Galilean-invariant effective field theory for the X(3872) at next-to-leading order

XEFT is a low-energy effective field theory for charm mesons and pions that provides a systematically improvable description of the $X(3872)$ resonance. To simplify calculations beyond leading order, we introduce a new formulation of XEFT with a dynamical field for a pair of charm mesons in the resonant channel. We simplify the renormalization of XEFT by introducing a new renormalization scheme that involves the subtraction of amplitudes at the complex ${D}^{*0}{\overline{D}}^{0}$ threshold. The new formulation and the new renormalization scheme are illustrated by calculating the complex pole energy of $X$ and the ${D}^{*0}{\overline{D}}^{0}$ scattering amplitude to next-to-leading order using Galilean-invariant XEFT.