Effect of strong magnetic field on competing order parameters in two-flavor dense quark matter

We study the effect of strong magnetic field on competing chiral and diquark order parameters in a regime of moderately dense quark matter. The inter-dependence 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 $G_D/G_S\sim 1.1$ 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 non-zero. However, a strong magnetic field $B\gtrsim 10^{18}$ 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.


I. INTRODUCTION
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 µ and large magnetic field using the NJL model, similar in some respects to previous works [8,9,[39][40][41], which treat the quark mass non-perturbatively. Instanton-based calculations and random-matrix methods have also been employed in studying the interplay of condensates [42][43][44]. In essence, smearing of the Fermi surface by diquark pairing can affect the onset of chiral symmetry restoration, which happens at µ ∼ M q , where M q is the constituent quark mass scale [45]. Since M q 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 selfconsistent 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 10 19 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 II, we state the NJL model Lagrangian for the 2SC quark matter. In Section III, we recast the partition function and thermodynamic potential in terms of interpolating bosonic variables. In Section IV, 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, T star ≪ µ). In Section V, 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 VI.

II. LAGRANGIAN FOR 2SC QUARK MATTER
The Lagrangian density for two quark flavors (N f = 2) applicable to the scalar and pseudoscalar mesons and scalar diquarks is where q ≡ q ia is a Dirac spinor which is a doublet (where i = {u, d}) in flavor space and triplet (where a = {1, 2, 3}) in color space. The charge-conjugated fields are defined as q C = −q T C and q C = Cq T with charge-conjugation matrix C = −iγ 0 γ 2 . The components of the τ = (τ 1 , τ 2 , τ 3 ) are the Pauli matrices in flavor space and, (ǫ f ) ij and (ǫ c ) αβ3 are the antisymmetric matrices in flavor and color spaces respectively. The common quark chemical potential is denoted asμ 1 andm = diag(m u , m d ) is the current quark mass matrix in the flavor basis. We take the exact isospin symmetry limit, m u = m d = m 0 = 0.
The U(1) and SU(3) c gauge fields are denoted by A µ and G µ respectively. Here, e is the electromagnetic charge of an electron and g is the SU(3) c coupling constant. The electromagnetic charge matrix for quark is defined 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 β-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 M u = M d = M s [9]. G D respectively. In general, one can extend the NJL Lagrangian considered in Eq. 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 are σ = (qq) ; π = (qiγ 5 τ q) ; ∆ = q C iγ 5 ǫ f ǫ c q ; ∆ * = qiγ 5 ǫ f ǫ c q C ; (2) and after the HS transformation, the bosonized Lagrangian density becomes where m = m 0 + σ. We set π = 0 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 non-vanishing vacuum expectation values (VEVs) for σ and ∆. The diquark condensates of u and d 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 U (1) 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 the Eq. (4) all quantities are rotated. In f lavor ⊗ color space in units of the rotated charge of an electronẽ = √ 3ge/ 3g 2 + e 2 the rotated charge matrix is The other diagonal generator T 3 c plays no role here because the degeneracy of color 1 and 2 ensures that there is no long range gluon 3-field. We take a constant rotated background U(1) magnetic field B = Bẑ along +z axis. The gapped 2SC phase isQ-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.

III. THERMODYNAMIC POTENTIAL
The partition function in the presence of an external magnetic field B in the mean field approximation is given by where N is the normalization factor, β = T −1 is the inverse of the temperature T , B is the external magnetic field andL is the Lagrangian density in terms of the rotated quantities.
The full partition function Z can be written as a product of three parts, Z = Z c Z 1,2 Z 3 .
Here, Z c serves as a constant multiplicative factor, Z 1,2 denotes the contribution for quarks with color "1" and "2" and Z 3 is for quarks with color "3". These three parts can be expressed as The kinetic operators L kin (q, q c ) now read (i ∂ + µγ 0 − M) where M = m 0 + σ and we use the notation Here, 1 c and 1 f are unit matrix on color and flavor spaces respectively. In our case, this translates toQ charges u 1,2 = 1/2, d 1,2 = −1/2, u 3 = 1 and d 3 = 0. With s-quarks as inert background, we also have s 1,2 = −1/2 and s 3 = 0. Imposing the charge neutrality and β-equilibrium conditions is known to stress the pairing and lead to gluon condensation and a strong gluomagnetic 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, Ω = −T ln Z/V (where V is the volume of the system) is facilitated by introducing eight-component where G and G 0 are the quark propagators and inverse of the propagators are given by with The determinant computation is simplified by reexpressing theQ-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 find where E ± ∆,a = (E ± p,a ) 2 + ∆ 2 with E ± p,a = E p,a ± µ and a = {0, 1, ±1/2}. The energy E p,a is defined as E p,a = p 2 ⊥,a + p 2 z + m 2 , if a = 0 then p 2 ⊥,0 = p 2 x + p 2 y else p 2 ⊥,a = 2|a|ẽBn. The sum over p 0 = iω k denotes the discrete sum over the Matsubara frequencies, n labels the Landau levels in the magnetic field which is taken in theẑ direction.

IV. GAP EQUATIONS AND SOLUTION
Using the following identity we can perform the discrete summation over the Matsubara frequencies Then we go over to the 3-momentum continuum using the replacement p → V (2π) −3 d 3 p, where V is the thermal volume of the system. Finally, the zero-field thermodynamic potential can be expressed as, In presence of a quantizing magnetic field, discrete Landau levels suggest the following where α n = 2 − δ n0 is the degeneracy factor of the n-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 (zerotemperature) thermodynamic potential Ω obtained in presence of magnetic field.

Chiral gap equation :
Since the above equations involve integrals that diverge in the ultra-violet region, we must regularize in order to obtain physically meaningful results. We choose to regulate these functions using a sharp cut-off (step function in |p|), 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 2 .
The momentum cut-off restrict the number of completely occupied Landau levels n max which can be determined as follows We use the fact that p 2 z ≥ 0 to compute n max . For magnetic field B 0.02 GeV 2 (∼ 10 17 G, conversion to Gauss is given by 1 GeV 2 = 5.13 × 10 19 G), n max 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 B = 0 as well as large B. Before discussing our numerical results, we note the origin of the interdependence of the condensates. The chiral gap equation contains only G S which is determined by vacuum physics, but also depends indirectly on G D /G S (a free parameter) through ∆, which is itself dependent on the constituent m = m 0 + σ. Our numerical results can be understood as a consequence of this coupling and the fact that a large magnetic field stresses theqq pair (sameQ charge, opposite spins implies anti-aligned magnetic moments) while strengthening the qq pair (oppositeQ charge and opposite spins implies aligned magnetic moments).

V. 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 (17)  • Fermi-Dirac type [56]: where α is a smoothness parameter.
where p a = p 2 ⊥,a + p 2 z , with p 2 ⊥,0 = p 2 x +p 2 y for a = 0 and p 2 ⊥,a = 2|a|ẽBn for a = 1, ±1/2. Cutoff functions become smoother for larger values of α, or N in case of the Lorenzian 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 α = 0.01Λ throughout numerical analysis. where ρ is a free parameter. Although Fierz transforming one gluon exchange implies ρ = 0.75 for N c = 3 and fitting the vacuum baryon mass gives ρ = 2.26/3 [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 G D 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 ρ (= G D /G S ) 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 ∆ = 0 case (equivalently ρ = 0) for different magnitudes ofẽB. The choice ofẽB is made to see the effects of the inclusion of different Landau levels in the system. In Table I, we show the values of n max 1 and n max We follow this method to locate the first order phase transition point. In Fig. 2b, we plot µ c as a function ofẽB. We observe that µ c oscillates withẽB with dips whenever Λ 2 / (2|a|ẽB) takes an integer value, following the Shubnikov de Haas-van Alphen effect.
Similar oscillations in the density of states and various thermodynamic quantities are observed in metals in presence of magnetic field at very low temperature. The magnitude of oscillations becomes more pronounced as we increase the magnetic field. IfẽB 0.21 GeV 2 (∼ 10 19 G), only the zeroth Landau level is completely occupied as evident from Table I.
In Fig. 3, we show m and ∆ as functions of µ for different ρ in presence of strong magnetic field. In [8], the competition of chiral and diquark gaps without any magnetic field was discussed in great detail. We observe that m increases with the increase ofẽB. In [8], it was shown that with the increase of ρ, the first order transition of the chiral and diquark gaps becomes crossover through a second-order phase transition. When a strong magnetic field is present, we find that the crossover becomes a first order transition. This is an important finding of this work, which has several implication for neutron star physics as discussed in the conclusion. The critical chemical potential µ c is almost same for both the chiral and diquark phase transition, but takes on smaller values as we increase ρ for The curves with square, circle, triangle and diamond represent ρ = 0.75, 1, 1.25, 1.5 respectively.
The discontinuities in gaps signify a first-order phase transition.
In the weak (or zero) magnetic field limit, ∆ appears at a smaller µ with increasing ρ and rises smoothly from zero, until it becomes discontinuous at µ c . At µ c , the chiral gap m also changes discontinuously, with the jumps in the gaps decreasing with increasing ρ. For instance, in Table II we     further increase of ρ, and µ p c moves to the left with increasing ρ. These results forẽB ≈ 0 agree qualitatively with the zero field results of [8] with minor quantitative differences at less than a few percent level. The region where the condensates coexist was termed by them as the "mixed broken phase", since both chiral and (global) color symmetries are broken here. While it should not be confused with a genuine mixed phase, since the free energy admits a unique solution to the gap equations in this regime, it is clear that the width of this overlap region increases with increasing ρ.
The competition between the condensates is driven by the strong magnetic field, which in the case of m is a stress, since the chiral condensate involves quark spinors of opposite spin and sameQ-charge. On the other hand, the diquark condensate, with opposite spin andQ-charge, is strengthened by the strong magnetic field. Thus, we expect a strengthening of the competition between the two condensates, resulting in a qualitative change from the zero-field case. With increasing ρ, similar to theẽB = 0 case, δ m and δ ∆ decrease and the transition is first order in nature. The dramatic effect we observe is that the mixed broken phase for large ρ atẽB = 0 is no longer present in case of strong magnetic field case and the crossover region is replaced by a first-order transition.
Specifically, in Table II, we see for ρ = 1.25, a smooth crossover in theẽB = 0 case at µ p c ∼ 0.255 GeV becomes a first order transition with δ m = 0.185 GeV and δ ∆ = 0.122 GeV at µ c ∼ 0.284 GeV forẽB = 0.1 GeV 2 . The simultaneous appearance of the discontinuity in the gaps for large magnetic field case, at almost the same µ = µ c where both the condensates have their most rapid variation in theẽB = 0 case, is a physical feature and is also cutoff insensitive. We have checked that magnetic fieldẽB 5×10 17 G does not notably alter the competition between the condensates from the zero magnetic field case.

VI. CONCLUSIONS
We study the effects of a strong homogeneous magnetic field on the chiral and diquark condensates in a two-flavor superconductor using the NJL model. We implement a selfconsistent scheme to determine the condensates, by numerically iterating the coupled (integral) equations for the chiral and superconducting gaps. We obtain results for the nature of the competition between these condensates in two cases, at weak magnetic field limit where our results are qualitatively same as zero magnetic field results [8] and at strong magnetic field, where we find the competition between the gaps increases strongly causing a discontinuity in the gaps and disrupting the "mixed broken phase". This is a result of the modified free energy of the quarks in the condensate when subjected to a strong magnetic field. For magnetic fields as large as B ∼ 10 19 G, the anti-aligned magnetic moments of the quarks in the chiral condensate change the smooth crossover of the chiral transition to a sharp first order transition. The diquark gap also becomes discontinuous at this point. For magnetic fields B 10 18 G, there is no significant effect of the magnetic field on the competition between the condensates and zero-field results apply.
These findings can impact the physics of hybrid stars (neutron stars with quark matter) or strange quark stars in several ways. Firstly, the structure of neutron stars is strongly affected by a first-order phase transition, with the possibility of a third family of compact stars in addition to neutron stars and white dwarfs [65] that is separated from conventional neutron stars by a radius gap of a few km. We can speculate that strange stars or hybrid stars with superconducting quark cores inside them belong to this third family.
Since we find that a strong magnetic field increases the likelihood of a first-order phase transition and hence a mixed phase, magnetars could also possibly belong to this category of compact stars since they permit quark nucleation [66] and carry large interior magnetic fields which modify their mass-radius relationship [67]. Secondly, it was pointed out in [27] that for large values of the local magnetic field and in the small density window of the metastable region, it is possible to realize domains or nuggets of superconducting regions with different values for the gap. Charge neutrality can also disrupt the mixed broken phase, but the oscillations of the chiral gap remain, leading to nucleation of chirally restored droplets. Such kinds of nucleation and domain formation will release latent heat that might be very large owing to the large value of the magnetic field, serving as an internal engine for possible energetic events on the surface of the neutron star [68,69]. Such internal mechanisms are unlikely to occur in a pure neutron star without a quark core. Thirdly, strong magnetic fields and quark cores affect the radial and non-radial oscillation modes of neutron stars, which could be a discriminating feature in the gravitational wave signal from vibrating neutron stars. The frequency of the fundamental radial mode shows a kink at the density characterizing the onset of the mixed phase, and the frequencies depend on the magnetic field [70]. Non-radial modes such as g-modes can probe the density discontinuity arising as a result of the phase transition in neutron stars [71] or strange quark stars [72], although the effect of magnetic fields in this context is as yet unexplored. Another important aspect of rotating compact stars are the r-modes [73], which could be responsible for spinning down neutron stars or strange quark stars from their Kepler frequency down to the observed values seen in low-mass X-ray binaries. The effects of a strong magnetic field on the r-mode driven spin down of neutron stars have been studied in [74,75], while r-modes in crystalline quark matter are discussed in [76]. The even-parity counterpart for the r-modes, which include non-radial oscillation modes such as the f -and p-modes have also been explored for the case of strange quark stars in [77,78]. Our findings give additional motivation to the study of such interesting effects associated with a first-order transition in neutron stars with strong magnetic fields, and a systematic study of these effects in the new era of gravitational waves and neutron star observations may finally reveal the presence of quark matter in the core of neutron stars.