Comparison of the mixed-fermion-action Effects using different fermion and gauge actions with 2+1 and 2+1+1 flavors

The leading-order low-energy constant ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ in mixed-action chiral perturbation theory is calculated using $2+1+1$-flavor gauge ensembles with HISQ fermions and a tadpole-improved Symanzik gauge action at four lattice spacings $a\in [0.048,0.111]$ fm. By comparing our results to those from different actions and a $2+1$-flavor case, We find that the fermion action has the dominant impact, the gauge action has a secondary but measurable effect, and the contribution from charm quark loops is negligible within our current uncertainties.