  There are indications that some theories with spontaneous symmetry breaking
also feature a light scalar in their spectrum, with a mass comparable to the
one of the Goldstone modes. In this paper, we perform the one-loop
renormalization of a theory of Goldstone modes invariant under a chiral
$SU(n)\times SU(n)$ symmetry group coupled to a generic scalar singlet. We
employ the background field method, together with the heat kernel expansion, to
get an expression for the effective action at one loop and single out the
anomalous dimensions, which can be read off from the second Seeley-DeWitt
coefficient. As a relevant application, we use our master formula to
renormalize chiral-scale perturbation theory, an alternative to $SU(3)$ chiral
perturbation theory where the $f_0(500)$ meson is interpreted as a dilaton.
Based on our results, we briefly discuss strategies to test and discern both
effective field theories using lattice simulations.
