Neutral pseudoscalar and vector meson masses under strong magnetic fields in an extended NJL model: mixing effects

Mixing effects on the mass spectrum of light neutral pseudoscalar and vector mesons in the presence of an external uniform magnetic field $\vec B$ are studied in the framework of a two-flavor NJL-like model. The model includes isoscalar and isovector couplings both in the scalar-pseudoscalar and vector sectors, and also incorporates flavor mixing through a 't Hooft-like term. Numerical results for the $B$ dependence of meson masses are compared with present lattice QCD results. In particular, it is shown that the mixing between pseudoscalar and vector meson states leads to a significant reduction of the mass of the lightest state. The role of chiral symmetry and the effect of the alignment of quark magnetic moments in the presence of the magnetic field are discussed.


I. INTRODUCTION
It is well known that the presence of a large background magnetic field B has a significant impact on the physics of strongly interacting particles, leading to important effects on both hadron properties and QCD phase transition features [1][2][3]. By a "large" field it is understood here that the order of magnitude of B competes with the QCD confining scale Λ QCD squared, i.e. |eB| Λ 2 QCD , |B| 10 19 G. Such huge magnetic fields can be achieved in matter at extreme conditions, e.g. at the occurrence of the electroweak phase transition in the early Universe [4,5] or in the deep interior of compact stellar objects like magnetars [6,7].
Moreover, it has been pointed out that values of |eB| ranging from m 2 π to 15 m 2 π (|B| ∼ 0.3 to 5 × 10 19 G) can be reached in noncentral collisions of relativistic heavy ions at RHIC and LHC experiments [8,9]. Though these large background fields are short lived, they should be strong enough to affect the hadronization process, offering the amazing possibility of recreating a highly magnetized QCD medium in the lab.
From the theoretical point of view, the study of strong interactions in the presence of a large magnetic field includes several interesting phenomena, such as the chiral magnetic effect [10][11][12], which entails the generation of an electric current induced by chirality imbalance, and the so-called magnetic catalysis [13,14] and inverse magnetic catalysis [15,16], which refer to the effect of the magnetic field on the size of chiral quark-antiquark condensates and on the restoration of chiral symmetry. Yet another interesting issue is the possible existence of a phase transition of the cold vacuum into an electromagnetic superconducting state. For a sufficiently large external magnetic field, this transition would be induced by the emergence of quark-antiquark vector condensates that carry the quantum numbers of electrically charged ρ mesons [17,18]. The presence of such a superconducting (anisotropic and inhomogeneous) QCD vacuum state has been discussed in the past few years and still remains as an open problem [19][20][21][22].
It is clear that the study of the properties of light hadrons, in particular π and ρ mesons, comes up as a crucial task towards the understanding of the above mentioned phenomena.
In this work we study the mass spectrum of light neutral pseudoscalar and vector mesons in the presence of an external uniform magnetic field B, considering a two-flavor NJL-like model [60][61][62]. In general, in this type of model the calculations involving quark loops for nonzero B include the so-called Schwinger phases [63], which are responsible for the breakdown of translational invariance of quark propagators. However, in the particular case of neutral mesons these phases cancel out, and one is free to take the usual momentum basis to diagonalize the corresponding polarization functions [23][24][25][26][27]. One also has to care about the regularization procedure, since the presence of the external field can lead to spurious results, such as unphysical oscillations of various observables [64,65]. We consider here a magnetic field independent regularization (MFIR) method [25,26,32,66], which has been shown to be free from these effects and reduces the dependence of the results on model parameters. In addition, in our work we consider two mixing effects that have been mostly neglected in previous analyses. The first one is flavor mixing in the spin zero sector; while we restrict to a two-flavor model (keeping a reduced number of free parameters, and assuming that strangeness does not play an essential role), we consider quark-antiquark interactions both in I = 1 and I = 0 scalar and pseudoscalar channels, introducing a 't Hooft-like effective interaction [67]. The second one is the mixing between pseudoscalar and vector mesons, which arises naturally in the context of the NJL model. These mixing contributions are usually forbidden by isospin and angular momentum conservation, but they arise (and may become important) in the presence of the external magnetic field. In fact, our analysis shows that π 0 -η -ρ 0 -ω mixing has a substantial effect on the B dependence of the lowest mass state. As a additional ingredient, we consider the case of Bdependent effective coupling constants; this possibility -inspired by the magnetic screening of the strong coupling constant occurring for large B [68]-has been previously explored in effective models [37,[69][70][71][72] in order to reproduce the inverse magnetic catalysis effect observed at finite temperature in LQCD calculations.
In the case of the neutral vector mesons, we consider both states with quantum numbers S z = 0 and S z = ±1, where S z is the spin projection in the direction of the magnetic field (it is worth noticing that only S z = 0 states can mix with pseudoscalar states). Most LQCD results and effective model calculations agree in the finding that the masses of S z = ±1 states get monotonically enhanced with the magnetic field, while results for S z = 0 mesons are still not conclusive [31,45,49,50,52,57,58]. In our framework, which lacks a description of confinement, for large magnetic fields the masses of some of the S z = 0 states are found to grow beyond the qq pair production threshold; therefore our results in this region should be taken just as qualitative ones.
The paper is organized as follows. In Sec. II we introduce the theoretical formalism used to obtain neutral pseudoscalar and vector 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 Appendixes A, B and C to provide some technical details of our calculations.

A. Effective Lagrangian and mean field properties
Let us start by considering the Euclidean Lagrangian density for an extended NJL twoflavor model in the presence of an electromagnetic field. We have where ψ = (u d) T , τ a = (1, τ ), τ being the usual Pauli-matrix vector, and m c is the current quark mass, which is assumed to be equal for u and d quarks. The model includes isoscalar and isovector vector couplings, and also a 't Hooft-like flavor-mixing term where The interaction between the fermions and the electromagnetic field A µ is driven by the covariant derivative whereQ = diag(Q u , Q d ), with Q u = 2e/3 and Q d = −e/3, e being the proton electric charge.
We consider the particular case in which one has a homogenous stationary magnetic field B orientated along the 3, or z, axis. Then, choosing the Landau gauge, we have A µ = B x 1 δ µ2 .
Since we are interested in studying meson properties, it is convenient to bosonize the fermionic theory, introducing scalar, pseudoscalar and vector fields σ a (x), π a (x) and ρ aµ (x), with a = 0, 1, 2, 3, and integrating out the fermion fields. The bosonized Euclidean action can be written as with where a direct product to an identity matrix in color space is understood. Note that for convenience we have introduced the combinations so that the flavor mixing in the scalar-pseudoscalar sector is regulated by the constant α.
For α = 0 quark flavors u and d get decoupled, while for α = 0.5 one has maximum flavor mixing, as in the case of the standard version of the NJL model.
We proceed by expanding the bosonized action in powers of the fluctuations of the bosonic fields around the corresponding mean field (MF) values. We assume that the fields σ a (x) have nontrivial translational invariant MF values given by τ aσa = diag(σ u ,σ d ), while vacuum expectation values of other bosonic fields are zero; thus, we write The MF piece is diagonal in flavor space. One has with where M f = m c +σ f is the quark effective mass for each flavor f .
The MF action per unit volume is given by where tr D stands for the trace in Dirac space, and S MF, f x,x = D MF, f x,x −1 is the MF quark propagator in the presence of the magnetic field. As is well known, the explicit form of the propagators can be written in different ways [2,3]. For convenience we take the form in which S MF, f x,x is given by a product of a phase factor and a translational invariant function, is the so-called Schwinger phase. We have introduced here the shorthand notation NowS f p can be expressed in the Schwinger form [2, 3] where we have used the following definitions. The perpendicular and parallel gamma matrices are collected in vectors γ ⊥ = (γ 1 , γ 2 ) and γ = (γ 3 , γ 4 ), and, similarly, we have defined p ⊥ = (p 1 , p 2 ) and p = (p 3 , p 4 ). Note that we are working in Euclidean space, where The integral in Eq. (12) is divergent and has to be properly regularized. As stated in the Introduction, we use here the magnetic field independent regularization (MFIR) scheme: for a given unregularized quantity, the corresponding (divergent) B → 0 limit is subtracted and then it is added in a regularized form. Thus, the quantities can be separated into a (finite) "B = 0" part and a "magnetic" piece. Notice that, in general, the "B = 0" part still depends implicitly on B (e.g. through the values of the dressed quark masses M f ), hence it should not be confused with the value of the studied quantity at vanishing external field.
To deal with the divergent "B = 0" terms we use here a 3D cutoff regularization scheme.
In the case of the quark-antiquark condensates where The expression of I 1f for the 3D cutoff regularization is given by Eq. (A3) of App. A [61], while the B-dependent function I mag 1f reads [13,64] I mag The corresponding gap equations, obtained from ∂S MF bos /∂σ f = 0, can be written as As anticipated, for α = 0 these equations get decoupled. For α = 0.5 the right hand sides become identical, thus one has in that case M u = M d .

B. Neutral meson system
As expected from charge conservation, it is easy to see that the contributions to the bosonic action that are quadratic in the fluctuations of charged and neutral mesons decouple from each other. In this work we concentrate on the neutral meson sector. For notational convenience we will denote isospin states by M = σ a , π a , ρ aµ , with a = 0, 3. Here σ 0 , π 0 and ρ 0 correspond to the isoscalar states σ, η and ω, while σ 3 , π 3 and ρ 3 stand for the neutral components of the isovector triplets a 0 , π and ρ, respectively. Thus, the corresponding quadratic piece of the bosonized action can be written as The functions G M M (x, x ) can be separated in two terms, namely where δ M M is an obvious generalization of the Kronecker δ, and the constants g M are given by The polarization functions J M M (x, x ) can be separated into u and d quark pieces, Here ε M = 1 for the isoscalars M = σ 0 , π 0 , ρ 0µ and ε M = −1 for M = σ 3 , π 3 , ρ 3µ , while the with As stated, since we are dealing with neutral mesons, the contributions of Schwinger phases associated with the quark propagators in Eq. (10) cancel out, and the polarization functions depend only on the difference x − x , i.e., they are translationally invariant. After a Fourier transformation, the conservation of momentum implies that the polarization functions turn out to be diagonal in the momentum basis. Thus, in this basis the neutral meson contribution to the quadratic action can be written as Now we have and the associated polarization functions are given by The where we have defined p ± = p ± q/2, and the quark propagatorsS f p in the presence of the magnetic field have been given in Eq. (12).
It is relatively easy to see that the functions J σaπ b (q) are zero for either a or b equal to 0 or 3. However, the remaining polarization functions do not vanish in general. Since we are interested in the determination of meson masses, we consider here the particular case in which mesons are at rest, i.e. we take q = 0, q 2 4 = −m 2 , where m stands for the corresponding meson mass. In that situation the nondiagonal polarization functions that mix the neutral scalar and vector mesons also vanish, i.e. for a, b = 0, 3 one hasĴ σaρ bµ = 0, where the notationĴ indicates that the polarization function is evaluated at the meson rest frame. In this way, the scalar meson sector gets decoupled at this level; we will not take into account these mesons in what follows. It can also be shown thatĴ ρaµρ bν , with a, b = 0, 3, vanish for µ = ν, while the functionsĴ πaρ bµ , with a, b = 0, 3, turn out to be proportional to δ µ3 .
It is found that all nonvanishing polarization functions are in general divergent. As done at the MF level, we consider the magnetic field independent regularization scheme, in which we subtract the corresponding "B = 0" contributions and then we add them in a regularized form. Thus, for a generic polarization functionĴ M M we havê The regularized "B = 0" piecesĴ 0,reg M M are given in App. A; it is easy to see that all nondiagonal polarization functionsĴ 0, reg M M , M = M , are equal to zero. In the case of the "magnetic" contributionsĴ mag M M , after a rather long calculation it is found that they can be expressed in the form given by Eq. (25), viz.
where the functionsF f, mag

M M
are given bŷ with 1 ⊥ = diag(1, 1, 0, 0). The expression for I mag 1f has been given in Eq. (15), whereas the integrals I mag nf for n = 2, . . . , 5 read For m < 2M f these integrals are well defined. In fact, in the case of I mag 3f (−m 2 ) one can even get the analytic result In the case of I mag 4f (−m 2 ), it is worth noticing that in the limit m 2 → 0 the second term on the r.h.s. of the corresponding expression in Eqs. (30) is found to be equal to I mag 1f . Thus, in this limit one has I mag 4f → 0, as it is required in order to avoid a nonzero contribution to the photon mass coming from the "magnetic piece" of the polarization function. On the other hand, for m ≥ 2M f (i.e., beyond the qq production threshold) the integrals are divergent.
To obtain finite results we perform in this case analytic extensions. The corresponding expressions, as well as some technical details, are given in App. B.
The vector fields ρ 0µ and ρ 3µ can be written in a polarization vector basis. Since we assume that the mesons are at rest, we can choose polarization vectors It is convenient to distinguish between states ρ a and ρ a⊥ , with a = 0, 3, corresponding to polarization vectors parallel and perpendicular to the magnetic field (spin projections S z = 0 and S z = ±1), respectively. From Eqs. (29) it is seen that for nonzero B pseudoscalar while those of the S z = ±1 vector mesons, m (k) Once the masses are determined, the spin-isospin composition of the physical meson states |k is given by the corresponding eigenvectors c (k) . Thus, one has It is also useful to consider the flavor basis π f , ρ f , where f = u, d. Isospin states can be written in terms of flavor states using the relations In the S z = ±1 sector, where there is no mixing between pseudoscalar and vector mesons, the states |ρ u⊥ and |ρ d⊥ turn out to be the mass eigenstates that diagonalize G ⊥ . This can be easily understood noticing that the external magnetic field distinguishes between quarks that carry different electric charges, and this is what breaks the u-d flavor degeneracy. In the flavor basis one has where the expression for I 2f (q 2 ) can be found in App. A, and I mag 4f (−m 2 ) has been given in Eqs. (30). A similar situation occurs in the S z = 0 sector if one has α = 0. In this particular case there is no flavor mixing either in the pseudoscalar or vector meson sectors, hence the 4 × 4 matrix G can be written as a direct sum of 2 × 2 flavor matrices G u and (this also holds for the implicit dependence on Q f and B through the quark effective masses M f ). If one has α = 0 this relation is no longer valid, and in general G cannot be separated into flavor pieces. In fact, as we discuss below, in the pseudoscalar sector it is seen that chiral symmetry largely dominates over flavor symmetry; for the range of values of B considered in this work, we find that even for α 1 the lightest S z = 0 mass eigenstates are very close to isospin states π 3 and π 0 , instead of approximating to flavor states π f .

A. Model parametrization and mean field results
To obtain numerical results for the dependence of meson masses on the external magnetic field, one first has to fix the parameters of the model.  these constants we use from now on the notation Finally, as stated in Sec. II A, the amount of flavor mixing induced by the 't Hooft-like interaction is controlled by the parameter α. In this work we choose to take as a reference value α = 0.1, since it leads (at B = 0) to an approximate η meson mass m η 520 MeV, in reasonable agreement with the physical value m phys η = 548 MeV. In fact, this mass is very sensitive to minor changes in α. An alternative estimate for this parameter can be obtained from the η − η mass splitting within the 3-flavor NJL model [73], which leads to α 0.2 [74]. In any case, to obtain a full understanding of the effects of flavor mixing we will also consider the values α = 0 and α = 0.5, corresponding to the situation in which flavors are decoupled and in which there is full flavor mixing, respectively. It is easily seen that for α = 0 the π and η mesons have equal (finite) masses, while when α approaches 0.5 the mass of the pion stays finite and that of the η meson becomes increasingly large.
In Fig. 1   it is seen that for M u (M d ) the slope becomes larger (smaller) as α decreases from α = 0.5 -where both masses coincide-to α = 0. Next, in Fig. 2, we show the dependence of normalized light quark-antiquark condensates on B. Following Ref. [16], we introduce the MeV being a phenomenological normalization constant. In the left and right panels of Fig. 2 we plot the values of ∆Σ and Σ − , respectively, as functions of eB. The gray bands correspond to LQCD values taken from Ref. [16], whereas the red bands cover our results for the range α = 0 to α = 0.5. We observe from this figure that the model reproduces properly the zero-temperature magnetic catalysis found in LQCD calculations. Moreover, it is seen that the dependence on the flavor mixing parameter α is rather mild.  Table I). In Fig. 3 we also  show the qq production thresholds m = 2M d and m = 2M u (dotted and short-dotted lines, respectively), beyond which some of the matrix elements of G get absorptive parts. The presence of these absorptive parts implies that for the statesρ andω there are certain values of the magnetic field above which the associated particles are unstable with respect to an unphysical decay into a qq pair. In fact, the existence of such decays is a well known feature of the NJL model, even in the absence of an external field [60][61][62]; it arises as a consequence of the lack of a confinement mechanism, which is a characteristic of this type of model.
In the presence of the magnetic field, one also has to deal with new poles that may arise from the thresholds related to the Landau level decomposition of the intermediate quark propagators. As customary, we will assume that the widths associated to these unphysical decays are small. Then, to determine the values of the corresponding masses, we consider an extremum condition for the meson propagators, similar to the method discussed e.g. in Ref. [75]. It has to be kept in mind, however, that these predictions for the meson masses are less reliable in comparison to those obtained for the states lying below the quark pair production threshold, and should be taken just as qualitative results. For this reason, in It is interesting at this stage to discuss the spin-isospin composition of mass states and their variation with the external field. As mentioned at the end of Sec. II B, the magnetic field tends to separate the states according to the charges of the quark components. In the case α = 0, although there is no flavor mixing, flavor degeneracy gets broken due to the magnetic field. Therefore, mass eigenstates turn out to be separated into particles with pure u or d quark content. If we use the labels k = 1, 3 and k = 2, 4 for "u" and "d" states respectively, we get [see Eqs. (36)] For definiteness, let us take m Moreover, according to the previous discussion, the latter must be pure u and d-states. Since they should be approximate mass eigenstates, one expects to find (c πu , c In Table I we quote the composition of the mass eigenstatesM described in Fig. 3, for some representative values of the magnetic field. For completeness, the coefficients corresponding to both spin-isospin and spin-flavor basis are included. We note that while the mass eigenvalues do not depend on whether B is positive or negative, the corresponding eingenvectors do. The relative signs in Table I   Let us first discuss the composition of theπ state (k = 1), which is the one that has the lowest mass. We see that even though α is relatively small, the effect of flavor mixing is already very strong; the spin-isospin composition is clearly dominated by the π 3 component, which is given by an antisymmetric equal-weight combination of u and d quark flavors.
Thus, the mass states are far from satisfying the flavor disentanglement expected for the case α = 0 [see Eqs. (39)], in which one has two approximate Goldstone bosons. In fact, once α is turned on, explicitly breaking the U(1) A symmetry, π 3 is the only state that remains being a pseudo-Goldstone boson; this forces the lowest-mass stateπ to be dominated by the π 3 component. As discussed above, the presence of the magnetic field distinguishes between flavor components π u and π d instead of isospin states. However, it is found that even for values of α as small as 0.01 the mass stateπ is still dominated by the π 3 component (|c (1) π 3 | 2 0.9) for the full range of values of eB considered here. In other words, extremely large magnetic fields would be required in order to rule the composition of light mass eigenstates, which is otherwise dictated by the invariance under U(1) T 3 , A transformations. Coming back to the case α = 0.1, we see that, although relatively small, the effect of the magnetic field on the composition of theπ state can be observed from the values in Table I. When eB gets increased, it is found that there is a slight decrease of the component π 3 in favor of the others. In addition, a larger weight is gained by the u-flavor components, as one can see by looking at the entries corresponding to the spin-flavor states (last four columns of Table I): one has |c  Turning now to the composition of theη state (k = 2) in Table I, we see that, as expected from the above discussion, it is dominated by the π 0 (I = 0) component for values of eB up to 1 GeV 2 . Regarding the flavor composition, in this case the d-quark content is the one that increases as eB does, with |c (2) π d | 2 + |c ranging from 0.02 at eB = 0.05 GeV 2 to 0.17 at eB = 1.0 GeV 2 . This is probably due to the fact that for α = 0.1 theη mass is closer to vector meson masses.
Finally, let us comment on the composition of theω andρ states (k = 3 and k = 4, respectively). As mentioned above, the masses of these states reach the threshold for qq decay for rather low values of the magnetic field, hence our predictions for these quantities should be taken as qualitative ones for a major part of the eB range considered here. It is worth noticing that there is a multiple number of thresholds, which get successively opened each time the meson mass is sufficiently large so that the quark and antiquark meson components can populate a new Landau level. The first thresholds in theūu and thedd sectors are reached at meson masses equal to 2M u and 2M d , respectively. It is important to realize that they do not correspond to a free quark together with a free antiquark, but to the quark and antiquark in their lowest Landau levels. Taking B > 0, if both the quark and the antiquark have vanishing z component of the momentum, the corresponding spin configurations are In both cases, the magnetic dipole moments of the quark and the antiquark are parallel to the magnetic field; the difference between both configurations arises from the opposite signs of the quark electric charges.
We only quote in Table I  M should be more reliable. Interestingly, we note that even at this low value of the magnetic field the composition of the vector meson mass states is clearly flavor-dominated: from Table I one Table I it is seen that these combinations correspond to c  observe that for α = 0 (no flavor mixing) there are two light mesons having similar masses; as stated above, these are pure flavor states and can be identified as approximate Goldstone bosons. For α = 0, the mass of theπ state is still protected owing to its pseudo-Goldstone boson character, whereas theη state becomes heavier when α gets increased, and disappears from the spectrum in the limit α = 0.5.
From Fig. 4 it is also seen that, for all values of α, the mixing between pseudoscalar and vector meson states produces a significant decrease in the mass of the lightest state. This might be surprising, since -as shown above-the vector meson components of theπ state are found to be very small even for large values of eB. The explanation of this puzzle is discussed in detail in App. C, where it is shown that these two facts are indeed consistent.
Moreover, for α = 0.5 it is shown that if the pseudoscalar-vector meson mixing is treated perturbatively, one can derive a simple formula for the B dependence of theπ mass, viz.
where κ = 5N 2 c g g v /(18π 4 m c ), f is either u or d, andmπ stands for theπ mass when no mixing is considered. Taking into account thatmπ is very weakly dependent on B (see dotted lines in Fig. 4), it follows that mπ basically depends on the magnetic field through Ref. [53] (improved staggered quarks) and Refs. [15,52,76] (dynamical staggered quarks).
We first note that in those calculations the authors neglect disconnected diagrams as well as the associated mixing, and work with the individual flavor states instead. In our calculation this can be achieved by setting α = 0. In any case, as seen from the above analysis, the mass of the lightest meson is approximately independent of the value of α; therefore, it is reasonable to compare the mentioned LQCD results with those obtained using the reference value α = 0.1 that leads to an acceptable value for the η meson mass at vanishing external field. We also note that LQCD results have been obtained using different methods and values of the pion mass at B = 0. In particular, the most recent ones (i.e. those in Ref. [53]) are based on a highly improved staggered quark action that uses m π (B = 0) = 220 MeV, while the calculations in Refs. [15,52,76] take the physical value of m π within a staggered simulation setup. Anyway, in our model we see that when the pseudoscalar-vector meson mixing is included, the values for theπ meson mass lie in general below LQCD predictions.
We have checked that this general result is quite insensitive to a reasonable variation of the model parameters. In addition, we have verified that the situation does not change significantly if the B = 0 expressions are regularized using the Pauli-Villars scheme, as proposed e.g. in Ref. [57]. The numerical results for the meson masses as functions of the magnetic field for the case α = 0.1 are shown in Fig. 6, where it is seen that both m ρ u⊥ and m ρ d⊥ get increased with B.
The enhancement is larger in the case of the ρ u⊥ mass; this can be understood from the larger (absolute) value of the u-quark charge, which measures the coupling with the magnetic field.
As in the case of S z = 0 mesons, there are multiple mass thresholds for qq pair production  It should be pointed out that even though there is no direct flavor mixing in this sector, ρ u⊥ and ρ d⊥ meson masses still depend on α. This is due to the fact that the values of M u and M d obtained at the MF level get modified by flavor mixing. We recall that M u (eB) = M d (2eB) for α = 0, while for α = 0.5 one has M u (eB) = M d (eB). The effect of flavor mixing is illustrated in Fig. 7, where we show the B dependence of ρ u⊥ and ρ d⊥ meson masses for α = 0, 0.1 and 0.5. As expected from the aforementioned relations between M u and M d , it is seen that the curves for both masses tend to become more similar as α increases. However, the overall effect is found to be relatively weak. As a reference we also plot (full black line) the situation in which the mixing between S z = ±1 vector states is neglected, and, therefore, the masses of both states coincide. We see that even in the case α = 0.5 there is a certain non-negligible repulsion between states when the mixing term is turned on. It is also interesting at this stage to analyze the impact of the regularization procedure on the predictions of the model. In Fig. 8 we show our results for the ρ 3⊥ mass together with those obtained in Ref. [22] and Ref. [57]. To carry out a proper comparison, in our model we have taken α = 0.5 and have set to zero the ρ 0⊥ − ρ 3⊥ mixing contributions, as done in those works (in which the ρ 0⊥ state is not included). Notice that this case corresponds to the solid line in the right panel of Fig. 7. In Ref. [22], divergent integrals are regularized through the introduction of Lorenztian-like form factors, both for vacuum and B-dependent contributions. On the other hand, in Ref. [57] the regularization is carried out using the MFIR method, as in the present work. However, to deal with vacuum-like terms the authors of Ref. [57] choose a Pauli-Villars regularization, instead of the 3D-cutoff scheme considered here. From Fig. 8 it is seen that our results for m ρ 3⊥ (black solid line) are quite similar to those found in Ref. [57] (red dotted line), indicating that they are not too much sensitive to the prescription used for the regularization of vacuum-like terms, once the MFIR method is implemented. Meanwhile, the ρ 3⊥ mass obtained by means of a form factor regularization (blue dashed line) shows a much stronger dependence on the magnetic field, specially for large values of eB. These results are consistent with those found in Ref. [65] for the regularization scheme dependence of the condensates in the presence of the magnetic field. Finally, in Fig. 9 we compare our results for the case α = 0.1 (dashed and dotted lines in the central panel of Fig. 7) with those quoted in Ref. [52] for the ρ u⊥ mass using LQCD calculations. In fact, these lattice results are obtained for a large vacuum pion mass of about 400 MeV; the comparison still makes sense, however, since we have checked that our results are rather robust under changes in the current quark masses leading to such a large value of m π . Considering the large error bars, from the figure one observes that LQCD results seem to indicate an enhancement of m ρ u⊥ when the magnetic field is increased, in agreement with the predictions from the NJL model. This qualitative behavior has been also found in previous LQCD studies [45,49,50,58].

D. B-dependent four-fermion couplings
As mentioned in the Introduction, while local NJL-like models are able to reproduce the magnetic catalysis (MC) effect at vanishing temperature, they fail to describe the so-called inverse magnetic catalysis (IMC) observed in lattice QCD. Among the possible ways to deal with this problem, one of the simplest approaches is to allow the model coupling constants to depend on the magnetic field. With this motivation, we explore in this subsection the possibility of considering a magnetic field dependent coupling g(eB). For definiteness, we adopt for this function the form proposed in Ref. [26], namely where with κ 1 = 0.321, κ 2 = 1.31 GeV −2 . Assuming this form for g(eB), the effective quark masses are found to be less affected by the presence of the magnetic field than in the case of a constant g. In fact, they show a non-monotonous behavior for increasing B, resembling the results found in Refs. [38,72]. It should be stressed that in spite of the rather different behavior of the dynamical quark masses, a similar zero-temperature magnetic catalysis effect is obtained both for a constant g and for a variation with B of the form given by Eq. (42).
Regarding the vector meson sector, one has to choose some assumption for the B dependence of the vector coupling constant. One possibility is to suppose that, due to their common gluonic origin, the vector couplings are affected by the magnetic field in the same way as the scalar and pseudoscalar ones. That is to say, one could take g v (eB) = g v F v (eB), with F v (eB) = F(eB). Under these assumptions, we have obtained numerical results for the behavior of meson masses with the magnetic field. The curves for the case α = 0.1 are given in Fig. 10, where we also show the qq production thresholds (dotted lines).
By comparison with the results in Fig. 3 and Fig. 6, it can be observed that the B It is important to note that the presence of an electromagnetic field allows for isospin mixing. In addition, the axial character of the magnetic field together with the loss of rotational invariance lead to pseudoscalar-vector mixing. These mixing contributions are usually forbidden by isospin and angular momentum conservation. However, they arise and may become important in the presence of the external magnetic field. Although full rotational invariance is broken, invariance under rotations around the magnetic field direction survives.
Therefore, the projection of the vector meson spin in the field direction, S z , is the observable that organizes the obtained results. Our analysis shows that for the determination of the masses (i.e., if particles are taken at rest), the scalar mesons, which in our case include the f 0 (or σ) and a 0 0 states, mix with each other but decouple from other mesons. Thus, they can be disregarded in the analysis of the pseudoscalar and vector meson masses. The remaining meson space can be separated into three subspaces: pseudoscalar and vector mesons with S z = 0, including π 0 , η, ρ 0 and ω, which mix with each other; vector mesons with S z = +1, including ρ 0 and ω mesons; same as before, with S z = −1.
Regarding the S z = 0 sector, we observe two different behaviors for the meson masses.
The masses of the two lightest mesons, which we have calledπ andη, are determined by the underlying symmetries and their breaking pattern. In the presence of the magnetic field, with α = 0, one has a "residual" U(1) T 3 ⊗U(1) T 3 ,A ⊗U(1) A chiral symmetry, explicitly broken only by a (small) current mass term, m c = 0, which guarantees the pseudo-Goldstone character of these two states. We have shown that flavor degeneracy gets broken by the magnetic field and mass eigenstates are separated into particles with pure u or d quark content. For α = 0.1, which leads to a reasonable value for the η mass in the absence of the magnetic field, the U(1) A symmetry is broken and only one pseudo-Goldstone boson,π, survives. From our results, we can conclude that even for magnetic field values as large as eB = 1 GeV 2 , theπ state is mostly a pseudoscalar isovector (third component) andη is mostly a pseudoscalar isoscalar. Increasing the magnetic field intensity from a low value of eB = 0.05 GeV 2 to eB = 1 GeV 2 we observe that the u content of theπ and the d content of theη get enhanced.
On the other hand, regarding the quark structure of the two heaviest mesons, which we callω andρ, it is found that even for a low value of the magnetic field, eB = 0.05 GeV 2 , the mass eigenstates turn out to be clearly dominated by the quark flavor content and spin orientation. This is what we could expect, since the magnetic field tends to separate quarks according to their electric charges, and favors that their magnetic moments be orientated parallel to the field direction.
The lack of confinement in the NJL model implies that the polarization functions get absorptive contributions, related with qq pair production, beyond certain thresholds. In the presence of the magnetic field, the position of each threshold is flavor and spin dependent, in such a way that for S z = 0 we have thresholds for meson mass values m f = 2 M f , while for S z = ±1 the thresholds rise to higher values m f = M f + M 2 f + 2 B f . As a consequence, we find thatω andρ states with S z = 0 enter into the continuum for values of the magnetic field around eB ∼ 0.1 GeV 2 , whereas S z = ±1 meson masses always lie under qq production thresholds for the considered range of values of eB. A common result for all these states is that their masses show an appreciable growth when the magnetic field varies from zero to eB = 1 GeV 2 . In the case of S z = ±1, the model reproduces reasonably well present LQCD results for ρ u ⊥ , taking into account the uncertainties in LQCD simulations. We have observed that the mass of the lightest state,π, gets reduced as the magnetic field increases. This behavior reproduces the trend of existing LQCD results. However, our results overestimate the mass reduction as compared to the one found in LQCD simulations.
It is seen that this reduction is significantly affected by the mixing between pseudoscalar and vector components, a fact that turns out to be independent of the value of the flavor mixing parameter α. From an analytical perturbative analysis, we have carefully studied how a small value of the vector components in theπ state can lead to a significant reduction of its mass. It is seen that both the mixture of the π channel with the ω and ρ channels contribute to this mass shrinkage.
While local NJL-like models are able to reproduce the magnetic catalysis effect at vanishing temperature, they fail to lead to the so-called inverse magnetic catalysis. One of the simplest ways to deal with this problem is to allow that the model coupling constants depend on the magnetic field. With this motivation, we have explored the possibility of considering magnetic field dependent couplings g(eB) and g v (eB). For definiteness we take the same dependence on B for both couplings; in that case, our results show that, for any value of α, the mass of theω state with S z = 0 is the only one that becomes significantly modified with respect to the case in which g and g v do not depend on the magnetic field. In particular, the B-dependence of the ratio r π = mπ(eB)/m π (0) is almost identical to that obtained when the couplings g and g v are kept constant. If one allows for different B dependences for g and g v it is possible to improve on the agreement with LQCD results for this ratio. However, this implies a rather strong enhancement in the masses of S z = ±1 vector meson states, leading to a rather large discrepancy with LQCD results in Ref. [52].
For simplicity, in the present work we have not taken into account the axial vector interactions. The influence of these degrees of freedom in the magnetic field dependence of light neutral meson masses, and, in particular, on the ratio r π , is certainly an issue that deserves further investigation. It would be also interesting to study the effect of the inclusion of quark anomalous magnetic moments. We expect to report on these issues in future publications.
with p ± = p ± q/2. Within the 3D-cutoff regularization scheme used in this work, the first of these integrals is given by where we have defined r Λf ≡ 1 + M 2 f /Λ 2 . In the case of I 2f (q 2 ), we note that in order to determine the meson masses, the external momentum q has to be extended to the region q 2 < 0. Hence, we find it convenient to write q 2 = −m 2 , where m is a positive real number.
Then, within the 3D-cutoff regularization scheme, the regularized real part of I 2f (−m 2 ) can be written as where .
For the regularized imaginary part we get Here, we have used the definition α n = 2 − δ 0n , together with is that given by Eq. (15), and we have introduced the functions T ± f (−m 2 ) given by with γ = 2 − β 2 0 = 1 + 4M 2 f /m 2 , and The integers N ± have been defined as In this appendix we present a simplified model to analyze the mass and composition of the lowest state of the S z = 0 meson sector. As seen in Sec. III B (see the discussion concerning Fig. 4), the mass of this state, while almost independent of the value of α, is significantly affected by the existence of a mixing between pseudoscalar and vector meson states. Thus, to simplify the analysis we consider the case α = 0.5, in which the relevant basis is only composed by the states π 3 , ρ 3 and ρ 0 . In addition, one has M u = M d ≡ M for any value of eB. Assuming as in the main text g v 0 = g v 3 = g v , it is easy to see that the ratio between the off-diagonal π 3 ρ 3 and π 3 ρ 0 mixing matrix elements is given by G π 3 ρ 3 /G π 3 ρ 0 = (B u − B d )/(B u + B d ) = 1/3. Hence, to simplify the problem even further, in what follows we only consider the π 3 -ρ 0 system (see, however, discussion at the end of this appendix). To check whether we are capturing the main effect of pseudoscalar-vector meson mixing on theπ mass it is useful to consider the ratio r π = mπ(eB)/m π (0). Assuming that g and g v do not depend on the magnetic field, and taking α = 0.1, for eB = 1 GeV 2 we get r π = 0.32 for the full π 0 -π 3 -ρ 0 -ρ 3 system, to be compared with the values r π = 0.38, obtained when we consider only the π 3 -ρ 0 system, and r π = 0.92, obtained for the case in which there is no mixing at all. These values clearly support our approximation of the full system by the much simpler π 3 ρ 0 one. It should be stressed that even in this simplified situation the lowest mass state is still found to be strongly dominated by the π 3 contribution. In fact, for eB = 1 GeV 2 we get c (1) ρ 0 = −0.083, close to the value −0.0797 obtained for the full system (see Table I). Defining a mixing angle θ by tan θ = c The strong dominance of the π 3 contribution to theπ state suggests that one should be able to determine the mixing effect on mπ using first order perturbation theory. On the other hand, this appears to be in contradiction with the aforementioned significant reduction of theπ mass. To get a better understanding of the situation, it is convenient to carry out some further approximations. The relevant mixing matrix elements to be considered are We note that here the gap equation has been used to get the expression for a. Given the model parameters, these coefficients can be easily computed for a given value of B e .
Keeping terms up to the leading order in m 2 , one gets in this way In addition, it can be seen that a/d = (g v 0 /g)(m c /M ) 1, and consequently ab bd.
Using this approximation we obtain wheremπ = a/b is the mass of the lightest state if there is no mixing at all. Within the same approximation, the coefficient of the ρ 0 piece of the lightest state is given by The numerical values for the above quantities can be calculated from Eqs. (C3), (C4). For a large magnetic field eB = 1 GeV 2 , assuming that the coupling constants are independent of B, we get r π = 0.39 and c (1) ρ 0 = −0.084, in excellent agreement with the results quoted above for the π 3 ρ 0 system. This confirms the validity of the approximations made so far.
It is also interesting to note that the expression for mπ given in Eq.
This expression, together with the one for c (1) ρ 0 in Eq. (C7), are the relations that one would obtain from a first-order perturbation analysis of the system described by the matrix in Eq. (C2) when d a + (b − b)m 2 , a condition that is always well satisfied in our case. One can observe that the somewhat unexpectedly "large" value of the mass shift arises from the small value of the coefficient b, which is found to be about 0.034 for eB = 1 GeV 2 (assuming B-independent couplings). In a conventional eigenvalue problem, one would have b = 1.
Finally, we note that the effect on this game of the ρ 3 meson, so far neglected, can be easily taken into account at this stage. Since, as shown above, the G π 3 ρ 0 matrix element can be treated perturbatively, and G π 3 ρ 3 is even 3 times smaller, one can account for the ρ 3 meson just replacing the factor c 2 in Eq. (C6) by 10/9 c 2 . The resulting expression for mπ can be rewritten as where κ = 5N 2 c g g v /(18π 4 m c ). To obtain this expression we have made use of Eq. (C3) together with the relation b = a/m 2 π = m c /(2gMm 2 π ). It should be emphasized that although the numerical values quoted in this appendix correspond to the case in which the couplings g and g v are kept fixed, Eq. (C9) can be shown to be approximatively valid also when they depend on B as considered in Sec. III D.