Equation of State of Strange Quark Matter in a Strong Magnetic Field

EQUATION OF STATE OF STRANGE QUARK MATTER IN A STRONG MAGNETIC FIELD

A. A. Isayev1,2, J. Yang3

1 National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine 2 V. N. Karazin Kharkiv National University, Kharkiv, Ukraine 3 Department of Physics and the Institute for the Early Universe Ewha Womans University, Seoul, Korea

Thermodynamic properties of strange quark matter (SQM) in strong magnetic fields H up to 1020 G are considered at zero temperature within the MIT bag model. The effects of the pressure anisotropy, exhibiting in the difference > between the pressures along and perpendicular to the field direction, become essential at H H th , with the estimate 17 < 18 10 H th 110 G. The longitudinal pressure vanishes in the critical field Hc , which can be somewhat less or larger than 1018 G, depending on the total baryon number density and bag pressure. As a result, the longitudinal instability occurs in strongly magnetized SQM. The appearance of such instability sets the upper bound on the magnetic field strength which can be reached in the interior of a neutron star with the quark core. The longitudinal and transverse pressures as well as the anisotropic equation of state of SQM are determined under the conditions relevant for the cores of magnetars.

1. Introduction

Strange quark matter (SQM), composed of deconfined ud, and s quarks, can be the true ground state of matter, as was suggested in Refs. [1, 3]. This conjecture, if will be confirmed, would have important astrophysical implications. In particular, strange quark stars can exist in the form of stable SQM self-bound by strong interactions [4]. Also, if SQM is metastable at zero pressure, it can appear in the high-density core of a neutron star as a result of the deconfinement phase transition. In this case, the stability of SQM is provided by the gravitational pressure from the outer hadronic layers. Then a relevant astrophysical object is a hybrid star having a quark core and the crust of hadronic matter. Another important aspect related to the physics of compact stars is that they are endowed with the magnetic field. For magnetars, the field strength at the surface can reach the values of about 1014 -1015 G. In the interior of a magnetar the magnetic field strength can reach even larger values of about 1020 G [5]. In such ultrastrong magnetic fields, the effects of the O(3) rotational symmetry breaking by the magnetic field become important [5]-[8]. In particular, the longitudinal (along the magnetic field) pressure is less than the transverse pressure resulting in the appearance of the longitudinal instability of the star’s matter if the magnetic field exceeds some critical value. The effects of the pressure anisotropy should be accounted for in the consistent study of structural and polarization properties of a strongly magnetized stellar object. In this research, we consider the effects of the pressure anisotropy in SQM under the presence of a strong magnetic field within the framework of the MIT bag model.

2. General formalism

In the simplest version of the MIT bag model, quarks are considered as free fermions moving inside a finite region of space called a “bag”. The effects of the confinement are accomplished by endowing the finite region with a constant energy per unit volume, the bag constant B . The energy spectrum of free relativistic fermions ( u , d , s quarks and electrons) in an external magnetic field has the form

i 22 1 s ε=ν km + +ν2| qH | , ν=+ n − sgn(), q iudse = ,,,, (1) zi i22 i where ν = 0,1,2... enumerates the Landau levels, n is the principal quantum number, s =+12 corresponds to a fermion with spin up, and s =−12 to a fermion with spin down. The lowest Landau level with ν = 0 is single degenerate and other levels with ν > 0 are double degenerate. Further we will consider magnetized SQM at zero temperature. In the zero temperature case, the thermodynamic potential for an ideal gas of relativistic fermions of i th species in the external magnetic field reads [9]

νi i max + μ ||qgH ⎪⎧ k ν ⎪⎫ Ω =−ii (2 −δ ) μkmi − 2 ln Fi, , (2) iiFi2 ∑ ννν,0⎨ , , ⎬ 4π ν= m 0 ⎪⎩⎭i,ν ⎪

−δ where the factor (2ν,0 ) takes into account the spin degeneracy of Landau levels, gi is the remaining degeneracy = = μ factor [ g f 3 for quarks (number of colors), and ge 1 for electrons], i is the chemical potential, and

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=+ν222i =μ− mmii,,,ν 2||, qHk i Fiiνν m.

μ−22m ν=i ii In Eq. (2), summation runs up to max I[], I[...] being an integer part of the value in the brackets. The 2|qHi | ∂Ω number density =−()i of fermions of i th species is given by iT∂μ i

νi ||qgHmax ρ=ii (2 −δ )k i . (3) iFπ2 ∑ ν,0 ,ν 2 ν=0

In order to find the chemical potentials of all fermion species, we will use the following constraints:

1 ()ρ + ρ + ρ = ρ , (4) 3 uds B

230ρ − ρ − ρ − ρ − = , (5) uds e

μ = μ + μ − , (6) due

μ = μ ds (7)

being the conditions of the total baryon number conservation, Eq. (4) (B is the total baryon number density), charge neutrality, Eq. (5), and chemical equilibrium, Eqs. (6), (7), with respect to the weak processes occurring in the quark core of a neutron star [4]. =+μ At zero temperature, the energy density EiiiiΩ for fermions of i th species reads

νi i max + μ ||qgH ⎪⎧ k ν ⎪⎫ Ekm=−δμ+ii (2 )i 2 ln Fi, . (8) iiFi2 ∑ ννν,0⎨ , , ⎬ 4π ν= m 0 ⎪⎩⎭i,ν ⎪

In the MIT bag model, the total energy density E , longitudinal pl and transverse pt pressures in quark matter are given by [5]

H 2 EE=++ B, (9) ∑ i π i 8

HH22 pBpHMB=−Ω − −, =−Ω − + − , (10) li∑∑ππ ti ii88

∂ Ωi where B is the bag constant, and MM==−()μ is the total magnetization. It is seen that the magnetic field ∑∑i ∂ i iiH strength enters differently to the longitudinal and transverse pressures that reflects the breaking of the O(3) rotational symmetry in a magnetic field. In a strong enough magnetic field, the quadratic on the magnetic field strength term (the Maxwell term) will be dominating, leading to increasing the transverse pressure and to decreasing the longitudinal pressure. Hence, there exists a critical magnetic field H c , at which the longitudinal pressure vanishes, resulting in the longitudinal instability of SQM. In the astrophysical context, this means that in the magnetic fields H.H c a neutron star with the quark core will be subject to the gravitational collapse along the magnetic field.

3. Numerical results and conclusions

As was mentioned in Introduction, SQM can be in absolutely stable state (strange quark stars), or in metastable state, which can be stabilized by high enough external pressure (hybrid stars). Note that the analysis of the absolute stability window in the parameter space for magnetized superconducting color-flavor-locked strange matter in Ref. [10] shows that the maximum allowed bag pressure decreases with the magnetic field strength (cf. Eq. (32) of that work). The same holds true for magnetized nonsuperconducting SQM because the arguments of Ref. [10] can be reiterated in the given case with the only change that in Eq. (32) of Ref. [10] one should use the potential ΩΩ= with Ω given ∑i i i

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by Eq. (2) of the present study. In numerical calculations, we adopt two values of the bag constant, B = 100 MeV/fm 3 and B = 120 MeV/fm 3 , which are slightly larger than the upper bound B 90 MeV/fm 3 from the absolute stability window at zero magnetic field strength [3]. The core densities corresponding to these bag pressures are chosen equal to = = = −3 B 3 0 and B 4 0 , respectively, which are, in principle, sufficient to produce deconfinement (0 0.16 fm being the nuclear saturation density). Therefore, in the astrophysical context, we assume a scenario in which SQM can be == formed in the core of a strongly magnetized neutron star. For the quark masses, we use the values mmud5 MeV, = and ms 150 MeV [9].

In order to calculate the energy density E of the system, transverse pt and longitudinal pl pressures, we, first, find the chemical potentials of all fermion species from the self-consistent Eqs. (4) - (7) and then determine the quantities

Ep,,tl p from Eqs. (9), (10). Fig. 1 shows the longitudinal pl and transverse pt pressures at zero temperature as functions of the magnetic field strength. It is seen that, first, the transverse and longitudinal pressures stay practically constant and indistinguishable from each other. This behavior of the pressures pt and pl corresponds to the isotropic regime. Beyond some threshold magnetic field Hth , the transverse pressure pt increases with H while the longitudinal pressure pl decreases with it, clearly reflecting the anisotropic nature of the total pressure in SQM in such strong magnetic fields (anisotropic regime). In the critical magnetic field H c , the longitudinal pressure pl vanishes. ≈ 17 == 3 ≈ 18 == 3 This happens at H c 7.4·10 G for B 3,0 B 100MeV/fm, and at H c 1.4·10 G for B 4,0 B 120MeV/fm. Above the critical magnetic field, the longitudinal pressure is negative leading to the longitudinal instability of SQM. < Therefore, the thermodynamic properties of SQM should be considered in the magnetic fields H H c .

Let us now make the estimate of the threshold magnetic field Hth at which the anisotropic regime enters. Fig. 2 δ= − shows the normalized difference between the transverse and longitudinal pressures ( pptl) p0 as a function of the magnetic field strength for the cases under consideration ( p0 being the isotropic pressure corresponding to the weak == field limit with ppplt0 ). Following Refs. [5, 7, 8], for finding the threshold field Hth one can use the approximate δ ≈ 17 == 3 ≈ 17 criterion 1. Then anisotropic regime enters at Hth 5.5·10 G for B 30 ,B 100MeV/fm , and at Hth 9.9·10 == 3 δ δ G for B 4,0 B 120MeV/fm. The anisotropy parameter reaches its maximum c ~2 in the critical field H c , corresponding to the onset of the longitudinal instability in SQM. Thus, as follows from the previous discussions, the << effects of the pressure anisotropy are important at HthHH c .

400 30 3 3 ρ = 3ρ , B = 100 MeV/fm ρ = 3ρ , B = 100 MeV/fm B 0 B 0 3 3 ρ = 4ρ , B = 120 MeV/fm ρ = 4ρ , B = 120 MeV/fm B 0 200 B 0 ]

3 20

0 δ

P [MeV/fm 10 -200

-400 0 15 16 17 18 19 10 10 10 10 10 1016 1017 1018 1019 H [G] H [G] Fig. 1. Transverse (ascending branches) and longitudinal Fig. 2. Same as in Fig. 1, but for the normalized difference δ= − (descending branches) pressures in magnetized SQM at ( pptl) p0 between the transverse and longitudinal zero temperature as functions of the magnetic field pressures. The vertical arrows show the maximum strength for ==3,B 100MeV/fm3 (solid lines) and δ B 0 normalized splitting c at the corresponding critical field ==4,B 120MeV/fm3 (dashed lines). B 0 H c .

Fig. 3 shows the energy density E of the system as a function of the magnetic field strength at zero temperature H 2 without the pure magnetic field energy contribution E = (left panel) and with account of it (right panel). It is seen f 8π that the energy density of solely magnetized SQM decreases with H . However, the overall effect of the magnetic field, with account of the Maxwell term, is to increase the energy density of the system. Nevertheless, this effect of the magnetic field is, in fact, insignificant because the magnetic field in SQM is bound from above by the critical magnetic field H c .

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Because of the pressure anisotropy, the equation of state of SQM in a strong magnetic field is also anisotropic.

Fig. 4 shows the dependence of the energy density E of the system on the transverse pressure pt (left panel) and on the longitudinal pressure pl (right panel) after excluding the dependence on H in these quantities. In particular, the anisotropic character of the pressure is reflected in the fact that the energy density is the increasing function of pt while it decreases with pl . This is because the dominant Maxwell term enters the transverse pressure pt and the energy density E with positive sign while it enters the longitudinal pressure pl with negative sign. In the right panel, the > physical region corresponds to pl 0 .

1400 1400

ρ = 3ρ , B=100 MeV/fm3 ρ = 3ρ , B=100 MeV/fm3 1200 B 0 1200 B 0 ρ = 4ρ , B=120 MeV/fm3 ρ = 4ρ , B=120 MeV/fm3 B 0 B 0 ] ] 3

3 1000 1000

800 800 [MeV/fm

f 600 600 E [MeV/fm E

E-E 400 400

200 200 (a) (b) 0 0 1016 1017 1018 1019 1020 1016 1017 1018 1019 H [G] H [G] Fig. 3. Same as in Fig. 1, but for the energy density E of the system at zero temperature (a) without the magnetic field =π2 energy contribution EHf 8 and (b) with account of E f . The vertical arrows indicate the points corresponding to the critical field H c .

1000 1100 ρ = 3ρ , B=100 MeV/fm3 ρ = 3ρ , B=100 MeV/fm3 B 0 B 0 900 ρ = 4ρ , B=120 MeV/fm3 1000 ρ = 4ρ , B=120 MeV/fm3 B 0 B 0 900 800 ] ] 3 3 800 700 700 600

E [MeV/fm E [MeV/fm 600

500 500 (b) (a) 400 400 0 100 200 300 -300 -200 -100 0 100 3 p [MeV/fm ] 3 T p [MeV/fm ] L

Fig. 4. The energy density E of the system at zero temperature as a function of: (a) the transverse pressure pt and (b) the longitudinal pressure pl . The meaning of the vertical arrows in the left panel is the same as in Fig. 3. In the right > panel, the physical region corresponds to pl 0 .

In summary, we have considered the impact of strong magnetic fields up to 1020 G on the EoS of SQM at zero temperature under additional constraints of total baryon number conservation, charge neutrality and chemical equilibrium with respect to various weak processes occurring in the system. The study has been done within the framework of the MIT bag model. In the numerical calculations, we have adopted two sets of the total baryon number == 3 == 3 density and bag pressure, B 30 ,B 100MeV/fm and B 40 ,B 120MeV/fm . It has been shown that the longitudinal pressure decreases with the magnetic field (contrary to the transverse pressure increasing with H ) and vanishes in the critical field H c , resulting in the longitudinal instability of SQM. The value of the critical field H c depends on the total baryon number density of SQM and the bag pressure B , and it turns out to be somewhat less or larger than 1018 G for the two sets of the parameters, considered in the given study. The longitudinal and transverse pressures as well as the anisotropic EoS of magnetized SQM have been determined at the total baryon number densities and magnetic field strengths relevant to the interiors of magnetars.

J.Y. was supported by grant 2010-0011378 from Basic Science Research Program through NRF of Korea funded by MEST and by grant R32-10130 from WCU project of MEST and NRF.

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REFERENCES

1. Bodmer A. Collapsed Nuclei // Phys. Rev. D - 1971. - Vol. 4. - P. 1601 - 1606. 2. Witten E. Cosmic separation of phases // Phys. Rev. D - 1984. - Vol. 30. - P. 272 - 285. 3. Farhi E., Jaffe R. L. Strange matter // Phys. Rev. D - 1984. - Vol. 30. - P. 2379 - 2390. 4. Alcock C., Farhi E., Olinto A. V. Strange stars // Astrophys. J. - 1986. - Vol. 310. - P. 261 - 272. 5. Ferrer E.J., de la Incera V., Keith J.P. et al. Equation of state of a dense and magnetized fermion system // Phys. Rev. C - 2010. - Vol. 82. - P. 065802. 6. Khalilov V.R. Macroscopic effects in cold magnetized nucleons and electrons with anomalous magnetic moments // Phys. Rev. D - 2002. - Vol. 65. - P. 056001. 7. Isayev A.A., Yang J. Anisotropic pressure in dense neutron matter under the presence of a strong magnetic field // Phys. Lett. B - 2012. - Vol. 707. - P. 163 - 168. 8. Isayev A.A., Yang J. Finite temperature effects on anisotropic pressure and equation of state of dense neutron matter in an ultrastrong magnetic field // Phys. Rev. C - 2011. - Vol. 84. - P. 065802. 9. Chakrabarty S. Quark matter in a strong magnetic field // Phys. Rev. D - 1996. - Vol. 54. - P. 1306 - 1316. 10. Paulucci L., Ferrer E. J., de la Incera V., Horvath J. E. Equation of state for the magnetic-color-flavor-locked phase and its implications for compact star models // Phys. Rev. D - 2011. - Vol. 83. - P. 043009.

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