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We calculate the direct $CP$ violation for the decay process of${B}^{0}\rightarrow \pi^{0}\pi^{0}$, ${B}^{+}\rightarrow \pi^{0}\pi^{+}$,${B}^{0}\rightarrow K^{0}\pi^{0}$ and ${B}^{+}\rightarrow K^{+}\pi^{0}$ viaisospin symmetry breaking effects from the $\pi^{0}-\eta-\eta'$ mixingmechanism in perturbative QCD approach. Isospin symmetry breaking originatesfrom the electroweak interaction and the u-d quark mass difference through thestrong interaction which are known to be tiny. However, we find that isospinsymmetry breaking at the leading order changes the $CP$ violation from the newstrong phases. Our calculation results for the $CP$ violation are within orincluding the range of experimental results. We also compare our results withthose from the QCD factorization and the perturbative QCD schemes withoutregard to isospin symmetry breaking effects.

Isospin symmetry breaking effects for the mode ${B}\rightarrow \pi\pi(K)$ in Perturbative QCD