Jet quenching parameter of the quark-gluon plasma in a strong magnetic field: Perturbative QCD and $\mathrm{AdS}/\mathrm{CFT}$ correspondence

We compute the jet quenching parameter $\widehat{q}$ of a quark-gluon plasma in the presence of a strong magnetic field using perturbative QCD (pQCD) in the weakly coupled regime, and $\mathrm{AdS}/\mathrm{CFT}$ correspondence in the strongly coupled regime of $\mathcal{N}=4$ super Yang-Mills. In the weakly coupled regime, we compute $\widehat{q}$ in pQCD at complete leading order (that is, leading log as well as the constant under the log) in the QCD coupling constant ${\alpha }_s$, assuming the hierarchy of scales ${\alpha }_{s}eB\ll T^2\ll eB$. We consider two cases of jet orientations with respect to the magnetic field: 1) the case of a jet moving parallel to the magnetic field; 2) the case of a jet moving perpendicular to the magnetic field. In the former case, we find $\widehat{q}\sim {\alpha }_s^2(eB)T\log (1/{\alpha }_s)$, while in the latter we have $\widehat{q}\sim {\alpha }_s^2(eB)T\log (T^2/{\alpha }_{s}eB)$. In both cases, this leading-order result arises from the scatterings with thermally populated lowest-Landau-level quarks. In the strongly coupled regime described by the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, we find $\widehat{q}\sim \sqrt{\lambda }(eB)T$ or $\widehat{q}\sim \sqrt{\lambda }\sqrt{eB}{T}^2$ in the same hierarchy of $T^2\ll eB$ depending on whether the jet is moving parallel or perpendicular to the magnetic field, respectively, which indicates a universal dependence of $\widehat{q}$ on $(eB)T$ in both regimes for the parallel case, the origin of which should be the transverse density of lowest-Landau-level states proportional to $eB$. Finally, the asymmetric transverse momentum diffusion in the case of a jet moving perpendicular to the magnetic field may give an interesting azimuthal asymmetry of the gluon bremsstrahlung spectrum in the Baier-Dokshitzer-Mueller-Peigne-Schiff and Zakharov (BDMPS-Z) formalism.

Figure 1

The imaginary cut of the jet self-energy is equal to the damping rate, that is, the total scattering rate with thermal (hard) particles, especially the lowest-Landau-level quarks. The exchanged gluon line is Debye screened by the same hard LLL states.

Figure 2

Three Feynman diagrams for $\mathcal{M}$. The first diagram dominates in the large-$P$ limit in the gauge $\epsilon ·P={\epsilon }^{\prime }·P=0$.