Heavy quark medium polarization at next-to-leading order
aa r X i v : . [ h e p - ph ] F e b BI-TP 2008/40arXiv:0812.2105
Heavy quark medium polarization at next-to-leading order
Y. Burnier, M. Laine, M. Veps¨al¨ainen
Faculty of Physics, University of Bielefeld, D-33501 Bielefeld, Germany
Abstract
We compute the imaginary part of the heavy quark contribution to the photon polarizationtensor, i.e. the quarkonium spectral function in the vector channel, at next-to-leading orderin thermal QCD. Matching our result, which is valid sufficiently far away from the two-quarkthreshold, with a previously determined resummed expression, which is valid close to thethreshold, we obtain a phenomenological estimate for the spectral function valid for all non-zero energies. In particular, the new expression allows to fix the overall normalization of theprevious resummed one. Our result may be helpful for lattice reconstructions of the spectralfunction (near the continuum limit), which necessitate its high energy behaviour as input,and can in principle also be compared with the dilepton production rate measured in heavyion collision experiments. In an appendix analogous results are given for the scalar channel.January 2009 . Introduction
Heavy fermion vacuum polarization, i.e. the contribution of a massive fermion species tothe (imaginary part of the) photon polarization tensor, or to the spectral function of theelectromagnetic current, is one of the classic observables of relativistic quantum field theory:the result has been known up to 2-loop, or next-to-leading, or O ( α s ) level already since the1950s [1]. Nevertheless, significant new insights were still obtained in the 1970s [2] andeven in the 1990s [3]. By now a lot of information is also available concerning correctionsof O ( α s ) and O ( α s ) (for recent work and references, see ref. [4]). The physics motivationfor the continued interest is related, for example, to determining the heavy quark productioncross section, σ ( e − e + −→ c ¯ c ), often expressed through the R -ratio, as well as to computingthe heavy quarkonium decay width.In the present paper, we consider essentially the same observable as in the classic works,but in a situation where the heavy quarks live at a finite temperature, T , rather than inthe vacuum. We refer to this observable as the “heavy quark medium polarization”. Againthe result has direct physical significance, in that it determines the heavy quark contributionto the production rate of lepton–antilepton pairs from the thermal plasma [5] (cf. eq. (2.2)below). There has been considerable phenomenological interest particularly in what a finitetemperature does to the resonance peaks in the spectral function, given that this might yielda gauge for the formation of a deconfined partonic medium [6]. Some recent work on theresonance region within the weak-coupling expansion, taking steps towards a systematic useof effective field theory techniques to resum appropriate classes of higher loop orders, canbe found in refs. [7]–[11] (see ref. [12] for a review and ref. [13] for an alternative approachwith similar results), and recent reviews on some of the phenomenological approaches on themarket can be found in refs. [14, 15] (possible pitfalls of ad hoc potential models at finitetemperatures have been reviewed in ref. [16], and underlined from a different perspectivein ref. [13]). Analogous spectral functions can also be determined for theories with gravityduals [17].Unfortunately, it appears that ultimately weak-coupling (and related) techniques will beinsufficient for determining quantitatively the shape of the spectral function around the res-onance region. The reason is that field theory at finite temperatures suffers from infraredproblems, implying that the weak-coupling series goes in powers of ( α s /π ) / rather than α s /π , often with large (sometimes non-perturbative [18]) coefficients; see, e.g., ref. [19] andreferences therein. Therefore, particularly for the case of charmonium where even at zero tem-perature weak-coupling computations can hardly be trusted, it appears that non-perturbativetechniques are a must. Even though the situation should be somewhat better under control The older computations were formulated within QED, but at this order the results carry over directly toQCD, whose notation we adopt. can be obtained througha certain analytic continuation of the Euclidean correlator; however, as a mathematical op-eration, at finite temperatures such an analytic continuation is unique only if the asymptoticbehaviour for large Minkowskian arguments is known (see, e.g., ref. [20]). In a practicalsetting, many further problems arise because lattice data is not analytic in nature; yet theneed to input outside information (“priors”) to the analysis certainly remains a central issue(see refs. [21] for recent lattice results, and ref. [22] for an overview).It is at this point that weak-coupling techniques may again become helpful. The goalwould now be not so much to determine the spectral function around the resonance region,but to determine it at very high energies, which information should be relatively reliable,thanks to asymptotic freedom. Indeed, the free thermal spectral function has been studiedin great detail previously, even at a finite lattice spacing [23]. Nevertheless, loop correctionsare expected to remain substantial even up to energy scales of several tens of GeV, so it isimportant to account for them, and this is the basic goal of the present work. As a morerefined goal, we wish to demonstrate that thermal corrections are small far away from thethreshold; thereby knowledge of the asymptotic behaviour could be taken to higher orders,by employing well-known numerically-implemented results from zero temperature [24] (thisassumes, of course, that a continuum extrapolation can be carried out on the lattice).Apart from this lattice-related goal, we also wish to pursue the complementary goal oftreating the bottomonium spectral function without any exposure to the often hard-to-controlsystematic uncertainties of lattice simulations. This can be achieved by constructing aninterpolation between the asymptotic result determined in the present paper, and the near-threshold behaviour estimated within a resummed framework in ref. [8].The plan of the paper is the following. In sec. 2 we define the observable to be computed.The general strategy of the computation is discussed in sec. 3, and the main results aresummarized in sec. 4. A phenomenological reconstruction of the spectral function in thewhole energy range is carried out in sec. 5, while sec. 6 lists our conclusions. In appendixA, we display in some detail the intermediate steps entering the determination of one ofthe “master” sum-integrals appearing in the computation; in appendix B, we list the finalresults for all the master sum-integrals; and in appendix C we provide results for the spectralfunction in the scalar channel, discussing briefly also the ambiguities that hamper this case.2 . Basic definitions
The heavy quark contribution to the spectral function of the electromagnetic current can bedefined as ρ V ( ω ) ≡ Z ∞−∞ d t e iωt Z d − ǫ x (cid:28)
12 [ ˆ J µ ( t, x ) , ˆ J µ (0 , )] (cid:29) , (2.1)where ˆ J µ ≡ ˆ¯ ψ γ µ ˆ ψ ; ˆ ψ is the heavy quark field operator in the Heisenberg picture; h . . . i ≡Z − Tr [( ... ) e − β ˆ H ] is the thermal expectation value; β ≡ /T is the inverse temperature; andwe assume the metric convention (+ −−− ). This spectral function determines the productionrate of muon–antimuon pairs from the system [5],d N µ − µ + d x d q = − e Z π ) q (cid:18) m µ q (cid:19)(cid:18) − m µ q (cid:19) n B ( ω ) ρ V ( ω ) , (2.2)where Z is the heavy quark electric charge in units of e , and n B is the Bose-Einstein distri-bution function. In defining eq. (2.1) and the argument of ρ V in eq. (2.2), we have assumedthat the muon–antimuon pair is at rest with respect to the thermal medium, i.e. q ≡ ( ω, ). The pole mass of the heavy quark (charm, bottom) is denoted by M .The parametric temperature range we concentrate on in this paper is the one where the“quarkonium” resonance peak disappears from the spectral function ρ V [8]: g M < T < gM . (2.3)This implies that in any case T ≪ M , so that exponentially small corrections, ∼ exp( − βM ),can well be omitted. The thermal effects come thereby exclusively from the gluonic sector,where no exponential suppression takes place.In order to compute the spectral function ρ V of eq. (2.1), we start by determining thecorresponding Euclidean correlator, C E ( ω n ) ≡ Z β d τ e iω n τ Z d − ǫ x D ˆ J µ ( τ, x ) ˆ J µ (0 , ) E , (2.4)for which a regular path-integral expression can be given (i.e., hats can be removed from thedefinition). Here ω n ≡ πnT , n ∈ Z , denotes bosonic Matsubara frequencies. The spectralfunction is then given by the discontinuity (see, e.g., refs. [25, 26]) ρ V ( ω ) = Disc h C E ( − iω ) i ≡ i h C E ( − i [ ω + i + ]) − C E ( − i [ ω − i + ]) i . (2.5)In the following we denote Euclidean four-momenta with capital letters, in particular Q ≡ ( ω n , ). Moreover, Σ R K ≡ T P k n µ ǫ R d d k / (2 π ) d stands for a sum-integral over bosonic Mat-subara four-momenta, while Σ R { P } signifies a sum-integral over fermionic ones. The space-timedimensionality is denoted by D ≡ − ǫ , and the space dimensionality by d ≡ − ǫ . For a non-zero total spatial momentum q , with 0 < | q | ≪ M , the main modification of our results wouldbe a shift of the two-particle threshold from ω ≈ M to ω ≈ M + q / M . . Details of the computation At a finite temperature T it is not clear, a priori , whether the result of the computation willbe infrared finite, given that (after analytic continuation) the gluon propagator contains theBose-enhanced factor n B ( k ) ≈ T /k , for | k | ≪ T . For this reason, we carry out the analysisby using the Hard Thermal Loop resummed [27, 28] form of the gluon propagator, whichtakes into account Debye screening, and thereby shields (part of) the infrared divergences.Introducing (see, e.g., refs. [25, 26]) P T ( K ) = P T i ( K ) = P Ti ( K ) ≡ , P Tij ( K ) ≡ δ ij − k i k j k , (3.1) P Eµν ( K ) ≡ δ µν − K µ K ν K − P Tµν ( K ) , (3.2)where K = ( k n , k ), k n = 2 πnT , the Euclidean gluon propagator can be written as h A aµ ( x ) A bν ( y ) i = δ ab PZ K e iK · ( x − y ) (cid:20) P Tµν ( K ) K + Π T ( K ) + P Eµν ( K ) K + Π E ( K ) + ξ K µ K ν ( K ) (cid:21) , (3.3)where ξ is a gauge parameter. The projector P T is transverse both with respect to K and tothe four-velocity of the heat bath and, in the static limit, describes colour-magnetic modes;the projector P E is transverse only with respect to K and, in the static limit, describescolour-electric modes. The self-energies Π T , Π E are well-known [27, 28] functions of the form m f ( k n / | k | ), where m D = ( N c / N f / gT is the Debye mass parameter; we will not needtheir explicit expressions in the following, apart from knowing that f is an even function ofits argument and regular on the real axis. The fermion propagator has the free form, h ψ ( x ) ¯ ψ ( y ) i = PZ { P } e iP · ( x − y ) − i /P + M B P + M B , (3.4)where M B is the bare heavy quark mass. The first step of the computation is to carry out the Wick contractions and the Diractraces. At 1-loop level, omitting Q -independent terms which are killed by the discontinu-ity in eq. (2.5), we get = [ Q − indep.] + 2 C A PZ { P } ( D − Q − M ∆( P )∆( P − Q ) . (3.5)Here C A ≡ N c , and ∆( P ) ≡ P + M . (3.6)4t next-to-leading order (NLO), we have to evaluate the counterterm graph as well asgenuine 2-loop graphs. The counterterm graph can be deduced from the 1-loop expression ineq. (3.5), by re-interpreting the mass parameter as the bare one, M B , and then expanding itin terms of the pole mass: M B = M − g C F M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19) + O ( g ) , (3.7)where C F ≡ ( N − / N c , and ¯ µ is the scale parameter of the MS scheme. This yields= [ Q − indep.] + 24 g C A C F M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19) × PZ { P } (cid:20) ( D − Q − M ∆ ( P )∆( P − Q ) + 2∆( P )∆( P − Q ) (cid:21) . (3.8)For the genuine 2-loop graphs, we make use of the identities K µ P Tµν ( K ) = Q µ P Tµν ( K ) = 0 , P Tµµ ( K ) = D − , P µ P ν P Tµν ( K ) = p − ( p · ˆ k ) , (3.9)where ˆ k ≡ k / | k | , and the second equality follows from the fact that Q is aligned with theheat bath. We then complete squares in the numerator, and note that PZ K { P } Q · K [ K + Π( K )]∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) = 0 , (3.10)as can be shown with the shifts P → − P + Q, K → − K . Thereby we arrive at+ = [ Q − indep.] + 4 g C A C F PZ K { P } (cid:26) (3.11) (cid:18) K + Π T − K + Π E (cid:19) [ p − ( p · ˆ k ) ] ×× (cid:20) − − D ) Q + 4 M ]∆ ( P )∆( P − Q )∆( P − K ) − − D ) Q + 4 M ] + 4 K ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) + D − K + Π T (cid:20) − P )∆( P − Q ) + (2 − D ) Q + 4 M ∆ ( P )∆( P − Q )+ 2∆( P )∆( P − Q − K ) + − D − Q · K + 4 K ∆( P )∆( P − Q )∆( P − K ) − [(2 − D ) Q + 4 M ] K ∆ ( P )∆( P − Q )∆( P − K ) − [(6 − D ) Q / M ] K + K ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) + 1 K + Π E (cid:20) − − D ) Q + 4 M ]∆( P )∆( P − Q )∆( P − K ) + 4[(2 − D ) Q + 4 M ] M ∆ ( P )∆( P − Q )∆( P − K )+ (2 − D ) Q + (8 − D ) Q M + 8 M + [(2 − D ) Q + 4 M ] K ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) (cid:27) . ξ has disappeared; thus P Eµν could havebeen replaced with δ µν − P Tµν . Given eq. (3.11), we need to carry out the Matsubara sums and the spatial momentum inte-grals. More concretely, the steps (specified in explicit detail for one example in appendix A)are as follows: • Writing the gluon propagator in a spectral representation, the Matsubara sums T P k n and T P { p n } can be carried out exactly in all cases. • The result after these steps contains many appearances of the Fermi distributions, n F ( E ) ≡ / [exp( βE ) + 1], where the energy E is that of a heavy quark, E ≥ M . Allsuch terms are suppressed by at least e − M/T ≪
1, and can be omitted. • The remaining temperature dependence appears as Bose distributions with the gluonenergy, n B ( k ). Here the issue is the opposite: in the small energy range, | k | ≪ T ,there is an enhancement factor T /k , which could lead to infrared divergences. This isan important point, so we devote a separate subsection to it (sec. 3.4). The upshot isthat there are no infrared divergences at the present order. • Having verified the absence of infrared divergences, we can forget about the HTL re-summation in the gluon propagators, i.e. set Π T = Π E = 0 in eq. (3.11), and insertthe free spectral function for the gluons. Thereby the integral over the gluon energy k is trivially carried out. (In practice, we first insert the free gluon spectral function,integrate over k , and verify the absence of infrared divergences a posteriori for eachindependent (“master”) sum-integral separately.) • The remaining spatial integrals, over k and p , are effectively three-dimensional (overthe absolute values of k , p and over the angle between k and p ). Some of them areultraviolet divergent, and require regularization. The integrals come in two forms, whichwe call “phase space” and “factorized”. We are able to carry out two of the integrationsin all cases; for the zero-temperature parts entering the final result, all three integrationsare doable [1]–[3], while for the finite-temperature parts an exponentially convergentintegral over k = | k | remains to be carried out numerically.The results obtained after these steps are listed for all the master sum-integrals appearing ineq. (3.11) in appendix B. 6 .4. Absence of infrared divergences Inserting the free gluon spectral function, which sets k = k , into any of the master sum-integrals, there remains an integral over the gluon momentum k to be carried out. In principlethis integral could be infrared divergent. This turns out indeed to be the case for the “phasespace” and “factorized” parts (for definitions, see appendix A.2) of the integrals separately;in fact, the integrals denoted by S and S (cf. eqs. (B.20), (B.32)), have logarithmicallydivergent infrared parts even at zero temperature, which were an issue in the 1970s [2].However, the infrared divergences were found to cancel in the sum of the phase space andfactorized parts. In our case, the logarithmic divergences turn into linear ones, due to theadditional factor n B ( k ) ≈ T /k ; nevertheless, when the phase space and factorized parts areadded together, we find that both powerlike and logarithmic divergences cancel, and theintegrals become finite, for each master sum-integral separately. This can clearly be seenin eqs. (B.24) and (B.35) for S and S , respectively. The same is true for the integralsdenoted by ˆ S , ˆ S , ˆ S (eqs. (B.25), (B.36), (B.45)), appearing in the first term of eq. (3.11)and disappearing if the HTL self-energies are set to zero from the outset. Therefore, weconclude that there are no infrared problems in our observable at the next-to-leading order.(It is to be expected, though, that there are some at higher orders.)
4. Final result
Given the considerations in sec. 3.4, showing the absence of infrared divergences, we are freeto set Π T = Π E = 0 in eq. (3.11). Noting furthermore that the factorized gluon tadpole reads PZ K K = T
12 + O ( ǫ ) , (4.1)and employing the notation of appendix B for the sum-integrals S ji ( ω ), the full result can bewritten as ρ V ( ω ) | raw = − C A ( ω + 2 M ) S ( ω ) + 8 g C A C F (cid:26)(cid:20) T − M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19)(cid:21)h − S ( ω ) + ( ω + 2 M − ǫ ω ) S ( ω ) i +2 S ( ω ) − ω + 2 M − ǫ ω ) S ( ω ) − − ǫ ) S ( ω ) + 4(1 − ǫ ) S ( ω )+2( ω + 2 M − ǫ ω ) h M S ( ω ) − (1 − ǫ ) S ( ω ) i − ( ω − M ) S ( ω )+ h (2 − ǫ ) ω + 2 ǫM i S ( ω ) − (1 − ǫ ) S ( ω ) (cid:27) + O ( ǫ ) . (4.2)We have set here ǫ → M → M + g T C F , (4.3)i.e. δM = g T C F / M . Note that this term multiplies the function S ( ω ) = θ ( ω − M ) / [16 πω ( ω − M ) / ](1 + O ( ǫ, e − βM )) (cf. eq. (B.6)), which diverges at the threshold,while the sum of all the other terms turns out to remain finite. Thereby the thermal cor-rection would completely dominate the result close enough to the threshold, were it not tobe resummed into a mass correction `a la eq. (4.3). On the other hand, once it has been re-summed, this term is in general small: in the range that we are interested in, g M < T < gM ,it corresponds parametrically to a higher order contribution. Therefore, for simplicity, wedrop this term in the following (of course, if desired, it is trivial to include it as an overallmass shift), and reinterpret the result as ρ V ( ω ) = − C A ( ω + 2 M ) S ( ω ) + 8 g C A C F (cid:26)(cid:20) − M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19)(cid:21)h − S ( ω ) + ( ω + 2 M − ǫ ω ) S ( ω ) i +2 S ( ω ) − ω + 2 M − ǫ ω ) S ( ω ) − − ǫ ) S ( ω ) + 4(1 − ǫ ) S ( ω )+2( ω + 2 M − ǫ ω ) h M S ( ω ) − (1 − ǫ ) S ( ω ) i − ( ω − M ) S ( ω )+ h (2 − ǫ ) ω + 2 ǫM i S ( ω ) − (1 − ǫ ) S ( ω ) (cid:27) + O ( ǫ ) . (4.4)Nevertheless, it is perhaps appropriate to stress that only the part of the thermal correctionmultiplying the function S ( ω ) can be unambiguously resummed on the grounds that theresult would otherwise diverge at the threshold, while the term ∼ T S ( ω ) could in principlebe kept explicit, and would then have an O (1) effect on the thermal part of the result.Inserting the explicit expressions for the functions S ji ( ω ) from appendix B into eq. (4.4),the final result for the vacuum part becomes ρ V ( ω ) | vac = − θ ( ω − M ) C A ( ω − M ) ( ω + 2 M )4 πω + θ ( ω − M ) 8 g C A C F (4 π ) ω (cid:26) (4 M − ω ) L (cid:18) ω − √ ω − M ω + √ ω − M (cid:19) + (7 M + 2 M ω − ω ) acosh (cid:18) ω M (cid:19) + ω ( ω − M ) (cid:20) ( ω + 2 M ) ln ω ( ω − M ) M −
38 ( ω + 6 M ) (cid:21)(cid:27) + O ( ǫ, g ) , (4.5)8here the function L is defined as L ( x ) ≡ ( x ) + 2 Li ( − x ) + [2 ln(1 − x ) + ln(1 + x )] ln x . (4.6)The result in eq. (4.5) agrees with the classic result from the literature [1]–[3]. The thermalcorrection, in turn, reads, ρ V ( ω ) | T = 8 g C A C F (4 π ) ω Z ∞ d k n B ( k ) k (cid:26) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17)(cid:20) ω k s − M ω ( ω + 2 k )+( ω + 2 M ) p ω ( ω + 2 k ) p ω ( ω + 2 k ) − M − (cid:16) ω − M + 2 ωk ( ω + 2 M ) + 2 ω k (cid:17) acosh r ω ( ω + 2 k )4 M (cid:21) + θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17)(cid:20) ω k s − M ω ( ω − k )+( ω + 2 M ) p ω ( ω − k ) p ω ( ω − k ) − M − (cid:16) ω − M − ωk ( ω + 2 M ) + 2 ω k (cid:17) acosh r ω ( ω − k )4 M (cid:21) + θ ( ω − M ) (cid:20) − ω + 2 M ) ω p ω − M +4 (cid:16) ω − M + 2 ω k (cid:17) acosh (cid:18) ω M (cid:19)(cid:21)(cid:27) + O ( e − βM , g ) , (4.7)where we have restricted to ω > ω < ρ V ( − ω ) = − ρ V ( ω )).Eq. (4.7) is our main result.A numerical evaluation of eq. (4.7), compared with the vacuum part in eq. (4.5), is shown infig. 1. We note that even though the thermal part is not exponentially suppressed for ω > M ,it still only amounts to a small correction at phenomenologically interesting temperatures.On the other hand, the thermal part does possess the new qualitative feature that the result isnon-zero below the threshold as well, where it is then the dominant effect; this can be tracedback to reactions where a heavy quark and anti-quark annihilate into a gluon remaining insidethe thermal medium, and a photon escaping from it.As an amusing remark, we note that while the next-to-leading order vacuum part is dis-continuous at the threshold, the next-to-leading order thermal part appears to be continuous.A similar pattern holds also for the scalar channel (fig. 6): then the next-to-leading ordervacuum part is continuous, while the next-to-leading order thermal part appears to have acontinuous first derivative. These features are perhaps a manifestation of the fact that a non-zero temperature in general “smoothens” the spectral function; in a resummed framework,9 .0 10.0 ω /M - ρ V - l oop / ω g C A C F ρ V | vac ρ V | T at T = 0.4M ρ V | T at T = 0.2M ρ V | T at T = 0.1M Figure 1:
The vacuum and thermal parts of the next-to-leading order correction in the vector channel,normalized by dividing with − ω g C A C F . The vacuum part remains finite for ω → ∞ (in units ofthe figure, its asymptotic value is 3 / π ), while the thermal part disappears fast for ω/M ≫ it may then not be surprising if any resonance peak of the vacuum result should disappearfrom the spectral function at high enough temperatures.To summarize, the characteristic feature of fig. 1 is a significant “threshold enhancement”,due mostly to the vacuum part at T ≪ M . Within perturbation theory, this is to beinterpreted as a first term of a series which, when summed to all orders, builds up possiblequarkonium resonance peaks at ω < M . At the same time, the result of a resummedcomputation (to be discussed in more detail at the beginning of the next section) shouldextrapolate towards the perturbative one at some ω > M .
5. Phenomenological implications
We would now like to combine our result with that obtained within an NRQCD [30, 31] andPNRQCD [32, 33] inspired resummed framework in ref. [8]. In order to do this, we need topay attention to the correct normalization of the resummed result. In fact, the well-known(vacuum) normalization factor can be read off from eq. (4.5): denoting v ≡ √ ω − M ω , (5.1)10 .0 2.5 3.0 ω /M - ρ V / ω resummedNLO QCDfree theory M = 2 GeV, T = 400 MeV
Figure 2:
A comparison of the near-threshold “resummed” result of ref. [8], matched to the “NLOQCD” expression of the present paper through an overall normalization factor, as discussed in thetext. The difference of the NLO QCD result and the free theory result contains both the vacuum partand the thermal part; the magnitude of the latter is reflected by how much the curve deviates fromzero below the threshold. the leading order vacuum expression can be expanded near the threshold as − ρ V ( ω ) ω (cid:12)(cid:12)(cid:12)(cid:12) LO = θ ( ω − M ) (cid:20) C A v π + O ( v ) (cid:21) , (5.2)while the next-to-leading order vacuum result becomes − ρ V ( ω ) ω (cid:12)(cid:12)(cid:12)(cid:12) NLO = 8 g C A C F θ ( ω − M ) (cid:20) π − v π + O ( v ) (cid:21) . (5.3)Since radiative corrections within a non-relativistic framework always contain a power of v , itis possible to account for the second term in eq. (5.3), equalling − g C F /π times the leadingterm in eq. (5.2), only by a multiplicative correction of the current, J µ QCD = J µ NRQCD (cid:18) − g C F π + ... (cid:19) . (5.4)In principle the coupling here should be evaluated at the scale ∼ M [35], but in practice ourresolution is low enough that we follow a simpler recipe (cf. next paragraph). In any case, The same relation is valid both for NRQCD and PNRQCD [34]. .0 2.5 3.0 ω /M - ρ V / ω T = 250 MeVT = 300 MeVT = 350 MeVT = 400 MeVT = 450 MeVT = 500 MeV
M = 2 GeV ω /M - ρ V / ω T = 250 MeVT = 300 MeVT = 350 MeVT = 400 MeVT = 450 MeVT = 500 MeV
M = 4 GeV ω /M - ρ V / ω T = 250 MeVT = 300 MeVT = 350 MeVT = 400 MeVT = 450 MeVT = 500 MeV
M = 6 GeV
Figure 3:
The phenomenologically assembled vector channel spectral function ρ V ( ω ), in units of − ω ,for M = 2 , , M is the heavy quark pole mass.Note that for better visibility, the axis ranges are different in the rightmost figure. the normalization factor is numerically significant, and its precise treatment plays a role;we actually do not impose it exactly, but rather search for a value minimizing the squareddifference of the two results in the range ( ω − M ) /M = 0 . − .
4, thereby also accountingfor thermal corrections. This results in a normalization factor in the range 0 . − .
9, whichindeed is the same ballpark as suggested by (the square of) eq. (5.4), given our choice of g (cf. next paragraph). The “interpolated”, or rather “assembled” result, is subsequentlydefined as ρ (assembled) V ≡ max( ρ (QCD) V , ρ (resummed) V ). An example for how the interpolation worksin practice is shown in fig. 2.As far as the value of g goes, no systematic choice is possible in the absence of NNLO com-putations at finite temperature. We follow here a purely phenomenological recipe, whereby g is taken from another context where a sufficient level has been reached [36], and take [37] g ≃ π . T / Λ MS ) , for N c = N f = 3 . (5.5)We also fix Λ MS ≃
300 MeV to be compatible with ref. [8]. It should be obvious that thesubsequent results contain unknown uncertainties; still, the situation could in principle besystematically improved upon through higher order computations.The resulting full spectral function is shown in figs. 3, 4 for various masses and tempera-tures, as a function of ω . The corresponding dilepton production rate from eq. (2.2) is shownin fig. 5. Compared with the results in ref. [8], the absolute magnitude of the rate has de-creased by about 10 – 30%, due to the inclusion of the normalization factor. We should againstress that particularly the charmonium case contains large uncertainties, and our results areto be trusted on the qualitative level only. 12 .0 2.5 ω /M - ρ V / ω M = 1 GeVM = 2 GeVM = 3 GeVM = 4 GeVM = 5 GeVM = 6 GeV
T = 250 MeV ω /M - ρ V / ω M = 1 GeVM = 2 GeVM = 3 GeVM = 4 GeVM = 5 GeVM = 6 GeV
T = 350 MeV ω /M - ρ V / ω M = 1 GeVM = 2 GeVM = 3 GeVM = 4 GeVM = 5 GeVM = 6 GeV
T = 450 MeV
Figure 4:
The phenomenologically assembled vector channel spectral function ρ V ( ω ), in units of − ω ,for T = 250 , ,
450 MeV (from left to right). To the order considered, M is the heavy quark polemass. Note that for better visibility, the axis ranges are different in the leftmost figure.
6. Conclusions
The purpose of this paper has been to compute the heavy quark contribution to the spectralfunction of the electromagnetic current at next-to-leading order in thermal QCD. The resultconsists of a well-known vacuum part, eq. (4.5), and a new thermal part, eq. (4.7). Thethermal part is illustrated numerically in fig. 1 in comparison with the vacuum part.The thermal corrections in our result arise exclusively from the gluons with which theheavy quarks interact. Although these contributions are not exponentially suppressed, theyturn out to be power-suppressed at large energies ω ≫ M : their general magnitude is O ( g T ), and given that T < gM is the phenomenologically interesting temperature range(cf. eq. (2.3)), they can in principle be omitted in comparison with the next-to-leading orderzero-temperature corrections, of O ( g M ). This also means that the asymptotic behaviourof the spectral function, needed as input for lattice studies, could (in the continuum limit)be extracted from the well-studied zero-temperature computations (see, e.g., ref [24]). Atzero temperature the next-to-leading order correction could, perhaps, even be worked out ata finite lattice spacing.On the other hand, decreasing the energy towards the threshold, the thermal correctionsbecome increasingly important. In fact, at next-to-leading order, the vacuum spectral func-tion vanishes at ω < M , while the thermal correction stays finite. The result emerges fromphase space integrals associated with the energy constraint δ ( ω + k − E − E ), where ω is thephoton energy; k is the gluon energy; and E , E are the energies of a heavy quark and anti-quark. Graphically, the process corresponds to the annihilation of quarkonium into a gluonand a photon, the former of which remains within the thermal medium. Since large values of k are Boltzmann suppressed, the thermal corrections are substantial only for | ω − M | < ∼ T .13 .6 1.8 2.0 2.2 2.4 ω /M -15 -14 -13 -12 -11 -10 d N µ − µ + / d x d Q M = 1.5 ... 2.0 GeV T = M e V T = M e V T = M e V ω /M -26 -24 -22 -20 -18 -16 d N µ − µ + / d x d Q M = 4.5 ... 5.0 GeV T = M e V T = M e V T = M e V T = M e V T = M e V Figure 5:
The physical dilepton production rate, eq. (2.2), from charmonium (left) and bottomonium(right), as a function of the energy, for various temperatures. The mass M corresponds to the polemass, and is subject to uncertainties of several hundred MeV; we use the intervals 1.5...2.0 GeV and4.5...5.0 GeV to illustrate the uncertainties. The low mass corresponds to the upper edge of eachband. Compared with ref. [8], the main change is a 10 – 30% reduction of the overall magnitude. Combining our new results, valid far enough away from the threshold, with previously de-termined resummed expressions, valid close to the threshold, we have subsequently assembledphenomenological estimates for the spectral function in a macroscopic energy range (figs. 3,4). The corresponding dilepton production rate is shown in fig. 5. Analogous results andplots for the spectral function in the scalar channel have been given in appendix C. The com-putations of the present paper play an important role in these plots particularly in that theyfix the overall normalization of the assembled curves. We hope that these results can even-tually be incorporated in a simulation including an expanding and cooling thermal fireball,which would then allow for a direct comparison with the dilepton production rate measuredin heavy ion collision experiments.We note, finally, that we have restricted to ω > ω ≈
0, related to the heavy quarkdiffusion coefficient. However, that structure is suppressed by exp( − βM ), and a non-trivialresult also only arises at the order O ( α s ) [38], so that our present computation at O ( α s )cannot add anything to the known results [39].14 cknowledgements We thank S. Caron-Huot, D. B¨odeker and Y. Schr¨oder for useful discussions, and are grate-ful to the BMBF for financial support under project
Hot Nuclear Matter from Heavy IonCollisions and its Understanding from QCD . Appendix A. Intermediate steps for a master sum-integral
We elaborate in this appendix on the steps outlined in sec. 3.3. The starting point is theexpression in eq. (3.11).
A.1. Matsubara sums and the spectral function
The first step is to carry out the Matsubara sums T P k n , T P { p n } . The sum T P k n iscomplicated by the appearance of the functions Π E , Π T in the gluon propagators. The reasonfor their introduction was that there could in principle be infrared divergences associated withthe gluons; in the Euclidean formalism, these would come from small spatial momenta k forthe Matsubara zero mode k n = 0, and could then be regulated by the fact that Π E (0 , k ) = m >
0. Our strategy in the following will be to assume that there are no infrared divergences,whereby we can set Π T = Π E = 0; the absence of divergences will be verified a posteriori .Nevertheless, it has still been important to keep Π E = Π T in eq. (3.11), because it couldhappen that the structure multiplied by 1 / ( K + Π T ) − / ( K + Π E ), which vanishes in thefree limit, contains infrared sensitive parts which do not completely cancel against each otherin the presence of Π E = Π T .In order to allow for an eventual introduction of Π E and Π T , we write for the moment thegluon propagators in the spectral representation,1 K + Π( K ) = Z ∞−∞ d k π ρ ( k , k ) k − ik n , (A.1)and carry out the Matsubara sum T P k n with the kernel 1 / ( k − ik n ). In the free case, whenthe spectral function reads ρ free ( k , k ) = π k h δ ( k − k ) − δ ( k + k ) i , (A.2)with k ≡ | k | , the whole procedure is obviously just a rewriting of the decomposition1 k n + k = 12 k (cid:20) k − ik n + 1 k + ik n (cid:21) . (A.3)15ote that the procedure is rather versatile and could also be interpreted as1 K + Π( K ) = (cid:20) − Π( K ) K + Π( K ) (cid:21) K (A.4)= Z ∞−∞ d k π (cid:26) ρ free ( k , k ) k − ik n + ¯ ρ ( k , k )2 k (cid:18) k + ik n − k + ik n + 1 k − ik n − k − ik n (cid:19) k − k (cid:27) , where ¯ ρ ( k , k ) is the spectral function corresponding to − Π / ( K + Π), and we made use ofthe spectral function’s antisymmetry in k → − k . All the sums over k n are now with thesame kernel as the one following from eq. (A.1). The representation in eq. (A.4) would berelevant for Π E , in which case ρ E would have a pole at k = k [25].After this lengthy introduction, we are ready to carry out the sums. We describe theprocedure in some detail for one of the master sum-integrals appearing in eq. (3.11); for theothers, the results are listed in appendix B.The case we choose to consider in detail is S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n P )∆( P − Q )∆( P − K ) (cid:21) Q =( ω n →− iω, ) . (A.5)Denoting E p ≡ p p + M , E p − k ≡ p ( p − k ) + M , (A.6)we can rewrite the sums as T X k n T X { p n } k − ik n ][ p n + E p ][( p n − ω n ) + E p ][( p n − k n ) + E p − k ]= T X k n X { p n } X { r n } X { s n } βδ r n − p n + ω n , βδ s n − p n + k n , [ k − ik n ][ p n + E p ][ r n + E p ][ s n + E p − k ]= Z β d τ Z β d σ e iω n τ T X k n X { p n } X { r n } X { s n } e ik n σ k − ik n e − ip n ( τ + σ ) p n + E p e ir n τ r n + E p e is n σ s n + E p − k , (A.7)where in the last step we used a representation of the Kronecker delta-function, βδ t n , = R β d τ e it n τ . The sums have factorized and can now be carried out: T X k n e ik n σ k − ik n = n B ( k ) e ( σ mod β ) k , < σ mod β < β , (A.8) T X { r n } e ± ir n τ r n + E p = n F ( E p )2 E p h e ( β −| τ mod 2 β | ) E − e | τ mod 2 β | E i , − β ≤ τ mod 2 β ≤ β , (A.9)where n F ( ω ) ≡ / [exp( βω ) + 1] and n B ( ω ) ≡ / [exp( βω ) − The subsequent integrals over τ and σ are elementary; we simply need to split R β d σ = R β − τ d σ + R ββ − τ d σ , and note that in The sum in eq. (A.8) is discontinuous at σ = 0 mod β , and defining its value at the discontinuity re-quires care; although of no importance in the present context, we note that the expression with the correctantisymmetry in k corresponds to the “average”, n B ( k )(1 + e βk ) / n B ( k ) + . | τ + σ mod 2 β | = 2 β − τ − σ . Setting e iω n β ≡ ω n -dependence of the result appears only in structures like 1 / ( iω n + P i E i ), and we can readoff the discontinuity: Disc (cid:20) iω n + P i E i (cid:21) ω n →− iω = − πδ ( ω + X i E i ) . (A.10)Implementing these steps in practice, and restricting the k -integral to positive values bymaking use of the antisymmetry of ρ ( k , k ), we arrive at S ( ω ) = Z ∞ d k π Z k , p ρ ( k , k ) π E p E p − k (cid:26) (A.11)12 E p h δ ( ω − E p ) − δ ( ω + 2 E p ) i (1 − n F1 ) ×× h (∆ − + ∆ − − + )(1 + n B0 − n F2 ) − (∆ − −− + ∆ − − )( n B0 + n F2 ) i − h δ ( ω − ∆ ++ ) − δ ( ω + ∆ ++ ) i ∆ − ∆ − − + h (1 + n B0 )(1 − n F1 − n F2 ) + n F1 n F2 i − h δ ( ω − ∆ −− ) − δ ( ω + ∆ −− ) i ∆ − −− ∆ − − h − n B0 (1 − n F1 − n F2 ) + n F1 n F2 i − h δ ( ω − ∆ + − ) − δ ( ω + ∆ + − ) i ∆ − −− ∆ − − h n B0 n F1 − (1 + n B0 ) n F2 + n F1 n F2 i − h δ ( ω − ∆ − + ) − δ ( ω + ∆ − + ) i ∆ − ∆ − − + h n B0 n F2 − (1 + n B0 ) n F1 + n F1 n F2 i(cid:27) . To simplify the expression somewhat, we have introduced the shorthands∆ στ ≡ k + σE p + τ E p − k , σ, τ = ± , (A.12) n B0 ≡ n B ( k ) , n F1 ≡ n F ( E p ) , n F2 ≡ n F ( E p − k ) . (A.13)Note that the result in eq. (A.11) is antisymmetric in ω → − ω , as must be the case.Inspecting eq. (A.11), we note the appearance of structures in the denominator, ∆ + − etc,which look like they might vanish for some k , p . In fact, in one of the other master sum-integrals, even the structure 1 / ( E p − k − E p ) appears, which certainly vanishes, for 2 p · k = k .It can be verified, however, that such poles always cancel between the various types of terms inthe expression, and do not hinder the actual integration. (If integration variables are changedin a subset of the expression, p → − p + k , to remove an apparent symmetry in E p ↔ E p − k ,then such terms do not in general cancel any more; nevertheless their contribution remainsfinite and correct if the poles are interpreted as principal values.) A.2. Spatial momentum integrals
The result so far, eq. (A.11), contains integrals with two types of delta-functions: ones with δ ( ω ± E p ), which we call “factorized” (fz) integrals, because the gluon momentum k does17ot appear inside the δ -functions; and ones with more complicated δ -functions, which we call“phase space” (ps) integrals. In both cases, our strategy is to first carry out the integralover the quark momentum p ≡ | p | and over the angle between p and k ; the integral overthe gluon momentum k ≡ | k | is left for later (it is this integral which could potentially sufferfrom infrared divergences).We start by considering the phase space integrals, which are ultraviolet finite, so that wecan set d = 3. In order to simplify the task, we ignore from now on terms suppressed byexp( − βM ) ≪
1. This means that all appearances of n F ( E p ) and n F ( E p − k ) can be omitted.Furthermore, restricting to ω >
0, we note that the delta-function δ ( ω − ∆ −− ) = δ ( ω + E p + E p − k − k ) can only be realized for k > M , and will then lead to an exponentiallysmall contribution due to the appearance of the Bose distribution n B ( k ). The delta-function δ ( ω + ∆ ++ ) = δ ( ω + k + E p + E p − k ) does not get realized at all. Thereby only two of theeight delta-functions in eq. (A.11) remain non-zero, and the integral simplifies to S ( ω ) (cid:12)(cid:12) ps = Z ∞ d k π Z d k (2 π ) ρ ( k , k ) Z d p (2 π ) πE p E p − k (cid:26) δ ( ω − k − E p − E p − k )[1 + n B ( k )] φ ( k ) + δ ( ω + k − E p − E p − k ) n B ( k ) φ ( − k ) (cid:27) , (A.14)where φ ( k ) ≡ − k + E p + E p − k )( k − E p + E p − k ) . (A.15)Fixing k and denoting z ≡ − p · k /pk , so that E p − k = p p + k + 2 pkz + M , we can changeintegration variables from p, z to E p , E p − k : Z d p = 2 π Z ∞ d p p Z +1 − d z = 2 π Z ∞ M d E p Z E + p − k E − p − k d E p − k E p E p − k k , (A.16)where E ± p − k ≡ p p ± pk + k + M . The hard task is to figure out when the δ -functions getrealized. For δ ( ω − k − E p − E p − k ) this happens provided that E − p − k < ω − k − E p < E + p − k ,which leads to ω > M , k < ω − M , k < p ( ω − k ) − M , (A.17) ω − k − k s − M ( ω − k ) − k < E p < ω − k k s − M ( ω − k ) − k . (A.18)In the case of the free gluon spectral function, with k = k , these simplify to ω > M , k < ω − M ω , (A.19) ω − k − k s − M ω ( ω − k ) < E p < ω − k k s − M ω ( ω − k ) . (A.20)18or δ ( ω + k − E p − E p − k ), we simply need to set k → − k in eqs. (A.17), (A.18); puttingsubsequently k = k , the explicit expressions read ω > , k > max (cid:18) , M − ω ω (cid:19) , (A.21) ω + k − k s − M ω ( ω + 2 k ) < E p < ω + k k s − M ω ( ω + 2 k ) . (A.22)Note also that the function φ evaluates to − / [4 ω ( ω − E p )] after integration over E p − k , forboth delta functions in eq. (A.14).Inserting the free gluon spectral function from eq. (A.2) and using the simplified formulaefrom eqs. (A.19)–(A.22), the integrals over E p − k and E p can now be carried out. For thethermal part, i.e. the one proportional to n B0 , this yields S ( ω ) (cid:12)(cid:12) T ps = 1(4 π ) ω (cid:26)Z ∞ d k n B ( k ) (cid:20) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17) acosh r ω ( ω + 2 k )4 M (A.23) − θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17) acosh r ω ( ω − k )4 M (cid:21)(cid:27) + O ( e − βM ) . The vacuum part, on the other hand, is given by the latter row of eq. (A.23), but just withoutthe function n B ( k ); then the final k -integral is doable as well, and we end up with S ( ω ) (cid:12)(cid:12) vacps = 1(4 π ) θ ( ω − M ) (cid:20) ( ω − M ) ω + 2 M − ω ω acosh (cid:18) ω M (cid:19)(cid:21) . (A.24)Consider next the factorized integrals, i.e. the first term inside the curly brackets ineq. (A.11). Again we start by integrating over p, z , and leave the integration over k forlater. This time it is useful to view z as part of the k -integral, i.e. µ ǫ Z d d k (2 π ) d = 4 µ ǫ (4 π ) d +12 Γ( d − ) Z ∞ d k k d − Z − d z (1 − z ) ( d − / , (A.25)where d ≡ − ǫ . The factorized integrals are, in general, ultraviolet divergent, and necessitatekeeping track of ǫ = 0. As always, a helpful strategy is to add and subtract a simple infraredfinite regulator, such as 1 / ( k + M ) α , where α is so chosen that the complicated expressionbecomes ultraviolet finite after the subtraction, and can be worked out at ǫ = 0, while theultraviolet divergent integral with the measure of eq. (A.25) is taken over the simple regulator.In the complicated but ultraviolet finite integral, it is useful to change integration variablesfrom z to E p − k , using Z +1 − d zE p − k = Z E + p − k E − p − k d E p − k pk . (A.26)19e should remark that in our particular example, S , the trick of adding and subtractinga regulator is superfluous, given that the divergent integral can be directly identified as aknown case, but in the general case we have found it to be very helpful.Now, because of the constraint δ ( ω − E p ) (for ω >
0) in the factorized integrals, theintegral over p can be carried out trivially. In fact, comparing eq. (A.11) with (B.3), whichdefines a corresponding 1-loop integral (denoted by S ( ω ) and given explicitly in eq. (B.4)),we arrive at S ( ω ) (cid:12)(cid:12) fz = S ( ω ) Z ∞ d k π µ ǫ Z d d k (2 π ) d E p − k ρ ( k , k ) (cid:26)h (∆ − + ∆ − − + )(1 + n B0 − n F2 ) − (∆ − −− + ∆ − − )( n B0 + n F2 ) i(cid:27) p = √ ω − M / . (A.27)Let us first inspect the vacuum ( T = 0) part hereof, i.e. the term without n B0 or n F2 . Insertingthe free gluon spectral function from eq. (A.2), the multiplier of S ( ω ) becomes B ≡ µ ǫ Z d d k (2 π ) d kE p − k (cid:20) k + E p + E p − k + 1 k − E p + E p − k (cid:21) p = √ ω − M / . (A.28)This can be compared with the integral B ( P ; 0 , M ) ≡ µ ǫ Z d D K (2 π ) D K [( P − K ) + M ] (A.29)= µ ǫ Z d d k (2 π ) d kE p − k (cid:20) ip + k + E p − k + 1 − ip + k + E p − k (cid:21) , (A.30)where we denoted K = ( k , k ) and carried out the integral over k . In other words, B = B ( P E ; 0 , M ), where P E ≡ ( − iE p , p ˆ e ) | p = √ ω − M / , P E = − M , (A.31)and ˆ e is a unit vector; the value of this standard vacuum integral reads B ( − M ; 0 , M ) = 1(4 π ) (cid:20) ǫ + ln ¯ µ M + 2 + O ( ǫ ) (cid:21) . (A.32)Combining this with eq. (B.4), the factorized vacuum part becomes S ( ω ) (cid:12)(cid:12) vacfz = S ( ω ) B ( − M ; 0 , M )= θ ( ω − M ) ( ω − M ) ω (4 π ) tanh (cid:16) βω (cid:17)(cid:20) ǫ + ln ¯ µ M ( ω − M ) + 4 + O ( ǫ ) (cid:21) . (A.33)For completeness, we have even kept exponentially small thermal terms in the coefficient of1 /ǫ , given that it is useful to crosscheck the exact cancellation of ultraviolet poles; after thischeck, we set tanh( βω/
4) = 1 + O (exp( − βM )), given that ω ≥ M .20onsider then the thermal part of eq. (A.27). Again, we omit exponentially small terms ∼ exp( − βM ), and use the free gluon spectral function. Because of the remaining factor n B0 ,the k -integral is exponentially convergent, and we can set ǫ = 0. Employing eq. (A.26) thethermal part becomes S ( ω ) (cid:12)(cid:12) T fz = S ( ω )(4 π ) Z ∞ d k k n B ( k ) Z +1 − d zE p − k ×× (cid:20) k + E p + E p − k + 1 k − E p + E p − k − k − E p − E p − k − k + E p − E p − k (cid:21) p = √ ω − M / = S ( ω )(4 π ) p Z ∞ d k n B ( k ) ×× ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( k + E p + E + p − k )( k − E p + E + p − k )( k − E p − E + p − k )( k + E p − E + p − k )( k + E p + E − p − k )( k − E p + E − p − k )( k − E p − E − p − k )( k + E p − E − p − k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = √ ω − M / . (A.34)Making use of ( k + σE p ) − ( E τp − k ) = 2 k [ σE p − τ p ] , σ, τ = ± , (A.35)it can be seen that the argument of the logarithm evaluates to unity. Hence, S ( ω ) (cid:12)(cid:12) T fz = 0.To summarize, combining eqs. (A.23), (A.24), (A.33), we get S ( ω ) = S ( ω ) (cid:12)(cid:12) T ps + S ( ω ) (cid:12)(cid:12) vacps + S ( ω ) (cid:12)(cid:12) vacfz . (A.36)The other master sum-integrals can be worked out in the same way, and the final results arelisted in appendix B. 21 ppendix B. General results for all master sum-integrals We collect in this appendix the results for all the master sum-integrals entering the computa-tion, obtained with the methods explained in appendix A. In each case, we list the definitionof the sum-integral; an intermediate result obtained after carrying our the Matsubara sumsand taking the discontinuity; and the final result, obtained after restricting to the free gluonspectral function, omitting terms suppressed by exp( − βM ) (except from the ultraviolet di-vergences), and carrying out the final spatial integrations. As before, the integration measurefor the spatial integrations is defined as Z p ≡ µ ǫ Z d − ǫ p (2 π ) − ǫ , (B.1)and ¯ µ = 4 πµ e − γ E denotes the MS scale parameter. To simplify the expressions somewhat,we also make use of the shorthands listed in eqs. (A.12), (A.13). The subscripts “ps” and“fz” denote “phase space” and “factorized” integrations, respectively, in the sense of sec. A.2. B.1. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)PZ { P } P )∆( P − Q ) (cid:21) Q =( − iω, ) . (B.2)Carrying out the Matsubara sum and taking the discontinuity leads to S ( ω ) = Z p π E p h − n F ( E p ) ih δ ( ω − E p ) − δ ( ω + 2 E p ) i . (B.3)The remaining integral is trivial due to the δ -function and, restricting to ω >
0, we arrive at S ( ω ) = θ ( ω − M ) ( ω − M ) πω tanh (cid:16) βω (cid:17)(cid:20) ǫ (cid:18) ln ¯ µ ω − M + 2 (cid:19) + O ( ǫ ) (cid:21) . (B.4) B.2. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)PZ { P } ( P )∆( P − Q ) (cid:21) Q =( − iω, ) . (B.5)It is easy to see that S = − d S / d M . Therefore, from eq. (B.4), we obtain S ( ω ) = θ ( ω − M ) ( ω − M ) − πω tanh (cid:16) βω (cid:17)(cid:20) ǫ ln ¯ µ ω − M + O ( ǫ ) (cid:21) . (B.6)22 .3. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n P )∆( P − Q − K ) (cid:21) Q =( − iω, ) . (B.7)Performing the Matsubara sums, taking the discontinuity, and making use of the antisym-metry of ρ ( k , k ) yields S ( ω ) = Z ∞ d k π Z k , p ρ ( k , k ) π E p E p − k (cid:26) (B.8) h δ ( ω − ∆ ++ ) − δ ( ω + ∆ ++ ) ih (1 + n B0 )(1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ −− ) − δ ( ω + ∆ −− ) ih − n B0 (1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ + − ) − δ ( ω + ∆ + − ) ih n B0 n F1 − (1 + n B0 ) n F2 + n F1 n F2 i + h δ ( ω − ∆ − + ) − δ ( ω + ∆ − + ) ih n B0 n F2 − (1 + n B0 ) n F1 + n F1 n F2 i(cid:27) . Inserting the free gluon spectral function, and omitting exponentially small terms, yields S ( ω ) = 1(4 π ) (cid:26) θ ( ω − M ) (cid:20) ( ω − M ) ( ω + 2 M )8 ω + M ( M − ω ) ω acosh (cid:18) ω M (cid:19)(cid:21) + Z ∞ d k k n B ( k ) (cid:20) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17)s − M ω ( ω + 2 k )+ θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17)s − M ω ( ω − k ) (cid:21)(cid:27) + O ( e − βM ) . (B.9) B.4. S The sum-integral S is defined in eq. (A.5); its value after the Matsubara sums is given ineq. (A.11); the result after the phase space integrals is the sum of eqs. (A.23), (A.24), (A.33). B.5. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n Q · K ∆( P )∆( P − Q )∆( P − K ) (cid:21) Q =( − iω, ) . (B.10)23erforming the Matsubara sums, taking the discontinuity, and making use of the antisym-metry of ρ ( k , k ) yields S ( ω ) = Z ∞ d k π Z k , p ρ ( k , k ) πk ω E p E p − k (cid:26) (B.11)12 E p h δ ( ω − E p ) + δ ( ω + 2 E p ) i (1 − n F1 ) ×× h (∆ − − ∆ − − + )(1 + n B0 − n F2 ) + (∆ − −− − ∆ − − )( n B0 + n F2 ) i + h δ ( ω − ∆ ++ ) + δ ( ω + ∆ ++ ) i ∆ − ∆ − − + h (1 + n B0 )(1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ −− ) + δ ( ω + ∆ −− ) i ∆ − −− ∆ − − h − n B0 (1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ + − ) + δ ( ω + ∆ + − ) i ∆ − −− ∆ − − h n B0 n F1 − (1 + n B0 ) n F2 + n F1 n F2 i + h δ ( ω − ∆ − + ) + δ ( ω + ∆ − + ) i ∆ − ∆ − − + h n B0 n F2 − (1 + n B0 ) n F1 + n F1 n F2 i(cid:27) . Inserting the free gluon spectral function, the ultraviolet divergent factorized vacuum partreads S ( ω ) (cid:12)(cid:12) vacfz = − θ ( ω − M ) ω ( ω − M ) π ) tanh (cid:16) βω (cid:17)(cid:20) ǫ + ln ¯ µ M ( ω − M ) + 3 + O ( ǫ ) (cid:21) , (B.12)where in the coefficient of the divergence we have accounted even for exponentially smallterms. The vacuum part from the phase space integrals reads S ( ω ) (cid:12)(cid:12) vacps = θ ( ω − M )(4 π ) (cid:20) ω − M ) (2 M − ω )32 ω + ω − ω M + 6 M ω acosh (cid:18) ω M (cid:19)(cid:21) , (B.13)while the thermal parts amount to S ( ω ) (cid:12)(cid:12) T fz = 1(4 π ) (cid:26)Z ∞ d k k n B ( k ) θ ( ω − M ) (cid:20) − (cid:18) ω M (cid:19) (cid:21)(cid:27) + O ( e − βM ) , (B.14) S ( ω ) (cid:12)(cid:12) T ps = 1(4 π ) (cid:26)Z ∞ d k k n B ( k ) (cid:20) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17) acosh r ω ( ω + 2 k )4 M (B.15)+ θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17) acosh r ω ( ω − k )4 M (cid:21)(cid:27) + O ( e − βM ) . .6. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)PZ K { P } K K ∆( P )∆( P − Q )∆( P − K ) (cid:21) Q =( − iω, ) . (B.16)Because of the ultraviolet divergent factor in the numerator, the use of the spectral repre-sentation requires care in this case, and we rather proceed directly with the sum, havinggone over into free gluons to start with. Carrying out the shift K → P − K , the summationfactorizes, S ( ω ) = Disc (cid:20)PZ { P } P )∆( P − Q ) (cid:21) Q =( − iω, ) × PZ { K } K ) = S ( ω ) I ( M ) , (B.17)where S ( ω ) is given in eq. (B.4), while I is a basic tadpole integral generalized to finitetemperature. In fact, the finite temperature effects in I are exponentially small and can beomitted: I ( M ) = Z k k h − n F ( E k ) i = − M (4 π ) (cid:20) ǫ + ln ¯ µ M + 1 (cid:21) + O ( ǫ, e − βM ) . (B.18)Keeping exponentially small terms in the coefficient of the divergence, though, we arrive at S ( ω ) = − θ ( ω − M ) ( ω − M ) M ω (4 π ) tanh (cid:16) βω (cid:17)(cid:20) ǫ + ln ¯ µ M ( ω − M ) + 3 + O ( ǫ, e − βM ) (cid:21) . (B.19) B.7. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n ( P )∆( P − Q )∆( P − K ) (cid:21) Q =( − iω, ) . (B.20)25erforming the Matsubara sums, taking the discontinuity, and making use of the antisym-metry of ρ ( k , k ) yields S ( ω ) = Z ∞ d k π Z k , p ρ ( k , k ) π E p E p − k (cid:26) (B.21)12 E p h δ ( ω − E p ) − δ ( ω + 2 E p ) i (1 − n F1 ) ×× h (∆ − + ∆ − − + )(∆ − − ∆ − − + + E − p )(1 + n B0 − n F2 )+(∆ − −− + ∆ − − )(∆ − −− − ∆ − − − E − p )( n B0 + n F2 ) i − β E p h δ ( ω − E p ) − δ ( ω + 2 E p ) i (1 − n F1 ) n F1 ×× h (∆ − + ∆ − − + )(1 + n B0 − n F2 ) − (∆ − −− + ∆ − − )( n B0 + n F2 ) i + 12 E p h δ ′ ( ω − E p ) + δ ′ ( ω + 2 E p ) i (1 − n F1 ) ×× h (∆ − + ∆ − − + )(1 + n B0 − n F2 ) − (∆ − −− + ∆ − − )( n B0 + n F2 ) i + h δ ( ω − ∆ ++ ) − δ ( ω + ∆ ++ ) i ∆ − ∆ − − + (∆ − − + − ∆ − ) ×× h (1 + n B0 )(1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ −− ) − δ ( ω + ∆ −− ) i ∆ − −− ∆ − − (∆ − −− − ∆ − − ) ×× h − n B0 (1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ + − ) − δ ( ω + ∆ + − ) i ∆ − −− ∆ − − (∆ − −− − ∆ − − ) ×× h n B0 n F1 − (1 + n B0 ) n F2 + n F1 n F2 i + h δ ( ω − ∆ − + ) − δ ( ω + ∆ − + ) i ∆ − ∆ − − + (∆ − − + − ∆ − ) ×× h n B0 n F2 − (1 + n B0 ) n F1 + n F1 n F2 i(cid:27) . In the factorized part, it is useful to carry out a partial integration in order to remove thestructure δ ′ ( ω − E p ) + δ ′ ( ω + 2 E p ): Z d d p (2 π ) d δ ′ ( ω − E p ) g ( p, E p , E p − k ) (B.22)= Z d d p (2 π ) d δ ( ω − E p ) (cid:26) ( d − E p g p + E p p ∂g∂p + 12 E p ∂ ( E p g ) ∂E p + ( p + kz ) E p pE p − k ∂g∂E p − k (cid:27) . The subsequent steps proceed as described in appendix A.In contrast to S , S , however, it is not possible to give separate closed expressions forthe factorized and phase space vacuum parts of S , because the integrals are logarithmi-26ally divergent at the lower limit of the k -integration (in the thermal case, they are linearlydivergent). Yet the sum is finite, and inserting the free gluon spectral function, we get S ( ω ) (cid:12)(cid:12) vac = θ ( ω − M )4 ω (4 π ) ( ω − M ) (cid:26) tanh (cid:16) βω (cid:17)(cid:20) ǫ + ln ¯ µ M ( ω − M ) + 2 (cid:21) (B.23)+ ω − M M ln ω ( ω − M ) M + ( ω − M ) (4 M − ω ) ωM acosh (cid:18) ω M (cid:19)(cid:27) + O ( ǫ ) , where in the coefficient of the divergence we have accounted even for exponentially smallthermal corrections. For the thermal part proper we obtain S ( ω ) (cid:12)(cid:12) T = 14 ω M (4 π ) Z ∞ d k n B ( k ) k (cid:20) (B.24) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17)p ω ( ω + 2 k ) p ω ( ω + 2 k ) − M + θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17)p ω ( ω − k ) p ω ( ω − k ) − M − θ ( ω − M ) × ω p ω − M (cid:21) + O ( e − βM ) . The last line, which originates from the factorized integrals, subtracts the values of the firsttwo lines at k = 0 (for ω > M ), rendering the integral infrared finite. B.8. ˆ S The sum-integral ˆ S is defined asˆ S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n p − ( p · ˆ k ) ∆ ( P )∆( P − Q )∆( P − K ) (cid:21) Q =( − iω, ) . (B.25)Carrying out the Matsubara sums proceeds precisely like for S , and leads to an expressionlike eq. (B.21); it is also again useful to carry out the partial integration in eq. (B.22). Thesubsequent steps lead to the vacuum partˆ S ( ω ) (cid:12)(cid:12)(cid:12) vac = θ ( ω − M )( ω − M ) ω (4 π ) (cid:26) tanh (cid:16) βω (cid:17)(cid:20) ǫ + ln ¯ µ M ( ω − M ) + 1 (cid:21) − ω ( ω − M ) M + 2(7 ω − M ) ω ( ω − M ) acosh (cid:18) ω M (cid:19) + 2 ω ( ω − M ) α (cid:18) √ ω − M ω (cid:19)(cid:27) + O ( ǫ ) . (B.26)27ere the function α ( v ) ≡ Z ∞ d xx (cid:20) θ ( v − x )(1 − x ) ln √ − x + √ v − x √ − x − √ v − x + ln (1 + x + √ vx + x )( − x + √ − vx + x )(1 + x + √ − vx + x )( − x + √ vx + x ) (cid:21) , (B.27)where the integration variable x is related to k through k = xω/
2, is finite, but we have notbothered to work out its analytic expression, given that it does not appear in our final result.The thermal part readsˆ S ( ω ) (cid:12)(cid:12)(cid:12) T = 12 ω (4 π ) Z ∞ d k n B ( k ) k (cid:26) (B.28) θ ( ω ) θ (cid:18) k − M − ω ω (cid:19) ×× (cid:20) − p ω ( ω + 2 k ) p ω ( ω + 2 k ) − M + ω ( ω + 2 k ) acosh r ω ( ω + 2 k )4 M (cid:21) + θ ( ω − M ) θ (cid:18) ω − M ω − k (cid:19) ×× (cid:20) − p ω ( ω − k ) p ω ( ω − k ) − M + ω ( ω − k ) acosh r ω ( ω − k )4 M (cid:21) + θ ( ω − M ) × (cid:20) ω p ω − M − ω acosh (cid:18) ω M (cid:19) (cid:21)(cid:27) + O ( e − βM ) . The last line, which originates from the factorized integrals, subtracts the values of the firsttwo lines at k = 0 (for ω > M ), rendering the integral infrared finite. B.9. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)PZ K { P } K K ∆ ( P )∆( P − Q )∆( P − K ) (cid:21) Q =( − iω, ) . (B.29)Because of the ultraviolet divergent factor in the numerator, the use of the spectral represen-tation requires care in this case, and we rather proceed directly with the sum, as in the caseof S . Carrying out the shift K → P − K , the summation factorizes, S ( ω ) = Disc (cid:20)PZ { P } ( P )∆( P − Q ) (cid:21) Q =( − iω, ) × PZ { K } K ) = S ( ω ) I ( M ) , (B.30)28here S ( ω ) is given in eq. (B.6), while I is given in eq. (B.18). Keeping exponentially smallterms in the coefficient of the divergence, we arrive at S ( ω ) = − θ ( ω − M ) M ω ( ω − M ) (4 π ) tanh (cid:16) βω (cid:17)(cid:20) ǫ + ln ¯ µ M ( ω − M ) + 1 + O ( ǫ, e − βM ) (cid:21) . (B.31) B.10. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) Q =( − iω, ) . (B.32)Performing the Matsubara sums, taking the discontinuity, and making use of the antisym-metry of ρ ( k , k ) yields S ( ω ) = Z ∞ d k π Z k , p ρ ( k , k ) π E p E p − k (cid:26) (B.33)18 E p h δ ( ω − E p ) − δ ( ω + 2 E p ) i (1 − n F1 ) ×× (cid:20)(cid:18) ∆ − −− E p + E p − k + ∆ − − E p − E p − k (cid:19) ( n B0 + n F2 ) − (cid:18) ∆ − E p + E p − k + ∆ − − + E p − E p − k (cid:19) (1 + n B0 − n F2 ) (cid:21) + 18 E p − k h δ ( ω − E p − k ) − δ ( ω + 2 E p − k ) i (1 − n F2 ) ×× (cid:20)(cid:18) ∆ − −− E p − k + E p + ∆ − − + E p − k − E p (cid:19) ( n B0 + n F1 ) − (cid:18) ∆ − E p − k + E p + ∆ − − E p − k − E p (cid:19) (1 + n B0 − n F1 ) (cid:21) + h δ ( ω − ∆ ++ ) − δ ( ω + ∆ ++ ) i ∆ − ∆ − − ∆ − − + h (1 + n B0 )(1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ −− ) − δ ( ω + ∆ −− ) i ∆ − −− ∆ − − ∆ − − + h − n B0 (1 − n F1 − n F2 ) + n F1 n F2 i + h δ ( ω − ∆ + − ) − δ ( ω + ∆ + − ) i ∆ − − ∆ − ∆ − −− h n B0 n F1 − (1 + n B0 ) n F2 + n F1 n F2 i + h δ ( ω − ∆ − + ) − δ ( ω + ∆ − + ) i ∆ − − + ∆ − ∆ − −− h n B0 n F2 − (1 + n B0 ) n F1 + n F1 n F2 i(cid:27) . In the factorized part, a change of integration variables p → k − p allows trivially to changethe structure with δ ( ω − E p − k ) − δ ( ω + 2 E p − k ) into the familiar one with δ ( ω − E p ) − ( ω + 2 E p ). (The only complication is that then the difference 1 / ( E p − E p − k ) needs to beinterpreted as a principal value.) The subsequent steps proceed as described in appendix A.Like with S , it is again not possible to give separate closed expressions for the factorizedand phase space vacuum parts of S , because the integrals are logarithmically divergent atthe lower limit of the k -integration (in the thermal case, they are linearly divergent). Thesum is infrared finite, however, and inserting the free gluon spectral function, yields [3] S ( ω ) (cid:12)(cid:12) vac = θ ( ω − M ) ω (4 π ) L (cid:16) ω − √ ω − M ω + √ ω − M (cid:17) + O ( ǫ ) , (B.34)where the function L is defined in eq. (4.6). For the thermal parts we obtain, omittingexponentially small terms, S ( ω ) (cid:12)(cid:12) T = 2 ω (4 π ) Z ∞ d k n B ( k ) k (cid:20) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17) acosh r ω ( ω + 2 k )4 M + θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17) acosh r ω ( ω − k )4 M − θ ( ω − M ) × (cid:18) ω M (cid:19) (cid:21) + O ( e − βM ) . (B.35)The last line, which originates from the factorized integrals, subtracts the values of the firsttwo lines at k = 0 (for ω > M ), rendering the integral infrared finite. B.11. ˆ S The sum-integral ˆ S is defined asˆ S ( ω ) ≡ Disc (cid:20)Z ∞−∞ d k π PZ K { P } ρ ( k , k ) k − ik n p − ( p · ˆ k ) ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) Q =( − iω, ) . (B.36)Carrying out the Matsubara sums proceeds precisely like for S , and leads to an expressionlike eq. (B.33). The subsequent steps lead to the vacuum partˆ S ( ω ) (cid:12)(cid:12)(cid:12) vac = θ ( ω − M ) ω (4 π ) (cid:20) − M L (cid:16) ω − √ ω − M ω + √ ω − M (cid:17) + ω β (cid:18) √ ω − M ω (cid:19)(cid:21) + O ( ǫ ) . (B.37)Here the function β ( v ) ≡ Z ∞ d xx (cid:20) θ ( v − x ) √ − x p v − x − v + 34 (cid:16)p vx + x − p − vx + x (cid:17) + x −
48 ln (cid:12)(cid:12)(cid:12)(cid:12) (1 + √ vx + x )( − √ − vx + x )(1 + √ − vx + x )( − √ vx + x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) (B.38)30here the integration variable x is related to k through k = xω/
2, is finite, but we have notbothered to work out its analytic expression, given that it does not appear in our final result.The thermal part readsˆ S ( ω ) (cid:12)(cid:12)(cid:12) T = 12 ω (4 π ) Z ∞ d k n B ( k ) k (cid:26) (B.39) θ ( ω ) θ (cid:18) k − M − ω ω (cid:19) ×× (cid:20)p ω ( ω + 2 k ) p ω ( ω + 2 k ) − M − M acosh r ω ( ω + 2 k )4 M (cid:21) + θ ( ω − M ) θ (cid:18) ω − M ω − k (cid:19) ×× (cid:20)p ω ( ω − k ) p ω ( ω − k ) − M − M acosh r ω ( ω − k )4 M (cid:21) + θ ( ω − M ) × (cid:20) − ω p ω − M + 8 M acosh (cid:18) ω M (cid:19) (cid:21)(cid:27) + O ( e − βM ) . The last line, which originates from the factorized integrals, subtracts the values of the firsttwo lines at k = 0 (for ω > M ), rendering the integral infrared finite. B.12. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)PZ K { P } K K ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) Q =( − iω, ) . (B.40)Like with S and S , we proceed directly with free gluons rather than using the spectralrepresentation. Carrying out the shift K → P − K , the summation factorizes, S ( ω ) = Disc (cid:26)(cid:20)PZ { P } P )∆( P − Q ) (cid:21) Q =( − iω, ) × (cid:20)PZ { K } K )∆( K − Q ) (cid:21) Q =( − iω, ) (cid:27) = 2 Re (cid:20)PZ { P } P )∆( P − Q ) (cid:21) Q =( − iω, ) × Disc (cid:20)PZ { K } K )∆( K − Q ) (cid:21) Q =( − iω, ) , (B.41)where Re[ ... ] denotes the regular (non-discontinuous) part, while the discontinuous part can beidentified with the function S ( ω ), given in eq. (B.4). The Matsubara sum in the regular partcan be carried out as before; the only difference with respect to the procedure in appendix Ais that taking the regular part after the substitution ω n → − iω yields a principle value rather31han a delta-function:Re (cid:20)PZ { P } P )∆( P − Q ) (cid:21) Q =( − iω, ) = Z p E p (cid:20) P (cid:18) ω + 2 E p (cid:19) − P (cid:18) ω − E p (cid:19)(cid:21)h − n F ( E p ) i . (B.42)It is seen that the finite-temperature effects continue to be exponentially suppressed. Thezero-temperature part, on the other hand, equals the real part of the general function B ,another special case of which was met in eq. (A.29):Re (cid:20) µ ǫ Z d D P (2 π ) D P + M )[( P − Q ) + M ] (cid:21) Q =( − iω, ) = Re h B ( − ω ; M , M ) i = 1(4 π ) (cid:20) ǫ + ln ¯ µ M + 2 − ω − M ) ω acosh (cid:18) ω M (cid:19) + O ( ǫ ) (cid:21) , ω > M . (B.43)Combining this with S ( ω ), and keeping the exponentially small terms in the coefficient ofthe divergence, we arrive at S ( ω ) = θ ( ω − M ) ( ω − M ) ω (4 π ) tanh (cid:16) βω (cid:17) × (B.44) × (cid:20) ǫ + ln ¯ µ M ( ω − M ) + 4 − ω − M ) ω acosh (cid:18) ω M (cid:19) + O ( ǫ, e − βM ) (cid:21) . B.13. ˆ S The sum-integral ˆ S is defined asˆ S ( ω ) ≡ Disc (cid:20)PZ K { P } K K [ p − ( p · ˆ k ) ]∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) Q =( − iω, ) . (B.45)The summation factorizes into two independent parts like for S ; however, the spatial in-tegrations do not factorize due to the additional structure in the numerator. Therefore theevaluation is somewhat more involved, yet the general techniques introduced in appendix Ayield a solution:ˆ S ( ω ) = θ ( ω − M ) ( ω − M ) ω (4 π ) tanh (cid:16) βω (cid:17) × (B.46) × (cid:20) ǫ + ln ¯ µ M ( ω − M ) + 4 + ω + 2 M ω − M ) − ω − ω M + 12 M ) ω ( ω − M ) acosh (cid:18) ω M (cid:19) + O ( ǫ, e − βM ) (cid:21) . .14. S The sum-integral S is defined as S ( ω ) ≡ Disc (cid:20)PZ K { P } K ( K ) ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) Q =( − iω, ) . (B.47)Like with S , S and S , we proceed directly with free gluons rather than using the spectralrepresentation. Cancelling one K and carrying out the shift K → P − K , we get S ( ω ) = Disc (cid:20)PZ { K,P } ∆( P ) + ∆( K ) − M + P · K )∆( P )∆( P − Q )∆( K )∆( K − Q ) (cid:21) Q =( − iω, ) . (B.48)Another change of integration variables, P → Q − P , shows that PZ { P } P ∆( P )∆( P − Q ) = Q PZ { P } P )∆( P − Q ) , (B.49)and similarly for the term with Σ R { K } . Thereby we arrive at S ( ω ) = 2 S ( ω ) + 12 ( ω − M ) S ( ω ) (B.50)= θ ( ω − M ) ( ω − M ) ω (4 π ) tanh (cid:16) βω (cid:17) ×× (cid:26) ( ω − M ) (cid:20) ǫ + ln ¯ µ M ( ω − M ) + 3 (cid:21) +( ω − M ) (cid:20) − ω − M ) ω acosh (cid:18) ω M (cid:19)(cid:21) + O ( ǫ, e − βM ) (cid:27) . (B.51)33 ppendix C. Spectral function in the scalar channel For completeness, we have worked out the spectral function corresponding to the scalarchannel with the same methods as described above for the vector channel. It seems, though,that the physical significance is not clear in the scalar case: no direct relation to an observable,in the sense of eq. (2.2), has been worked out, as far as we know, and the computation assuch appears to possess a number of ambiguities. In particular, the scalar density requiresrenormalization, and the renormalization factor cannot be uniquely specified; moreover theresummation of the spectral function within a potential model near the threshold appears tolead to ambiguities [8]. Nevertheless, on the lattice the correlator of (bare) scalar densitiescan be treated on the same footing as that of the vector currents [21, 22].Concerning the first of the issues, namely renormalization, the method we choose is toconsider the object ˆ
S ≡ M ( δ ) B ˆ¯ ψ ˆ ψ , (C.1)where M ( δ ) B is essentially the bare quark mass defined in eq. (3.7), only with a possibleadditional constant as a “probe”, (cid:16) M ( δ ) B (cid:17) ≡ M − g C F M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 + δ (cid:19) + O ( g ) . (C.2)We then define ρ S ( ω ) ≡ Z ∞−∞ d t e iωt Z d − ǫ x (cid:28)
12 [ ˆ S ( t, x ) , ˆ S (0 , )] (cid:29) , (C.3)which turns out to be finite. Starting again at 1-loop level, and omitting Q -independentterms which are killed by the discontinuity in eq. (2.5), we get= [ Q − indep.] − C A M PZ { P } Q + 4 M ∆( P )∆( P − Q ) . (C.4)The counterterm graph yields= [ Q − indep.] + 12 g C A C F M (4 π ) (cid:26)(cid:18) ǫ + ln ¯ µ M + 43 (cid:19) (C.5) × PZ { P } (cid:20) Q + 8 M ∆( P )∆( P − Q ) − M ( Q + 4 M )∆ ( P )∆( P − Q ) (cid:21) + δ PZ { P } Q + 4 M ∆( P )∆( P − Q ) (cid:27) . Q − indep.] + 4 g C A C F M PZ K { P } (cid:26)(cid:18) K + Π T − K + Π E (cid:19) [ p − ( p · ˆ k ) ] ×× (cid:20) − Q + 4 M )∆ ( P )∆( P − Q )∆( P − K ) − Q + 4 M )∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) + D − K + Π T (cid:20) Q + 4 M ∆ ( P )∆( P − Q ) + 2 Q · K ∆( P )∆( P − Q )∆( P − K ) − ( Q + 4 M ) K ∆ ( P )∆( P − Q )∆( P − K ) − ( Q + 4 M ) K ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) + 1 K + Π E (cid:20) − Q + 4 M )∆( P )∆( P − Q )∆( P − K ) + 4( Q + 4 M ) M ∆ ( P )∆( P − Q )∆( P − K )+ ( Q + 2 M )( Q + 4 M ) + Q K ∆( P )∆( P − Q )∆( P − K )∆( P − Q − K ) (cid:21) (cid:27) . (C.6)Again any dependence on the gauge parameter ξ has disappeared, and P Eµν of eq. (3.2) couldhave been replaced with δ µν − P Tµν .We note that the master sum-integrals appearing in eq. (C.6) are a subset of those ineq. (3.11). Therefore the discussion in sec. 3.4 continues to hold, and there are no infrareddivergences in the result, so that we can set Π T = Π E = 0 in eq. (C.6). The full result cannow be written as ρ S ( ω ) | raw = 2 C A M ( ω − M ) S ( ω ) + 4 g C A C F M (cid:26)(cid:20) − π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19)(cid:21) ( ω − M ) S ( ω ) − δ (4 π ) ( ω − M ) S ( ω ) − (cid:20) T − M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19)(cid:21) ( ω − M ) S ( ω )+ 4( ω − M ) S ( ω ) + 4(1 − ǫ ) S ( ω ) − ω − M ) h M S ( ω ) − (1 − ǫ ) S ( ω ) i + ( ω − M )( ω − M ) S ( ω ) − h ǫω + 4(1 − ǫ ) M i S ( ω ) (cid:27) + O ( ǫ ) . (C.7)We have set here ǫ → T vanish as θ ( ω − M )( ω − M ) at the threshold,35eing thus subdominant with respect to the leading thermal corrections which remain non-zero. Nevertheless, we would like to apply a “universal” thermal resummation, i.e. preciselyeq. (4.3); however, it may be questioned whether it is valid to do this also in the term M ,coming from the (“ultraviolet-completed”) definition of the scalar current, or only in moreinfrared sensitive parts. It seems to us that this question can be fully settled only through anext-to-next-to-leading order computation; in the following, we assume that the resummationof eq. (4.3) is only carried out in the Lagrangian, not in the definition of the scalar density.If so, a redefinition of the mass according to eq. (4.3) leads to the modified result ρ S ( ω ) = 2 C A M ( ω − M ) S ( ω ) + 4 g C A C F M (cid:26) T S ( ω ) − δ (4 π ) ( ω − M ) S ( ω ) − π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19)h ( ω − M ) S ( ω ) − M ( ω − M ) S ( ω ) i + 4( ω − M ) S ( ω ) + 4(1 − ǫ ) S ( ω ) − ω − M ) h M S ( ω ) − (1 − ǫ ) S ( ω ) i + ( ω − M )( ω − M ) S ( ω ) − h ǫω + 4(1 − ǫ ) M i S ( ω ) (cid:27) + O ( ǫ ) . (C.8)Unfortunately, the issue of what is resummed is not insignificant in the sense that the differ-ence between eqs. (C.7) and (C.8) is numerically of O (1) for ω > M .Inserting the explicit expressions for the functions S ji ( ω ) from appendix B into eq. (C.8),the final result for the vacuum part reads ρ S ( ω ) | vac = θ ( ω − M ) C A M ( ω − M ) πω + θ ( ω − M ) 4 g C A C F M (4 π ) ω (cid:26) (C.9)( ω − M )( ω − M ) L (cid:18) ω − √ ω − M ω + √ ω − M (cid:19) + (cid:18) ω − ω M − M (cid:19) acosh (cid:18) ω M (cid:19) − ω ( ω − M ) (cid:20) ( ω − M ) (cid:18) ln ω ( ω − M ) M + 34 δ (cid:19) −
38 (3 ω − M ) (cid:21)(cid:27) + O ( ǫ, g ) , where the function L is defined in eq. (4.6). Let us note that although similar to thevector channel spectral function in eq. (4.5) at first sight, eq. (C.9) has also some significantdifferences; in particular, logarithms of ω/M do not cancel at ω ≫ M any more, but theasymptotic behaviour becomes ρ S ( ω ) | vac ω ≫ M ≈ − g C A C F ω M (4 π ) (cid:18) ln ω M + δ − (cid:19) . (C.10)As the dependence on δ and δ ’s definition through eq. (C.2) show, the logarithm is in somesense related to the need to renormalize the scalar density and its correlators.36 .0 10.0 ω /M -1e-040e+001e-042e-043e-044e-045e-04 ρ S - l oop / ω M g C A C F ρ S | vac ρ S | T at T = 0.4M ρ S | T at T = 0.2M ρ S | T at T = 0.1M δ = - . δ = . δ = + . Figure 6:
The vacuum and thermal parts of the next-to-leading order correction in the scalar channel,normalized by dividing with 4 ω M g C A C F . The vacuum part can become negative because thebare scalar correlator is multiplied with a bare mass parameter, cf. eqs. (C.1), (C.2); the constant δ illustrates how strong the dependence on the renormalization convention is. The thermal correction, in turn, reads, ρ S ( ω ) | T = 4 g C A C F M (4 π ) ω Z ∞ d k n B ( k ) k (cid:26) (C.11) θ ( ω ) θ (cid:16) k − M − ω ω (cid:17)(cid:20) − ( ω − M ) p ω ( ω + 2 k ) p ω ( ω + 2 k ) − M + 2 (cid:16) ( ω − M )( ω − M ) + 2 ωk ( ω − M ) + 2 ω k (cid:17) acosh r ω ( ω + 2 k )4 M (cid:21) + θ ( ω − M ) θ (cid:16) ω − M ω − k (cid:17)(cid:20) − ( ω − M ) p ω ( ω − k ) p ω ( ω − k ) − M + 2 (cid:16) ( ω − M )( ω − M ) − ωk ( ω − M ) + 2 ω k (cid:17) acosh r ω ( ω − k )4 M (cid:21) + θ ( ω − M ) (cid:20) ω − M + 4 k ) ω p ω − M − (cid:16) ( ω − M )( ω − M ) + 2 ω k (cid:17) acosh (cid:18) ω M (cid:19)(cid:21)(cid:27) + O ( e − βM , g ) , where we represented T as π T = 6 R ∞ d k k n B ( k ).37 .0 2.5 3.0 ω /M ρ S / ω Μ T = 250 MeVT = 300 MeVT = 350 MeVT = 400 MeVT = 450 MeVT = 500 MeV
M = 2 GeV ω /M ρ S / ω Μ T = 250 MeVT = 300 MeVT = 350 MeVT = 400 MeVT = 450 MeVT = 500 MeV
M = 4 GeV ω /M ρ S / ω Μ T = 250 MeVT = 300 MeVT = 350 MeVT = 400 MeVT = 450 MeVT = 500 MeV
M = 6 GeV
Figure 7:
The phenomenologically assembled scalar channel spectral function ρ S ( ω ), in units of ω M , for M = 2 , , M is the heavy quark polemass. Note that for better visibility, the axis ranges are different in the rightmost figure. As discussedafter eq. (C.14), we are not confident that these plots have a definite physical significance; the figuresare meant for illustration only. A numerical evaluation of this result, compared with the vacuum part of eq. (C.9), isshown in fig. 6. For small ω the thermal part appears to be somewhat more significantthan in the case of the vector channel; this is because there is a cancellation of positive andnegative contributions in the vacuum part, before the negative terms take over at large ω (cf.eq. (C.10)). The thermal part, in contrast, stays positive and vanishes rapidly at large ω .We wish to draw attention to the amusing feature, already mentioned at the end of sec. 4,that while the next-to-leading order vacuum part is continuous, the next-to-leading orderthermal part appears even to have a continuous first derivative. In the vector channel, incontrast, the next-to-leading order vacuum part is discontinuous at the threshold, while thenext-to-leading order thermal part appears to be continuous (cf. fig. 1). In other words, thethermal part seems always to be one degree smoother than the vacuum part.In order to now combine our result with that obtained within a resummed framework inref. [8], we need to match the normalizations, in analogy with eq. (5.4). Indeed, employingthe notation of eq. (5.1), the leading order vacuum result in eq. (C.9) becomes ρ S ( ω ) ω M (cid:12)(cid:12)(cid:12)(cid:12) LO = θ ( ω − M ) C A v π , (C.12)while the next-to-leading order result can be expanded as ρ S ( ω ) ω M (cid:12)(cid:12)(cid:12)(cid:12) NLO = 4 g C A C F θ ( ω − M ) (cid:20) v π − v π (cid:18) δ (cid:19) + O ( v ) (cid:21) . (C.13)Since radiative corrections within a non-relativistic potential model always involve a power of v , it is possible to account for the second term in eq. (C.13), equalling − g C F (1 + 3 δ/ / π .0 2.5 ω /M ρ S / ω Μ M = 1 GeVM = 2 GeVM = 3 GeVM = 4 GeVM = 5 GeVM = 6 GeV
T = 250 MeV ω /M ρ S / ω Μ M = 1 GeVM = 2 GeVM = 3 GeVM = 4 GeVM = 5 GeVM = 6 GeV
T = 350 MeV ω /M ρ S / ω Μ M = 1 GeVM = 2 GeVM = 3 GeVM = 4 GeVM = 5 GeVM = 6 GeV
T = 450 MeV
Figure 8:
The phenomenologically assembled scalar channel spectral function ρ S ( ω ), in units of ω M , for T = 250 , ,
450 MeV (from left to right). To the order considered, M is the heavy quarkpole mass. Note that for better visibility, the axis ranges are different in the leftmost figure. Asdiscussed after eq. (C.14), we are not confident that these plots have a definite physical significance;the figures are meant for illustration only. times the leading term in eq. (5.2), only by a multiplicative correction of the scalar density, S QCD = S NRQCD (cid:20) − g C F π (cid:18) δ (cid:19) + ... (cid:21) . (C.14)Even though closer to unity than in eq. (5.4), the normalization factor could be numericallysignificant. In fact, if we leave the normalization factor open, and search for a value mini-mizing the squared difference of the resummed and QCD results (with δ = 0) in the range( ω − M ) /M = 0 . − . . − .
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