application/xmlElectroweak effects in the double Dalitz decay Bs→l+l−l′+l′−Yusuf DinçerL.M. SehgalPhysics Letters B 556 (2003) 169-176. doi:10.1016/S0370-2693(03)00131-XjournalPhysics Letters BCopyright © 2003 Elsevier Science B.V. All rights reserved.Elsevier B.V.0370-26935563-420 March 20032003-03-20169-17616917610.1016/S0370-2693(03)00131-Xhttp://dx.doi.org/10.1016/S0370-2693(03)00131-Xdoi:10.1016/S0370-2693(03)00131-Xhttp://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB19589S0370-2693(03)00131-X10.1016/S0370-2693(03)00131-XElsevier Science B.V.PhenomenologyFig. 1Invariant mass distribution dΓ/dx12dx34 for Bs→e+e−μ+μ− in electroweak theory.Fig. 2Ratio (dΓ/dx12dx34)EW/(dΓ/dx12dx34)QED showing influence of electroweak parameter C9,C10 and the form factor F(x12,x34).Fig. 3Ratio (dΓ/dx12dx34)EW/(dΓ/dx12dx34)QED in the limit of a constant form factor (F(x12,x34)=1), illustrating specific effect of electroweak parameters η9=C9/(2C7) and η10=C10/(2C7).Table 1Numerical values of the integrals I1,…,I6 for Bs→ll̄l′l̄′ in QEDBs→eeμμBs→eeeeBs→μμμμI17.75432.5011.772I27.80632.5561.821I315.55665.0533.589I417.55858.4165.115I517.64158.4995.199I60.05480.05560.0540Table 2Numerical values of the integrals Ĩ1,…,Ĩ6 for Bs→ll̄l′l̄′ in electroweak theoryBs→eeμμBs→eeeeBs→μμμμĨ19.33635.4913.856Ĩ29.47735.6434.002Ĩ318.79371.1147.837Ĩ420.41163.4575.784Ĩ520.63763.6856.003Ĩ60.1480.1520.146Electroweak effects in the double Dalitz decay Bs→l+l−l′+l′−YusufDinçerdincer@physik.rwth-aachen.deL.M.Sehgalsehgal@physik.rwth-aachen.deInstitute of Theoretical Physics, RWTH Aachen, D-52056 Aachen, GermanyEditor: P.V. LandshoffAbstractWe investigate the double Dalitz decays Bs→l+l−l′+l′− on the basis of the effective Hamiltonian for the transition bs̄→l+l−, and universal form factors suggested by QCD. The correlated mass spectrum of the two lepton pairs in the decay Bs→e+e−μ+μ− is derived in an efficient way, using a QED result for meson decays mediated by two virtual photons: Bs→γ∗γ∗→e+e−μ+μ−. A comment is made on the correlation between the planes of the two lepton pairs. The conversion ratios ρlll′l′=Γ(Bs→l+l−l′+l′−)Γ(Bs→γγ) are estimated to be ρeeee=3×10−4,ρeeμμ=9×10−5 and ρμμμμ=3×10−5, and are enhanced relative to pure QED by 10–30%.1IntroductionIn a recent paper [1] we investigated the decay Bs→l+l−γ(l=e,μ), using the effective Hamiltonian for the transition bs̄→l+l−, and obtained a prediction for the conversion ratio (1)ρll=Γ(Bs→l+l−γ)Γ(Bs→γγ) in terms of the Wilson coefficients C7,C9 and C10. An essential ingredient of the calculation was the use of a universal form factor characterising the matrix elements 〈γ|s̄iσμν(1+γ5)b|Bs〉 and 〈γ|s̄γμ(1±γ5)b|Bs〉, as suggested by recent work [2] on QCD in the heavy quark limit (mb⪢ΛQCD). It was found that the ratio ρll was significantly higher than one would expect from a QED calculation of Dalitz pair production Bs→γ∗γ→l+l−γ, the difference reflecting the presence of the short-distance coefficients C9,C10, as well as the universal 1/Eγ behaviour of the QCD-motivated form factor. The purpose of the present Letter is to apply the same considerations to the “double Dalitz decay” Bs→l+l−l′+l′−, to determine whether there is similar enhancement of the double conversion ratio (2)ρlll′l′=Γ(Bs→l+l−l′+l′−)Γ(Bs→γγ), compared to what one would obtain from the QED process Bs→γ∗γ∗→l+l−l′+l′−. 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 Bs→γγ vertex.2Matrix element and invariant mass spectrumWe begin with the effective Hamiltonian for bs̄→l+l−[3](3)Heff=αGF2πVtbV∗tsCeff9(s̄γμPLb)l̄γμl+C10(s̄γμPLb)l̄γμγ5l−2C7q2s̄iσμνqν(mbPR+msPL)bl̄γμl, where PL,R=(1∓γ5)/2 and q is the sum of the l+ and l− momenta. Ignoring small q2-dependent corrections in Ceff9, the values of the Wilson coefficients are (4)C7=−0.315,C9=4.334,C10=−4.624. Then, as shown in [4], the matrix element for Bs→l+l−γ has the form (5)MBs→l+l−γ=αGF2πeVtbV∗ts1MBsϵμνρσϵ∗νqρkσ(A1l̄γμl+A2l̄γμγ5l)+iϵ∗(k·q)−(ϵ∗·q)kμ(B1l̄γμl+B2l̄γμγ5l), where (6)A1=C9fV+2C7M2Bsq2fT,A2=C10fV,B1=C9fA+2C7M2Bsq2f′T,B2=C10fA. The form factors fV,fA,fT,f′T, defined in Ref. [1], will be taken to have the universal form (7)fV=fA=fT=f′T=13fBsΛs1xγ+OΛ2QCDE2γ, predicted in the heavy quark approximation (mb⪢ΛQCD,mb⪢ms) in QCD [2]. Here, Λ̄s=mBs−mb≈0.5 GeV, xγ=2Eγ/MBs=1−q2/M2Bs, and fBs≈200 MeV is the Bs decay constant. The essential feature for our purpose will be the universal 1/xγ behaviour, the absolute normalization dropping out in the calculation of the conversion ratio. (Corrections to universality are discussed in Ref. [5].)To obtain the matrix element for Bs→l+l−l′+l′− we treat the second lepton pair l′+l′− as a Dalitz pair associated with internal conversion of the photon in Bs→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 (8)MBs→e+e−μ+μ−∼ek2a+q2L+μ(q1,q2)+a−q2L−μ(q1,q2)Lνem(k1,k2)ϵμνρσqρkσ+i(gμνk·q−kμqν), 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 Lem are given by (9)L±μ(q1,q2)=ū(q1)γμ(1±γ5)v(q2),Lemμ(k1,k2)=ū(k1)γμv(k2), where k1+k2=k,q1+q2=q. The coefficients a±(q2) are related to those in Eq. (6) by (10)a±q2=A1q2±A2q2, where we have used the fact that for universal form factors, B1,2=A1,2.At this stage, it is expedient to compare the matrix element (8) with the matrix 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 (11)MBarker=const·ek2eq2Lμem(q1,q2)Lνem(k1,k2)ξPϵμνρσqρkσ+ξS(gμνk·q−kμqν). The coefficients ξP and ξS are normalized so that |ξP|2+|ξS|2=1. (In Ref. [6] they are denoted by ξP=cosζ,ξS=sinζeiδ.)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) (12)1Γγγd2Γdx12dx34Barker=2α29π2λ12λ34λw23−λ2123−λ234|ξP|2λ2+|ξS|2λ2+3w22. The variables entering the above formula are defined as follows: (13)x12=(q1+q2)2/M2=q2/M2,x34=(k1+k2)2/M2=k2/M2,x1=x2=m21M21x12,x3=x4=m23M21x34,z=1−x12−x34,λ12=(1−x1−x2)2−4x1x2,λ34=(1−x3−x4)2−4x3x4,w2=4x12x34,λ=z2−w2. Here m1 and m3 denote the masses of the electron and muon, and M the mass of the decaying meson. The phase space in the variables x12 and x34 is defined by x034<x34<(1−x12)2, x012<x12<(1−x34)2, where x012=4m21/M2,x034=4m23/M2.We can now adapt the QED result (12) to the process Bs→e+e−μ+μ−, by 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 Bs decay corresponds to the QED matrix element considered by Barker et al., if we put ξP=1/2,ξS=i/2. This allows us to obtain the invariant mass spectrum for the double Dalitz decay Bs→e+e−μ+μ− in electroweak theory: (14)1ΓγγdΓdx12dx34EW=η9+1x122+η210+η9+1x342+η210x212x234x212+x234F(x12,x34)21ΓγγdΓdx12dx34QED, where (15)1ΓγγdΓdx12dx34QED=α29π2λ12λ34λw23−λ1223−λ3422λ2+32w2. Here we have used the abbreviation η9=C9/(2C7) and η10=C10/(2C7), introduced in Ref. [1]. The electroweak formula (14) reduces to the QED result in the limit η9=η10=0,F(x12,x34)=1.The form factor F(x12,x34) is chosen to have the universal form (16)F(x12,x34)=1(1−x12)1(1−x34) (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−x12) that occurs in the single Dalitz pair process Bs→e+e−γ.In Fig. 1 we plot the correlated invariant mass spectrum for Bs→e+e−μ+μ− in electroweak theory. The ratio of the electroweak and QED spectra is shown in Fig 2, and indicates the effects associated with the coefficients η9 and η10, and the form factor F(x12,x34). One notes a slight depression in the region x12=−2C7C9 or x34=−2C7C9, connected with the vanishing of the term (C9+2C7x12)2 or (C9+2C7x34)2. There is also a general enhancement for increasing values of x12,x34, because of the form factor (16). If the form factor F(x12,x34) is set equal to one, the ratio of the electroweak and QED spectra has the structure plotted in Fig. 3, illustrating the effects which depend specifically on the electroweak parameters η9,η10.The absolute value of the conversion ratio ρeeμμ is obtained by integrating (1ΓγγdΓ/dx12dx34)EW over the range of x12 and x34. In the QED case, this ratio is conveniently expressed in terms of the integrals I1…6 introduced in Ref. [6]: (17)I1=23∫∫dx12dx34λ312λ334λ3w2,I2=23∫∫dx12dx34λ312λ334λz2w2,I3=43∫∫dx12dx34λ312λ334λ2zw2,I4=∫∫dx12dx34λ12λ34λ3w23−λ212−λ234,I5=∫∫dx12dx34λ12λ34λz2w23−λ212−λ234,I6=16∫∫dx12dx34λ12λ34λ3−λ2123−λ234. These integrals are listed in Table 1 (where, for completeness, we have also given the values for the final states eēeē and μμ̄μμ̄). These integrals allow us to calculate the QED double conversion ratio (18)(ρeeμμ)QED=α26π2I1+I2+2(I4+I5+I6)=7.6×10−5. The corresponding result for electroweak theory, based on the differential decay rate (14), can be expressed in terms of the integrals (19)Ĩ1=23∫∫dx12dx34λ312λ334λ3w2G(x12,x34),Ĩ2=23∫∫dx12dx34λ312λ334λz2w2G(x12,x34),Ĩ3=43∫∫dx12dx34λ312λ334λ2zw2G(x12,x34),Ĩ4=∫∫dx12dx34λ12λ34λ3w23−λ212−λ234G(x12,x34),Ĩ5=∫∫dx12dx34λ12λ34λz2w23−λ212−λ234G(x12,x34),Ĩ6=16∫∫dx12dx34λ12λ34λ3−λ2123−λ234G(x12,x34). The factor G(x12,x34) in the integrand of Eq. (19) contains the effects of the electroweak coefficients η9,η10 and the universal form factor F(x12,x34): (20)G(x12,x34)=η9+1x122+η102+η9+1x342+η102x212x234x212+x234F(x12,x34)2. The integrals Ĩ1,…,Ĩ6 are given in Table 2. The electroweak conversion ratio, analogous to the QED result (18), is given by (21)(ρeeμμ)EW=α26π2Ĩ1+Ĩ2+2Ĩ4+Ĩ5+Ĩ6=9.1×10−5. In comparison to the QED result (18), the double conversion ratio for B→eēμμ̄ in electroweak theory is enhanced by ∼20%.A calculation of the spectra for the channels eēeē and μμ̄μμ̄ is complicated by interference between the exchange and direct amplitudes. The conversion ratio for these channels takes the form (22)ρ=ρ1+ρ2+ρ12, where ρ1 and ρ2 denote the “direct” and “exchange” contribution, and ρ12 an interference term. Numerical calculations of the decays π0→e+e−e+e− and KL→e+e−e+e− suggest that ρ12 is small and ρ1≈ρ2. Thus a rough estimate of the double conversion ratio can be obtained using the formula (21), with an extra factor (14)·2 where (14) is the statistical factor for two identical fermion pairs, and 2 comes from adding direct and exchange contributions. This yields, using the numbers in Table 2(23)(ρeēeē)EW≈2.9×10−4,(ρμμ̄μμ̄)EW≈2.8×10−5. For comparison, the QED results, using Table 1, are (24)(ρeēeē)QED≈2.7×10−4,(ρμμ̄μμ̄)QED≈2.2×10−5. Thus the enhancement in the case of eēeē is ∼10% and that in μμ̄μμ̄ about 30%. Combining (21) and (23), the ratio of the channels eēeē,eēμμ̄ and μμ̄μμ̄ is approximately (25)eēeē:eēμμ̄:μμ̄μμ̄=3:1:0.3.To obtain the absolute branching ratios, we note that the decay rate of Bs→γγ, derived from the effective Hamiltonian (3), involves the Wilson coefficient C7 and the universal form factor fT(xγ=1) (see Eq. (7)). Using nominal values for fBs and Λ̄s, and evaluating C7 at the renormalization scale μ=mb, Ref. [7] finds Br(Bs→γγ)=1.23×10−6. Using this as a reference value, we obtain: (26)BrBs→eēeē=3.6×10−10,BrBs→eēμμ̄=1.1×10−10,BrBs→μμ̄μμ̄=3.5×10−11.3Correlation of e+e− and μ+μ− planes in Bs→eēμμ̄One of the distinctive features of the electroweak Bs→γγ matrix element is that the coefficients ξS and ξP (normalized to |ξS|2+|ξP|2=1) are given by ξS=i2 and ξP=12. The equality |ξS|2=|ξP|2 leads to the simplification that the factor |ξP|2λ2+|ξS|2(λ2+32w2) appearing in the spectrum (12) could be written as 12[2λ2+32w2] in going over to the electroweak case (Eq. (15)). A further interesting consequence is the distribution of the angle φ between the e+e− and μ+μ− planes in Bs→eēμμ̄. Generalising the QED result given in Ref. [6] to the electroweak case, the correlation in φ is given by (27)1ΓγγdΓdφeēμμ̄EW=α26π3Ĩ1sin2φ+Ĩ2cos2φ+Ĩ4+Ĩ5+Ĩ6. The fact that Ĩ2 is so close to Ĩ1 means that the spectrum dΓ/dφ is essentially independent of φ. Furthermore, the fact that arg(ξS/ξP)=π/2 reflects itself in 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 Bs→γγ matrix element (associated, for example, with the elementary process bs̄→cc̄→γγ) which cause the superposition of scalar and pseudoscalar terms to deviate slightly from the ratio ξS/ξP=i[7,8]. From the work of Bosch and Buchalla [7], we find (28)ξSξP=i1−23C1+NC2C7λBmBg(zc)−1, where (29)g(z)≈−2+−2ln2z+2π2−4πilnzz+Oz2, and zc=mc2/m2b∼0.1,C1=1.1,C2=−0.24,N=3. There is thus a small correction to the equality |ξP|=|ξS|. More interestingly, the phase δ=arg(ξP/ξS) is not exactly 90°, implying that a term of the form Ĩ3sinφcosφcosδ could appear in dΓ/dφ. These corrections are, however, too small, to have a measurable impact on the spectrum and branching ratio of the decay Bs→eēμμ̄ calculated above.4ConclusionsWe have calculated the spectrum and rate of the double Dalitz decay Bs→e+e−μ+μ−, using the effective Hamiltonian for the flavour-changing neutral current reaction bs̄→l+l−, and form-factors motivated by the heavy quark limit of QCD. A method is given for obtaining the correlated mass spectrum dΓ/dx12dx34 from the known results for the QED process Bs→γ∗γ∗→e+e−μ+μ−. The conversion ratios ρll̄l′l̄′=Γ(Bs→l+l−l′+l′−)/Γ(Bs→γγ) show an enhancement over the QED result, ranging from 10% for the channel e+e−e+e− to 30% for the channel μ+μ−μ+μ−. Our best estimate of the branching ratios, using the QCD estimate Br(Bs→γγ)=1.23×10−6 given in [7], is Br(Bs→eēeē)=3.6×10−10, Br(Bs→eēμμ̄)=1.1×10−10,Br(Bs→μμ̄μμ̄)=3.5×10−11. These branching ratios may have a chance of being observed at future hadron machines producing up to 1012Bs mesons.AcknowledgementsWe wish to thank A. Chapovsky for a useful discussion. One of us (Y.D.) acknowledges the award of a Doctoral stipend from the state of Nordrhein-Westphalen.References[1]Y.DinçerL.M.SehgalPhys. Lett. B52120017[2]G.KorchemskyD.PirjolT.-M.YanPhys. Rev. D612000114510[3]G.BuchallaA.J.BurasM.E.LautenbacherRev. Mod. Phys.6819961125[4]Y. Dinçer, L.M. Sehgal, Ref. [1]T.M.AlievA.ÖzpineciM.SavciPhys. Rev. D5519977059C.Q.GengC.C.LihW.M.ZhangPhys. Rev. D622000074017G.EilamC.D.LüD.X.ZhangPhys. Lett. B3911997461[5]G. Korchemsky, D. Pirjol, T.-M. Yan, Ref. [2]F.KrügerD.MelikhovS.Descotes-GenonC.T.Sachrajda[6]A.R.BarkerH.HuangP.A.ToaleJ.EngleSee also earlier workZ.E.S.UyPhys. Rev. D431972802T.MiyazakiE.TakasugiPhys. Rev. D819732051N.M.KrollW.WadaPhys. Rev.9819551355[7]S.BoschG.BuchallaJHEP02082002054[8]C.V.ChangG.L.LinY.P.YaoPhys. Lett. 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