Charm-loop effect in $B \to K^{(*)} \ell^{+} \ell^{-}$ and $B\to K^*γ$

Abstract

We calculate the long-distance effect generated by the four-quark operators with $c$-quarks in the $B\to K^{(*)} \ell^+\ell^-$ decays. At the lepton-pair invariant masses far below the $\bar{c}c$-threshold, $q^2\ll 4m_c^2$, we use OPE near the light-cone. The nonfactorizable soft-gluon emission from $c$-quarks is cast in the form of a nonlocal effective operator. The $B\to K^{(*)}$ matrix elements of this operator are calculated from the QCD light-cone sum rules with the $B$-meson distribution amplitudes. As a byproduct, we also predict the charm-loop contribution to $B\to K^*γ$ beyond the local-operator approximation. To describe the charm-loop effect at large $q^2$, we employ the hadronic dispersion relation with $ψ=J/ψ,ψ(2S), ...$ contributions, where the measured $ B\to K^{(*)}ψ$ amplitudes are used as inputs. Matching this relation to the result of QCD calculation reveals a destructive interference between the $J/ψ$ and $ψ(2S)$ contributions. The resulting charm-loop effect is represented as a $q^2$-dependent correction $ΔC_9(q^2)$ to the Wilson coefficient $C_9$. Within uncertainties of our calculation, at $q^2$ below the charmonium region the predicted ratio $ΔC_9(q^2)/C_9$ is $\leq 5% $ for $B\to K \ell^+\ell^-$, but can reach as much as 20% for $B\to K^*\ell^+\ell^-$, the difference being mainly caused by the soft-gluon contribution.

A few comments added, version to appear in JHEP