The ${\mathrm{\Delta }}_{mix}$ parameter in the overlap on domain-wall mixed action

A direct calculation of the mixed-action parameter ${\mathrm{\Delta }}_{mix}$ 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: ${24}^3\times 64$ ($a{m}_l=0.005$, $a=0.114\text{fm}$) and $32^3 \times 64$($a m_l = 0.004, 0.006, 0.008$,$a=0.085\fm$). For pion masses close to $300\text{MeV}$ we find \hbox{${\mathrm{\Delta }}_{mix}=0.030(6){\text{GeV}}^4$} at $a=0.114\text{fm}$ and ${\mathrm{\Delta }}_{mix}=0.033(12){\text{GeV}}^4$ at $a=0.085\text{fm}$. 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 ${\mathrm{\Delta }}_{mix}=0.030(6)(5){\text{GeV}}^4$, where the first error is statistical and the second is the systematic error associated with the fitting method.