Partial summation of the nonlocal expansion for the gravitational effective action in 4 dimensions

The vacuum action for the gravitational field admits a known expansion in powers of the Ricci tensor with nonlocal operator coefficients (form factors). We show that going over to a different basis of curvature invariants makes a partial summation of this expansion possible. Only the form factors of the Weyl-tensor invariants need to be calculated. The full action is then uniquely recovered to all orders from the knowledge of the trace anomaly. We present an explicit expression for the partially summed action, and point out simplifications resulting in the vertex functions. An application to the effect of the vacuum gravitational waves is discussed.