Jet transport coefficient $\widehat{q}$ in lattice QCD

We present the first calculation of the jet transport coefficient $\widehat{q}$ in quenched and ($2+1$)-flavor QCD on a 4D Euclidean lattice. The lightlike propagation of an energetic parton is factorized from the mean square gain in momentum transverse to the direction of propagation, which is expressed in terms of the thermal field-strength field-strength correlator. The leading-twist term in its operator product expansion is calculated on the lattice. Continuum extrapolated quenched results, and full QCD estimates based on unrenormalized lattice data, over multiple lattice sizes, are compared with (non)perturbative calculations and phenomenological extractions of $\widehat{q}$. The lattice data for $\widehat{q}$ show a temperature dependence similar to the entropy density. Within uncertainties, these are consistent with phenomenological extractions, contrary to calculations using perturbation theory.

Figure 1

A forward scattering diagram for the hard quark undergoing a single scattering off the gluon field in the plasma. The vertical dashed line represents the cut-line.

Figure 2

Temperature dependence of vacuum-subtracted field-strength correlators on quenched and unquenched SU(3) lattices. The operators are unrenormalized and have been computed for lattice sizes $n_{\tau }=4$, 6 and 8.

Figure 3

Bare and renormalized leading-twist operator $⟨\mathrm{T}r[F^{3i}{F}^{3i}-F^{4i}{F}^{4i}]⟩/T^4$ in pure SU(3) gauge theory. Our result agrees with the entropy density obtained in shifted boundary condition approach [55] after rescaling by $T_F=1/2$.

Figure 4

Comparison of the tadpole-factor rescaled leading-twist operator $⟨\mathrm{Tr}[F^{3j}{F}^{3j}-F^{4j}{F}^{4j}]/(T^4{u}_0^4)⟩$ with $T_F$-rescaled entropy $s/T^3$ in ($2+1$)-flavor QCD [50, 60] scaled by the gluon fraction in the Stefan-Boltzmann (SB) limit ($R_{\mathrm{SB}}$) or in EQCD ($R_{\mathrm{EQCD}}$) [61].

Figure 5

Lattice determination of $\widehat{q}$ for a highly energetic hard quark traversing a pure glue (blue) and ($2+1$)-flavor QCD (red) plasma. Error bars show statistical errors, solid bands represent scale variation. The pure SU(3) result is renormalized and continuum extrapolated. For the QCD estimate, systematic errors due to estimated renormalization (via $u_0$) and lattice cutoff ($n_{\tau }=6$) are indicated by the checkered band. Also plotted are the phenomenological extractions from the JET [3] and JETSCAPE [65] collaborations, nonperturbative results from an $N_f=2$, 3D lattice calculation [36], an $N_f=0$ stochastic vacuum model calculation [39], and LO-HTL calculations from Ref. [66], with ${\alpha }_S$ evaluated at $2\pi T\le \mu \le 4\pi T$.

Figure 6

Forward scattering diagrams for future outlook. (a) Transverse broadening due to flavor changing process. (b) A typical single scattering and single emission diagram.