Charge Asymmetry and Photon Energy Spectrum in the Decay $B_s \to l^+ l^- \gamma$

We consider the structure-dependent amplitude of the decay $B_s \to l^+ l^- \gamma$ $(l=e,\mu)$ in a model based on the effective Hamiltonian for $b \bar{s} \to l^+ l^-$ containing the Wilson coefficients $C_7,C_9$ and $C_{10}$. The form factors characterising the matrix elements $<\gamma | \bar{s} \gamma_\mu (1 \mp \gamma_5) b | \bar{B}_s>$ and $<\gamma | \bar{s} \sigma_{\mu\nu} (1 \mp \gamma_5) b | \bar{B}_s>$ are taken to have the universal form $f_V \approx f_A \approx f_T \approx f_{B_s} M_{B_s} R_s / (3 E_\gamma)$ suggested by recent work in QCD, where $R_s$ is a parameter related to the light cone wave function of the $B_s$ meson. Simple expressions are obtained for the charge asymmetry $A(x_\gamma)$ and the photon energy spectrum $d \Gamma/ d x_\gamma (x_\gamma = 2 E_\gamma/M_{B_s})$. The decay rates are calculated in terms of the decay rate of $B_s \to \gamma \gamma$. The branching ratios are estimated to be $Br(B_s \to e^+ e^- \gamma) = 2.0 \times 10^{-8}$ and $Br(B_s \to \mu^+ \mu^- \gamma) = 1.2 \times 10^{-8}$, somewhat higher than earlier estimates.


Introduction
The rare decay B s → l + l − γ is of interest as a probe of the effective Hamiltonian for the transition bs → l + l − , and as a testing ground for form factors describing the matrix elements γ|sγ µ (1 ∓ γ 5 )b|B s and γ|siσ µν (1 ∓ γ 5 )b|B s [1,2]. The branching ratio for B s → l + l − γ can be sizeable in comparison to the non-radiative process B s → l + l − , since the chiral suppression of the latter is absent in the radiative transition. We will be concerned mainly with the structure-dependent part of the matrix element, since the correction due to bremsstrahlung from the external leptons is small and can be removed by eliminating the end-point region s l + l − ≈ M 2 Bs . (For related studies of radiative B decays, we refer to the papers in Ref. [3].) Our objective is to calculate the decay spectrum of B s → l + l − γ using form factors suggested by recent work in QCD [4]. These form factors have the virtue of possessing a universal behaviour 1/E γ for large E γ , as well as a universal normalization. These features can be tested in measurements of B + → µ + νγ and B s → γγ. We derive simple formulae for the photon energy spectrum dΓ/dx γ , x γ = 2E γ /m Bs , and the charge asymmetry A(x γ ), defined as the difference in the probability of events with E + > E − and E + < E − , E ± being the l ± energies. This asymmetry is large over most of the x γ domain. Predictions are obtained for the branching ratios Br(B s → e + e − γ) and Br(B s → µ + µ − γ) which are somewhat higher than those estimated in previous literature [1,2].

Matrix Element and Differential Decay Rate
The effective Hamiltonian for the interaction bs → l + l − has the standard form [5] where P L,R = (1 ∓ γ 5 )/2 and q is the sum of the l + and l − momenta. For the purpose of this paper, we will neglect the small q 2 −dependent terms in C eff 9 , arising from one-loop contributions of four-quark operators, as well as long-distance effects associated with cc resonances. The Wilson coefficients in Eq.(1) will be taken to have the constant values To obtain the amplitude for B s → l + l − γ, one requires the matrix elements γ|sγ µ (1 ∓ γ 5 )b|B s and γ|siσ µν (1 ∓ γ 5 )b|B s . We parametrise these in the same way as in Ref. [1,2] The form factors f V , f A , f T and f ′ T are dimensionless, and related to those of Aliev et al [1] The matrix element forB s → l + l − γ can then be written as (neglecting terms of order m s /m b ) where Bs (In the coefficient of C 7 , we have approximated m b M Bs by M 2 Bs ). The Dalitz plot density in the energy variables E ± is where [1,2,6] spin It is convenient to introduce dimensionless variables . Taking x γ and ∆ as the two coordinates of the Dalitz plot, phase space is defined by In terms of x γ and ∆, the differential decay width takes the form The last term is linear in ∆ and produces an asymmetry between the l + and l − energy spectra.
We will derive from Eq. (10) two distributions of interest: (i) The charge asymmetry A(x γ ) defined as To proceed further, we must introduce a model for the form factors which appear in the functions A 1,2 and B 1,2 defined in Eq. (5).

Model for Form Factors
First of all, we note that the form factors f T and f ′ T defined in Eq. (3) are necessarily equal, by virtue of the identity This was pointed out by Korchemsky et al [4]. We therefore have to deal with three independent form factors f V , f A and f T . These have been computed in Ref. [4] using perturbative QCD methods combined with heavy quark effective theory. For the vector and axial vector form factors of the radiative decay B + → l + νγ, and their tensor counterpart, defined as in Eq. (3), these authors obtain the remarkable result where R is a parameter related to the light-cone wave-function of the B meson, with an order of magnitude R −1 ∼Λ = M B − m b , where the binding energyΛ is estimated to be between 0.3 and 0.4 GeV. Applying the same reasoning to the form factors forB s → l + l − γ, we conclude that In what follows, we will neglect the term Q b /m b , and approximate the form factors by whereΛ s = M Bs − m b will be taken to have the nominal value 0.5 GeV. Several of our results will depend only on the universal form f V,A,T (E γ ) ∼ 1/E γ , independent of the normalization. As pointed out in [4], a check of the behaviour f V,A ∼ 1/E γ in the case of B + → µ + νγ is afforded by the photon energy spectrum, which is predicted to be In the case of the reaction B s → l + l − γ, the normalization of the tensor form factor f T (E γ ) at E γ = M B /2 (i.e. x γ = 1) can be checked by appeal to the decay rate of B s → γγ. To see this connection, we note that the matrix element of B s → γ(k, ǫ) + γ(k ′ , ǫ ′ ) can be obtained from that of B s → l + l − γ by putting C 9 = C 10 = 0, and replacing the factor (ef T C 7 /q 2 ) (lγ µ l) by f T (x γ = 1)ǫ ⋆′ µ . This yields the matrix element The result for A ± coincides with that obtained in Refs. [7,8,9] Λs . ( In Ref. [8,9], the role of the parameter Λ s is played by the constituent quark mass m s . ) Thus the decay width of B s → γγ, serves as a test of the normalization factor f T (x γ = 1). We remark, parenthetically, that the calculation of B s → γγ, based on an effective interaction for b → sγγ, produces the amplitudes A + and A − given in Eq. (18) in the limit of retaining only the 'reducible' diagrams related to the transition b → sγ. Inclusion of 'irreducible' contributions like bs → cc → γγ introduces a correction term in A − causing the ratio |A + /A − | to deviate from unity. Estimates in Ref. [7,8] yield values for this ratio between 0.75 and 0.9. The branching ratio Br(B s → γγ) is estimated at 5 × 10 −7 , with an uncertainty of about 50%.
Having specified our model for the form factors f V (x γ ), f A (x γ ) and f T (x γ ), we proceed to present results for the spectrum and branching ratio of B s → l + l − γ.

Charge Asymmetry
With the assumption of universal form factors f V = f A = f T ∼ 1 xγ , the asymmetry A(x γ ) in Eq. (11) assumes the simple form This is plotted in Fig.1, and is clearly large and negative over most of the x γ domain, changing sign at x γ = 1 + 2C 7 C 9 . (A negative asymmetry corresponds to l − being more energetic, on average, than l + in the decayB s (= bs) → l + l − γ.) The average charge asymmetry is and has the numerical value A e = −0.28 , A µ = −0.47 for the modes l = e, µ, the difference arising essentially from the end-point region x γ ≈ 1 − 4 r.

Photon Energy Spectrum
With the form factors of Eq. (16), the photon energy spectrum simplifies to where the constant factor N is defined after Eq. (10). It is expedient to write this distribution in terms of the decay rate ofB s → γγ. We then obtain the prediction The first factor (in curly brackets { }) is the QED result expected if the decaȳ B s → l + l − γ is interpreted as a Dalitz pair reactionB s → γγ ⋆ → γl + l − , without form factors. The factor (1/x γ ) 2 results from the universal behaviour f V,A,T ∼ 1/x γ given in Eq. (10), while the last factor is the electroweak effect associated with the coefficients η 9 = C 9 /(2C 7 ) and η 10 = C 10 /(2C 7 ). This distribution is plotted in Figs.  2 and 3, where the QED result is shown for comparison.

Rates and Branching Ratios
From the photon spectrum given in Eq. (23), we derive the 'conversion ratios' The numerical values are R e = 4.0% and R µ = 2.3%. These are to be contrasted with the QED result given by which yields R e (QED) = 2.3%, R µ (QED) = 0.67%. The absolute branching ratios ofB s → l + l − γ, obtained by taking Br(B s → γγ) = 5 × 10 −7 [7,8] are Our results for the average charge asymmetry A l , the conversion ratios R l and the branching ratios are summarized in Table 1.

Comments
(i) The branching ratios calculated by us are somewhat higher than those obtained in previous work [1,2], which used a different parametrization of the form factors f V , f A , f T , f T ′ based on QCD sum rules [1] and light-front models [2]. In particular these parametrizations do not satisfy the relation f T = f ′ T which, as noted in [4], follows from the identity σ µν = i 2 ǫ µναβ σ αβ γ 5 .
(ii) Our predictions for the charge asymmetry A and the conversion ratio Γ(B s → l + l − γ)/Γ(B s → γγ) are independent of the parameterΛ s which appears in the form factor in Eq. (16). The branching ratios in Table 1 assume Br(B s → γγ) = 5×10 −7 , and can be rescaled when data on this channel are available.
(iii) A full analysis of the decayB s → l + l − γ requires inclusion of the bremsstrahlung amplitude corresponding to photon emission from the leptons in B s → l + l − . This contribution is proportional to f Bs m l and affects the photon energy spectrum in the small x γ region. We have calculated the corrected spectrum for B s → l + l − γ, following the procedure in [11], and the result is shown in Fig. 4 for the case l = µ. As anticipated, the correction is limited to small x γ , and can be removed by a cut at small photon energies.
(iv) The QCD form factors in Eq. (16) are valid up to corrections of order (Λ QCD /E γ ) 2 .
In the small x γ region, arguments based on heavy hadron chiral perturbation theory suggest form factors dominated by the B ⋆ pole with the appropriate quantum numbers, for example, Defining M B ⋆ s − M Bs = ∆M, this form factor has the behaviour f V (x γ ) ∼ 1 xγ+δ , with δ ≈ 2∆M/M Bs ≈ 0.02. We have investigated the effect of replacing the QCD form factor of Eq. (16) by a different universal form f V,A,T (x γ ) = f Bs /(3Λ s (x γ + δ)), and found only minor changes in the numbers given in Table 1. In general, one must expect some distortion in the spectrum at low x γ , compared to that shown in Figs. 1-4. (v) We will examine separately the predictions for A(x γ ) and dΓ/dx γ in the reaction B s → τ + τ − γ, in which the bremsstrahlung part of the matrix element plays a significant role [11]. We will consider also refinements due to the q 2 −dependent term in C eff 9 , and the effects of cc resonances. In view of their clear signature, non-negligible branching ratios and interesting dynamics, the decays B s → l + l − γ could form an attractive domain of study at future hadron colliders.