We derive the quantum kinetic equation for a pure gluon plasma, applying the
background field and closed-time-path method. The derivation is more general
and transparent than earlier works. A term in the equation is found which, as
in the classical case, corresponds to the color charge precession for partons
moving in the gauge field.Comment: RevTex 4, 4 pages, no figure, PRL accepted versio

B. S. DeWitt

C. Greiner

D. F. Litim

E. Calzetta

G. 't Hooft

G. Z. Zhou

H. Kluberg-Stern

H. Satz

H. Stöcker

H.-Th. Elze

J. Jalilian-Marian

J. Schwinger

J.-P. Blaizot

K. Geiger

K. J. Eskola

K. Redlich

L. F. Abbott

L. McLerran

L. V. Keldysh

M. Gyulassy

P. R. Kelly

Q. Wang

R. B. Sohn

R. Baier

S. A. Bass

S. K. Wong

S. Mrowczynski

U. Heinz

W. Greiner

X.-F. Guo

English

arXiv

Crossref

Physical Review Letters

Kinetic Equation for Gluons in the Background Gauge of QCD

We derive the kinetic equation for pure gluon QCD plasma, applying the background field method. The derivation is quite general and is not limited to the vicinity of equilibrium as it was assumed in earlier work. We show that the quantum kinetic equation contains a term as in the classical case, that describes a color charge precession of partons moving in the gauge field. We emphasize that this new term is necessary for the gauge covariance of the resulting equation.We derive the quantum kinetic equation for a pure gluon plasma, applying the background field and closed-time-path method. The derivation is more general and transparent than earlier works. A term in the equation is found which, as in the classical case, corresponds to the color charge precession for partons moving in the gauge field

Wang, Q.

Redlich, K.

Stoecker, Horst

Greiner, W.

CERN Document Server

arXiv:nucl-th/0111040   13 Nov 2001CERN-TH/2001-313KineticEquationforGluonsintheBackgroundGaugeofQCDQ.Wang,1,K.Redlich,2,3,yH.Stocker,1,zandW.Greiner1,x1InstitutefurTheoretischePhysik,J.W.Goethe-Universitat,D-60054Frankfurt,Germany2TheoryDivision,CERN,CH-1211Geneva23,Switzerland3InstituteforTheoreticalPhysics,UniversityofWroclaw,PL-50204Wroclaw,Poland(Dated:January29,2002)WederivethekineticequationforpuregluonQCDplasma,applyingthebackgroundeldmethod.Thederivationisquitegeneralandisnotlimitedtothevicinityofequilibriumasitwasassumedinearlierwork.Weshowthatthequantumkineticequationcontainsatermasintheclassicalcase,thatdescribesacolorchargeprecessionofpartonsmovinginthegaugeeld.Weemphasizethatthisnewtermisnecessaryforthegaugecovarianceoftheresultingequation.PACSnumbers:12.38.Mh,25.75.-q,24.85.+p,11.15.KcHeavyioncollisionsatultra-relativisticenergiesarewidelyexpectedtobealaboratorytostudytheforma-tionandpropertiesofhighlyexcitedQCDmatterinthedeconnedQuark-GluonPlasma(QGP)phase[1].TheQGPisconsideredasapartonicsystembeingat(orcloseto)localthermalequilibrium.Thus,tostudythecondi-tionsforthepossibleformationofQGPinheavyioncolli-sionsoneneedstoaddressthequestionofthermalizationoftheinitiallyproducedpartonicmedium[2].Onthetheoreticallevelthisrequirestheformulationofkineticequations[3]involvingcolordegreesoffreedomandthenon-AbelianstructureofQCDdynamics.Dierentmod-elsfortheinitialconditionsinultra-relativisticheavyioncollisionssuggestthatattheearlystagethemediumisdominatedbygluondegreesoffreedom[4];thekineticequationforthepuregluonplasmaisthusofspecialin-terest.Thekineticsofaparticlemovinginanon-AbeliangaugeeldwasrststudiedforSU(2)symmetry[5].Theequationofmotionderivedin[5]madeitclearthatinthepresenceofthegaugeeldthecolorchargeofthepar-ticlekeepsrotatingorundergoingprecessionwithtime.Consequently,thereisanexplicittermintheclassicalki-neticequation,whichdescribesthetimeevolutionofthecolorchargeofpartons.Applyingatypicalmethodtoderiveasemi-classicaltransportequationfromthequan-tumtheory,whichiscontainedintheformulationoftheequationofmotionfortheWignerfunctionanditssub-sequentgradientexpansion[6,7],oneeasilylosestheexplicitpresenceofthecolorchargeprecession.Theusualtreatmentofthegluontransportequationisbasedonthedecompositionofthegluoneldintoameaneldandaquantumuctuation,orintoasoftandahardpart.Underthisapproximationthegluontrans-portequationthendescribesthekineticsofthequantaintheclassicalmeaneld[6,8].ThispictureissomehowElectronicaddress:qwang@th.physik.uni-frankfurt.deyElectronicaddress:redlich@violet.ift.uni.wroc.plzElectronicaddress:stoecker@th.physik.uni-frankfurt.dexElectronicaddress:greiner@th.physik.uni-frankfurt.desimilartotheoneusedwhilestudyingtheenergylossofafastpartonmovinginthesoftmeaneld[9,10].Sincethisdecompositionisnotrigorous,theexplicitgaugein-varianceoftheformalismremainsunclear.ToincludetheclassicalchromoeldintoQCDinaproperway,oneusesthebackgroundeldmethodofQCD(BG-QCD)introducedbyDeWittand'tHooft[11,12,13].Thead-vantageofBG-QCDisthatitisformulatedinanexplicitgaugeinvariantmanner.Oneoftherstattemptstoderivethegluontrans-portequationinBG-QCDwaspresentedin[14].Buttheobtainedequationisnottransparent.ThemostrecentworkhasbeendonebyBlaizotandIancu[15,16]inthecontextoftheclose-time-path(CTP)formalism.There,however,theauthorsfocusonformulatingthetransportequationinthevicinityofequilibrium.Inthispaper,weusetheCTPmethodtoderivethekineticequationforthegluonplasmainBG-QCD.Ourderivationisgoingbeyondpreviousresultsasitisquitegeneralandnotlimitedtothevicinitytoequilibrium.Oneofthemostimportantfeaturesofthekineticequa-tionderivedhereisthatacolorchargeprecessiontermisexplicitlypresent,similartotheoneintheclassicalequation[5,6,7].Wedemonstratethatthistermisnec-essarytoguaranteethegaugecovarianceoftheresultingVlasovequation.Inthefollowingweuseg=diag(1; 1; 1; 1)asthemetrictensor,andwewriteLorentzindicesassub-scriptsandcoloronesassuperscriptsfortherelevantquantities.Forthegaugeeldanditseldtensorwealsodenote:AAaTaandF[A]Fa[A]Ta,where(Ta)ij=ifiajarethegeneratorsoftheSU(3)cadjointrepresentation.Thetwo-pointGreenfunction(GF)orself-energy(SE)aretreatedasmatrices,andthusthecolorand/orLorentzindicesaresometimesomitted.Applyingthebackgroundeldmethod,wedecomposetheconventionalgluoneldintothesumofaclassicalbackgroundpartAandaquantumuctuationQ.In-cludingtheappropriategaugexingandghosttermforthebackgroundgaugeDij[A]Qj=0,theBG-QCDLa-2grangian reads [13]L =  14Fi[A+ Q]Fi[A+ Q] 12(Dij[A]Qj)2+CiDij[A]Djk[A+Q]Ck; (1)where Dij[A(x)] = [@x  igA(x)]ijis the covariantderivative, Ci/Ciare the ghost/anti ghost elds and  isthe gauge xing parameter.The above Lagrangian is invariant under the localgauge transformation of type I (type II transformationis irrelevant to the current problem) where the back-ground eld transforms as a conventional gauge elds,A0= UAU 1+ig 1U@U 1, while the gluon and ghosteld transform as a matter eld, Q0= UQU 1[13].Here U (x) = exp[ig!a(x)Ta] is the transformation ma-trix.The non-equilibrium dynamics is usually described inthe CTP formalism [17]. Here the action can be writtenas SCTP= S+  S +K(A; Q) where all elds in Sare dened on the positive/negative time branches andK(A; Q) is the kernel incorporating initial state cor-relations. The GF for the gluon has four components:[G++; G+ ; G +; G  ]  [GF; G<; G>; GF], marked bythe positive or the negative time, respectively. At the treelevel there are only vertices with all positive or all nega-tive time variables, which dier by a sign. The non-localkernel K(A; Q) appearing in SCTPis non-vanishingonly at the initial time and plays the role of the initialdensity matrix. If the initial time is not in the remotepast, it introduces a time dependence to the distributionfunction, and consequently the time-translational invari-ance is lost [18]. It is interesting to note that the situationhere is similar to the pinch singularity discussed in [19].In the CTP formalism, choosing the physical repre-sentation for the GF and SE [20], the Dyson-Schwingerequation (DSE) for GCreadsD(x1)GC(x1; x2) =  Zd4x0[C(x1; x0)GA(x0; x2)+R(x1; x0)GC(x0; x2)] ; (2)GC(x1; x2)Dy(x2) =  Zd4x0[GR(x1; x0)C(x0; x2)+GC(x1; x0)A(x0; x2)] ; (3)where the dierential operators D and Dyin the Feynmangauge [21] ( = 1) are expressed by:Dhi= gDha[A]Dai[A] + 2gfhaiFa[A] ; (4)Dy;hi= gDy;ha[A]Dy;ai[A] + 2gfhaiFa[A] ; (5)where Dy;ij[A(x)]  [ @x+igA(x)]ijis the conjugatecovariant derivative where the dierential operator actson the function on its left.In the evolution of the gluon eld one distinguishes dif-ferent scales, which characterize quantum and soft collec-tive motion. We introduce a mass parameter, , as theseparation point of the quantum and the kinetic scale.In the weak coupling limit g  1, the scale of collectiv-ity  1=(g) is much larger than the typical extensionof hard quantum uctuations  1=. The eect of theclassical eld A on the hard quanta involves the cou-pling gA to the hard propagator and is of the size ofthe soft wavelength  1=(g). The above separation ofscales is the basis for the gradient expansion where oneexpresses all 2-point GF;s in terms of the relative y andthe central X coordinate. Here are some typical scales:y = x1 x2 1=, X = (x1+x2)=2  1=(g), A(X)  and F [A(X)]  g2.In order to obtain the gauge-covariant kinetic equationone uses the gauge-covariant Wigner function~G(X; y)(~GC,~G>or~G<) dened byG(x1; x2) = V (x1; X)~G(X; y)V (X;x2) ; (6)where V (z1; z2) = TPexp(igRz1P ;z2dzA) denotes a Wil-son link with respect to the classical background eld.One can also dene the Wilson link as a functional ofA + Q, but it is a much more complicated case and be-yond the scope and subject of this work.The covariant Wigner function~G(X; y) transforms asU (X)~G(X; y)U 1(X); only the transformation at a sin-gle point X is thus relevant. For G(x1; x2), however, thegauge transformation involves two points and thereforeis not gauge-covariant.The DSE given by Eqs.(2) and (3) can be expressedin terms of covariant Wigner functions. To evaluateD(x1)G(x1; x2) one needs to know the derivatives of aWilson link. Following the results obtained in [7]ÆV (z1; z2) = igA(z1)dz1V (z1; z2) igV (z1; z2)A(z2)dz2 igZ10ds V z1; z(s)F z(s)V z(s); z2(z1  z2) dz2+ (dz1  dz2)s; (7)one nds,Dx1(V1~GV2)  V1@y~G+12DX~G+12ig~GA ig38yF~G  ig18~GyFV2; (8)D2x1(V1~GV2)  V114D2X~G+ @yDX~G+ @2y~G ig34yF@y~G  ig14(@y~G)yF+ig(@y~G)A  ig38yFDX~G+ig12(DX~G)A+ ig14~G@XA ig18(DX~G)yF 14g2~GA2V2;(9)3where we have used: V1 V (x1; X), V2 V (X;x2),~G ~G(X; y), DX D[A(X)], D2t DtDt, AA(X), A2 A(X)A(X) and F F(X). Theconjugate expressions (V1~GV2)Dyx2and (V1~GV2)Dy2x2canbe obtained from Eqs.(8) and (9) by taking their Hermi-tian conjugates and then interchanging x1and x2. Usingthese results together with Eq.(6) we derive the followinggauge conditions for~G [22]:@X~G+ ig[~G; A] = 0 ; @y~G= 0 ; (10)which are valid up to O(g). The second equation is atransversality condition for~G.Taking the dierence between Eq.(2) and Eq.(3) andusing Eq.(9) and its conjugate expression, we derive thefollowing kinetic equation for the gluon plasma:q  @X~GC+ ig(~GCq A  q A~GC)+12gqF(@q~GC) +12g(@q~GC)qF+g(F~GC ~GCF) = 0 ; (11)where~GC~GC(X; q).The above equation is located at the collective coor-dinate X and is gauge-covariant under the local gaugetransformation U (X), i.e. it transforms as U (  )U 1.Indeed, noting that both Fand~GCare gauge-covariant and @qdoes not aect U , it is obvious thatthe last three terms are gauge-covariant. To verify thatthe rst two terms also preserve gauge covariance, weexplicitly write down their transformationq  @X~GC! U (q  @X~GC)U 1+ (q  @XU )~GCU 1+U~GCq  @XU 1;ig~GCq A ! igU~GCq A U 1  U~GCq  @XU 1; igq A~GC!  igUq A~GCU 1 (q  @XU )~GCU 1:With the above gauge transformations one clearly seesthat the sum of the rst two terms in Eq.(11) indeedtransforms as U (  )U 1and therefore preserves thegauge covariance.The quantum kinetic equation Eq.(11) is derived in aquite general manner without requirements or assump-tions that the QCD plasma appears close to equilibrium.In the following we show that Eq.(11) is a natural quan-tum generalization of the classical Vlasov equation. Inparticular the color charge precession will be explicitlyidentied in the quantum description of color charge ki-netics given by Eq.(11).The classical kinetic equation for the color singletdistribution function f(x; p;Q) has the following form[5, 6, 7]:p[@  gQaFa@p gfabcAb(x)Qc@Qa]f(x; p;Q) = 0 ; (12)where Qais the classical color charge and a = 1; ::; N2c 1.Comparing Eq.(12) with the quantum expressionEq.(11) it is clear that the color singlet distribution func-tion f is replaced by the gauge-covariantWigner function~GCwhich is a color matrix in the adjoint representation.One can also recognize that the rst, third and fourthterms of Eq.(11) are the quantum generalization of therst two terms in Eq.(12). The last term in Eq.(11) ap-pears from the covariant operators [23] and hence is notpresent in the classical equation.Particularly interesting is the appearance of the secondterm in Eq.(11). We have seen that its presence is crucialto assure the gauge covariance of the Vlasov equation. Inaddition this term has an interesting physical meaning.It is the quantum analogue to the color charge precessionin the classical kinetic equation. To see this more clearly,one can expand~GC(X; q) with respect to the generatorsTaof the adjoint representation as follows~GC(X; q) = N(X; q) + TaNa(X; q)+TaTbNab(X; q) +    ; (13)where Taare quantum analogues to the classical colorcharges Qa; N(X; q) is the color singlet and Na(X; q)the color octet function, etc. This expansion is similar tothe multipole expansion of an electromagnetic source inelectrodynamics. In the lowest order approximation, onecan keep only the rst two terms in Eq.(13) and neglectall others. Then the second term of Eq.(11) becomesig(~GCq A  q A~GC) '  gfabcqAbTc@Ta~GC(14)which is just the classical color precession term inEq.(12).Since we know that DX gA(X)  g, the rst twoterms of Eq.(11), i.e. q  @X~GCand the color precessionterm, are at leading order O(g2), while other terms areat subleading order O(g22). In the vicinity of equilib-rium the natural scale in the system is the temperatureT . The mean distance between particles is of the orderof  1=T , while 1=(gT ) characterizes the scale of collec-tive excitations [15, 16]. For small coupling constant gthese two scales are well separated. The covariantWignerfunctions can be expanded around their equilibrium val-ues:~G =~G(0)+ Æ~G, where the equilibrium function~G(0)is a color singlet and the uctuation Æ~G  g2~G. Typicalscales are q  T , DX g2T , gF  (DX)2 g4T2. Thus,at leading order, only the rst term of Eq.(11) survivesand the precession term vanishes due to the color-singletnature of~G(0).The analogy and dierences of the quantum and theclassical Vlasov equation can be made even more trans-parent by formulating the equations for the color mo-ments. Corresponding to Eq.(11) one getsq  @Xh+ gqFa@qha+g(Faha  haFa) = 0 ; (15)q  @Xha+ gfabcq Abhc4+gqFb@q12[hab+ hba]+g(Fbhab  hbaFb) = 0 ; (16)where we dened h= Tr(~GC), ha= Tr(Ta~GC) andhab= Tr(TaTb~GC). The classical equations for colormoments of f(x; p;Q) arep  @xf(x; p)  gpFa@pfa(x; p) = 0 ; (17)p  @xfa(x; p) + gfabcp Ab(x)fc(x; p) gpFb@pfab(x; p) = 0 ; (18)where we dened fab(x; p) =RdQQaQbf(x; p;Q) andf(x; p) =RdQf(x; p;Q), fa(x; p) =RdQQaf(x; p;Q).Comparing Eq.(15) with Eq.(17) and Eq.(16) withEq.(18), one sees that apart from the term (Fh hF), which comes from the covariant operators, thequantum and classical equation have similar structure.The identication of the color precession term in Eq.(11)is then straightforward.In summary, by applying the closed-time-path formal-ism we have derived the gluon kinetic equation in thebackground gauge of QCD. Equation (11) is quite gen-eral and is not limited to the vicinity of equilibrium. Itis a kinetic equation with respect to a gauge-covariantWigner function, which is a matrix in adjoint color space.Therefore it contains many non-Abelian features whichare absent from the well known classical equation. A no-table feature is that, as in the classical case, it containsa term that corresponds to the color precession, the non-Abelian analogue to the Larmor precession for particleswith magnetic moments in a magnetic eld. This termwas not explicitly shown before in the formulation of aBoltzmann equation for the QCD plasma. We nd thatthis term is necessary to the gauge covariance of the ki-netic equation.One of us Q.W. acknowledges nancial support of theAlexander von Humboldt-Foundation (AvH) and appre-ciated help from D. Rischke. This work is partiallyfunded by DFG, BMBF and GSI. K.R. acknowledges apartial support of the Polish Committee for Scientic Re-search (KBN-2P03B 03018). Stimulating comments anddiscussions with R. Baier, J.-P. Blaizot, E. Iancu andS. Leupold are acknowledged. Our special thanks go toX.-N. Wang for his help and interest in this work.[1] For a recent review, see eg. S.A. Bass, M. Gyulassy, H.Stocker and W. Greiner; J. Phys. G25, R1 (1999); H.Satz, Rep. Prog. Phys. 63, 1511 (2000).[2] R. Baier, A.H. Mueller, D. Schi and D.T. Son, Phys.Lett. B502, 51 (2001).[3] For a recent review, see eg. J.-P. Blaizot and E. Iancu,hep-ph/0101103, Phys. Rept., to appear.[4] L. McLerran and R. Venugopalan, Phys. Rev. D49, 2233(1994); D49, 3352 (1994); D50, 2225 (1994); K.J. Eskola,K. Kajantie, P.V. Ruuskanen and K. Tuominen, Nucl.Phys. B570, 379 (2000).[5] S. K. Wong, Nuovo Cimento A65, 689 (1970).[6] H.-Th. Elze and U. Heinz, Phys. Rep. 183, 81 (1989); U.Heinz, Ann. Phys. 161, 48 (1985); 168, 148 (1986).[7] H.-Th. Elze, M. Gyulassy and D. Vasak, Nucl. Phys.B276, 706 (1986).[8] K. Geiger, Phys. Rev. D56, 2665 (1997).[9] M. Gyulassy and X.-N. Wang, Nucl. Phys. B420, 583(1994); M. Gyulassy, X.-N. Wang and M. Plumer, Phys.Rev. D51, 3436 (1995); X.-F. Guo, and X.-N. Wang,Phys. Rev. Lett. 85, 3591 (2000).[10] R. Baier et al., Nucl. Phys. B483, 291 (1997); B484, 265(1997); R. Baier et al., Nucl. Phys. B531, 403 (1997).[11] B.S. DeWitt, Phys. Rev. D162, 1195 and 1239 (1967).[12] G. 't Hooft, Nucl. Phys. B62, 444 (1973).[13] L.F. Abbott, Nucl. Phys. B185, 189 (1981); R.B. Sohn,Nucl. Phys.B273, 468 (1986); H. Kluberg-Stern and J.B.Zuber, Phys. Rev. D12, 482 (1975).[14] H.-Th. Elze, Z. Phys. C47, 647 (1990).[15] J.-P. Blaizot, and E. Iancu, Nucl. Phys. B557, 183(1999).[16] J.-P. Blaizot and E. Iancu, Nucl. Phys.B417, 608 (1994);B570, 326 (2000); Phys. Rev. Lett. 70, 3376 (1993).[17] J. Schwinger, J. Math. Phys. 2, 407 (1961); L.V. Keldysh,Sov. Phys. JETP 20, 1018 (1965); G.Z. Zhou, Z.B. Su,B.L. Hao and L. Yu, Phys. Rep. 118, 1 (1985); E.Calzetta and B.L. Hu, Phys. Rev. D37, 2878 (1988).[18] Q. Wang, K. Redlich, H. Stocker and W. Greiner, inpreparation.[19] T. Altherr and D. Seibert, Phys. Lett. B333, 149 (1994);R. Baier, M. Dirks, K. Redlich and D. Schi, Phys. Rev.D56, 2548 (1977); C. Greiner and S. Leupold, Eur. Phys.J. C8, 517 (1999); I. Dadic, Phys. Rev. D59, 125012(1999); D63, 025011 (2001).[20] In the physical representation the elements of GF andSE matrices are expressed by the symmetric (C), theretarded (R) and the advanced (A) components. Theyare related to the original components by the followingtransformations: GA= GF  G>, GR= GF  G<,GC= G>+ G<(the same relations hold for the SEC=R=Aup to a sign in front of >and <).[21] In fact due to gauge conditions: Dij[A(x1)]Gjk(x1; x2) =0 and Gij(x1; x2)Dy;jk[A(x2)] = 0, Eqs. (2,3) are inde-pendent of the gauge parameter .[22] To derive these results one needs the following gauge con-ditions for G (G>, G<or GC) with respect to x1and x2:D[A(x1)]G(x1; x2) = 0 and G(x1; x2)Dy[A(x2)] =0. They are consequences of the background gauge con-dition D[A(x)]Q(x) = 0.[23] This term can be written in dierent form by using gen-erators of the Lorentz transformation in vector represen-tation. Similar term can be found for the quark, but ex-pressed through generators in spinor representation.

Stöcker, H.

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Kinetic Equation for Gluons in the Background Gauge of QCD

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