Distance between various discretized fermion actions

We present the leading order mixed-action effect ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{x}}\equiv m_{\pi ,\mathrm{v}\mathrm{s}}^2-\frac{m_{\pi ,\mathrm{v}\mathrm{v}}^2+m_{\pi ,\mathrm{s}\mathrm{s}}^2}{2}$ using HISQ, clover or overlap valence fermion actions on gauge ensembles using various sea fermion actions across a widely-used lattice spacing range $a\in [0.04,0.19]$~fm. The results suggest that ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ decreases as the fourth order of the lattice spacing on the gauge ensembles with dynamical chiral sea fermions, such as Domain wall or HISQ fermions. When a clover sea fermion action which has explicit chiral symmetry breaking is used in the ensemble, ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ can be much larger regardless of the valence fermion action used.