  The study of heavy-light meson masses should provide a way to determine
renormalized quark masses and other properties of heavy-light mesons. In the
context of lattice QCD, for example, it is possible to calculate hadronic
quantities for arbitrary values of the quark masses. In this paper, we address
two aspects relating heavy-light meson masses to the quark masses. First, we
introduce a definition of the renormalized quark mass that is free of both
scale dependence and renormalon ambiguities, and discuss its relation to more
familiar definitions of the quark mass. We then show how this definition enters
a merger of the descriptions of heavy-light masses in heavy-quark effective
theory and in chiral perturbation theory ($\chi$PT). For practical
implementations of this merger, we extend the one-loop $\chi$PT corrections to
lattice gauge theory with heavy-light mesons composed of staggered fermions for
both quarks. Putting everything together, we obtain a practical formula to
describe all-staggered heavy-light meson masses in terms of quark masses as
well as some lattice artifacts related to staggered fermions. In a companion
paper, we use this function to analyze lattice-QCD data and extract quark
masses and some matrix elements defined in heavy-quark effective theory.

High Energy Physics - Phenomenology;High Energy Physics - Lattice;High Energy Physics - Theory