Quark mass dependence of the vacuum electric conductivity induced by the magnetic field in $SU(2)$ lattice gluodynamics

We study the electric conductivity induced by the magnetic field $B$ in quenched $SU(2)$ lattice gauge theory at a finite temperature below the deconfinement phase transition as a function of the bare quark mass $m_q$ in the range $m_q=55\mathrm{MeV}\dots 540\mathrm{MeV}$. At fixed quark mass, the conductivity grows linearly with the magnetic field strength $|B|$. The proportionality coefficient in this dependence increases towards smaller quark masses and seems to saturate at some finite value in the zero-mass limit. The nonanalytic dependence on the field strength might result from the mixing between vector mesons and pions in an external magnetic field. We discuss the implications of our results for dilepton angular distributions in heavy-ion collisions.

Figure 1

Current-current correlators at different quark masses. Above on the left: correlators of the longitudinal components of the current at $\sqrt{qB}=0.45\mathrm{GeV}$ and different quark masses. Above on the right: correlators of the longitudinal and transverse components of the current at $\sqrt{qB}=0.45\mathrm{GeV}$ and different quark masses. Below: correlators of the longitudinal and transverse components of the current at $m_q=100\mathrm{MeV}$ (on the left) and at $m_q=540\mathrm{MeV}$ (on the right) and different magnetic fields.

Figure 2

A comparison of current-current correlators (1) calculated at ${14}^3\times 14$ and ${20}^3\times 14$ lattices at equal values of the lattice spacing and magnetic field strength $\sqrt{qB}=0.45\mathrm{GeV}$.

Figure 3

Induced conductivity in the direction of the magnetic field as a function of field strength at different quark masses. Solid lines are the linear fits (4) of the data. For $m_q=55\mathrm{MeV}$ the data points are very close to those for $m_q=100\mathrm{MeV}$, and thus were moved a little along the $qB$ axis in order to make the plot more illustrative.

Figure 4

Optimal values of the coefficients $A(m_q)$ in the linear fits (4) for different quark masses.