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 ${\kappa }_{\perp }^{\mathrm{LO}}\propto {\alpha }_s^{2}TeB$ 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 nonzero leading order contributions to ${\kappa }_{\parallel }^{\mathrm{LO}}$ 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 ${\kappa }_{\parallel }^{\mathrm{LO},\text{gluon}}\propto {\alpha }_s^2{T}^3$ and the latter to ${\kappa }_{\parallel }^{\mathrm{LO},\text{massive}}\propto {\alpha }_s({\alpha }_{s}eB{)}^{1/2}{m}_q^2$. 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.

Figure 1

Momentum diffusion of a heavy quark (double line) due to Coulomb scatterings with thermal quarks and gluons (collectively denoted as a dashed line).

Figure 2

One-loop gluon self-energy due to the polarization of a pair of quark and antiquark LLL states. Indices “$a$” and “$r$” denote the Schwinger-Keldysh basis.

Figure 3

Integral (35) for the transverse momentum diffusion coefficient. Lines show the analytic expressions given in between the brackets in Eqs. (36), which are confirmed by the numerical integration shown by filled circles.

Figure 4

Schematic illustration for the mechanism to derive an additional contribution to the elliptic flow of heavy quarks. Because of ${\kappa }_{\perp }\gg {\kappa }_{\parallel }$ heavy quarks are more dragged along the in-plane than along the out-of-plane.

Figure 5

Schematic illustration for the effects of the magnetic field on the heavy quark elliptic flow.