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

Abstract

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 thecc-threshold, q 2 ≪ 4m 2 c , we use OPE near the light-cone. The nonfactorizable softgluon 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 charmloop 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 ∆C 9 (q 2 ) to the Wilson coefficient C 9 . Within uncertainties of our calculation, at q 2 below the charmonium region the predicted ratio ∆C 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.

B s → Kℓν ℓ and B (s) → π(K)ℓ + ℓ − decays at large recoil and CKM matrix elements

“…Furthermore, the interval for our updated result for the B → K vector form factor in table 2 lies somewhat above the previous LCSR prediction [11], f + BK (0) = 0.34 +0.05 −0.02 , mainly due to the smaller value of f B from the two-point sum rule used here and due to the slightly smaller value of the effective threshold in LCSR used in [11]. On the other hand, in the LCSR for f T BK (q 2 ) some minor corrections, implemented here in the subleading twist-4 terms, largely compensate the shift caused by the B-decay constant, so that our result in table 2 is close to f T BK (0) = 0.39 +0.05 −0.03 obtained in [11].…”

“…twist-3 and twist-4 DAs, not shown in table 1 for brevity, are taken from [21], they were also used in [11,15,18]. Furthermore, in LCSRs the renormalization scale µ and the Borel parameters M for the sum rules with B (B s ) interpolating current quoted in table 1 are chosen, largely following [15].…”

“…These hadronic matrix elements in the B → π + − decay amplitude are multiplied by the CKM parameters other than V td , making the determination of the latter not straightforward. We take into account the nonlocal hadronic effects in B → K + − and B → π + − , employing the methods used in [12,13] and originally suggested in [11]. The nonlocal hadronic matrix elements are calculated at spacelike q 2 , using OPE, QCD factorization [14] and LCSRs, and are then matched to their values at timelike q 2 via hadronic dispersion relations.…”

“…In refs. [11,12], this part of hadronic input was cast in the form of an effective (process-and q 2 -dependent) addition ∆C BP 9 (q 2 ) to the Wilson coefficient C 9 . Here, as in ref.…”

Perturbative corrections to B → D form factors in QCD

“…Employing the massive quark propagator near the light cone from the background gluon field method [46][47][48] S(x, 0, m c ) = 0|T {c(x), c(0)}|0 (4.1)…”