The unbearable smallness of magnetostatic QCD corrections

One loop corrections to the magnetostatic QCD action are evaluated to dimension six in the magnetostatic fields.The result is remarkably simple $${1\over 4}\int d\vec x (F^a_{ij})^2 - {g_M^2 N \over {32\pi m_E^3}} \int d\vec x\Bigg({1\over{60}}(D_m^{ab} F^b_{mr})^2 + {1\over 180} f^{abc} F^a_{mn} F^b_{nr }F^c_{rm} \Bigg)(1+\mathcal{O}(g_M^2)).$$ In all of the deconfined phase the correction is quite small. The term of $\mathcal{O}(g^2_M)$ is the dimension eight contribution and is only presented in a covariant derivative basis. We discuss their physical relevance for the pressure and the spatial Wilson loop.