Charm-loop effect in B → K ^(*) ℓ ⁺ ℓ ⁻ and B → K ^* γ

Abstract

We calculate the long-distance effect generated by the four-quark operators with c-quarks in the B → K ^(*) ℓ ⁺ ℓ ⁻ decays. At the lepton-pair invariant masses far below the $\bar{c}c$-threshold, q ² ≪ 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 → 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 → 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 → 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 ≤5% for B → Kℓ ⁺ ℓ ⁻, but can reach as much as 20% for B → K ^* ℓ ⁺ ℓ ⁻, the difference being mainly caused by the soft-gluon contribution.