Renormalization of entanglement entropy and the gravitational effective action

The entanglement entropy of quantum fields across a spatial boundary is UV divergent, its leading contribution proportional to the area of this boundary. We demonstrate that the Callan-Wilczek formula provides a renormalized geometrical definition of this entanglement entropy for a class of quantum states defined by a path integral over quantum fields propagating on a curved background spacetime. In particular, UV divergences localized on the spatial boundary do not contribute to the entanglement entropy, the leading contribution to the renormalized entanglement entropy is given by the Bekenstein-Hawking formula, and subleading UV-sensitive contributions are given in terms of renormalized couplings of the gravitational effective action. These results hold even if the UV-divergent contribution to the entanglement entropy is negative, for example, in theories with non-minimal scalar couplings to gravity. We show that subleading UV-sensitive contributions to the renormalized entanglement entropy depend nontrivially on the quantum state. We compute new subleading UV-sensitive contributions to the renormalized entanglement entropy, finding agreement with the Wald entropy formula in all cases. We speculate that the entanglement entropy of an arbitrary spatial boundary may be a well-defined observable in quantum gravity.