On the resilience of the gravitational variational principle under renormalization

A well-defined variational principle for gravitational actions typically requires to cancel boundary terms produced by the variation of the bulk action with a suitable set of boundary counterterms. This can be achieved by carefully balancing the coefficients multiplying the bulk operators with those multiplying the boundary ones. A typical example of this construction is the Gibbons-Hawking-York boundary action that needs to be added to the Einstein-Hilbert one in order to have a well-defined metric variation for General Relativity with Dirichlet boundary conditions. Quantum fluctuations of matter fields lead to a renormalization of these coefficients which may or may not preserve this balance. Indeed, already at the level of General Relativity, the resilience of the matching between bulk and boundary constants is far from obvious and it is anyway incomplete given that matter generically induces quadratic curvature operators. We investigate here the resilience of the matching of higher-order couplings upon renormalization by a non-minimally coupled scalar field and show that a problem is present. Even though we do not completely solve the latter, we show that it can be greatly ameliorated by a wise splitting between dynamical and topological contributions. Doing so, we find that the bulk-boundary matching is preserved up to a universal term (present for any Weyl invariant matter field content), whose nature and possible cancellation we shall discuss in the end.