Testing thermal photon and dilepton rates

We confront the thermal NLO vector spectral function (both the transverse and longitudinal channel with respect to spatial momentum, both above and below the light cone) with continuum-extrapolated lattice data (both quenched and with $N_{\rm f} = 2$, at $T \sim 1.2 T_{\rm c}$). The perturbative side incorporates new results, whose main features are summarized. The resolution of the lattice data is good enough to constrain the scale choice of $\alpha_{\rm s}$ on the perturbative side. The comparison supports the previous indication that the true spectral function falls below the resummed NLO one in a substantial frequency domain. Our results may help to scrutinize direct spectral reconstruction attempts from lattice QCD.


Introduction
Photons and lepton-antilepton pairs produced in a heavy ion collision are experimentally measurable (cf., e.g., refs. [1][2][3]) and, given that they do not interact after production, offer for a probe of the inner dynamics of strong interactions in this environment. To leading order in the electromagnetic fine-structure constant α em , the thermal parts of both production rates can be related to the spectral function ρ V , associated with the QCD vector current [4][5][6], Here n B is the Bose distribution; M ≡ √ ω 2 − k 2 , ω and k are the invariant mass, energy, and momentum, respectively, of a virtual photon; Q i is the charge of a quark of flavour i in units of the elementary charge; disconnected contributions proportional to ( i Q i ) 2 have been omitted; and we have simplified eq. (1.2) by considering energies 2m ℓ ≪ M ≪ m Z .
There is a long history of perturbative determinations of ρ V in various kinematic domains. Focussing first on massless quarks, a next-to-leading order (NLO) computation at vanishing momentum (k = 0) initially suggested that perturbation theory works well [7][8][9]. However, pushing the energy towards a soft regime (ω ≪ πT , k = 0) and implementing Hard Thermal Loop (HTL) resummation, a large enhancement was found [10,11]. Subsequently the focus shifted to the more typical hard momenta (k ∼ πT ), where a logarithmic singularity, shielded by HTL-resummation, was identified when approaching the light cone (M ≪ πT ) [12][13][14]. In addition, there are non-logarithmic terms of similar magnitude [15], originating from amongst others multiple scatterings with collinear enhancement (the so-called LPM effect [16]), whose systematic handling necessitated a major effort [17][18][19][20]. By now these resummed results have been extended up to NLO close to the light cone [21,22]. With different methods, the NLO level has also been reached above the light cone (M ∼ πT ) [23,24], and the corresponding results have been shown to permit for a smooth interpolation towards the light-cone ones [25]. Far above the light cone, the spectral function is considerably simpler [26], and can in fact be determined to a high precision [27], by making use of N 4 LO vacuum results [28,29]. Finally, quark mass effects have been included up to the NLO level at finite temperature, both for m ≫ πT [30] and for m < ∼ πT [31].
Diverse as the progress is, it should be clear that eventually we need to go beyond perturbation theory in the determination of ρ V . Lattice QCD entails the measurement of an imaginary-time correlation function G V (τ, k), which is related to ρ V through The inversion of this relation is notoriously challenging (cf., e.g., ref. [32]). A recent attempt was made in ref. [33], for continuum-extrapolated quenched QCD. It is clear from eq. (1.3) that, apart from the physical domain ω > k, lattice results are also affected by the spacelike domain ω < k. However it can be argued that, in infinite volume, ρ V should be smooth across the light cone [34]. Thus ref. [33] made use of perturbative information at M > ∼ πT and a fitted interpolating polynomial at 0 ≤ ω < ∼ k 2 + (πT ) 2 . A subsequent work considered N f = 2 data [35], noting that for the photon channel the contribution of a longitudinal polarization can be subtracted and replacing the interpolating polynomial through a Padé ansatz. Further ideas at implementing analytic continuation have also been put forward [36,37]. The purpose of the present paper is to scrutinize the spectral reconstructions of refs. [33,35]. With this aim we improve the status of perturbative predictions in two respects: we incorporate full NLO results for ω < k [38], and consider separately the transverse and longitudinal polarizations as proposed in ref. [35]. After implementing proper resummation close to light cone, these expressions can be inserted on the right-hand side of eq. (1.3), and subsequently the left-hand side can be compared with lattice data. The perturbative results depend on a parameter, namely the value of the renormalized gauge coupling, and these comparisons permit to "calibrate" the choice made.
Our presentation is organized as follows. In sec. 2 we define the basic quantities considered. In sec. 3 we consider various limits, theoretical constraints, and resummations that pertain to their perturbative determination. Comparisons with quenched and unquenched lattice data comprise sec. 4, whereas conclusions are offered in sec. 5.
We are mostly interested in a spectral function, which can be obtained as an imaginary part of the Euclidean correlator, (2.2) Its argument is the Minkowskian four-momentum K ≡ (ω, k), with K 2 ≡ M 2 . Following ref. [35], we are particularly interested in the linear combinations where repeated indices are summed over. Here D ≡ 4 − 2ǫ is the dimension of spacetime. On the light cone, ρ V and ρ H coincide, so that we may replace ρ V through ρ H in eq. (1.1). At leading order (cf., e.g., ref. [39]), where we have defined k ± ≡ (ω ± k)/2 and introduced the polylogarithms Denoting by g 2 = 4πα s the gauge coupling, by N c the number of colours, by C F ≡ (N 2 c − 1)/(2N c ) the quadratic Casimir coefficient, and by Σ {P } a sum-integral with fermionic Matsubara momenta, the NLO expressions for Π V ≡ Π µµ and Π 00 can be cast in the forms The spectral functions corresponding to all structures here are worked out in ref. [38].

OPE limit
We now take an imaginary part of eqs. (2.6) and (2.7) according to eq. (2.2). Analytic results can be obtained by considering |ω ±k| ≫ πT [26]. Limiting values for the "master" structures in eq. (2.6) were given in appendix B of ref. [44]. The additional ones appearing in eq. (2.7) can be determined by making use of techniques described in ref. [45], and are listed in ref. [38].
Inserting the expansions, we find that all 1/ǫ-divergences, the corresponding logarithms, as well as thermal corrections proportional to p n B 16πp or p n F 16πp , cancel (n B and n F are the Bose and Fermi distributions, respectively). The remainders read Thereby, in accordance with the general argument in ref. [35], the combination in eq. (2.3) displays only a thermal correction: The integrals evaluate to p p n B π = πT 4 30 and p p n F π = 7πT 4 240 , so that ρ H approaches zero from the positive side. We note, however, that the Operator Product Expansion (OPE) shows poor convergence; the actual ρ H switches from negative to positive only around ω ∼ 20T .

LPM limit
We next consider an "opposite" limit to that in sec. 3.1, namely M 2 → 0 ± . The spatial momentum is kept fixed, with a value k ∼ πT . In this limit the spectral function needs to be resummed in order to account for the Landau-Pomeranchuk-Migdal (LPM) effect.
Close to the light cone, it is often convenient to represent the two polarizations in a basis different from that in eq. (2.3). Specifically, we define the "transverse" and "longitudinal" spectral functions as where ⊥ and refer to the components perpendicular and parallel to k. Current conservation implies that ρ L = −(M 2 /k 2 )ρ 00 , and in this basis eq. (2.3) becomes Following ref. [19], the LPM-resummed spectral functions ρ i , with i = T, L, read where È stands for a principal value, and g and f are Green's functions satisfying The operatorĤ acts in the plane transverse to light-like propagation, where m 2 ∞ is an "asymptotic" quark thermal mass, given in eq. (3.15), whereas g 2 ) are parameters of a dimensionally reduced effective theory [40][41][42].

Prediction for IR-singularities around the light cone
An interesting application of eqs. (3.6)-(3.8) is that by re-expanding them as a power series in g 2 , we can find out what kind of singularities the strict 2-loop results [38] should contain close to the light cone. For this purpose, we follow a procedure described in sec. 5.1 of ref. [25]. At zeroth order in g, the expressions become where The polylogarithms appearing here were defined in eq. (2.5). Even though I 1,2 are not analytic around the light cone, eq. (3.9) vanishes there. Given that the last term in eq. (3.8) is of O(g 4 ), the corrections of O(g 2 ) are proportional to the parameter m 2 ∞ . For ρ L , we find no such correction: For ρ T , a correction is found which contains a well-known logarithmic divergence as well as a finite part which is discontinuous across the light cone: The integral on the last row is defined in the sense of a principal value at large |ǫ|, where terms ∼ 1/ǫ cancel due to contributions from negative and positive ǫ. Eq. (3.13) predicts that the strict 2-loop spectral function is discontinuous across the light cone, specifically where we inserted the definition of m 2 ∞ from eq. (3.15).

Matching of IR-singularities around the light cone
It is a basic premise of LPM resummation that close to the light cone it eliminates the IR singularities that plague the perturbative series. In other words, when eq. (3.13) is subtracted from the 2-loop expression, the remainder should be non-singular. 1 The logarithmic singularities and discontinuities originate from two structures, both contained in eq. (2.6). The first source are the factorized terms on the second line. Setting D → 4 and identifying the discontinuity from the second line is . (3.16) Carrying out the Matsubara sum and taking the cut, we find The discontinuity of this expression precisely matches the terms ∝ 1/k in eq. (3.14). The other terms of eq. (3.14) match the spectral function denoted by which in ref. [24] was shown to reproduce the logarithmic singularity shown on the first row of eq. (3.13). Here we focus on the discontinuity. The expression obtained after carrying out the Matsubara sums is given in eq. (B.84) of ref. [44], with σ 1 = σ 2 = σ 4 = −, σ 5 = +.
The discontinuity comes from the "virtual" part of ρ I h' (the last lines of eq. (B.84)). If we define and denote for brevity δ x ≡ δ(x), the virtual part reads Now, the δ constraints in eq. (3.20) are equivalent to those emerging from eq. (3.16). Recalling ǫ pk ≡ |p − k|, a key observation is that if we approach the light cone from above (ω → k + ), only the first channel contributes, and the contribution emerges from the domain ǫ pk ≈ k − ǫ p , i.e. p k and ǫ p < k. If we approach the light cone from below (ω → k − ), there is a contribution from the second channel, which emerges from the domain ǫ pk ≈ ǫ p − k, i.e. p k and ǫ p > k. Below the light cone there is also a contribution from the fourth channel, but now it emerges from the domain ǫ pk ≈ ǫ p + k, i.e. −p k and ǫ p > 0. In total we get (3.21) Carrying out the angular integral in eq. (3.19) and setting subsequently ω and k · p to the values required by eq. (3.21), it can be verified that the UV-divergent vacuum term and the IR-sensitive 2 thermal terms drop out. Moreover, the integral over q yields Going over to a variable ǫ = ±ǫ p for convenience, we subsequently find Multiplying by −16g 2 N c C F from eq. (2.6), the part ∝ −1/ǫ of eq. (3.14) is reproduced.

Sum rules
A traditional further constraint on spectral functions is offered by sum rules (cf., e.g., ref. [46] and references therein). Unlike the OPE and LPM limits, the sum rules are sensitive to the complete frequency domain. However, for ρ V they are of limited value, as they require the subtraction of poorly known vacuum parts (containing a dense spectrum of resonances). In contrast, a nice and convergent sum rule can be obtained for ρ H [35]: We have used our perturbative results in order to test which frequency domain gives a contribution to eq. (3.24). It must be noted that ρ H displays a highly non-trivial structure, changing sign twice: ρ H is positive at ω ≤ k, becomes negative at ω > ∼ k as is necessary for the cancellation required by eq. (3.24), but then again becomes positive when |ω − k| ≫ πT , as shown by eq. (3.3). While we have verified that the sum rule is satisfied within numerical uncertainties by our strict 2-loop result and can also be imposed once resummations are included (cf. below), we also see that the asymptotics plays an important role, with the domain ω ≥ 20T giving a substantial contribution to the absolute value of the integral.

Summary: resummed spectral functions
Having discussed various limits and crosschecks of the spectral functions, we are now ready to put together estimates for phenomenological purposes. The full resummed spectral functions (i ∈ {V, H, T, L}) are defined as where ρ i | strict 2-loop is from ref. [38]; ρ i | full LPM is from sec. 3.2; and ρ i | expanded LPM is from sec. 3.3. The function φ, which should be unity if resummations were implemented "exactly", and must in any case equal unity in the IR domain, can be used to correct for the fact that kinematic simplifications pertinent only to the IR domain have been employed in order to implement the resummation. Outside of this domain, we can use φ to switch off the resummation more rapidly than it would switch off otherwise. We find it practical to define φ LO ≡ θ(ω * − ω), where ω * is chosen so that the second structure of eq. (4.1) satisfies eq. (3.24) (just like the first structure does). The superscript LO stands for leading-order LPM resummation, as described in secs. 3.2 and 3.3, and we find that numerically ω * ∼ 15...25T , depending on k. We also incorporate NLO LPM-resummed results from ref. [22], however for these the "expanded" version is not available, and we thus impose a faster cutoff away from the light cone, inspired by discussions in ref. [22], In order to display the practical effect of the resummation, consider the difference ρ i | full LPM − ρ i | expanded LPM at leading order. Results are shown in fig. 1. Prominent features are a logarithmic divergence around light cone, cancelling the one from ρ i | strict 2-loop , as well as the vanishing of the correction when ω → 0 or ω → ∞ (in fig. 1 the spectral function is divided by ω).
A practical evaluation of the spectral function necessitates a choice of the renormalization scale for the gauge coupling. Motivated by the arguments in ref. [34], we may expect that the physics of the IR domain is represented by a dimensionally reduced description, whereby a fastest apparent convergence criterion suggests [47,48] µ (N f = 0) opt = 6.74T ,μ Away from the IR domain, the scale should be set by virtuality. In order to smoothly interpolate between these two possibilities, we choosē taking ξ = 1 for N f = 0 and a larger ξ = 2 for N f = 2. As these are on the low side compared with eq. (4.3), we varyμ in the range (1.0...2.0) ×μ opt , noting that the gauge coupling grows uncontrollably large forμ = 0.5μ opt (α s > 0.5). The gauge coupling is solved for from 5-loop evolution [49][50][51]. We have verified that the results are stable if resorting to lower-order running or modifying the interpolation in eq. (4.4) while keeping the limits at πT ≪ |M | and πT ≫ |M | fixed. At very large ω, we let ρ V continuously cross into vacuum-like N 4 LO perturbative behaviour [27]. Such results can be inserted into eq. (1.3), in order to construct G V . For ρ H the vacuum tail is absent, nevertheless the results for G H are quite sensitive to a broad frequency range 0 ≤ ω < ∼ 30T .

Comparison with lattice data for
We start the lattice comparison with the data that were produced and analyzed in ref. [33]. The correlator measured was where V µ is the (Minkowskian) vector current and ... c stands for the connected contractions.
In the continuum limit this correlator diverges at small τ and is conveniently normalized to the free result For scale setting, we use T c /Λ MS ≃ 1.24, which has ∼ 10% uncertainty [52]. Resummed NLO spectral functions ρ V are shown for three momenta in fig. 2(left), and the corresponding imaginary-time correlators G V obtained from eq. (1.3) in fig. 2(right), where they are also compared with lattice data. Despite the low temperature, we observe a remarkable agreement. On close inspection, the perturbative curves are above the lattice ones, requiring a non-perturbative suppression of ρ V . The same qualitative features persist at T = 1.3T c (not shown), however the difference between the perturbative and lattice results is slightly smaller, as may be expected from a gradually decreasing α s . The conclusions that we draw from these observations are summarized in sec. 5.

4.3.
Comparison with lattice data for N f = 2 [35,53] Finally we move on to unquenched lattice data, obtained recently for N f = 2 in refs. [35,53]. In this case we concentrate on the ultraviolet finite correlator (k ≡ ke z ) 1T c for N f = 0, the latter normalized to eq. (4.6). LPM LO refers to results from secs. 3.2 and 3.3, employing the two scale choicesμ =μ opt andμ = 2μ opt (cf. eq. (4.4)). The notation LPM NLO indicates that the contribution from ref. [22] has been added; in this case we useμ =μ opt . The black squares are lattice results from ref. [33]. The spectral function can become negative at very small ω due to the subtraction of ρ 00 (cf. eq. (4.5)); the related physics is discussed in more detail around eq. (5.1).
Let us stress again that the spectral functions corresponding to G V and G H agree on the light cone but are substantially different away from it (cf. fig. 2(left) vs. fig. 3(left)).
The spectral function ρ H is shown in fig. 3(left), and the corresponding imaginary-time correlator G H in fig. 3(right). Like in fig. 2(right), the lattice correlators fall in general below the perturbative curves. The uncertainties of the perturbative imaginary-time correlators, as reflected by the scale dependence and the difference between LPM LO and LPM NLO resummations, are relatively speaking larger for N f = 2, a manifestation of the fact that the dominant vacuum UV tail is absent and therefore the data is more sensitive to IR physics. Nevertheless it is comforting that the qualitative pattern remains similar. The conclusions drawn from the comparison are discussed in sec. 5.

Conclusions
Motivated by a comparison with lattice data, unresummed NLO (2-loop) vector spectral functions have recently been extended into two new domains [38]: below the light cone (ω < k), and to a longitudinal polarization that vanishes at the light cone but is non-zero elsewhere. Even if the spacelike domain, corresponding to deep inelastic scattering off a thermal medium, sounds academic, it is essential for a comparison with lattice data, given that imaginary-time measurements get a large contribution from this region (cf. eq. (1.3)). The longitudinal polarization, in turn, is useful in the UV domain, as it permits to subtract the short-distance singularities from the lattice measurement (cf. eq. (3.3)) [35].
With the 2-loop results at hand, they can be resummed close to the light cone as specified in eq. (4.1) (parametrically, this is needed for |ω − k| < ∼ α s T 2 /k). Making use of methods developed in ref. [34], this resummation has been worked out to NLO by now [21,22], implying in this context corrections suppressed by √ α s . We have incorporated the latter corrections in our results, switching them off away from the light cone when they lose their validity.
The comparison of the imaginary-time correlators following from the resummed NLO spectral functions against lattice data can be viewed as the inspection of many separate "sum rules", one for each τ . Put together, this constrains the spectral function in a non-trivial way. In particular, we find that the correlators are affected by the choice of the renormalization scale of α s (cf. figs. 2 and 3). Reasonable agreement is obtained by scale choices reminiscent of those originating from dimensional reduction (cf. eq. (4.3)).
After fixing the renormalization scale, the perturbative results lie in general somewhat above the lattice data. Such a non-perturbative suppression confirms the previous finding based on a polynomial interpolation of ρ V [33]. At the same time the comparison of figs. 2(right) and 3(right) testifies to the improved resolution power of the correlator G H [35], so we are looking forward to final results from Padé fits of ρ H [53].
It would be interesting to investigate if resummed NLO rates embedded in hydrodynamical simulations of heavy ion collisions also overshoot the experimental results at small virtualities. To our knowledge this exercise has been implemented only on a rough level so far [58], supporting however this type of an overall trend. Nevertheless, it could still be that the physical photon rate is well predicted or even underestimated by the NLO result, if there is a large suppression of the spectral weigth in some other domain. The general expectation is that strong interactions should suppress thermal fluctuations particularly at small ω and k.