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Advances in High Energy Physics

Volume 2017 (2017), Article ID 6472909, 11 pages

https://doi.org/10.1155/2017/6472909

## Effect of Strong Magnetic Field on Competing Order Parameters in Two-Flavor Dense Quark Matter

^{1}Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden^{2}Department of Physics and Astronomy, California State University Long Beach, Long Beach, CA 90840, USA

Correspondence should be addressed to Tanumoy Mandal

Received 29 November 2016; Accepted 9 January 2017; Published 14 February 2017

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2017 Tanumoy Mandal and Prashanth Jaikumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the effect of strong magnetic field on competing chiral and diquark order parameters in a regime of moderately dense quark matter. The interdependence of the chiral and diquark condensates through nonperturbative quark mass and strong coupling effects is analyzed in a two-flavor Nambu-Jona-Lasinio (NJL) model. In the weak magnetic field limit, our results agree qualitatively with earlier zero-field studies in the literature that find a critical coupling ratio below which chiral or superconducting order parameters appear almost exclusively. Above the critical ratio, there exists a significant mixed broken phase region where both gaps are nonzero. However, a strong magnetic field G disrupts this mixed broken phase region and changes a smooth crossover found in the weak-field case to a first-order transition for both gaps at almost the same critical density. Our results suggest that in the two-flavor approximation to moderately dense quark matter strong magnetic field enhances the possibility of a mixed phase at high density, with implications for the structure, energetics, and vibrational spectrum of neutron stars.

#### 1. Introduction

The existence of deconfined quark matter in the dense interior of a neutron star is an interesting question that has spurred research in several new directions in nuclear astrophysics. On the theoretical side, it has been realized that cold and dense quark matter must be in a superconductor/superfluid state [1–6] with many possible intervening phases [7–14] between a few times nuclear matter density and asymptotically high density, where quarks and gluons interact weakly. The observational impact of these phases on neutron star properties can be varied and dramatic [15–22]. Therefore, it is of interest to situate theoretical ideas and advances in our understanding of dense quark matter in the context of neutron stars, which serve as unique astrophysical laboratories for such efforts. The phase structures of hot quark matter have been probed in experiments such as in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC). It is estimated in [23–25] that the magnetic field originating from off-central nucleon-nucleon collisions at these colliders can be as large as 10^{18}–10^{20} G. On the astrophysical side, the strength of the magnetic field in some magnetars is of the order 10^{14}–10^{15} G [26], while in the core of such objects, magnetic field might reach up to 10^{18}–10^{19} G. Therefore, it is not surprising that many recent works have stressed the role of strong magnetic fields on hot or dense quark matter [27–35].

At very high density (i.e., , where is the baryon chemical potential and is the scale of quantum chromodynamics) and for number of flavors , the preferred pairing pattern is a flavor and color democratic one termed as the color-flavor-locked (CFL) phase [7]. This idealized phase, while displaying the essentially novel features of the color superconducting state, is unlikely to apply to the bulk of the neutron star matter, since even ten times nuclear matter saturation density () only corresponds to a quark chemical potential MeV. At these densities, quark mass and strong coupling effects can be important and must be treated nonperturbatively. It is reasonable to think that the strange quark current mass, being much larger than that of the up and down quarks, inhibits pairing of strange quarks with light quarks. For the purpose of this work, we therefore adopt the scenario of quark matter in the two-flavor superconducting phase, which breaks the color symmetry to , leaving light quarks of one color (say “3”) and all colors of the strange quark unpaired. Although this phase initially appeared to be disfavored in compact stars [36, 37] once constraints of neutrality were imposed within a perturbative approach to quark masses, the NJL model where masses are treated dynamically still allows for the 2SC phase. Since the issue is not settled, we proceed by adopting the NJL model which best highlights the competition between the chiral and diquark condensates in a straightforward way. Also, our results will be qualitatively true for the 2SC+s phase [9, 10], which can be studied similarly by simply embedding the strange quark, which is inert with respect to pairing, in the enlarged three-flavor space. The additional complications of compact star constraints have been examined before [27, 35, 38] and do not change the main qualitative conclusions of the present work, namely, that strong magnetic field alters the competition between the chiral and diquark order parameters from the weak-field case.

Our objective in this paper is a numerical study of the competition between the chiral and diquark condensates at moderately large and large magnetic field using the NJL model, similar in some respects to previous works [8, 9, 39–41], which treat the quark mass nonperturbatively. Instanton-based calculations and random-matrix methods have also been employed in studying the interplay of condensates [42–44]. In essence, smearing of the Fermi surface by diquark pairing can affect the onset of chiral symmetry restoration, which happens at , where is the constituent quark mass scale [45]. Since appears also in the (Nambu-Gorkov) quark propagators in the gap equations, a coupled analysis of chiral and diquark condensates is required. This was done for the two-flavor case with a common chemical potential in [8], but for zero magnetic field. We use a self-consistent approach to calculate the condensates from the coupled gap equations and find small quantitative (but not qualitative) differences from the results of Huang et al. [8] for zero magnetic field. This small difference is most likely attributed to a difference in numerical procedures in solving the gap equations. We also address the physics of chiral and diquark condensates affected by large in-medium magnetic field that are generated by circulating currents in the core of a neutron or hybrid star. Magnetic field in the interior of neutron stars may be as large as G, pushing the limits of structural stability of the star [46, 47]. There is no Meissner effect for the rotated photon, which has only a small gluonic component; therefore, magnetic flux is hardly screened [48], implying that studies of magnetic effects in color superconductivity are highly relevant. Note that the rotated gluonic field, which has a very small photonic component, is essentially screened due to the 2SC phase. Including the magnetic interaction of the quarks with the external field leads to qualitatively different features in the competition between the two condensates, and this is the main result of our work.

In Section 2, we state the NJL model Lagrangian for the 2SC quark matter. In Section 3, we recast the partition function and thermodynamic potential in terms of interpolating bosonic variables. In Section 4, we obtain the gap equations for the chiral and diquark order parameters by minimizing the thermodynamic potential (we work at zero temperature throughout since typical temperature in stars ). In Section 5, we discuss our numerical results for the coupled evolution of the condensates as functions of a single ratio of couplings, chemical potential, and magnetic field before concluding in Section 6.

#### 2. Lagrangian for 2SC Quark Matter

The Lagrangian density for two quark flavors () applicable to the scalar and pseudoscalar mesons and scalar diquarks iswhere is a Dirac spinor which is a doublet (where ) in flavor space and triplet (where ) in color space. The charge-conjugated fields are defined as and with charge-conjugation matrix . The components of are the Pauli matrices in flavor space and and are the antisymmetric matrices in flavor and color spaces, respectively. The common quark chemical potential is denoted as (for simplicity we assume a common chemical potential for all quarks; in an actual neutron star containing some fraction of charge neutral 2SC or 2SC+s quark matter in -equilibrium, additional chemical potentials for electric charge and color charges must be introduced in the NJL model; furthermore, there can be more than one diquark condensate and in general [9]) and is the current quark mass matrix in the flavor basis. We take the exact isospin symmetry limit, . The and gauge fields are denoted by and , respectively. Here, is the electromagnetic charge of an electron and is the coupling constant. The electromagnetic charge matrix for quark is defined as with (in unit of ). The couplings of the scalar and diquark channels are denoted as and , respectively. In general, one can extend the NJL Lagrangian considered in (1) by including vector and ’t Hooft interaction terms which can significantly affect the equation of state of the compact stars with superconducting quark core [49, 50]. In this paper, our main aim is to investigate the competition between chiral and diquark condensates and therefore we do not consider other interactions in our analysis.

We introduce auxiliary bosonic fields to bosonize the four-fermion interactions in Lagrangian (1) via a Hubbard-Stratonovich (HS) transformation. The bosonic fields areand after the HS transformation, the bosonized Lagrangian density becomeswhere . We set in our analysis, which excludes the possibility of pion condensation for simplicity [51]. Order parameters for chiral symmetry breaking and color superconductivity in the 2SC phase are represented by nonvanishing vacuum expectation values (VEVs) for and . The diquark condensates of and quarks carry a net electromagnetic charge, implying that there is a Meissner effect for ordinary magnetism, while a linear combination of the photon and gluon leads to a “rotated" massless field which is identified as the in-medium photon. We can write the Lagrangian in terms of rotated quantities using the following identity:In the* r.h.s.* of (4) all quantities are rotated. In space in units of the rotated charge of an electron the rotated charge matrix isThe other diagonal generator plays no role here because the degeneracy of colors and ensures that there is no long range -field gluon. We take a constant rotated background magnetic field along axis. The gapped 2SC phase is -neutral, requiring a neutralizing background of strange quarks and/or electrons. The strange quark mass is assumed to be large enough at the moderate densities under consideration so that strange quarks do not play any dynamical role in the analysis.

#### 3. Thermodynamic Potential

The partition function in the presence of an external magnetic field in the mean field approximation is given bywhere is the normalization factor, is the inverse of the temperature , is the external magnetic field, and is the Lagrangian density in terms of the rotated quantities. The full partition function can be written as a product of three parts, . Here, serves as a constant multiplicative factor, denotes the contribution for quarks with colors “1” and “2,” and is for quarks with color “3.” These three parts can be expressed asThe kinetic operators now read , where , and we use the notation . In space in units of the rotated charge matrix is given by . Here, and are unit matrix on color and flavor spaces, respectively. In our case, this translates to charges and . With -quarks as inert background, we also have and . Imposing the charge neutrality and -equilibrium conditions is known to stress the pairing and lead to gluon condensation and a strong gluo-magnetic field [52]. The role of such effects has been studied in [27], but here our focus is on the interdependence of the condensates and their response to the strong magnetic field.

Evaluation of the partition function and the thermodynamic potential, (where is the volume of the system), is facilitated by introducing eight-component Nambu-Gorkov spinors for each color and flavor of quark, leading towhere and are the quark propagators and inverse of the propagators is given bywith . The determinant computation is simplified by reexpressing the -charges in terms of charge projectors in the color-flavor basis, following techniques applied for the CFL phase [53]. The color-flavor structure of the condensates can be unraveled for the determinant computation by introducing energy projectors [8] and moving to momentum space, whereby we findwhere with and . The energy is defined as ; if then ; else . The sum over denotes the discrete sum over the Matsubara frequencies; labels the Landau levels in the magnetic field which is taken in the direction.

#### 4. Gap Equations and Solution

Using the following identity we can perform the discrete summation over the Matsubara frequenciesThen we go over to the 3-momentum continuum using the replacement , where is the thermal volume of the system. Finally, the zero-field thermodynamic potential can be expressed asIn presence of a quantizing magnetic field, discrete Landau levels suggest the following replacement:where is the degeneracy factor of the th Landau level (all levels are doubly degenerate except the zeroth level). The thermodynamic potential in presence of magnetic field is given by

In either case, we can now solve the gap equations obtained by minimizing the (zero-temperature) thermodynamic potential obtained in presence of magnetic field.

Since the above equations involve integrals that diverge in the ultraviolet region, we must regularize the divergences in order to obtain physically meaningful results. We choose to regulate these functions using a sharp cutoff (step function in ), which is common in effective theories such as the NJL model [39, 40], although one may also employ a smooth regulator [7, 53] without changing the results qualitatively for fields that are not too large (e.g., a smooth cutoff was employed in [53] to demonstrate the de Haas-van Alphen oscillations in the gap parameter at very large magnetic field). The momentum cutoff restrict the number of completely occupied Landau levels which can be determined as follows:We use the fact that to compute . For magnetic field (G, conversion to Gauss is given by G), is of the order of 50, and the discrete summation over Landau levels becomes almost continuous. In that case, we recover the results of the zero magnetic field case as described in the next section. For fixed values of the free parameters, we were able to solve the chiral and diquark gap equations self-consistently, for as well as large . Before discussing our numerical results, we note the origin of the interdependence of the condensates. The chiral gap equation contains only which is determined by vacuum physics but also depends indirectly on (a free parameter) through , which is itself dependent on the constituent . Our numerical results can be understood as a consequence of this coupling and the fact that a large magnetic field stresses the pair (same charge and opposite spins implied antialigned magnetic moments) while strengthening the pair (opposite charge and opposite spins implied aligned magnetic moments).

#### 5. Numerical Analysis

In order to investigate the competition between the chiral and the diquark condensates, in this section, we solve the two coupled gap equations (15) numerically. These gap equations involve integrals that have diverging behavior in the high-energy region (this is an artifact of the nonrenormalizable nature of the NJL model). Therefore, to obtain physically meaningful behavior, one has to regularize the diverging integrals by introducing some cutoff scale . A sharp cutoff function sometimes leads to unphysical oscillations in thermodynamical quantities of interest and especially for a system with discrete Landau levels. A novel regularization procedure called “Magnetic Field Independent Regularization” (MFIR) scheme [54, 55] can remove the unphysical oscillations completely even if a sharp cutoff function is used within MFIR. To reduce the unphysical behavior, it is very common in literature to use various smooth cutoff functions although they cannot completely remove the spurious oscillations. Here, we list a few of them:(i)Fermi-Dirac type [56]: , where is a smoothness parameter.(ii)Woods-Saxon type [38]: , where is a smoothness parameter.(iii)Lorentzian type [57]: , where is a positive integer,where , with for and for . Cutoff functions become smoother for larger values of or in case of the Lorentzian type of regulator. We have checked our numerical results for different cutoff schemes like sharp cutoff (Heaviside step function) and various smooth cutoff parameterizations as mentioned above and found that our main results are almost insensitive for different cutoff schemes. We therefore use a smooth Fermi-Dirac type of regulator with throughout numerical analysis.

One can fix various NJL model parameters, the bare quark mass , the momentum cutoff , and the scalar coupling constant by fitting the pion properties in vacuum, namely, the pion mass MeV, the pion decay constant MeV, and the constituent quark mass GeV. Similarly, one can fix the diquark coupling constant by fitting the scalar diquark mass (~600 MeV) to obtain vacuum baryon mass of the order of ~900 MeV [58]. There are some factors that can, in principle, alter those model parameters, for example, strength of the external magnetic field, temperature, and choice of the cutoff functions. Assuming that those factors have only small effects on the parameters and expecting that our numerical results would not change qualitatively, we fix the parameters in the isospin symmetric limit as follows (a discussion of the parameter choice can be found in [59]):where is a free parameter. Although Fierz transforming one gluon exchange implies for and fitting the vacuum baryon mass gives [58], the underlying interaction at moderate density is bound to be more complicated; therefore we choose to vary the coupling strength of the diquark channel to investigate the competition between the condensates.

We investigate the behavior of the chiral and diquark gaps along the chemical potential direction in presence of magnetic field for different magnitudes of the coupling ratio () at zero temperature. Before we discuss the influence of diquark gap on the chiral phase transition, we first demonstrate the behavior of the chiral gap for case (equivalently ) for different magnitudes of . The choice of is made to see the effects of the inclusion of different Landau levels in the system. In Table 1, we show the values of and and the corresponding values of the transition magnetic field . For example, if , then the number of fully occupied Landau levels . In Figure 1, we show as functions of in absence of diquark gap for different choices of . In Figures 1(a) and 1(b), we show in absence of magnetic field () and in the weak magnetic field limit ( or equivalently ~2.5 × 10^{17}G), respectively. One can see that these two figures look almost identical. The reason is that the number of completely occupied Landau levels, , becomes very large (e.g., for ) in the weak magnetic field limit, making the discrete Landau level summation quasicontinuous. As we increase the magnetic field, noticeable deviations appear in the behavior of the chiral gap as seen in Figures 1(c) to 1(f).