Finite-volume and magnetic effects on the phase structure of the three-flavor Nambu--Jona-Lasinio model

In this work we analyze the finite-volume and magnetic effects on the phase structure of a generalized version of Nambu--Jona-Lasinio model with three quark flavors. By making use of mean-field approximation and Schwinger's proper-time method in a toroidal topology with antiperiodic conditions, we investigate the gap equation solutions under the change of the size of compactified coordinates, strength of magnetic field, temperature and chemical potential. The 't Hooft interaction contributions are also evaluated. The thermodynamic behavior is strongly affected by the combined effects of relevant variables. The findings suggest that the broken phase is disfavored due to both increasing of temperature and chemical potential, and the drop of the cubic volume of size $L$, whereas it is stimulated with the augmentation of magnetic field. In particular, the reduction of $L$ (remarkably at $L\approx 0.5 - 3 $~fm) engenders a reduction of the constituent masses for $u,d,s$-quarks through a crossover phase transition to the their corresponding current quark masses. On the other hand, the presence of a magnetic background generates greater values constituent quark masses, inducing smaller sizes and greater temperatures at which the constituent quark masses drop to the respective current ones.


I. INTRODUCTION
In the recent decades we have witnessed experimental and theoretical progresses in the assessment of strongly interacting matter under extreme conditions. One of the most interesting predictions of its underlying theory, Quantum Chromodynamics (QCD), is that it experiences a phase transition to a deconfined state at sufficiently high temperatures. This mentioned state composed of quarks and gluons is commonly referred to as the quark-gluon plasma (QGP) and it can now be observed in heavy-ion collisions [1,2]. Its thermodynamic and transport properties continues to the forefront of endeavour of community to outline its rich phase structure [2][3][4].
In the papers cited above, important questions arise in order to have a detailed description of limited-size physical systems, like the mentioned QGP. One example is the spontaneous symmetry breaking of chiral symmetry, which occurs for systems in the bulk approximation, but is disfavored for finite volume. Thus, this context gives rise to the natural debate about what are the conditions in which an ideal bulk system seems a good approximation for confined systems. In other words, one may wonder about the range of the size of the system V = L 3 in which the finite-volume effects affect the phase structure and the spontaneous symmetry breaking, making modifications in the thermodynamical properties of quarkionic matter.
Specially, in a very recent paper by Wang et al. [23] is discussed the influence of the finite-volume effects on the chiral phase transition of quark matter at finite temperature, based on a two-flavour NJL model approach with a proper-time regularization and use of a stationary-wave condition. It is found that when the cubic volume size L is larger than 500 fm, the chiral quark condensate is indistinguishable from that at L = ∞, which is far greater than the estimates of the size of QGP produced at laboratory and the lattice QCD simulations. Also, they found that when the space size L is less than 0.25 fm, the spontaneous symmetry breaking concept is no longer valid.
On the other hand, the importance of a magnetic background on the phase diagram of strongly interacting matter has also been a subject of great interest. The motivation comes from both phenomenology of heavy ion collisions and compact stars, in which a strong magnetic background is produced [39][40][41][42][43][44][45][46][47][48][49][50]. Taking into account the case of heavy-ion collisions at RHIC and LHC, a notable dependence of phase diagram with the strength of magnetic field eH is estimated in the hadronic scale, i.e. eH ∼ 1 − 10 m 2 π (m π = 140 MeV is the pion mass). In this scenario, intriguing distinct phenomena appearing for different ranges of temperatures have been proposed recently by use of effective approaches: for sufficiently small magnetic field, magnetic catalysis (enhancement of broken phase) occurs at smaller temperatures, while the inverse magnetic catalysis effect (stimulation of the restoration of chiral symmetry) takes place at higher temperatures [43,45,[47][48][49][50]. It should be also noted that inverse magnetic catalysis can also occur even at T = 0, in the region of the phase diagram involving chemical potential versus eH for intermediate/high values of field strength.
Consequently, a natural discussion emerges about the combined effects on the phase structure of a thermal gas of quark matter (which might be described as a first approximation by the NJL model) restricted to a reservoir and under the effect of a magnetic background.
Hence, in view of these recent theoretical and experimental advances on the physics of strongly interacting matter, we believe that there is still room to other contributions. To this end, the main interest of this paper is to extend the analysis performed in Ref. [23]. In the present work we will address the questions raised above under the framework of a generalized version of NJL model. We investigate the finite-size effects on the phase structure of three-flavor NJL model with 't Hooft interaction, which engenders a six-fermion interaction term, without and with the presence of a magnetic background. We make use of mean-field approximation and Schwinger's proper-time method in a toroidal topology with antiperiodic conditions, manifested by the utilization of generalized Matsubara prescription for the imaginary time and spatial coordinate compactifications. To deal with the nonrenormalizable nature of the NJL model, the ultraviolet cut-off regularization procedure is employed.
The behavior of constituent quark masses M u , M d and M s , engendered by the solutions of gap equations, are studied under the change of the size L of compactified coordinates, temperature T and chemical potential µ. The 't Hooft interaction contributions are also evaluated.
We organize the paper as follows. In Section II, we calculate the (T, L, µ, H)-dependent effective potential and gap equations obtained from the three-flavor NJL model in the meanfield approximation, using Schwinger's proper-time method generalized Matsubara prescription. The phase structure of the system and the behavior of constituent quark masses M u , M d and M s are shown and analyzed in Section III, without and with the presence of a magnetic background. Finally, Section IV presents some concluding remarks.

A. Lagrangian and thermodynamics
Let us start by introducing the N f -flavor version of the NJL model, whose most commonly used Lagrangian density reads [51][52][53][54][55][56], where q is the quark field carrying N f flavors; L Mass denotes the mass term, withm = diag(m 1 , . . . , m N f ) being the corresponding mass matrix; L 4 represents the fourfermion-interaction term, with G being the respective coupling constant and λ a the generators of U(N f ) in flavor space; and L 6 designates the so-called 't Hooft interaction, with κ being the corresponding coupling constant.
We which appears in the QCD context from the gluon sector, in terms of a tree-level interaction in the present pure quark model. As discussed in Refs. [54][55][56], the 't Hooft interaction is phenomenologically relevant to yield the mass splitting of η and η ′ mesons. Moreover, it can be shown that in the chiral limit, m u = m d = m s = 0, the η ′ acquires a finite value of mass due to L 6 , whereas the other pseudoscalar mesons stay massless.
The focus of the present analysis concerns the lowest-order estimate of the phase structure of this model. In this sense, the calculations for obtaining relevant quantities will be performed within the mean-field (Hartree) approximation. To this end, the non-vanishing quark condensates, (i = u, d, s), are assumed to be the only allowed expectation values which are bilinear in the quark fields. In this approximation, the interaction terms in L NJL are linearized in the presence of φ i , that is, which means that terms quadratic in the fluctuations will be neglected. Moreover, terms in channels without condensate or nondiagonal in flavor space, like (qλ b q), b = 0, and (qiγ 5 λ a q), are are excluded. Therefore, in this context the NJL Lagrangian density can be rewritten where we have introduced M as the constituent quark mass matrix, which is diagonal in Notice that the constant terms in L MF have been neglected, since they give trivial contributions.
At this point we can present the thermodynamic potential density at temperature T and quark chemical potential µ, which is defined by where Z is the grand canonical partition function, β = 1/T , H the Hamiltonian density (the Euclidean version of Lagrangian density L MF ) and Tr the functional trace over all states of the system (spin, flavor, color and momentum). Thus, the integration over fermion fields generates the mean-field thermodynamic potential in the form where Ω M i (T, µ i ) is the free Fermi-gas contribution, Here the sum over n τ denotes the sum over the fermionic Matsubara frequencies, p 0 = iω nτ = (2n τ + 1) π/β; (n τ = 0, ±1, ±2, . . .).
Then, the minimization of the thermodynamic potential in Eq. (10) with respect to quark condensates allows us to obtain the following gap equations, The physical solutions of Eq. (11) are determined from the stationary points of the thermodynamic potential, which lead to the standard expression for the quark condensates, where N c = 3 andω We have assumed for simplicity the quark chemical potentials as µ i ≡ µ. Eq. (11) contains a non-flavor mixing term proportional to the coupling constant G (coming from the four-point interaction contribution) and a flavor-mixing term proportional to the coupling constant κ (coming from the six-point interaction). Hence, the role of the 't Hooft term is to engender flavor-mixing contributions in the constituent masses.

B. Generalized Matsubara prescription and proper-time formalism
To take into account finite-size effects on the phase structure of the model, we denote the Euclidean coordinate vectors by , with L j being the length of the compactified spatial dimensions. As a consequence, the Feynman rules explicited in the arguments of the sum-integral mixing in Eq. (12) must be replaced according to the generalized Matsubara prescription [57][58][59], i.e., such that where n τ , n α = 0, ±1, ±2, . . . Due to the KMS conditions [57,58,60,61], which determines the thermal Matsubara frequencies according to the field statistics, the fermionic nature of the system under study imposes an antiperiodic condition in the imaginary-time coordinate.
However, in the case of spatial compactified coordinates x j there are no restrictions with respect to the periodicity. So, the choice of the boundary conditions of the fermion fields in the spatial directions results in a relevant issue, especially because it affects the meson masses and other relevant observables, as claimed in studies mentioned in Introduction.
Current lattice QCD simulations usually employ periodic quark boundary conditions [37], thanks to its tendency to reduce the finite volume effects, favoring the obtention of the thermodynamic limit. Nevertheless, as has been supported by a large number of studies that make use of effective QCD models [10,23,24,37], the fields should take the same boundary condition in their spatial and temporal directions. One important aftermath of this selection is the symmetry in the formalism between the spatial and temporal directions, we can fit the model parameters at zero temperature and infinite volume and hold them in the finite temperature and volume analysis. Thus, following this assumption, we also adopt antiperiodic boundary conditions for the compactified spatial coordinates: b j in Eq. (15) assumes the value 1/2.
In the present work the thermodynamic potential and the gap equations will be treated within the Schwinger proper-time method [62,63]. Accordingly, the kernel of the propagator in Eq. (12) can be rewritten as where τ is the so-called proper time. Therefore, the application of Eq. (16) and the generalized Matsubara prescription (15) into (12) yields the quark chiral condensates expressed In addition, to perform the necessary manipulations in a relatively simple and more tractable way, it is convenient to employ the Jacobi theta functions [64]. Noticing the very useful property of the second Jacobi theta function, then the (T, L j , µ)-dependent chiral quark condensate in Eq. (17) can be expressed in the following way: It is worthy noticing that in order to regularize the proper-time approach, we introduce a ultraviolet cutoff Λ, namely, We conclude this section by remarking that the bulk form of the system can be studied straightforwardly. To do this, the continuum limit is performed by reverting the Matsubara prescription in Eq. (14), which yields in Eq. (19) Gaussian integrals in momentum space.
So, the expression for the chiral quark condensate acquires the form Hence, we have obtained above the thermodynamic potential and gap equations whose solutions produce the (T, L j , µ)-dependent constituent quark masses. Other thermodynamic quantities such as pressure, entropy and others can be obtained by similar procedures to those described above.
In the next section, we will discuss the thermodynamic behavior of the present model.
A fundamental consequence of presence of magnetic background is the modification of Feynman rules: the four-momentum integrals suffer the dimensional reduction, i.e.
When combined with finite-temperature and finite-size effects, the Euclidean coordinate vectors must be rewritten with x 0 ∈ [0, β] and x z ∈ [0, L], L being the size of the compactified spatial dimension. Then, taking the Matsubara prescription discussed in Eq. (14) the integration over momenta space is modified accordingly, Hence, the thermodynamic potential and gap equations given by Eqs. (10) and (11) must be analyzed keeping in mind the thermodynamic relations and chiral condensates with the modified integration over momenta as in Eq. (24). This means that the expressions for φ i in Eq. (19) should be replaced by which can be rewritten, after performing the proper time regularization procedure and the sum of geometrical series and spin polarization, as In the next section, we will discuss the (T, L j , µ, ω)-dependence of thermodynamic quantities introduced above.

III. PHASE STRUCTURE
We devote this section to the analysis of the phase structure of the system, focusing on how it behaves with the change of the relevant parameters of the model and, in special, the influence of the boundaries on the behavior of constituent quark masses M u and M s , which are solutions of expressions given by Eq. (11); explicitly, they are with M i = M i (T, L, µ, ω). We simplify the present study by fixing L 1 = L 2 = L 3 = L, which means that the system consists in a (u, d, s)-quark gas constrained in a cubic box.

A. Absence of magnetic field
Once the regularization procedure is chosen, one must determine the complete set of input parameters that provides a satisfactory description of hadron properties at zero temperature and density. In this sense, we use the model parameters reported in Ref. [21], which have The contribution of the 't Hooft interaction will be always taken into account unless explicitly stated otherwise, as in the case of Fig. 2, where the parameters have been refitted in order to obtain the same dressed masses at zero temperature and density that correctly reproduce the relevant observables. Moreover, since we work in the limit of isospin symmetry for current u and d quark masses, in absence of magnetic field the solutions of Eq.
Hence, throughout this subsection we will show and discuss the results keeping this fact in mind.
We start, for completeness, by studying the behavior of the constituent quark masses under a change of parameters but without the presence of boundaries, which is basically the scenario described in Refs. [56,65]. In Fig. 1  Therefore the broken phase is inhibited and a crossover transition takes place. Above a given temperature value, the dressed quark masses approach the magnitudes of the corresponding current quark masses, i.e. m u ≈ 7.0 MeV ; m s ≈ 195.6 MeV. It is also worth mentioning that the constituent mass for the s-quark has a smoother drop than the one for the u-quark, falling to the m s magnitude at larger T . Another feature is that at higher temperatures the system tends faster toward the symmetric chiral phase as the chemical potential increases. We now investigate in detail the influence of boundaries on the phase structure of the system, considering the (u, d, s)-quark gas in a region delimited by a cubic box. In Fig. 2 we pronounced than the situation of a vanishing κ. Also, we notice that below a given value of L, the dressed quark masses tend to the corresponding current quark masses. In particular, this happens at L ≈ 1-1.5 fm for u, d-quarks. So, analogously to the behavior described in the previous figures, M s has a smoother falloff to the current quark mass than M u . Thus, the dependence of the phase structure of the system on the inverse length 1/L is qualitatively similar to the one found for the temperature, which is expected, due to the equivalent nature between 1/L and T , manifested in the generalized Matsubara prescription in Eq (15).
The plot in Fig. 3 is the same as in Fig. 2 for the case κ = 0, but with the solutions have only one spatial compactified coordinate. We remember that in gap equations (27) of present situation we use a different expression for the condensates, i.e. Eq. (26) rather than (19), in which a dimensional reduction of four-momentum integrals in Feynman rules took place. The parameter set used is shown in Eq. (28), which gives the correct hadronic observables at T, µ, 1/L, ω = 0. respectively, which can be compared with Figs. 5 and 6. Again, they manifest the smaller values of constituent quark masses reached in the case of finite size and greater temperatures.
Since the magnetic field increases the constituent quark masses, the critical size L c at which M i stand with the corresponding values of current quarks masses in all range of temperature should be even smaller than the situation without magnetic field (0.5-1.0 fm).
We conclude this section by discussing some issues. We notice that the relevance of the parameters that provides a satisfactory description of hadron properties at T, µ, 1/L, ω = 0 (according to Ref. [21]), the comparison with existing literature only makes sense with those works that have proceeded in the same way.
Notwithstanding the points raised above, we stress the main results of this work as follows.
In the end, we see that the phase structure of the system is strongly affected by the combined variation of relevant variables, depending on the competition among their respective effects.
At smaller temperatures, with the system in the broken phase, the bulk approach seems a good approximation in the range of greater values of the size of the system L, since the constituent masses do not suffer modification. Nevertheless, keeping T fixed, the reduction of L from a given value engenders a reduction of M i via a crossover phase transition to their corresponding current quark masses. Yet at the same temperature and at the range We must emphasize, however, that in our understanding it seems more appropriate to adopt the same boundary condition in both spatial and temporal directions, as remarked in subsection II B and supported by a large number of studies using effective models [10,23,24,37].

IV. CONCLUDING REMARKS
In this work we have analyzed the finite volume and magnetic effects on the phase struc- For higher values of magnetic field strength, we remarked that the approximate isospin symmetry for the constituent masses of the u and d quarks is no longer valid, with M u being bigger than M d due to the larger coupling of u quark to magnetic background.
Finally, we emphasize that the results outlined above can give us insights about the finitevolume and magnetic effects on the critical behavior of quark matter produced in heavy-ion collisions. In particular, estimations have been done on the range of size of the system at which the bulk approximation looks a reasonable one. Further studies are necessary in order to verify the efficacy of the proposed framework.