Within classical transport theory, an approach is presented to derive
systematically effective collision terms and noise sources for effective
kinetic equations of non-Abelian plasmas. This procedure amounts to the
`integrating-out' of fluctuations, and is applicable for in- and
out-of-equilibrium situations. When applied to a hot non-Abelian plasma close
to equilibrium we recover the collision integral and the noise source first
obtained by B\"odeker. It is also shown that our approach, and hence
B\"odeker's effective theory, is consistent with the fluctuation-dissipation
theorem.Comment: 7 pages, needs sprocl.sty, invited talk given by DFL at 5th workshop
  on QCD (QCD2000), Villefranche-sur-Mer, France, 3-7 Jan 200

Litim, Daniel F.

Manuel, Cristina

English

arXiv

Within classical transport theory, an approach is presented to derive systematically effective collision terms and noise sources for effective kinetic equations of non-Abelian plasmas. This procedure amounts to the `integrating-out' of fluctuations, and is applicable for in- and out-of-equilibrium situations. When applied to a hot non-Abelian plasma close to equilibrium we recover the collision integral and the noise source first obtained by B\"odeker. It is also shown that our approach, and hence B\"odeker's effective theory, is consistent with the fluctuation-dissipation theorem.Within classical transport theory, an approach is presented to derive systematically effective collision terms and noise sources for effective kinetic equations of non-Abelian plasmas. This procedure amounts to the `integrating-out' of fluctuations, and is applicable for in- and out-of-equilibrium situations. When applied to a hot non-Abelian plasma close to equilibrium we recover the collision integral and the noise source first obtained by B\"odeker. It is also shown that our approach, and hence B\"odeker's effective theory, is consistent with the fluctuation-dissipation theorem

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Deriving eective transport equations for non-Abelian plasmasDaniel F. Litim a;1 and Cristina Manuel a;21Inst. f. Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg, Germany.2Theory Division, CERN, CH-1211 Geneva 23, Switzerland.Within classical transport theory, an approach is presented to derive systematicallyeective collision terms and noise sources for eective kinetic equations of non-Abelian plasmas. This procedure amounts to the ‘integrating-out’ of fluctuations,and is applicable for in- and out-of-equilibrium situations. When applied to ahot non-Abelian plasma close to equilibrium we recover the collision integral andthe noise source rst obtained by Bo¨deker. It is also shown that our approach,and hence Bo¨deker’s eective theory, is consistent with the fluctuation-dissipationtheorem.HD-THEP-00-23CERN-TH 2000/0981 IntroductionUnderstanding the physics of a quark-gluon plasma at high temperature ordensity { which is expected to be detected in the up-coming heavy ion ex-periments at RHIC and LHC { requires reliable theoretical tools to describetransport phenomena in non-Abelian plasmas in- or out-of-equilibrium. Suchmethods are also of relevance for the physics of the early universe, if the baryonasymmetry could nally be understood within an electroweak framework.In this talkb an approach is discussed which allows for a systematic deriva-tion of eective kinetic equations for non-Abelian plasmas.1-3 The interest indeveloping a formalism based on a classical transport theory for non-Abeliangauge elds 4 is that main properties of a hot quantum plasma can already beunderstood within simple classical terms. Indeed, the soft non-Abelian gaugeelds { those having a huge occupation number { can be treated as classicalelds, while the hard gauge eld modes can be treated as classical particles.This method, based on ‘integrating-out’ fluctuations about some mean elds,has been known since long for Abelian plasmas 5, but a consistent extensionto the non-Abelian case was longtime missing.We apply our formalism to a hot plasma close to equilibrium. While theHTL eective theory is recovered in the Vlasov approximation 6, the inclusionof leading order corrections due to fluctuations results in Bo¨deker’s eectivetheory.7 For a more extensive discussion of this and related eective theories werefer to the discussion in ref. 8. Some further approaches to study quark-gluondynamics are considered in ref. 9.aE-Mails: D.Litim@thphys.uni-heidelberg.de, Cristina.Manuel@cern.chbInvited talk given by DFL at 5th workshop on QCD, Villefranche-sur-Mer, 3-7 Jan 2000.12 Microscopic transport equationsThe starting point for a classical transport theory is to consider an ensembleof classical point particles carrying a non-Abelian charge Qa, where the colourindex runs from a = 1 to N2 − 1 for a SU(N) gauge group. These particlesinteract self-consistently amongst each others, that is, through the classicalgauge elds created by the particles. Their classical equations of motion areknown as the Wong equations 10,mdx^d= p^ ; mdp^d= gQ^aFa p^ ; mdQ^ad= −gfabcp^AbQ^c : (1)Here, A denotes the gauge eld, F a = @Aa − @Aa + gfabcAbAc the cor-responding eld strength and z^  (x^; p^; Q^)() the world line of a particle. Anensemble of such particles can be described by their one-particle distributionfunction f  Pi R d(z − z^i). Making use of (1), its kinetic equation is 4p@@x− gfabcAbQc@@Qa− gQaF a@@pf(x; p; Q) = 0 : (2)This Boltzmann equation is collisionless, since df=d = 0 (Liouville’s theorem).However, it contains eectively collisions due to the long range interactionsas shall become clear in the sequel. Eq. (2) is completed by the Yang-Millsequation DF = J , with the current J [f ] obtained self-consistently fromthe particles. In this classical picture, f is a deterministic quantity once allinitial conditions have been xed. This is, however, not possible, and f has tobe considered as a stochastic (fluctuating) quantity instead.3 Coarse grainingOf particular interest for most physical applications are the macroscopic prop-erties of such a plasma of particles. In this case, a more appropriate quantityto study would be a coarse-grained distribution function f , whose dynamicalequation is expected to be of the Boltzmann-Langevin type,p( D − gQa F a@p  f = C[ f ] +  : (3)(We introduced the shorthands D[A]f  [@−gfabcQcA;b@Qa ]f , @  @=@x,@p  @=@p and @Qa  @=@Qa). A coarse-grained distribution function obtainsfrom the microscopic one after a suitable smearing-out over some characteristicvolume, which, when applied to (2), results in a collision integral C[f ] anda related source for stochastic noise . Physically speaking, these terms are2expected because the coarse-graining integrates-out the modes within a coarse-graining volume. The interactions of such modes can result into additionaleective interactions for the remaining long range modes. Also, the stochasticfluctuations of f are smeared-out, resulting in a noise term .4 Eective transport equationsAn eective kinetic equation, like (3), can be derived systematically withinclassical transport theory, starting with (2). To that end, we perform anensemble average in phase space which allows to split f and the gauge eldsinto their mean eld part and a fluctuating partf(x; p; Q) = f(x; p; Q) + f(x; p; Q) ; (4)A(x) = A(x) + a(x) ; (5)where f = hfi and A = hAi. The mean value of the statistical fluctua-tions vanish by denition, hfi = 0 and hai = 0. This separation into meanelds and statistical, random fluctuations corresponds eectively to a splitinto long/short wavelength modes associated to the mean elds/fluctuations.Performing the ensemble average of (2) and the Yang-Mills equations gives 1;2p( D − gQa F a@p  f = hi+ hi : (6)D F + hJfluci = J : (7)In (6) and (7), we collected all terms quadratic or cubic in the fluctuations into  gQa p( Da − Da)a + gfabcabac@p f(x; p; Q) ; (8)  gpfabcQc @Qa ab f(x; p; Q) + gaaab@p f(x; p; Q) ; (9)Ja;fluc  gfdbc Dadab;ac + fabcab;(( Da − Da)c + gf cdeadae: (10)In addition, we need the dynamical equations for the fluctuations. They arederived from subtracting (2) from (6) asp( D − gQa F a@p  f = gQa( Da − Da)ap@p f+gpab;fabcQc@Qa f +  +  − h + i (11)( D2a − D( Da)a + 2gfabc Fb ac; + Ja;fluc − hJa;fluc i = Ja; : (12)Eqs. (11) and (12) are the master equations for all higher order correlationfunctions of fluctuations. (This hierarchy of dynamical equations is similarto the BBGKY hierarchy.) All the equal-time correlators of f can be de-rived from the basic denition of the Gibbs ensemble average in phase space.23The most important correlator is the quadratic one hffit=t′ which can beexpressed in terms of f and a two particle correlation function.2;11Some few comments. (i) The set of dynamical eqs. (6) - (12) is equivalent tothe original set of microscopic equations. It is exact in that no approximationshave been performed so far, and can be applied to both in- or out-of-equilibriumsituations. (ii) The original equations as well as their split into mean gaugeelds and fluctuations have been shown to be consistent with the non-Abelianmean eld gauge symmetry.2;6 (iii) In order to obtain eective equations forthe mean elds only, one has to integrate-out the fluctuations. This amountto solving explicitly the dynamical equations of f and the gauge eld fluctua-tions for given initial conditions and to express the correlator terms explicitlyas functions of the mean quantities, making use of the basic equal-time cor-relators. In this light, our procedure can be seen as a derivation of collisionintegrals. (iv) The dynamical equations for the fluctuations are non-linear asthey involve fluctuations up to cubic order. Solving them will require someapproximations. For small gauge coupling an expansion in powers of g can beperformed. If the fluctuations remain small within a coarse-graining volume,the second moment approximation can be used, which amount ultimately toa linearisation of the dynamics of the fluctuations.5 These types of approxi-mations are systematic and consistent with the non-Abelian gauge symmetryof the mean elds. (v) In the Abelian limit hi vanishes identically, while hireduces to the well-known Balescu-Lennard collision integral.5 (vi) In the limitwhere fluctuations are fully neglected, the set of equations are known as the(non-Abelian) Vlasov equations.4;65 The hot non-Abelian plasma close to equilibriumWe now turn to the specic example of a hot thermal plasma close to equi-librium. ‘Hot’ implies that particle masses can be neglected to leading orderm(T )  T , and that the gauge coupling, as a function of temperature, is verysmall g(T )  1. The non-Abelian plasma is characterised by two dimensionlessparameters, the gauge coupling g and the plasma parameter , which obtainsas the ratio of the volume per particle over a Debye volume. The Debye (orscreening) length describes the polarisation of the plasma. A small plasmaparameter is mandatory for a kinetic description to be viable. Physically, thismeans that a large number of particles interact coherently within a Debye vol-ume (quasi-particle behaviour). For a classical plasma,  is an independentparameter related to the mean particle number and the gauge coupling.2 For aquantum plasma, the mean particle number can not be xed arbitrarily, and scales as   g3  1. This explains why a kinetic description, for quantum4plasmas, is intimately linked to the weak coupling limit.To simplify our analysis we shall perform some approximations, all of whichcan be understood as a systematic expansion in g. After ensemble averaging,the distribution function f is eectively coarse-grained over a Debye volume.Fluctuations of f within a Debye volume are parametrically suppressed bypowers of g. We expand f about the equilibrium distribution function f eq toleading order in g asf = f eq + g f (1) + f : (13)Solving (6) using (13) for f = 0 reproduces the HTL eective theory.6 Twofurther approximations are employed beyond the HTL level, when f 6= 0. Weshall discard cubic correlator terms in (6), which are additionally suppressedby a power in g, in favour of quadratic ones. This corresponds to neglectingthree-particle collisions in favour of binary ones. We also employ the secondmoment approximation { setting  = hi and  = hi in (11) { which meansthat the eects of collisions are neglected for the dynamics of the fluctuations.All these approximations can be systematically improved to higher order.Let us consider colour excitations, described by the colour current densityJ a (x;v) = g vZdQ dp0 djpjp2 (p0) (p2)Qa f(x; p; Q) ; (14)and v  p=p0 = (1;v);v2 = 1. The current J is obtained after an angleaverage over the directions of v as J(x) =RdΩv4 J (x;v). The colour mea-sure dQ is normalisedRdQ = 1 and contains the Casimir constraints of thegauge group, such asRdQQaQb = C2ab with C2 the quadratic group Casimir(for more details, see ref. 2;6). The approximate dynamical equations for thefluctuations become(v D J a= −m2Dvv( Da0 − D0aa − gfabcvab J c; ; (15)( D2a − D( Da)a+ 2gfabc Fb ac; = Ja ; (16)where the Debye mass mD obtains from the equilibrium distribution functionand the group representation of the particles. Solving for the fluctuations inthe present approximations yields a closed expression for J , and an iterativeexpansion in powers of the background elds for a.1;2 The seeked-for dynamicalequation for the mean quasi-particle colour excitations isv D J 0 + m2Dv F0 = Clin[ J 0] + (x;v) ; (17)where the linearised collision integral Clin can now be evaluated explicitly, usingthe expressions for a and J . The Yang-Mills equation remains unchanged at5this order. For the collision integral, one nally obtains to linear order in themean current, and to leading logarithmic order (LLO) in (ln 1=g)−1  1Clin[ J 0a ](x;v) = gfabc〈ab(x) J c;(x;v)A=0; linear in J; LLO= −γZdΩ04I(v;v0) J 0a (x;v0) (18)with the kernel I(v;v0) = 2(v − v0) − 4 (v:v0)2=p1− (v:v0)2. Notice thatthe collision integral is local in coordinate space, but non-local in the anglevariables. The rate γ = g2T4 ln 1=g is (twice) the hard gluon damping rate.In obtaining (18), we introduced an infra-red cut-o  gmD for the elsewiseunscreened magnetic sector. This result has been rst obtained by Bo¨deker inref. 7. In addition, the source for stochastic noise  in (15) can be identied asa(x;v) = gfabcab(x) J c;(x;v)A=0; J=0(19)ha(x;v)b(y;v0)i = 2γ m2D T I(v;v0) ab (x − y) ; (20)and the self-correlator (20) has been evaluated from (19) to LLO. Notice thatits strength is determined by the kernel of the dissipative process. Next, weshow the close relationship to the fluctuation-dissipation theorem (FDT).6 Fluctuation-dissipation theoremIt is well-known that dissipation in a (quasi-stationary) plasma is intimatelylinked to the fluctuations, a link which is given by the FDT. For the coarse-grained kinetic equation (3) one may ask how the spectral functions of  and fhave to be related to C[f ] in order to satisfy the FDT. This question has beenconsidered in ref.3. Within classical transport theory, the FDT is implementedin a straightforward way.11 The pivotal element is the kinetic entropy S[f ]. Weconsider small deviations of f from the equilibrium, f = f eq+f . The entropy,stationary at equilibrium, is expanded to quadratic order in f , S = Seq+S.We dene the thermodynamical force as F (z) = −(S)=(f(z)), wherez  (x;p; Q). Given C[f ](z), it can be shown 3;11 thath(z)(z0)i = −C(z)F (z0)+C(z0)F (z)(21)is the noise self-correlator for  compatible with the FDT. For the particularexample studied above, (20) follows from inserting (18) into (21). This provesthat Bo¨deker’s eective theory is compatible with the FDT. Furthermore, thecorrelator hffijt=t′ can be derived along the same lines and agrees, to lead-ing order, with hffijt=t′ from the Gibbs ensemble average. This guaranteesthat our formalism is consistent with FDT.67 DiscussionWe have presented a systematic approach to derive eective transport equa-tions for non-Abelian plasmas. The procedure amounts to the ‘integrating-out’of fluctuations about some mean elds. Most importantly, the formalism is ap-plicable for both in- and out-of-equilibrium situations, simply because the mainstatistical information as encoded in the equal-time correlators of fluctuationsdoes not depend on the mean distribution function being thermal, or not. Atthe same time, the formalism { and certain approximations to it { respectsthe underlying non-Abelian mean eld symmetry, which is at the basis for anyreliable computation.For the close-to-equilibrium plasma, we have shown how Bo¨dekers eec-tive theory emerges to leading logarithmic accuracy. The collision integralas well as the necessary noise source have been derived explicitely from themicroscopic theory. In addition, it is established on general grounds thatthe formalism (and hence Bo¨deker’s eective theory) is consistent with thefluctuation-dissipation theorem.As a nal comment we point out that the eective theory is the same for aclassical or a semi-classical/quantum plasma, diering only in the equilibriumdistribution function (Maxwell-Boltzmann vs. Bose-Einstein or Fermi-Dirac),and hence in the corresponding value for the Debye mass. The sole ‘quantum’eect which entered our computation resides in the non-classical statistics ofthe particles, which is all that is needed to correctly describe a hot quantumnon-Abelian plasma close to equilibrium at the present order of accuracy.References1. D.F. Litim and C. Manuel, Phys. Rev. Lett. 82 (1999) 4981.2. D.F. Litim and C. Manuel, Nucl. Phys. B 562 (1999) 237.3. D.F. Litim and C. Manuel, Phys. Rev. D (to appear), hep-ph/9910348.4. U. Heinz, Phys. Rev. Lett. 51 (1983) 351; Ann. Phys. 161 (1985) 48.5. Yu. L. Klimontovich, \Statistical Physics", Harwood Publ, (1986).6. P.R. Kelly, Q. Liu, C. Lucchesi and C. Manuel, Phys. Rev. Lett 72(1994) 3461; Phys. Rev. D 50 (1994) 4209.7. D. Bo¨deker, Phys. Lett. B426 (1998) 351.8. D. Bo¨deker, these proceedings.9. U. Heinz, these proceedings.10. S. Wong, Nuovo Cim. 65A (1970) 689.11. L. Landau and E.M. Lifshitz, \Statistical Physics part 1", Oxford (1981).7

Deriving effective transport equations for non-Abelian plasmas

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