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Thermal quark production in pure glue and quark gluon plasmas

arXiv:hep-ph/9305227v1 7 May 1993CERN-TH.6882/93Thermal quark production in pure glue and quark gluon plasmasTanguy Altherr∗and David Seibert†Theory Division, CERN, CH-1211 Geneva 23, SwitzerlandAbstractWe calculate production rates for massless (u, d) and massive (s, c, b) quarks in pureglue and quark gluon plasmas to leading order in the strong coupling constant g. Theleading contribution comes from gluon decay into q¯q pairs, using a thermal gluon prop-agator with finite thermal mass and damping rate. The rate behaves as α2S(ln 1/αS)2T 4when m, αS →0 and depends linearly on the transverse gluon damping rate for all valuesof the quark mass m. The light quark (u, d, s) chemical equilibration time is approx-imately 10-100 T −1 for g =2-3, so that quarks are likely to remain far from chemicalequilibrium in ultrarelativistic nuclear collisions.Submitted to Physics Letters BCERN-TH.6882/93April 1993∗On leave of absence from L.A.P.P., BP110, F-74941 Annecy-le-Vieux Cedex, France.Internet: taltherr@vxcern.cern.ch.†On leave until October 12, 1993 from: Physics Department, Kent State University,Kent, OH 44242 USA.

Internet: seibert@surya11.cern.ch.

Ultrarelativistic nuclear collisions may present a unique opportunity to study the hightemperature behaviour of hadronic matter in the laboratory. It is possible that in theseexperiments, the produced energy densities might be high enough and last for longenough for bulk, locally equilibrated, deconfined matter, or quark gluon plasma (QGP),to be created and studied.

However, recent work [1,2] has shown that, while the hotmatter should thermalize in about 0.3 fm/c, it may be largely a gluon plasma (GP),with few quarks. Chemical equilibration is expected to take much longer than thermalequilibration, if it occurs at all.One of the proposed signals for QGP is copious production of strange particles [3], asthe strangeness production rate is much higher in QGP than in normal hadronic matter.Similarly, it has also been argued that thermal charm production in a QGP could belarge [4].

However, these rates have never been calculated using thermal field theory.In this paper, we calculate the thermal quark production rate to leading order in αSfor GP and for QGP. This gives the chemical equilibration rates for both the massless(u, d) and massive (s, c, b) quark components of QGP.

We find that these rates aresmall; the light quarks (u, d, s) chemical equilibration time is approximately 10-100T −1 for g =2-3, so that quarks are likely to remain far from chemical equilibrium inultrarelativistic nuclear collisions.We take the idealized situation of a GP or QGP in thermal equilibrium. If such aplasma does exist, the massless gluons evolve into quasi-particles with effective massesof order gT [5].

Being massive, these quasi-gluons decay into q¯q pairs. This situation issimilar to the plasmon decay into ν¯ν pairs that is the dominant cooling mechanism forhigh density stars [6].For gauge field theories at high temperature, there exists a resummation programdeveloped by Braaten and Pisarski [7] that describes these quasi-particles.

It is easy tocheck that this resummation program is almost the same for the cases of QGP and GP.The difference is that the quark contribution inside each hard thermal loop is omittedin the case of a GP, because there are no thermal quarks. For instance, the plasmafrequency isω20 =N + Nf2 g2T 29,(1)for SU(N) gauge theory at temperature T, where g is the strong coupling constant andNf is the number of massless fermion flavours.

Thermal effects in QGP and GP are thusidentical, except that Nf = 0 for GP.The discussion of the different QCD corrections to the production of q¯q pairs is bestillustrated by using “cut” diagrams [8,9]. The lowest order diagram is shown in Fig.

1.To disentangle the contributions from different graphs and from different discontinuitieswithin the same graph, we consider only the leading term in the strong coupling constantg as g →0.Following the usual treatment of Braaten and Pisarski [7] in such a situation, weconsider the two cases when the gluon momentum is soft, Q ∼gT, and hard, Q ∼T. Ifit is soft, one has to use the effective quark-gluon vertices and gluon propagators whichresum the hard thermal loops [7].

One is therefore left with a quite involved calculation asseveral discontinuities can be taken in such a case. In principle, this situation should be1

very similar to previous calculations such as photon emission from a QGP [10]. Indeed,the na¨ıve lowest order diagrams for the q¯q production are the two-gluon fusion processesshown in Fig.

2.However, it is well known that these diagrams have a logarithmicsingularity when the quark mass, m, goes to zero [11]. The cross section for this processisσgg→q¯q = πα2S3s ln sm2whens ≫m2.

(2)The divergence being of kinematic origin, it is clear that in the case of unthermalizedand zero mass quarks, this singularity can be screened only by the thermal gluon mass,of order gT. As typical gluon energies are of order T, the quark production rate mustthen behave as g4T 4 ln 1/g.

When the unthermalized outgoing quarks are massive, thecross section for the processes in Fig. 2 is no longer singular and leads to a contributionof O(g4T 4) to the production rate.A consistent calculation would require a division of the processes into two pieces bythe introduction of an intermediate cutoffgT ≪q∗≪T on the gluon line (as in thephoton emission case).

Then the hard part, calculated from the first two graphs shownin Fig. 2, gives a contribution of order g4T 4 ln(T/q∗).

On the other hand, the soft part,calculated from the discontinuity of the graph in Fig. 1 together with effective verticesand gluon propagator, must give g4T 4 ln(q∗/ω0) with the same coefficient in front sothat, by adding the two contributions, the q∗dependence cancels.

To be honest, we havenot checked that the coefficient in front of the logarithm is the same for the soft and thehard pieces but we do not see any reason why it should differ.If the gluon momentum is hard and if one is using a bare gluon propagator, the ratevanishes for obvious kinematic reasons. However, as in the γ →ν¯ν process, one couldalso use a resummed propagator for the hard gluon, leading to a finite answer (in facta contribution of order g4T 4 as we shall see).

But unlike the photon, the gluon has ananomalously large damping rate and if the hard thermal gluon mass starts to be relevant,one must also include the damping rate inside the hard gluon propagator. We show thatthe rate behaves then as g4T 4(ln 1/g)2 for massless quarks and as g4T 4 ln 1/g for massivequarks.

It is therefore larger than the soft gluon contribution and than the na¨ıve lowestorder diagrams shown in Fig. 2.

The precise reasons for this unusual behaviour willbecome clear as the calculation proceeds.The gluon propagator that we use is therefore−iPµνT(L)δabQ2 −ReΠT(L)(ω, q) + 2iq0γT(L),(3)where PµνT(L) is the transverse (longitudinal) projector [5] and a and b are colour indices.Here (and for the rest of this paper) the subscript T(L) refers to transversely (longitudi-nally) polarized gluons. We have included the thermal gluon mass at hard momentum,ReΠT(L)(ω, q), and also the anomalously large gluon damping rate, γT(L) [12].Using the cutting rules of Kobes and Semenoff[9], the quark (antiquark) productionrate due to transverse gluon decay is given byRqT = dNqTd4x = g2Zd4Q(2π)4Zd4K(2π)4 (2π)2 δ[K2 −m2] δ[(Q −K)2 −m2](nB(ω) + θ(−q0))2

×Re2iQ2 −ReΠT (Q) + 2iq0γT(T abc)2 Tr [(̸K + m)γµ(̸K−̸Q + m)γν] PµνT . (4)Here K and Q are the quark and gluon four-momenta respectively, m is the quarkmass, and nB is the Bose-Einstein distribution function.

The transverse gluon projectionoperator is PijT = −δij + qiqj/q2, with all other components zero, and T abc are the colourSU(3) matrices. We use the standard high energy conventions that ¯h = c = kB = 1.The integrals over q0 and cos θ = ˆk · ˆq can be evaluated by using the two δ-functions,while the remaining angular integrals are all trivial:RqT = g26π4Z ∞2m dω nB(ω)Z √ω2−4m20dq qωγT[ω2 −q2 −ReΠT(Q)]2 + 4ω2γ2T×ZdE Tr [(̸K + m)γµ(̸K−̸Q + m)γν] PµνT .

(5)The trace over colour indices gives a factor 4/3, the quark energy k0 = E = (k2 +m2)1/2,and the gluon energy q0 = ω. The transverse component of the trace isTr [(̸K + m)γµ(̸K−̸Q + m)γν] PµνT= 4 [2KµKν −KµQν −QµKν + K · Qgµν] PµνT= 4hω2 −q2 −2k2(1 −cos2 θ)i,(6)wherecos θ = (2ωE −ω2 + q2)/(2kq).

(7)We then integrate the trace over the allowed range of E, where |ω2 −q2 −2Eω| ≤2kq:RqT = 4g29π4Z ∞2m dω ωnB(ω)Z √ω2−4m20dq q2γT (ω2 −q2 + 2m2)h1 −4m2ω2−q2i1/2[ω2 −q2 −ReΠT(Q)]2 + 4ω2γ2T. (8)The integral in eq.

(8) is cut offby nB when ω/T ≫1. The crucial point of ouranalysis is that the leading contribution comes when ω and q are hard, of order T. Thesoft Q contribution is perfectly regular and subleading by O(αS).

Because the dominantgluon momenta are hard, we make the subsequent approximations: i) γT is independentof the momentum and is given by [12]γT=g2NT8π lnω20m2mag + 2mmagγT+ 1.09681 . .

. !when mmag ≫γT,=αSNT2ln 1αS+ O(αST);(9)ii) as we are only interested in the leading order, we take ReΠT = 3ω20/2, as changing theδ-function δ(Q2 −ReΠT) by a Lorentzian with width γT changes ReΠT = 3ω20/2 only byO(g4T 2) corrections.

The gluon damping rate depends on the magnetic mass which is,for dimensional reasons, a number times g2T [13]. Fortunately, this uncertainty entersonly at the logarithmic level.3

When m = 0, we evaluate the integral over q analytically, obtaining a simple result:RqT = 2g2γT9π4Z ∞0dω ω2 nB(ω)(ln64ω49ω40 + 16γ2Tω2 + 3ω202γTω arctan 3ω204γTω + arctan 2ωγT!−4),(10)where terms of order higher than g4 have been dropped. Notice that the limit γT →0reproduces the result using a gluon propagator without a finite damping rate.The final result isRqT = 4ζ(3)3π3 α2Sln 1αS2T 4 + Oα2S ln 1αST 4.

(11)It is in fact the Q2 dependence of the rate that makes it so large. Should we use abare gluon propagator, δ(Q2), the rate would identically vanish.

With just the hardthermal mass, δ(Q2 −3ω20/2), the rate is of order g4T 4 with no logarithmic dependence.Taking into account the anomalously large damping rate γT shifts the gluon on-shellnessQ2 ∼ω20 by a logarithmic correction. The additional logarithm has a kinematic origin.The processes that lead to our result are of the kind shown in Fig.

3. The enhancementcomes both from the soft space-like gluon exchange (a Landau-damping term) and fromthe pole of the gluon propagator that is almost on shell, hence the two logarithms.

Thisdiagram is not suppressed kinematically. Notice also that using a resummed propagatorfor a hard line is not a new feature.

For the calculation of the damping rate itself, oneneeds to use the same propagator [12,14].It is easy to see that the longitudinal decay is subleading. The only difference (asidefrom the self-energies) between the longitudinal and transverse modes is a factor of two,as there are two transverse modes and only one longitudinal mode.

At hard momentum,the longitudinal thermal mass is exponentially suppressed, ReΠL = 0, so we have in thecase of zero bare quark massRqL = g2γL9π4Z ∞0dω ω2 nB(ω)(ln 2ωγL−4). (12)However, the longitudinal gluon damping rate is of order g2T with no log.

The quarkproduction due to longitudinal gluon decay is therefore of the same order as the neglectedterms in eq. (11).The discussion for massive quarks is the same through eq.

(8).When m ≫ω0,the rate behaves perturbatively as g2γT ∝g4 ln 1/g, and is therefore larger than othercontributions.We show in Fig. 4 the production rate as a function of m/T for GP(Nf = 0); the curves are very slightly altered for Nf = 2.We stress again that our result is the leading order perturbatively (when g →0),containing all relevant contributions to order α2S(ln 1/αS)2.The fact that the quarkgluon plasma is not really a system of weak coupling is a different problem, and shouldin principle be solved by adding higher order contributions to our result.

One shouldwork in this perturbative picture to make a reliable prediction.In any case, there seems to be a huge uncertainty in the quark production rate whenconsidering large values of g (g > 1). As the rate is linearly dependent on the damping4

rate, which itself depends on the value of the magnetic mass, which is unknown, it isdifficult at present to evaluate the effect of higher order terms that are important forlarge g. We believe that the rate can easily vary by an order of magnitude. One shouldbear this in mind when making predictions for ultrarelativistic heavy ion collisions.

Thedisadvantage of this uncertainty can be turned into a gain, if one is able to measure thethermal charm or bottom quark production. This would give a direct measurement ofthe gluon damping rate, and hence of the magnetic mass.Having computed the quark production rate, we also discuss the quark equilibrationtime given by τq = Neq/RqT , where Neq is the quark density at equilibrium.

In Fig. 5we plot τq as a function of m/T.

In calculating Neq, we take into account the completemass (bare plus thermal corrections) at hard momentum,m2β=m2 + g2T 23whenm ≪T,m2β=m2 + 2g2T 29whenm ≫T. (13)We cannot rigorously calculate the thermal mass between these limits, so we use theansatzm2β = m2 + (2 + e−m/T )g2T 29.

(14)For vanishing quark mass, we obtainτq =27π16α2S(ln 1/αS)2T −1. (15)For comparison, we also give the gluon equilibration time.

By detailed balance, thethermal gluon formation rate must be equal to the decay rate (twice the damping rate),so thatτg =13αS ln 1/αST −1. (16)We see that gluons equilibrate faster than even the lightest quarks.

This stems from thefact that quarks are always produced by pairs, which is a relatively slow process, whilegluons can be copiously emitted by radiative processes.Again, our results strongly depend on the value of the coupling constant. From thenumbers that we obtain, it seems clear that, in ultrarelativistic heavy ion collisions, onlythe light quarks (u, d, and s) may have time to come to chemical equilibrium, while theheavy quarks will not.

Even taking g = 3, τq > 10 T −1 for massless quarks, so chemicalequilibration will proceed very slowly.In conclusion, we have calculated the quark production rate from pure glue and quark-gluon plasma in equilibrium. The leading contribution comes from the decay of trans-verse gluons, which are mainly characterized by an anomalously large damping rate γT.The analogy of this process with the plasmon decay into ν¯ν is very close but mislead-ing.

In the latter case, the photon damping rate is smaller, γγ = O(e3T ln(1/e)), sothe plasmon is really described by a quasi-particle with an effective thermal mass. Ina QCD plasma, this is clearly not the case, and one must be very careful in computing5

the thermal rates by incorporating the damping rate inside the gluon propagator. Ourresult is another example of a physical quantity that is sensitive to the non-perturbativemagnetic mass scale [12].Although our calculation is perturbatively correct, there are large uncertainties whenmaking predictions for ultra-relativistic heavy ion collisions for the following reasons: i)the rate is linearly dependent on γT, which depends on the unknown magnetic mass,and ii) the quark gluon plasma, which could be formed in such collisions is not really asystem of weak coupling, so higher order terms should be calculated.This material is based upon work supported by the North Atlantic Treaty Organizationunder a grant awarded in 1991.References1.K.

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D47 (1993) 703.Figure captions1.The lowest-order “cut” diagram for thermal quark production.2.The na¨ıve lowest order Feynman diagrams for quark-antiquark production.3.A Feynman diagram contributing to leading order for thermal quark production.4.The quark production rate, RqT , in a GP (Nf = 0) as a function of m/T.5.The quark chemical equilibration time, τq, in a GP as a function of m/T.6

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