Heavy Quark Diffusion in Strong Magnetic Fields at Weak Coupling and Implications for Elliptic Flow

We compute the momentum diffusion coefficients of heavy quarks, ${\kappa }_{\parallel }$ and ${\kappa }_{\perp }$, in a strong magnetic field $B$ along the directions parallel and perpendicular to $B$, respectively, at the leading order in QCD coupling constant ${\alpha }_s$. We consider a regime relevant for the relativistic heavy ion collisions, ${\alpha }_{s}eB\ll T^2\ll eB$, so that thermal excitations of light quarks are restricted to the lowest Landau level (LLL) states. In the vanishing light-quark mass limit, we find ${\kappa }_{\perp }^{\mathrm{L}\mathrm{O}}\propto {\alpha }_s^{2}TeB$ in the leading order that arises from screened Coulomb scatterings with (1+1)-dimensional LLL quarks, while ${\kappa }_{\parallel }$ gets no contribution from the scatterings with LLL quarks due to kinematic restrictions. We show that the first non-zero leading order contributions to ${\kappa }_{\parallel }^{\mathrm{L}\mathrm{O}}$ come from the two separate effects: 1) the screened Coulomb scatterings with thermal gluons, and 2) a finite light-quark mass $m_q$. The former leads to $\kappa_\parallel^{\rm LO,,gluon} \propto \alpha_s^2 T^3$andthelatterto$\kappa_\parallel^{\rm LO,,massive}\propto \alpha_s (\alpha_s eB)^{1/2} m_q^2$. Based on our results, we propose a new scenario for the large value of heavy-quark elliptic flow observed in RHIC and LHC. Namely, when ${\kappa }_{\perp }\gg {\kappa }_{\parallel }$, an anisotropy in drag forces gives rise to a sizable amount of the heavy-quark elliptic flow even if heavy quarks do not fully belong to an ellipsoidally expanding background fluid.