Multipole expansion at the level of the action in d-dimensions

In this paper we study the multipole expansion of the long-wavelength effective action for radiative sources in (d & thorn; 1) spacetime dimensions. We present detailed expressions for the multipole moments for the case of scalar-, electromagnetic-, and (linearized) gravitational-wave emission. For electromagnetism and gravity, we derive expressions for the odd-parity, magnetic-type moments as SO(d) duals of the ones traditionally used in the literature. The d-dimensional case features a novel set of "Weyl-type" moments, coupling to the spatial part of the Weyl tensor, which are absent in three dimensions. Agreement is found in the overlap with previous known results, notably in the d -> 3 limit. Because of its reliance on dimensional regularization, the results presented here play a crucial role for the further development of the effective field theory approach to gravitational dynamics, and in particular for the computation of the gravitational-wave flux, starting at the third post-Newtonian order.