Mofiz, U A

Murtaza, G

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Available at: http://www.ictp.trieste.it/~pub off IC/2000/113United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSON THE LANGMUIR OSCILLATIONAND LANDAU DAMPING IN QUARK GLUON PLASMAU.A. MozSchool of Science and Technology, Bangladesh Open University,Gazipur-1705, BangladeshandThe Abdus Salam International Centre for Theoretical Physics, Trieste, ItalyandG. MurtazaySalam Professor of Physics, Government College Lahore, Pakistan,Department of Physics, Quaid-i-Azam University, Islamabad 45320, PakistanandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.AbstractOn the basis of semiclassical kinetic equations for quark-gluon plasma (QGP) and Yang-Millsequation in covariant gauge, Langmuir oscillation and linear Landau damping is investigated.It is found that plasma eigen modes are directly related with the wave number and it is highlycoupled with the thermal part of QGP. The linear Landau damping also exists in QGP, whichshows that plasma modes heavily damp for j k j! 0.MIRAMARE { TRIESTEAugust 2000Regular Associate of the Abdus Salam ICTP. E-mail: mozua@bou.bangla.net; amberm@bangla.netySenior Associate of the Abdus Salam ICTP. E-mail: murtaza qau@yahoo.comI. INTRODUCTIONRecently, there has been much interest in theoretical and experimental study of quark-gluonplasma (QGP) [1-3]. Because of asymptotic freedom of QCD, quarks are liberated at suecientlyhigh temperature and densities. There is a phase transition which occurs at temperatures ofseveral hundred MeV. The thermodynamic properties of a quark-gluon plasma were consideredin detail in Ref.[4], in which fairly good expressions were obtained for its free energy. Kineticproperties of the quark-gluon plasma and its collective excitations are investigated in Ref. [5].Kalashnikov and Klimov [6] investigated the properties of the polarization operator calculatedin QCD at nite temperatures and densities in the "one-loop" approximation and the spectrumof the elementary excitations in such a system is found explicitly.Most of the theoretical analyses of the dynamics and signatures of QGP plasma phase haverelied on the assumption of local thermal and chemical equilibrium. These assumptions havepromoted the construction of a kinetic frame work for plasmas with non-abelian interactions,which allow discussions of nonequilibrium phenomenon with the approach to equilibrium.Heinz and Siemens [7] carried out an analyses of colored collective modes in a QGP on thebasis of 'quark-gluon transport theory' near equilibrium. They found that two optical modes(one longitudinal and one transverse) exist starting for k=0 at the plasma frequency, while thereis no acoustic mode starting at ! = 0. An important conclusion is done here, that linear Landaudamping is absent in QGP due to the contribution of massless gluon in the collective modes.Further, Markov and Markova [8] developed the theory of nonlinear Landau damping on thebasis of hard thermal loop approximation. Linear Landau damping is abandoned on the basisof an earlier paper [7].In this paper, we re-examine the Linear Landau damping in QGP on the basis of semi-classical kinetic equations and Yang-Mills [SU(3)] equation in a covariant gauge. In Sec.II. weformulate the linearized system of equations for quark-gluon plasma with thermal and chemicalequilibrium. The regular distribution functions are bosonic for quarks and anti quarks, whileit is fermionic for the gluons with the global equilibrium in QGP. In Sec.III. we do the Fouriertransformation of the linearized system and a generalized expression for the conductivity (polar-ization) tensor is obtained. The Langmuir oscillation and linear Landau damping is studied indetail. It is found that the eigen modes in QGP is strongly related to the wave number and alsoto the thermal part of the system. Linear Landau damping, depending stongly on the wave num-ber and temperature, also exists in QGP. For smaller j k j! 0; we nd that !p j k j13;, 1jkj:II. EQUATIONS FOR DISTRIBUTION FUNCTIONS AND GAUGE FIELDSWe consider an ultrarelativistic quark-gluon plasma in, or, close to thermal equilibrium, at atemperature T . We use the natural units, c = kB= 1 and the metric g;= diag(1; 1; 1; 1).We consider a SU(Nc) gauge theory with Nfavors of quarks. The color indices a; b; :: run from1 to N2c  1. The generators of the gauge group are denoted by taand Ta, respectively, for thefundamental and the adjoint representations, and are normalized such that Tr(tatb) =12abandTr(TaTb) = Ncab. It follows that (Ta)bc=  ifabc, and tata= Cf; where Cf= (Nc  1)=(2Nc)is the Casimir of the fundamental representation and fabcare the structure constants of the2group: [ta; tb] = ifabctc. Furthermore, Dand^Dare the covariant derivatives which act asD= @  ig[A(x); ];^D= @  ig[^A(x); ]; [; ] denotes the commutator, f; g denotes theanticommutator, and A= Aatafor the fundamental representation and^A= AaTafor theadjoint representation. The eld F= FatawithFa= @Aa  @Aa+ gfabcAaAa; (1)obeys the Yang-Mills (YM) equation in a covariant gauge@F(x)  ig[A(x); F(x)]   1@@A(x) =  J(x); (2)where,  is a gauge parameter and g is the dimensionless coupling parameter. Jis the colorcurrentJ= gtaZd4pp[Trta(fq  fq) + Tr(Tafg)]: (3)We are neglecting the spin eects. Thus the distribution functions of quarks fq, antiquarksfq, and gluons fgsatisfy the semiclassical kinetic equationspDfq;q12gpfF;@fq;q@pg = 0;p^Dfg+12gpf^F;@fg@pg = 0; (4)where, the upper sign refers to quarks and the lower one to antiquarks, and^F= FaTa.Initially, we are interested with the linear response in the QGP plasma. Therefore, wedecompose the distriburion functions into two parts, namely, regular and random (turbulent)onesfs= fRs+ fTs; s = q; q; g; (5)so that< fs>= fRs; < fTs>= 0; (6)where angular brackets < : > indicate a statistical ensamble of averaging. Further we setA= AR+AT; < AT>= 0; (7)by denition. For simplicity, the regular part of the eld ARis considered equal to zero. Simi-larly,J= JR+ JT; < J>= JR; < JT>= 0: (8)LetfTs= fT (1)s+ fT (2)s+ :::JT= JT (1)+ JT (2)+ ::: (9)Substituting Eqs. (5) - (9) into Eqs.(2) - (4), and collecting only the rst order of perturba-tions, we obtain the following linearized system of equations for the QGP plasma:3@(FT)L   1@@AT=  JT (1)=  gtaZd4pp[Trta(fT (1)q  fT (1)q) + Tr(TafT (1)g)]: (10)p@fT (1)q;q= 12g12gpf(FT)L;@fRq;q@pg = 0; (11)p^DfT (1)g=  12gpf(^FT)L;@fRg@pg = 0; (12)where,(FT)L= @AT  @AT= (@ATa  @ATa)ta (FTa)ta: (13)(^FT)L= @^AT  @^AT= (@^ATa  @^ATa)Ta (^FTa)Ta: (14)We suppose that the characteristic time of relaxation of oscillations is small as comparedto the time of relaxation of the fRs. Therefore, we neglect the variation of the regular part ofthe distribution functions in space and time, assuming that these functions are specied anddescribe the global equilibrium in QGP [8]fRq;q f0q;q= 22Nf(p0)(2)3(p2)1e(pu)=T+ 1;fRg f0g= 22(p0)(2)3(p2)1e(pu)=T  1: (15)III. LANGMUIR OSCILLATION AND LANDAU DAMPINGTaking the Fourier transformation of the linearized equations (11) and (12), we ndf(1)q;q(k; p) = gp  k + ip0A(k)@f(0)q;q@p; (16)f(1)g(k; p) =  gp  k + ip0^A(k)@f(0)g@p; (17)where, (p  kg  pk): (18)Thus, the Fourier transformation of equation (10) takes the formk(kA0  k0A)   1k0kA=  J0(k)= g2Zd4pp0p  k + ip0fA(k)(@f(0)q@p+@f(0)q@p) +^A(k)@f(0)g@pg: (19)We may write4J(1)0(k) = g2Zd4pp0p  k + ip0(pk @@p  p  k@@p)[f(0)q+ f(0)q+Ncf(0)g]A(k): (20)In the above, for simplicity, we omit the sux " T " for the gauge eld.Now, we can write the linear current asJ0(k) = 0A(k); (21)where,0= g2Zd4pp0p  k + ip0(pk @@p  p  k@@p)Neq; (22)with Neq=12(f(0)q+ f(0)q) +Ncf(0)g, is the conductivity (polarization) tensor of QGP.We are interested in the study of oscillation and Landau damping of Langmuir mode inQGP. The term 00represents such oscillation, thus we study 00in detail.00= g2Zd4pp0p  k + ip0(p0~k @@~p  ~p ~k@@p0)Neq(23)For an easy calculation of the integral, we begin with the one dimensional case of the problem.Therefore, we consider ~p k~k k ~u. Then00=  g2T2Nf(2)34Z10dpp0ku+ u0kpp0k0  kp+ ip0F (p): (24)where,F (p) = sse(p0u0 pu s)=T(e(p0u0 pu s)=T+ 1)2; (25)with q= q= 1; g= 2Nc; q= ; q=  q; g= 0; represents a combined distribution ofquarks, antiquarks and gluons.The principal value of the integral related with the combined distribution function F (p) isgiven by the integral typeZ10dpp0ku+ u0kpp0k0  kp+ ip0e(p0u0 pu )=T(e(p0u0 pu )=T+ 1)2; (26)which is very much complicated to yield any physical results. Therefore, we chose thatF (p)  G(p) = e (p W )2; (27)where ; ;W are some chosen parameters as function of free parameters (; T; u; u0). Thefunction F (P ) and G(p) may be close to identical with the proper selection of the parameterssatisfying at least the following conditions:i) F (0) = G(0); ii) F0(W ) = G0(W ); iii) F (1) = G(1):We plot the functions F (p) and G(p) with the xed values of parameters u0= P0= T =u = 1;  = :01; Nc= 8; with the chosen parameters  = 4;  = :1;W = :8. These functionsare shown in Fig.1a and Fig.1b, respectively. The gures validate our approximation to someextent.5FIGURES0 2 4 6 8 10p01234f0 2 4 6 8 10p01234gFIG. 1. Dependence of functions F(p) and G(p) for the xed values free parametersThus, the integral equation (24 ) may be written approximately as00  g2T8Nf(2)3Z10dpp0ku+ u0kpp0k0  kp+ ip0e (p W )2: (28)6LetI =Z10dpp0ku+ u0kpp0k0  kp+ ip0e (p W )2=  p0u[IG1(p)p  presdp+ iG1(p = pres)]  u0[IG2(p)p  presdp+ iG2(p = pres); (29)where G1(p) = e (p W )2; G2(p) = pe (p W )2; pres=p0k0resk, respectively.Now, expanding1p  pres=  1presf1 + (ppres) + (ppres)2+ ::::::::g; (30)and keeping up to 2nd order in (ppres), from the principal and residual values of the integral, wend00=g2T8Nf(2)3[p0upr+ (p0up2r+u0pr)(e p202+W ) + (p0up3r+u0p2r)f1(W  p02)e p20+1 + 22W2g if(p0u+ pru0)e (pr W )2g]; (31)From equation (19) we nd the dielectric tensor for the Langmuir oscillation"l= (1   1)k20+00: (32)To extract Langmuir oscillation and damping of longitudinal spectrum, we split k0= k0r+ik0i.In fact, k0r= !rand k0i= !i.Thus the dielectric tensor splits into two parts"l= "lr+ i"li; (33)where"lr(k; k0r) = 0; (34)determines the Langmuir spectrum andk0i=  "li(k; k00)@"lr(k;k0r)@k0jk0=k0r; (35)gives the Landau damping. Accordingly, from equations (28) and (29), we nd that"lr(!r; k) = !2r Ak!r Bk2!2r Ck3!3r; (36)and"li(!r; k) =g2T82Nf(2)3(1 +  1) 1p0(u+!rk)e (p0!rk W )2: (37)Then the equation"lr(!r; k) = 0; (38)yields7!2r=Ak!r+Bk2!2r+Ck3!3r; (39)whereA =g2T8Nf(2)3(1 +  1) 1[u+u0p0(e p202+W )];B =g2T8Nf(2)3(1 +  1) 1[up0(e p202+W ) +u0p20f1(W  p02)e p20+1 + 22W2g];C =g2T8Nf(2)3(1 +  1) 1[u0p30f1(W  p02)e p20+1 + 22W2g]: (40)Equation (35) determines the Landau damping, which is given by, = !i=  g2T82Nf(2)3(1 +  1) 1p0(u+!rk)e (p0!rk W )22!r+Ak!2r+Bk2!3r+Ck3!4r: (41)For k ! 0, we nd!r fg2T8Nf(2)3(1 +  1) 1[u+u0p0(e p202+W )]kg13; (42)and,   g2T82Nf(2)3(1 +  1) 1p0(u+!rk)e (p0!rk W )23!r(43)For u = 0 ( thermal bath)!rj k j13; , 1j k j: (44)ACKNOWLEDGEMENTSThis work was done within the framework of the Associateship Scheme of the Abdus SalamInternational Centre for Theoretical Physics, Trieste, Italy. Financial support from the SwedishInternational Development Cooperation Agency is acknowledged. The authors also acknowledgethe Abdus Salam ICTP for hospitality where this work was done. They would also like to thankProfessor Riazuddin for fruitful discussions.8REFERENCES[1] J. I. Kapusta, S. M. H. Wong, Phys. Rev. D62, 037301 (2000).[2] Quark-Gluon Plasma 2, Edited by R. C. Hwa, World Scientic, 1995,[3] Quark-Gluon Plasma, Edited by B. Sinha, S. Pal ans S. Raha, Springer-Verlag, 1990.[4] J. I. Kapusta, Nucl. Phys. B148, 461 (1979).[5] J. P. Blaizot and J.-Y. Ollitraut, Phys. Rev. D48, 1390 (1993).[6] O. K. Kalashnikov and V. V. klimov, Sov. J. Nucl. Phys. 31, 699 (1981). V. V. Klimov, Sov.Phys. JETP 55, 199 (1982).[7] U. Heinz and P. J. Siemens, Phys. Lett, 158B, 11 (1985).[8] Yu. A. Markov and M. A. Markova, ph/9902397 v2 27 Apr 1999.9

On the Langmuir oscillation and Landau damping in quark gluon plasma

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