Abstract
Motivated by the first search for the rare charged-current B decay to four leptons, ell bar{nu }_ell ell ^{(prime )} bar{ell }^{(prime )}, we calculate the decay amplitude with factorization methods. We obtain the Brightarrow gamma ^* form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in Lambda _mathrm{QCD}/m_b to next-to-leading order in alpha _s, and at mathcal {O}(alpha _s^0) at next-to-leading power. Our calculations predict branching fractions of a few times 10^{-8} in the ell ^{(prime )} bar{ell }^{(prime )} mass-squared bin up to q^2=1~GeV^2 with n_+q>3~GeV. The branching fraction rapidly drops with increasing q^2. An important further motivation for this investigation has been to explore the sensitivity of the decay rate to the inverse moment lambda _B of the leading-twist B meson light-cone distribution amplitude. We find that in the small-q^2 bin, the sensitivity to lambda _B is almost comparable to B^- rightarrow mathrm {ell }^- bar{nu }_{mathrm {ell }}gamma when lambda _B is small, but with an added uncertainty from the light-meson intermediate resonance contribution. The sensitivity degrades with larger q^2.
Highlights
We obtain the B → γ ∗ form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in QCD/mb to next-to-leading order in αs, and at O(αs0) at next-to-leading power
The radiative decay B− → −νγ has been extensively studied in the context of QCD factorization (QCDF) [1,2,3,4,5] when the energy of the photon Eγ is large compared to the scale of the strong interaction QCD
We focus on the kinematic region, where the γ ∗, respectively the lepton pair, has large energy but small invariant mass q2 6 GeV2.1 The four-lepton decays have not been observed up to now, but the LHCb experiment [8] established an upper bound of Br (B+ → μ+νμμ−μ+) < 1.6 · 10−8 on the branching fraction of the muonic mode under the assumption that the smaller of the two possible μ+μ− invariant masses is below 980 MeV, which is close to, somewhat below, theoretical expectations [9,10]
Summary
The radiative decay B− → −νγ has been extensively studied in the context of QCD factorization (QCDF) [1,2,3,4,5] when the energy of the photon Eγ is large compared to the scale of the strong interaction QCD. ∞ 0 dω φ+B (ω)/ω of φ+B (ω), the leadingtwist B meson light-cone distribution amplitude (LCDA) This strategy is difficult to implement in the hadronic B experiment LHCb, since the photon in the radiative decay cannot be reconstructed. Since the four-lepton final state is produced from a virtual W boson and photon, an extension of previous calculations is required to B → γ ∗ form factors. With these kinematic restrictions the differential branching fraction of the four-lepton decay is expressed, at LP, in terms of generalized inverse moments of the B meson LCDA, which can be related to λB.
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