Renormalization of chiral perturbation theory with spinless matter field in curved spacetime

We generalize chiral perturbation theory with spinless matter fields in the fundamental representation of $\mathrm{S}\mathrm{U}(N)$ to curved spacetime in the presence of an external gravitational field. This work is motivated by recent interest in investigating energy-momentum tensor matrix elements of matter fields. The complete chiral Lagrangian is constructed, including both minimal extensions from flat spacetime and new curvature-induced terms, up to the next-to-next-to-leading order in the chiral expansion, $\mathcal{O}(p^3)$. We perform a systematic one-loop renormalization of the generating functional using the background field method combined with the heat-kernel technique, and explicitly compute the resulting ultraviolet divergences.