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

In rare $B$ meson decays $B\rightarrow K \ell^+ \ell^-$, 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 $\bar{b} \rightarrow \bar{s} +γ\rightarrow \bar{s}+\ell^+ + \ell^-$. If such a contribution is sizable compared with the standard estimation of $B\rightarrow K \ell^+ \ell^-$, we will observe visible difference between $d Γ(B^0\rightarrow K^0 \ell^+ \ell^-)/d q^2$ and $d Γ(B^+\rightarrow K^+ \ell^+ \ell^-)/d q^2$ 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 $d Γ(B\rightarrow K \ell^+ \ell^-)/d q^2$ 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 $q$, $d \rightarrow d+γ$ ($u \rightarrow u+γ$) 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.