-modes of strange quark stars with strong magnetic fields

Anisotropic hydrodynamics, bulk viscosities, and r-modes of strange quark starswith strong magnetic fields

Xu-Guang Huang,1,2,3,4 Mei Huang,3 Dirk H. Rischke,1,2 and Armen Sedrakian2

1Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am Main, Germany2Institut fur Theoretische Physik, J. W. Goethe-Universitat, D-60438 Frankfurt am Main, Germany

3Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, China4Physics Department, Tsinghua University, Beijing 100084, China

(Received 19 October 2009; published 18 February 2010)

In strong magnetic fields the transport coefficients of strange quark matter become anisotropic. We

determine the general form of the complete set of transport coefficients in the presence of a strong

magnetic field. By using a local linear response method, we calculate explicitly the bulk viscosities �? and

�k transverse and parallel to the B field, respectively, which arise due to the nonleptonic weak processes

uþ s $ uþ d. We find that for magnetic fields B < 1017 G, the dependence of �? and �k on the field is

weak, and they can be approximated by the bulk viscosity for the zero magnetic field. For fields B >

1018 G, the dependence of both �? and �k on the field is strong, and they exhibit de Haas–van Alphen–

type oscillations. With increasing magnetic field, the amplitude of these oscillations increases, which

eventually leads to negative �? in some regions of parameter space. We show that the change of sign of �?signals a hydrodynamic instability. As an application, we discuss the effects of the new bulk viscosities on

the r-mode instability in rotating strange quark stars. We find that the instability region in strange quark

stars is affected when the magnetic fields exceed the value B ¼ 1017 G. For fields which are larger by an

order of magnitude, the instability region is significantly enlarged, making magnetized strange stars more

susceptible to r-mode instability than their unmagnetized counterparts.

DOI: 10.1103/PhysRevD.81.045015 PACS numbers: 12.38.�t, 11.10.Wx, 12.38.Mh, 21.65.Qr

I. INTRODUCTION

Neutron stars provide a natural laboratory to study ex-tremely dense matter. In the interiors of such stars, thedensity can reach up to several times the nuclear saturationdensity, n0 ’ 0:16 fm�3. At such high densities quarkscould be squeezed out of nucleons to form quark matter[1–3]. The true ground state of dense quark matter at highdensities and low temperatures remains an open problemdue to the difficulty of solving nonperturbative quantumchromodynamics (QCD). It has been suggested thatstrange quark matter that consists of comparable numbersof u, d, and s quarks may be the stable ground state ofnormal quark matter [4]. This has led to the conjecture thatthe family of compact stars may have members consistingentirely of quark matter (so-called strange stars) and/ormembers featuring quark cores surrounded by a hadronicshell (hybrid stars) [5].

Observationally, it is very challenging to distinguish thevarious types of compact objects, such as the strange stars,hybrid stars, and ordinary neutron stars. Their early coolingbehavior is dominated by neutrino emission which is auseful probe of the internal composition of compact stars.Thus, cooling simulations provide an effective test of thenature of compact stars [6–16]. However, many theoreticaluncertainties and the current amount of data on the surfacetemperatures of neutron stars leave sufficient room forspeculations [17–19]. Another useful avenue for testingthe internal structure and composition of compact stars is

astroseismology, i.e., the study of the phenomena related tostellar vibrations [20–25]. In particular, there are a numberof instabilities which are associated with the oscillations ofrotating stars. Here we will be concerned with the so-calledr-mode instability (see Refs. [22,23] for reviews). Thisinstability is known to limit the angular velocity of rapidlyrotating compact stars. The r-mode and related instabilitiesin rotating neutron stars are damped by the shear and bulkviscosities of matter; therefore these are important ingre-dients of theoretically modeling rapidly rotating stars.Such models and their microscopic input can then be con-strained via the observations of rapidly rotating pulsars,such as the Crab pulsar and the millisecond pulsars.For quark matter in chemical equilibrium, the shear

viscosity is dominated by strong interactions betweenquarks. The bulk viscosity, however, is dominated byflavor-changing weak processes, whereas strong interac-tions play a secondary role. For normal (nonsuperconduct-ing) strange quark matter, the bulk viscosity is dominatedby the nonleptonic process [26–30]

uþ s ! uþ d; (1a)

uþ d ! uþ s; (1b)

since the contributions of the leptonic processes uþ e $dþ � and uþ e $ sþ � are suppressed due to muchsmaller phase spaces. The bulk viscosity of various phasesof quark matter has been studied extensively; seeRefs. [24,26–41].

PHYSICAL REVIEW D 81, 045015 (2010)

1550-7998=2010=81(4)=045015(19) 045015-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.045015

Compact stars are strongly magnetized. Neutron starobservations indicate that the magnetic field is of the orderof B� 1012–1013 G at the surface of ordinary pulsars.Magnetars—strongly magnetized neutron stars—may fea-ture even stronger magnetic fields of the order of1015–1016 G [42–48]. An upper limit on the magnetic fieldcan be set through the virial theorem. Gravitational equi-librium of stars is compatible with magnetic fields of theorder of 1018–1020 G [49–51]. In such a strong magneticfield, not only the thermodynamical but also the hydro-dynamical properties of stellar matter will be significantlyaffected. In particular, due to the large magnetization ofstrange quark matter the fluid will be strongly anisotropicin a strong magnetic field (we note here that the magneti-zation of ordinary neutron matter is small [52]). Therefore,there is a need to develop an anisotropic hydrodynamictheory to describe strongly magnetized matter in compactstars. As we show below, the matter is completely de-scribed in terms of eight viscosity coefficients, whichinclude six shear viscosities and two bulk viscosities.

In this paper, we will carry out a theoretical study of theanisotropic hydrodynamics of magnetized strange quarkmatter and will calculate the two bulk viscosities. We willalso discuss the implications of the anisotropic bulk vis-cosities on the r-mode instability in rotating quark stars.

The paper is organized as follows. The formalism ofanisotropic hydrodynamics for magnetized strange quarkmatter is developed in Sec. II. In Sec. III we apply the locallinear response method to derive explicit expressions forbulk viscosities. The stability of the fluid under a strongmagnetic field is analyzed in Sec. IV. Section V containsour numerical results for the bulk viscosities. The dampingof the r-mode instability in rotating quark stars by the bulkviscosity is studied in Sec. VI. Section VII contains oursummary. We use natural units @ ¼ kB ¼ c ¼ 1. The met-ric tensor is g�� ¼ diagð1;�1;�1;�1Þ. We will use the SIsystem of units in our equations involving electromagne-tism; however, we will quote the strength of the magneticfield in centimeter-gram-second units (Gauss), as is com-mon in the literature on compact stars.

II. ANISOTROPIC HYDRODYNAMICS

A. Ideal hydrodynamics

Hydrodynamics arises as an effective theory valid in thelong-wavelength, low-frequency limit where the energy-momentum tensor T��, the conserved baryon current n�B ,the conserved electric current n�e , the entropy density fluxs�, etc., are expanded in terms of gradients of the four-velocity u� and the thermodynamic parameters of thesystem, such as the temperature T, baryon chemical po-tential �B, etc. The hydrodynamic equations can be ex-pressed as conservation laws for the total energy-momentum tensor T��, as well as baryon and electriccurrents, n�B and n�e . The zeroth-order terms in the expan-sion correspond to an ideal fluid and we shall use the index

0 to label them. In the presence of an electromagnetic field,the zeroth-order terms can be generally written as [53,54]

T��0 ¼ T��

F0 þ T��EM;

T��F0 ¼ "u�u� � P�� 1

2ðM��F�� þM��F�

�Þ;n�B0 ¼ nBu

�; n�e0 ¼ neu

�; s�0 ¼ su�; (2)

where ", P, nB, ne, and s are the local energy density,thermodynamic pressure, baryon number density, electriccharge density, and entropy density, respectively, measuredin the rest frame of the fluid. �� g�� u�u� is theprojector on the directions orthogonal to u�.Here T��

EM ¼ �F��F�� þ g��F��F��=4 is the energy-

momentum tensor of the electromagnetic field. F�� is thefield-strength tensor which can be decomposed into com-ponents parallel and perpendicular to u� as

F�� ¼ F��u�u� � F��u�u

� þ��F

��

� E�u� � E�u� þ 12

��ðu�B� � u�B�Þ; (3)

where in the second line we have introduced the four-vectors E� � F��u� and B� � ��F��u�=2 with

�� being the totally antisymmetric Levi-Civita tensor.In the rest frame of the fluid, u� ¼ ð1; 0Þ, we have E0 ¼B0 ¼ 0, Ei ¼ Fi0, and Bi ¼ �ijkFjk=2, which are pre-

cisely the electric and magnetic fields in this frame.Therefore, E� and B� are nothing but the electric andmagnetic fields measured in the frame where the fluidmoves with a velocity u�.The antisymmetric tensorM�� is the polarization tensor

which describes the response to the applied field strengthF��. For example, if � is the thermodynamic potential ofthe system, M�� @�=@F��. For later use, we also

define the in-medium field-strength tensor H�� F�� M��. In analogy to F�� we can decompose M�� and H��

as

M�� ¼ ðP�u� � P�u�Þ þ 12

��ðM�u� �M�u�Þ;H�� ¼ ðD�u� �D�u�Þ þ 1

2��ðH�u� �H�u�Þ;

(4)

with P� � �M��u�, M� � ��M��u�=2, D� �H��u�, and H� � ��H��u�=2.

In the rest frame of the fluid, the nontrivial componentsof these tensors are ðF10; F20; F30Þ ¼ E, ðF32; F13; F21Þ ¼B, ðM10;M20;M30Þ ¼ �P, ðM32;M13;M21Þ ¼ M,ðH10; H20; H30Þ ¼ D, and ðH32; H13; H21Þ ¼ H. Here Pand M are the electric polarization vector and magnetiza-tion vector, respectively. In the linear approximation theyare related to the fields E and B by P ¼ eE and M ¼mB, with e and m being the electric and magneticsusceptibilities. The four-vectors E�; B�; � � � are all space-like, E�u� ¼ 0; B�u� ¼ 0; � � � , and normalized as

E�E� ¼ �E2; B�B� ¼ �B2; � � � , where E � jEj and

B � jBj.

HUANG et al. PHYSICAL REVIEW D 81, 045015 (2010)

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Since the electric field is much weaker than the magneticfield in the interior of a neutron star, we will neglect it inmost of the following discussion. Upon introducing thefour-vector b� � B�=B, which is parallel to B� and isnormalized by the condition b�b� ¼ �1, and the antisym-

metric tensor b�� b�u�, we can write

F�� ¼ �Bb��; M�� ¼ �Mb��;

H�� ¼ �Hb��;(5)

with M � jMj and H � jHj.The Maxwell equation ��@�F�� ¼ 0 takes the form

@�ðB�u� � B�u�Þ ¼ 0: (6)

Its nonrelativistic form, which is known as the inductionequation, is given by

@B

@t¼ r� ðv� BÞ; r �B ¼ 0; (7)

where v is the three-velocity of the fluid. ContractingEq. (6) with b� gives

�þD lnB� u�b�@�b� ¼ 0; (8)

where � � @�u� and D � u�@�. The second Maxwell

equation can be written as

@�H�� ¼ n�e ; (9)

whose nonrelativistic form is

r �D�H � r � v ¼ n0e;

r� ðH�D� vÞ � @D

@t�H� @tv ¼ ne;

(10)

where n0e is the electric charge density and ne is thecorresponding current. It is useful to rewrite the energy-momentum tensor in the following form [55–57],

T��F0 ¼ "u�u� � P?�� þ Pkb�b�;

T��EM ¼ 1

2B2ðu�u� �� b�b�Þ;

�� þ b�b�;

(11)

where �� is the projection tensor on the direction per-pendicular to both u� and b�. We have defined the trans-verse and longitudinal pressures P? ¼ P�MB andPk ¼ P relative to b�; here P is the thermodynamic pres-

sure. In the absence of a magnetic field, the fluid is iso-tropic and P? ¼ Pk ¼ P. In the local rest frame of fluid,

we have b� ¼ ð0; 0; 0; 1Þ (without loss of generality, wechoose the z axis along the direction of the magnetic field);hence the electromagnetic tensor takes the usual form,while T��

F0 ¼ diagð"; P?; P?; PkÞ.Next we would like to check the consistency of the terms

that appear in T��F0 with the formulas of standard thermo-

dynamics involving electromagnetic fields. By using thethermodynamic relation

" ¼ Tsþ�BnB þ�ene � P; (12)

and the conservation equations for n�B0, n�e0, and s

�0 in ideal

hydrodynamics, one can show that the hydrodynamicequation u�@�T

��0 ¼ 0 together with the Maxwell equa-

tion (8) implies

D" ¼ TDsþ�BDnB þ�eDne �MDB; (13)

which is consistent with the standard thermodynamic rela-tion

d" ¼ Tdsþ�BdnB þ�edne �MdB: (14)

One should note that the potential energy �MB has al-ready been included in our definition of ". Otherwise, newterms �MB, �DðMBÞ, and �dðMBÞ should be added tothe left-hand sides of Eqs. (12)–(14), respectively. Thus,we conclude that our hydrodynamical equations are con-sistent with well-known thermodynamic relations.

B. Navier-Stokes-Fourier-Ohm theory

By keeping the first-order terms of the derivative expan-sion of conserved quantities, one obtains the Navier-Stokes-Fourier-Ohm theory. In this theory, T��, n�B , n

�e ,

and s� can be generally expressed as

T�� ¼ T��0 þ h�u� þ h�u� þ ��;

n�B ¼ nBu

� þ j�B ;

n�e ¼ neu� þ j�e ;

s� ¼ su� þ j�s ;

(15)

where h�, ��, j�B , j�e , and j�s are the dissipative fluxes.

They all are orthogonal to u�; this reflects the fact that thedissipation in the fluid should be spatial. We shall assumethat j

�s can be expressed as a linear combination of h�, j

�B ,

and j�e [58,59]. This allows us to incorporate the fact that

the entropy flux is determined by the energy-momentumand baryon number diffusion fluxes. Thus,

j�s ¼ h� � �Bj�B � �ej

�e ; (16)

with the coefficients , �e, and �B being functions ofthermodynamic variables.Next, the hydrodynamic equations are specified by uti-

lizing the conservation laws of the total energy-momentumT��, the baryon number density flow n

�B , electric current

n�e , and the second law of thermodynamics,

@�T�� ¼ 0; @�n

�B ¼ 0;

@�n�e ¼ 0; T@�s

� � 0:(17)

To discuss the dissipative parts, let us first define thefour-velocity u�, since it is not unique when energy ex-change by thermal conduction is allowed for. We will usethe Landau-Lifshitz frame in which u� is chosen to beparallel to the energy density flow, so that h� ¼ 0. Uponprojecting the first equation of Eq. (17) on u� and after

ANISOTROPIC HYDRODYNAMICS, BULK VISCOSITIES, . . . PHYSICAL REVIEW D 81, 045015 (2010)

045015-3

some straightforward manipulations, we find

ð"þ PÞ�þD"� ��@�u� þMDB ¼ j�eu�F��: (18)

Combining Eq. (18) and (12), and the second equation inEq. (17), we arrive at

T@�s� ¼ ��w�� þ ð�B � T�BÞ@�j�B � Tj

�Br��B

þ ð�e � T�eÞ@�j�e � j�e ðTr��e þ E�Þ;(19)

where r� � ��@� and w�� 1

2 ðr�u� þr�u�Þ. For athermodynamically and hydrodynamically stable system,Eq. (19) should be non-negative. This implies

�B ¼ ��B; �e ¼ ��e; �� ¼ ��w��;

j�B ¼ ��Tr��B; j�e ¼ ��ðTr��e þ E�Þ;(20)

where � � 1=T, �� is the rank-four tensor of viscositycoefficients, and �� and �� are thermal and electricalconductivity tensors with respect to the diffusion fluxes ofbaryon number density and electric charge density. Bydefinition, �� is symmetric in the pairs of indices �,� and �, �. It necessarily satisfies the condition��ðB�Þ ¼ ��ð�B�Þ, which is Onsager’s symmetryprinciple for transport coefficients. Similarly, the tensors�� and �� should satisfy the conditions ��ðB�Þ ¼��ð�B�Þ and ��ðB�Þ ¼ ��ð�B�Þ. Furthermore, allthe tensors of transport coefficients ��, ��, and ��

must be orthogonal to u� by definition.As we have seen, the appearance of the magnetic field

makes the system anisotropic. Such anisotropy is specifiedby the vector b�, so that the tensors ��, ��, and ��

should be in general expressed in terms of u�, b�, g��, andb��. All independent irreducible tensor combinations hav-ing the symmetry of �� and which are orthogonal to u�

are [60](see also the Appendix)

ðiÞ��;

ðiiÞ�� þ��;

ðiiiÞ��b�b� þ ��b�b�;

ðivÞb�b�b�b�;ðvÞ��b�b� þ ��b�b� þ ��b�b� þ ��b�b�;

ðviÞ��b�� þ ��b�� þ ��b�� þ��b��;

ðviiÞb��b�b� þ b��b�b� þ b��b�b� þ b��b�b�;

ðviiiÞb��b�� þ b��b��: (21)

All independent irreducible tensor combinations havingthe symmetry of �� and �� and which are orthogonalto u� are

ðiÞ��;

ðiiÞb�b�;ðiiiÞb��:

(22)

In accordance with the number of tensors (21) and (22), afluid in a magnetic field in general has eight independentviscosity coefficients, three independent thermal conduc-tion coefficients, and three independent electrical conduc-tivities. They may be defined as the coefficients in thefollowing decompositions for the viscous stress tensor,heat flux, and electric charge flux:

�� ¼ 2�0ðw�� =3Þ þ �1ð�� 32�

��Þð�� 32�Þ

� 2�2ðb��b� þ b��b�Þw��

� �3ð2b��b��w�� w�� w�

�Þ� 2�4ð��b�� þ��b��Þw��

þ 2�5ðb��b�b� þ b��b�b�Þw��

þ 32�?�

��þ 3�kb�b�’; (23)

j�B ¼ �Tr��B � �1b�b�Tr��B � �2b

��Tr��B; (24)

j�e ¼ �ðr��e þ E�Þ � �1b

�b�ðr��e þ E�Þ� �2b

��ðr��e þ E�Þ; (25)

where � � ��w��, ’ � b�b�w��, and �� is con-

structed so that the �’s are the coefficients of its tracelessparts; i.e., they can be regarded as shear viscosities. �’s arethe coefficients of the parts with nonzero trace and can beconsidered as bulk viscosities. The �’s and �’s are thermaland electrical conductivities, respectively.Now the divergence of entropy density flux (19) can be

explicitly written as

T@�s� ¼ 2�0ðw�� 1

3��Þðw�� 1

3��Þ þ �1ð�� 32�Þ2 þ 2�2ðb�b�w�� b�b�w

��Þðb�b�w�� b�b�w��Þ

þ �3ðb��w�� b��w�

�Þðb��w�� b��w

��Þ þ 3

2�?�2 þ 3�k’2 � �T2r��Br��B þ �1T

2ðb�r��BÞ2� �ðTr��e þ E�ÞðTr��e þ E�Þ þ �1ðTb�r��e þ E�b�Þ2: (26)


045015-4

One should note that the terms corresponding to the transport coefficients �4, �5, �2, and �2 in Eqs. (23)–(25) do notcontribute to the divergence of the entropy density flux. For stable systems, all the other transport coefficients must bepositive definite according to the second law of thermodynamics. In Sec. IV we will demonstrate explicitly that negativebulk viscosities �? or/and �k indeed cause an instability in the hydrodynamic evolution of strange stars.

To conclude this section, we compare our definition of the viscosity coefficients in Eq. (23) with the definition given inRef. [60] for nonrelativistic fluid, which reads

�ij ¼ 2~�ðwij � �ij�=3Þ þ ~��ij�þ ~�1ð2wij � �ij�þ �ijwklbkbl � 2wikbkbj � 2wjkbkbi þ bibj�þ bibjwklbkblÞþ 2~�2ðwikbkbj þ wjkbkbi � 2bibjwklbkblÞ þ ~�3ðwikbjk þ wjkbik � wklbikbjbl � wklbjkbiblÞþ 2~�4ðwklbilbjbk þ wklbjlbibkÞ þ ~�1ð�ijwklbkbl þ bibj�Þ; (27)

where bij � ijkbk and the remaining notations are self-explanatory.

Our viscosity coefficients in Eq. (23) are related to thecoefficients in Eq. (27) by

�0 ¼ ~�þ ~�1; �1 ¼ 34 ð~�1 þ 1

2~�1 � 3

2~�Þ;

�2 ¼ ~�2 � ~�1; �4 ¼ 12 ~�3; �5 ¼ ~�4;

�? ¼ ~� þ 13~�1; �k ¼ ~� þ 4

3~�1:

(28)

In Ref. [60] there is no term that corresponds to our�3. Thereason is that Ref. [60] considers the combination ofvectors (viii) in Eq. (21) as dependent on the others; inthe Appendix we will show that (at least for relativisticfluids) all the combinations (i)–(viii) are linearly indepen-dent. Note that the transport coefficients in Eq. (26) appearas prefactors of quadratic forms; therefore the second lawof thermodynamics requires that these coefficients must bepositive definite for stable ensembles. This is not manifestin Eq. (27).

III. BULK VISCOSITIES

The typical oscillation frequency of neutron stars is ofthe order of magnitude of the rotation frequency, 1 s�1 &! & 103 s�1. The most important microscopic processeswhich dissipate energy on the corresponding time scalesare the weak processes.

The compression and expansion of strange quark matterwith nonzero strange quark mass will drive the system outof equilibrium. The processes (1a) and (1b) are the mostefficient microscopic processes that restore local chemicalequilibrium. Therefore, the bulk viscosities are determinedmainly by the processes (1a) and (1b). In this section wewill derive analytical expressions for the bulk viscosities�? and �k [61].

Let us imagine an isotropic flow vðtÞ � ei!t which char-acterizes the stellar oscillation. If there are no dissipativeprocesses, such an oscillation will drive the system fromone instantaneous equilibrium state to another instanta-neous equilibrium state. The appearance of dissipationchanges the picture: during the oscillations the thermody-namic quantities will differ from their equilibrium values.

Let us explore how the thermodynamic quantities evolveduring the flow oscillation.In general, we can write the change of baryon density

nB � ðnu þ nd þ nsÞ=3 induced by the oscillation of thefluid as

nBðtÞ ¼ nB0 þ �nBðtÞ; �nB ¼ �neqB þ �n0B; (29)

where nB0 is the static (time-independent) equilibriumvalue, �neqB denotes the equilibrium value shift from nB0due to the volume change, and �n0B denotes the instanta-neous departure from the equilibrium value. Because pro-cesses (1a) and (1b) conserve baryon number, �n0BðtÞ canbe set to zero, if we neglect other microscopic processes.Then �nB can be determined through the continuity equa-tion of ideal hydrodynamics,

�nBðtÞ ¼ �nB0i!

�: (30)

Since processes (1a) and (1b) also conserve the sum nd þns, a similar argument leads to the relation

�nd þ �ns ¼ �nd0 þ ns0i!

�: (31)

When the system is driven out of chemical equilibrium,the chemical potential of the s quark will be slightly differ-ent from that of the d quark. Let us denote this differenceby �� ¼ �s ��d ¼ ��s � ��d, with ��f being the

deviation of �f from its static equilibrium value. Up to

linear order in the deviation we find

��ðtÞ ’�@�s

@ns

�0�ns �

�@�d

@nd

�0�nd; (32)

where nf denotes the number density of quarks of flavor f

and the subscript 0 indicates that the quantity in the bracketis computed in static equilibrium state. �ns and �nd are thedeviations of s-quark and d-quark densities from theirstatic equilibrium value. In the final expressions theyshould be functions of �.The instantaneous departure from equilibrium is re-

stored by the weak processes (1a) and (1b). Adopting thelinear approximation, this can be described by


045015-5

�d � �s ¼ ��; � > 0; (33)

where �d and �s are the rates of processes (1a) and (1b),respectively. If the weak processes are turned off, oneshould have

_�neqf ¼ �nf0� ¼ nf0

_�nBnB0

; (34)

where the dot denotes the time derivative. After turning onthe weak processes, we have

_�n u ¼ _�nB; _�nd ¼ nd0_�nBnB0

þ ��ðtÞ;

_�ns ¼ ns0_�nBnB0

� ��ðtÞ:(35)

This system of coupled linear first-order equations isclosed by substituting Eqs. (30)–(32). It is then easy toobtain the solution,

�nu ¼ �nu0�

i!;

�nd ¼ � i!nd0 þ �ð@�s=@nsÞ0ðnd0 þ ns0Þi!þ �A

�

i!;

�ns ¼ � i!ns0 þ �ð@�d=@ndÞ0ðnd0 þ ns0Þi!þ �A

�

i!;

(36)

where the coefficient A is defined by

A ¼�@�s

@ns

�0þ

�@�d

@nd

�0: (37)

The parallel and transverse components of the pressurePk and P? can be written as

Pk ¼ Pkeq þ �Pk

0; P? ¼ P?eq þ �P?

0; (38)

and

Pk=?eq ’ Pk=?

0 þXf

�@Pk=?@nf

�0�neqf þ

�@Pk=?@B

�0�B;

�Pk=?0 ’ X

f

�@Pk=?@nf

�0�n0f; (39)

where

�n0f � �nf � �neqf : (40)

The small departure of the magnetic field �B can becalculated by the variation of Eq. (8). One finds

�B ¼ � 2

3

B

i!�: (41)

A direct calculation then gives

�n0u ¼ 0; �n0d ¼�Ck

i!þ �A

�

i!;

�n0s ¼ � �Cki!þ �A

�

i!;

(42)

where we introduced the coefficient Ck as

Ck ’ nd0

�@�d

@nd

�0� ns0

�@�s

@ns

�0: (43)

Now we obtain

�Pk=?0 ¼ ��CkCk=?

i!þ �A

�

i!; (44)

with C? defined as

C? ’ Ck � XB; X ¼�@M@nd

�0�

�@M@ns

�0: (45)

Then the deviation of T�� from its equilibrium value canbe written as

�T�� ¼ �Re�P?0�� þ Re�Pk

0b�b�: (46)

For isotropic flows, we have

�� ¼ �?�� kb�b��: (47)

By comparing the above two expressions, we obtain

�k ¼�Ck

2

!2 þ �2A2; (48)

and

�? ¼ �C?Ck!2 þ �2A2

: (49)

Expressions (48) and (49) show that the bulk viscosities �?and �k are functions of the perturbation frequency !, the

weak rate �, and the thermodynamic quantities Ck, C?, A.From the derivation above we can convince ourselves thatthese expressions should be valid also in the case of color-superconducting matter. For the zero magnetic field,Eq. (49) reduces to Eq. (48), which, with parameters Ck,A, and � taken in the absence of magnetic field, gives theexpression for the usual bulk viscosity �0 defined in iso-tropic hydrodynamics.Both �? and �k attain their maxima in the limit of zero

frequency, �maxk ¼ C2

k=ð�A2Þ, �max? ¼ CkC?=ð�A2Þ; and

the maxima are inversely proportional to the weak inter-action rate. At high frequency, ! � �A, �? and �k fall offas 1=!2. For practical applications to cold strange stars,where the chemical potential is much larger than thetemperature, the quantities Ck, C?, and A can be evaluated

in the zero-temperature limit. Their dependence on tem-perature is weak. Contrary to this, the coefficient � de-pends strongly on temperature: for normal quark matter, �has a power-law dependence on T (see Sec. V); for a fullypaired color-superconducting phase, the weak rate is ex-


045015-6

ponentially suppressed by a Boltzmann factor e��=T with� being the superconducting gap. Consequently, �? and �kdepend exponentially on T [24,32–37,40,41].

Before we find the numerical values for the bulk vis-cosities, we first need to analyze the stability of magnetizedstrange quark matter. We observe that according to Eq. (49)negative values of �? are a priori not excluded. In thefollowing we analyze the consequences and implicationsof negative �? on the stability of the system.

IV. STABILITYANALYSIS

A. Mechanical stability

Stable equilibrium in a self-gravitating fluid, such as instrange quark stars, is attained through the balance ofgravity and pressure. The gravitational equilibrium re-quires that both components of the pressure Pk and P?should be positive (otherwise the star will undergo a gravi-tational collapse). At zero temperature, the one-loop ther-modynamic pressure P ¼ T lnZ=V of noninteractingstrange quark matter, where Z is the grand partition func-tion, is given by

P¼ Xf¼u;d;s

NcqfB

4�2

Xnfmax

n¼0

�n

��f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

q

�ðm2f þ 2qfBnÞ ln

�f þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2qfBn

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

f þ 2qfBnq �

; (50)

where qf is the absolute value of electric charge, �f and

mf are the chemical potential and the mass of quark of

flavor f, n labels the Landau levels, �n ¼ 2� �0n is the

degree of degeneracy of each Landau level, and nfmax ¼Int½ð�2

f �m2fÞ=ð2qfBÞ� is the highest Landau level for

quarks of flavor f. By differentiating Eq. (50) with respectto B one can easily get the magnetization as

M¼Xf

Xnfmax

n¼0

�n

Ncqf

4�2

��f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

q

�ðm2f þ 4nqfBÞ ln

�f þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

f þ 2nqfBq �

: (51)

In Fig. 1 we illustrate the magnetization as a function ofmagnetic field at zero temperature. The parameters arechosen as

�u ¼ �d ¼ �s ¼ 400 MeV; ms ¼ 150 MeV;

mu ¼ md ¼ 5 MeV:(52)

On average, the magnetization increases when B grows andeventually becomes constant when B> Bc, where

Bc � Maxffð�2f �m2

fÞ=ð2qfÞg: (53)

However, the detailed structure of the magnetization ex-hibits strong de Haas–van Alphen oscillations [63]. Thisoscillatory behavior is of the same origin as the de Haas–van Alphen oscillations of the magnetization in metals andoriginates from the quantization of the energy levels asso-ciated with the orbital motion of charged particles in amagnetic field. The irregularity of this oscillation shown inFig. 1 is due to the unequal masses and charges of u, d, ands quarks.When B> Bc, all quarks are confined to their lowest

Landau level and their transverse motions are frozen. Inthis case, the longitudinal pressure Pk / B, so the magne-

FIG. 1. The magnetization of strange quark matter as functionof the magnetic field B.

FIG. 2 (color online). The parallel Pk (dashed black curve) andtransverse P? (solid red curve) pressures of strange quark matteras functions of the magnetic field B in units of the pressure P0 forzero magnetic field.


045015-7

tization M � @P=@B ¼ P=B is independent of B, and thetransverse pressure P? ¼ P�MB of the system vanishes.This behavior is evident in Fig. 2.

Thus, we conclude that when all the quarks are confinedto their lowest Landau level, the transverse pressure van-ishes. The system therefore becomes mechanically un-stable and would collapse due to the gravity [64]. Thisphenomenon establishes an upper limit on the magneticfield sustained by a quark star. Given �d � 400 MeV, thisupper limit is roughly Bc � 1020 G as shown in Fig. 2.

B. Thermodynamic stability

The thermodynamic stability requires that the local en-tropy density should reach its maximum in the equilibriumstate [65]. The total energy density of the fluid and themagnetic field is

"total ¼ Tsþ�fnf � Pþ B2

2; (54)

and the corresponding first law of thermodynamics, invariational form, is

�"total ¼ T�sþ�f�nf þH�B; (55)

where H is the strength of the magnetic field and � standsfor a small departure of a given quantity from its equilib-rium value. Varying Eq. (55) on both sides and taking intoaccount that "total, nf, and B are independent variational

variables, one obtains

�2s ¼ � 1

T��f�nf � 1

T�H�B ¼ � 1

T�xT�x; (56)

where �x ¼ ð�nu; �nd; �ns; �BÞ and

¼

@�u

@nu0 0 @�u

@B

0 @�d

@nd0 @�d

@B

0 0 @�s

@ns

@�s

@B@H@nu

@H@nd

@H@ns

@H@B

0BBBBB@

1CCCCCA

0

: (57)

The thermodynamical stability criteria require that �s ¼ 0,�2s 0, or, equivalently, is positive definite. Taking intoaccount the relation ð@H=@nfÞ0 ¼ �ð@�f=@BÞ0, it is easyto show that these criteria are equivalent to the requirement�

@nf@�f

�0� 0;

�@M

@B

�0 1: (58)

From Eq. (50) we obtain

nf0 ¼NcqfB

2�2

Xnfmax

n¼0

�n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

q;

@nf0@�f

¼ NcqfB

2�2

Xnfmax

n¼0

�n

�fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

q ;

(59)

then it is evident that the condition ð@nf=@�fÞ0 � 0 is

always satisfied. However, ð@M=@BÞ0 is divergent whenB approaches a n � 0 Landau level for each flavor quarkfrom below,

�@M

@B

�0! NcqfB

�2

ðnqfÞ2�f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

q ;

when B ! Bfn � �2

f �m2f

2nqf� 0þ: (60)

This shows that strange quark matter will be thermody-

namically unstable just below each Landau level Bfn (n �

0) for every flavor f. The first three thermodynamicallyunstable windows (TUWs) associated, respectively, withBd1 , B

u1 , and Bs

1 are illustrated in the logB�� plane inFig. 3 for our parameters (52). The TUW is actually verynarrow. One may conjecture that such an instability maylead to formation of magnetic domains [63]. The presenceof possible magnetic domains in neutron star crusts wasdiscussed in Ref. [66]; furthermore, such a possibility forcolor-flavor-locked quark matter was pointed out inRef. [67]. We will not pursue here the study of domainstructure and related physics, since among other things,this will require us to specify the geometry of the system.

C. Hydrodynamic stability

In this subsection we address the problem of hydrody-namic stability within the theory presented in Sec. II. Thefluid is said to be stable if it returns to its initial state after atransient perturbation. Otherwise, i.e., when the perturba-tion grows and takes the fluid into another state, the fluid isunstable. Our particular goal here is to determine whether a

FIG. 3 (color online). The first three thermodynamically un-stable windows for strange quark matter in the logB��d planefor T ¼ 0 and �u ¼ �d.


045015-8

small, plane-wave perturbation around a homogeneousequilibrium state grows for nonzero �? and �k [58]. All

the other transport coefficients are set to zero. For thispurpose, it is sufficient to solve the hydrodynamic andMaxwell equations that are linearized around the homoge-neous equilibrium state

@��T�� ¼ 0; @��n

�B ¼ 0; @��n

�e ¼ 0;

@��H�� ¼ 0; ��@��F�� ¼ 0;

(61)

where

�T�� ¼ �T��F0 þ �T��

EM þ ��;

�T��F0 ¼ �"u�u� � �P?�� þ �Pkb�b�

þ ð"þ P?Þð�u�u� þ u��u�Þþ ðPk � P?Þð�b�b� þ b��b�Þ;

�T��EM ¼ B�Bðu�u� �� b�b�Þ

þ B2ð�u�u� þ u��u� � �b�b� � b��b�Þ;�� ¼ �?��þ �kb�b��’þ �?��

þ �k�ðb�b�Þ’;�n

�B ¼ �nBu

� þ nB�u�;

�n�e ¼ �neu

� þ ne�u�;

�H�� ¼ �Hb�� þH�b��;

�F�� ¼ �Bb�� þ B�b��;

�� ¼ �b�b� þ b��b� � �u�u� � u��u�;

�� ¼ 12�

��ð@��u� þ @��u�Þ;�’ ¼ 1

2b�b�ð@��u� þ @��u�Þ: (62)

Upon linearizing the normalization conditions u�u� ¼ 1,

b�b� ¼ �1, u�b� ¼ 0, one finds that the perturbed var-

iables need to satisfy the constraints

�u�u� ¼ �b�b� ¼ u��b� þ �u�b� ¼ 0: (63)

In the equations above the perturbations are assumed tohave the form �Q ¼ �Q0 expðikxÞ, where �Q0 is constant,and the unperturbed quantities are independent of spaceand time. The hydrodynamic and Maxwell equations needto be supplemented by an equation of state in order to closethe system. The linearized equation of state is given by

�P ¼ c2s�"þM�B; (64)

where c2s � ð@P=@"ÞB is the speed of sound.In the most general case Eqs. (61)–(64) constitute 15

independent equations, but the equations associated withthe conservation of nB and ne are decoupled from theothers if the equation of state is taken in the form (64).Therefore, we are left with 13 equations. We work in therest frame of the equilibrium fluid, u� ¼ ð1; 0; 0; 1Þ andb� ¼ ð0; 0; 0; 1Þ and choose as independent variables

�Yi ¼ f�u1; �u2; �u3; �b0; �b1; �b2; �"; �Pk; �P?;

�B; �M; ��; �’g: (65)

(N.B. One can choose other independent variables, but theresults do not change). The 13 linear equations can becollected into the following matrix form:

Gij�Yj ¼ 0: (66)

The matrix G has the following form:

G¼

�k2 k1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 k2 �k1 0 0 0 0 0 0 0

�Hk0 �Hk0 0 0 Hk3 Hk3 0 0 0 k1þ k2 �k1� k2 0 0k3 k3 0 0 �k0 �k0 0 0 0 0 0 0 0Bk1 Bk2 0 0 0 0 0 0 0 �k0 0 0 0�hk1 �hk2 �hk3 �HBk3 0 0 k0 0 0 Bk0 0 0 0�hk0 �hk0 0 0 HBk3 HBk3 0 0 k1þ k2 Bðk1þ k2Þ 0 ��?ðk1þ k2Þ 00 0 �hk0 �HBk0 HBk1 HBk2 0 k3 0 �Bk3 0 0 �kk30 0 1 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 �1 1 M B 0 0ik1 ik2 0 0 0 0 0 0 0 0 0 1 00 0 �ik3 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 �c2s 1 0 �M 0 0 0

0BBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCA

;

(67)

where h ¼ "þ PþHB is the total enthalpy. The expo-nential plane-wave solutions for frequencies k0 and wavevectors k satisfy the dispersion relations given by

detG ¼ 0: (68)

For the modes propagating parallel (longitudinal modes) or


045015-9

perpendicular (transverse modes) to the magnetic field,Eq. (68) has simple solutions.

1. Transverse modes, k3 ¼ 0. There are two types oftransverse modes. One is solely determined by theMaxwell equations and has the following dispersion rela-tion,

k0 ¼ k?; (69)

where k2? ¼ k21 þ k22, and describes simply an electromag-

netic wave. Another solution has the dispersion relation

k0 ¼i�?k2?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ð"þ PkÞð"þ P?Þc2sk2? � k4?�

2?

q2ð"þ PkÞ

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"þ P?"þ Pk

scsk? þ i�?k2?

2ð"þ PkÞ ; (70)

where the second approximate relation is valid in the long-wavelength limit. This solution represents a sound wavepropagating perpendicular to the magnetic field. The speed

of this sonic wave isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið"þ P?Þ=ð"þ PkÞ

qcs and is smaller

than the speed cs of a ordinary sound wave. It is seen thatpositive �? implies dissipation of the sonic wave, i.e., adecay of the initial disturbance. We conclude that the fluidflow is stable in the case. However, we see that for negative�?, the initial disturbance grows and the fluid is unstable.Thus, we conclude that negative transverse bulk viscosityimplies hydrodynamic instability via growth of transversesound waves.

2. Longitudinal modes, k1 ¼ k2 ¼ 0. We find three typesof longitudinal modes. The first one is again the electro-magnetic wave with the dispersion relation

k0 ¼ k3: (71)

The second one is a transverse wave oscillating perpendic-ularly to the magnetic field, but traveling along the mag-netic field lines. It has the dispersion relation

k0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

BH

"þ Pk þ BH

sk3: (72)

This mode is the Alfven wave whose speed is equal toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBH=ð"þ Pk þ BHÞ

q. The third longitudinal mode has

the following dispersion relation:

k0 ¼i�kk23

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ð"þ PkÞ2c2sk23 � k43�

2k

q2ð"þ PkÞ

� csk3 þ i�kk232ð"þ PkÞ : (73)

This mode represents an ordinary sound wave with dis-

sipation due to the longitudinal bulk viscosity �k. It isobvious that if �k < 0 this mode will not decay, but rather

grow, thus leading to hydrodynamic instability.In the next section, we will show that for certain values

of the parameters, the transverse bulk viscosity �? could beindeed negative. We emphasize here that this does notimply a violation of the second law of thermodynamics,but rather this manifests a hydrodynamic instability of theground state; i.e., small perturbations will take the systemvia this hydrodynamic instability to a new state. A candi-date state is the one which has inhomogeneous (domain)structure. Both the structure of the new state and thetransition from the homogeneous to the inhomogeneousstate are interesting problems which are beyond the scopeof this study. However, wewould like to point out a numberanalogous cases where a negative transport coefficientindicates instability towards formation of a new statewith domain structure. One such case is the negativeresistivity (also known as the Gunn effect) in certain semi-conducting materials [68,69]. Another case is the negative(effective) shear viscosity, which is extensively studied inthe literature [70–73]. Finally, negative bulk viscosity hasbeen investigated in different contexts in [74,75].

V. RESULTS FOR THE BULK VISCOSITIES

In order to calculate the bulk viscosities, we need todetermine the coefficients A, Ck, C?, and �. From

Eqs. (50), (51), and (59) we obtain in a straightforwardmanner

A ¼�NcqsB

2�2

Xnsmax

n¼0

�n

�sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

s �m2s � 2nqsB

p ��1

þ�NcqdB

2�2

Xndmax

n¼0

�n

�dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

d �m2d � 2nqdB

q ��1;

Ck ¼ nd0

�NcqdB

2�2

Xndmax

n¼0

�n

�dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

d �m2d � 2nqdB

q ��1

� ns0

�NcqsB

2�2

Xnsmax

n¼0

�n

�sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

s �m2s � 2nqsB

p ��1;

C? ¼ Ck ��@M

@�d

=@nd0@�d

� @M

@�s

=@ns0@�s

�B; (74)

where

@M

@�f¼ Ncqf

2�2

Xnfmax

n¼0

�n

�2f �m2

f � 3nqfBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

f �m2f � 2nqfB

q : (75)

The rate � of the weak processes (1a) and (1b) shouldalso be affected by a strong magnetic field. The majoreffect of a magnetic field on � is to modify the phase space


045015-10

of weak processes (1a) and (1b) [76,77]. Taking this intoaccount, one obtains

� ¼ 64�5

5~G2�dT

2

�quB

2�2

Xnumax

n¼0

�n

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

u �m2u � 2nquB

p �2

��qdB

2�2

Xndmax

n¼0

�n

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

d �m2d � 2nqdB

q �

��qsB

2�2

Xnsmax

n¼0

�n

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

s �m2s � 2nqsB

p �; (76)

where ~G2 � G2Fsin

2�Ccos2�C ¼ 6:46� 10�24 MeV�4 is

the Fermi constant.(1) When the magnetic field is much smaller than the

typical chemical potential, say, qdB � �2d, its effect

on the bulk viscosities is negligible. For typicalparameters (52), this condition holds up to B�1017 G. In this case, the system is practically iso-tropic; �k and �? are effectively degenerate with the

isotropic �0, the bulk viscosity of unmagnetizedmatter. The zero magnetic field limit results can be

obtained easily by replacingPnfmax

n¼0 qfB ! 2k2Ff, and

B ! 0. The bulk viscosity for zero magnetic field �0as function of oscillation frequency ! for varioustemperatures is shown in Fig. 4. The ‘‘shoulder’’structure and the temperature dependence of �0 areeasily understood from Eq. (48) and have beenwidely discussed in the literature [24,28–37,40,41].

(2) When the magnetic field is extremely large, say,B � Bc � 1020 G, for our choice of parameters(52), all the quarks are confined in their lowestLandau level. In this case we obtain

A ¼ 2�2

Ncqd�dBðkFd þ kFsÞ; Ck ¼ m2

s �m2d

�d

;

C? ¼ 0; � ¼ 4G2q2uq2dB

4T2

5�3�2ukFs

; (77)

and therefore

�? ¼ 0; �k � 45m4s�

2ukFs

16� ~G2q2uB2T2ðkFs þ kFdÞ2

:

(78)

We used the parameters (52) and assumed physi-cally interesting frequencies !< 104 s�1. The bulkviscosity �? vanishes as a consequence of vanishingP? when B> Bc. Since �k is now inversely propor-

tional to B2 it approaches zero for large B.Therefore, both �? and �k are suppressed for large

B. In constrast, Pk is enhanced by the extremely

large magnetic field (see Fig. 2).(3) When the magnetic field is strong, but not strong

enough to confine all the quarks to their lowestLandau level, the situation becomes complicated.For our chosen parameters (52), this situationroughly corresponding to the interval 1017 G<B<1020 G. In this case, a finite number of Landaulevels is occupied, and the essential observation isthat Ck andC? can be negative. The behaviors of Ckand C? are shown in Fig. 5 as functions of B. Let usconcentrate on the few levels just above the value1019 G. When B grows passing over Bd

n or Bsn for

each n, both Ck and C? change their sign. More

importantly, they have always opposite signs.Therefore, in this region, �? is negative, which leadsto hydrodynamic instability (see the analysis inSec. IVC).

FIG. 4 (color online). The isotropic bulk viscosity �0 at zeromagnetic field as a function of the oscillation frequency ! for�u ¼ �d ¼ 400 MeV at T ¼ 0:01 (solid blue curve) 0.1(dashed red curve), and 1 (dotted black curve) MeV.

FIG. 5 (color online). Coefficients Ck (dashed blue curve) andC? (dotted red curve) as functions of B at zero temperature.


045015-11

The numerical values of the bulk viscosities �k and �?are shown in Fig. 6 as functions of B. The parameters arethose given in Eq. (52). We also fix the temperature T ¼0:1 MeV and oscillation frequency ! ¼ 2�� 103 s�1.Both �k and �? have ‘‘quasiperiodic’’ oscillatory depen-

dence on the magnetic field. The two boundaries of each

‘‘period’’ correspond to a pair of neighboring Bfn, f ¼ u, d,

s, and n ¼ 0; 1; 2 � � � , and hence the period is roughly�B� 2qfB

2=k2Ff for large B. Therefore, on average, the

period increases as B grows. The amplitude of these oscil-lations also grows with increasing magnetic field until B ’Bc. Thereafter all the quarks are confined to their lowestLandau levels and �? vanishes. From Fig. 6 we see that themagnitudes of �? and �k can be 100 to 200 times larger

than their zero-field value �0. Because of the unequal

masses and charges of u, d, and s quarks, �k and �? behave

very irregularly. We illustrate the zoomed-in curves aroundB ¼ 1017 and B ¼ 1018 G in the subpanels, which lookmore regular. The quasiperiodic structures are more evi-dent in these subpanels.The most unusual feature seen in Fig. 6 is that for a wide

range of field values, the transverse bulk viscosity �? isnegative. Therefore, strange quark matter in this region ishydrodynamically unstable. Besides this hydrodynamical

instability, near each Bfn, there is a narrowwindow in which

thermodynamical instability arises. We depict �k and �? in

these unstable regions by dashed red curves. The solid bluecurves correspond to the stable regime.The magnetic field in a compact star need not be homo-

geneous and may have a complicated structure with poloi-dal and toroidal components. Furthermore, the fields willbe functions of position in the star because of the densitydependence of the parameters of the theory. Furthermore,the instabilities, described above, may lead to fragmenta-tion of matter and formation of domain structures, wherethe regions with magnetic fields are separated from thosewithout magnetic field by domain walls. Accordingly, onlythe averaged viscosities over some range of magnetic fieldshave practical sense for assessing the large-scale behaviorof matter. Averaging over many oscillation periods in thestable region, we find that the averaged values of �k and �?are much more regular, with their magnitudes restrictedfrom 0 to several �0 (see Fig. 7). In obtaining the curves inFig. 7, we have eliminated the viscosities lying in theunstable regime. The solid black curves are obtained byaveraging over a short period �log10ðB=GÞ ¼ 0:05. Theperiod was chosen such that the most rapid fluctuations aresmeared out, but the oscillating structures over a largerscale are intact. The dashed red curves correspond toaveraging over an even longer period, �log10ðB=GÞ ¼0:5. The result of long-period averaging is that �k first

increases slowly and then drops down quickly once B>1018:5 G; similarly, �? first slowly decreases and thendrops down very fast for B� 1018:5 G. Such a droppingbehavior reflects the fact that a large number of quarks arebeginning to occupy the lowest Landau level.We note that the appearances of thermodynamic, me-

chanical, and hydrodynamical instabilities are all inducedby the Landau quantization of the quark levels, i.e., arequantum mechanical in nature. More precisely, they are alldue to the interplay between the Landau levels and the

Fermi momentum (reflected in the quantity Bfn).

Additionally, the hydrodynamical instability requires thatthe quark matter is paramagnetized. Although we did ouranalysis by using the free quark gas approximation,Eq. (50), it should be valid as long as there are sharpFermi surfaces (low temperature), quantized Landau levels(high magnetic field), and paramagnetization. The appear-ances of these instabilities are expected to be a robustfeature for such systems.

FIG. 6 (color online). Bulk viscosities �k and �? scaled by theisotropic bulk viscosity �0 as functions of the magnetic field B atfixed frequency ! ¼ 2�� 103 s�1 and temperature T ¼0:1 MeV. The dashed red curves correspond to viscosities lyingin the unstable regions and would be not physically reachable.Subpanels show the amplifications around 1017 and 1018 G. Ourparameters are given in Eq. (52).


045015-12

We also checked that if one imposes the neutralitycondition, i.e., the condition 2nu ¼ nd þ ns, there is onlyminor quantitative change, while the qualitative conclu-sions are almost unchanged. Our choice of chemical po-tential �u ¼ �d ¼ 400 MeV roughly corresponds to thechoice of nB � 4� 5n0 for neutral strange quark matter.

VI. r-MODE INSTABILITY WINDOW

The purpose of this section is to discuss the damping ofthe r-modes of Newtonian models of strange stars bydissipation driven by the bulk viscosities �? and �k. As iswell-known, rotating equilibrium configurations of self-gravitating fluids are susceptible to instabilities at highrotation rates. Starting from the mass-shedding limit andgoing down with the rotation rate, the first instability point

corresponds to the dynamical instability of the l ¼ 2 andm ¼ 2 mode. This bar-mode instability is independent ofthe dissipative processes inside the star and occurs atvalues of the kinetic to potential energy ratio T=W �0:27 [78]. For smaller rotation rates two secular instabil-ities with l ¼ 2 arise, each corresponding to a sign of m ¼2. For incompressible fluids at constant density the T=Wvalues for the onset of secular instabilities coincide. Oneinstability is driven by the viscosity; the other instability isdriven by the gravitational radiation. For realistic stars theT=W values for the onset of these instabilities do notcoincide; relativity and other factors shift the viscosity-driven instability to higher values of T=W. At the sametime the gravitational radiation instability is shifted tolower values of T=W. The gravitational radiation instabil-ity arises for the modes which are retrograde in the corotat-ing frame, while prograde in the (distant) laboratory frame.The underlying mechanism is the well establishedChandrasekhar-Friedman-Schutz (CFS) mechanism[79,80]. The bulk and shear viscosities can prevent thedevelopment of the CFS instabilities, except in a certainwindow in the rotation and temperature plane.In the following we shall concentrate on axial modes of

Newtonian stars, the so-called r-modes, which are knownto undergo a CFS-type instability. Our main goal will be toassess the role of strong magnetic fields and bulk viscosityon the stability of these objects. We shall adopt the formal-ism of Refs. [22,81,82] for our study of the damping of ther-modes by bulk viscosity. For the sake of simplicity weshall describe both fluid mechanics and gravity in theNewtonian approximation.The equations that describe the dynamical evolution of

the star are

@t�þr � ð�vÞ ¼ 0; (79a)

@tvþ v � rv ¼ �rðh��Þ � �rU; (79b)

r2� ¼ �4�G�; (79c)

where h is defined by the integral

hðPÞ �Z P

0

dP0

�ðP0Þ : (80)

The quantity � is the mass density of the fluid which isassumed to satisfy a barotropic equation of state, � ¼�ðPÞ. � is the gravitational potential and G is the gravita-tional constant. The potential U is used to determine thevelocity field v.The oscillation modes of a uniformly rotating star can be

completely described in terms of two perturbation poten-tials �U � U�U0 and �� 0, where U0 and�0

are the potentials that correspond to the equilibrium con-figuration of the star. We assume that the time and azimu-thal angular dependence of any perturbed quantity isdescribed by / ei ~!tþim’, where m is an integer and ~! isthe frequency of the mode in the laboratory frame. Let �

FIG. 7 (color online). The averaged bulk viscosities �k and �?scaled by the isotropic bulk viscosity �0 as functions of themagnetic field B at fixed frequency ! ¼ 2�� 103 s�1 andtemperature T ¼ 0:1 MeV. The solid black curve correspondsto averaging over a short period [�log10ðB=GÞ ¼ 0:05], whilethe dashed red curve corresponds to averaging over a long period[�log10ðB=GÞ ¼ 0:5]. The viscosities lying in the unstable re-gions have been eliminated in the averaging.


045015-13

denote the rotation frequency of the star and ! denote thefrequency of the perturbed quantity measured in the coro-tating frame [which corresponds to the ! in Eqs. (48) and(49) because we will work in the corotating frame]. Forsmall � there is a simple relation between �, ~!, and ![81,82],

! ¼ ~!þm�: (81)

By linearizing the Euler equation (79b) around the equi-librium configuration, the velocity perturbation �va isdetermined by [81,82]

�va ¼ iQabrb�U: (82)

The tensor Qab is a function of ~! and the rotation fre-quency � of the star,

Qab ¼ 1

ð ~!þm�Þ2 � 4�2

�ð ~!þm�Þ�ab

� 4�2

~!þm�zazb � 2iravb

0

�; (83)

where z is a unit vector pointing along the rotation axis ofthe equilibrium star, which we assume to be parallel to themagnetic field, i.e., zi ¼ bi in Cartesian coordinate system.Here v0 ¼ r�sin�’ is the fluid velocity of the equilibriumstar.

Having the linearized Euler equation, one proceeds tothe linearization of the mass continuity equation (79a) andthe equation for the gravitational potential (79c); one finds

rað�Qabrb�UÞ ¼ �ð ~!þm�Þð�Uþ ��Þd�=dh;r2�� ¼ �4�Gð�Uþ ��Þd�=dh: (84)

These equations, together with the appropriate boundaryconditions at the surface of the star for �U and at infinityfor ��, determine the potentials �U and ��.

For slowly rotating stars, Eq. (84) can be solved order byorder in �,

�U ¼ R2�2

��U0 þ �U2

�2

�G�þOð�4Þ

�;

�� ¼ R2�2

��0 þ ��2

�2

�G�þOð�4Þ

�;

(85)

where R is the radius of a nonrotating star. Since we needonly the perturbed velocity, we will focus on �U in thefollowing discussion. The zeroth-order contribution to ther-mode is generated by the following form of the potential�U0,

�U0 ¼ �

�r

R

�mþ1

Pmmþ1ðcos�Þei ~!tþim’; (86)

~! ¼ �ðm� 1Þðmþ 2Þmþ 1

�; (87)

where � is an arbitrary dimensionless constant and Pml ðxÞ

are the associated Legendre polynomials. It has beenshown that the most unstable mode is the one with m ¼2 [22,83]; therefore we shall consider only this case in thefollowing discussion. Substituting �U0 into Eq. (82) oneobtains the first-order perturbed velocity,

�v0 ¼ �0R��r

R

�mYB

mmð�; ’Þei!t; (88)

where �0 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ðmþ 1Þ3ð2mþ 1Þ!=mp

and YBlmð�; ’Þ is

the magnetic-type spherical harmonic function,

Y Blmð�;’Þ ¼

r�rYlmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ 1Þp : (89)

It is straightforward to check that the first-order per-turbed velocity satisfies

@�v0z

@z¼ 0; r � �v0 ¼ 0; (90)

therefore it does not contribute to the dissipation due to thebulk viscosities �? and �k. In order to see how the bulk

viscosities damp the r-mode instability one must considernext-to-first order, i.e., the third-order perturbed velocitywhich is generated by the potential �U2. One cannotdetermine analytically �U2 from Eqs. (84) and (85), butthe angular structure of �U2 can be well represented by thefollowing spherical harmonics expansion [82],

�U2 ¼ �f1ðrÞP1mþ1ðcos�Þei ~!tþim’

þ �f2ðrÞPmmþ3ðcos�Þei ~!tþim’: (91)

The functions f1ðrÞ and f2ðrÞ have been determined nu-merically in Ref. [82]. A useful approximation is providedby the following simple expressions,

f1ðrÞ ¼ �0:1294

�r

R

�3 � 0:0044

�r

R

�4 þ 0:1985

�r

R

�5

� 0:0388

�r

R

�6; (92)

f2ðrÞ ¼ �0:0092

�r

R

�3 þ 0:0136

�r

R

�4 � 0:0273

�r

R

�5

� 0:0024

�r

R

�6; (93)

which excellently fit the numerical result. We will useEqs. (92) and (93) in the following numerical calculation.The energy of r-modes comes both from the velocity

perturbation and the perturbation of the gravitational po-tential. For slowly rotating stars, the main contributioncomes from the velocity perturbation [22,23,83,84].Then, the energy of the r-mode measured in the corotatingframe is

~E ¼ 1

2

Z��v � �vd3r: (94)


045015-14

Assuming spherical symmetry, we have

~E ¼ 1

2�02�2R�2mþ2

Z R

0�r2mþ2dr: (95)

This energy will be dissipated both by gravitational radia-tion and by the thermodynamic transport in the fluid[22,23],

d ~E

dt¼

�d ~E

dt

�Gþ

�d ~E

dt

�T: (96)

The dissipation rate due to gravitational radiation is givenby [22,23,85]�

d ~E

dt

�G¼ � ~!ð ~!þm�ÞX

l�2

Nl!2l½j�Dlmj2 þ j�Jlmj2�;

(97)

where

Nl ¼ 4�Gðlþ 1Þðlþ 2Þlðl� 1Þ½ð2lþ 1Þ!!�2 : (98)

�Dlm and �Jlm are the mass and current multipole mo-ments of the perturbation,

�Dlm ¼Z

��rlY lmd

3r;

�Jlm ¼ 2

ffiffiffiffiffiffiffiffiffiffiffil

lþ 1

s Zrlð��vþ ��vÞ � YB

lmd3r:

(99)

Taking into account Eq. (87) one obtains

~!ð ~!þm�Þ ¼ � 2ðm� 1Þðmþ 2Þðmþ 1Þ2 �2 < 0; (100)

which implies that the total sign of ðd ~E=dtÞG is positive:gravitational radiation always increases the energy of ther-modes.

In order to compare the relative strengths of differentdissipative processes, it is convenient to introduce thedissipative time scales defined by

�i � � 2 ~E

ðd ~E=dtÞi; (101)

where the index i labels the dissipative process.The lowest-order contribution to ðd ~E=dtÞG comes from

the current multipole moment �Jll. For the most importantcase l ¼ m ¼ 2, this leads to the following time scale(derived for a simple polytropic equation of state P / �2)[82,83]:

1

�G¼ � 1

3:26

��2

�G�

�3s�1: (102)

The bulk viscosities �? and �k dissipate the energy of ther-mode according to

�d ~E

dt

��?

¼ � 3

2

Z�?

��@�vx

@xþ @�vy

@y

��2

d3r;

�d ~E

dt

��k¼ �3

Z�k��@�vz

@z

��2

d3r:

(103)

Accordingly, the time scales ��? and ��k are given by

��?;�k ¼ � 2 ~E

ðd ~E=dtÞ�?;�k: (104)

Currently, the shear viscosities �1–�5 of strange quarkmatter are not known. In order to determine the damping ofthe r-mode by shear viscosity, we take as a crude estimatethe value of �0 in the absence of a magnetic field [62]

�0 ’ � ¼ 5:5� 10�3��5=3s �14=3

d � T�5=3; (105)

where �s is the coupling constant of strong interaction. Wewill choose the value �s ¼ 0:1 and apply Eq. (105) tohighly degenerate 3-flavor quark matter with equal chemi-cal potentials of all flavors (�u ’ �d ’ �s). The contribu-tion to the energy dissipation rate ~E due to shear viscosity� now becomes�

d ~E

dt

��¼ �

Z�jwij � �ij�=3j2d3r: (106)

Assuming a uniform mass density star, the time scale ��can be simply expressed as [85]

1

��¼ 7�

�R2: (107)

The total time scale �ð�; TÞ is given by the followingsum,

1

�� 1

�Gþ 1

��?þ 1

��kþ 1

��; (108)

which characterizes how fast the r-mode decays. Mostimportantly, if the sign of � is negative the amplitude ofthe r-mode will not decay; rather it will increase with time.Thus, it is important to determine the critical angularvelocity �c for the onset of instability

1

�ð�c; TÞ ¼ 0: (109)

At a given temperature, stars with�>�c will be unstabledue to gravitational radiation.Figure. 8 shows the critical angular velocity �c of a

strange quark star with mass M ¼ 1:4M� and radius R ¼10 km as a function of the magnetic field B near 1018 G.The temperature is fixed as T ¼ 0:001 MeV and otherparameters are taken according to Eq. (52). In obtainingFig. 8, we have taken into account the thermodynamicaland hydrodynamical stability conditions. The solid bluecurves correspond to the thermodynamically and hydro-dynamically stable region, while the dashed red curves


045015-15

correspond to unstable regions. The critical angular veloc-ity is strongly oscillating with increasing B. This behavioris due to the oscillating nature of the bulk viscosities �k and�? as shown in Fig. 6. Thus, this macroscopic behaviororiginates from a purely quantum mechanical effect,namely, the Landau quantization of the energy levels ofquarks. As discussed for the bulk viscosities �k and �?shown in Fig. 7, averaging is needed to obtain physicallyrelevant quantities. In Fig. 9 we show the averaged criticalangular velocity at various temperatures. The solid blackcurves are obtained by averaging over a short period� logðB=GÞ ¼ 0:05, whereas the dashed red curves corre-spond to averaging over a long period � logðB=GÞ ¼ 0:5.It is seen that after short-period averaging, the criticalangular velocity (solid black curves) shows regular oscil-lation, the amplitude of which is growing as the B fieldincreases. The critical angular velocity�c displays a sharpdrop for fields B 1018:5 G (dashed red curves), which isthe consequence of the sharp drop of �k and �? shown in

Fig. 7. Thus, we conclude that for extremely large mag-netic fields, the critical angular velocity at which ther-mode instability sets in could be significantly lowerthan in the absence of magnetic field.

Figure. 10 shows the window of the r-mode instability inthe�� log10T plane for a strange quark star of massM ¼1:4M� and radius R ¼ 10 km. The regions above therespective curves correspond to the parameter space wherethe r-mode oscillations are unstable; i.e., a star in thisregion will rapidly spin down by emission of gravitationalwaves. The dashed green curve corresponds to vanishingbulk viscosities �k ¼ �? ¼ 0. The solid black curve rep-

resents the (in)stability window of an unmagnetizedstrange quark star. The curves with symbols show thetypical instability window for magnetic fields around

FIG. 8 (color online). The critical angular velocity �c of astrange quark star as a function of magnetic field B at tempera-ture T ¼ 0:001 MeV. The dashed red lines correspond to theunstable, while the solid blue lines correspond to the stableregime.

FIG. 9 (color online). The averaged critical angular velocity�c of a strange quark star as a function of magnetic field B atvarious temperatures T ¼ 0:001, 0.1, and 10 MeV. The solidblack curve corresponds to averaging over a short period[� logðB=GÞ ¼ 0:05], while the dashed red curve correspondsto averaging over a long period [� logðB=GÞ ¼ 0:5].


045015-16

1017 G (the red curve marked by triangles) and 1018:8 G(the blue curve, marked by circles). The symboled curvesare obtained by using the bulk viscosities averaged over theperiod �log10ðB=GÞ ¼ 0:5. For low temperatures, T <0:3 keV, the r-mode instability is suppressed mainly bythe shear viscosity; at these low temperatures the bulkviscosities are an insignificant source of damping, inde-pendent of how large the magnetic field is. However, forlarger temperatures the bulk viscosities dominate thedamping of r-mode oscillations. For magnetic fields belowB� 1017 G, the critical rotation frequency is almost inde-pendent of the B field. The r-mode instability windowincreases as the magnetic field grows. For fields B>1018 G it is a very sensitive function of the field, as aconsequence of the rapid variation of the bulk viscositieswith the field. Asymptotically, the instability window canbecome significantly larger than the window at the zeromagnetic field (see also Fig. 9). For completeness, Fig. 10also shows the observed distribution of low mass x-raybinaries (LMXBs) by the shadowed box, which corre-sponds to the typical temperatures (2� 107–3� 108 K)and rotation frequencies (300–700 Hz) of the majority ofobserved LMXBs [86]. It is seen that even in the case ofextremely large magnetic fields, our instability window isconsistent with the current LMXB data.

VII. SUMMARY

In this paper we have studied anisotropic hydrodynamicsof strongly magnetized matter in compact stars. We findthat there are in general eight viscosity coefficients: six of

them are identified as shear viscosities, and the other two,�? and �k, are bulk viscosities [see Eq. (23)]. We applied

our formalism to magnetized strange quark matterand gave explicit expressions for the bulk viscosities[Eqs. (48) and (49)] due to the nonleptonic weak reactions(1a) and (1b). Because of the Landau quantization of theenergy levels of charged particles in a strong magneticfield, the magnetic field dependence of �k and �? is very

complicated and exhibits ‘‘quasiperiodic’’ de Haas–van Alphen–type oscillations (see Fig. 6). For a magneticfield B 1017 G the effect of the magnetic field on thetransport coefficients is small and the bulk viscosities canbe well approximated by their zero-field values. For largefields 1017 B 1020 G the viscosities are substantiallymodified; �? may even become negative for some values ofthe B field. We showed that negative �? render the fluidhydrodynamically unstable.For a number of reasons (density dependence of parame-

ters along the star profile, formation of domains, intrinsicmulticomponent nature of the magnetic field) the depen-dence of the transport coefficients on the magnetic fields isneeded at different resolutions (i.e., they require somesuitable averaging over a range of magnetic fields). Wehave provided such averages over an increasingly largerscale. We find that if the averaging period is small the bulkviscosities show regular oscillations, the amplitudes ofwhich increase with magnetic field (see the solid blackcurves in Fig. 7). These oscillations are smoothed out if wefurther increase the averaging scale. At this larger scale themost interesting feature is the rapid drop in the bulkviscosity of the matter due to the confinement of quarksto the lowest Landau level; this occurs for magnetic fieldsin excess of B> 1018:5 G (see the dashed red curves inFig. 7).As an application, we utilized our computed anisotropic

bulk viscosities to study the problem of damping of r-modeoscillations in rotating Newtonian stars. We find that theinstability window increases as the magnetic field is in-creased above the value B> 1017 G. By increasing thefield one covers the entire range of parameter space whichlies between the two extremes: the case when bulk viscos-ity vanishes (dashed green curve in Fig. 10, which corre-sponds to extremely large magnetic fields B * 1019 G forwhich the bulk viscosity drops to zero), and the case whenthe magnetic field is absent (solid black curve in Fig. 10).The found novel dependence of the r-mode instabilitywindow on the magnetic field may help to distinguishquark stars from ordinary neutron stars with strong mag-netic fields, since the latter are much more difficult tomagnetize. It would be interesting to see whether theobjects that lie in between these extremes, e.g., hybridconfigurations featuring quark cores and hadronic enve-lopes (see Ref. [87] and references therein), may interpo-late smoothly between the physics of ordinary and strangecompact objects.

FIG. 10 (color online). The r-mode instability window for astrange quark star. The star is stable below the respective curves.The dashed green curve corresponds to vanishing bulk viscos-ities �k ¼ �? ¼ 0. The solid black curve represents the window

of unmagnetized strange quark matter. The curves with symbolsshow the typical behavior of the instability window when themagnetic fields are around 1017 G (red curve) and 1018:8 G (bluecurve). The shadowed box represents typical temperatures (2�107–3� 108 K) and rotation frequencies (300–700 Hz) of themajority of observed LMXBs [86].


045015-17

ACKNOWLEDGMENTS

We thank T. Brauner, T. Koide, B. Sa’d, A. Schmitt, andI. Shovkovy for helpful discussions. This work is sup-ported, in part, by the Helmholtz Alliance Program of theHelmholtz Association, Contract No. HA216/EMMI‘‘Extremes of Density and Temperature: Cosmic Matterin the Laboratory’’ and the Helmholtz International Centerfor FAIR within the framework of the LOEWE(Landesoffensive zur Entwicklung Wissenschaftlich-Okonomischer Exzellenz) program launched by the Stateof Hesse. M.H. is supported by CAS program‘‘Outstanding Young Scientists Abroad Brought-In’’,CAS key Projects No. KJCX3-SYW-N2,No. NSFC10735040, No. NSFC10875134, and by K. C.Wong Education Foundation, Hong Kong.

APPENDIX: LINEAR INDEPENDENCE OFCOMPONENTS IN EQ. (21)

The purpose of this appendix is to show explicitly thatthe eight different decompositions in Eq. (21) are linearlyindependent. To this end, let us write down a general linearcombination of the eight different decompositions,

a1ðiÞ þ a2ðiiÞ þ � � � þ a8ðviiiÞ ¼ 0: (A1)

If (i)–(viii) are linearly independent, the coefficients a1–a8should all vanish for any values of the vectors u� and b�.First, it is easy to see that the components (vi) and (vii) areindependent of the other components, because they haveodd parity under reflection b� ! �b�, whereas the othersix components have even parity under this transformation.

Besides that, it is obvious that (vi) and (vii) are indepen-dent of each other. Therefore, we only need to treat thelinear equation

a1ðiÞ þ � � � þ a5ðvÞ þ a8ðviiiÞ ¼ 0: (A2)

By contracting the indices � and �, we obtain the follow-ing three conditions:

3a1 þ 2a2 � a3 þ 2a8 ¼ 0;

3a3 � a4 þ 4a5 þ 2a8 ¼ 0;

3a1 þ 2a2 � 12a3 þ a4 � 4a5 ¼ 0:

(A3)

Contracting Eq. (A1) with b� and b� we find the followingtwo additional conditions:

a1 þ a3 ¼ 0; 2a2 � a3 þ a4 � 4a5 ¼ 0: (A4)

Contracting the indices � and � in Eq. (61), we obtain onefurther condition:

a1 þ 4a2 � 2a3 þ a4 � 6a5 ¼ 0: (A5)

The nontrivial solution of the set of Eqs. (A2)–(A5) is

a1 ¼ a3 ¼ 0; a2 ¼ a4=2 ¼ a5 ¼ �a8: (A6)

Then, we have the following condition,

a2ðb��b�� þ b��b�� Þ ¼ 0:

(A7)

The only possible solution is a2 ¼ 0, which thus proves theindependence of (i)–(viii).

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