Δmixparameter in the overlap on domain-wall mixed action

A direct calculation of the mixed action parameter ${\ensuremath{\Delta}}_{\mathrm{mix}}$ with valence overlap fermions on a domain-wall fermion sea is presented. The calculation is performed on four ensembles of the $2+1$ flavor domain-wall gauge configurations: ${24}^{3}\ifmmode\times\else\texttimes\fi{}64$ ($a{m}_{l}=0.005$, $a=0.114\text{ }\text{ }\mathrm{fm}$) and ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ ($a{m}_{l}=0.004$, 0.006, 0.008, $a=0.085\text{ }\text{ }\mathrm{fm}$). For pion masses close to 300 MeV we find ${\ensuremath{\Delta}}_{\mathrm{mix}}=0.030(6)\text{ }\text{ }{\mathrm{GeV}}^{4}$ at $a=0.114\text{ }\text{ }\mathrm{fm}$ and ${\ensuremath{\Delta}}_{\mathrm{mix}}=0.033(12)\text{ }\text{ }{\mathrm{GeV}}^{4}$ at $a=0.085\text{ }\text{ }\mathrm{fm}$. The results are quite independent of the lattice spacing and they are significantly smaller than the results for valence domain-wall fermions on asqtad sea or those of valence overlap fermions on clover sea. Combining the results extracted from these two ensembles, we get ${\ensuremath{\Delta}}_{\mathrm{mix}}=0.030(6)(5)\text{ }\text{ }{\mathrm{GeV}}^{4}$, where the first error is statistical and the second is the systematic error associated with the fitting method.