  We consider the mutual Renyi information I^n(A,B)=S^n_A+S^n_B-S^n_{AUB} 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 I^n(A,B)\sim C^n_AC^n_B(R_AR_B/r^2)^a, where a the smallest sum of the
scaling dimensions of operators whose product has the quantum numbers of the
vacuum, and the constants C^n_{A,B} depend only on the shape of the regions and
universal data of the CFT.
  For a free massless scalar field, where 2x=d-1, we show that C^2_AR_A^{d-1}
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 Renyi entropies of a
single spherical region for a scalar field theory with a small mass.
