Distance between various discretized fermion actions

We present the leading-order mixed-action effect ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{mix}}\ensuremath{\equiv}{m}_{\ensuremath{\pi},\mathrm{vs}}^{2}\ensuremath{-}\frac{{m}_{\ensuremath{\pi},\mathrm{vv}}^{2}+{m}_{\ensuremath{\pi},\mathrm{ss}}^{2}}{2}$ using highly improved staggered quarks (HISQ), clover, or overlap valence fermion actions on gauge ensembles using various sea fermion actions across a widely used lattice spacing range $a\ensuremath{\in}[0.04,0.19]\text{ }\text{ }\mathrm{fm}$. The results suggest that ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{mix}}$ 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 that has explicit chiral symmetry breaking is used in the ensemble, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{mix}}$ can be much larger regardless of the valence fermion action used.