Lepton-number violating four-body decays of heavy mesons
Han Yuan, Tianhong Wang, Guo-Li Wang, Wan-Li Ju, Jin-Mei Zhang
aa r X i v : . [ h e p - ph ] J u l Lepton-number violating four-body decays of heavy mesons
Han Yuan ∗ , Tianhong Wang † , Guo-Li Wang ‡ , Wan-Li Ju § Department of Physics, Harbin Institute of Technology, Harbin 150001, China
Jin-Mei Zhang ¶ Xiamen Institute of Standardization, Xiamen 361004, China
Abstract
Neutrinoless hadron Lepton Number Violating (LNV) decays can be induced by virtualMajorana neutrino, which in turn indubitably show the Majorana nature of neutrinos. Manythree-body LNV processes and Lepton Flavour Violating (LFV) processes have been studiedextensively in theory and by experiment. As a supplement, we here study 75 four-body LNV(LFV) processes from heavy pseudoscalar B and D decays. Most of these processes have notbeen studied in theory and searched for in experiment, while they may have sizable decayrates. Since the four-body decay modes have the same vertexes and mixing parameters withthree-body cases, so their branching fractions are comparable with the corresponding three-body decays. We calculate their decay widths and branching fractions with current boundson heavy Majorana neutrino mixing parameters, and estimate some channels’ reconstructionevents using the current experimental data from Belle. In the Standard Model (SM), neutrinos are strictly massless, yet non-zero neutrino masses havebeen detected in experiment [1–4]. So given the physics of neutrinos, extension of the SM isnecessary. But by now, the nature of neutrinos is still puzzling, because it is still not clearwhether neutrinos are Dirac or Majorana particles. So before determine how to extend thephysics of SM, we have to clarify the neutrinos type, Dirac or Majorana.There is strong theoretical motivation for Majorana mass term to exist since it could naturallyexplain the smallness of the observed neutrino masses [5, 6]. As is known, though not derivedfrom first principle, the SM conserves the lepton number, but Majorana mass term violateslepton number by two units (∆ L = 2). In which case the neutrinoless hadron LNV decays withlike sign dilepton final state are crucial for the existence of Majorana neutrinos. The possibleLepton Flavor Violating (LFV) meson decays could be induced either by Majorana neutrino orneutrino oscillation in which case the neutrino is a Dirac neutrino. However, neutrino oscillation ∗ [email protected] † [email protected] ‡ gl [email protected] § wl ju [email protected] ¶ [email protected]
1t loop level would be suppressed by powers of m ν m W and thus the branching fraction could notbe brought to an observable level. As a result any direct observation of LFV (LNV) processindicates the existence of Majorana neutrino.Many efforts have been made to determine the Majorana nature of neutrinos by studying theLNV and LFV processes. As the neutrinos in the final state are undetectable to the detectors,therefore neutrinoless processes are preferred, e.g. , the neutrinoless double β nuclei decay (0 νββ )has long been advocated as a premier demonstration of possible Majorana nature of neutrinos[7, 8]; the Majorana neutrino exchanges in τ lepton three-body or four-body decays [9–11]; theLNV process pp → ℓ ± ℓ ± + X or pp → ℓ ± ℓ ± jj at LHC [12, 13]; the top-quark or W-bosonfour-body decay [14, 15]; the LNV or LFV meson decays with like sign dilepton in the final state[16–21], et al .Recently, Atre et al. [22] have studied K , D , D s and B decays via a fourth massive Majorananeutrino. They demonstrated if the exchanged Majorana neutrino is resonant, which means it ison mass-shell. Then the corresponding branching fractions can be enhanced by several orders,in which case the fractions can be reached by the current experiments. Inspired by the effect ofresonant neutrino, various three body meson decays M +1 → ℓ +1 ℓ +2 M − where ∆ L = 2 have beenstudied in [22–27] and so have four-body decays B → Dℓℓπ in [28].In the experiment, some of these LNV (LFV) processes have been searched. For example,Fermilab E791 Collaboration reported their results of searching for the LNV and LFV decays of D into 3 and 4-bodies, they presented upper limits on the branching fractions at 90% confidencelevel (CL) [29]. Recently, using 772 × B ¯ B pairs accumulated at Υ(4 S ) resonance with thesame CL, the Belle Collaboration set the upper limits on the LNV (LFV) B + → D − ℓ + ℓ ′ + decays [30]. Using a sample of 471 ± B ¯ B events, the BABAR Collaboration searchedfor the LNV processes B → K − ( π − ) ℓ + ℓ + and placed upper limits on their branching fractionsalso with 90% CL [31]. The LHCb Collaboration, using 0 .
41 fb − of data collected with theLHCb detector in proton-proton collisions at a center-of-mass energy of 7 TeV, reported theirupper limits on the branching fractions of B − decays to D ( ∗ )+ µ − µ − , π + µ − µ − and D s µ − µ − at95% CL. They also searched for the 4-body decay B − → D π + µ − µ − and set upper limit onits branching fraction [32] for the first time. The experimental situation of searching for theLNV and LFV processes can be found in Refs. [11, 33]. Though these LNV and LFV processesare still unobservable, the upper limits for branching fraction have been obtained, which also inturn limit the mixing parameters between Majorana neutrino and charged lepton.Though lots of LNV (LFV) processes have been studied by experiment and in theory, there2re still many channels which have not been considered, especially the four-body LNV (LFV)meson decays, most of which are still absent in literature. Some channels of that kind mayhave considerable branching fractions and may be accessible in current experiment. LNV four-body decays also offer complementary information about the masses and heavy mixings of sucha heavy (resonant) Majorana neutrino, so they are worth studying deeply. In this paper, westudy 75 four-body LNV (LFV) processes of dilepton decays B ( D ) → M ℓ ℓ M , where M stands for a pseudoscalar meson, M can be a pseudoscalar or a vector meson and ℓ ( ℓ ) = e, µ .These ∆ L = 2 LNV (LFV) 4-body meson decays are induced by a Majorana neutrino, andthe possible lowest order diagrams are illustrated in Figure 1 (a-b). Some processes, such asthe decays of ¯ B → D − ℓ +1 ℓ +2 M − , where M − stands for π − , K − , ρ − , K ∗− , D − or D − s , arerepresented by an exclusive Feynmann diagram shown in Figure 1 (a); but some decays, like B + → ¯ D ℓ +1 ℓ +2 M ′ − , where M ′ − denotes D − or D − s , have both decay modes shown in Figure 1 (a)and (b). In Figure 1 (a), if the Majorana neutrino mass lies between a few hundred MeV to 4.4GeV (since it is heavy, it may be a fourth generation neutrino), the neutrino could be on mass-shell (resonance), and the corresponding decay rate will be much enhanced due to the effect ofneutrino-resonance. The contribution of Figure 1 (a) will be much greater than that of neutrino-exchange diagram in Figure 1 (b), which is suitable for a continuous neutrino mass. So we willfocus on the neutrino-resonance of diagram figure 1 (a). The contribution of neutrino-exchangediagram figure 1 (b) and the interference between two diagrams will be ignored.There are two key points to calculate these 4-body decay modes. One is the selection ofthe mixing parameters, since most of these Majorana neutrino induced 4-body decay modesdo not have experiment results and we cannot extract the mixing parameters by these decays.So we followed Atre et al ’s method in which the parameters are determined by experimentaldata [22]. We choose the strongest constrains which were abstracted from the current data asthe input in our paper [22, 26] to guarantee the accuracy. The other point is the calculationof the hadronic matrix element between initial meson B ( D ) and final meson M . We use theMandelstam formalism [34] which the hadronic matrix element is described as an overlappingintegral over the wave functions of the initial and final states [35]. The wave functions areobtained by solving the relativistic Bethe-Salpeter (BS) equation [36].This paper is organized as followed, in section 2, we outline the formulas of the transitionmatrix element. In section 3, we present the details of how to calculate the hadronic matrixelement. In section 4, we show the results and conclude the branching fraction of heavy meson4-body decays as a function of the heavy neutrino mass.3 Theoretical Details
The leading order Feynman diagrams for the LNV (LFV) 4-body decays of heavy meson M : M ( P ) → M ( P ) ℓ +1 ( P ) ℓ +2 ( P ) M − ( P ) (1)are shown in Figure 1 (a-b). Here M is the pseudoscalar B or D with momentum P , two chargedleptons ℓ +1 , ℓ +2 have momentum P and P , pseudoscalar meson M with momentum P denotes π , K or D , meson M with momentum P can be a pseudoscalar meson π , K , D and D s et al ,or a vector meson ρ , K ∗ , et al .Such LNV (LFV) process can occur through a Majorana neutrino, and the vertex betweenthis Majorana neutrino and charged lepton is beyond the SM. Following previous studies [22, 37],we assumed that there is only one heavy Majorana neutrino which may be a fourth generationneutrino. It can be kinematically accessible in the range we are interested in. Then the gaugeinteraction lagrangian responsible for the LNV (LFV) decay can be written as: L = − g √ W + µ τ X ℓ = e V ∗ ℓ N c γ µ P L ℓ + h . c ., (2)where P L = (1 − γ ), N is the mass eigenstate of the fourth generation Majorana neutrinoand V ℓ is the mixing matrix between the charged lepton ℓ and heavy Majorana neutrino N . M, P p m γ µ (1 − γ ) p ′ m ′ M , P p m p ′ m N ℓ +1 , P ℓ +2 , P M , P (a) M,P p m p ′ m ′ p m p ′ m M ,P m M ,P m N ℓ +1 ,P ℓ +2 ,P (b) Figure 1: Feynman diagram of the four-body decay of heavy mesonThe transition amplitude for the 4-body decay M ( P ) → M ( P ) ℓ +1 ( P ) ℓ +2 ( P ) M − ( P ) shownin Figure 1 (a) can be written as: M = g V q q V q q M W h M ( P ) | ¯ q γ µ (1 − γ ) q | M ( P ) i × M µν × h M ( P ) | ¯ q γ ν (1 − γ ) q | i , (3)where the momentum dependence in the propagator of W boson has been ignored since it ismuch smaller than the W mass; g is the weak coupling constant; V q q ( V q q ) is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element between quarks q and q ( q and q ); M µν is thetransition amplitude of the leptonic part. 4s mentioned before, only the contribution of the diagram in Figure 1 (a) is considered,where the Majorana neutrino is on mass shell, and the effective narrow-width approximationrelated to the resonant contribution can enhance the decay rate substantially. In this case,according to Ref. [22, 26], the leptonic matrix element M µν can be given as: M µν = g V ℓ V ℓ m " ¯ u γ µ γ ν P R ν q N − m + i Γ N m + ¯ u γ ν γ µ P R ν q ′ N − m + i Γ N m , (4)where V ℓ is the mixing parameter between the heavy Majorana neutrino and charged lepton, P R = (1 + γ ); q N is the momentum of heavy Majorana neutrino ( q ′ N is the case of exchangethe two final charged leptons), m is the mass of the heavy Majorana neutrino and Γ N is thetotal decay width of the heavy neutrino.Mesons M and M are pseudoscalar mesons and the corresponding hadronic matrix elementin Eq. (3) can be described as a function of form factors: h M ( P ) | ¯ q γ µ (1 − γ ) q | M ( P ) i = P µ ( f + + f − ) + P µ ( f + − f − ) , (5)The method to calculate the form factors f + , f − will be shown in section 3.The last part h M | h ν | i in Eq. (3) is related to the decay constant of the meson M . If M is a pseudoscalar with momentum P , we obtain the following relation: h M ( P ) | ¯ q γ ν (1 − γ ) q | i = iF M P ν , (6)where F M is decay constant of meson M . If M is a vector with momentum P and polarizationvector ǫ , the corresponding relation will become: h M ( P , ǫ ) | ¯ q γ ν (1 − γ ) q | i = M F M ǫ ν , (7)here we use the same symbol M to denote the meson and its mass.By combining Eq. (4), Eq. (5) and Eq. (6), we rewrite the decay amplitude Eq. (3) in thecase of meson M as a pseudoscalar: M = 2 G F V ℓ V ℓ V q q V q q F M m × ¯ u (cid:20) P P ( f + + f − ) + P P ( f + − f − )( P + P ) − m + i Γ N m + P P ( f + + f − ) + P P ( f + − f − )( P + P ) − m + i Γ N m (cid:21) P R ν , (8)where G F is Fermi constant. If meson M is a vector, we just replace P with M ǫ in numeratorin Eq. (8). With the numerical values of form factors f + and f − obtained in section 3, thecalculation of this decay amplitude is not complicated.5 Hadronic transition matrix element
In order to calculate the hadronic matrix element and get the numerical value of form factors f + , f − , we use the Mandelstam formalism [34], in which the transition amplitude between twomesons is described as a overlapping integral over the Bethe-Salpeter wave functions of initialand final mesons [35]. Using this method with further instantaneous approximation [38], in thecenter of mass system of initial meson, in leading order, we write the hadronic matrix elementas [39]: h M ( P ) | ¯ q γ µ (1 − γ ) q | M ( P ) i = Z d ~q (2 π ) Tr (cid:20) ¯ ϕ ++ P ( ~q ) γ µ (1 − γ ) ϕ ++ P ( ~q ) PM (cid:21) , (9)where P and P are the momenta of initial and final mesons; M in denominator is the massof initial meson; q is relative momentum between quark and antiquark inside the initial meson; ~q = ~q + m m ′ + ma ′ ~r is the relative momentum inside the final meson M , m ′ ( m ′ ) is mass ofantiquark (quark) in final meson M , ~r is three dimension momentum of meson M ; ϕ ++ is thepositive wave function for a meson in the BS method; for the final state, we have define thesymbol ¯ ϕ ++ P = γ ( ϕ ++1 P ) + γ .Table 1: Mass of quark in unit of GeV.quark b c s d u mass 4 .
96 1 .
62 0 . .
311 0 . ϕ ++ for a pseudoscalar meson can be writtenas [40]: ϕ ++ P = A (cid:18) B + PM + q ⊥ C + q ⊥ PM D (cid:19) γ , (10)where q ⊥ = (0 , ~q ), and A = M (cid:20) f ( ~q ) + f ( ~q ) m + m ω + ω (cid:21) ,B = ω + ω m + m ,C = − m − m m ω + m ω , (11) D = ω + ω m ω + m ω . In Eq. (11), m and m are the masses of quark and antiquark inside the meson, and we listtheir values in Table 1; ω i is defined as ω i = q m i + ~q , i = 1 , f ( ~q ) and f ( ~q ) are the wavefunction of the meson. 6ith Eq. (10) and Eq. (11), we take the integral on the right side of Eq. (9), then the formfactor f + , f − can be expressed as: f + = 12 (cid:18) T M + T M + M − E M T (cid:19) ,f − = 12 (cid:18) T M − T M − M + E M T (cid:19) , (12)where M and E = p M + ~r are the mass and energy of final meson M ; and T = Z d ~q (2 π ) A At ,T = Z d ~q (2 π ) A At ,T = 1 | ~r | Z d ~q (2 π ) A At | ~q | cos θ,t = C m m + m E − C~q · ~qM + BD M (cid:18) ~q · ~q + m m + m ~q + m m + m E (cid:19) − CC (cid:18) ~q + m m + m ~q · ~q (cid:19) − BB − DD m m + m E M ~q · ~q,t = − − m m + m C M − m m + m BD E − DD ~q ,t = − C − B D − BD E M − CE M + DD M (cid:18) ~q · ~q + m m + m ~q (cid:19) , (13)where A , B , C and D have the same meanings as those in Eq. (11), while the parametersare replaced by the one of final pseudoscalar.Numerical values of wave functions f ( ~q ) and f ( ~q ) can be obtained by solving the coupledSalpeter equations [40]:( M − ω ) (cid:20) f ( ~q ) + f ( ~q ) m ω (cid:21) = − Z d~k (2 π ) ω n ( V s − V v ) h f ( ~k ) m + f ( ~k ) m ω i − ( V s + V v ) f ( ~k ) ~k · ~q o , ( M + 2 ω ) (cid:20) f ( ~q ) − f ( ~q ) m ω (cid:21) = − Z d~k (2 π ) ω n ( V s − V v ) h f ( ~k ) m − f ( ~k ) m ω i − ( V s + V v ) f ( ~k ) ~k · ~q o . (14)where we have chosen the Cornell potential, which is a linear potential plus a single gluonexchange reduced vector potential, and in momentum space the expression is: V s ( ~q ) = − (cid:18) λα + V (cid:19) δ ( ~q ) + λπ ~q + α ) ,V v ( ~q ) = − π α s ( ~q )( ~q + α ) ,α s ( ~q ) = 12 π
27 1log( a + ~q Λ QCD ) , (15)7here a = e = 2 . λ = 0 .
21 GeV is the string constant; α = 0 .
06 GeV is a parameterfor the infrared divergence compensation; the QCD scale Λ
QCD = 0 .
27 GeV characterizes therunning strong coupling constant α s ; the constant V is a parameter by hand in potential modelto match the experimental data, whose values for different mesons are listed in Table. 2.Table 2: Parameters V in unit of GeVmeson B D K πV -0.091 -0.375 -0.962 -0.999With these parameters, we solved the full Salpeter equation Eq. (14), and obtained thenumerical values of wave functions f ( ~q ) and f ( ~q ) for pseudoscalar mesons B , D , K and π .Meanwhile, the meson masses of these pseudoscalar mesons are also obtained which agree withexperimental data. Besides the parameters appearing in potential, there are other parameters whose values need tobe determined. We choose the CKM matrix elements [41]: V ud = 0 . V us = 0 . V cd = 0 . V cs = 0 . V cb = 40 . × − , V ub = 3 . × − . The decay constants of pseudoscalar andvector mesons used in our calculation are listed in Table 3.Table 3: Decay constants F M of pseudoscalar and vector mesons in unit of MeV.meson π ρ K K ∗ D D s F M | V ℓ V ℓ | and the heavy neutrinomass m in Eq. (4). Following the approaches in Refs. [22, 26], we take the mixing parameter V ℓ and the mass m as phenomenological parameters. Since the mixing parameters are commonconstant, we take some decay modes with the same | V ℓ V ℓ | into our consideration and havemixing parameters numerical upper bounds in experiment, thus we extract the numerical valuesof mixing parameters from these processes. Details can be found in Refs. [22, 26]. We choose thestrongest constrains on mixing as input in this paper to guarantee accuracy. For the value of m ,since we only consider the case in which the heavy neutrino is on mass shell, we determine themass of neutrino by kinematics. With numerical values of mixing parameters and neutrino mass m , the neutrino total decay width Γ N is calculated, which covers all possible decay channels8f Majorana neutrino at the mass m [22]. So in our calculation, Γ N is not fixed but mass andmixing parameter dependent.With these parameters and the limits on mixing parameters, 75 LNV (LFV) decay widthsand branching fractions of the heavy mesons D + , D , B + , and B are calculated. Among theseprocesses, there are some channels where the meson M is a light meson, π or K . We mustpoint out that since we have made instantaneous approximation to Bethe-Salpeter equation,the result of the hadronic matrix element including a light meson may not be accurate in theheavy meson case. Since all these decays are beyond the SM, accurate calculation is not theissue, and we also take the results including these decays. In the calculation of decay rate, weperform a Monte Carlo sampling of the branching fractions and the mass of heavy neutrino. Forexample we calculate the excluded region of the branching fractions as a function of the heavyneutrino mass m and plot the results in Figures 2-7. The regions inside and above the curveare excluded by current experiment data, while the region below the curve is allowed in theory.The curve is not smooth, which is caused by two reasons. First, we choose different mixingparameters | V ℓ V ℓ | according to different ranges of heavy neutrino mass m . Since the currentlimits on mixing parameters are related to heavy Majorana neutrino mass, depending on todifferent neutrino mass range, we choose different LNV (LFV) processes to get the strongestconstrains on mixing parameters. For example, in process B → D − e + e + M − , we choose threeprocesses K + → e + e + π − , D + → e + e + π − and B + → e + e + π − to limit | V e | . Second, asdiscussed above, the value of neutrino total decay width Γ N is mass and mixing parameterdependent, whose values change with respect mixing parameter | V ℓ | and neutrino mass m .Because the mixing parameters is piecewise, the branching fractions are also piecewise as afunction of the neutrino mass. The difference of value choices of mixing parameters may be themain reason for the difference between our results and those in Ref. [28, 44], which calculatedthe branching fractions of ¯ B → D + e − e − π + and B − → D µ − µ − π + . We mention that, incalculations of the decay modes B + → π ℓ +1 ℓ +2 M − , we lack the information of mixing parameter | V e V µ | when neutrino mass m > | V e V µ | = 0 in these cases, that is, there are no predictions when neutrino mass m is largerthan 4 GeV in Figure 6 (b).There is another point that seems unusual in the results of some branching fractions. Forexample in Fig. 5 (b), we show the branching fraction for the decay mode D + → ¯ K e + µ + K − .At two edges of the curve, which are the points of the allowed smallest and largest neutrinomasses separately, the values of the branching fractions are very small. The small rates is notunusual actually, because it happens due to the restriction of the phase space. The very small9inematic phase space at edges lead to those small branching fractions.Because some 4-body decays of mesons have broader phase space than the corresponding3-body processes and the resonance neutrino mass is determined kinematically, one of the ad-vantages of these 4-body decays is that we can detect much wider range of Majorana neutrinomass. For example, we can study the heavy neutrino if its mass is in the range of 2 GeV ∼ B − → e − e − D + [26]. While durning the 4-body decay B → D − e + e + π − or B + → ¯ D e + e + π − , we can reach the range of possible neutrino mass from0.2 GeV to 3.4 GeV. Another advantage is that the branching fraction is not small comparedwith the corresponding 3-body decay [22, 26]. Because, in some cases, they have same vertexes,mixing parameters | V ℓ V ℓ | and CKM matrix elements.We have mentioned that the dominant factors of the branching fractions comes from themixing parameter | V ℓ V ℓ | , which are limited by the current experimental data. Besides theseparameters, there are other important parameters: CKM matrix elements, which are also de-terminant factors to the values of branching fractions. We note that if the final mesons are D − and π − , there are two decay modes, B → D − ℓ +1 ℓ +2 π − and B → π − ℓ +1 ℓ +2 D − . In thefirst decay mode, the CKM matrix elements are | V cb V ud | , while for the second are | V ub V cd | ,as | V ub V cd | / | V cb V ud | ∼ × − , so we ignore the decay B → π − ℓ +1 ℓ +2 D − and its interfer-ence with B → D − ℓ +1 ℓ +2 π − . For the same reason we only consider the contribution of decay D → K − ℓ +1 ℓ +2 π − and ignore the decay mode D → π − ℓ +1 ℓ +2 K − .In some particular channels, there is an additional contribution coming from intermediatemesons resonance [44]. For example, in the decay channel B + → ¯ D µ + µ + π − , besides the CKMfavored diagram in Figure 1 (a), there is another CKM dis-favored diagram (see Figure 1 (b) inRef. [44]), where the two final mesons can be induced by a intermediate resonance D ∗− (2010),in range of 2 . ≤ m ≤ . D ∗− (2010) may resultsin a considerable contribution in decay B + → ¯ D µ + µ + π − , but we do not take into considerationthese cases.Some channels with large branching ratios are detectable by the current experiments. Forexample, the Belle Collaboration produced 772 million B ¯ B events per year [45], which can beused to study the four-body B meson LNV and LFV decays. For Belle detector, the recon-struction efficiencies of π , ρ , K ∗ , D , D and D s are 65%, 61%, 58%, 78%, 83% and 74%,respectively; the identification efficiencies of π ± and K ± are 95% and 86% [46]; the electronsand muons efficiency rates both approximate 90% [45]. With these efficiencies, we choose maxi-mum branching fractions interval in each process, and estimate the reconstruction events shown10n Table 4, Table 5 and Table 6. Of particular note, all the results do not include the influenceof Geometrical Acceptance. The reason why we do not calculate K and D reconstruction eventsin Table 6, is that the branching fractions of K and D are too small, which is less than the B ¯ B events.There are 3 million D ¯ D events [47] and 2 . × D + D − events [48] produced in CLEOCollaboration every year. From the Fig. 4, Fig. 5 and Fig. 7, we can find that the maximumbranching fractions of D + and D decays approach 10 − . But if the detection efficiency isconsidered, the decay modes of D would be difficult to detect.Table 4: Branching Fraction of B → D − ℓ + ℓ + M − and corresponding Reconstruction Eventsestimated using Belle’s data.Branching Fraction Reconstruction Events M e + e + e + µ + µ + µ + e + e + e + µ + µ + µ + π − ∼ − − ∼ − − ∼ − ∼
460 4600 ∼
460 4600 ∼ K − ∼ − − ∼ − − ∼ − ∼
42 420 ∼
42 420 ∼ ρ − ∼ − − ∼ − − ∼ − ∼ ∼ ∼ K ∗ − ∼ − − ∼ − − ∼ − ∼
28 280 ∼
28 280 ∼ D − − − D s − ∼ − − − ∼
36 360 360Table 5: Branching Fraction of B + → ¯ D ℓ + ℓ + M − and corresponding Reconstruction Eventsestimated using Belle’s data.Branching Fraction Reconstruction Events M e + e + e + µ + µ + µ + e + e + e + µ + µ + µ + π − ∼ − − ∼ − − ∼ − ∼
493 4930 ∼
493 4930 ∼ K − ∼ − − ∼ − − ∼ − ∼
45 450 ∼
45 450 ∼ ρ − ∼ − − ∼ − − ∼ − ∼ ∼ ∼ K ∗ − ∼ − − ∼ − − ∼ − ∼
30 300 ∼
30 300 ∼ D − ∼ − − ∼ − − ∼ ∼ D s − − − ∼ −
380 380 380 ∼ B + → π ℓ + ℓ + M − and corresponding Reconstruction Events es-timated using Belle’s data.Branching Fraction Reconstruction Events M e + e + e + µ + µ + µ + e + e + e + µ + µ + µ + π − ∼ − − ∼ − − ∼ − ∼ ∼ ∼ ρ − ∼ − − ∼ − − ∼ − ∼
25 250 ∼
25 250 ∼ B and D , since the 4-body decays share the same vertexes and mixing parametersas well as the CKM matrix elements with the corresponding 3-body decays. Relatively largebranching fractions which are comparable with the 3-body decays are obtained, some channelscan be reached by current experiments, especially the processes B → Dℓ + ℓ + M when M are π , K and ρ . Acknowledgments
We would like to thank Tao Han for his suggestions to carry out this research and providingthe FORTRAN codes Hanlib for the calculations. We are also very grateful to Yoshi Sakaifor offering the data of particle reconstruction efficiency in Belle Collaboration. This work wassupported in part by the National Natural Science Foundation of China (NSFC) under grantNo. 11175051.
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D 84 (2011) 032001 arXiv:1101.119514 (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on B → Dee π B → DeeKB → Dee ρ B → DeeK * B → DeeDB → DeeDs (a) B → D − e + e + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on B → De µπ B → De µ KB → De µρ B → De µ K * B → De µ DB → De µ Ds (b) B → D − e + µ + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on B → D µµπ B → D µµ KB → D µµρ B → D µµ K * B → D µµ DB → D µµ Ds (c) B → D − µ + µ + M − Figure 2: Theoretically excluded regions inside the curve for the branching fraction of B → D − ℓ + ℓ + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on B → D ee π B → D eeKB → D ee ρ B → D eeK * B → D eeDB → D eeDs (a) B + → ¯ D e + e + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on B → D e µπ B → D e µ KB → D e µρ B → D e µ K * B → D e µ DB → D e µ Ds (b) B + → ¯ D e + µ + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on B → D µµπ B → D µµ KB → D µµρ B → D µµ K * B → D µµ DB → D µµ Ds (c) B + → ¯ D µ + µ + M − Figure 3: Theoretically excluded regions inside the curve for the branching fraction of B + → ¯ D ℓ + ℓ + M − (GeV)1e-161e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D → Kee π D → KeeKD → Kee ρ (a) D → K − e + e + M − (GeV)1e-161e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D → Ke µπ D → KeµKD → Ke µρ (b) D → K − e + µ + M − (GeV)1e-161e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D → K µµπ D → K µµ KD → K µµρ (c) D → K − µ + µ + M − Figure 4: Theoretically excluded regions inside the curve for the branching fraction of D → K − ℓ + ℓ + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D → K ee π D → K eeKD → K ee ρ (a) D + → ¯ K e + e + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D → K e µπ D → K e µ KD → K e µρ (b) D + → ¯ K e + µ + M − (GeV)1e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D → K µµπ D → K µµ kD → K µµρ (c) D + → ¯ K µ + µ + M − Figure 5: Theoretically excluded regions inside the curve for the branching fraction of D + → ¯ K ℓ + ℓ + M − (GeV)1e-161e-141e-121e-101e-081e-060.0001 B r a n c h i ng F r ac ti on B →π ee π B →π eeKB →π ee ρ B →π eeD (a) B + → π e + e + M − (GeV)1e-161e-141e-121e-101e-081e-060.0001 B r a n c h i ng F r ac ti on B →π e µπ B →π e µ KB →π e µρ B →π e µ D (b) B + → π e + µ + M − (GeV)1e-161e-141e-121e-101e-081e-060.0001 B r a n c h i ng F r ac ti on B →π µµπ B →π µµ KB →π µµρ B →π µµ D (c) B + → π µ + µ + M − Figure 6: Theoretically excluded regions inside the curve for the branching fraction of B + → π ℓ + ℓ + M − (GeV)1e-161e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D →π ee π D →π eeKD →π ee ρ (a) D + → π e + e + M − (GeV)1e-161e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D →π e µπ D →π e µ KD →π e µρ (b) D + → π e + µ + M − (GeV)1e-161e-141e-121e-101e-081e-060.00010.011 B r a n c h i ng F r ac ti on D →π µµπ D →π µµ KD →π µµρ (c) D + → π µ + µ + M − Figure 7: Theoretically excluded regions inside the curve for the branching frcation of D + → π ℓ + ℓ + M −2