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The $Δ_{mix}$ parameter in the overlap on domain-wall mixed action

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