Electroweak Effects in the Double Dalitz Decay B l+l l +l s − ′ ′− 3 → 0 0 Yusuf Dinc¸er and L.M. Sehgal 2 ∗ † n Institute of Theoretical Physics, RWTH Aachen a J D-52056 Aachen, Germany 7 2 2 v Abstract 6 5 We investigate the double Dalitz decays B l+l l+l on the basis of 0 s − ′ ′− → 1 the effective Hamiltonian for the transition bs¯ l+l , and universal form − 0 factors suggested by QCD. The correlated mass→spectrum of the two lepton 3 pairs in the decay B e+e µ+µ is derived in an efficient way, using a 0 s − − → / QEDresultfor meson decays mediated by two virtualphotons: B γ γ h s → ∗ ∗ → p e+e−µ+µ−. A comment is made on the correlation between the planes of the p- two lepton pairs. The conversion ratios ρlll′l′ = Γ(BsΓ→(Bls+l−γlγ′+)l′−) are estimated e to be ρ = 3 10 4,ρ = 9 10 5 and ρ →= 3 10 5, and are h eeee × − eeµµ × − µµµµ × − enhanced relative to pure QED by 10 30%. : v − i X r a ∗e-mail:[email protected] †e-mail:[email protected] 1 1 Introduction In a recent paper [1] we investigated the decay B l+l γ(l = e,µ), using the s − → effective Hamiltonian for the transition bs¯ l+l , and obtained a prediction for the − → conversion ratio Γ(B l+l γ) s − ρ = → (1) ll Γ(B γγ) s → in terms of the Wilson coefficients C ,C and C . An essential ingredient of the 7 9 10 calculation was the use of a universal form factor characterising the matrix elements γ s¯iσ (1+γ )b B¯ and γ s¯γ (1 γ )b B¯ , as suggested by recent work [2] on µν 5 s µ 5 s | | | ± | QCD in the heavy quark limit (m Λ ). It was found that the ratio ρ b QCD ll (cid:10) (cid:11) (cid:10) ≫ (cid:11) was significantly higher than one would expect from a QED calculation of Dalitz pair production B γ γ l+l γ, the difference reflecting the presence of the s ∗ − → → short-distance coefficients C ,C , as well as the universal 1/E behaviour of the 9 10 γ QCD-motivated form factor. The purpose of the present paper is to apply the same considerations to the “double Dalitz decay” B l+l l+l , to determine whether s − ′ ′− → there is similar enhancement of the double conversion ratio Γ(B l+l l+l ) s − ′ ′− ρlll′l′ = → , (2) Γ(B γγ) s → compared to what one would obtain from the QED process B γ γ l+l l+l . s ∗ ∗ − ′ ′− → → We examine also the correlation in the invariant mass of the two lepton pairs, and the nature of the angular correlation between the l+l and l+l planes, which is a − ′ ′− crucial test of the B γγ vertex. s → 2 Matrix Element and Invariant Mass Spectrum We begin with the effective Hamiltonian for bs¯ l+l [3] − → αG = FV V Ceff(s¯γ P b)¯lγ l Heff √2π tb t∗s 9 µ L µ (cid:26) ¯ + C (s¯γ P b)lγ γ l (3) 10 µ L µ 5 C 2 7s¯iσ qν(m P +m P )b¯lγ l µν b R s L µ − q2 (cid:27) where P = (1 γ )/2 and q is the sum of the l+ and l momenta. Ignoring small L,R 5 − ∓ q2 dependent corrections in Ceff, the values of the Wilson coefficients are − 9 C = 0.315,C = 4.334,C = 4.624. (4) 7 9 10 − − 2 Then, as shown in [4], the matrix element for B¯ l+l γ has the form s − → αG 1 (B¯ l+l γ) = FeV V M s → − √2π tb t∗sM · Bs ǫ ǫ νqρkσ(A ¯lγ l+A ¯lγ γ l) (5) µνρσ ∗ 1 µ 2 µ 5 · ¯ ¯ + i(ǫ (k q) (ǫ q)k )(B lγ l+B lγ γ l) (cid:2) ∗ ∗ µ 1 µ 2 µ 5 · − · where (cid:3) M2 A = C f +2C Bsf , 1 9 V 7 q2 T A = C f , 2 10 V (6) M2 B = C f +2C Bsf , 1 9 A 7 q2 T′ B = C f . 2 10 A Theformfactorsf ,f ,f ,f ,definedinRef. [1],willbetakentohavetheuniversal V A T T′ form 1f 1 Λ2 f = f = f = f = Bs + ( QCD), (7) V A T T′ 3 Λ x O E2 s γ γ predicted in the heavy quark approximation (m Λ ,m m ) in QCD [2]. b QCD b s Here, Λ¯ = m m 0.5GeV,x = 2E /M =≫1 q2/M2 ,≫and f 200MeV s Bs − b ≈ γ γ Bs − Bs Bs ≈ is the B decay constant. The essential feature for our purpose will be the universal s 1/x behaviour, the absolute normalization dropping out in the calculation of the γ conversion ratio. (Corrections to universality are discussed in Ref. [5]). To obtain the matrix element for B l+l l+l we treat the second lepton pair s − ′ ′− → l+l as a Dalitz pair associated with internal conversion of the photon in B ′ ′− s → l+l γ. From this point on, we will specialise to the final state e+e µ+µ , consisting − − − of two different lepton pairs. This avoids the complications due to the exchange diagram that occurs in dealing with two identical pairs. The matrix element then has the structure e (B¯ e+e µ+µ ) (a (q2)Lµ(q ,q )+a (q2)Lµ(q ,q ))Lν (k ,k ) M s → − − ∼ k2 + + 1 2 − − 1 2 em 1 2 (8) [ǫ qρkσ +i(g k q k q )] µνρσ µν µ ν · · − where k and q are the four-momenta of the two lepton pairs, k2 and q2 being the corresponding invariant masses. The currents L and L are given by em ± Lµ(q ,q ) = u¯(q )γµ(1 γ )v(q ), 1 2 1 5 2 ± ± (9) Lµ (k ,k ) = u¯(k )γµv(k ). em 1 2 1 2 3 where k +k = k,q + q = q. The coefficients a (q2) are related to those in Eq. 1 2 1 2 ± (6) by a (q2) = A (q2) A (q2), (10) 1 2 ± ± where we have used the fact that for universal form factors, B = A . 1,2 1,2 At this stage, it is expedient to compare the matrix element (8) with the ma- trix element for double Dalitz pair production in QED. We will make use of the recent analysis of Barker et al.[6], who have studied the reaction Meson γ γ ∗ ∗ → → l+l l+l , using a vertex for Meson γγ that is a general superposition of scalar − ′ ′− → and pseudoscalar forms, the matrix element being e e = const. Lµ (q ,q )Lν (k ,k ) MBarker · k2q2 em 1 2 em 1 2 (11) [ξ ǫ qρkσ +ξ (g k q k q )] . P µνρσ S µν µ ν · · − The coefficients ξ and ξ are normalized so that ξ 2+ ξ 2 = 1. (In Ref. [6] they P S P S | | | | are denoted by ξ = cosζ,ξ = sinζeiδ.) P S From this matrix element, Barker et al. have derived the correlated invariant mass spectra for the decay into e+e µ+µ (ignoring form factors at the Mγ γ − − ∗ ∗ vertex) 1 d2Γ 2α2 λ λ λ = 12 34 (3 λ2 )(3 λ2 ) Γ dx dx 9π2 w2 − 12 − 34 (cid:20) γγ (cid:18) 12 34(cid:19)(cid:21)Barker (cid:18) (cid:19) (12) 3w2 ξ 2λ2 + ξ 2(λ2 + ) . P S · | | | | 2 (cid:20) (cid:21) The variables entering the above formula are defined as follows: x = (q +q )2/M2 = q2/M2, 12 1 2 x = (k +k )2/M2 = k2/M2, 34 1 2 m2 1 x = x = 1 , 1 2 M2 x 12 m2 1 x = x = 3 , 3 4 M2 x 34 (13) z = 1 x x , 12 34 − − λ = (1 x x )2 4x x , 12 1 2 1 2 − − − λ = p(1 x x )2 4x x , 34 3 4 3 4 − − − w2 = 4x x , p12 34 λ = √z2 w2. − 4 Here m and m denote the masses of the electron and muon, and M the mass of the 1 3 decaying meson. The phase space in the variables x and x is defined by x0 < 12 34 34 x < (1 √x )2,x0 < x < (1 √x )2, where x0 = 4m2/M2,x0 = 4m2/M2. 34 − 12 12 12 − 34 12 1 34 3 We can now adapt the QED result (12) to the process B e+e µ+µ , by s − − → comparing the matrix element (11) with that in Eq. (8). The essential observation is that in the approximation of neglecting lepton masses, the vector and axial vector parts of the chiral currents Lµ contribute equally and independently to the invariant ± mass spectrum. In addition, the matrix element for B decay corresponds to the s QED matrix element considered by Barker et al., if we put ξ = 1/√2,ξ = i/√2. P S This allows us to obtain the invariant mass spectrum for the double Dalitz decay B e+e µ+µ in electroweak theory: s − − → 1 dΓ 1 1 = (η + )2 +η2 + (η + )2 +η2 Γ dx dx 9 x 10 9 x 10 (cid:20) γγ (cid:18) 12 34(cid:19)(cid:21)EW (cid:26)(cid:20) 12 (cid:21) (cid:20) 34 (cid:21)(cid:27) (14) x2 x2 1 dΓ 12 34 F(x ,x ) 2 , · x2 +x2 | 12 34 | · Γ dx dx 12 34 (cid:20) γγ (cid:18) 12 34(cid:19)(cid:21)QED where 1 dΓ α2 λ λ λ 3 = 12 34 (3 λ2 )(3 λ2 )(2λ2 + w2). (15) Γ dx dx 9π2 w2 − 12 − 34 2 (cid:20) γγ (cid:18) 12 34(cid:19)(cid:21)QED Here we have used the abbreviation η = C /(2C ) and η = C /(2C ), introduced 9 9 7 10 10 7 in Ref. [1]. The electroweak formula (14) reduces to the QED result in the limit η = η = 0,F(x ,x ) = 1. 9 10 12 34 The form factor F(x ,x ) is chosen to have the universal form 12 34 1 1 F(x ,x ) = . (16) 12 34 (1 x )(1 x ) 12 34 − − (a possible normalization factor drops out in the calculation of the conversion ratio). This is a plausible (but not unique) generalization of the universal QCD form factor 1/(1 x ) that occurs in the single Dalitz pair process B e+e γ. 12 s − − → In Fig. 1 we plot the correlated invariant mass spectrum for B e+e µ+µ s − − → in electroweak theory. The ratio of the electroweak and QED spectra is shown in Fig 2, and indicates the effects associated with the coefficients η and η , and the 9 10 form factor F(x ,x ). One notes a slight depression in the region x = 2C7 or 12 34 12 − C9 x = 2C7, connected with the vanishing of the term (C + 2C7)2 or (C + 2C7)2. 34 − C9 9 x12 9 x34 There is also a general enhancement for increasing values of x ,x , because of the 12 34 5 form factor (16). If the form factor F(x ,x ) is set equal to one, the ratio of the 12 34 electroweak and QED spectra has the structure plotted in Fig. 3, illustrating the effects which depend specifically on the electroweak parameters η ,η . 9 10 The absolute value of the conversion ratio ρ is obtained by integrating eeµµ ( 1 dΓ/dx dx ) over the range of x and x . In the QED case, this ratio is Γγγ 12 34 EW 12 34 conveniently expressed in terms of the integrals I introduced in Ref. [6]: 1...6 2 λ3 λ3 λ3 I = dx dx 12 34 , 1 3 12 34 w2 Z Z 2 λ3 λ3 λz2 I = dx dx 12 34 , 2 3 12 34 w2 Z Z 4 λ3 λ3 λ2z I = dx dx 12 34 , 3 12 34 3 w2 Z Z (17) λ λ λ3 I = dx dx 12 34 (3 λ2 λ2 ), 4 12 34 w2 − 12 − 34 Z Z λ λ λz2 I = dx dx 12 34 (3 λ2 λ2 ), 5 12 34 w2 − 12 − 34 Z Z 1 I = dx dx λ λ λ(3 λ2 )(3 λ2 ). 6 6 12 34 12 34 − 12 − 34 Z Z These integrals are listed in Table 1 (where, for completeness, we have also given the values for the final states ee¯ee¯and µµ¯µµ¯). These integrals allow us to calculate the QED double conversion ratio α2 (ρ ) = (I +I +2(I +I +I )) eeµµ QED 6π2 1 2 4 5 6 (18) = 7.6 10 5. − × The corresponding result for electroweak theory, based on the differential decay rate 6 (14), can be expressed in terms of the integrals 2 λ3 λ3 λ3 I˜ = dx dx 12 34 G(x ,x ), 1 3 12 34 w2 12 34 Z Z 2 λ3 λ3 λz2 I˜ = dx dx 12 34 G(x ,x ), 2 3 12 34 w2 12 34 Z Z 4 λ3 λ3 λ2z I˜ = dx dx 12 34 G(x ,x ), 3 3 12 34 w2 12 34 Z Z (19) λ λ λ3 I˜ = dx dx 12 34 (3 λ2 λ2 )G(x ,x ), 4 12 34 w2 − 12 − 34 12 34 Z Z λ λ λz2 I˜ = dx dx 12 34 (3 λ2 λ2 )G(x ,x ), 5 12 34 w2 − 12 − 34 12 34 Z Z 1 I˜ = dx dx λ λ λ(3 λ2 )(3 λ2 )G(x ,x ). 6 6 12 34 12 34 − 12 − 34 12 34 Z Z The factor G(x ,x ) in the integrand of Eq.(19) contains the effects of the elec- 12 34 troweak coefficients η ,η and the universal form factor F(x ,x ): 9 10 12 34 2 2 1 1 G(x ,x ) = η + +η2 + η + +η2 12 34 9 x 10 9 x 10 · ("(cid:18) 12(cid:19) # "(cid:18) 34(cid:19) #) (20) x2 x2 12 34 F(x ,x ) 2. · x2 +x2 ·| 12 34 | 12 34 The integrals I˜,...,I˜ are given in Table 2. The electroweak conversion ratio, 1 6 analogous to the QED result (18), is given by α2 (ρ ) = I˜ +I˜ +2(I˜ +I˜ +I˜) eeµµ EW 6π2 1 2 4 5 6 (21) = 9.1 (cid:16)10 5. (cid:17) − × In comparison to the QED result (18), the double conversion ratio for B ee¯µµ¯ in → electroweak theory is enhanced by 20%. ∼ A calculation of the spectra for the channels ee¯ee¯ and µµ¯µµ¯ is complicated by interference between the exchange and direct amplitudes. The conversion ratio for these channels takes the form ρ = ρ +ρ +ρ , (22) 1 2 12 7 where ρ and ρ denote the “direct” and “exchange” contribution, and ρ an in- 1 2 12 terference term. Numerical calculations of the decays π0 e+e e+e and K − − L → → e+e e+e suggest thatρ issmall andρ ρ . Thus a roughestimate of thedouble − − 12 1 2 ≈ conversion ratio can be obtained using the formula (21), with an extra factor (1) 2 4 · where (1) is the statistical factor for two identical fermion pairs, and 2 comes from 4 adding direct and exchange contributions. This yields, using the numbers in Table 2 (ρ ) 2.9 10 4, ee¯ee¯ EW − ≈ × (23) (ρ ) 2.8 10 5. µµ¯µµ¯ EW − ≈ × For comparison, the QED results, using Table 1, are (ρ ) 2.7 10 4, ee¯ee¯ QED − ≈ × (24) (ρ ) 2.2 10 5. µµ¯µµ¯ QED − ≈ × Thus the enhancement in the case of ee¯ee¯ is 10% and that in µµ¯µµ¯ about 30%. ∼ Combining (21) and (23), the ratio of the channels ee¯ee¯,ee¯µµ¯ and µµ¯µµ¯ is approx- imately ee¯ee¯: ee¯µµ¯ : µµ¯µµ¯ (25) = 3 : 1 : 0.3 To obtain the absolute branching ratios, we note that the decay rate of B¯ γγ, s → derived from the effective Hamiltonian (3), involves the Wilson coefficient C and 7 the universal form factor f (x = 1) (see Eq. (7)). Using nominal values for f T γ Bs and Λ¯ , and evaluating C at the renormalization scale µ = m , Ref. [7] finds s 7 b Br(B γγ) = 1.23 10 6. Using this as a reference value, we obtain: s − → × Br(B¯ ee¯ee¯) = 3.6 10 10, s − → × Br(B¯ ee¯µµ¯) = 1.1 10 10, (26) s − → × Br(B¯ µµ¯µµ¯) = 3.5 10 11. s − → × ¯ 3 Correlation of e+e and µ+µ planes in B − − s → ¯ ¯ eeµµ One of the distinctive features of the electroweak B¯ γγ matrix element is that s → the coefficients ξ and ξ (normalized to ξ 2 + ξ 2 = 1) are given by ξ = S P S P S | | | | 8 i and ξ = 1 . The equality ξ 2 = ξ 2 leads to the simplification that the √2 P √2 | S| | P| factor ξ 2λ2+ ξ 2(λ2+ 3w2) appearing in the spectrum (12) could be written as | P| | S| 2 1[2λ2 + 3w2] in going over to the electroweak case (Eq.(15)). A further interesting 2 2 consequence is the distribution of the angle φ between the e+e and µ+µ planes in − − B¯ ee¯µµ¯. Generalising the QED result given in Ref. [6] to the electroweak case, s → the correlation in φ is given by 1 dΓ ee¯µµ¯ α2 = I˜ sin2φ+I˜ cos2φ+(I˜ +I˜ +I˜) . (27) Γ dφ 6π3 1 2 4 5 6 (cid:18) γγ (cid:19)EW h i The fact that I˜ is so close to I˜ means that the spectrum dΓ/dφ is essentially 2 1 independent of φ. Furthermore, the fact that arg(ξ /ξ ) = π/2 refelects itself in S P the absence of a term proportional to sinφcosφ, the presence of which would lead to an asymmetry between events with sinφcosφ > 0 and < 0. It may be remarked that there are corrections to the B γγ matrix element s → (associated, for example, with the elementary process bs¯ cc¯ γγ) which cause → → the superposition of scalar and pseudoscalar terms to deviate slightly from the ratio ξ /ξ = i [7, 8]. From the work of Bosch and Buchalla [7], we find S P 1 ξS 2C1 +NC2 λB − = i 1 g(z ) (28) c ξ − 3 C m P (cid:20) 7 B (cid:21) where g(z) 2+( 2ln2z +2π2 4πilnz)z + (z2), (29) ≈ − − − O and z = m2/m2 0.1,C = 1.1,C = 0.24,N = 3. There is thus a small c c b ∼ 1 2 − correction to the equality ξ = ξ . More interestingly, the phase δ = arg(ξ /ξ ) P S P S is not exactly 900,implying| th|at a| te|rm of the form I˜ sinφcosφcosδ could appear 3 in dΓ/dφ. These corrections are, however, too small, to have a measurable impact on the spectrum and branching ratio of the decay B ee¯µµ¯ calculated above. s → 4 Conclusions WehavecalculatedthespectrumandrateofthedoubleDalitzdecayB¯ e+e µ+µ , s − − → using the effective Hamiltonian for the flavour-changing neutral current reaction bs¯ l+l , and form-factors motivated by the heavy quark limit of QCD. A method − → is given for obtaining the correlated mass spectrum dΓ/dx dx from the known 12 34 results for the QED process B¯ γ γ e+e µ+µ . The conversion ratios ρ = s ∗ ∗ − − l¯ll′¯l′ → → 9 Γ(B l+l l+l )/Γ(B γγ) showanenhancement over theQEDresult, ranging s − ′ ′− s → → from 10%for the channel e+e e+e to 30% for the channel µ+µ µ+µ . Our best es- − − − − timate of the branching ratios, using the QCD estimate Br(B γγ) = 1.23 10 6 s − → × given in [7], is Br(B ee¯ee¯) = 3.6 10 10, Br(B ee¯µµ¯) = 1.1 10 10,Br(B s − s − s → × → × → µµ¯µµ¯) = 3.5 10 11. These branching ratios may have a chance of being observed − × at future hadron machines producing up to 1012 B mesons. s Acknowledgments We wish to thank A. Chapovsky for a useful discussion. One of us (Y.D.) acknowl- edges the award of a Doctoral stipend from the state of Nordrhein-Westphalen. References [1] Y. Dinc¸er and L.M. Sehgal, Phys. Lett. B521 (2001) 7-14, [hep-ph/0108144] [2] G. Korchemsky, D. Pirjol and T.-M. Yan, Phys.Rev. D61 (2000) 114510, [hep-ph/9911427] [3] G.Buchalla, A.J. Buras and M.E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 1125-1144, [hep-ph/9512380] [4] Y. Dinc¸er and L.M. Sehgal, Ref. [1]; T.M. Aliev, A. O¨zpineci and M. Savci, Phys. Rev. D55 (1997) 7059-7066, [hep-ph/9611393]; C.Q. Geng, C.C. Lih and W.M. Zhang, Phys. Rev. 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