Effective Valence Quark Model and a Possible Dip in $dBr(B\to K\ell \bar{\ell })/d{q}^2$

Abstract

In rare $B$ meson decays $B\to K{\ell }^{+}{\ell }^{-}$, a possible contribution of ${\ell }^{+}{\ell }^{-}$ emission via photon from the "spectator" quark $q$ ($q=u,d$) in the $B$ meson $(q\bar{b})$ is investigated in addition to the conventional one $\bar{b}\to \bar{s}+\gamma \to \bar{s}+{\ell }^{+}+{\ell }^{-}$. If such a contribution is sizable compared with the standard estimation of $B\to K{\ell }^{+}{\ell }^{-}$, we will observe visible difference between $d\Gamma (B^0\to K^0{\ell }^{+}{\ell }^{-})/d{q}^2$ and $d\Gamma (B^{+}\to K^{+}{\ell }^{+}{\ell }^{-})/d{q}^2$ in $q^2$ dependence ($q^2\equiv m_{\ell \ell }^2$). Besides, as a result of the interference between the conventional one and a new one, a dip appears in $d\Gamma (B\to K{\ell }^{+}{\ell }^{-})/d{q}^2$ at a small region of $q^2$. The interference effect in the $B^0$ decay will also be observed differently from that in the $B^{+}$ decay. The calculation is done based on a semi-classical approach, a valence quark model. In the present model, the photon emission from the spectator quark $q$, $d\to d+\gamma$ ($u\to u+\gamma$) is independent of the $b$-$s$ transition mechanism, and the characteristic results are due to a straightforward estimate of the quark propagator which cannot be incorporated into the factorization method. The model is not a valence quark "dominant" model, so that, for example, the valence quarks in the final state carry only $24\%$ of the energy-momentum of the kaon.

Comments: 26 pages, 10 figures, version accepted by IJMPA