Jet-quenching parameter $\hat{q}$ in the nonperturbative region

Abstract

We investigate the jet-quenching parameter $\hat{q}$ for the quark-gluon plasma (QGP) for $N_c=3$, defined nonperturbatively with the Wilson loop in the light-cone (LC) coordinate, at finite temperature $(T)$. Considering the effective static (heavy) quark-antiquark potential $V_{\bar{Q}Q}=\sigma L+C-A/L$, where $L$ indicates the LC transverse separation between the quarks, we obtain $\hat{q}\approx8V_{\bar{Q}Q}/L^2$. The $T$ dependences for $L$ and other relevant parameters are extracted from the $T$-dependent instanton (trivial-holonomy caloron) length parameters and the lattice QCD data. By choosing $L\approx{a}\approx\bar{\rho}_T$, in which $a$ and $\bar{\rho}_T$ denote the lattice spacing and the $T$-dependent average (anti)instanton size, respectively, we acquire numerically that $\hat{q}=(5\sim25)\,\mathrm{GeV/fm}$ for $T=(0\sim0.6)$ GeV, and these values are well consistent with other estimations from AdS/CFT and experimental analyses. It turns out that $\hat{q}$ is produced almost by the Coulomb and constant potentials of $V_{\bar{Q}Q}$. We also observe that the ratio $T^3/\hat{q}$ turns into being saturated to $\sim4.45\times10^{-2}$ for $T\gtrsim0.4$ GeV, indicating the strongly-coupled QGP, and $\hat{q}$ behaves proportionally to $T^3$ for high $T$.