We compute the momentum diffusion coefficients of heavy quarks,$\kappa_\parallel$ and $\kappa_\perp$, in a strong magnetic field $B$ along thedirections parallel and perpendicular to $B$, respectively, at the leadingorder in QCD coupling constant $\alpha_s$. We consider a regime relevant forthe relativistic heavy ion collisions, $\alpha_s eB\ll T^2\ll eB$, so thatthermal excitations of light quarks are restricted to the lowest Landau level(LLL) states. In the vanishing light-quark mass limit, we find$\kappa_\perp^{\rm LO}\propto \alpha_s^2 T eB$ in the leading order that arisesfrom screened Coulomb scatterings with (1+1)-dimensional LLL quarks, while$\kappa_\parallel$ gets no contribution from the scatterings with LLL quarksdue to kinematic restrictions. We show that the first non-zero leading ordercontributions to $\kappa_\parallel^{\rm LO}$ come from the two separateeffects: 1) the screened Coulomb scatterings with thermal gluons, and 2) afinite light-quark mass $m_q$. The former leads to $\kappa_\parallel^{\rmLO,\,gluon} \propto \alpha_s^2 T^3$ and the latter to $\kappa_\parallel^{\rmLO,\,massive}\propto \alpha_s (\alpha_s eB)^{1/2} m_q^2$. Based on our results,we propose a new scenario for the large value of heavy-quark elliptic flowobserved in RHIC and LHC. Namely, when $\kappa_\perp\gg\kappa_\parallel$, ananisotropy in drag forces gives rise to a sizable amount of the heavy-quarkelliptic flow even if heavy quarks do not fully belong to an ellipsoidallyexpanding background fluid.
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