Comment on “Damping of energetic gluons and

arXiv:hep-ph/9203211v1 16 Mar 1992CERN-TH.6375/92Comment on “Damping of energetic gluons andquarks in high-temperature QCD”Anton RebhanTheory Division, CERNCH-1211 Geneva 23, SwitzerlandABSTRACTBurgess and Marini have recently pointed out that the leading contribu-tion to the damping rate of energetic gluons and quarks in the QCD plasma,given by γ = cg2 ln(1/g)T, can be obtained by simple arguments obviatingthe need of a fully resummed perturbation theory as developed by Braatenand Pisarski. Their calculation confirmed previous results of Braaten andPisarski, but contradicted those proposed by Lebedev and Smilga.Whileagreeing with the general considerations made by Burgess and Marini, I cor-rect their actual calculation of the damping rates, which is based on a wrongexpression for the static limit of the resummed gluon propagator.

The effectof this, however, turns out to be cancelled fortuitously by another mistake,so as to leave all of their conclusions unchanged. I also verify the gauge in-dependence of the results, which in the corrected calculation arises in a lessobvious manner.CERN-TH.6375/92January 1992

It has been established by Braaten and Pisarski [1] that a perturbation theory forthe dispersion relations of quasi-particles in high-temperature QCD requires at leastresummation of the leading-order terms, called “hard thermal loops”, whose characteristicscale is given by gT, where g is the coupling constant and T the temperature. By now, anumber of applications exist [2–5] which employ the resummation techniques developed inRef.

[1] to explore the physics of the hot QCD plasma at the scale g2T. Complete resultscan be obtained, if they are not sensitive to a further resummation of the correctionsof order g2T, which would have to include the perturbatively incalculable screening ofstatic magnetic fields [6].Burgess and Marini [7] have recently discussed the case where resummation of thehard thermal loops leaves logarithmic infrared divergences, and they have made precisethe notion [2] that the resummation procedure still allows to reliably extract terms ∝g2T ln(mel./mmagn.) ∼g2 ln(1/g)T, if not those of order g2T.The particular example considered in Ref.

[7] is the evaluation of the leading contribu-tions to the damping rate of gluons or quarks with momenta |p| ≫gT. This kinematicalregion leads to an enormous simplification of the resummation program, because onlythe leading corrections to one internal propagator carrying soft integration momentumneed be resummed, with no complications from the vertices.

The similar case of verymassive quarks has previously been discussed in Refs. [2] and [5].

Burgess and Marinifurther noticed that in such processes, which are dominated by the subleading scale g2T,only the static limit of the resummed gauge propagator is needed.The calculation thus becomes technically similar to the well-known resummation of“ring diagrams” in thermodynamical potentials, which goes under the name of “plasmoneffect” [8]. However, this term is somewhat misleading, as only the static limit of inter-nal lines with multiple self-energy insertions is relevant, which thus resums the electricDebye screening mass rather than the (different) plasmon mass corresponding to long-wavelength plasma oscillations.

The latter is determined by the long-wavelength limit ofthe gluon self-energylimq→0 Πµν(q0, q) = m2(ηµν −δ0µδ0ν) + O(gmq0),(1)with m2 = 19(Ca + 12nq)(gT)2, whereas the static limit islimq0→0 Πµν(q0, q) = m2el.δ0µδ0ν + O(gmq),(2)with m2el. = 3m2 [9].

Evidently, these limits do not commute.1

In the calculation carried out in Ref. [7], Eq.

(1) was used instead of Eq. (2) for theresummed gluon propagator at zero frequency, which led the authors of Ref.

[7] to using∆∗wrongµνq0=0 = −" 1q2δ0µδ0ν +1q2 −m2 ηµν −δ0µδ0ν −qµqνq2!+ ξqµqν(q2 −ξm2)q2#(3)in place of the correct one∆∗µνq0=0 = −"1q2 −m2el.δ0µδ0ν + 1q2 ηµν −δ0µδ0ν −qµqνq2!+ ξ qµqν(q2)2#. (4)In the latter only the spatially longitudinal mode is screened, leaving both the spatiallytransverse mode and the (4-D longitudinal) gauge mode massless.Recalculation of the “hard” (|q| > λ ≫g2T) contributions to the damping rate γ ofenergetic (|p| ≫gT) transverse gluons considered in Ref.

[7] leads toγhard=g2CaT 14π2ImZ 1−1 dzZ ∞λdq qz + q/2|p| −iε" 1q2 −1q2 + m2el.−(1 −ξ)z2q2#+O(g2Tλ0),(5)where the terms in the large brackets correspond to the contributions of spatially trans-verse, spatially longitudinal, and gauge modes, respectively. [In the case of quarks, itturns out that the only change consists in replacing Ca by Cf.

]On the other hand, with the wrong propagator of Eq. (3) used in Ref.

[7], these termswould read"1q2 + m2 −1q2 −(1 −ξ)z2q2(q2 + m2)(q2 + ξm2)#. (6)The leading contribution to γ can be extracted from the logarithmic dependence ofγhard on the cutoffλ ≪gT, together with the assumption that the inherent scale of theundetermined soft contribution is given by g2T (through the non-perturbative magneticmass or through dynamical screening at this scale).

The spatially transverse and spatiallylongitudinal contributions in Eq. (5) thus lead toγ ≈g2CaT4π ln mel.λ+ ln λg2T!= g2CaT4πln 1g + O(g2T),(7)with the transverse mode being responsible for the dominant term proportional to ln(g2T),and therefore for the positive sign of γ.

The latter is a consequence of the positivity ofthe transverse density in a spectral representation of the resummed gluon propagator [2].2

The wrong result of Eq. (6), on the other hand, should have led to a result of equalmagnitude, but with a reversed sign, as the roles of spatially longitudinal and transversemodes happen to be interchanged.

(The difference between m and mel. =√3m onlyaffects the terms of O(g2).) The fact that in Ref.

[7] also a positive result was reportedis due to the additional mistake of a reversed sign of iε in their Eq. (11) compared withEq.

(5) above. With the usual sign convention γ = −Im E|pole, the correct analyticalcontinuation is given by k0 →k0 + iε.A more conspicuous difference between the correct and the wrong results, Eq.

(5) andEq. (6), respectively, concerns the contributions from the gauge modes.

With the wrongexpression for the static gluon propagator, Eq. (3), the gauge modes obviously would notcontribute to the infrared singular part, whereas in the corrected result, Eq.

(5), theyseem to do so by superficial power counting. However, performing the angular integrationZ 1−1 dzz2z + q/2|p| −iε = O q|p|!

(8)reveals that they indeed do not contribute to the leading logarithms in Eq. (7), as ex-pected from general arguments [10] for the gauge independence of dispersion relations infinite-temperature QCD.Thus, all the results on γ presented in Ref.

[7], its magnitude, its sign, and its gaugeindependence, remain, somewhat fortuitously, unchanged, and continue to confirm theresults by Braaten and Pisarski [11], while contradicting those proposed by Lebedev andSmilga [12].Acknowledgements: I should like to thank Tanguy Altherr and Rob Pisarski for usefuldiscussions.3

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