Some results on the mutual information of disjoint regions in higher dimensions

We consider the mutual Rényi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R A , B . We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S d − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C ( n ) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Rényi entropies of a single spherical region for a scalar field theory with a small mass.