Thermal dilepton rates and electrical conductivity of the QGP from the lattice

We investigate the temperature dependence of the thermal dilepton rate and the electrical conductivity of the gluon plasma at temperatures of 1.1, 1.3, and $1.5T_c$ in quenched QCD. Making use of nonperturbatively clover-improved Wilson valence quarks allows for a clean extrapolation of the vector meson correlation function to the continuum limit. We found that the vector correlation function divided by $T^3$ is almost temperature independent in the current temperature window. The spectral functions are obtained by ${\chi }^2$ fitting of phenomenologically inspired Ansätze for the spectral function to the continuum extrapolated correlator data, where the correlations between the data points have been included. Systematic uncertainties arising from varying the Ansätze motivated from strong coupling theory as well as perturbation theory are discussed and estimated. We found that the electrical conductivity of the hot medium, related to the slope of the vector spectral function at zero frequency and momentum, is $0.2C_{em}\lesssim \sigma /T\lesssim 0.7C_{em}$ for $T=1.1T_c$ and $0.2C_{em}\lesssim \sigma /T\lesssim 0.4C_{em}$ for the higher temperatures. The dilepton rates and soft photon rates, resulting from the obtained spectral functions, show no significant temperature dependence, either.

Figure 1

Left and top right: All three lattice correlators and the resulting continuum extrapolated correlator for the data sets at $T=1.1T_c$, $T=1.3T_c$, and $T=1.5T_c$, respectively. Note that the finest lattice agrees with the continuum extrapolation down to $\tau T\sim 0.2$ in all cases. The single black data point at $\tau T=0$ indicates the continuum extrapolated result for the inverted quark number susceptibility. Bottom right: The continuum extrapolations for all three temperatures.

Figure 2

Top: The extrapolation of ${\chi }_q/T^2$. The leftmost data points are the continuum extrapolated values. Bottom: Continuum extrapolated ratios of renormalized correlation functions $G_{ii}/G_V^{\mathrm{free}}$ for all three temperatures, not normalized by ${\chi }_q/T^2$. The solid line is the corresponding noninteracting ratio $G_{ii}^{\mathrm{free}}/G_V^{\mathrm{free}}$.

Figure 3

Top: A heat map of the entries of the estimated continuum correlation matrix for all points $\tau T>0.1$ at $1.1T_c$. The axes label the row and column entry, respectively. Hence, the midpoint $\tau T_j=\tau T_k=0.5$ is located in the bottom right corner. Bottom: The eigenvalues of the covariance matrices of the data. Note that they decrease in a regular fashion, without strong fluctuations.

Figure 4

The necessary extrapolation in $\tau T$ to obtain ${\mathrm{\Delta }}_{ii}$ for the case $T=1.1T_c$. The fit interval is $\tau T\in [0.2,0.45]$, i.e., the point at the far right is not included in the fit.

Figure 5

The spectral functions resulting from the fit for all temperatures. The dotted lines are the Breit-Wigner and the free contributions separately to guide the eye. Note the consistently higher intercept of the spectral functions with the cut applied. Bottom right: The final results for the electrical conductivity for all three temperatures as resulting from Ansatz ${\rho }_{\mathrm{ans}}$.

Figure 6

The spectral function resulting from the fit of the (coarse) model ${\rho }_{\mathrm{flat}}$ for all temperatures.

Figure 7

Fit of a real delta peak in the low frequency region. The points at $\tau T>0.5$ are the second thermal moments and their fit results, respectively. Left: fit including the covariance of the data. Note how the second thermal moments are described much worse than the corresponding correlator data points. Right: Fit without the covariance of the data. The correlator is described nicely; the thermal moments are a bit off, but less than in the fit including the covariance. The curves are offset for visibility.

Figure 8

The increase of electrical conductivity upon the increase of the cutoff ${\omega }_0/T$ in (27). It reaches its maximum around ${\omega }_0/T\simeq 3$ for all three temperatures. The smoothing parameter is fixed to ${\mathrm{\Delta }}_0/T=0.5$ throughout the analysis.

Figure 9

The resulting spectral function when utilizing perturbative input.

Figure 10

Top left: The solutions of different Ansätze compared for $T=1.1T_c$. Note that the difference between ${\rho }_{\mathrm{R}}$ and ${\rho }_{\mathrm{ans}}$ is most pronounced at $T=1.1T_c$. Top right: Integrating $\rho /(\omega T)$ up to ${\omega }_{\max }$, i.e., numerically computing its primitive function for $T=1.1T_c$. Bottom: The thermal moments for all $T$ and their respective reconstructions from the fit, shown for all three models ${\rho }_{\mathrm{ans}}$, ${\rho }_{\mathrm{flat}}$, and ${\rho }_{\mathrm{R}}$.

Figure 11

Left: The thermal dilepton rate as obtained from ${\rho }_{\mathrm{R}}$ as a function of $\omega /T$, accompanied by the HTL rate and the noninteracting Born rate. Right: The final results for the electrical conductivity. They incorporate the full systematics, i.e., the minimum and maximum conductivities, respectively, of ${\rho }_{\mathrm{ans}}$ and ${\rho }_{\mathrm{R}}$.