---

THE DAMPING OF ENERGETIC GLUONS AND QUARKS

arXiv:hep-th/9109051v1 26 Sep 1991McGill–91/24July 1991THE DAMPING OF ENERGETIC GLUONS AND QUARKSIN HIGH-TEMPERATURE QCDC.P. Burgess and A.L.

Marini∗Physics Department, McGill University, 3600 University St.Montr´eal, Qu´ebec, Canada, H3A 2T8.ABSTRACTWhen a gluon or a quark is sent through the hot QCD plasma it can be absorbed into theambient heat bath and so can acquire an effective lifetime. At high temperatures and forweak couplings the inverse lifetime, or damping rate, for energetic quarks and transversegluons, (those whose momenta satisfy |p| ≫gT) is given by γ(p) = c g2 log1gT +O(g2T).

We show that very simple arguments suffice both to fix the numerical coefficient, c,in this expression and to show that the O(g2T) contribution is incalculable in perturbationtheory without further assumptions. For QCD with Nc colours we find (expressed in termsof the casimir invariants Ca = Nc and Cf = (N 2c −1)/(2Nc)): cg = + Ca4π for gluons andcq = + Cf4π for quarks.

These numbers agree with the more detailed calculations of Pisarskiet.al. but disagree with those of Lebedev and Smilga.

The simplicity of the calculationalso permits a direct verification of the gauge-invariance and physical sign of the result.∗internet: cliff@physics.mcgill.ca and alex@physics.mcgill.cabitnet: py30@mcgilla

The behaviour of nonabelian gauge theories at finite temperature is of theoreticalinterest due to the surprisingly rich structure they exhibit even within the perturbativeregime of weak coupling and high temperatures. They may also have phenomenologicalapplications to the interpretation of the collisions of heavy nuclei at high energies.In recent years much attention has been directed towards understanding the propertiesthat gluons and quarks acquire as they propagate through a quark-gluon plasma.1 Thishas been at least partially due to the sometimes contradictory and confusing results of theearliest one-loop calculations.

In these calculations the gluon damping constant, γg(p),was found in the static (|p| →0) limit to depend on the gauge on which it was computedand, for some gauges, to be negative. If taken seriously such a negative damping constantwould indicate an instability of the thermal state towards gluon emission.

The source ofthe confusion with these calculations2 is that they neglect higher-loop contributions thatare of the same order in the gauge coupling constant, g, as are the terms that are kept.This is because in perturbation theory (in three space dimensions) at finite temperaturessuccessive terms in the loop expansion need not be suppressed relative to fewer loops byadditional factors of the gauge coupling.The failure of the loop expansion arises at finite temperatures because of the occurenceof severe infrared divergences. These infrared divergences are more troublesome than theyare at zero temperature due to the singular behaviour of the Bose-Einstein distributionfunction at low energies: n(E) = (eE/T −1)−1 ≈TE +· · ·.

This behaviour causes quantitiesto blow up like a power of the infrared cutoffrather than simply logarithmically as they doat zero temperature.3 Indeed, a simple power-counting argument shows that for QCD attemperature T a generic ℓ-loop graph can contribute an amount proportional to (g2T/λ)ℓrelative to the tree-level result. λ in this expression is an infrared cutoff.

Higher loopsclearly need not be suppressed once λ is as low as g2T.In fact the loop expansion already fails even for an infrared cutoffas large as λ ≈gTsince a subclass of diagrams can be even more infrared singular than is indicated by thegeneric power-counting argument. The dangerous graphs are those such as the ‘ring’ graphswithin which multiple self-energy insertions are made along a single internal line.4Inreference [5] Braaten and Pisarski argue that these last contributions may be resummed by2

dressing all ‘soft’ lines—those carrying momenta less than or of order gT—by the calculablecontributions of ‘hard thermal loops’.In ref. [2] they then compute the implicationsfor the particular case of the damping rate for static gluons, arguing that the failureof perturbation theory is in this case completely cured by such a resummation.Thislast conclusion is, however, disputed by Lebedev and Smilga,6who argue that othercontributions beyond the hard thermal loops can contribute equally large effects to thestatic gluon damping constant.The purpose of this letter is to point out that for at least some physical quantitiesthe dominant part of the result for small g can be identified quite simply without makinguse of the complete resummation formalism.

We take by way of illustration the dampingconstant, γ(p), for transverse gluons (and for quarks) but evaluated for momenta |p| ≫gTrather than in the static limit. For these momenta the dominant contribution for smallcoupling to γ(p) is momentum independent and has the form c g2 log1gT + b g2T + · · ·.We show that the coefficient c can be computed by a simple, analytic calculation thatonly involves wavelengths that are within the perturbative regime: λ ≫g2T.

The same isnot true for the coefficient b which can receive contributions for λ ≈g2T. We can in thisinstance therefore explicitly verify that γ(p) is indeed gauge-independent and positive.Our results, given in and immediately following eq.

(12), may be compared to moredetailed calculations. For fermions they agree with those of ref.

[7] but disagree by afactor of 3 with the real-time calculations of ref. [6].

For transverse gluons we agree withthe result of an as-yet-unpublished version of a full resummation calculation,8 but do notagree with the real-time estimate of ref. [6].We now turn to a description of our calculation.

We focus first on the purely gluonictheory since this is the case for which there is the most simplification over a full calculation.The dispersion relation relating the energy and momentum for a relativistic transversegluon traversing the plasma may be determined by the position, E(p) = ω(p) −iγ(p), inthe complex energy plane of the zero of the transverse part of the inverse gluon propaga-tor. Using the usual one-parameter family of covariant gauges for which the bare inverseFeynman propagator is:G−1bareµν (p) = p2ηµν +ξ−1 −1pµpν,(1)3

and the full thermal inverse propagator is:G−1fullµν (p) =G−1bareµν (p) + Πµν(E, p),(2)the dispersion relation for transverse gluons becomes:E2(p) −p2 = 12Πii −pipjΠijp2. (3)Here i and j are to be summed over the three spatial directions, i, j = 1, 2, 3, in theplasma rest frame.

Our task is to compute the leading contributions to Πµν(E, p) and to,in particular, identify the origin of the g2 log g terms.In order to sort out the size of the various contributions to Πµν it is useful to dividethe loop integrations according to whether or not they involve only energies and momentathat are greater than some infrared cutoff, λ. That is:Πµν(E, p) = Πsoftµν (E, p, λ) + Πhardµν (E, p, λ),(4)in which all loop momenta in Πhardµν (E, p, λ) are cut offin the infrared at λ.

Provided thatλ is chosen sufficiently high Πhardµνis calculable within the loop expansion. Πsoftµνis notcomputable in this way and must be obtained by other means.

An important issue to beaddressed is how much any desired quantity depends on the largely undetermined Πsoftµν .If λ is chosen much greater than gT then the lowest-order contribution to Πhardµνarisesat O(g2) due to one-loop graphs. There are four such one-loop graphs: one tadpole andone vacuum-polarization graph having either an internal gluon, quark or ghost loop.

Forexample, using the Matsubara imaginary-time technique,9 the gluon vacuum-polarizationgraph contributes an amount:Πhardµν (E, p, λ) = g2CaT2∞Xn=−∞Zq2>λ2d3q(2π)4 Vµλρ(p, −p −q, q) Gλκ(p + q)× Gρσ(q) Vνσκ(−p, −q, p + q). (5)Here the gluon propagator is the inverse of eq.

(1) and the three-point gluon vertex is givenby the zero-temperature expression: Vµλρ(p, q, k) = (p−q)ρηµλ +(q−k)µηρλ +(k −p)ληµρ.The quadratic invariant in the adjoint representation is Ca = Nc (= 3) for the gauge group4

SU(Nc). We work in Euclidean signature and the time component of every four-vectoris an integer times 2πT.

The summation is over the integer corresponding to the loopmomentum, q. The lower limit on the momentum integration is meant as a reminder ofthe infrared cutoff.If the infrared cutoffshould instead be chosen to satisfy g2T ≪λ ≪gT then higher-loop graphs can contribute to the same order in g as do the one-loop graphs in whichthe loop momenta are of order gT.After resumming these higher-loop contributionsthe leading result for Πhardµνis given by the same one-loop graphs as before, with theproviso5 that each propagator (or vertex) is to be replaced with an ‘effective’ resummedpropagator (or vertex): Gµν →G∗µν and Vµλρ →V ∗µλρ.

Each of these resummed quantitiesagrees (up to higher powers of g) with its bare counterpart unless all of the momentaentering the line (or vertex) in question are themselves O(gT). For such soft momenta,however, the resummed items are relatively complicated functions of the ratios of theenergies and momenta that flow through that part of the graph.The main technicaldifficulty with working with the effective propagators and vertices lies in manipulating thismore complicated form.Some of this extra complication simplifies if the external momentum is itself chosen tobe much larger than gT, as is the case for when the dispersion relation, E(p), is evaluatedfor hard momenta such as |p| ≈T.

The reason for this simplification is that since theresummed and bare quantities agree if any of the relevant momenta are hard, |p| ≫gT, itis only necessary to work with the resummed versions if all of the momenta passing througha particular line (or vertex) are soft. For hard external momenta many of the internal linesand vertices must also carry hard momenta and so may be represented by the usual bareFeynman rules.

For example, in the vacuum-polarization graphs both of the vertices andone propagator must necessarily carry hard momenta if the external momentum is itselflarge in comparison to gT. This leaves at most a single internal propagator to be dressed.Now comes the main point: the dominant part of Πµν for small coupling—i.e.

theg2 log g terms—can be determined with no knowledge of Πsoftµν , and using virtually none ofthe complications of the resummation formalism in Πhardµν .The principal observation is that all of the terms in Πµν that are proportional to5

g2 log g are completely determined by the infrared divergent part of Πhardµν . To see howthis works consider the lowest order contribution to the dispersion relation, E(p), fortransverse gluons with momenta |p| ∼T ≫gT.

This is determined from eq. (3) given thevacuum polarization, Πµν(E, p) evaluated at hard, on-shell momenta: E = |p| ∼T.

Asis established in more detail below, the contribution of Πhardµνto the right-hand-side of eq. (3) diverges logarithmically with λ for g2T ≪λ ≪gT:F hard(T, |p|, λ) ≡12"(Πhard)ii −pipjΠhardijp2#= g2A logλµhard+ B + O λT+ O(g3).

(6)In this expression A and B are purely functions of |p| and T and µhard is a calculable energyscale of the high-frequency part of the theory which turns out below to be µhard ≈gT.In order to extract information about F(T, |p|) ≡F hard(T, |p|, λ) + F soft(T, |p|, λ)from eq. (6) it is necessary to say something about the behaviour of F soft(T, |p|, λ).

Theonly property of F soft(T, |p|, λ) that is required is that its λ−dependence must cancel thatof F hard(T, |p|, λ):λ∂F soft∂λ≡−λ∂F hard∂λ= g2−A + O λT+ O(g3). (7)This determines F soft to have the form:F soft = g2A logµsoftλ+ C + O λT+ O(g3),(8)in which A is the same function as in eq.

(6). The constant µsoft that appears within thelogarithm in this equation is the constant of integration that arises in passing from eq.

(7)to eq. (8).

It has dimensions of mass and is chosen to be of order g2T since this is thelargest mass scale present in the soft part of the problem.Adding the results of eqs. (6) and (8) therefore gives:F = −Ag2 logµhardµsoft+ g2(B + C) + O λT+ O(g3)= −A g2 log1g+ O(g2),(9)6

which determines the coefficient of the g2 log g term completely in terms of the calculablecoefficient A.The next point is that since A is determined by the infrared divergent part of F hard, itis insensitive to most of the complications of the resummation. To illustrate the simplicitywith which A may be determined we now outline its calculation for the damping constant.Inspection of the Feynman rules shows that the only potentially infrared-divergentpart of the leading contribution to F hard comes purely from the vacuum-polarization graphfor Πhardµνin which it is a gluon which circulates around the loop.Furthermore, evenfor this graph an infrared divergence can arise only from the term for which the integern = q0/(2πT) for the soft gluon line in eq.

(5) vanishes, and even then only if the externalfour-momentum is on shell: E = |p|. Since only n = 0 contributes, it is sufficient to knowthe form for the resummed propagator at zero frequency, where it reduces to:5(G∗)−1µν (p) =G−1bareµν (p) + m2Pµν.

(10)Here Pµν is the projection matrix onto the rest frame of the plasma, and so is given inthis frame by the matrix diag(0, 1, 1, 1). m denotes the lowest order gluon mass, or plasmafrequency, which is given in terms of the number of quarks, nq, by m2 = 19(gT)2 Ca + nq2.Substituting this into eq.

(5), evaluating on the lowest-order mass shell, E = |p| ≈T,and recognizing that at most one internal line can be soft at a time (and so need be dressed)then gives an infrared-divergent contribution:F hard(T, |p|)div = −g2CaTm2|p|2π2Z ∞λdq1q(q2 + m2) logq −2|p| + iǫq + 2|p|= +ig2CaT|p|2πlog λm,(11)from which we read A = (iCaT|p|)/(2π) and µsoft = m. Notice that the real part of F isinfrared finite so only the lifetime acquires a g2 log g contribution. Using this result in themass-shell condition gives our main result:γg(p) = −Im F2|p|= +g2CaT4πlog1g+ O(g2).

(12)7

This entire argument when repeated for the quark self-energy similarly gives a g2 log gcontribution to the branch of the fermion spectrum that survives at large momenta. Asimple calculation gives its coefficient as γq(p) = g2Cf T4πlog1g+ O(g2).

Cf denotes thequadratic invariant in the fundamental representation: Cf = (N 2c −1)/(2Nc) (= 43).There are several features of this calculation that bear emphasis: (i) First, as is re-quired for a good approximation to a physical quantity, γ is independent of the gauge pa-rameter ξ. All terms that depend on the gauge-parameter contribute only an infrared-finiteresult.

(ii) The sign of γ is positive, indicating stability. (iii) Notice that a determinationof the subleading O(g2) contributions would require knowledge of both of the coefficientsB of eq.

(6) and C of eq. (8).

Although B is calculable using the complete resummationformulation, C is not and can only at present be determined by making some assumptionsconcerning the behaviour of the plasma in the low-frequency regime λ ≈g2T. It followsthat the coefficient B need not a priori by itself be gauge-independent (or positive).

(iv)Also, since it is a logarithmic infrared-divergence that is responsible for the logarithmic de-pendence on g, its coefficient is insensitive to the details of how the cutoffis implemented.Finally, (v) since the infrared-divergent term in F hard is explicitly proportional to m2 (c.f.eq. (11)) it only receives contributions from the two-loop and higher graphs that serveto dress the soft propagator in the gluon vacuum-polarization graph.

It also follows thatthe imaginary part arises only from the self-energy of internal lines which carry soft loopmomenta |q| < gT since the mass m may be taken to be zero for larger momenta. Thisagrees with what is expected physically from unitarity given the constraints of energy andmomentum conservation in the plasma.We conclude that for some quantities in which infrared divergences in the perturbativeexpansion introduce a logarithmic dependence on the gauge coupling, g, it is possible tovery simply identify the dominant contributions.

This simplicity allows a check on morecomplete and more involved calculations.ACKNOWLEDGEMENTS: The authors would like to acknowledge helpful conversationswith Charles Gale, Joe Kapusta, Randy Kobes, Gabor Kunstatter and Rob Pisarski, aswell as funding by the Natural Sciences and Engineering Research Council of Canada andles Fonds pour la Formation de Chercheurs et l’Aide `a la Recherche du Qu´ebec.8

REFERENCES[1] O.K. Kalashnikov and V.V.

Klimov, Sov. J. Nucl.

Phys. 31 (1980) 699; D.J.

Gross,R.D. Pisarski and L.G.

Yaffe, Rev. Mod.

Phys. 53 (1981) 43; U. Heinz, K. Kajantieand T. Toimela, Phys.

Lett. 183B (1987) 96, Ann.

Phys. (NY) 176 (1987) 218;K. Kajantie and J. Kapusta, Ann.

Phys. (NY) 160 (1985) 477; J.C. Parikh, P.J.Siemens and J.A.

Lopez, unpublished; T.H. Hansson and I. Zahed, Phys.

Rev. Lett.58 (1987) 2397, Nucl.

Phys. B292 (1987) 725; H.-Th.

Elze, U. Heinz, K. Kajantieand T. Toimela, Zeit.Phys.C37 (1988) 305; H.-Th.Elze, K. Kajantie and T.Toimela, Zeit. Phys.

C37 (1988) 601; R. Kobes and G. Kunstatter, Phys. Rev.

Lett.61 (1988) 392; S. Nadkarni, Phys. Rev.

Lett. 61 (1988) 396; M.E.

Carrington, T.H.Hansson, H. Yamagishi and I. Zahed, Ann. Phys.

(NY) 190 (1989) 373; J. Milana,Phys. Rev.

D39 (1989) 2419; G. Gatoffand J. Kapusta, Phys. Rev.

D41 (1990) 611;J. Kapusta and T. Tomeila, Phys. Rev.

D39 (1989) 3197; R. Kobes, G. Kunstatterand K.W. Mak, Phys.

Lett. 223B (1989) 433; S. Catani and E. d’Emilio, Phys.

Lett.238B (1990) 373, Nucl. Phys.

B (19) ; K.A. James, Nucl.

Phys. B (19) .

[2] R.D. Pisarski, Phys.

Rev. Lett.

63 (1989) 1129, Nucl. Phys.

A525 (1991) 175c; E.Braaten and R.D. Pisarski, Phys.

Rev. Lett.

64 (1990) 1338. [3] S. Weinberg, in the proceedings of the International School of Subnuclear Physics,Ettore Majorana, Erice, Sicily, Jul 23 - Aug 8, 1976 (QCD161:I65:1976).

[4] For a discussion see, for example, J. Kapusta, Finite Temperature Field Theory, (Cam-bridge University Press, 1985). [5] E. Braaten and R.D.

Pisarski, Phys. Rev.

Lett. 64 (1990) 1338, Nucl.

Phys. B337(1990) 569.

[6] V.V. Lebedev and A.V.

Smilga, University of Berne preprints BUTP-89/25 andBUTP-90/38 (unpublished). [7] R.D.

Pisarski, Phys. Rev.

Lett. 63 (1989) 1129; and E. Braaten and R.D.

Pisarski,unpublished. [8] E. Braaten and R.D.

Pisarski, unpublished. [9] T. Matsubara, Prog.

Theor. Phys.

14 (1955) 351.9

---

출처: arXiv:9109.051 • 원문 보기