Finite-Volume and Magnetic Effects on the Phase Structure of the Three-Flavor Nambu–Jona-Lasinio Model

PHYSICAL REVIEW D 99, 076001 (2019)

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

Luciano M. Abreu* Instituto de Física, Universidade Federal da Bahia, 40170-115 Salvador, Bahia, Brazil † Emerson B. S. Corrêa Faculdade de Física, Universidade Federal do Sul e Sudeste do Pará, 68505-080 Marabá, Pará, Brazil ‡ Cesar A. Linhares Instituto de Física, Universidade do Estado do Rio de Janeiro, 20559-900 Rio de Janeiro, Rio de Janeiro, Brazil

Adolfo P. C. Malbouisson§ Centro Brasileiro de Pesquisas Físicas/MCTI, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil

(Received 15 January 2019; revised manuscript received 27 February 2019; published 5 April 2019)

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 the 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 ≈ 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 of constituent quark masses, inducing smaller sizes and greater temperatures at which the constituent quark masses drop to the respective current ones.

DOI: 10.1103/PhysRevD.99.076001

I. INTRODUCTION heavy-ion collisions [1,2]. The understanding of this rich phase structure via thermodynamic and transport properties In recent decades, we have witnessed experimental and continues to the forefront of endeavor of community [2–4]. theoretical progress in the assessment of strongly interact- In particular, one interesting feature on the phase structure ing matter under extreme conditions. One of the most of strongly interacting matter is the finite-volume effects. interesting predictions of its underlying theory, QCD, is There are estimations indicating that QGP-like systems that it experiences a phase transition to a deconfined state at yielded in heavy-ion collisions are of the order of units or sufficiently high temperatures. This mentioned state com- – posed of quarks and gluons is commonly referred to as the dozens offemtometer [5 7]. In this sense, the influence of the quark-gluon plasma (QGP), and it can now be observed in size of the system on its thermodynamic behavior and phase diagram has been widely analyzed in the literature through distinct and effective approaches, such as Dyson-Schwinger *[email protected] – † equations of QCD [8 10], the quark-meson model [11,12], [email protected] ‡ extensions and generalizations of the Nambu–Jona-Lasinio [email protected] § – – [email protected] (NJL) model [13 23], and others [24 38]. The main findings of these works point out that the finite volume, combined Published by the American Physical Society under the terms of with other relevant variables, acts on the thermodynamic the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to behavior of strongly interacting matter. the author(s) and the published article’s title, journal citation, In the papers cited above, important questions arise and DOI. Funded by SCOAP3. about how to have a detailed description of limited-size

2470-0010=2019=99(7)=076001(12) 076001-1 Published by the American Physical Society ABREU, CORRÊA, LINHARES, and MALBOUISSON PHYS. REV. D 99, 076001 (2019) physical systems, like the mentioned QGP. One example is generalized version of the NJL model. We investigate the the spontaneous symmetry breaking of chiral symmetry, finite-size effects on the phase structure of three-flavor which occurs for systems in the bulk approximation but is NJL model with ’t Hooft interaction, which engenders a disfavored for finite volume. Thus, this context gives rise to six-fermion interaction term, without and with the presence the natural debate about what the conditions in which an of a magnetic background. We make use of mean-field ideal bulk system seems a good approximation for confined approximation and Schwinger’s proper-time method in a systems are. In other words, one may wonder about the toroidal topology with antiperiodic conditions, manifested range of the size of the system V ¼ L3 in which the finite- by the utilization of a generalized Matsubara prescription for volume effects affect the phase structure and the sponta- the imaginary time and spatial coordinate compactifications. neous symmetry breaking, making modifications in the To deal with the nonrenormalizable nature of the NJL model, thermodynamical properties of quarkionic matter. the ultraviolet cutoff regularization procedure is employed. Especially, a very recent paper by Wang et al. [23] The behavior of constituent quark masses Mu, Md, and Ms, discusses the influence of the finite-volume effects on the engendered by the solutions of gap equations, is studied chiral phase transition of quark matter at finite temperature, under the change of the size L of compactified coordinates, based on a two-flavor NJL model approach with a proper- temperature T, and chemical potential μ.The’t Hooft time regularization and use of a stationary-wave condition. interaction contributions are also evaluated. It is found that when the cubic volume size L is larger than We organize the paper as follows. In Sec. II, we calculate 500 fm the chiral quark condensate is indistinguishable the ðT;L;μ;HÞ-dependent effective potential and gap from that at L ¼ ∞, which is far greater than the estimates equations obtained from the three-flavor NJL model in of the size of QGP produced in a laboratory and the lattice the mean-field approximation, using Schwinger’sproper- QCD simulations. Also, they found that when the space time method generalized Matsubara prescription. The phase size L is less than 0.25 fm the spontaneous symmetry structure of the system and the behavior of constituent quark breaking concept is no longer valid. masses Mu, Md,andMs are shown and analyzed in Sec. III, On the other hand, the importance of a magnetic back- without and with the presence of a magnetic background. ground on the phase diagram of strongly interacting matter Finally, Sec. IV presents some concluding remarks. has also been the subject of great interest. The motivation comes from the phenomenology of both heavy-ion colli- II. MODEL sions and compact stars, in which a strong magnetic back- A. Lagrangian and thermodynamics ground is produced [39–50]. Taking into account the case of N heavy-ion collisions at the Relativistic Heavy Ion Collider Let us start by introducing the f-flavor version of the and LHC, a notable dependence of the phase diagram with NJL model, the most commonly used Lagrangian density – the strength of magnetic field eH is estimated in the hadronic of which reads [51 56] 2 scale, i.e., eH ∼ 1–10mπ (mπ ¼ 140 MeV is the pion mass). L ¼ q¯ði=∂Þq þ L þ L4 þ L6; ð1Þ In this scenario, intriguing distinct phenomena appearing NJL Mass for different ranges of temperatures have been proposed q N L where is the quark field carrying f flavors; Mass recently by the use of effective approaches: for sufficiently denotes the mass term, small magnetic field, magnetic catalysis (enhancement of L ¼ −q¯ mq;ˆ ð Þ broken phase) occurs at smaller temperatures, while the Mass 2 inverse magnetic catalysis effect (stimulation of the restora- mˆ ¼ ðm ; …;m Þ tion of chiral symmetry) takes place at higher temperatures with diag 1 Nf being the corresponding mass [43,45,47–50]. It should be also noted that inverse magnetic matrix; L4 represents the four-fermion-interaction term, catalysis can also occur even at T ¼ 0, in the region of the N2 −1 phase diagram involving chemical potential vs eH for Xf 2 2 intermediate/high values of field strength. L4 ¼ G ½ðq¯λaqÞ þðqi¯ γ5λaqÞ ; ð3Þ Consequently, a natural discussion emerges about the a¼0 combined effects on the phase structure of a thermal gas of with G being the respective coupling constant and λa the quark matter (which might be described as a first approxi- generators of UðNfÞ in flavor space; and L6 designates the mation by the NJL model) restricted to a reservoir and so-called ’t Hooft interaction, under the effect of a magnetic background. Hence, inview of these recent theoretical and experimental L6 ¼ −κ½det ðq¯ð1 − γ5ÞqÞþdet ðq¯ð1 þ γ5ÞqÞ; ð4Þ advances on the physics of strongly interacting matter, we believe that there is still room for other contributions. To this with κ being the corresponding coupling constant. end, the main interest of this paper is to extend the analysis We remark some features of the model introduced above. performed in Ref. [23]. In the present work, we will address The version of the NJL model presented here is generalized the questions raised above under the framework of a to any number of Nf flavors, but we are particularly

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N ¼ 3 L interested in the case of f . This means that the quark in MF have been neglected, since they give trivial field has its components related to the lightest quark flavors: contributions. T q ¼ðu; d; sÞ . This allows us to identify the generators λa of At this point, we can present the thermodynamic T Uð3Þ in flavor space as follows:pffiffiffiffiffiffiffiffi ðλ1; …; λ8Þ are the Gell- potential density at temperature and quark chemical μ Mann matrices, and λ0 ¼ 2=3I3. Also, the corresponding potential , which is defined by mˆ ≡ ðm ;m ;m Þ mass matrix becomes diag u d s . Besides, T henceforth, the isospin symmetry on the Lagrangian level ΩðT;μÞ¼− Z V ln is assumed, i.e., mu ¼ md, while SUð3Þ-flavor symmetry is Z m ≠ m 1 explicitly broken, generating s u. ¼ − ln Tr exp −β d3xðH − μq†qÞ ; ð8Þ From the perspective of UðNfÞL × UðNfÞR symmetry in βV Lagrangian terms, we see that L4 is a symmetric four-point interaction. On the other hand, the ’t Hooft interaction L6 is where Z is the grand canonical partition function, β ¼ 1=T, a determinant in flavor space, which means that it is a H is the Hamiltonian density (the Euclidean version of L flavor-mixing 2Nf-point interaction, involving an incom- Lagrangian density MF), and Tr is the functional trace ing and an outgoing quark of each flavor. Thus, for three over all states of the system (spin, flavor, color, and flavors, it engenders a six-fermion-interaction term and is momentum). Thus, the integration over fermion fields SUðNfÞL × SUðNfÞR symmetric but breaks the UAð1Þ generates the mean-field thermodynamic potential in the symmetry, which was left unbroken by L4. Thus, L6 form expresses the UAð1Þ anomaly, which appears in the ΩðT;μÞ¼2Gðϕ2 þ ϕ2 þ ϕ2Þ − 4κϕ ϕ ϕ QCD context from the gluon sector, in terms of a tree- Xu d s u d s level interaction in the present pure quark model. As þ Ω ðT;μ Þ; ð Þ Mi i 9 discussed in Refs. [54–56], the ’t Hooft interaction is i¼u;d;s phenomenologically relevant to yield the mass splitting of η 0 and η mesons. Moreover, it can be shown that in the chiral where ΩM ðT;μiÞ is the free Fermi-gas contribution, m ¼ m ¼ m ¼ 0 η0 i limit, u d s , the acquires a finite value of Z L X 3 mass due to 6, whereas the other pseudoscalar mesons 1 d p 0 ΩM ðT;μiÞ¼− tr ln½p=1i − μiγ − Mi: ð10Þ stay massless. i β ð2πÞ3 The focus of the present analysis concerns the lowest- nτ order estimate of the phase structure of this model. In this Here, the sum over nτ denotes the sum over the fermionic sense, the calculations for obtaining relevant quantities will 0 Matsubara frequencies, p ¼ iωn ¼ð2nτ þ 1Þπ=β; be performed within the mean-field (Hartree) approxima- τ ðn ¼ 0; 1; 2; …Þ tion. To this end, the nonvanishing quark condensates, τ . Then, the minimization of the thermodynamic potential in Eq. (10) with respect to quark condensates allows us to ϕ ≡ hq¯ q iðÞ i i i 5 obtain the following gap equations:

(i ¼ u, d, s), are assumed to be the only allowed expect- Mi ¼ mi − 4Gϕi þ 2κϕjϕkði ≠ j ≠ kÞ: ð11Þ ation values which are bilinear in the quark fields. In this L approximation, the interaction terms in NJL are linearized The physical solutions of Eq. (11) are determined from the in the presence of ϕi, that is, stationary points of the thermodynamic potential, which lead to the standard expression for the quark condensates, ðqq¯ Þ2 ≃ 2ϕ ðq¯ q Þ − ϕ2; ð Þ Z i i i i 6 1 X d3p 1 ϕ ≡ hq¯ q i¼−4N M ; i i i c i 3 2 2 2 β ð2πÞ ω˜ n þ p⃗ þ Mi which means that terms quadratic in the fluctuations will be nτ τ neglected. Moreover, terms in channels without a conden- ð12Þ sate or nondiagonal in flavor space, like ðq¯λbqÞ, b ≠ 0, and ðqi¯ γ5λaqÞ, are excluded. Therefore, in this context, the NJL where Nc ¼ 3 and Lagrangian density can be rewritten as 2π 1 μβ 2 2 2 ω˜ ¼ n þ − i : ð Þ L ¼ q¯ði=∂ − MÞq − 2Gðϕ þ ϕ þ ϕ Þþ4κϕ ϕ ϕ ; ð Þ nτ τ 13 MF u d s u d s 7 β 2 2π where we have introduced M as the constituent We have assumed for simplicity the quark chemical quark mass matrix, which is diagonal in flavor space: potentials as μi ≡ μ. Equation (11) contains a non–flavor M ≡ diagðMu;Md;MsÞ. Notice that the constant terms mixing term proportional to the coupling constant G

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(coming from the four-point interaction contribution) and a However, in the case of spatial compactified coordinates flavor-mixing term proportional to the coupling constant κ xj, there are no restrictions with respect to the periodicity. (coming from the six-point interaction). Hence, the role of So, the choice of the boundary conditions of the fermion the ’t Hooft term is to engender flavor-mixing contributions fields in the spatial directions results in a relevant issue, in the constituent masses. especially because it affects the meson masses and other relevant observables, as claimed in studies mentioned in the B. Generalized Matsubara prescription and Introduction. Current lattice QCD simulations usually proper-time formalism employ periodic quark boundary conditions [37], thanks To take into account finite-size effects on the phase to tendency to reduce the finite-volume effects, favoring the structure of the model, we denote the Euclidean coordinate obtention of the thermodynamic limit. Nevertheless, as has vectors by xE ¼ðxτ;x1;x2;x3Þ, where xτ ∈ ½0; β and been supported by a large number of studies that make use of xj ∈ ½0;Ljðj ¼ 1; 2; 3Þ, with Lj being the length of the effective QCD models [10,23,24,37], the fields should take compactified spatial dimensions. As a consequence, the the same boundary condition in their spatial and temporal Feynman rules explicited in Eq. (12) must be replaced directions. One important aftermath of this selection is the according to the generalized Matsubara prescription [57– symmetry in the formalism between the spatial and temporal 59], i.e., directions, through the exchange Li ↔ β, causing T;Li- G κ Z independent coupling constants and . So, we can fit the 1 X∞ d3p model parameters at zero temperature and infinite volume fðω˜ ; p⃗Þ β 3 nτ and hold them in the finite-temperature and -volume n ¼−∞ ð2πÞ τ analysis. Thus, following this assumption, we also adopt ∞ 1 X antiperiodic boundary conditions for the compactified → fðω˜ n ; ω¯ n ; ω¯ n ; ω¯ n Þ; ð14Þ βL L L τ 1 2 3 spatial coordinates: bj in Eq. (15) assumes the value 1=2. 1 2 3 n ;n ;n ;n ¼−∞ τ 1 2 3 In the present work, the thermodynamic potential and the such that gap equations will be treated within the Schwinger proper- time method [62,63]. Accordingly, the kernel of the 2π propagator in Eq. (12) can be rewritten as p → ω¯ ≡ ðn þ b Þ; ð Þ j nj j j 15 Lj Z ∞ 1 2 2 2 ¼ dτ exp½−τðω˜ n þ p⃗ þ M Þ; ð16Þ ω˜ 2 þ p⃗2 þ M2 τ i where nτ, nα ¼ 0; 1; 2; … Because of the Kubo-Martin- nτ i 0 Schwinger conditions [57,58,60,61], which determine the thermal Matsubara frequencies according to the field where τ is the so-called proper time. Therefore, the statistics, the fermionic nature of the system under study application of Eq. (16) and the generalized Matsubara imposes an antiperiodic condition in the imaginary-time prescription (15) into (12) yields the quark chiral con- coordinate. densates, expressed as

Z ∞ Xþ∞ 4NcMi 2 2 2 2 2 ϕi ¼ − dτ exp½−τðMi − μ Þ exp ½−τð4π =β Þðnτ þ 1=2Þ exp½ið2nτ þ 1Þ2πμτ=β βLxLyLz 0 nτ;nx;ny;nz¼−∞ 2 2 2 2 2 2 2 2 2 × exp ½−τð4π =LxÞðnx þ 1=2Þ exp ½−τð4π =LyÞðny þ 1=2Þ exp ½−τð4π =Lz Þðnz þ 1=2Þ : ð17Þ

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, θ2½b; q,

Xþ∞ Xþ∞ 2 1=4 nðnþ1Þ exp ½−aðn þ 1=2Þ exp ½ið2n þ 1Þb¼2½expð−aÞ ½expð−aÞ cos ½ð2n þ 1Þb¼θ2½b; expð−aÞ; ð18Þ n¼−∞ n¼0 then, the ðT;Lj; μÞ-dependent chiral quark condensate in Eq. (17) can be expressed in the following way: Z ∞ 4NcMi 2 2 2πμτ 2 2 ϕiðT;Lj; μÞ¼− dτ exp½−τðMi − μ Þθ2 ; expð−4π τ=β Þ βL1L2L3 0 β 2 2 2 2 2 2 × θ2½0; expð−4π τ=L1Þθ2½0; expð−4π τ=L2Þθ2½0; expð−4π τ=L3Þ: ð19Þ

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It is worth noticing that, in order to regularize the proper- magnetic field in the z direction. Therefore, the field time approach, we introduce an ultraviolet cutoff Λ, namely, operators are given by the set of normalized eigenfunctions Z Z of the Landau basis, which means that the energy eingen- ∞ ∞ fðτÞdτ → fðτÞdτ: ð20Þ values associated to the solutions of the equation of motion 0 1=Λ2 read

2 2 2 We conclude this section by remarking that the bulk form Ei ¼ pz þ Mi þjQijωð2l þ 1 − sÞ; ð22Þ of the system can be studied straightforwardly. To do this, the continuum limit is performed by reverting the Matsubara where ω ≡ eH is the so-called cyclotron frequency; s ¼ prescription in Eq. (14), which yields in Eq. (19) Gaussian 1 is the spin variable; and l ¼ 0; 1; 2; …, denotes the so- integrals in momentum space. So, the expression for the called Landau levels. chiral quark condensate acquires the form A fundamental consequence of the presence of magnetic ϕ ðT;L → ∞; μÞ background is the modification of Feynman rules: the four- i j Z momentum integrals suffer the dimensional reduction, i.e., N M ∞ dτ → − pcffiffiffiffiffii ½−τðM2 − μ2Þθ Z Z 3 3=2 exp i 2 4 X X∞ 2 2 π β 1=Λ2 τ d p jQijω d p fðpÞ → fðp ;p ; l; sÞ: ð2πÞ4 2π ð2πÞ2 0 z 2πμτ s¼ l¼0 × ; expð−4π2τ=β2Þ : ð21Þ β ð23Þ Hence, we have obtained above the thermodynamic When combined with finite-temperature and finite-size potential and gap equations of which the solutions produce effects, the Euclidean coordinate vectors must be rewritten the ðT;Lj; μÞ-dependent constituent quark masses. Other with x0 ∈ ½0; β and xz ∈ ½0;L, L being the size of the thermodynamic quantities such as pressure, entropy, and compactified spatial dimension. Then, taking the Matsubara others can be obtained by similar procedures to those prescription discussed in Eq. (14), the integration over described above. momenta space is modified accordingly, In the next section, we will discuss the thermodynamic behavior of the present model. Z d4p fðpÞ 4 C. Presence of a magnetic background ð2πÞ XX∞ X∞ Considering now the system in the presence of a uniform jQijω 1 → fðω˜ n ;ω¯ n ;jQijω;l;sÞ: ð24Þ 2π βL τ z external magnetic field, we redefine the Lagrangian density s¼ l¼0 n ;n ¼−∞ in Eq. (1) with the derivative in kinetic term replaced by the τ z D˜ ¼ ∂ þ iQAˆ ext modified covariant derivative, μ μ μ , where Hence, the thermodynamic potential and gap equations Aext μ denotes the 4-potential associated to the magnetic given by Eqs. (10) and (11) must be analyzed, keeping in ˆ background and Q is the quark charge electric matrix, mind the thermodynamic relations and chiral condensates ˆ Q¼diag½ðQu;Qd;QsÞe, with Qu ¼ −2Qd ¼ −2Qs ¼ 2=3. with the modified integration over momenta as in Eq. (24). Aμ ¼ ϕ We choose for convenience the Landau gauge, ext This means that the expressions for i in Eq. (19) should be ð0; −Hx2; 0; 0Þ, where H is the intensity of the external replaced by

Z 4N M jQ jω X X∞ ∞ 2πμτ ϕ ðT;L;μ; ωÞ¼− c i i dτ ½−τðM2 − μ2Þθ ; ð−4π2τ=β2Þ i 2πβL exp i 2 β exp z s¼ l¼0 0 2 2 × θ2½0; expð−4π τ=Lz Þ exp ½−τjQijωð2l þ 1 − sÞ; ð25Þ which can be rewritten, after performing the proper-time regularization procedure and the sum of geometrical series and spin polarization, as Z ∞ 4NcMijQijω 2 2 2πμτ 2 2 ϕiðT;L;μ; ωÞ¼− dτ exp½−τðMi − μ Þθ2 ; expð−4π τ=β Þ 2πβL 1=Λ2 β z 2 2 exp ð2jQijωτÞ 1 × θ2½0; expð−4π τ=Lz Þ þ : ð26Þ −1 þ exp ð2jQijωτÞ −1 þ exp ð2jQijωτÞ

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In the next section, we will discuss the ðT;Lj; μ; ωÞ 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, especially, the influence of the boundaries on the behavior of constituent quark masses Mu and Ms, which are solutions of expressions given by Eq. (11); explicitly, they are

Mu ¼ mu − 4Gϕu þ 2κϕdϕs;

Md ¼ md − 4Gϕd þ 2κϕsϕu;

Ms ¼ ms − 4Gϕs þ 2κϕuϕd; ð27Þ with Mi ¼ MiðT;L;μ; ωÞ. We simplify the present study by fixing L1 ¼ L2 ¼ L3 ¼ L, which means that the system consists of a ðu; d; sÞ-quark gas constrained in a cubic box.

A. Absence of magnetic field FIG. 1. Constituent quark masses Mu (top panel) and Ms Once the regularization procedure is chosen, one must (bottom panel) as functions of the temperature T, in the bulk determine the complete set of input parameters that limit ðL → ∞Þ for different values of chemical potential μ. 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 functions of the temperature T, in the bulk limit ðL → ∞Þ been estimated by the fitting in light of the following and taking different values of the chemical potential μ.It m ¼ 138 f ¼ 92 m ¼ observables: π MeV, π MeV, K can be seen that the set of parameters given by Eq. (28) 495 MeV, and mη0 ¼ 958 MeV. Our choice of the set is engenders the constituent quark masses Mu ≈ 216 MeV the following: and Ms ≈ 451 MeV at vanishing temperature. In the range of smaller temperatures, there is no relevant modification. m ≈m ¼ 7 0 ;m¼ 195 6 ; Λ ¼ 924 1 ; u d . MeV s . MeV . MeV This phenomenon remains up to a certain temperature, GΛ2 ¼ 3.059; κΛ5 ¼ 85.50: ð28Þ where the masses start to decrease with the augmentation of T. Therefore, the broken phase is inhibited, and a crossover The contribution of the ’t Hooft interaction will be always transition takes place. Above a given temperature value, the taken into account unless explicitly stated otherwise, as dressed quark masses approach the magnitudes of the in the case of Fig. 2, in which the parameters have been corresponding current quark masses, i.e., mu ≈ 7.0 MeV; refitted in order to obtain the same dressed masses at zero ms ≈ 195.6 MeV. It is also worth mentioning that the temperature and density that correctly reproduce the constituent mass for the s quark has a smoother drop than relevant observables. Moreover, since we work in the limit the one for the u quark, falling to the ms magnitude at larger of isospin symmetry for current u and d quark masses, in T. Another feature is that at higher temperatures the system the absence of magnetic field, the solutions of Eq. (27) tends faster toward the symmetric chiral phase as the satisfy Mu ≈ Md. Hence, throughout this subsection, we chemical potential increases. will show and discuss the results while keeping this fact We now investigate in detail the influence of boundaries in mind. on the phase structure of the system, considering the We start, for completeness, by studying the behavior of ðu; d; sÞ-quark gas in a region delimited by a cubic box. the constituent quark masses under a change of parameters In Fig. 2, we have plotted the values of Mi that are solutions but without the presence of boundaries, which is basically of the gap equations in Eq. (27) as a function of the the scenario described in Refs. [56,65]. In Fig. 1,wehave inverse length 1=L, at vanishing temperature and chemi- plotted the values of Mu (top panel) and Ms (bottom panel) cal potential. For the sake of comparison with the that are solutions of the gap equations in Eq. (27) as existing literature (e.g., Refs. [23,56,65]), we have taken

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FIG. 3. Constituent quark masses Mu and Ms as functions of the inverse length 1=L at vanishing temperature and chemical potential, taking the chiral limit for the u, d quarks, i.e., setting mu ¼ 0.

different decreasing more 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, analo- gously to the behavior described in the previous figures, FIG. 2. Constituent quark masses Mu (top panel) and Ms Ms has a smoother falloff to the current quark mass than (bottom panel) as functions of inverse of length L at vanishing Mu. Thus, the dependence of the phase structure of the temperature and chemical potential. we have taken into account system on the inverse length 1=L is qualitatively similar the situations with and without the ’t Hooft interaction term (i.e., to the one found for the temperature, which is expected, κ ≠ 0 or κ ¼ 0, respectively). due to the equivalent nature between 1=L and T, manifested in the generalized Matsubara prescription in Eq (15). ’ into account the situations with and without the tHooft The plot in Fig. 3 is the same as in Fig. 2 for the case κ ≠ 0 κ ¼ 0 interaction term (i.e., and , respectively). κ ≠ 0, but with the solutions for Mu and Ms obtained at κ Note that in the case of a vanishing the parameters in vanishing temperature and chemical potential taking the Λ Eq (28) have been readjusted, keeping the values of chiral limit for the u, d quarks, i.e., setting mu ¼ 0. Once m G m and u fixed and varying and s to obtain the same more, as in the previous situation, in the chiral limit, we dressed masses estimated at zero temperature and density have refitted the parameters to give a good prescription of κ ≠ 0 for the situation with . Then, in this context, we the hadronic observables at zero temperature and density worked with the set and in the bulk. Therefore, keeping the value of Λ and ms fixed and varying G and κ to obtain the same dressed T ¼ μ ¼ 1=L ¼ 0 mu ≈ md ¼ 7.0 MeV;ms ¼ 167.8 MeV; masses estimated at , we have employed the set Λ ¼ 924.1 MeV;GΛ2 ¼ 3.900; κ ¼ 0: ð29Þ

mu ≈ md ¼ 0 MeV; ms ¼ 195.6 MeV; We can observe that the bulk approach appears as a good Λ ¼ 924.1 MeV; GΛ2 ¼ 2.903; κΛ5 ¼ 114.64: approximation in the range of greater values of L (up to 3–4 fm). The dressed masses are affected by the pres- ð30Þ ence of boundaries as the size of the system decreases, acquiring smaller values smoothly. In this region of Again, the restoration of chiral symmetry is induced at lower L, we see that the presence of the ’tHooft smaller sizes of the system, lost in the bulk. In this context, interaction, which engenders a system of coupled gap the results suggest the existence of a critical volume Vc ¼ 3 3 3 3 equations shown in Eq. (27), keeps the essential features ðLcÞ ≈ ð1.5Þ − ð2.0Þ fm at which the constituent mass unchanged when the parameter set is properly chosen to for the u quark reaches a zero value via a second-order reproduce the masses Mi in the bulk. Only in the case phase transition. On the other hand, Ms goes down to the of Ms do the contributions coming from κ ≠ 0 yield a value of ms softly.

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M M FIG. 4. Constituent quark masses u (top panel) and s FIG. 5. Constituent quark masses Mu (top panel) and Ms 1=L (bottom panel) as functions of the inverse length at (bottom panel) as functions of temperature T at L ¼ 1 fm, taking T ¼ 150 μ MeV, for different values of the chemical potential . different values of chemical potential μ.

Let us concentrate at this point on the evaluation of B. Magnetic background effects the combined effects of boundaries, finite temperature, and Now, we investigate the combined effects of finite finite chemical potential on the system. They can be better volume and magnetic background on the phase structure described from Fig. 4, in which we plotted the values of of the thermal gas of quark matter, taking into account that effective masses that are solutions of the gap equation in here we have only one spatial compactified coordinate. We Eq. (27) as function of inverse length x ¼ 1=L for remember that in gap equations (27) of the present situation different values of μ, keeping the temperature fixed at we use a different expression for the condensates, i.e., 150 MeV. As suggested in previous figures, in the bulk Eq. (26) rather than (19), in which a dimensional reduction (1=L → 0 or L → ∞), Mu and Ms suffer a decrease with of four-momentum integrals in Feynman rules took place. augmentation of chemical potential and drop of L, tending The parameter set used is shown in Eq. (28), which gives to the respective current quark masses. Also, at smaller L, the correct hadronic observables at T, μ, 1=L, ω ¼ 0. the dressed masses associated to different values of μ Furthermore, noticing that the coupling of the quarks converge to the values of the corresponding current quark to the electromagnetic field depends on the quark flavor masses. (see the discussion in Sec. II C), then, in principle, we This analysis is completed with Figs. 5 and 6, in which cannot adopt the solutions of Eq. (27) satisfying Mu ≈ Md, we plotted the values of Mu and Ms as functions of the as was done previously. That being so, we solve the system temperature T, taking different values of chemical potential of three gap equations in (27) to assess the differences μ and size L, respectively. When compared to Fig. 1, the between Mu and Md engendered by the coupling to plots in Fig. 5 manifest the smaller values of constituent magnetic background. quark masses reached in the case of finite volume. Figure 6 In Fig. 7, we plotted the values of constituent quark shows the values of the size of the cubic box in which Mu masses as functions of the inverse of thickness x ¼ 1=L for and Ms stand with the corresponding values of current different values of ω, keeping the temperature and chemical quarks masses in all ranges of temperature. Hence, our potential fixed. For higher values of the considered range of findings suggest that the presence of boundaries disfavors magnetic field strength, we remarked that the approximate the maintenance of long-range correlations, inducing the isospin symmetry for the constituent masses of the u and d inhibition of the broken phase. quarks is broken, with Mu being bigger than Md due to the

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FIG. 6. Constituent quark masses Mu (top panel) and Ms (bottom panel) as functions of the temperature T at μ ¼ 100 MeV, taking different values of size L. larger coupling of the u quark to magnetic background. Also, we see that in the bulk (x → 0 or L → ∞) the presence of an external magnetic field engenders the augmentation of Mi. In other words, the rise of field strength enhances the broken phase. This feature is expected and called in the literature a magnetic catalysis effect. This phenomenon remains in all considered ranges of L. On the other hand, as previously, the reduction of L FIG. 7. Constituent quark masses Mu (top panel), Md (middle engenders a reduction of constituent masses, but more panel), and Ms (bottom panel) as functions of the inverse of slightly than the situation without magnetic field. Also, in length 1=L at T, μ ¼ 0 MeV, taking different values of external the range of smaller values of L, the dressed masses magnetic field. associated to different values of ω converge to the values of corresponding current quark masses. The interesting point is that the growth of the field strength induces smaller the constituent quark masses, the critical size Lc at which values for L at which the system stands with the values mi. Mi stand with the corresponding values of current quarks Thus, the combination of finite-size and magnetic effects masses in all range of temperatures should be even smaller generates a competition between them, since the former than the situation without magnetic field (0.5–1.0 fm). inhibits the broken phase whereas the last one yields its We conclude this section by discussing some issues. stimulation. We notice that the relevance of the regularization proce- We complement this analysis with Figs. 8 and 9,in dure and parametrization in effective models cannot be which we plotted the values of Mi as functions of the underestimated. The findings obtained above are naturally temperature T, taking different values of field strength ω dependent on the set of parameters considered as input in and size L, respectively, which can be compared with Eq. (27). Modifications in these choices will obviously Figs. 5 and 6. Again, they manifest the smaller values of alter the magnitudes of the constituent quark masses and constituent quark masses reached in the case of finite size the ranges of (T, L, μ, ω), which engender changes on and greater temperatures. Since the magnetic field increases their behavior, making them drop faster or slower and

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M M FIG. 8. Constituent quark masses Mu (top panel), Md (middle FIG. 9. Constituent quark masses u (top panel), d (middle M T panel), and Ms (bottom panel) as functions of the temperature T panel), and s (bottom panel) as functions of the temperature μ ¼ 0 at μ ¼ 0 MeV, taking different values of external magnetic field at MeV, taking different values of external magnetic field z L ¼ 1 for the finite size in the z fixed at L ¼ 2 fm. for the finite size in the fixed at fm. accordingly reshape the curves plotted in previous figures. bulk approach seems a good approximation in the range Thus, since in the present approach we have chosen the of greater values of the size of the system L, since set of input parameters that provides a satisfactory descrip- the constituent masses do not suffer modification. tion of hadron properties at T, μ, 1=L, ω ¼ 0 (according to Nevertheless, keeping T fixed, the reduction of L from a Ref. [21]), the comparison with existing literature only given value engenders a reduction of Mi via a crossover makes sense with those works that have proceeded in the phase transition to their corresponding current quark same way. masses. Yet at the same temperature and at the range Notwithstanding the points raised above, we stress the L

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Finally, we should point out the combined effect of an The thermodynamic behavior is strongly affected by the external magnetic field and the choice of boundary combined effects of relevant variables, depending on the conditions in this model. The boundary effect in the range of their change. context of the antiperiodic condition has been studied in Looking at the influence of a finite volume, we can detail and summarized in the previous paragraph. Then, remark the main results of this work. At smaller temper- the contribution of the compactified dimension length in atures, with the system in the broken phase, the bulk the situation of periodic boundary conditions deserve approach seems a good approximation in the range of some comments. After the compactification of the z greater values of the volume of the system V ¼ L3, since coordinate in the energy eigenvalues in Eq. (22),it the constituent masses are not modified. Nevertheless, can be seen that for the periodic condition we have while keeping T fixed, there is a range in which the p → ω¯ ≡ 2πn =L the prescription z nz z z, and no energy gap reduction of L (remarkably at L ≈ 0.5–3 fm) engenders a is yielded between the vacuum and the lowest Landau reduction of the constituent masses for u, d, s quarks level l ¼ 0 at zero momentum n ¼ 0 in the chiral limit. through a crossover phase transition to the their corre- This makes the vacuum condensation of fermion pairs in sponding current quark masses. After all, the broken the infrared range of spectrum possible (see Ref. [13] for phase is disfavored due to the increasing of both the a detailed discussion). In this sense, the presence of temperature and chemical potential, and the drop of L. magnetic field catalyzes the dynamical breaking of chiral However, in the region in which the broken phase is symmetry. Another perspective to see this is looking at inhibited, the current quark masses are not affected by the chiral condensate in Eq. (26) and gap equations in the change of variables. (27), in which the Jacobi θ function θ2 must be replaced Looking at the magnetic background influence, only by the θ3 in the scenario of the periodic condition. phenomena associated to the magnetic catalysis take place; Thanks to the different behaviors of the θ functions, the an enhancement of the broken phase occurs in all consid- consequence of this replacement is relevant: while in the ered ranges of temperature and size. When a magnetic antiperiodic case, the magnetic catalysis is inhibited, with background is present, constituent quark masses acquire the length L playing a role similar to the inverse of greater values, and the increase of the field strength induces β temperature , in the periodic situation, the boundary has smaller values for L and greater values of T at which the Mi the opposite effect (symmetry breaking is enhanced, and tend to the respective values of current quark masses. For accordingly the constituent quark masses Mi acquire higher values of magnetic field strength, we remarked that higher values as L diminishes). Thus, there is not a the approximate isospin symmetry for the constituent critical value of the size Lc at which the symmetry is masses of the u and d quarks is no longer valid, with restored, and the combined effect of magnetic back- Mu being bigger than Md due to the larger coupling of the u ground and boundary conditions in the periodic case quark to magnetic background. strengthens the broken phase. Finally, we emphasize that the results outlined above We must emphasize, however, that in our understanding can give us insight about the finite-volume and magnetic it seems more appropriate to adopt the same boundary effects on the critical behavior of quark matter produced in condition in both the spatial and temporal directions, as heavy-ion collisions. In particular, we have been estimated remarked in Sec. II B and supported by a large number of the range of size of the system at which the bulk studies using effective models [10,23,24,37]. approximation looks reasonable. Further studies are necessary in order to verify the efficacy of the proposed IV. CONCLUDING REMARKS framework. In this work, we have analyzed the finite-volume and magnetic effects on the phase structure of a generalized ACKNOWLEDGMENTS version of the Nambu–Jona-Lasinio model with three quark flavors. By making use of mean-field approxima- The authors would like to thank the Brazilian funding tion and the Schwinger proper-time method in a toroidal agencies CNPq and CAPES. L. M. A. would like to thank the topology with antiperiodic conditions, we have inves- Brazilian funding agencies CNPq (Contracts No. 308088/ tigated the gap equation solutions under the change of 2017-4 and No. 400546/2016-7) and FAPESB (Contract the size of compactified coordinates, temperature, chemi- No. INT0007/2016) for financial support. E. B. S. C. would cal potential, and strength of external magnetic field. like to thank E. Cavalcanti for helpful discussions.

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