Exact Mutual Information Difference: Scalar vs. Maxwell Fields

We compute, for any R\'enyi index $n$, the exact difference between the mutual R\'enyi informations of a pair of free massless scalars and that of a Maxwell field in $d=4$ dimensions. Using the standard dimensional reduction method in polar coordinates, the problem is mapped to that of a single scalar field in $d=2$ with Dirichlet boundary conditions, which in turn can be conveniently related to the algebra of a chiral current on the full line. This latter identification, which maps algebras on an interval to two-interval algebras, yields exact results that clarify the structure of the long-distance OPE perturbative expansion of the mutual information. We find that this series has a finite radius of convergence only for integer $n>1$, while it becomes only asymptotical for $n=1$ and general non-integer values of $n$.