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Motivated by the recent dilaton-based proof of the 4d a-theorem, we study thedilaton effective action for RG flows in d dimensions. When d is even, theaction consists of a Wess-Zumino (WZ) term, whose Weyl-variation encodes thetrace-anomaly, plus all Weyl-invariants. For d odd, the action consists ofWeyl-invariants only. We present explicit results for the flat-space limit ofthe dilaton effective action in d-dimensions up to and including 8-derivativeterms. GJMS-operators from conformal geometry motivate a form of the actionthat unifies the Weyl-invariants and anomaly-terms into a compact general-dstructure.  A new feature in 8d is the presence of an 8-derivative Weyl-invariant thatpollutes the O(p^8)-contribution from the WZ action to the dilaton scatteringamplitudes; this may challenge a dilaton-based proof of an a-theorem in 8d.  We use the example of a free massive scalar for two purposes: 1) it allows usto confirm the structure of the d-dimensional dilaton effective actionexplicitly; we carry out this check for d=3,4,5,...,10; and 2) in 8d wedemonstrate how the flow (a_UV - a_IR) can be extracted systematically from theO(p^8)-amplitudes despite the contamination from the 8-derivativeWeyl-invariant. This computation gives a value for the a-anomaly of the 8d freeconformal scalar that is shown to match the value obtained from zeta-functionregularization of the log-term in the free energy.

RG flows in d dimensions, the dilaton effective action, and the a-theorem