Direct $CP$ violation from isospin symmetry breaking effects in PQCD

We investigate the direct $CP$ violation for the decay process of ${\bar{B}}_s\to P(V){\pi }^0$ (P,V refer to the pseudoscalar meson and vector meson, respectively) via isospin symmetry breaking effects from the ${\pi }^0-\eta -{\eta }^{\mathrm{\prime }}$ mixing mechanism in PQCD factorization approach. Isospin symmetry breaking arises from the electroweak interaction and the u-d quark mass difference by the strong interaction which are known to be tiny. However, we find that isospin symmetry breaking at the leading order shifts the $CP$ violation due to the new strong phases.