Physics Letters B 548 (2002) 29–34

www.elsevier.com/locate/npe

Bulk viscosity of interacting strange quark matter

Zheng Xiaopinga,b, Yang Shuhuaa, Li Jiarongb, Cai Xub

a Department of Physics, Huazhong Normal University, PR Chinab Institute of Particle Physics, Huazhong University, PR China

Received 2 July 2002; received in revised form 18 September 2002; accepted 10 October 2002

Editor: W. Haxton

Abstract

The quarks are regarded as quasiparticles, which acquire an effective mass generated by the interaction with the other quarksof the dense system. The bulk viscosity is re-expressed in the case of interacting strange quark matter. We find that the viscosityis a few∼tens times larger than that of non-interacting quark gas due to the small correction of the effect of the medium to theequation of state. The limit on the spins of strange stars will rise so significantly that the pulsars with shorter period may exist. 2002 Published by Elsevier Science B.V.

PACS: 97.60.Jd; 12.38.Mh; 97.60.Gb

As the hypothesis: compact objects, strange stars,have not been proven by observation yet, probingstrange stars and then identifying them are significantchallenges in nuclear physics and astrophysics. It hasbeen shown in a series of papers, that the emission ofgravitational radiation due tor-mode instabilities inhot, young neutron stars severely limits the rotationperiod of these stars [1–6]. Whereas Madsen foundthat ther-mode instability does not play any role in ayoung strange star contrary to young neutron stars [7].The author reasonably concluded that finding a youngpulsar with a rotation period below 5–10 millisecondswould be a strong indication of the existence of strangestars. In another article [8], the author also discussedthe evolution of pulsars as strange stars due tor-modeinstabilities in rapidly rotating stars. Interestingly,strange stars can explain the apparent lack of very

E-mail address: zhxp@phy.ccnu.edu.cn (X. Zheng).

rapid pulsars with frequencies above 642 Hz since theexisting data are consistent with pulsars being strangestars under some circumstances. All above involve theviscosities of strange quark matter contained in theinterior of these stars. This is because viscosity dampsthe r-modes. As known, the shear viscosity of quarkmatter [7,9] is roughly comparable to that of neutronstar matter for the parameter range of our interestwhere the bulk viscosity of quark matter, which dampsthe r-modes, is decisive for the localization of theinstability, differs from neutron star matter. The bulkviscosity of strange quark matter mainly arises fromthe non-leptonic weak interactions [10]

(1)u + d ↔ s + u.

The importance of dissipation due to the reaction(1) was first stressed by Wang and Lu [11] in thecase of neutron stars with quark cores. Afterward,Sawyer [12] expressed the damping as a function oftemperature and oscillation frequency. Furthermore,

0370-2693/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(02)02840-X

30 X. Zheng et al. / Physics Letters B 548 (2002) 29–34

Madsen in 1992 [13] considered the non-linearity ofthe reaction rate and corrected the factor error of therate in [12] in order to obtain the exact formula ofviscosity of strange matter. All these works showeda huge bulk viscosity of strange matter relative tonuclear matter. Possible implications of a pulsar beingstrange star will result from the discrepancy of bulkviscosities between strange quark matter and normalnuclear matter.

However, up to now, the viscosity has been cal-culated basing on the equation of state (EOS) de-scribed as a free Fermi gas of quarks at zero tempera-ture. In fact, the coupling among quarks (i.e., so-calledmedium effect) due to strong interaction is very im-portant for quark–gluon plasma (QGP). Many investi-gations have been devoted to the field. We here hopeto uncover the medium effect on the bulk viscosityof strange matter. As known, one of the most impor-tant medium effects are effective masses generated bythe interaction of the quarks with the system. Schertlerwith his colleagues had studied the contribution of ef-fective mass effect to EOS of strange matter [14] andapplied the EOS to astrophysical cases for understand-ing the strange star’s structure [15], where the authorsapplied the idea of an ideal gas of quasiparticles witheffective masses to the case of cold strange matter.

We follow Ref. [14]: At finite chemical potential(µi ), the effective (quark) mass of quasiparticle isobtained from zero momentum limit of the quarkdispersion following from the hard dense loop (HDL)approximation [16] of the quark self energy. In thecase of a vanishing current quark mass, as assumedfor up and down quarks, the effective mass is given asfollows [17]

(2)m∗2q = g2µ2

q

6π2.

For strange quarks due to non-vanishing current quarkmasses, we incorporate the current mass into theeffective one [18]

(3)m∗s = ms

2+

√m2

s

4+ g2µ2

s

6π2 ,

where g is coupling constant of strong interaction.An upper limit for the coupling constant is given byg = 7.7 in [14]. As was used in [14], we here takegas a parameter ranging from 0 to 5. Furthermore we

replace the dispersion relation of the collective quarksbranch byεk = √

k2 +m∗2.Evidently, transport coefficients depend mainly on

EOS for given reaction (1) [11–13]. Medium effectsaffect the EOS, then the viscosity will be changed.Although effective mass effect on structure (or EOS)of a strange star is small [14], the viscosity is verysensitive to the EOS. In the following, we will showthat this is indeed the case.

We continue to adopt the quasiparticle pictureto treat an ensemble of quasiparticles as a free,degenerate Fermi gas of particles with a total energy

(4)Heff =d∑

i=1

∑k

√k2 + m∗2

i (µ)nk + B∗V.

Due to the µ-dependence ofm∗(µ) the functionE∗(µ) = B∗(µ)V defines a necessary energy coun-terterm in order to maintain thermodynamic self-consistency [14].d denotes the degree of degeneracyand

∑k nk = N is particle number of system. Thus the

partition function has

(5)Z = e−βE∗∏

k

[1+ e−β(εk−µ)

]d

,

where β = 1/T ,T is the temperature. The grandthermodynamic potentialΩ in a volume is easilyderived.

We immediately get the pressure

P = −∂Ω

∂V

∣∣∣∣µ

= d

48π2

[µkF

(2µ2 − 5m∗2) + 3m∗4 ln

(kF + µ

m∗

)](6)− B∗(µ).

We also obtain other static quantities such as particlenumber density and energy density according to ther-modynamic relations

(7)ρ(µ) = dP

dµ

∣∣∣∣m∗

,

(8)ε(µ) = µdP

dµ

∣∣∣∣m∗

− P(µ).

To maintain these fundamental relations above, weshould require constraint condition [14]

(9)

(∂P

∂m∗

)µ

= 0.

X. Zheng et al. / Physics Letters B 548 (2002) 29–34 31

B∗ will be uniquely defined following from the con-straint equation (9).

Therefore, the particle and energy densities areexplicitly expressed as

(10)ρ = d

6π2k3F,

ε = d

16π2

[µkF

(2µ2 − m∗2) − m∗4 ln

(kF + µ

m∗

)](11)+ B∗(µ).

This, combining (6) with (10) and (11), is the EOSof the strange matter derived by Schertler, Greiner andThoma in quasiparticle description, together with MITbag constantB0, in β-equilibrium [14].

Now we start with the bulk viscosity of SQM inquasiparticle description. We follow the method ofderivation of the bulk viscosity which was presentedby Wang and Lu [11] and then developed by Mad-sen [13].

We still keep the Wang and Lu’s notation forconvenience, definingv, the volume per unit massandni , the quark numbers per unit mass. Assume thevolume vibrates around the equilibrium volumev0,with the periodτ and the perturbation amplitude#v.

When the amplitude is small, i.e.,vv0

1, weexpress the pressureP(t) through a expansion nearequilibriumP0

P(t) = P0 +(∂P

∂v

)0δv +

(∂P

∂nd

)0δnd

(12)+(∂P

∂ns

)0δns .

According to the reaction (1), we shall have

(13)δnd = −δns ≡t∫

0

dnd

dt.

If the net rate of reaction (1) dnd/dt and EOS inequilibrium state were given, the mean dissipation rateof the vibrational energy per unit mass can be obtainedfrom following expression

(14)

(dw

dt

)ave

= −1

τ

τ∫0

P(t)dv

dtdt .

It has been certified by Wang and Lu [11] that thefirst two terms in (13) do not contribute to the energy

dissipation rate (15). This conclusion is independentof concreteP0. The derivative of the third and fourthterms in (12) can be obtained from thermodynamicalrelation

(15)∂P

∂ni

= −∂µi

∂v.

We need slightly change Eq. (10) in place ofρ with ni

(16)ni = 1

πk3

Fiv = 1

π

(µ2

i − m∗2i

)3/2v.

It immediately reads

(17)

(∂µi

∂v

)0= − k2

Fi

3v0Ci

,

(18)

(∂µi

∂ni

)0= π2

3v0kFiCi

,

whereCi = µi − m∗i

∂m∗i

∂µi. One gets(

∂P

∂nd

)0δnd +

(∂P

∂ns

)0δns

(19)= 1

3v0

(k2

Fd

Cd

− k2Fs

Cs

) t∫0

dnd

dt.

We continue to use the formula of the reaction rateadopted by Madsen [13]

(20)

dnd

dt≈ 16

5π2G2

F sin2 θc cos2 θcµ5dδµ

[δµ2 + 4π2T 2]v0

with

(21)16

5π2G2F sin2 θc cos2 θc = 6.76× 10−26 MeV−4.

The chemical potential differenceδµ can be derivedlike the pressure in Eq. (12)

(22)

δµ(t) =(∂δµ

∂v

)0δv +

(∂δµ

∂nd

)0δnd +

(∂δµ

∂ns

)0δns .

Substituting Eqs. (17) and (18) into above formula, wehave

δµ = 1

3

(k2

Fd

Cd

− k2Fs

Cs

)v

vsin

(2πt

τ

)

(23)− π2

3

1

v

(1

kFdCd

− 1

kFsCs

) t∫0

dnd

dt.

32 X. Zheng et al. / Physics Letters B 548 (2002) 29–34

The definition of the bulk viscosity is from [12,13].Finally, it reads

ζ = 1

πv0

(v0

v

)1

3

(k2

Fd

Cd

− k2Fs

Cs

)

(24)×τ∫

0

dt

[ t∫0

dnd

dt

]cos

(2πt

τ

).

Our derivations are made in quasiparticle description.We find from these equations thatB∗ modifying theEOS due to the medium effect do not contribute tothe viscosity, but the effective mass play an importantrole on it. The prefactors in (19), (23) and (24) differfrom that in [7,8,13]. They depend not only on thechemical potentialµ but also on coupling constantgdue tom∗-dependence ofkFi and Ci . When takingg = 0, the mass ofu, d-quark vanishes, we havekFd = µd, kFs = (µ2

d − m2s )

1/2,Cd = Cs = µd , andthen the results above go back to Madsen’s. It hasbeen realized that the medium effect of EOS expressedin Eqs. (10) and (11) is small, see the study [14],but the small contribution to EOS greatly influencesthe dynamical quantity (24). Hence the viscosityζ

is sensitive to the massm∗ although the EOS isweakly mass-dependent under the assumption of weakcoupling.

Combining Eq. (20) with Eq. (22), the quantitiesδµ, dnd/dt can be calculated in principle, but ageneral solution cannot appear analytically because ofstrong coupling of the two equations. We will give thesolution in the high temperature approximation andthen the numerical solution for general case similar tothat made by Madsen.

When the temperature is high enough, i.e., 2πT δµ, the cubic termδµ3 in Eq. (20) can be neglected.Eqs. (20) and (22) are easily solved analytically. WithMadsen’s expression, the bulk viscosity reads

ζ = α∗T 2

ω2 + β∗T 4

(25)

×[1− [

1− exp(−β∗1/2T 2τ

)]2β∗1/2T 2/2

ω2 + β∗T 4

].

The symbol∗ is used to distinguish our result fromMadsen’s. Hereω = 2π/τ , and

(26)

α∗ = 9.39× 1022µ5d

(k2

Fd

Cd

− k2Fs

Cs

)2 (g cm−1s−1),

(27)

β∗ = 7.11× 10−4[µ5

d

2

(1

kFdCd

− 1

kFsCs

)]2 (s−2).

These factors depend not only on the current massms but also the couplingg when the chemical potentialis given.α,β in [13] is independent of the couplingconstant. This is because of inclusion of mediumdependent mass effect in our study.

As Madsen has pointed out, the dominating termin rate (20) is proportional toδµ3 at low temperature(2πT δµ). Therefore, in general Eqs. (20), (22)and (24) must be solved numerically. The resultsof such calculations for the given parametersg,ms

and µ are shown in Fig. 1. The viscosity is muchlarger than that of non-interacting quark gas, wherethe maximum increase of the viscosity is larger by

Fig. 1. A comparison of the our bulk viscosity (solid curvesg = 4)with non-interacting SQM (dotted curves),ms = 80 MeV,µd = 470 MeV, τ = 0.001 s. From bottom to top:T = 10−5,10−4,10−3,100,10−2,10−1 MeV. The viscosityζ in g cm−1 s−1.

X. Zheng et al. / Physics Letters B 548 (2002) 29–34 33

Fig. 2. Bulk viscosity as function of perturbation forms = 150 MeV,µd = 470 MeV,τ = 0.001 s,T = 10−4 MeV. From bottom to top:g = 0,1,2,3,4,5. The viscosityζ in g cm−1 s−1.

Fig. 3. Bulk viscosity as function of temperature forms = 150 MeV,µd = 470 MeV, τ = 0.001 s,#v/v = 10−4. From bottom to top:g = 0,1,2,3,4,5. The viscosityζ in g cm−1 s−1.

about two orders of magnitude. To understand deeplythe medium effect, we also give theg-dependenceof the curves of the viscosity vs.#v/v,T , shown inFigs. 2 and 3. In principle, the viscosity become largerwith increasingg. Analogous to Madsen’s result, thelinear effect, i.e., the constant viscosity in Fig. 2,dominates at tiny vibrations while the viscosity isquickly enlarged when there exist the somewhat largervibrations. Fig. 3 shows that the effect of interactingmedium on the viscosity is rather huge except in theextreme high temperature limit, over a few×109 K.To understand the observed pulsar’s data, the viscosityfor the parameter range of interest is a few×105 ∼a few×107 K because Madsen finds that the viscosityin the range plays an important role if pulsars arestrange stars [8]. Because the medium effect canenlarge the bulk viscosity of SQM, the critical spinfrequency curves in frequency-temperature space (seethe figures in [8]) will be modified. Fig. 3 in [8] showsthat a 2-flavor color superconducting phase (2SC)seems marginally ruled out by the observation data,but we can expect that 2SC quark matter stars may besafe absolutely if the medium effect is included.

In summary, the bulk viscosity of SQM includingmedium effect is derived. The influence of interactingmedium on the viscosity should not be neglected. Wefind that there is a maximal discrepancy of over an or-der of magnitude. We may expect that the limiting an-gular velocity curves in frequency-temperature spacewill change due to the larger bulk viscosity, which willbe discussed in detail in another study.

Acknowledgements

This work was supported in part by the NationalNatural Science Foundation of China under GrantNos. 10175026, 10005002.

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