Two loop renormalization of the magnetic coupling in hot QCD

Well above the critical temperature hot QCD is described by 3d electrostatic QCD with gauge coupling $g_E$ and Debye mass $m_E$. We integrate out the Debye scales to two loop accuracy and find for the gauge coupling in the resulting magnetostatic action $g_M^2=g_E^2(1-{1\over{48}}{g_E^2N\over{\pi m_E}} -{17\over{4608}}({g_E^2N\over{\pi m_E}})^2+O(({g_E^2N\over{\pi m_E}})^3))$.