoscalar (ψiγ 5 λ a ψ) terms present in S E are replaced by s a (x) and p a (x) and the functional integration on the fermionic fields ψ andψ can be performed by standard methods. To perform the integration over the auxiliary fields we use the stationary phase approximation (SPA), choosings a (x) andp a (x) in order to minimize the integrand of the partition function. This yields a set of coupled equations among the bosonic fields; at the end,s a (x) andp a (x) are to be considered as implicit functions of σ a (x) and π a (x). Finally, we use the mean field approximation by expanding the bosonized action in powers of field fluctuations around the corresponding translationally invariant mean field valuesσ a and π a , i. e., σ a (x) =σ a + δσ a (x) and π a (x) =π a + δπ a (x). Due to charge conservation, onlyσ 0 ,σ 3 andσ 8 are different from zero, while the vacuum expectation values of pseudoscalar boson fields are zero,π a = 0. For convenience, we introduceσ = diag(σ u ,σ d ,σ s ) = λ 0σ0 + λ 3σ3 + λ 8σ8 . At the mean field level, the Euclidean action per unit volume reads where tr D stands for the trace in Dirac space while S f represents the inverse mean field quark propagator for each flavor with effective mass M f = m f + σ f . Moreover,s f =s f (σ a ) represent the auxiliary fields at the mean field level within the SPA approximation (note thatp f = 0). From the condition δS bos E /δσ f = 0 it follows thats f = 2φ f where φ f is the chiral condensate for each flavor given by As is well known, the quark propagator can be written in different ways [16,17]. For convenience we take the following one is the so-called Schwinger phase. We have introduced here the shorthand notation We expressS f p in the Schwinger form [16,17] 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 ). Similarly, p ⊥ = (p 1 , p 2 ) and p = (p 3 , p 4 ). Note that in our convention {γ µ , γ ν } = −2δ µν and we have introduced the no- The integral in Eq. (7) is divergent and has to be properly regularized. We will use the MFIR scheme, where one subtracts from the unregulated integral the B = 0 limit and then adds it in a regulated form. We obtain The expression of I vac 1f for the 3D cutoff regularization scheme we use in this work can be found in Eq. (A5) of App. A. The expression of I mag 1f , given in Eq. (B5) of App. B, reads where Finally, by combining the equations from the SPA together with the gap equations, we obtain that the regularized form of the set of coupled equations for the effective quarks masses read

B. Meson sector
For the calculation of meson masses, we consider the second-order correction to the mean field bosonized Euclidean action S E . At the quadratic level we get for the pseudoscalar sector where the sum indexes run over the nonet of pseudoscalar mesons. Namely, P, P ′ = π 3 , π ± , K 0 ,K 0 , K ± , η 0 , η 8 . The inverse meson propagator in coordinate space can be written as For P, P ′ = π ± , K ± , K 0 ,K 0 this operator is diagonal where In these expressions On the other hand, the two-point function G P,P ′ (x, x ′ ) is non-diagonal but symmetric in the P, P ′ = π 3 , η 0 , η 8 subspace. The corresponding matrix elements of T P,P ′ are where In turn, the polarization function elements can be expressed as where the coefficients γ f P,P ′ are given by

Neutral mesons
For neutral mesons the contributions of Schwinger phases associated with the quark propagators in Eq. (17) cancel out. Therefore, the polarization functions depend only on the difference (x − x ′ ), which leads to the conservation of momentum, since they are translationally invariant. If we take the Fourier transform of neutral meson fields to the momentum basis, the corresponding transform of the polarization functions will be diagonal in momentum space. Thus, the neutral meson contribution to the quadratic action in the momentum basis can be written as Here, the inverse neutral kaon propagator is given by while for P, P ′ = π 3 , η 0 , η 8 we have The values of T P,P ′ and γ f P,P ′ can be found in Eqs. (18) and (21), respectively. In the neutral case, the functions c f f ′ in momentum space are given by where p ± = p ± q/2. We remark here that these functions are divergent. Within the MFIR scheme they can be regularized as where the first term in the right-hand side correspond to the vacuum contribution while the second term to the magnetic one. In this work we regularize the otherwise divergent vacuum term through a 3D cutoff; the corresponding expression is given in Eq. (A13) of App. A. For the calculation of the masses we can set q 2 ⊥ = 0, while the Euclidean parallel components are to be evaluated at the negative real space q 2 = −m 2 P , with m P > 0. Then, assuming that m P < M f + M f ′ , the magnetic contribution can be written as (see App. B) The function I mag 1f has already been expressed in Eq. (9) while where ψ(x) is the digamma function and we have defined For m P < M f + M f ′ we have thatx f f ′ > 0 for all values of y within the integration range of the integral of Eq. (28). Thus, the limit ǫ → 0 can be directly taken.
On the other hand, for η ′ we expect that m P > M f + M f ′ . In this case one has to have special care sincex f f ′ can be negative within the interval 0 < y < 1. We proceed by taking the analytic continuation of both the digamma and logarithm functions. This implies that the inverse propagators become complex functions. Thus, we assume that q develops an imaginary part where Γ P is associated with the decay width of the meson. Following the customary method introduced in Ref. [73], we assume that the width is not too large and neglect its contribution inside I mag 2f f ′ function (this also applies to the equivalent vacuum contribution) Note that in Eq. (28) where g n = 2 − δ n0 and N = Floor m 2 P β 2 0 /8B f . Moreover, For the neutral kaons, we expect m K 0 = mK0 < M d + M s . In this case the polarization function is real and I mag 2f f ′ is well defined in the ǫ → 0 limit of Eq. (28). Therefore, the pole-mass will be given by the solution of In the P, P ′ = π 3 , η 0 , η 8 subspace, the corresponding quadratic action can be expressed in matrix notation through the following inverse matrix propagator which is actually symmetric. The physical meson pole-masses and widths will be given by the roots where the three pair of roots are to be associated with the π 0 , η, η ′ . Of course, one expects to get Γ π 0 = Γ η = 0 while Γ η ′ is expected to be non-vanishing. Note that when B = 0, π 3 (and therefore π 0 ) decouples from the η 0 , η 8 states due to isospin symmetry. However, in the presence of an external magnetic field this symmetry breaks down due to different quark electric charges.
In this case, the π 0 , η, η ′ neutral mesons consist of a mix of π 3 , η 0 , η 8 states, reflected by the fact that non-diagonal terms are present in the inverse propagator of Eq. (35).

Charged mesons
In this case the contributions of Schwinger phases associated with the quark propagators do not cancel out, leading to a breakdown of translational invariance. In order to diagonalize the charged meson fields, we employ the Ritus-like formalism. We find it convenient to introduce the following notation convention withq = (k, q 2 , q 3 , q 4 ) where k labels the charged meson Landau level. The Ritus-like eigenfunctions are F P q (x) = N k e i(q 2 x 2 +q 3 x 3 +q 4 x 4 ) D k (ρ P ) .
Here D k (x) are the cylindrical parabolic functions. We have also defined N k = (4πB P ) 1/4 / √ k! and ρ P = √ 2B P x 1 − s P 2/B P q 2 , where B P = |Q P B| and s P = sign(Q P B). Note that in our case, for the π ± and K ± mesons these definitions reduce to B P = |eB| and s π ± = s K ± = ±1.
The corresponding transformed polarization functions will be diagonal inq,q ′ space. Thus, the charged meson contribution to the quadratic action in the Ritus basis can be written as where Π 2 = q 2 + (2k + 1)B P . The inverse propagators read where we have used These functions are divergent and need to be regularized. Within the MFIR scheme using a 3D cutoff they can be expressed as The regularized vacuum contribution is given in App. A. After a long but straightforward calculation (see [59] for details), we obtain the following expression for the magnetic contribution where we have introduced the definitions t f = tanh(B f zy), For pions and kaons we expect them to develop only a real pole-mass, i.e. Π 2 = −m 2 P with m P < M f + M f ′ . In that case the integrals in Eq. (43) are convergent and well-defined. Therefore, for each Landau level the charged mesons pole-masses will be given by the solutions of

III. NUMERICAL RESULTS
To obtain numerical results for the magnetic field dependence of the meson masses one has to fix the model parametrization. Here, following Ref. . We adopt the one proposed in Ref. [31] in the context of an SU(3) NJL model with the same parameters that we use. In that work the current quark masses, Λ and K were kept constant while for G(B) the form results obtained in Ref. [37]. In that work these quantities have been extracted from a LQCD calculation of the baryon masses using a simple minded approximation based on the constituent quark model.
It should be stressed that in spite of the rather different behavior between the dynamical quark masses, a magnetic catalysis effect at zero temperature is obtained independently on whether G depends on B or not. This is shown in Fig. 2, where we displayed the conveniently normalized light quark condensates. These quantities are defined as it follows. As in Ref. [19], for the case of vanishing temperature we are interested in, we first introduce where we have explicitly stated the magnetic field dependence of the quark condensate, defined in Eq.  We turn now to our results for the magnetic field dependence of the masses of the nonet of pseudoscalar mesons. They are shown in Fig. 3, where for charged mesons we instead display their lowest energy states, given by (note that both E P and m P depend on B although not explicitly stated). The left (right) panel corresponds to the case of constant coupling G (B-dependent G). We observe that, except for the η ′ -mass, the B-dependence is rather mild in the case of the neutral mesons. On the other hand a rather strong increase with growing B is found for charged meson masses. These results are analyzed in further detail in what follows.
The case of η ′ is somewhat special and, therefore, indicated in dashed lines in Fig. 3. In fact, already at B = 0 its mass is above the threshold for qq-decay and, thus, the associated qq polarization diagram receives an unphysical imaginary part. Following Ref. [73] we accept this as an unavoidable feature of the NJL model and define the η ′ -mass as the real part of the corresponding pole in the complex plane. We should keep in mind, however, that this fact makes To discuss our results for the other neutral mesons (π 0 , K 0 ,K 0 and η) in more detail we display in Fig. 4 the corresponding masses taken with respect to their values at B = 0. We show results using a constant and a B-dependent coupling G together with LCQD simulations from Refs. [41,42] for comparison. It should be noticed that these LQCD calculations correspond to non-physical pion masses i.e. 415 and 220 MeV, respectively, for vanishing magnetic field. In both cases they point to a stronger decrease of the π 0 mass with increasing B than the one found in our calculation with (black squares) and Ref. [42] (magenta circles) are added for comparison.
constant G. On the other hand, the results obtained using a B-dependent G are in reasonable good agreement with LQCD ones. A similar observation have been made in Ref. [34] in the context of a two-flavor NJL model. This seems to also provide further support to the relation between the IMC effect and the reduction of the neutral pion mass at finite B mentioned in Ref. [42]. In the case of K 0 andK 0 masses (central panel), the only LQCD result that has been reported is that of Ref. [42]. We observe that, once again, a much better agreement with these results are obtained when a B-dependent coupling G is used in the NJL model. Finally, in the right panel we show our predictions for the behavior of the normalized η-meson mass. They turn out to be quite similar to the ones obtained for the K 0 andK 0 relative masses.
Finally, we consider the masses of charged pseudoscalar mesons π ± and K ± . In Fig. 5 we display the differences in their squared lowest energies from the case of a zero magnetic field, i.e. E 2 (B) − E 2 (B = 0). We also include their masses in the top graphs for completeness. We show results for G and G(B) as compared to a point-like charged meson and LQCD simulations from Refs. [41,42]. We observe that for both charged pion and kaons our results show a stronger increase with growing B as compared with the ones associated with point-like mesons. Those obtained using a B-dependent G are, however, somewhat closer to them. As for the comparison with LQCD results we note that in the case of charged pions there are significant differences between the results reported by the two different LQCD groups, specially at large magnetic fields.
Although our results seem to be more consistent with those of Ref. [41] it should be recalled that they correspond to a larger (unphysical) value of the B = 0 pion mass and have larger error bars. In any case, we see that, for both charged pions and kaons, our NJL results show no sign of the strong non-monotonous behavior found in the LQCD calculation of Ref. [42]. Results obtained within the SU(2) version of the model [30], seem to indicate that the inclusion of quarks anomalous magnetic moments does not modify the trend of the charged pion mass obtained in the present work. At the mean field level, effective quark masses, shown in Fig. 1, steadily increase with B for constant G but display a non-monotonous behavior for G(B) which resembles the one found in Ref. [24]. Moreover, our results for the difference and average of the condensates calculated for both constant G and G(B) and their comparisons with the available LQCD results, as shown in Fig. 2, support the fact that the parametrizations used in this paper are in a very reasonable agreement with LQCD results.
In order to study meson masses we go beyond the mean-field approximation, considering second order corrections to the bosonized Euclidean action of the SU(3) NJL model. Mesons are treated as quantum fluctuations in the random phase approximation. While for neutral mesons one can take the usual momentum basis to diagonalize the corresponding polarization functions, this is not possible for charged mesons since Schwinger phases do not cancel out. In that case, we have employed a method based on the Ritus eigenfunction approach to magnetized relativistic systems.
As discussed in Sec. II, at the quadratic level the inverse propagators corresponding to the neutral π 3 , η 0 and η 8 fields are arranged in terms of a symmetric 3x3 matrix; the pole-masses and widths of the physical mesons (π 0 , η and η ′ ) are obtained as the roots of this inverse propagator matrix determinant. Note that in the B = 0 case, besides the 't Hooft-Maekawa interaction which breaks the U A (1) symmetry and is responsible for the coupling between the η 0 and η 8 , the magnetic field also breaks the isospin symmetry, leading to a mixing between all three states [72]. This is in contrast to the B = 0 case where due to the isospin symmetry (M u = M d ) the π 3 field is decoupled leaving only η 0 and η 8 mixed in a symmetric 2x2 matrix.
As already known from the usual SU(3) NJL model at B = 0, the η ′ meson comes out in the model as a resonance or unstable particle. In this case, the propagator becomes a complex number and from the analysis of the complex pole, the mass of the resonance is obtained. In the presence of a finite magnetic field, the situation is more dramatic since the propagators may develop several poles depending on B, which have to be properly treated. We have developed in this paper a new formalism to deal with this situation. Of course, the results for the η ′ meson are less reliable and its calculation certainly approaches the limit of applicability of the NJL model, since this model does not include confinement. In fact, we find that using G(B) the coupling strength is not enough to form an η ′ -resonance for eB 0.5 GeV 2 .
Our results for the normalized π 0 mass in Fig. 4 show that, for constant G, the mass displays a non-monotonous behavior with B, which initially decreases but is afterwards enhanced for eB 0.5 GeV 2 . On the other hand, using a B-dependent coupling G(B) we recover the monotonous decreasing behavior found in LQCD results. Something similar happens with K 0 andK 0 masses.
For η, our prediction is similar to that of K 0 andK 0 . We thus conclude that incorporating the inverse magnetic catalysis in the NJL model, here through the G(B) coupling, is fundamental for qualitatively reproduce the available LQCD results. Concerning charged mesons, our results for the differences in their squared lowest energies from the B = 0 case are shown in Fig. 5, where a strong enhancement with B is seen. This increase even surpasses the one associated with a pointlike charged meson. Our NJL results are in reasonable agreement with LQCD results of Ref. [41] within error bars On the other hand, no sign of the non-monotonous behavior found in the LQCD calculation of Ref. [42] is observed. where Depending on the value of q m , this expression may still have a pole in a point of the integration line if r > 0. For those regions of q m where a pole exists, we proceed by employing a generalized version of the Sokhotski-Plemelj formula. Assuming there exists a function f (x) that has single poles at a set of values x j , for which exist two other functions g(x) and h(x) such that g(x j ) = 0 and h(x j ) = 0, then where PV denotes the Cauchy principal value of the integral. By using this property we can fully calculate the complex function I vac 2f f ′ in the most general case. For the regularized real part we get where for q m < q Here y ± = √ ±r, with r defined in Eq. (A7), and For the regularized imaginary part we get Putting all together, the regularized version of the vacuum c 0 f f ′ function defined in Eq. (A3) is given by When q m < M f + M f ′ we always have that x f f ′ > 0. Then function I mag 2f f ′ as given in Eq.(B6) is well-defined and can alternatively written as where ψ(x) is the digamma function. Note that in this case one can safely take the ǫ → 0 limit.
On the other hand, when q m > M f +M f ′ it happens thatx f f ′ can be negative in the integration domain. In this case, the integral in Eq. (B6) is not convergent. However, one can still proceed by considering the analytic extension of the form given in Eq. (B8). Sincex f f ′ is a positive quadratic function of y, it is immediate to see that ψ(x f f ′ ) has N + 1 poles, where To proceed we first isolate the poles by using the digamma recurrence relation ψ Expressed this way, the first term in the right-hand side is pole-free. Then where g n = 2 − δ n0 . The complex logarithm is defined by taking the principal branch. Lastly, the third term on the right-hand side of Eq. (B11) contains two simple poles, which once again can be handled using the generalization of the Sokhotski-Plemelj formula presented in Eq. (A8). After some algebra we finally obtain that for q m > M f + M f ′