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 $d=3+1$ dimensions. In spherical coordinates the problem decomposes into one-dimensional ones along the radial coordinate for each angular momentum. We show that the entanglement entropy of a Maxwell field is equivalent to one of two identical massless scalars from which the mode of $l=0$ has been removed. This shows the relation clogM=2(clogS-clog^Sₗ₌₀) between the logarithmic coefficient in the entropy for a Maxwell field clogM, the one for a $d=3+1$ massless scalar clogS, and the logarithmic coefficient clog^Sₗ₌₀ for a $d=1+1$ scalar with a Dirichlet boundary condition at the origin. Using the accepted values for these coefficients clogS=-1/90 and clog^Sₗ₌₀=1/6, we get clogM=-16/45, which coincides with Dowker's calculation, but does not match the coefficient -31/45 in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers clogM, clogS and clog^Sₗ₌₀, verifying the relation, as well as checked that they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.