Measuring B -> tau nu and B_c -> tau nu at the Z peak
aa r X i v : . [ h e p - ph ] J un TUM-HEP-683/08
Measuring B ± → τ ± ν and B ± c → τ ± ν at the Z peak A.G. Akeroyd , , Chuan Hung Chen , , S. Recksiegel Department of Physics, National Cheng Kung University, Tainan 701, Taiwan National Center for Theoretical Sciences, Taiwan and Physikdepartment T31, Technische Universit¨at M¨unchen, D-85747 Garching, Germany (Dated: October 30, 2018)
Abstract
The measurement of B ± → τ ± ν τ at the B factories provides important constraints on theparameter tan β/m H ± in the context of models with two Higgs doublets. Limits on this decayfrom e + e − collisions at the Z peak were sensitive to the sum of B ± → τ ± ν τ and B ± c → τ ± ν τ . Dueto the possibly sizeable contribution from B ± c → τ ± ν τ we suggest that a signal for this combinationmight be observed if the LEP L3 Collaboration used their total data of ∼ . × hadronic decaysof the Z boson. Moreover, we point out that a future Linear Collider operating at the Z peak (Giga Z option) could constrain tan β/m H ± from the sum of these processes with a precision comparableto that anticipated at proposed high luminosity B factories from B ± → τ ± ν τ alone. PACS numbers: 12.60Fr, 13.20He . INTRODUCTION In April 2006 the BELLE collaboration announced the first observation of the purelyleptonic decay B ± → τ ± ν τ [1] utilizing an integrated luminosity of 414 fb − . The measuredbranching ratio (BR) is in agreement with the Standard Model (SM) rate within theoreticaland experimental errors:BR( B ± → τ ± ν τ ) = (cid:0) . +0 . − . ( stat ) +0 . − . ( syst ) (cid:1) × − (1)Subsequently, the BABAR collaboration reported a measurement with an integrated lu-minosity of 346 fb − which is an average of separate analyses with semi-leptonic [2] andhadronic [3] tags: BR( B ± → τ ± ν τ ) = (1 . ± . ± . ± . × − (2)The average of the BELLE and BABAR measurements is [4]:BR( B ± → τ ± ν τ ) = (cid:0) . +0 . − . (cid:1) × − (3)Significantly improved precision for BR( B ± → τ ± ν τ ) would require a high luminosity L ≥ cm − s − B factory [5, 6, 7, 8, 9, 10, 11]. In the context of the SM the decay B ± → τ ± ν τ provides a direct measurement of the combination f B V ub , where f B is the decay constantwhich can only be calculated by non-perturbative techniques such as lattice QCD. ChargedHiggs bosons ( H ± ) present in the Two Higgs Doublet Model (2HDM) and the MinimalSupersymmetric SM (MSSM) would also mediate B ± → τ ± ν τ [12] with the New Physicscontribution being sizeably enhanced if tan β (the ratio of vacuum expectation values of thetwo Higgs doublets) is large [13]. The above measurements of B ± → τ ± ν τ now provide avery important constraint on the parameter tan β/m H ± in the context of the 2HDM andthe MSSM. Hence this decay is of much interest in both the SM and models beyond the SMand improved precision in the above measurements is certainly desirable.Prior to the era of the B factories three LEP collaborations searched for B ± → τ ± ν τ and obtained upper bounds within an order of magnitude of the SM prediction [14, 15, 16].Such limits were actually sensitive to the sum of B ± → τ ± ν τ and B ± c → τ ± ν τ [17] sincethe centre-of-mass energy ( √ s = 91 GeV) was above the B ± c production threshold (unlikethe B factories). The strongest limits were set by the L3 collaboration which obtainedBR( B ± → τ ± ν τ ) < . × − [14]. Since BR( B ± → τ ± ν τ ) has now been measured at the B factories, the L3 limit can now be used to provide a limit on the product of the transitionprobability f ( b → B c ) and BR( B ± c → τ ± ν τ ). A quantitative study of the magnitude ofthe contribution of B ± c → τ ± ν τ to the LEP limits was performed in [17]. We updatethis analysis using the significant improvements in the measurements of the CKM matrixand calculations of f B . Moreover, the measurements of the B ± c production cross-sectionat the Fermilab Tevatron [18, 19, 20, 21] provide the first measurements of the transitionprobability for b → B c and suggest much larger values than the theoretical estimations usedin the numerical analysis of [17]. The L3 limit on BR( B ± c → τ ± ν τ ) was obtained with ∼ . × hadronic Z boson decays, which is slightly less than half the full L3 data takenat the Z peak. We suggest that a search for B ± /B ± c → τ ± ν τ using the full L3 data sample( ∼ . × hadronic Z decays [22]) would not only strengthen the limit on the productof f ( b → B c ) and BR( B ± c → τ ± ν τ ) but also offer the possibility of a signal, which wouldbe an additional observation of B ± → τ ± ν τ and the first observation of B ± c → τ ± ν τ . It2 F = 1 . · − GeV − m e = 0 .
511 MeV m µ = 0 . m τ = 1 .
777 GeV | V ub | = 0 . | V cb | = 0 . m B u = 5 .
279 GeV τ B u = 1 . · − sm B c = 6 .
271 GeV τ B c = 0 . · − sf B u = 0 . f B c = 0 .
450 GeVTABLE I: Input parameters used in this paper, unless indicated otherwise in the text. BR ( B + q → τ + ν τ ) BR ( B + q → µ + ν µ ) BR ( B + q → e + ν e ) B u . · − . · − . · − B c .
022 9 . · − . · − TABLE II: Standard Model predictions for the branching ratios (central values). is also pointed out that a future e + e − Linear Collider operating at the Z peak (the Giga Z option [23, 24, 25, 26]) could offer similar sensitivity to the parameter tan β/m H ± fromthese leptonic decays as the proposed high luminosity B factories. This article is structuredas follows: in section II we present basic formulae for the decay rates for B ± /B ± c → τ ± ν τ and discuss the H ± contribution; we study the admixture of B ± c → τ ± ν and B ± → τ ± ν atthe Z peak in section III and give our conclusions in section IV. II. THE DECAYS B ± → τ ± ν AND B ± c → τ ± ν In the SM, the purely leptonic decays ( ℓ ± ν ℓ ) of B ± and B ± c proceed via annihilation toa W boson in the s -channel. The decay rate is given by (where q = u or c ):Γ( B + q → ℓ + ν ℓ ) = G F m B q m ℓ f B q π | V qb | − m ℓ m B q ! (4)Due to helicity suppression, the rate is proportional to m ℓ and one expects: BR ( B + q → τ + ν τ ) : BR ( B + q → µ + ν µ ) : BR ( B + q → e + ν e ) = m τ : m µ : m e (5)These decays are relatively much more important for B ± c than B ± u due to the enhancementfactor | V cb /V ub | ( f B c /f B u ) . Using the input parameters given in table I, we obtain the SMpredictions listed in table II.The effect of H ± in the 2HDM (Model II) on the decays B ± u → ℓ + ν ℓ was considered in[13] and the analogous analysis for B ± c → ℓ + ν ℓ was presented in [27]. In both cases the H ± contribution modifies the SM prediction by a global factor r qH where: r qH = [1 − tan β ( M B q /m H ± ) ] ≡ [1 − R M B q ] (6)The H ± contribution interferes destructively with that of W ± . There are two solutions for r qH = 1 which occur at R = 0 and R ∼ .
27 GeV − for B ± u → ℓ + ν ℓ ( R = 0 and R ∼ . − for B ± c → ℓ + ν ℓ ). This is shown in Fig. 1 where BR ( B u → τ + ν τ ) is plotted as a3 IG. 1: BR ( B u → τ + ν τ ) as a function of tan β and m H ± . The plotted range of the BR correspondsto the 1- σ range of the world average measurement (1 . ± . · − , and the line indicates thecentral value. function of tan β and m H ± . For tan β/m H ± ∼
0, the BR remains at its SM value (slightlyhigher than the thin line indicating the central value of the experimental measurement), butthis SM value can also be achieved along a line through the steep part of the surface where r H = (1 − = 1.If the b quark couples to both Higgs doublets at tree-level (which is referred to as thetype-III 2HDM), Eq. (6) is modified to [28]: r H = (cid:18) − tan β ǫ tan β m B m H ± (cid:19) (7)In the Minimal Supersymmetric SM (MSSM), the parameter ˜ ǫ does not appear at tree-levelbut is generated at the 1-loop level [29, 30] (with the main contribution originating fromgluino diagrams) and may reach values of 0.01. The redefinition of both the b quark Yukawacoupling and the CKM matrix element V ub are encoded in ˜ ǫ [31, 32]. The impact of ˜ ǫ = 0on r H has been developed in [33, 34, 35, 36]. In particular, the value of R where r H = 1shifts depending on the magnitude and sign of ˜ ǫ .In Fig. 2 we show the impact of the measurement of BR ( B u → τ + ν τ ) on the plane of[tan β , m H ± ] in the 2HDM (Type III) which updates the study of [28] (for a recent analogousplot with a somewhat lower value of f B see [37]). The white regions are excluded and theshaded areas correspond to BR ( B u → τ + ν τ ) within the 1- σ experimental range. We plotoverlapping bands for the 1- σ ranges of the input parameters and consider ˜ ǫ = 0 , . , − . ǫ , corresponding to positive values of the µ -parameter,are preferred in order to explain the ( g − µ anomaly [38], but in general both signs arepossible. 4he different values for ˜ ǫ result in significantly different allowed regions in the planeof [tan β , m H ± ]. Importantly, these constraints from BR ( B u → τ + ν τ ) are from a tree-level process and when applied to the MSSM are only sensitive to the assumptions for thesoft SUSY breaking sector via ˜ ǫ (recently emphasized in [39]), i.e., a higher order effect.In contrast, other important B physics observables such as b → sγ , B s − B s mixing and B d,s → µµ are all loop induced processes. Consequently, constraints on the plane [tan β , m H ± ]from such processes are very sensitive to the assumptions made for the sparticle masses, andin certain cases the constraints can be removed completely . In global studies of B physicsobservables in specific MSSM scenarios [34, 42] the measured BR ( B u → τ + ν τ ) also playsan important role. Certainly, improved precision for BR ( B u → τ + ν τ ) is desirable and veryrelevant in the era of the LHC in which the plane [tan β , m H ± ] will be probed via directproduction of Higgs bosons. Currently only High Luminosity B factories operating at theΥ(4 S ) are discussed when considering future facilities which could offer improved precisionfor BR ( B u → τ + ν τ ).Another promising approach to probe the plane [tan β , m H ± ] is via the tree-level H ± contribution to the semileptonic decays B → Dτ ν [33, 35, 36, 43]. We note here that H ± can mediate the analogous leptonic decays K ± → µ ± ν [13, 44] and D ± s → µ ± ν, τ ± ν [13, 45]but constraints on the plane [tan β , m H ± ] from these processes are not yet competitive.However, such processes might play a role in the future with increased experimental precisionand reduced theoretical uncertainties. III. AT THE Z PEAK
In this section we discuss the searches for B ± → τ ± ν using data from e + e − collisions atthe Z peak ( √ s ∼
91 GeV). It was pointed out in [17] that such searches would also besensitive to the decay B ± c → τ ± ν . Assuming that the detection efficiencies are the same the ratio of τ ± ν events originating from B ± → τ ± ν and B ± c → τ ± ν is given by: N c N u = (cid:12)(cid:12)(cid:12)(cid:12) V cb V ub (cid:12)(cid:12)(cid:12)(cid:12) f ( b → B ± c ) f ( b → B ± ) (cid:18) f B c f B (cid:19) M B c M B τ B c τ B (cid:16) − m τ M Bc (cid:17) (cid:16) − m τ M B (cid:17) (8)The largest uncertainty in the determination of N c is from the transition probability f ( b → B ± c ) and the decay constant f B c . The magnitude of N c is suppressed by the small f ( b → B ± c )but this can be compensated by the large ratio ( V cb f B c ) / ( V ub f B ) . Consequently N c canbe similar in magnitude to N u . In the analysis of [17] three scenarios were defined in orderto account for the error in the determination of N c /N u : “Central” and “Max/Min” ( ± σ above/below the central values of the input parameters). Since the analysis of [17] therehave been significant improvements in the measurements of V ub and V cb . In addition, thedecay constant f B has now been calculated in unquenched lattice QCD with smaller errorsand a central value considerably larger [46] than the values used in both [17] and the L3analysis [14]. We are unaware of an unquenched lattice QCD calculation of f B c and the In the non-SUSY 2HDM (Model II) b → sγ constrains m H ± independently of tan β . A recent study[40, 41] obtains m H ± >
295 GeV. In practice, the shorter lifetime of B ± c would result in a slightly inferior detection efficiency [17]. IG. 2: The constraint on the tan β - m H ± plane in the 2HDM (Type III) from the measurement of BR ( B ± → τ ± ν τ ). The coloured areas correspond to allowed ranges of tan β and m H ± for variousvalues of ˜ ǫ = 0 , . , − .
01 (green, blue and red, respectively), BR ( B u → τ + ν τ ) (1- σ range,overlapping) and f B (overlapping). error in this parameter has not been reduced significantly since [17]. The main uncertaintyin the ratio N c /N u is from f ( b → B ± c ), which in [17] was varied in the range suggested bytheoretical estimations [47]: 2 × − < f ( b → B ± c ) < × − . At that time B ± c was stillundiscovered and hence there was no measurement of f ( b → B ± c ).However, f ( b → B ± c ) can now be extracted (although with a large uncertainty) from themeasurement of ratio of B ± c → J/ Ψ ℓ + ν ℓ to B ± → J/ Ψ K ± which is defined by: R ℓ = σ ( B + c ) · BR( B c → J/ψℓ ± ν ℓ ) σ ( B + ) · BR( B → J/ψK + ) (9)Tevatron Run II data gives R e = 0 . ± .
07 [20], and the denominator in eq. (9) has beenmeasured precisely by various experiments. The transition probability f ( b → B c ) determines σ ( B + c ) and several theoretical calculations are available for BR( B c → J/ψℓ ± ν ℓ )). In Fig. 3we display contours of R e as a function of BR( B c → J/ Ψ e + ν e ) and f ( b → B c ), and theband denotes the prediction of the various theoretical calculations for BR( B c → J/ Ψ e + ν e )whose values lie in the range (2 . ∼ . R e = 0 . ± .
05 [18] is accommodated by f ( b → B c ) = 1 . × − .However, in order to satisfy the central value of the Run II measurement, the transitionprobability f ( b → B c ) needs to be 4 . × − . An even larger value for f ( b → B c ) was6 f ( b → B c )10 B R ( B c → J / Ψ e + ν e ) . . . . . . . . . . . . . FIG. 3: Contours of R e in the plane of BR( B c → J/ Ψ e + ν e ) and transition probability f ( b → B c ).The shaded region denotes the theoretical prediction for BR( B c → J/ Ψ e + ν e ). suggested in Ref. [49]. Such unexpectedly large values of f ( b → B c ), which are indicatedby Tevatron Run II data, would significantly enhance the contribution of B ± c → τ ± ν to theLEP searches for B ± → τ ± ν . Of course, f ( b → B c ) is dependent on the available centre ofmass energy (at higher energies there is more phase space to produce a charm quark insteadof a light quark), but the value of f ( b → B c ) is expected to be of comparable size at LEPand at the LHC [47]. In our numerical analysis in Sec. III A, we will consider values of f ( b → B ± c ) up to 5 · − . A. The LEP search for B ± → τ ± ν and the contribution of B ± c → τ ± ν Three LEP collaborations searched for the decay B ± → τ ± ν using data taken at the Z peak ( √ s = 91 GeV). L3 [14] used around 1 . × hadronic decays of the Z boson whichcorresponds to about half their total data [22]. DELPHI [15] and ALEPH [16] used theirfull data samples of around 3 . × hadronic decays of the Z boson. The best sensitivitywas from the L3 experiment which set the upper limit BR( B ± → τ ± ν ) < . × − . The L3limit is of particular interest since it could be improved if the full data sample of ∼ . × hadronic Z boson decays were used.The LEP searches were sensitive to τ ± ν events originating from both B ± → τ ± ν and B ± c → τ ± ν . Hence the published limits constrain the “effective branching ratio” defined by:BR eff = BR( B ± → τ ± ν ) (cid:18) N c N u (cid:19) (10)This expression applies to searches for B ± → τ ± ν at the Z peak. For searches at the Υ(4 S )clearly N c = 0 and BR eff =BR( B ± → τ ± ν ). In our numerical analysis in this section we willuse eq. (10) with the experimental value of BR( B ± → τ ± ν ) as input. The calculation of N c /N u in eq. (10) uses eq .(8) (i.e. the expression for the SM) with input parameters takenfrom Table I. Our analysis can be applied to any model for which N c /N u ∼ | N c /N u | SM ,which includes the 2HDM because the scale factors in eq. (6) are almost equal.7he “max” scenario of [17] showed that the current limit of BR eff < . × − would besensitive to the SM rate for BR( B ± → τ ± ν ). The measurements of BR( B ± → τ ± ν ) at the B factories are consistent with the SM prediction which suggests that the L3 search was notso far from observing a signal.In Fig. 4 we plot BR eff in the plane [ f ( b → B ± c ) , f B c ], for two values of BR( B ± → τ ± ν )corresponding to the central value and 1 σ above the world average. All other parametersare held at their central values from Tab. I. The region above the contour BR eff = 5 . × − (red/dark grey) is excluded by the L3 limit [14], while the contour BR eff = 4 × − representsthe hypothetical sensitivity if the full data of 3 . × hadronic decays of the Z bosonwere used. The green/light grey area between the two contours is the area where a signalwould be seen if the full dataset were studied. Depending on the other input parametersand the B ± → τ ± ν branching ratio, this area can cover a very significant part of the[ f ( b → B ± c ) , f B c ] parameter space. We therefore consider a re-analysis using the full L3dataset very worthwhile.A different way of studying the number of B c events was followed in [17]. The numberof B c events per B u event can be calculated as a function of f ( b → B ± c ), and the authorsobtained N c /N u = 1 . f ( b → B ± c ) / − for central values of the input parameters (“max”scenario: 2.3). With updated values for the input parameters, we now find N c N u = 0 . · f b → B c / − (central values)1 . · f b → B c / − (“optimistic” values) (11)where for the “optimistic” values of the parameters from Tab. I we have chosen that end ofthe 1- σ range that results in a higher value for N c /N u , and the “optimistic” f B c was chosento be 550 MeV.These numbers are lower than those of [17] mainly because the central value of V ub /V cb has increased in the last ten years. The inverse of this ratio enters N c /N u quadratically andtherefore reduces this quantity. On the other hand, experimental data does not precludevalues of f ( b → B ± c ) which are much higher (a few × − ) than the theoretical estimates,and so the admixture of B ± c → τ ± ν can still easily reach 100%. B. Giga Z option at a future e + e − Linear Collider
A future e + e − Linear Collider operating at the Z peak with a luminosity of 5 × cm − s − could produce 10 Z bosons in 50 −
100 days of operation [23, 24, 25, 26])This corresponds to roughly 1000 times the number of Z bosons recorded at each LEP de-tector. Historically, limits on B ± → τ ± ν from Z decays have been comparable to (if notstronger than) those at Υ(4 S ) for the same number of Z bosons and B mesons. For example,the CLEO collaboration obtained BR( B ± → τ ± ν ) < . × − with 9 . × B mesons[50], while L3 obtained BR( B ± → τ ± ν ) < . × − with 1 . × hadronic decays of the Z boson.High luminosity B factories [5, 6, 7, 8, 9, 10, 11] anticipate data samples of 10 B mesons.By the time of operation of a Giga Z the two main sources of uncertainty in N c (and henceBR eff ) will have been substantially reduced. The error in f ( b → B c ) will be reduced fromLHC-b measurements [51] of the cross-section in Eq. 9, and improved lattice calculations of f B c and/or ( f B c /f B ) would also reduce the error in N c . In Table III we present the requirednumber of B mesons and Z bosons for a precision of 20% and 4% in the measurement of8 IG. 4: The effective BR ( B ± → τ ± ν ) at the Z peak in the plane [ f ( b → B ± c ) , f B c ]. The publishedL3 limit and a possible stricter limit are indicated. In the upper (lower) panel BR( B ± → τ ± ν ) istaken to be the central value of the world average measurement (the 1- σ upper value). rror BR( B ± /B ± c → τ ± ν ) High Lum. B Factory ( B mesons) Giga Z ( Z bosons)20% 2 . × . ×
4% 8 . × × TABLE III: Required number of B mesons ( Z bosons) for a precision of 20% and 4% in themeasurement of BR( B ± /B ± c → τ ± ν ), assuming a signal of BR eff = 4 ± × − at L3. B ± → τ ± ν at a high luminosity B factory and BR eff at Giga Z . The numbers for a highluminosity B factory are taken from [10]. For the Giga Z precision we assume a signal ofBR eff = 4 ± × − (50% error) at L3 with 3 . × hadronic Z decays, and scale the errorby 1 / √ N , where N is the total number of Z bosons at Giga Z divided by the full L3 datasample of ∼ . × Z bosons.It is clear from Table III that a Giga Z facility might be capable of measuring BR eff in Eq. 10 with similar precision to that anticipated for B ± → τ ± ν at high luminosity B factories. We believe that this competitiveness of the Giga Z facility has not been pointedout for the leptonic B decays although it has been emphasized for the decay B → X s νν in [24]. If both facilities were realized this would enable competitive and complementaryconstraints on tan β/m H ± in the context of models with H ± . IV. CONCLUSIONS
The decay B ± → τ ± ν has been observed at the e + e − B factories and is recognized as animportant constraint on the parameter tan β/m H ± in the context of models with Two Higgsdoublets. We studied the contribution of B ± c → τ ± ν to the LEP searches for B ± → τ ± ν (first pointed out in [17]), whose main uncertainty is from the value for the transitionprobability b → B ± c which is now being measured at the Tevatron Run II. Using valuesof this transition probability which are consistent with the current Tevatron measurements(which accommodate values significantly larger than the theoretical estimations), we foundthat the contribution of B ± c mesons to the search for B ± /B ± c → τ ± ν can be as large as thatof B ± . We suggested that a re-analysis of the L3 search for B ± → τ ± ν [14] using all the datataken at the Z peak could provide a signal for the admixture of B ± /B ± c → τ ± ν . Finally, itwas pointed out that the Giga Z option of a future e + e − collider could offer measurementsof these leptonic B ± /B ± c decays which are comparable in precision and complementary withthose anticipated at the proposed high luminosity B factories. Acknowledgements
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