Entanglement entropy of a Maxwell field on the sphere

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 $c_{\log }^{}$, the one for a $d=4$ massless scalar $c_{\log }^{}$, and the logarithmic coefficient $c_{\log }^{}$ for a $d=2$ scalar with Dirichlet boundary condition at the origin. Using the accepted values for these coefficients $c_{\log }^{}=-1\mathrm{/}90$ and $c_{\log }^{}=1\mathrm{/}6$ we get $c_{\log }^{}=-16\mathrm{/}45$, 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 $c_{\log }^{}$, $c_{\log }^{}$ and $c_{\log }^{}$, verifying the relation, as well as checked they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.