Determination of $\gamma$ from the $\eta - \eta'$ mixing within QCD Factorization

Abstract

The charmless nonleptonic $B\to \eta K$ decay is an useful probe to test the strong interaction dynamics part of the Standard Model. Within the QCD factorization framework, we analyse this particular decay by using the most recent data of $BR(B^+ \to \eta K^+)$ and $A_{cp}(B^+ \to \eta K^+)$ available in the Heavy Flavour Averaging Group (HFAG) webcite. Using these data we constraint the unitarity angle $\gamma$ and $\eta-\eta'$ mixing angle $\theta$. We find that such constraint is scale dependent, e.g. for $\gamma = 70^o$, the data for $BR(B^+ \to \eta K^+)$ (= $(2.6 \pm 0.5)\times 10^{-6}$) suggests that $\theta$ should lie in between $-46^o$ and $-44^o$, $-30^o$ and $-26^o$ for $\mu = m_b/2$. For $\mu = m_b$, the same $BR$ data (with the same $\gamma$) suggests that $\theta$ should lie in between $-54^o$ and $-50^o$, $-30^o$ and $-26^o$. The allowed region followed from $A_{cp}=-0.25\pm 0.14$, is found to be rather wider than that obtained from the $BR(B^+ \to \eta K^+)$ data. For $\gamma=70^o$ and $\theta = -21.3^o$, we find $BR(B^-(+) \to \eta K^-(+))= 2.93(4.91) \times 10^{-6}$ and $A_{cp}=-0.252$ at $\mu=m_b$, and about $ 2.98(5.72) \times 10^{-6}$ and -0.315 corresponding to $\mu=m_b/2$. We investigate the role of the power corrections in such constraints.