Radiative Baryonic $B$ Decays

We study the structure-dependent contributions to the radiative baryonic $B$ decays of $B \to {\bf B}{\bf \bar B'}\gamma$ in the standard model. We show that the decay branching ratios of $Br(B \to {\bf B}{\bf \bar B'}\gamma)$ are $O(10^{-7})$, which are larger than the estimated values of $O(10^{-9})$ induced from inner bremsstrahlung effects of the corresponding two-body modes. In particular, we find that $Br(B^- \to \Lambda \bar p \gamma)$ is around $1 \times 10^{-6}$, which is close to the pole model estimation but smaller than the experimental measurement from BELLE.

The radiative baryonic B decays of B → BB ′ γ are of interest since they are threebody decays with two spin-1/2 baryons (B and B ′ ) and one spin-1 photon in the final states. The rich spin structures allow us to explore various interesting observables such as triple momentum correlations to investigate CP or T violation [1,2]. Moreover, since these radiative decays could dominantly arise from the short-distance electromagnetic penguin transition of b → sγ [3] which has been utilized to place significant constraints on physics beyond the Standard Model (SM) [4,5], they then appear to be the potentially applicable probes to new physics.
There are two sources to produce radiative baryonic B decays. One is the inner bremsstrahlung (IB) effect, in which the radiative baryonic B decays of B → BB ′ γ are from their two-body decay counterparts of B → BB ′ via the supplementary emitting photon attaching to one of the final baryonic states. Clearly, the radiative decay rates due to the IB contributions are suppressed by α em comparing with their counterparts. According to the existing upper bounds of B → BB ′ , given by [6,7,8] Br(B 0 → pp) < 2.7 × 10 −7 (BABAR) , Br(B 0 → ΛΛ) < 7.9 × 10 −7 (BELLE), one finds that Unfortunately, the above branching ratios are far from the present accessibility at the B factories of BABAR and BELLE. However, the other source, which is the structure-dependent (SD), is expected to enhance the decays of Br(B → BB ′ γ), such as B → Λpγ arising from b → sγ [1,9,10]. With the large branching ratio of b → sγ [11,12] in the range of 10 −4 we expect that Br(B − → BB ′ γ) could be as large as Br(B − → BB ′ ). In this report, we shall concentrate on the SD contributions to Br(B → BB ′ γ).
To start our study, we must tackle the cumbersome transition matrix elements in B → BB ′ . As more and more experimental data on three-body decays [13,14,15] in recent years, the theoretical progresses are improved to resolve the transition matrix element problems.
One interesting approach is to use the pole model [16,17] through the intermediated particles and another one is to rely on the QCD counting rules [18,19,20] by relating the transition In this paper, we handle the transition matrix elements according to the QCD counting rules.
We begin with the decay of B − → Λpγ. As depicted in Fig. 1, in the SM the relevant Hamiltonian due to the SD contribution for B − → Λpγ is with the tensor operator where V tb V * ts and c ef f 7 are the CKM matrix elements and Wilson coefficient, respectively, and the decay amplitude is found to be where we have used the condition m b ≫ m s such that the terms relating to m s are neglected.
We note that Eq. (5) is still gauge invariant.
In order to solve the encountered transition matrix elements in Eq. (5), we write the most general form where p = p B − p Λ − pp and a i (c i ) (i = 1, ..., 3) are form factors.
To find out the coefficients a i (c i ) in Eq. (6), we invoke the work of Chua, Hou and Tsai in Ref. [20]. In their analysis, three form factors F A , F P and F V are used to describe B → BB ′ transitions based on the QCD counting rules [18], that require the form factors to behave as inverse powers of t = (p B + pB′) 2 . The detail discussions can be referred to Refs. [19,20].
In this paper, we shall follow their approach. The representations of the matrix elements for the B − → pp transition are given by [20] pp|ū with a derived relation F S = F P . In terms of the approach of [19,20], those of the B − → Λp transition are given by where the form factors related to those of B − → pp in Eq. (7) are shown as The three form factors F A , F V and F P can be simply presented as [19,20] where C i (i = A, V, P ) are new parametrized form factors, which are taking to be real.
limit, the parameters a i (c i ) in Eq. (6) are associated with the scalar and pseudo-scalar matrix elements defined in Eq. (8). As a result, we get that The amplitude in Eq. (5) then becomes with three unknown form factors F Λp A , F Λp V and F Λp P . We note that the terms corresponding to a 2 disappear due to the fact of ε · p = 0. Even though c 2 can only be determined by experimental data, according to QCD counting rules, c 2 needs an additional 1/t than c 1 to flip the helicity, so that it is guaranteed to give a small contribution and can be neglected.
After summing over the photon polarizations and baryon spins, from Eq. (12), the decay rate of Γ is given by the integration of where It is important to note that, since the penguin-induced radiative B decays are associated with axial-vector currents shown in Eq. (5), we have used [21] λ=1,2 where n = (1, 0, 0, 0), to sum over the photon polarizations instead of the direct replacement of λ=1,2 ε * λ µ ε λ ν → −g µν which is valid in the QED-like theory due to the Ward identity. For the numerical analysis of the branching ratios, we take the effective Wilson coefficient fitting, we need 2 degrees of freedom (DOF) by ignoring the C P term since its contribution is always associated with one more 1/t over C A and C V ones, as seen in Eq. (10). We will take a consistent check in the next paragraph to this simplification. To illustrate our results, we fix the color number N C = 3 and weak phase γ = 54.8 • . The input experimental data and numerical values are summarized in Table I.  In Ref. [1], it was suggested that the reduced energy release can make the branching ratios of three-body decays as significant as their counterparts of two-body modes or even larger, and one of the signatures would be baryon pair threshold effect [1,20]. In Fig. 2, from Eq. (13) we show the differential branching ratio of dBr(B − → Λpγ)/dm Λp vs. m Λp representing the threshold enhancement around the invariant mass m Λp = 2.05 GeV, which is consistent with Fig. 2 in Ref. [24] of the BELLE result. Around the threshold, the baryon pair contains half of the B meson energy while the phone emitting back to back to the baryon pair with another half of energy which explains the peak at E γ ∼ 2 GeV in Fig. 3 of Ref. [24]. Such mechanism is similar to the two-body decays so that factorization method works [1] even in the three-body decays.
To discuss other radiative baryonic B − decays, we give form factors by relating them to F V,A,P in the B − → pp transition similar to the case of B − → Λpγ as follows: To calculate the branching ratio of B → BB ′ γ, we can use the formula in Eq. (13) by replacing Λ andp by B andB ′ , respectively. The two sets of predicted values for B → BB ′ γ with and without C P are shown in Table II, respectively. As a comparison, we also list the work of the pole model approach by Cheng and Yang [9] in the table. We note that, in  As seen in Table II, both our results and those of the pole model satisfy the relations because of the SU(3) symmetry. In the pole model, the decay branching ratios of B − → Λpγ and B − → Ξ 0Σ− γ are found to be large, around 1.2 × 10 −6 , since ther are intermediated through Λ b and Ξ b , which correspond to large coupling constants g Λ b →B − p and g Ξ 0 b →B − Σ + , respectively. However, in our work, the branching ratio of B − → Λpγ is about three times larger than that of B − → Ξ 0Σ− γ, which is O(10 −7 ). Regardless of these differences, both two methods are within the experimental data allowed ranges, such as those of from CLEO [25] and Br(B − → Σ 0p γ) < 3.3 × 10 −6 from BELLE [24]. due to the IB effects of their two-body counterparts. In particular, we have found that Br(B − → Λpγ) is (1.16 ± 0.31) × 10 −6 and (0.92 ± 0.20) × 10 −6 with and without C P , respectively, which are consistent with the pole model prediction [9] but smaller than the experimental data from BELLE [24]. More precise measurements are clearly needed.