Massive vector current correlator in thermal QCD
aa r X i v : . [ h e p - ph ] D ec Massive vector current correlator in thermal QCD
Y. Burnier and M. Laine
Institute for Theoretical Physics, Albert Einstein Center, University of Bern,Sidlerstrasse 5, CH-3012 Bern, Switzerland
Abstract
We present an NLO analysis of the massive vector current correlator at temperatures abovea few hundred MeV. The physics of this correlator originates from a transport peak, relatedto heavy quark diffusion, and from the quark-antiquark threshold, related to quarkoniumphysics. In the bottom case both can be studied with separate effective theories, but for charmthese may not be accurate, so a study within the full theory is needed. Working in imaginarytime, the NLO correlator can be computed in unresummed perturbation theory; comparingwith lattice data, we find good agreement. Subsequently we inspect how non-perturbativemodifications of the transport peak would affect the imaginary-time correlator. The massiveNLO quark-number susceptibility is also contrasted with numerical measurement.October 2012 . Introduction
Heavy (charm and bottom) quarks are excellent probes for the properties of the hot QCDplasma generated in heavy ion collision experiments. On the theoretical side, the existence ofa mass scale M ≫ πT ≫
200 MeV renders the heavy quarks relatively tractable, permittingfor an interpolation between the simple static dynamics of the infinite mass limit and thehigh mobility case manifested by lighter quarks. It is particularly fortunate that two heavyflavours are available, offering for a handle on the functional dependence on
M/πT . On theexperimental side, heavy quarks and quarkonia are readily tagged because of their distinctiveleptonic decays. Indeed thermal modifications of the bottomonia spectra were among thefirst spectacular results produced by the LHC heavy ion program [1].For the bottom quark case, recent years have seen significant progress in theoretical studiesof the main phenomena involved, namely single quark “transport” (diffusion, kinetic equi-libration) as well as physics near the quark-antiquark threshold (quarkonium dissociation,chemical equilibration) (cf. ref. [2] for a review and refs. [3]–[12] for recent contributions).Largely this progress has been achieved through the use of modern effective field theory meth-ods (Heavy Quark Effective Theory, or HQET, for single quarks; Non-Relativistic QCD, orNRQCD, for quarkonium physics). Once properly formulated the effective field theory ob-servables can be measured non-perturbatively with lattice Monte Carlo methods, and indeedfirst results suggest that these avenues may lead to substantial progress [13]–[17].In the charm quark case, however, it is not guaranteed that the heavy quark expansionconverges fast enough to yield quantitatively accurate results for all observables of interest.On the other hand, it no longer appears prohibitively expensive to treat charm quarks as“light” degrees of freedom in lattice QCD. Indeed, results have appeared concerning boththermodynamic quantities [18, 19, 20] and imaginary-time correlators relevant for determin-ing dynamical properties of the system [21] (earlier works can be found in refs. [22]–[25] andreferences therein). Yet, a modest scale hierarchy between πT and M does exist, and con-sequently systematic errors, both from lattice artifacts and from the unavoidable analyticcontinuation [26], are likely to be harder to control than in the light quark case. In fact,given that systematic errors related to analytic continuation remain substantial even for lightquarks [27], further crosschecks appear welcome.The goal of the current study is to derive results within next-to-leading order (NLO)perturbation theory which may help in the analysis of lattice data such as those in ref. [21].In order to allow for a direct appreciation of Euclidean measurements, we inspect how specificmodifications of transport properties and threshold features manifest themselves in imaginarytime. Ultimately, in accordance with the philosophy of ref. [28] and practical tests of refs. [27,29], the goal would be to subtract “trivial” ultraviolet features from continuum-extrapolatedlattice data, in order to allow for a model-independent extraction of real-time physics [30].1n the technical side, the current work represents a continuation of our earlier study [31],in which the massive vector current spectral function was computed at NLO in the domain M ≫ πT (keeping only those thermal effects which are not exponentially suppressed). Herewe keep the full mass dependence, permitting for an extrapolation also to the regime M ≪ πT ,as well as contributions from the transport peak at ω ≪ M which were omitted in ref. [31].Moreover we work directly in imaginary time , which has the benefit that the usual problemsof convergence at small ω are milder. In particular, at NLO infrared safe results can beobtained without resummations, similarly to what has previously been achieved for gluonicobservables [32, 29].The plan of this paper is the following. After specifying the observables considered anddiscussing the methods employed (sec. 2), we outline the qualitative structure of our find-ings in sec. 3. The detailed analytic and numerical results of the strict NLO analysis com-prise sec. 4, whereas in secs. 5 and 6 the effects of non-perturbative modifications of thetransport peak and quarkonium threshold, respectively, are inspected. Sec. 7 presents ourconclusions; appendix A results for all the “master” sum-integrals considered; and appendix Bdetails related to renormalization.
2. Observables and methods
The main quantity considered is the vector current correlator related to a massive flavour.Like in lattice QCD we work in Euclidean signature, with the usual thermal boundary con-ditions imposed across the time direction. Then the correlator is defined as G V ( τ ) ≡ d X µ =0 Z x D ( ¯ ψγ µ ψ )( τ, x ) ( ¯ ψγ µ ψ )(0 , ) E T (2.1) ≡ G ( τ ) − G ii ( τ ) , < τ < T , (2.2)where the Dirac matrices are Minkowskian, and a sum over spatial indices is implied. Be-cause of current conservation the charge correlator is independent of τ , and we denote thecorresponding “susceptibility” by χ ≡ Z β d τ G ( τ ) = β G (0) , β ≡ T . (2.3)For future reference let us also record the free massless results for these correlators [33]: G free ii ( τ ) ≡ N c T (cid:20) π (1 − τ T ) 1 + cos (2 πτ T )sin (2 πτ T ) + 2 cos(2 πτ T )sin (2 πτ T ) + 16 (cid:21) , (2.4) χ free ≡ N c T . (2.5)2ere C A ≡ N c = 3 refers to the number of colours. Later on the group theory factor C F ≡ ( N − / (2 N c ) will also appear. Spacetime dimension is denoted by D = d +1 = 4 − ǫ .At NLO, we find it convenient to compute the correlators G V and G . The spatial part G ii is then obtained from eq. (2.2). Verifying explicitly the τ -independence of G providesfor a nice crosscheck of the computation. The Wick contractions for G V are as given in ref. [31]. Denoting Q ≡ ( ω n , ) , ∆ P ≡ P + M , (2.6)the leading-order (LO) vector correlator reads, in momentum space,= 2 C A PZ { P } (cid:26) ( D − Q − M ∆ P ∆ P − Q − D − P (cid:27) . (2.7)Here { P } denotes fermionic Matsubara momenta, and Σ R { P } ≡ T P { p n } R p .At NLO, we have to decide on a meaning of the renormalized mass. Although conceptuallysubtle, it is technically convenient to employ a pole mass; then the bare mass parameter, M B ,can be expressed as M B = M + δM where at NLO δM = − g C F Z K (cid:26) ( D − (cid:20) K − ¯ P − K (cid:21) + 4 M K ∆ ¯ P − K (cid:27) ¯ P = − M (2.8)= − g C F Z k ǫ k E pk (cid:20) ( D − E pk − ǫ k ) + 4 M ( ǫ k + E pk )( ǫ k + E pk ) − E p (cid:21) (2.9)= − g C F M (4 π ) (cid:18) ǫ + ln ¯ µ M + 43 (cid:19) . (2.10)Here ¯ µ is the scale parameter of the MS scheme, terms of O ( ǫ ) were omitted, and ǫ k ≡ | k | , E p ≡ p p + M , E pk ≡ p ( p − k ) + M . (2.11)Because of Lorentz invariance the vector ¯ P in eq. (2.8) can be chosen at will, as long as weset p = ± iE p after carrying out the K -integral; this means that the vector p in eq. (2.9) isarbitrary. The corresponding counterterm contribution reads= 4 C A δM PZ { P } (cid:26) D − P − P ∆ P − Q + 4 M − ( D − Q ∆ P ∆ P − Q (cid:27) , (2.12)and it is often convenient to identify p of eq. (2.9) as the integration variable of eq. (2.12).3he “genuine” 2-loop graphs amount to+ = 4 g C A C F PZ K { P } (cid:26) ( D − K ∆ P − ( D − ∆ P ∆ P − K − D − K ∆ P ∆ P − K + 4( D − M K ∆ P ∆ P − K − D − K ∆ P ∆ P − Q + 4( D − M − ( D − Q K ∆ P ∆ P − Q + 2( D − K ∆ P − K ∆ P − Q + 4( D − P ∆ P − K ∆ P − Q − D − M − ( D − Q ∆ P ∆ P − K ∆ P − Q − M + 2( D − K · Q − D − Q K ∆ P ∆ P − K ∆ P − Q + 16 M − D − Q M K ∆ P ∆ P − K ∆ P − Q − D − M + ( D − − D ) Q + ( D − K ∆ P ∆ P − K ∆ P − Q ∆ P − K − Q + 8 M − D − M Q − ( D − Q K ∆ P ∆ P − K ∆ P − Q ∆ P − K − Q (cid:27) . (2.13) In the case of the zero components, viz. G , the LO correlator reads, in momentum space,= 2 C A PZ { P } (cid:26) P − Q + 4 E p ∆ P ∆ P − Q (cid:27) , (2.14)whereas the counterterm graph can be expressed as= 4 C A δM PZ { P } (cid:26) Q + 4 E p ∆ P ∆ P − Q − P ∆ P − Q − P (cid:27) . (2.15)The genuine 2-loop graphs amount to + = 4 g C A C F PZ K { P } (cid:26) − D − K ∆ P + D − P ∆ P − K + 2 K ∆ P ∆ P − K − M K ∆ P ∆ P − K The appearance of the “energy variables” in the numerators implies that there is a certain redundancy inthe basis:0 = PZ K { P } (cid:26) K · QK ∆ P ∆ P − K ∆ P − Q − Q ∆ P ∆ P − K ∆ P − Q ∆ P − K − Q + Q ǫ k K ∆ P ∆ P − K ∆ P − Q ∆ P − K − Q (cid:27) . (2.16)We have used this to eliminate terms containing Q ǫ k in the numerator. D − K ∆ P ∆ P − Q + ( D − E p + Q ) K ∆ P ∆ P − Q − K ∆ P − K ∆ P − Q + 4( D − P ∆ P − K ∆ P − Q − ( D − E p + Q )∆ P ∆ P − K ∆ P − Q − M − K · Q + 4 Q K ∆ P ∆ P − K ∆ P − Q + 4 M (4 E p + Q ) K ∆ P ∆ P − K ∆ P − Q − ( D − E p + E pk − ǫ k + K + Q ) − M ∆ P ∆ P − K ∆ P − Q ∆ P − K − Q + 4 M ( E p + E pk − ǫ k ) + 2 Q ( E p + E pk + M ) + Q K ∆ P ∆ P − K ∆ P − Q ∆ P − K − Q (cid:27) . (2.17) The next step is to convert the momentum-space expressions to configuration space. If wedenote the above result (eqs. (2.7), (2.12), (2.13)) by ˜ G V ( ω n ), then the conversion is obtainedas G V ( τ ) = T X ω n e − iω n τ ˜ G V ( ω n ) , (2.18)and similarly for G . At NLO we are thereby faced with a three-fold Matsubara sum. Makinguse of standard techniques, reviewed in some detail in ref. [31], these sums can be carriedout in a closed form, whereby we are left with integrals over at most two spatial momenta(i.e. two radial directions and one angle). Intermediate results at this stage are displayed forall individual master sum-integrals in appendix A, and for their sums in appendix B, in thelatter case with the cancellation of 1 /ǫ -divergences verified as well.Two interesting crosschecks are available. First of all, all τ -dependent terms disappearfrom G , cf. eq. (B.3). Second, individual parts of the expressions contain “contact terms” ∝ δ β ( τ ), where δ β denotes the periodic Dirac-delta. These arise from structures that areindependent of ω n , but also from sum-integrals in which Q appears in the numerator. It canbe verified, however, that all contact terms cancel, both at LO and at NLO.It remains to carry out the spatial integrals. We write Z p , k = Z p,k Z z , (2.19)where the normalization of the angular variable z = cos θ p , k is chosen so that R z Z z = 12 pk Z E + pk E − pk d E pk E pk = 12 pk Z ǫ + pk ǫ − pk d ǫ pk ǫ pk , (2.20)5here E ± pk ≡ p ( p ± k ) + M , ǫ ± pk ≡ | p ± k | . The angular integrals are doable in most cases.In addition, it is also possible to carry out partial integrations with respect to the radialdirections, which helps to reduce the number of independent terms (for the massless case, seeref. [34] for a recent discussion). Closed massless loop integrals are typically solvable, but ingeneral the integrations remain to be carried out numerically. Our final expressions are givenin the next two sections, cf. eqs. (3.2)–(3.5), (4.1), (4.4), (4.5).
3. Leading order results and qualitative pattern
In order to illustrate the qualitative structure of the results, we recall in this section the LOexpressions for the quantities considered. In general, two types of contributions appear: thosethat depend on τ , and those that are constant. To display the τ -dependence we introducethe periodic dimensionless function D E k +1 ··· E n E ··· E k ( τ ) ≡ e ( E + ··· + E k )( β − τ )+( E k +1 + ··· + E n ) τ + e ( E + ··· + E k ) τ +( E k +1 + ··· + E n )( β − τ ) [ e βE ± · · · [ e βE n ± . (3.1)In the denominator, the sign is chosen according to whether the particle is a boson or a fermion(at NLO, there is only one boson, with the on-shell energy denoted by ǫ k , cf. eq. (2.11)). Wealso denote D E p ≡ D E p E p . With these conventions, the LO result for G V reads G LOV ( τ ) | τ -dep. = − G LO ii ( τ ) | τ -dep. = − C A Z p (cid:16) M E p (cid:17) D E p ( τ ) , (3.2) G LOV ( τ ) | const. = − C A Z p M T n ′ F ( E p ) E p . (3.3)Here n F denotes the Fermi distribution ( n B denotes the Bose distribution). The susceptibilityonly contains a τ -independent part: T χ LO = G LO = − C A Z p T n ′ F ( E p ) . (3.4)Finally, the constant part of the spatial correlator is obtained through the use of eq. (2.2): G LO ii ( τ ) | const. = − C A Z p (cid:16) − M E p (cid:17) T n ′ F ( E p ) . (3.5)To our knowledge none of these leading-order expressions can be integrated in terms ofstandard elementary functions.In terms of the spectral function, viz. ρ ii ( ω ), the constant contribution in eq. (3.5) arisesfrom (an infinitely narrow) transport peak around ω = 0, whereas the “fast” τ -dependence ineq. (3.2) originates from the quark-antiquark continuum at | ω | ≥ M . Around the middle of6he Euclidean time interval, D E p ( β/
2) = − T n ′ F ( E p ). Therefore, both terms contribute ina comparable manner at large Euclidean time separations, even though they originate fromcompletely different types of physics (see also ref. [35]).At NLO, the same three structures appear as at LO: a constant G , as well as a spatialcorrelator G ii which has both a constant and a τ -dependent part. It is generally believed thatthe perturbative series for the constant part of G ii breaks down at some point and that in thefull theory, G ii has no projection to the Matsubara zero mode; the reason is that the spatialcomponents of the vector current do not couple to conserved charges. The corresponding“slow” τ -dependence then reflects the physics of a “smeared” transport peak. Yet it remainstrue that single quark physics from the transport peak and quark-antiquark physics fromthe threshold region are expected to contribute in a comparable manner to the Euclideancorrelator around τ = β/
2. (This is demonstrated explicitly in figs. 3, 4 below.)
4. NLO results
Proceeding to NLO, the result for the susceptibility is obtained from eq. (B.3) after partialintegrations:
T χ
NLO = G NLO = 4 g C A C F Z p T n ′ F ( E p ) p Z k (cid:20) n B ( ǫ k ) ǫ k + n F ( E k ) E k (cid:18) − M k (cid:19)(cid:21) . (4.1)This can be shown to agree with what can be extracted from ref. [36] as T ∂ µ p ( T, µ ) | µ =0 .(The substance of the information is already there in ref. [37].) The massless loop evaluatesto R k n B ( ǫ k ) /ǫ k = T /
12, and in the limit M ≫ πT its effect can be interpreted as an effectivemass correction to the LO result of eq. (3.4), M → M + g T C F / M ≪ πT eq. (4.1) reduces to the well-known correction (cf. [39] andreferences therein) G NLO M ≪ πT = − g N c C F T π + O ( g ) . (4.2)A numerical evaluation of eqs. (3.4), (4.1) is shown in fig. 1. For the gauge coupling we haveinserted the 2-loop value for the thermal coupling g E /T from ref. [40], and for comparison withquenched lattice data from refs. [41, 21] we assume that T c / Λ MS ≃ . T /T c ≃ .
45. In themassive case, where no continuum extrapolation is available in ref. [21], we choose to comparewith results obtained on a lattice 128 ×
48. The quark mass cited, m MS ( m c ) = 1 . viz. M ≃ m MS ( m c ) (cid:26) g MS ( m c ) C F (4 π ) + O ( g ) (cid:27) , (4.3)7 .0 1.0 2.0 3.0 4.0 5.0 M / T χ / χ fr ee LONLOlattice
T = 1.45 T c , T c = 1.25 Λ MS _ Figure 1:
The quark-number susceptibility, eqs. (3.4), (4.1), normalized to the free result fromeq. (2.5). The lattice result at M = 0 comes from ref. [41], that at M > N f = 0) and only the former represents the continuum limit. The uncertainty of M/T isour estimate (cf. the text). Lattice results for charm by other groups can be found in refs. [19, 20]. we get M ≃ . T c / √ σ =0 . √ σ = 428 MeV, so we estimate M/T ≃ . ± .
5, but with substantial systematicuncertainties, from lattice artifacts, string tension measurement [43], quenching, as well asperturbative input. With this in mind, the excellent agreement seen in fig. 1 is remarkable,and supports the long-held belief that all quarks, and heavy quarks in particular, are welldescribed by the weak-coupling expansion at surprisingly low temperatures. (In the masslesscase the good agreement is consistent with the recent unquenched study of ref. [44], in whicha similar “dimensionally reduced” resummation scheme was used as here [36], however analmost perfect match already at NLO may be somewhat coincidental.)
The vector current correlator is obtained from eqs. (B.1), (B.2) after partial integrations. Its τ -dependent part reads G NLOV | τ -dep. g C A C F = Z p D E p ( τ )4 π (cid:20)(cid:18) M E p (cid:19)(cid:18) − p E p ln E p + pE p − p (cid:19) − − (cid:18) M E p (cid:19) Z ∞ d k θ ( k ) k (cid:21) + Z p,k P (Z z D ǫ k E p E pk ( τ ) ǫ k E p E pk ∆ + − ∆ − + (cid:20) − E p + M (cid:18) ∆ −− ∆ ++ + 2 ǫ k ∆ + − ∆ − + (cid:19) + 4 ǫ k M ∆ ∆ + − ∆ − + (cid:21) Z z D ǫ k E p E pk ( τ ) ǫ k E p E pk ∆ + − ∆ − + (cid:20) − E p + M (cid:18) ∆ ++ ∆ −− + 2 ǫ k ∆ + − ∆ − + (cid:19) + 4 ǫ k M ∆ −− ∆ + − ∆ − + (cid:21) + Z z D E p ǫ k E pk ( τ ) ǫ k E p E pk ∆ ++ ∆ −− (cid:20) E p + E pk − M (cid:18) ∆ + − ∆ − + + 2 ǫ k ∆ ++ ∆ −− (cid:19) − ǫ k M ∆ − + ∆ ++ ∆ −− (cid:21) + D E p ( τ )2 ǫ k (cid:18) M E p (cid:19) (cid:20) − E p ( E + pk − E − pk ) − pǫ k ( E + pk + E − pk )2 p ( E p − ǫ k ) + ǫ k M ( E + pk − E − pk ) p ( E p − ǫ k ) E + pk E − pk + 2 E p − M pE p (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) ( E p + p )(2 p − ǫ k )( E p − p )(2 p + ǫ k ) (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) − ǫ k / ( E p + E − pk ) − ǫ k / ( E p + E + pk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + θ ( k ) (cid:21) + D E p ( τ ) n B ( ǫ k ) ǫ k (cid:20) M p E p + 1 pE p ln E p + pE p − p + 1 ǫ k (cid:18) M E p (cid:19)(cid:18) − E p − M pE p ln E p + pE p − p (cid:19) (cid:21) + D E p ( τ ) n F ( E k ) E k (cid:20) M p E p + 10 p + 16 p M + 3 M p − k ) p E p + M pkE p ln (cid:12)(cid:12)(cid:12)(cid:12) p + kp − k (cid:12)(cid:12)(cid:12)(cid:12) − E p + 2 E k E p − M pk ( E p − E k ) E p (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) p + kp − k (cid:12)(cid:12)(cid:12)(cid:12) + E k E p + E k ln (cid:12)(cid:12)(cid:12)(cid:12) M + E p E k + pkM + E p E k − pk (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:21)) , (4.4)where P denotes a principal value, and ∆ ++ etc are defined in eq. (A.2). The constantcontribution reads G NLOV | const. g C A C F = Z p T n ′ F ( E p ) Z k (cid:26) n B ( ǫ k ) ǫ k (cid:20) p − E p (cid:21) + n F ( E k ) E k (cid:20) p − E p − M p k − M E p E k + M (4 E k − M )2 pkE p E k ln (cid:12)(cid:12)(cid:12)(cid:12) p + kp − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) . (4.5)The spatial correlator subsequently follows from eq. (2.2), with the susceptibility insertedfrom eq. (4.1).Although the expression in eq. (4.4) is finite, its numerical evaluation is non-trivial. Thereare at least three separate challenges: at small k various parts of the expression are divergent,and care must be taken in order to avoid significance loss in their cancellation; at k = p thereis a pole which is defined in the sense of a principal value; and at large k there is a vacuumpart which decreases only slowly (although it is integrable). For the reader’s benefit, let usbriefly specify how we have dealt with these challenges. • The small- k divergence originates from the terms integrated over z in eq. (4.4) andfrom terms where the integral had already been carried out. For the latter type thedivergent part reads D E p ( τ ) ǫ k (cid:18) M E p (cid:19) (cid:18) − E p − M pE p ln E p + pE p − p (cid:19)(cid:18)
12 + n B ( ǫ k ) (cid:19) , (4.6)9ontaining both vacuum and Bose-enhanced structures. The integral R z needs to becarried out precisely enough such that the cancellation takes duly place. • The principal value integration can be handled for instance by reflecting the range p > k into the range p < k : Z ∞ d p p φ ( p ) = k Z d x h x φ ( kx ) + 1 x φ (cid:16) kx (cid:17) i . (4.7)Here φ has to be evaluated precisely enough for cancellations at x = 1 to take place. • A possible way to accelerate the convergence at large k is with the help of the function θ ( k ) in eq. (4.4). (Note that a power tail only appears in the vacuum part.) Thesimplest subtraction removes just the leading asymptotic behaviour − E p /k , e.g. θ ( k ) ≡ E p Θ( k − k min ) k + λ , Z ∞ d k θ ( k ) k = 3 E p λ ln (cid:16) λ k min (cid:17) , (4.8)but of course more refined choices can be envisaged. (By replacing λ by a “gluonmass” it would be possible to take k min → R ∞ d k θ ( k ) /k exactly,cf. eqs. (B.22), (B.23), but the price to pay is that then the small- k range has morestructure than before, with a would-be divergence only cut off at k < ∼ λ .)For a transparent representation of the numerical results, we consider two different nor-malizations. A simple and theoretically clean possibility is to normalize the results to the free massless expression from eq. (2.4). Another reference point is to make use of the fullNLO spectral function in vacuum [45]–[48]: ρ vac V ( ω ) = − 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 ( g ) , (4.9)where the function L is defined as L ( x ) ≡ ( x ) + 2 Li ( − x ) + [2 ln(1 − x ) + ln(1 + x )] ln x . (4.10)There is no transport peak in the vacuum expression, and recalling eq. (2.2), the correspond-ing spatial correlator can be obtained through G rec ii ( τ ) ≡ Z ∞ d ωπ ( − ρ vac V )( ω ) cosh (cid:16) β − τ (cid:17) ω sinh βω . (4.11)10 .0 0.1 0.2 0.3 0.4 0.5 τ T G ii / G ii fr ee LONLOlattice
T = 1.45 T c , T c = 1.25 Λ MS _ M/T = 1M/T = 5 τ T G ii / G ii r ec LONLOlattice
T = 1.45 T c , T c = 1.25 Λ MS _ M/T = 1 M/T = 5
Figure 2:
Left: The vector correlator, normalized to the free result from eq. (2.4), for
M/T =1 . , . , . , . , . N f = 0) and do not contain a continuum extrapolation ( N τ = 48, N s = 128). A normalization with respect to a similar “reconstructed” correlator G rec ii ( τ ) has been usedin ref. [21] (see also ref. [49]), and may be useful for phenomenological purposes, althoughfrom the theoretical perspective it induces new systematic uncertainties.In fig. 2 we show our results in both normalizations, compared with lattice data fromref. [21]. (The gauge coupling has been fixed as explained in connection with fig. 1; atvery small τ a different running would appear reasonable but in the absence of an NNLOcomputation and continuum-extrapolated lattice data, we stick to the simplest choice in thefollowing.) Like in fig. 1, an excellent agreement is found at large time separations, if a value M/T ≃ . τ is probably due to themissing continuum extrapolation.)
5. Modification of the transport peak
As mentioned in sec. 3, the correlator G ii has a constant ( τ -independent) part at LO and atNLO, but within the full dynamics this is expected to turn into a slowly evolving function.The purpose of this section is to estimate how precisely Euclidean data should be measuredin order to resolve the slow time dependence.11 .0 0.1 0.2 0.3 0.4 0.5 τ T G ii / G ii r ec M/T = 3.5, 2 π T D = ootransport peaklattice
T = 1.45 T c , T c = 1.25 Λ MS _ π T D = 5 2 π T D = 1 τ T G ii / G ii r ec M/T = 3.5, 2 π T D = ootransport peaklattice
T = 1.45 T c , T c = 1.25 Λ MS _ π T D = 5 2 π T D = 1
Figure 3:
Left: The effect of a modified transport peak, for
M/T = 3 . πT D = 1 .
0, 2 .
0, 3 . .
0, 5 . τ region. It is clear that a high precision isneeded for resolving the diffusion coefficient from the massive vector current correlator. (The curvescould be put on top of lattice data through a minor change of M/T , but we have refrained from doingthis in the absence of a continuum extrapolation of the reconstructed correlator.)
In order to reach this goal, we model the transport peak through a Lorentzian shape, ρ (L) ii ( ω ) ≡ Dχ ωη ω + η ω πT ) . (5.1)Here D corresponds to the heavy flavour diffusion coefficient. The Lorentzian shape can becorrect only at small frequencies, | ω | ≪ πT , cf. e.g. ref. [50]; we have chosen to cut it off atlarge frequencies through the same recipe that has been used in the massless case [27]. Thesusceptibility χ is fixed according to ref. [21], χ/T = 0 . D , in eachcase tuning the other parameter η so as to keep the area under the transport peak fixed atthe value predicted by the NLO expression, eqs. (3.5), (4.1), (4.5), (2.2):1 β Z β d τ G (L) ii ( τ ) = Z ∞ d ωπ ρ (L) ii ( ω ) βω ≡ [ G LO ii + G NLO ii ] const. . (5.2)Subsequently the “correct” G ii ( τ ) is obtained by replacing the part [ G LO ii + G NLO ii ] const. througha τ -dependent function, G (L) ii ( τ ), determined by ρ (L) ii via eq. (4.11). As a guideline, we recallthat 2 πT D ≃ − πT D ∼
2; and that in the massless case values down to 2 πT D ∼ πT D ∼ −
2, then there is hope ofresolving it with high enough precision of the lattice data (it appears that statistical errors12 .0 0.1 0.2 0.3 0.4 0.5 τ T G ii / G ii r ec M/T = 3.5added resonancelattice
T = 1.45 T c , T c = 1.25 Λ MS _ A = 0.1 A = 0.5 τ T G ii / G ii r ec M/T = 3.5added resonancelattice
T = 1.45 T c , T c = 1.25 Λ MS _ ∆ M / M = 0.02 ∆ M / M = 0.10
Figure 4:
Left: The effect of a modified amplitude of a resonance peak, for
M/T = 3 .
5, ∆
M/M = 0 . A = 0 .
1, 0 .
2, 0 .
3, 0 .
4, 0 . A = 0 . M/M = 0 . , . , . , . , . should probably be reduced to 20% of the current ones to be sensitive to the features of thetransport peak; obviously improvements in statistical accuracy need to be accompanied bya corresponding decrease in systematic uncertainties). In the case 2 πT D ∼ −
6. Modification of the threshold region
The second qualitative structure affecting the massive vector current correlator, the quarko-nium threshold region inducing a “fast” time dependence from the energy scale ∼ M , isalso expected to undergo drastic changes in the interacting theory. At low temperatures, thespectral function is characterized by quarkonium resonances; at high temperatures, these areexpected to move, broaden, and eventually dissolve into a mere threshold enhancement.Based on our earlier investigations [51, 52], we expect that in the temperature range ofinterest there is at most one resonance peak in the vector channel spectral function, placedslightly to the left from the free quark-antiquark threshold. Akin to eq. (5.1), we model this13y a skewed Breit-Wigner shape, ρ (BW) ii ( ω ) ≡ A ω γ ( ω ′ ) + γ ω ′ M ) , ω ′ ≡ ω − M + ∆ M , (6.1)constructed so as not to contribute to the transport coefficient. To reduce the number offree parameters to two, we set ∆ M ≡ γ in the following. The contribution from such apeak, determined through eq. (4.11), is added to the thermal NLO result as such, and tothe vacuum result of eq. (4.9) with A → A , γ → γ/
5, keeping the area under the peakroughly invariant. Obviously these choices are arbitrary, but they should nevertheless conveya qualitative impression on the importance of resonance contributions. Based on ref. [31],in which a resonance peak around a threshold was matched to the thermal NLO spectralfunction above the threshold, we expect that
A < ∼ . M/M < ∼ . τ thana change of the transport peak. In fact, if the resolution, after continuum extrapolation, werehigh enough that the lattice and perturbative curves could be subtracted from each other(rather than normalizing them to a function which diverges at short distances), then the twofeatures could be disentangled by inspecting intermediate distances, 0 . < τ < .
3, in whichthe threshold region contributes much more prominently than the transport peak. In contrastthe two features are difficult to tell apart if reliable data is only available for 0 . < τ < .
7. Conclusions
The main purpose of this paper has been to compute the massive quark-number susceptibilityand vector current correlator at next-to-leading order (NLO) in thermal QCD. The NLOresults are shown in eqs. (4.1), (4.4), (4.5), and illustrated numerically in figs. 1, 2.Our semi-analytic results can be directly compared with numerical lattice Monte Carlosimulations. Although no continuum limit has been reached for the massive vector currentcorrelator [21], the agreement as seen in figs. 1, 2 is quite remarkable. (We have comparedwith quenched data, because these are “closest” to the continuum limit, but our analyticresults are also valid for the unquenched case.)The good agreement suggests that resummed perturbative NLO computations, throughwhich the Minkowskian spectral function has been determined around the threshold re-gion [51, 52], might be more accurate than sometimes assumed. A similar observation has also The approach in this section is schematic; for possible other model shapes see, e.g., ref. [53]. (A generaldiscussion of the sum rule approach can be found in ref. [54].) πT D ∼ − τ -range. At this stage vacuum physicscan be subtracted, as outlined in ref. [27], and the non-divergent remainder subjected (atleast in principle) to a model-independent analytic continuation [30]. Acknowledgements
We thank H.-T. Ding for helpful discussions and providing us with lattice data from ref. [21],and M. Veps¨al¨ainen for collaboration at initial stages of this work. This work was partlysupported by the Swiss National Science Foundation (SNF) under grant 200021-140234.
Appendix A. Results for individual master sum-integrals
In this appendix we list results for the 2-loop sum-integrals in eqs. (2.13), (2.17), after carryingout the Matsubara sums as well as the Fourier transformation in eq. (2.18). For brevity werefer to the sum-integrals with the notation I m m m n n n n n ( τ ) ≡ T X ω n e − iω n τ PZ K { P } ( M ) m ( Q ) m (2 K · Q ) m ( K ) n ∆ n P ∆ n P − K ∆ n P − Q ∆ n P − K − Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q =( ω n , ) . (A.1)15he index m guarantees that all the masters have the same dimensionality. It is alsoconvenient to introduce the shorthand notations∆ στ ≡ ǫ k + σE p + τ E pk , ∆ σ = E p + σE pk , (A.2)where the energies are defined as in eq. (2.11). In some cases the remaining integrands canbe simplified via the symmetrization p ↔ k − p .Four of the masters appearing are independent of the external momentum Q , and thereforelead to contact terms: I ( τ ) = δ β ( τ ) PZ K { P } K ∆ P , (A.3) I ( τ ) = δ β ( τ ) PZ K { P } P ∆ P − K , (A.4) I ( τ ) = δ β ( τ ) PZ K { P } K ∆ P ∆ P − K , (A.5) I ( τ ) = δ β ( τ ) PZ K { P } M K ∆ P ∆ P − K . (A.6)No further details are given since these contact terms cancel against contributions frommasters with m > I ( τ ) = Z p , k n B ( ǫ k )8 ǫ k E p h D E p ( τ ) + 2 T n ′ F ( E p ) i , (A.7) I ( τ ) = Z p , k M [1 + 2 n B ( ǫ k )]32 ǫ k E p h D E p ( τ ) − E p ∂ E p D E p ( τ )+ 4 T n ′ F ( E p ) − T E p n ′′ F ( E p ) i , (A.8) I ( τ ) = I ( τ ) + Z p , k n B ( ǫ k )8 ǫ k E p h ∂ E p D E p ( τ ) i , (A.9) I ( τ ) = Z p , k ǫ k E p E pk h D ǫ k E p E pk ( τ ) + D ǫ k E p E pk ( τ ) − D E p ǫ k E pk ( τ ) i , (A.10) I ( τ ) = Z p , k M ǫ k E p E pk (cid:26) − D ǫ k E p E pk ( τ )∆ ++ ∆ − + − D ǫ k E p E pk ( τ )∆ + − ∆ −− + 2 ǫ k D E p ǫ k E pk ( τ )∆ ++ ∆ − + ∆ −− + D E p ( τ ) + 2 T n ′ F ( E p ) E p (cid:20) ǫ k + E pk ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) ++ ∆ −− + 1∆ + − ∆ − + (cid:17) − E pk n B ( ǫ k ) (cid:16) ++ ∆ + − + 1∆ − + ∆ −− (cid:17)(cid:21)(cid:27) , (A.11)16 ( τ ) = Z p , k E p E pk (cid:26) D ǫ k E p E pk ( τ )∆ − + + D ǫ k E p E pk ( τ )∆ + − − ǫ k D E p ǫ k E pk ( τ )∆ ++ ∆ −− − E p D E p ( τ ) (cid:20) ++ ∆ − + − n F ( E pk ) (cid:16) ++ ∆ − + + 1∆ + − ∆ −− (cid:17) + n B ( ǫ k ) (cid:16) ++ ∆ − + − + − ∆ −− (cid:17)(cid:21)(cid:27) , (A.12) I ( τ ) = I ( τ ) + Z p , k ǫ k E p E pk (cid:26) ∆ ++ ∆ − + D ǫ k E p E pk ( τ ) + ∆ −− ∆ + − D ǫ k E p E pk ( τ ) − (cid:16) ∆ − + ∆ ++ + ∆ − + ∆ −− (cid:17) D E p ǫ k E pk ( τ ) − E p D E p ( τ ) (cid:20) ǫ k + E pk ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) ++ ∆ −− + 1∆ + − ∆ − + (cid:17) − E pk n B ( ǫ k ) (cid:16) ++ ∆ + − + 1∆ − + ∆ −− (cid:17)(cid:21)(cid:27) , (A.13) I ( τ ) = Z p , k − n F ( E pk )8 E p E pk h D E p ( τ ) + 2 T n ′ F ( E p ) i , (A.14) I ( τ ) = Z p , k M [1 − n F ( E pk )]32 E p E pk h D E p ( τ ) − E p ∂ E p D E p ( τ )+ 4 T n ′ F ( E p ) − T E p n ′′ F ( E p ) i , (A.15) I ( τ ) = I ( τ ) + Z p , k − n F ( E pk )8 E p E pk h ∂ E p D E p ( τ ) i , (A.16) I ( τ ) = Z p , k M ǫ k E p E pk (cid:26) D ǫ k E p E pk ( τ )∆ ∆ − + + D ǫ k E p E pk ( τ )∆ − ∆ −− − D E p ǫ k E pk ( τ )∆ − + (cid:16) + 1∆ −− (cid:17) + D E p ( τ ) + 2 T n ′ F ( E p )2 E p (cid:20) ( ǫ k + E pk ) (cid:16) ∆ ++ ∆ − + − E p ∆ ∆ − + (cid:17) − ǫ k n F ( E pk ) (cid:16) ∆ ++ ∆ −− − E p ∆ + ∆ ∆ −− + ∆ + − ∆ − + − E p ∆ − ∆ − ∆ − + (cid:17) − E pk n B ( ǫ k ) (cid:16) ∆ ++ ∆ + − + 2 E p ( ǫ k + E p )∆ ∆ − + ∆ − + ∆ −− − E p ( ǫ k − E p )∆ − + ∆ −− (cid:17)(cid:21) − ∂ E p D E p ( τ ) + 2 T n ′′ F ( E p )4 E p (cid:20) ǫ k + E pk ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) ++ ∆ −− + 1∆ + − ∆ − + (cid:17) − E pk n B ( ǫ k ) (cid:16) ++ ∆ + − + 1∆ − + ∆ −− (cid:17)(cid:21)(cid:27) , (A.17)17 ( τ ) = I ( τ ) + Z p , k M ǫ k E p E pk (cid:26) − D ǫ k E p E pk ( τ )∆ − + − D ǫ k E p E pk ( τ )∆ − + D E p ǫ k E pk ( τ ) (cid:16) + 1∆ −− (cid:17) + 4 D E p ( τ ) (cid:20) ( ǫ k + E pk ) E p ∆ ∆ − + − ǫ k n F ( E pk ) (cid:16) ∆ + ∆ ∆ −− + ∆ − ∆ − ∆ − + (cid:17) + E pk n B ( ǫ k ) (cid:16) ǫ k + E p ∆ ∆ − − ǫ k − E p ∆ − + ∆ −− (cid:17)(cid:21) + ∂ E p D E p ( τ ) (cid:20) ǫ k + E pk ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) ++ ∆ −− + 1∆ + − ∆ − + (cid:17) − E pk n B ( ǫ k ) (cid:16) ++ ∆ + − + 1∆ − + ∆ −− (cid:17)(cid:21)(cid:27) , (A.18) I ( τ ) = Z p , k M E p E pk (cid:26) T n ′ F ( E p ) n ′ F ( E pk ) E pk − h − n F ( E pk ) i(cid:20) D E p ( τ )∆ + ∆ − − T n ′ F ( E p ) E pk (cid:21)(cid:27) , (A.19) I ( τ ) = Z p , k E pk h − n F ( E pk ) i D E p ( τ )∆ + ∆ − , (A.20) I − ( τ ) = Z p , k E p E pk (cid:26)(cid:16) ǫ k − E p − E pk (cid:17) T n ′ F ( E p ) n ′ F ( E pk ) E pk − h − n F ( E pk ) i(cid:16) ǫ k − E p + E pk (cid:17)(cid:20) D E p ( τ )∆ + ∆ − − T n ′ F ( E p ) E pk (cid:21)(cid:27) , (A.21) I ( τ ) = Z p , k M ǫ k E p E pk (cid:26)(cid:16) ǫ k − E p − E pk (cid:17) ǫ k E p E pk T n ′ F ( E p ) n ′ F ( E pk )∆ ++ ∆ + − ∆ − + ∆ −− + D ǫ k E p E pk ( τ )∆ ∆ + − ∆ − + + D ǫ k E p E pk ( τ )∆ −− ∆ + − ∆ − + − D E p ǫ k E pk ( τ )∆ − + ∆ ++ ∆ −− − D E p ( τ )2 E p (cid:20) E p ( ǫ k + 2 E pk )∆ + ∆ − ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) + ∆ ++ ∆ −− + 1∆ − ∆ + − ∆ − + (cid:17) + E pk n B ( ǫ k ) ǫ k (cid:16) ++ ∆ + − − − + ∆ −− (cid:17)(cid:21) + 2 T n ′ F ( E p ) E p (cid:20) E pk (cid:16) ∆ ++ + E pk ∆ + ∆ − + + E pk ∆ − + (cid:17) ǫ k n F ( E pk )2 E pk (cid:16) ǫ k + E p − E pk − ∆ ∆ −− + ǫ k + E p − E pk − − ∆ − ∆ − + (cid:17) + E pk n B ( ǫ k ) (cid:16) ∆ − + 1∆ − + ∆ −− (cid:17)(cid:21)(cid:27) , (A.22) I ( τ ) = Z p , k M ǫ k E p E pk (cid:26) − D ǫ k E p E pk ( τ )∆ + − ∆ − + − D ǫ k E p E pk ( τ )∆ + − ∆ − + + 2 D E p ǫ k E pk ( τ )∆ ++ ∆ −− + 2 D E p ( τ ) (cid:20) E p ( ǫ k + 2 E pk )∆ + ∆ − ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) + ∆ ++ ∆ −− + 1∆ − ∆ + − ∆ − + (cid:17) + E pk n B ( ǫ k ) ǫ k (cid:16) ++ ∆ + − − − + ∆ −− (cid:17)(cid:21)(cid:27) , (A.23) I ( τ ) = 2 I ( τ ) + Z p , k ǫ k E p E pk (cid:26) ∆ D ǫ k E p E pk ( τ )∆ + − ∆ − + + ∆ −− D ǫ k E p E pk ( τ )∆ + − ∆ − + − − + D E p ǫ k E pk ( τ )∆ ++ ∆ −− − E p D E p ( τ ) (cid:20) E p ( ǫ k + 2 E pk )∆ + ∆ − ∆ ++ ∆ − + − ǫ k n F ( E pk ) (cid:16) + ∆ ++ ∆ −− + 1∆ − ∆ + − ∆ − + (cid:17) + E pk n B ( ǫ k ) ǫ k (cid:16) ++ ∆ + − − − + ∆ −− (cid:17)(cid:21)(cid:27) . (A.24) Appendix B. Renormalization of the complete result
Inserting the expressions for the master sum-integrals from appendix A into eqs. (2.12),(2.13), (2.15), (2.17), we obtain renormalized results for the correlators considered. Like insec. 3, the result can be divided into τ -dependent and constant terms. The former reads G NLOV | τ -dep. g C A C F == Z p , k D ǫ k E p E pk ( τ ) ǫ k E p E pk ∆ + − ∆ − + (cid:26) − E p − E pk + M (cid:20) ∆ −− ∆ ++ + 2 ǫ k ∆ + − ∆ − + (cid:21) + 4 ǫ k M ∆ ∆ + − ∆ − + (cid:27) + Z p , k D ǫ k E p E pk ( τ ) ǫ k E p E pk ∆ + − ∆ − + (cid:26) − E p − E pk + M (cid:20) ∆ ++ ∆ −− + 2 ǫ k ∆ + − ∆ − + (cid:21) + 4 ǫ k M ∆ −− ∆ + − ∆ − + (cid:27) + Z p , k D E p ǫ k E pk ( τ ) ǫ k E p E pk ∆ ++ ∆ −− (cid:26) + E p + E pk − M (cid:20) ∆ + − ∆ − + + 2 ǫ k ∆ ++ ∆ −− (cid:21) − ǫ k M ∆ − + ∆ ++ ∆ −− (cid:27) + Z p D E p ( τ ) (cid:26) “eq. (B.4)” 19 Z k n B ( ǫ k ) ǫ k (cid:20) − E p + M E p + M (2 E p + M )2 E p E pk (cid:18) + 1∆ −− − − − − + (cid:19) − E p + E pk − M ) E p E pk (cid:18) ++ ∆ −− − + − ∆ − + (cid:19) + M (2 E p − M ) E p (cid:18) ++ ∆ −− + 1∆ + − ∆ − + (cid:19)(cid:21) + Z k n F ( E pk ) E pk P (cid:20) E p + M E p + M (2 E p + M )2 E p ǫ k (cid:18) − + + 1∆ −− − − − (cid:19) − E p + E pk − M ) E p (cid:18) + ∆ ++ ∆ −− + 1∆ − ∆ + − ∆ − + (cid:19) + 4( E p + E pk ) − ǫ k E p ∆ + ∆ − + E pk M (2 E p − M ) E p (cid:18) + ∆ ++ ∆ −− − − ∆ + − ∆ − + (cid:19)(cid:21)(cid:27) + Z p E p ∂ E p D E p ( τ ) (cid:26) Z k n B ( ǫ k ) ǫ k (cid:20) − E p − M E p + M (2 E p + M )2 E p (cid:18) ++ ∆ + − + 1∆ −− ∆ − + (cid:19)(cid:21) + Z k n F ( E pk ) E pk (cid:20) − E p − M E p + M (2 E p + M )2 E p (cid:18) ++ ∆ −− + 1∆ + − ∆ − + (cid:19)(cid:21)(cid:27) . (B.1)The constant contribution, in turn, can be expressed as G LOV | const. g C A C F == Z p , k T n ′ F ( E p ) n ′ F ( E pk )2 E p E pk (cid:26) − ǫ k + E p + E pk + 2 M (cid:18) ++ ∆ −− + 1∆ + − ∆ − + (cid:19)(cid:27) + Z p T n ′ F ( E p ) (cid:26) “eq. (B.5)”+ Z k n B ( ǫ k ) ǫ k (cid:20) − E p + M E p − ǫ k M E p (cid:18) ∆ − − −− ∆ − + (cid:19) + M (2 E p − M ) E p (cid:18) ++ ∆ + − + 1∆ −− ∆ − + (cid:19)(cid:21) + Z k n F ( E pk ) E pk (cid:20) − E p + M E p + ǫ k − E p − E pk E p E pk + 2 M E p E pk (cid:18) ∆ ∆ ∆ −− − ∆ − ∆ − ∆ − + (cid:19) + (cid:18) M (2 E p − M ) E p − M E p E pk (cid:19)(cid:18) ++ ∆ −− + 1∆ + − ∆ − + (cid:19)(cid:21)(cid:27) + Z p T E p n ′′ F ( E p ) (cid:26) Z k n B ( ǫ k ) ǫ k (cid:20) − M E p + M E p (cid:18) ++ ∆ + − + 1∆ −− ∆ − + (cid:19)(cid:21) + Z k n F ( E pk ) E pk (cid:20) − M E p + M E p (cid:18) ++ ∆ −− + 1∆ + − ∆ − + (cid:19)(cid:21)(cid:27) . (B.2)Finally, for the susceptibility, the NLO term amounts to G NLO g C A C F = Z p , k T n ′ F ( E p ) n ′ F ( E pk ) E p E pk (cid:26) − M (cid:18) ++ ∆ −− − + − ∆ − + (cid:19)(cid:27) + Z p T E p n ′′ F ( E p ) (cid:26) Z k n B ( ǫ k ) ǫ k (cid:20) − E p + M E p (cid:18) ++ ∆ + − + 1∆ −− ∆ − + (cid:19)(cid:21) + Z k n F ( E pk ) E pk (cid:20) − E p + M E p (cid:18) ++ ∆ −− + 1∆ + − ∆ − + (cid:19)(cid:21)(cid:27) . (B.3)The “0”s in eqs. (B.1), (B.2), (B.3) represent vacuum contributions that vanish after renor-malization. The coefficients of ∂ E p D E p ( τ ) and T n ′′ F ( E p ) vanish already when eqs. (2.12),(2.13) are summed together, but the coefficients of D E p ( τ ) and T n ′ F ( E p ) not. They can beexpressed as (a principal value prescription is implied where necessary)“eq. (B.4)” = Z k (cid:26) − ǫ E p E pk − (4 − ǫ + M E p )( ǫ k + E pk ) ǫ k E pk [( ǫ k + E pk ) − E p ] + 2(1 − ǫ ) E pk [( ǫ k + E pk ) − E p ] − − ǫ + M E p )( ǫ k + E pk ) M ǫ k E pk [( ǫ k + E pk ) − E p ] + 2(1 − ǫ )(2 + ǫ ) + ǫM E p E pk ( E pk − E p ) − (1 − ǫ )( ǫ k + E pk − E p )4 E p E pk ( E pk − E p ) − (4 − ǫ + ǫM E p − M E p )( ǫ k + 2 E pk ) E p ǫ k E pk [( ǫ k + E pk ) − E p ]( E pk − E p ) (cid:27) , (B.4)“eq. (B.5)” = Z k (cid:26) − ǫ E p E pk − ( ǫ k + E pk ) M ǫ k E p E pk [( ǫ k + E pk ) − E p ] − ǫ k + E pk ) M ǫ k E p E pk [( ǫ k + E pk ) − E p ] + ǫM E p E pk − (1 − ǫ )( ǫ k + E pk − E p )4 E p E pk + [( ǫ k + 2 E pk )( ǫ k + E pk ) − ǫ k E p ] M ǫ k E p E pk [( ǫ k + E pk ) − E p ] (cid:27) . (B.5)The various structures here can be identified as specific vacuum integrals: Z k ǫ k = Z K K , (B.6) Z k E pk = Z K P − K , (B.7)21 k ǫ k E pk ǫ k + E pk ( ǫ k + E pk ) − E p = (cid:20)Z K K ∆ P − K (cid:21) p = iE p , (B.8) Z k − E p E pk [( ǫ k + E pk ) − E p ] = (cid:20) p Z K k K ∆ P − K (cid:21) p = iE p , (B.9) Z k − E p ( ǫ k + E pk )2 ǫ k E pk [( ǫ k + E pk ) − E p ] = (cid:20) p Z K p − k K ∆ P − K (cid:21) p = iE p , (B.10) Z k E pk = (cid:20)Z K P − K (cid:21) p = iE p , (B.11) Z k E pk ( E pk − E p ) = (cid:20)Z K P − K ∆ P − K − Q (cid:21) Q =(2 iE p , ) , (B.12) Z k ǫ k + E pk − E p E pk = (cid:20)Z K K ∆ P − K (cid:21) p = iE p , (B.13) Z k ǫ k + E pk − E p E pk ( E pk − E p ) = (cid:20)Z K K ∆ P − K ∆ P − K − Q (cid:21) p = iE p ,Q =(2 iE p , ) , (B.14) Z k ( ǫ k + 2 E pk )( ǫ k + E pk ) − ǫ k E p ǫ k E pk [( ǫ k + E pk ) − E p ] = (cid:20)Z K K ∆ P − K (cid:21) p = iE p , (B.15) Z k ǫ k + 2 E pk ǫ k E pk [( ǫ k + E pk ) − E p ]( E pk − E p ) = (cid:20)Z K K ∆ P − K ∆ P − K − Q (cid:21) p = iE p ,Q =(2 iE p , ) . (B.16)After having expressed the integrals in these forms, we can make use of Lorentz invariancein order to remove redundant structures:(B.9) = E p M h (B.7) − (B.6) i , (B.17)(B.10) = E p M h (B.8) − (B.11) − M (B.15) i , (B.18)(B.13) = (B.7) − M (B.11) , (B.19)(B.14) = (B.7) + 2( E p − M ) (B.12) . (B.20)Inserting these relations, eq. (B.5) vanishes exactly. The coefficient of D E p ( τ ), eq. (B.4),does not vanish; after the transformation of eqs. (B.17)–(B.20) it can be written as“eq. (B.4)” = (cid:26)Z K P (cid:20) − ǫ ) M (cid:18) K − K (cid:19) − − ǫ ) K ∆ P − K + 2(1 + ǫ )∆ K + (cid:18) M E p (cid:19) K (cid:18) K − Q − K (cid:19) − (cid:18) M E p (cid:19) M K ∆ P − K − (cid:18) − M E p (cid:19) E p K ∆ P − K ∆ P − K − Q (cid:21)(cid:27) p = iE p ,Q =(2 iE p , ) + O ( ǫ )22 14 π (cid:20)(cid:18) M E p (cid:19)(cid:18) − p E p ln E p + pE p − p (cid:19) − (cid:21) + (cid:18) M E p (cid:19) Z k P (cid:26) M E pk (cid:20) ǫ k ( ǫ k + E p )∆ + 12 ǫ k ( ǫ k − E p )∆ − + − ǫ k − E p ) E pk (cid:21) + 2 E p − M ǫ k E p E pk (cid:18) + + 1∆ − − ++ + 1∆ − + (cid:19)(cid:27) + O ( ǫ ) , (B.21)where the first row represents the result of the ultraviolet sensitive integrals containing singleor double propagators. The remaining k -integral is infrared (IR) divergent and needs to beevaluated together with the other terms of eq. (B.1). Its form after angular integration canbe found in eq. (4.4), as the structure preceding θ ( k ).It may be interesting to note that regulating the IR through a “gluon mass” λ , we obtain (cid:26)Z K P (cid:20) M ( K + λ )∆ P − K (cid:21)(cid:27) p = iE p = 8(4 π ) ln Mλ + O ( λ ) , (B.22) (cid:26)Z K P (cid:20) E p ( K + λ )∆ P − K ∆ P ∗ − K (cid:21)(cid:27) p = iE p = 8(4 π ) E p p (cid:26) ln (cid:18) E p − pE p + p (cid:19) ln (cid:18) pλ (cid:19) + π
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