Estimates of Long - distance Contributions to the B s → γγ Decay

We present first calculations of new long-distance contributions to Bs → γγ decay due to intermediate Ds and D ∗ s meson states. The relevant γ vertices are estimated using charge couplings and transition moment couplings. Within our uncertainties, we find that these long-distance contributions could be comparable to the known short-distance contributions. Since they have different Cabibbo-KobayashiMaskawa matrix-element factors, there may be an interesting possibility of observing CP violation in this decay. CERN-TH/98-76 April 1998 [email protected] [email protected] A new era of experiments on rare B decays is about to dawn. The large data sets already obtained at LEP and CESR will be dwarfed by those provided by the ee B factories, and by the hadron experiments HERA-B, CDF, D0 and LHC-B. Among the rare B decays with particularly clean experimental signatures is Bs → γγ, whose branching ratio presently has the experimental upper limit B(Bs → γγ) < 1.48×10−4 [1]. Higher-sensitivity measurements of Bs decays are not among the highest priorities of the ee B factories, whose physics programmes are focussed initially on searches for CP violation in Bd decays, but the hadron experiments cannot avoid being sensitive to B(Bs → γγ) to levels several orders of magnitude below the present experimental upper limit [1]. The lowest-order short-distance contributions to the Bs → γγ decay arise from two sets of graphs: (i) box diagrams, and (ii) triangle diagrams with an external photon leg. These have been calculated [2–4], and yield, for mt ≈ 175 GeV, B(Bs → γγ) ∼ 3.8 × 10 , the next generation of experiments. Interestingly, the branching fraction can be enhanced substantially in extensions of the Standard Model such as a generic 2-Higgs scenario [5]. Emboldened by the fact that the short-distance QCD corrections are not too big [5–7], one may then even attempt to use such data as may be forthcoming as probes for new physics. However, it is not immediately obvious that this decay is dominated by such short-distance contributions. Estimates have been made of some long-distance contributions to the decay amplitudes, for example via intermediate charmonium states J/ψ and ψ [8]. These were found to be very small, indicating that the short-distance contribution might be correct in order of magnitude. But, before concluding this to be the case, one must examine other possible long-distance contributions. In this paper we study such contributions due to intermediate Ds and D ∗ s states via the diagrams shown in Figures 1 and 2. These include loops ofDs mesons alone, loops ofD ∗ s mesons alone, and diagrams involving radiative Ds → D s transitions. The vertices are estimated using data on B → DD,DD and DD decays, the electromagnetic charge couplings of the Ds and D s , and phenomenological estimates of the D ∗ s → Ds + γ decay rate for the inelastic transition-matrix element. The imaginary part of the diagrams can be calculated easily while the determination of the real part requires the use of dispersion relations. We find that these long-distance contributions to Bs → γγ decay may be comparable to the short-distance contributions calculated previously, within the considerable uncertainties inherent to the phenomenological inputs we use. We find no evidence that B(Bs → γγ) decay should occur at a rate very close to the present experimental upper limit, but even this possibility cannot be ruled out. Since the shortand long-distance contributions have different Cabibbo-Kobayashi-Maskawa matrix-element factors, the fact that they may be of comparable magnitudes offers the interesting possibility of observing CP violation in this decay mode. However, a detailed exploration of this possibility lies beyond the scope of this paper. At the quark level, the operators responsible for the B(Bs → γγ) decay are b̄γ5sFμνF μν Over a wide range of the top-quark mass, the partial width grows linearly with mt [4].