The files below contain perturbative thermal spectral functions corresponding to charmonium and bottomonium, at several temperatures above the deconfinement transition of quenched QCD. The basic philosophy for constructing such spectral functions was developed in 0812.2105, which built upon earlier works for the threshold region in 0704.1720 and 0711.1743. An improved approach for combining the threshold region and the ultraviolet asymptotics was worked out in 1709.07612, and this constitutes the implementation that was used for producing the data below.
For the moment we list data for quenched pseudoscalar and vector spectral functions. The scalar channel results may become available at a later date, once a proper update of the ultraviolet asymptotics to a multiloop level has been worked out. An extension to unquenched QCD, as already considered in 0812.2105, is likewise foreseen.
We start with the pseudoscalar data shown in fig. 7 of 1709.07612. The threshold region originates from 0711.1743. The file names indicate whether the temperature T = 1.1Tc, 1.3Tc, 1.5Tc, or 2.25Tc is considered, and whether the quark mass corresponds to charmonium or bottomonium. The letter P stands for the pseudoscalar channel, and nf0 serves as a reminder that these data apply to the quenched theory.
The columns of the files are:
1: ω / T
2: ρP /
[
T2
m2(
μ
ref )
]
Here μ ref ≡ 2 GeV is a traditional reference scale for MS quark masses; m( μ ref ) ≈ 1 GeV corresponds to charmonium in the quenched case; and m( μ ref ) ≈ 5 GeV corresponds to bottomonium. Here are the files:
For the vector channel the ultraviolet asymptotics is available up to 5-loop level, as explained in 1201.1994. The thermal physics of the threshold region is treated according to 0704.1720. The matching procedure between these two regimes, and a number of other refinements such as how the gauge coupling is chosen, were implemented according to 1709.07612.
The letter V stands for the vector channel in the files below (more precisely, for the sum over the spatial components of the vector current). The columns are:
1: ω / T
2: ρV /
T2
Here are the files:
An essential difference between the pseudoscalar and vector channels is that the vector channel gets a significant additional contribution from a "transport peak" located at very small frequencies (this region is not contained in the files above). The transport peak leads to an (almost) constant contribution to the corresponding imaginary-time correlator. These constant parts were worked up to NLO in 1210.1064. Here we list the constant parts for the temporal and spatial components of the vector correlator (Nf = 0):
| T / Tc | charm: G00 / T3 | charm: Gii / T3 | bottom: G00 / T3 | bottom: Gii / T3 |
|---|---|---|---|---|
| 1.1 | 0.247 | 0.125 | 0.809e-04 | 0.155e-04 |
| 1.2 | 0.284 | 0.149 | 0.168e-03 | 0.341e-04 |
| 1.3 | 0.324 | 0.176 | 0.348e-03 | 0.746e-04 |
| 1.5 | 0.416 | 0.245 | 0.146e-02 | 0.354e-03 |
| 1.6 | 0.455 | 0.276 | 0.248e-02 | 0.633e-03 |
| 2.0 | 0.579 | 0.386 | 0.117e-01 | 0.355e-02 |
| 2.25 | 0.637 | 0.444 | 0.229e-01 | 0.757e-02 |
The constant parts of G00 are trivially related to susceptibilities. Susceptibilities are exponentially sensitive to the ratio of the quark mass over the temperature, so we do expect modest differences to lattice measurements. Conceivably, susceptibilities measured on the lattice could be multiplied by the perturbative ratio Gii / G00, in order to estimate the integral over the transport peak affecting spatial imaginary-time correlators.
We have added rough estimates of the constant parts for the unquenched case (Nf = 3). However it should be stressed that in this case the gauge coupling is large and uncertainties are very substantial. We also note that the numerical values are much smaller than the quenched ones, because the temperatures are smaller in MeV, so that masses compared with the temperature are larger, for a given T/Tc. This leads to a big change in the suppression factor exp(-M/T). Here M corresponds to a pole mass, which is poorly determined (in fact inherently ambiguous).
In our opinion it only makes sense to use these data in the way explained above, i.e. by multiplying susceptibilities measured on the lattice by the perturbative ratio Gii / G00, in order to estimate the integral over the transport peak affecting spatial imaginary-time correlators. In these ratios, the exponential dependence on exp(-M/T) drops out. Physically, such ratios can be interpreted as the average velocity squared.
| T / Tc | charm: G00 / T3 | charm: Gii / T3 | bottom: G00 / T3 | bottom: Gii / T3 |
|---|---|---|---|---|
| 1.1 | 0.111e-02 | 0.262e-03 | 0.261e-13 | 0.201e-14 |
| 1.2 | 0.230e-02 | 0.583e-03 | 0.478e-12 | 0.400e-13 |
| 1.3 | 0.422e-02 | 0.114e-02 | 0.556e-11 | 0.501e-12 |
| 1.5 | 0.110e-01 | 0.331e-02 | 0.276e-09 | 0.283e-10 |
| 1.6 | 0.161e-01 | 0.509e-02 | 0.134e-08 | 0.146e-09 |
| 2.0 | 0.490e-01 | 0.181e-01 | 0.146e-06 | 0.194e-07 |
| 2.25 | 0.789e-01 | 0.315e-01 | 0.114e-05 | 0.169e-06 |