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

We compute the momentum diffusion coefficients of heavy quarks, $κ_\parallel$ and $κ_\perp$, in a strong magnetic field $B$ along the directions parallel and perpendicular to $B$, respectively, at the leading order in QCD coupling constant $α_s$. We consider a regime relevant for the relativistic heavy ion collisions, $α_s eB\ll T^2\ll eB$, 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 $κ_\perp^{\rm LO}\propto α_s^2 T eB$ in the leading order that arises from screened Coulomb scatterings with (1+1)-dimensional LLL quarks, while $κ_\parallel$ gets no contribution from the scatterings with LLL quarks due to kinematic restrictions. We show that the first non-zero leading order contributions to $κ_\parallel^{\rm LO}$ 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 $κ_\parallel^{\rm LO,\,gluon} \propto α_s^2 T^3$ and the latter to $κ_\parallel^{\rm LO,\,massive}\propto α_s (α_s eB)^{1/2} m_q^2$. 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 $κ_\perp\ggκ_\parallel$, 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.