Medium effect on anisotropic surface tension of magnetized quark matter

Over the years, many works have been dedicated to the effects of magnetic fields on the quantum chromodynamics (QCD) phase transition Bali:2011qj ; Kharzeev:2013jha and the equation of state in quark (neutron) stars Pal:1998jb ; Chakrabarty:1995uja ; Isayev:2011zz ; Menezes:2015fla ; Ferrer:2019xlr ; Chaudhuri:2022oru . The magnetic field modifies the microscopic properties of quark matter with the corresponding macroscopic implication in compact stars Chakrabarty:1996te ; Broderick:2000pe ; PerezMartinez:2005av ; Deb:2021ftm . It is well known that in the presence of a magnetic field, the anisotropic effect becomes significant and non-negligible in strong magnetic field, owing to the breaking of the spational rotational symmetry Canuto:1968apg ; Chaichian:1999gd ; Martinez:2003dz ; Felipe:2002wt ; Ferrer:2010wz ; Karmakar:2019tdp . To reflect the anisotropic structure with a rapid longitudinal expansion of QGP created in HICs, the anisotropic Coulomb potential can be produced through an angle-averaged screening mass Dong:2021gnb . Many theoretical works presented the analytic expression for anisotropic pressures under certain approximation Menezes:2015fla ; Ferrer:2019xlr ; Ferrer:2010wz ; Felipe:2007vb . Ferrer et al. showed the anisotropic pressure and estimated the threshold field that separates the isotropic and anisotropic regimes Ferrer:2010wz . Later, Isayev and Yang confirmed the splitting of the longitudinal and transverse pressure in their articles Isayev:2011zz ; Isayev:2011ug . The anisotropic pressure will affect the determination of the compressibility. The compressibility could manifest the anisotropic structure due to the breaking of the rotation symmetry. The discontinuity of longitudinal compressibility with the chemical potential and the temperature captures the signature of a first-order chiral phase transition Yang:2021rdo . With increasing temperature, the appearance of the longitudinal instability prevents the formation of a fully spin-polarized state in neutron matter and only the states with moderate spin polarization are accessible Isayev:2011zz . Recently, Lugones et. al. pushed the investigation of the surface tension in longitudinal and transverse components with respect to the magnetic field in the bag model Lugones:2018qgu ; Lugones:2016ytl ; Grunfeld:2020gnv . In this paper, our aim is to investigate the relevant anisotropic surface tension reflecting the breaking of the O(3) rotational symmetry in the deconfinement process.

In principle, the surface tension together with the QCD phase diagram should be investigated in the underlying fundamental theory, lattice QCD (LQCD). However, current LQCD methods are not sufficient to determine the matter structure at larger chemical potentials. The only available methods at relatively low energy are effective models. In literature, the phenomenological models overcome the difficulty of the QCD theory at finite chemical potentials. In order to interpret the chiral phase transition and dynamical symmetry breaking, the Nambu-Jona-Lasinio (NJL) model is widely used in the QCD-like investigation. The NJL model has proved to be very successful in the description of the spontaneous breakdown of chiral symmetry exhibited by the true (nonperturbative) QCD vacuum. It explains very well the spectrum of the low lying mesons which is intimately connected with chiral symmetry as well as many other low energy phenomena of strong interaction. The quark quasiparticle model, as an extended bag model, has been developed in studying the bulk properties of the dense quark matter at finite density and temperature. To describe the strong interaction effects in terms of effective fugacities, the effective fugacity quasiparticle model is proposed by Chandra and Ravisankar Chandra:2011en ; Chandra:2007ca . The advantage of the quasiparticle is the successful description of the confinement mechanism by the density- and/or temperature-dependent bag function, via which the first-order deconfining phase transition was constructed and the critical end point was determined Srivastava:2010xa . The aim of this work is to investigate the anisotropy of surface tension modified by the medium effect in strong magnetic fields. We also hope that the surface tension is helpful to investigate the deconfinement transition, since the surface tension is relevant for bubble nucleation of quark matter in supernovae Jimenez:2017fax . It can play an important role in the hadron-quark phase transition in the presence of a magnetic field when the anisotropic approach is followed as it was done recently Ferrer:2020tlz .

This paper is organized as follows. In Section II, we present the self-consistent thermodynamics of the magnetized quark matter in the quasiparticle model. The medium effect is included by introducing the effective bag function. The surface tension is modified by an additional term dependent on the bag function. In Section III, the numerical results for the confinement bag function and surface tension are shown in the strong magnetic field. The detailed discussions are focused on the anisotropy of the surface tension. The last section is a short summary.

The main purpose of this paper is to study the properties of the deconfined quark matter in strong magnetic fields. The nonzero quark masses are explored and the exact chiral symmetry are broken. The effective quasi-particle mass should be introduced to include the interaction effect in the quasiparticle approach. The total energy in the ensemble of quasiparticle as a free and degenerate Fermion gas can be written as

where $d$ denotes the degree of the degeneracy including the flavor, color and spin. The chemical potential-dependent bag function $B^{*}$ denotes the energy difference between the physical vacuum and the perturbative vacuum, which is necessary to ensure thermodynamic consistency.

For the medium dependence of the quark quasiparticle model, the effective quark mass $m_{i}^{*}$ is derived at the zero momentum limit of the dispersion relation following from the effective quark propagator by resuming one-loop self energy diagrams in the hard dense loop (HDL) approximation Schertler:1996tq . The in-medium effective mass of quarks can thus be expressed as Schertler:1996tq ; Schertler:1997vv ; Peshier:1999ww ; Bannur:2007tk ; Wen:2009zza

where $m_{i}$ is the current mass of corresponding quarks and the constant $g$ is related to the strong interaction constant $\alpha_{s}$ by the equation $g=\sqrt{4\pi\alpha_{s}}$. The quasi-particle idea can be recalled backward to the work by Fowler et.al.Fowler:1981rp that the particle mass may change with the environment parameters. Following the original ideas, the quark mass density dependent model is studied by Chakrabarty et.al.Chakrabarty:1989bq . Accordingly, as a phenomenological method, our quasiparticle model has a similar treatment. They are apparently different in approach but equally satisfactory in result. The in-medium screening mass in Eq.(2) is merely a model assumption on the quasiparticle mass in the present treatment, and can not be justified field-theoretically. The effective mass depends on the leading term of quark self-energy in the hard dense loop approximation. So the expression is only valid for a large chemical potential.

In order to investigate the finite size effect, we apply the multiple reflection expansion (MRE). It is originally proposed by Balian and Bloch in the distribution of eigenvalues of the wave equation inside the volume bounded by s closed surfaceBalian:1970fw . The eigenvalue density is smoothed by the asymptotic expansion to eliminate its fluctuation part due to the discrete eigenvalues on boundary condition. Later the MRE is developed in finite size quark matter by Madsen Madsen:1993ka ,Farhi and Jaffe Farhi:1984qu and Berger and Jaffe Berger:1986ps . In the MRE framework, the finite- size effects are considered in the modified density of state and the thermodynamic potential density is

where $T$ is the system temperature and $d_{i}=3$ is the color degeneracy factor for $i$-type quarks. The density of state for a spherical system is modified by the factor in the multi-expansion approach by Balian1970AP60

Here $x_{i}\equiv m_{i}^{*}/p$ is the ratio of the quark mass $m_{i}$ over the kinetic moment $p$. The dimensionless function $f_{\mathrm{S}}(x_{i})=-\frac{1}{4\pi^{2}}\mbox{arctan}(x_{i})$ would play an important role in the modification of the density of state Berger:1986ps ; Farhi1984PRD30 . In conventional form of MRE, the expansion includes the boundary surface and its curvature the density of states for a volume of arbitrary shape. In our work, the curvature term has no influence on our subject and is omitted. For the extremely relativistic particle with $m\ll p$, the density modification is not modified significantly Berger:1986ps . Generally, the modification of the density of state in the MRE framework constrain the low limit on the infrared cutoff due to the fact that $\rho_{\text{MRE}}$ becomes negative at small momenta Kiriyama:2005eh .

In the integrations of the thermodynamic potential, the following replacement should be applied

where the magnetic field strengths the degeneracy factor $|qB|$ together with the dimensional reduction. At zero temperature and strong magnetic fields, Eq.(3) is simplified as

where the single particle energy eigenvalue $E_{i}=\sqrt{{m_{i}^{*}}^{2}+p_{z}^{2}+2\nu_{i}|q_{i}B|}$ sensitively depends on the magnetic fields. The infrared cutoff of the momentum ${\Lambda_{\mathrm{IR}}}$ is required to obtain the non-negative density of state. At zero temperature, the upper limit $\nu_{i}^{\mathrm{max}}$ of the summation index $\nu_{i}$ can be understood from the positive value requirement on Fermi momentum and is defined by

In the present paper we take the current mass $m_{0}$=5.6 MeV and $\mu_{u}=\mu_{d}=\mu$ for isospin-symmetric quarks and the bag constant $B_{0}=(145$MeV)⁴ for bulk case. The main advantage of the quasiparticle model is the combination of the medium effect into both the effective quark mass and the bag function $B^{*}$. For the intensity of the field $eB_{m}<\mu^{2}$, the validity of the spherical geometry could be approximately available to some extent.

The chemical potential dependent bag function plays an important role in the description of the deconfinement transition. The bag function would play as a function of the chemical potential and the finite size. The magnetic field effect and the coupling constant are considered separately. In Fig. 1, the bag function decreases with the increase of the chemical potential, which indicates a signal of the deconfinement transition. As the coupling constant becomes larger and/or the magnetic field becomes stronger, the decreasing behavior of the bag function would happen at a smaller chemical potential, which indicates a critical chemical potential is similar to the temperature behavior characterized by the inverse magnetic catalysis effect Preis:2010cq . In Fig. 2, the effect of the finite size volume is shown on the bag function. The two horizontal dotted lines are the corresponding bag functions for bulk strange quark matter. The bag function increases as the spherical size decreases. As the spherical radius $R$ approach the infinite value, the bag function is gradually close to the constant $B_{0}$ for the bulk matter, which indicates the disappearance of the finite size effect.

At the larger chemical potential, a direct transition from the vacuum to quark matter happens possibly depending on the surface tension of bubble quark matter Fraga:2018cvr . The surface tension is plotted as a function of the magnetic field in Fig. 3. The chemical potential and the finite size are adopted as $\mu=300$ MeV and $V/S=10$ fm. The different coupling constants $g=1$, 2, and 3 are marked by the black, the red, and the blue curves from bottom to top. The longitudinal and transverse surface tensions are denoted by the solid and dashed curves respectively. It can be found that the longitudinal component increases with the magnetic field $eB_{m}$ and coupling constant $g$. In particular, the longitudinal surface tension is proportional to the magnetic field in the strong field limit. It can be simply understood that the magnetic field only has contribution to the coefficient in front of the integral of Eqs. (12) (13). On the contrary, the transverse surface tension would decrease as the increasing magnetic field. Furthermore, the transverse component feels the Landau level effect more sensitively, which results in a oscillation behavior. At weak magnetic field, the value of the surface tension is in agreement with the result estimated as the order as $(70~{}$MeV)³ Farhi:1984qu . As the magnetic field becomes much stronger, the anisotropy structure would be enhanced greatly.

In Fig. 4, the transverse surface tension is plotted as function of the magnetic field at four different chemical potentials $\mu=150$, 200, 250, 300 MeV and the fixed size $V/S=10$ fm. The oscillation behavior is shown clearly in the weaker magnetic field. The more quarks located in high Landau levels have a finite contribution to the motion perpendicular to the magnetic field. At the much stronger magnetic field, the transverse surface tension would drop down and vanish in the end, which can be understood that all quarks located in the LLL have no contribution to the transverse motion. It is suggested that the vanishing of the transverse surface tension is a signal of all charged particle occupied in the LLL. Moreover, the threshold value of the magnetic field for the vanishing of $\sigma^{\perp}$ would increase with the increase of the chemical potential.

To mimic a realistic magnetic field in compact stars, the magnetic field is recently suggested to increase polynomial instead of exponential as the chemical potential Dexheimer:2016yqu . In Fig. 5, the anisotropic surface tension is investigated as a function of the chemical potential. The nonuniform magnetic field $B_{\mathrm{m}}(\mu)$ and the fixed strength $B_{\mathrm{m}}(\mu_{0})$ are marked by the solid lines and the dashed lines, respectively. The initial point $\mu_{0}$ is associated to the surface chemical potential $\mu_{0}$ of compact stars Dexheimer:2016yqu ; Peterson:2023bmr . The parallel surface tension $\sigma^{\parallel}$ on left panel and transverse one $\sigma^{\perp}$ on right panel are calculated at different couplings $g=2$ and 3. The transverse surface tension is apparently smaller than the parallel one in the whole range of the chemical potential, which reflects the anisotropic structure. It should be emphasized that the transverse surface tension would decrease as the increasing the magnetic field at high densities, which is indicated in Fig. 4. However, the surface tension is enlarged by the increasing chemical potential. Therefore, the transverse surface tension is shown as an increasing function of the chemical potential. The tiny oscillation behavior occurs at low chemical potential due to the Landau level transition. The transverse surface tension under the magnetic field $B_{\mathrm{m}}(\mu)$ is smaller than that of the case $B_{\mathrm{m}}(\mu_{0})$. On the contrary, the parallel tension is larger and grows more rapidly under the the magnetic field profile $B_{\mathrm{m}}(\mu)$.

In this paper, the thermodynamics of magnetized quark matter in the finite size has been obtained in the quasiparticle model. The dense medium effect is included through the effective bag function, which depends on the chemical potential and the magnetic field. The bag function plays as an appropriately chosen vacuum energy constant ensuring thermodynamic consistency. On the other hand, it provides a measure for nonperturbative physics which cannot be described by the effective masses. Its variation would be helpful to investigate the deconfinement transition. As expected, it has been numerically shown that the bag function is a decreasing function of the chemical potential. The drop down behavior would be strengthened by the increase of both the magnetic field and the coupling interaction constant. It was verified that the bag function $B^{*}$ would gradually approach the bulk limit $B_{0}$ as the size becomes infinitely large. We have employed the extended quasiparticle model to the surface tension. It is found that the medium effect represented by the bag function would lead to an additional term in the surface tension. The anisotropy of the longitudinal and transverse surface tensions is enhanced by an increase of the magnetic field. The longitudinal surface tension is an increasing function of the magnetic field. But the transverse component would decrease and drop down to zero at extreme strong magnetic fields. The vanishing of transverse surface tension coincides with the condensation of all quarks in LLL. Finally, the new expression of the surface tension modified by the medium effect would be useful to generalize the current investigation on the bubble formation in the QCD transition. Last but no least, the infrared cutoff for avoiding the negative density of state is relevant with the confinement phenomenon. The more reasonable confinement mechanism should be produced by the self-consistent combination of the infrared cutoff and the bag constant in future work.