Computing $c$- and $a$-functions from entanglement

Abstract

We confirm the direct connection between entanglement entropy and the notion of irreversibility in the renormalization-group flow in the context of a simple theory for which a calculation from first principles is feasible. The change of the entanglement entropy for a spherical entangling surface as its radius grows from zero to infinity corresponds to the flow from the UV to the IR. Through analytical and numerical means, we compute the entanglement entropy for a free massive scalar theory, making use of the method of correlation functions. We deduce a $c$-function in $1+1$ dimensions and an $a$-function in $3+1$ dimensions. Both functions are monotonic and vary continuously between one and zero, as expected for this simple theory.