Multipole expansion at the level of the action in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>d</mml:mi></mml:math>-dimensions
Loris Amalberti, François Larrouturou, Zixin Yang
Abstract
In this paper we study the multipole expansion of the long-wavelength effective action for radiative sources in ($d+1$) 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 $d\ensuremath{\rightarrow}3$ 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.