  Mixed action lattice calculations allow for an additive lattice spacing
dependent mass renormalization of mesons composed of one sea and one valence
quark, regardless of the type of fermion discretization methods used in the
valence and sea sectors. The value of the mass renormalization depends upon the
lattice actions used. This mixed meson mass shift is an important lattice
artifact to determine for mixed action calculations; because it modifies the
pion mass, it plays a central role in the low energy dynamics of all hadronic
correlation functions. We determine the leading order, $\mathcal{O}(a^2)$, and
next to leading order, $\mathcal{O}(a^2 m_\pi^2)$, additive mass shift of
\textit{valence-sea} mesons for a mixed lattice action with domain-wall valence
fermions and rooted staggered sea fermions, relevant to the majority of current
large scale mixed action lattice efforts. We find that on the asqtad improved
coarse MILC lattices, this additive mass shift is well parameterized in lattice
units by $\Delta(am)^2 = 0.034(2) -0.06(2) (a m_\pi)^2$, which in physical
units, using $a=0.125$ fm, corresponds to $\Delta(m)^2 = (291\pm 8
\textrm{MeV})^2 -0.06(2) m_\pi^2$. In terms of the mixed action effective field
theory parameters, the corresponding mass shift is given by $a^2
\Delta_\mathrm{Mix} = (316 \pm 4 \textrm{MeV})^2$ at leading order plus
next-to-leading order corrections including the necessary chiral logarithms for
this mixed action calculation, determined in this work. Within the precision of
our calculation, one can not distinguish between the full next-to-leading order
effective field theory analysis of this additive mixed meson mass shift and the
parameterization given above.
