Entanglement entropy of a Maxwell field on the sphere

Abstract: We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in $d=4$ dimensions. In spherical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. We show the entanglement entropy of a Maxwell field is equivalent to the one of two identical massless scalars from which the mode of $l=0$ has been removed. This shows the relation $c^M_{\log}=2 (c^S_{\log}-c^{S_{l=0}}_{\log})$ between the logarithmic coefficient in the entropy for a Maxwell field $c^M_{\log}$, the one for a $d=4$ massless scalar $c_{\log}^S$, and the logarithmic coefficient $c^{S_{l=0}}_{\log}$ for a $d=2$ scalar with Dirichlet boundary condition at the origin. Using the accepted values for these coefficients $c_{\log}^S=-1/90$ and $c^{S_{l=0}}_{\log}=1/6$ we get $c^M_{\log}=-16/45$, which coincides with Dowker's calculation, but does not match the coefficient $-\frac{31}{45}$ in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers $c^M_{\log}$, $c^S_{\log}$ and $c^{S_{l=0}}_{\log}$, verifying the relation, as well as checked they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.