Jian-Feng Xu

Abstract The bulk viscosity of interacting strange quark matter in a strong external magnetic field Bm with a real equation of state is investigated. It is found that interquark interactions can significantly increase the bulk viscosity,and the magnetic field Bm can cause irregular oscillations in both components of the bulk viscosity, ζ‖ (parallel to Bm) and ζ⊥(perpendicular to Bm). A comparison with non-interacting strange quark matter reveals that when Bm is sufficiently large, ζ⊥is more affected by interactions than ζ‖. Additionally, the quasi-oscillation of the bulk viscosity with changes in density may facilitate the formation of magnetic domains. Moreover, the resulting rmode instability windows are in good agreement with observational data for compact stars in low-mass X-ray binaries. Specifically, the r-mode instability window for interacting strange quark matter in high magnetic fields has a minimum rotation frequency exceeding 1050 Hz, which may explain the observed very high spin frequency of a pulsar with ν=1122 Hz.

Keywords Strange quark matter · Bulk viscosity · Strong magnetic field · Strange star · R-mode instability window

1 Introduction

Since the first direct detection of gravitational waves(GWs)[1]emitted during the coalescence of a binary black hole (BH), dozens of GW events have been observed during the first and second observing runs of the advanced GW detector network[2].In addition to the observations of binary BH mergers, the first detection of GWs from a binary neutron star(NS)inspiral(the GW170817 event)[3]is extraordinarily significant,as the observation of the GWs emitted in this process, possibly combined with electromagnetic observation of the same source [4, 5], may yield insight into the structure of NSs and the equation of state(EOS) of matter under extreme conditions [6-10].

Although transient GWs originate from the coalescence of compact stellar objects, the principal sources of continuous gravitational emission are expected to be spinning NSs and/or quark stars (QSs), which need not be in binary systems. A comprehensive review of the mechanisms of continuous GW emission is given in Ref.[11].Continuous GWs can typically be generated by various processes that produce asymmetry[12].A pulsar with a mass quadrupole may emit GWs with a spin frequency equal to or twice that of the pulsar,whereas some NSs may radiate GWs strongly through a current quadrupole via r-modes, which oscillate at approximately four-thirds of the spin frequency. Unstable oscillation modes, in particular r-modes with a sufficiently large saturation amplitude, have attracted considerable attention as potential sources of detectable GWs.Methods of searching for GWs from the rmodes of known pulsars are described in Refs. [13-16].

The emission of GWs can generally drive r-mode oscillations of compact stars with a certain spin frequency and temperature via the Chandrasekhar-Friedman-Schutz mechanism [17, 18]. In addition, when r-mode oscillation with a sufficiently large saturation amplitude reaches an unstable state, it can in turn cause strong GW emission,which could carry away the angular momentum of compact stars, resulting in a sharp decrease in the spin frequency.This behavior suggests that r-mode instability is likely to play an important role in the evolution of the post-merger remnant[19].Moreover,the presence of r-mode instability results in theoretical difficulties in explaining the high spin frequencies of pulsars.

To solve this problem, different scenarios have been proposed [20-29]. One possible effective method emerged from research in recent decades, which indicated that interactions between quarks can increase the bulk viscosity of strange quark matter(SQM)by 1-2 orders of magnitude[30-34]. The large bulk viscosity can reduce the r-mode instability window; consequently, theoretical calculations are consistent with astrophysical observations[22].In Ref.[35], adopting a quark mass scaling with both linear confinement and perturbative interactions, we investigated the bulk viscosity of SQM in the equivparticle model. When we applied the resulting enhanced bulk viscosity,we found that the r-mode instability window for canonical strange stars with 1.4 M⊙is in good agreement with the observational frequencies and temperatures of pulsars in low-mass X-ray binaries (LMXBs).

Moreover,it is well known that the EOS of NS matter is still unclear and can be affected by many physical parameters such as the symmetry energy [36, 37] and the strong magnetic fields (on the order of approximately 1012-1013G) [38-43] that may be present on the surface of compact stars.For the so-called magnetars,the magnetic field can even be as large as 1014-1015G[44,45].In fact,the largest magnetic field that can be sustained by strange stars is estimated to be approximately 1.5×1020G [46].According to a previous study, SQM will be more stable when the magnetic field is included in the EOS[42].Additionally,a strong magnetic field can strongly suppress the reaction rate of the non-leptonic weak interaction u+d ?u+s, which is one source of the large bulk viscosity of SQM and is expected to affect the viscosity of SQM [47]. In addition, the bulk viscosity of magnetized NS matter was studied in Ref. [48].

Given the important role of bulk viscosity in the emission of continuous GWs by compact stellar objects, in this study we investigated the bulk viscosity of SQM with both strong interactions and magnetic field in the equivparticle model [42, 43, 49]. First, in Sect. 2, we illustrate the formulas for calculating the bulk viscosity of the interacting magnetized SQM. Next, in Sect. 3, we report and discuss the numerical results.Finally,a short summary is presented in Sect. 4.

2 Bulk viscosity of magnetized SQM in equivparticle model

In the equivparticle model, the quark masses mi， (i=u，d，and s)vary with the baryon number density nb,which effectively mimics the strong interactions between quarks.To study the effect of interactions on the bulk viscosity of magnetized SQM,we take the quark mass parameterized as follows [49]:

where mi0is the quark current mass, mI is the interacting part of the quark mass,and the model parameters C and D indicate the strength of perturbative interactions and confinement effects, respectively. Because electrons do not participate in strong interactions, their mass is me =me0 =0.511 MeV.

It is quite convenient to treat the thermodynamics in the equivparticle model, as the bare chemical potentials of quarks are replaced by effective potentials,that is,μi→μ*i.Therefore,the thermodynamic potential density for particle species i in a strong external magnetic field Bm has the same form as that of the free particles:

To investigate the bulk viscosity of SQM in a magnetic field, it is necessary to calculate the magnetization M of SQM. The contribution to magnetization from particle species i can be obtained by combining Eq. (2) with the relation Mi=-?Ωi/?Bm:

Further information about the equivparticle model of magnetized SQM is given in Ref. [43].

Like the pressure, the bulk viscosity of SQM becomes anisotropic in a strong magnetic field. By using a local linear response method,Huang et al.[28]presented explicit expressions for ζ⊥and ζ‖,which are transverse and parallel to the external strong magnetic field Bm,respectively.Both components of the bulk viscosity originate from the nonleptonic weak interaction u+d ?u+s and were studied using a simple bag model that did not include interactions and β-equilibrium between quarks [28]. However, as pointed out above, extensive investigations have shown that interactions between quarks also make an important contribution to the bulk viscosity of SQM [30-33, 35].

The bulk viscosities ζ‖and ζ⊥are given as [28]

from Eq. (14), which is exactly the same as the results reported in Ref. [28].

3 Numerical results and discussion

Fig.1 Anisotropic pressures P‖and P⊥as functions of magnetic field Bm.Regardless of whether interactions are included,when Bm?1018 G,P‖ and P⊥become distinguishable.Then,with increasing Bm,P⊥decreases, and P‖ first increases and then decreases. Moreover, in contrast to the case for non-interacting SQM (C =D=0), both P‖and P⊥are reduced when interactions between quarks are considered(C =0.7，D1/2 =129 MeV)

The bulk viscosity can be understood in some sense as the energy dissipation rate, which naturally is closely related to the pressure. Therefore, in Fig. 1, we show the anisotropic pressures P‖and P⊥as functions of magnetic field Bm with baryon number density nb =2n0, where n0=0.17 fm-3is the nuclear saturation density. Regardless of whether interactions are included, when Bm?1018G, P‖and P⊥become distinguishable. Then, with increasing Bm, P‖first increases and then decreases.However, P⊥decreases continuously. The reason is that with increasing Bm, increasing numbers of particles are confined to lower Landau levels. Moreover, when interactions between quarks are considered,both P‖and P⊥are reduced.

Figure 2 shows the magnetization M in three cases.The dashed line shows the magnetization of the SQM in the bag model with quarks in β-equilibrium, whereas the dotted line represents magnetized SQM in the equivparticle model with both interactions and β-equilibrium.Like the pressure in Fig. 1, the magnetization can be decreased significantly by interactions between quarks. Furthermore, for comparison,we show the results presented in Ref.[28](solid line),where the chemical potentials of u, d, and s quarks are μu=μd=μs=400 MeV.By contrast,here they are set to μ*u=μ*d=μ*s=300 MeV. From Fig. 2, for a sufficiently large magnetic field (e.g., Bm?1019.3G), the magnetization remains unchanged without β-equilibrium, whereas it decreases sharply when β-equilibrium is considered. The reason is that as the magnetic field Bm increases, an increasing number of particles transition from high Landau levels to low Landau levels, and thus νmax gradually becomes small until νmax =0 is reached. In this process,the oscillation in magnetization gradually becomes distinct and ultimately disappears,leaving the magnetization M as a function of only the chemical potentials, and thus a constant.

Fig. 2 Magnetization M as function of strong magnetic field Bm.Like the pressure, the magnetization can also be decreased by interactions between quarks.For comparison,we also show the results from Ref. [28] (solid line), in which the chemical potentials of u, d,and s quarks are assigned the same value

Fig. 3 Bulk viscosities as a function of baryon number density nb with constant magnetic field B=1018.5 G. Although strong interactions between quarks has little effect on ζ‖, on average, they obviously increase ζ⊥. In addition, the quasi-oscillation of the bulk viscosities with changes in density may facilitate the formation of magnetic domains, which may complicate the magnetic field distribution

Figure 3 shows the anisotropic bulk viscosities as functions of baryon number density nb under a constant magnetic field Bm =1018.5G.The dashed lines correspond to the bulk viscosities without interactions, whereas the solid lines show the bulk viscosities with both perturbative interactions and quark confinement effects. Although the oscillation of the parallel bulk viscosity ζ‖changes significantly, the magnitude of ζ‖does not change greatly on average. However, the magnitude of the transverse bulk viscosity ζ⊥increases significantly. Therefore, ζ⊥is likely to be more susceptible to interactions than ζ‖when Bm is sufficiently large. In fact, when interactions are not considered, the values of ζ⊥are even negative, which implies that QSs made of SQM in this state are hydrodynamically unstable[28].If the interactions are included,this situation can be improved greatly. Moreover, whenever a new Landau level appears,both ζ‖and ζ⊥suddenly decrease.In addition, these sudden decreases in the bulk viscosities with changes in density may result in the fragmentation of matter and the formation of a magnetic domain in QSs from the deep interior to the surface. Another possible reason for the formation of magnetic domains is the hydrodynamic instability caused by the negative bulk viscosity. After magnetic domain structure is formed, regions with a magnetic field become separated from those without a magnetic field by domain walls.Consequently,in a sense,only the averaged bulk viscosity has practical meaning for the large-scale behavior of matter over some range of magnetic fields.

Fig. 4 Parallel bulk viscosity ζ‖ and transverse bulk viscosity ζ⊥as functions of magnetic field strength at fixed baryon number density.When interactions are taken into account, both components of the bulk viscosity are increased,especially at low magnetic field strength

Figure 4 shows ζ‖and ζ⊥as functions of magnetic field.For fixed nb,at low magnetic field strength Bm,ζ‖and ζ⊥are clearly increased by interactions, in agreement with previous results where the effects of magnetic fields were not taken into account. As the magnetic field becomes stronger,the irregularity of the oscillation,which originates from the decrease in occupied Landau levels and the unequal masses and charges of different types of particles,becomes clear. The most severe problem that appears in Fig.4 is the negative values of ζ⊥(dashed line in the lower panel), which can be greatly improved by including interactions between quarks, except at extremely strong magnetic fields.

Furthermore, to study the properties of SQM in the stable state, the model parameters C and D should be constrained to the absolutely stable region of the stability window [49], where the approximate relationship between C and D can be roughly fitted as [35]

According to this relationship, when C increases, D1/2decreases,which is shown on the upper X axis in Fig.5.In addition, with increasing C and decreasing D, both ζ‖and ζ⊥decrease simultaneously. However, according to previous studies, the bulk viscosity should increase with increasing interquark interactions, including perturbative interactions and/or quark confinements effects. Therefore,the results shown in Fig. 5 imply that confinement effects may contribute more to the bulk viscosity than perturbative interactions for the parameters nb =2n0and Bm =1018.5G.Furthermore,the sudden decreases in both ζ‖and ζ⊥still originate from the variation of the occupied Landau levels.

Next, we discuss the calculations of the r-mode instability window of strange stars using the obtained bulk viscosities of magnetized SQM. To obtain the instability window, the following equation is generally solved:

Fig. 5 Bulk viscosities as functions of model parameters C and D. The relationship between C and D is constrained by the requirement of absolute stability of the SQM in the equivparticle model

where τgwis the characteristic time scale of GW emission;τsvand τbvrepresent the damping time scales of the shear and bulk viscosity, respectively; and the ellipse denotes other dissipation mechanisms, such as surface rubbing[16, 52-54]. Here, it should be stressed that the damping time scale of the bulk viscosity arises from both ζ‖and ζ⊥because of the magnetic field; that is,

where G=6.707×10-45MeV-2is the gravitational constant, ˉρ is the mean density of a compact star, and Ω is the angular rotation frequency. The damping time scale of the shear viscosity [59] is

Fig.6 R-mode instability window(the region below each curve)for a typical compact star with mass M =1.4 M⊙and radius R=10 km.Observational data on spin frequency and internal temperature of compact stars in LMXBs are also presented

Figure 6 shows the r-mode instability window for a typical compact star with mass M =1.4 M⊙and radius R=10 km.The observational data(solid dots with error bars)of the spin frequency ν=Ω/2π and internal temperature T of compact stars in LMXBs are also given for comparison [60]. The resulting instability window is in very good agreement with the observational data.All the stars appear in the stable region(the region below each curve). Compared with that of noninteracting SQM with low magnetic field strength (dotted lines), the stability window for interacting SQM with high magnetic field strength(solid lines)is much larger and yields a minimum rotation frequency that exceeds 1050 Hz,which may explain the recently observed very high spin frequency of a pulsar with ν=1122 Hz[61].Moreover,a comparison of the dashed lines (non-interacting SQM with high magnetic field strength) and dotted lines (non-interacting SQM with low magnetic field strength) reveals that although a strong magnetic field can enlarge the instability window, compact stars in LMXBs are still located well within the stable region.

4 Summary

The bulk viscosity of interacting magnetized SQM was investigated using the equivparticle model.

First,it was found that regardless of whether interactions are included, P‖and P⊥become distinguishable when Bm?1018G.Second,compared with that of non-interacting SQM,the magnetization M is significantly decreased by the effects of interquark interactions. In addition, the β-equilibrium condition can modify the behavior of M when Bm is extremely high. Then, the anisotropic bulk viscosities were studied at varying baryon number densities nb and magnetic fields Bm. The results showed that when Bm is sufficiently large,ζ⊥can be more susceptible to interactions than ζ‖,and the negative ζ⊥can be greatly improved by interquark interactions, which may result in stable QSs with strong magnetic fields. Moreover, the quasi-oscillation of the bulk viscosities with changes in density may facilitate the formation of magnetic domains, which may complicate the magnetic field distribution so that only the averaged bulk viscosity has practical meaning for the large-scale behavior of matter over some range of magnetic fields.Finally,the resulting rmode instability window for a typical compact star with mass M =1.4 M⊙and radius R=10 km was presented. The rmode instability windows are in good agreement with the observational data for compact stars in LMXBs.In particular,the instability window for interacting SQM with a high magnetic field has a minimum rotation frequency exceeding 1050 Hz, which may explain the observed very high spin frequency of a pulsar with ν=1122 Hz.