Galilean-Invariant XEFT

XEFT is a low-energy effective field theory for charm mesons and pions that can be used to describe the $X(3872)$ resonance. A Galilean-invariant formulation of XEFT is introduced to exploit the fact that mass is very nearly conserved in the transition $D^{*0} \to D^0 \pi^0$. The transitions $D^{*0} \to D^0 \pi^0$ and $D^{*0} \bar D^0 \to D^0 \bar D^0 \pi^0$ are described explicitly in XEFT. The effects of the decay $D^{*0} \to D^0 \gamma$ and of short-distance decay modes of the $X(3872)$, such as $J/\psi\, \pi^+ \pi^-$, can be taken into account by using complex on-shell renormalization schemes for the $D^{*0}$ and the $D^{*0} \bar D^0$ propagators. Galilean-invariant XEFT should facilitate accurate calculations of the properties of the $X(3872)$ resonance.