University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln
Kenneth Bloom Publications Research Papers in Physics and Astronomy
11-1-1997
Determination of the Michel parameters and the τ neutrino helicity in τ decay
J. P. Alexander Cornell University
Kenneth A. Bloom University of Nebraska-Lincoln, [email protected]
CLEO Collaboration
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Alexander, J. P.; Bloom, Kenneth A.; and Collaboration, CLEO, "Determination of the Michel parameters and the τ neutrino helicity in τ decay" (1997). Kenneth Bloom Publications. 170. https://digitalcommons.unl.edu/physicsbloom/170
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Kenneth Bloom Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. PHYSICAL REVIEW D VOLUME 56, NUMBER 9 1 NOVEMBER 1997
Determination of the Michel parameters and the neutrino helicity in decay
J. P. Alexander, C. Bebek, B. E. Berger, K. Berkelman, K. Bloom, D. G. Cassel, H. A. Cho, D. M. Coffman, D. S. Crowcroft, M. Dickson, P. S. Drell, K. M. Ecklund, R. Ehrlich, R. Elia, A. D. Foland, P. Gaidarev, R. S. Galik, B. Gittelman, S. W. Gray, D. L. Hartill, B. K. Heltsley, P. I. Hopman, J. Kandaswamy, P. C. Kim, D. L. Kreinick, T. Lee, Y. Liu, G. S. Ludwig, J. Masui, J. Mevissen, N. B. Mistry, C. R. Ng, E. Nordberg, M. Ogg,* J. R. Patterson, D. Peterson, D. Riley, A. Soffer, B. Valant-Spaight, and C. Ward Cornell University, Ithaca, New York 14853
M. Athanas, P. Avery, C. D. Jones, M. Lohner, C. Prescott, J. Yelton, and J. Zheng University of Florida, Gainesville, Florida 32611
G. Brandenburg, R. A. Briere, Y. S. Gao, D. Y.-J. Kim, R. Wilson, and H. Yamamoto Harvard University, Cambridge, Massachusetts 02138
T. E. Browder, F. Li, Y. Li, and J. L. Rodriguez University of Hawaii at Manoa, Honolulu, Hawaii 96822
T. Bergfeld, B. I. Eisenstein, J. Ernst, G. E. Gladding, G. D. Gollin, R. M. Hans, E. Johnson, I. Karliner, M. A. Marsh, M. Palmer, M. Selen, and J. J. Thaler University of Illinois at Champaign-Urbana, Urbana, Illinois 61801
K. W. Edwards Carleton University, Ottawa, Ontario, Canada K1S 5B6 and the Institute of Particle Physics, Canada
A. Bellerive, R. Janicek, D. B. MacFarlane, K. W. McLean, and P. M. Patel McGill University, Montre´al, Que´bec, Canada H3A 2T8 and the Institute of Particle Physics, Canada
A. J. Sadoff Ithaca College, Ithaca, New York 14850
R. Ammar, P. Baringer, A. Bean, D. Besson, D. Coppage, C. Darling, R. Davis, N. Hancock, S. Kotov, I. Kravchenko, and N. Kwak University of Kansas, Lawrence, Kansas 66045
S. Anderson, Y. Kubota, M. Lattery, S. J. Lee, J. J. O’Neill, S. Patton, R. Poling, T. Riehle, V. Savinov, and A. Smith University of Minnesota, Minneapolis, Minnesota 55455
M. S. Alam, S. B. Athar, Z. Ling, A. H. Mahmood, H. Severini, S. Timm, and F. Wappler State University of New York at Albany, Albany, New York 12222
A. Anastassov, S. Blinov,† J. E. Duboscq, K. D. Fisher, D. Fujino,‡ K. K. Gan, T. Hart, K. Honscheid, H. Kagan, R. Kass, J. Lee, M. B. Spencer, M. Sung, A. Undrus,† R. Wanke, A. Wolf, and M. M. Zoeller Ohio State University, Columbus, Ohio 43210
B. Nemati, S. J. Richichi, W. R. Ross, P. Skubic, and M. Wood University of Oklahoma, Norman, Oklahoma 73019
M. Bishai, J. Fast, E. Gerndt, J. W. Hinson, N. Menon, D. H. Miller, E. I. Shibata, I. P. J. Shipsey, and M. Yurko Purdue University, West Lafayette, Indiana 47907
L. Gibbons, S. Glenn, S. D. Johnson, Y. Kwon, S. Roberts, and E. H. Thorndike University of Rochester, Rochester, New York 14627
C. P. Jessop, K. Lingel, H. Marsiske, M. L. Perl, D. Ugolini, R. Wang, and X. Zhou Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309
0556-2821/97/56͑9͒/5320͑10͒/$10.0056 5320 © 1997 The American Physical Society 56 DETERMINATION OF THE MICHEL PARAMETERS AND... 5321
T. E. Coan, V. Fadeyev, I. Korolkov, Y. Maravin, I. Narsky, V. Shelkov, J. Staeck, R. Stroynowski, I. Volobouev, and J. Ye Southern Methodist University, Dallas, Texas 75275
M. Artuso, A. Efimov, F. Frasconi, M. Gao, M. Goldberg, D. He, S. Kopp, G. C. Moneti, R. Mountain, S. Schuh, T. Skwarnicki, S. Stone, G. Viehhauser, and X. Xing Syracuse University, Syracuse, New York 13244
J. Bartelt, S. E. Csorna, V. Jain, and S. Marka Vanderbilt University, Nashville, Tennessee 37235
R. Godang, K. Kinoshita, I. C. Lai, P. Pomianowski, and S. Schrenk Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
G. Bonvicini, D. Cinabro, R. Greene, L. P. Perera, and G. J. Zhou Wayne State University, Detroit, Michigan 48202
B. Barish, M. Chadha, S. Chan, G. Eigen, J. S. Miller, C. O’Grady, M. Schmidtler, J. Urheim, A. J. Weinstein, and F. Wu¨rthwein California Institute of Technology, Pasadena, California 91125
D. M. Asner, D. W. Bliss, W. S. Brower, G. Masek, H. P. Paar, S. Prell, and V. Sharma University of California, San Diego, La Jolla, California 92093
J. Gronberg, T. S. Hill, R. Kutschke, D. J. Lange, S. Menary, R. J. Morrison, H. N. Nelson, T. K. Nelson, C. Qiao, J. D. Richman, D. Roberts, A. Ryd, and M. S. Witherell University of California, Santa Barbara, California 93106
R. Balest, B. H. Behrens, K. Cho, W. T. Ford, H. Park, P. Rankin, J. Roy, and J. G. Smith University of Colorado, Boulder, Colorado 80309-0390 ͑CLEO Collaboration͒ ͑Received 15 May 1997͒ Using the CLEO II detector at the Cornell Electron Storage Ring operated at ͱsϭ10.6 GeV, we have determined the Michel parameters , , and ␦ in ϯ lϯ¯ decay as well as the neutrino helicity parameter → h in ϯ ϯ0 decay. From a data sample of 3.02ϫ106 produced pairs we analyzed events of the → topologies eϩeϪ ϩϪ (lϮ¯)(ϯ0) and eϩeϪ ϩϪ (Ϯ0¯)(ϯ0). We obtain ϭ0.747 → → → → ϭ0.747Ϯ0.010Ϯ0.006, ϭ1.007Ϯ0.040Ϯ0.015, ␦ϭ0.745Ϯ0.026Ϯ0.009, and h ϭϪ0.995Ϯ0.010 Ϯ0.003, where we have used the previously determined sign of h ͓ARGUS Collaboration, H. Albrecht et al., Z. Phys. C 58,61͑1993͒; Phys. Lett. B 349, 576 ͑1995͔͒. We also present the Michel parameters as determined from the electron and muon samples separately. All results are in agreement with the standard model VϪA interaction. ͓S0556-2821͑97͒06819-7͔
PACS number͑s͒: 13.35.Dx, 14.60.Fg
I. INTRODUCTION G l ␥ ¯ Ϫ ¯ ¯ ␥ Ϫ ϭ4 ͚ g⑀͓u⑀͑l ͒⌫␥vj͑l͔͓͒ui͑͒⌫ u͑ ͔͒, M & ␥ϭS,V,T The most general, local, derivative-free, and lepton- ⑀,ϭR,L number-conserving four-fermion point interaction ͓1–3͔ for ͑1͒ leptonic decays yields in the helicity projection form ͓4͔ the matrix element where Gl parametrizes the total strength of the interaction and l refers to e or . The matrices ⌫␥ define the properties of the two currents under a Lorentz transformation with ␥ϭS,V,T for scalar, vector, and tensor interactions. The in- dices ⑀ and label the right or left handedness (R,L) of the charged leptons. For a given ⑀, , and ␥, the handedness of *Permanent address: University of Texas, Austin, TX 78712. the neutrinos labeled by j and i are fixed. Only ten of the † ␥ Permanent address: BINP, RU-630090 Novosibirsk, Russia. twelve complex coupling constants g⑀ are linearly indepen- ‡Permanent address: Lawrence Livermore National Laboratory, dent. In the standard model VϪA interaction, the only non- V Livermore, CA 94551. zero coupling constant is gLLϭ1. 5322 J. P. ALEXANDER et al. 56
The interaction described by Eq. ͑1͒ is fully determined Despite the progress made in recent years ͓7–18͔, the by 19 real parameters. Without measuring the neutrinos and determination of the space-time structure in leptonic and the spin of the outgoing charged lepton, only the four Michel semihadronic decays is still an order of magnitude less parameters ͓1–3͔ , , , and ␦ are experimentally acces- precise than in decay. This indicates a need for high- sible. They are bilinear combinations of the coupling con- precision measurements of the Michel parameters in leptonic ␥ stants g⑀ and appear in the predicted energy spectrum of the decays as well as of the neutrino helicity in semihadronic charged lepton lϯ emitted in the decay ϯ lϯ¯. In the decays. In this paper, we present measurements of , , ␦, → and the neutrino helicity parameter h from an analysis of rest frame, neglecting radiative corrections and terms propor- 2 2 ϩ Ϫ ϩ Ϫ Ϯ ϯ 0 Ϯ 0 ϯ 0 tional to ml /m , this spectrum is given by e e (l ¯)( ) and ( ¯)( ) events.→ → ϯ ϯ 2 5 Ϯ ϯ 0 d⌫͑ l ¯͒ GFm The (l ¯)( ) sample used here is correlated with → ϭ x2 3͑1Ϫx͒ϩ 2 ͑4xϪ3͒ d⍀dx 1924 ͩ 3 that of Ref. ͓17͔. There, the Michel parameters and were determined with emphasis on a precise measurement of ,in ml 1Ϫx which the sensitivity comes mainly from the low-momentum ϩ6 ϯ cos ͑1Ϫx͒ m x P ͫ part of the muon spectrum. Here, the emphasis lies on the determination of the spin-dependent Michel parameters and 2 ␦. ϩ ␦͑4xϪ3͒ , ͑2͒ 3 ͬͪ
where xϭ2El /m is the scaled charged lepton energy, II. METHOD OF THE MEASUREMENT the polarization, and the angle between spin and leptonP momentum. The standard model VϪA charged weak current Equation ͑2͒ shows that the measurement of and ␦ re- ϩ Ϫ is characterized by ϭ3/4, ϭ0, ϭ1, and ␦ϭ3/4. quires the knowledge of the spin orientation. In e e an- A measurement of the Michel parameters allows one to nihilation at ͱsϷ10 GeV the average polarization is zero ␥ and no information on and ␦ can be obtained from single limit the coupling constants g⑀ . For example, and ␦ de- decays. However, spin-spin correlations exist between the termine the probability PR for a right-handed lepton to ϩ Ϫ ϩ Ϫ participate in leptonic decays: two leptons in e e , leading to correlations be- tween kinematical properties→ of the decay products. These 1 S 1 S V V T correlations have been used before ͓8,10,12͔ for the determi- P ϭ ͉g ͉2ϩ ͉g ͉2ϩ͉g ͉2ϩ͉g ͉2ϩ3͉g ͉2 R 4 RR 4 LR RR LR LR nation of , ␦, and h , where leptonic as well as semihad- 1 1 ronic decays served as spin analyzers. Here we use the semi- ϭ 2 ͓1ϩ 9 ͑3Ϫ16␦͔͒. ͑3͒ hadronic decay ϯ ϯ0 as spin analyzer. Its → In semihadronic decays the strong interaction imposes advantages are a large branching fraction, a very well under- constraints on the general form of the matrix element. De- stood hadronic current, and an experimentally clean signal. pending on the quantum numbers of the intermediate had- In the Born approximation, the matrix element for ronic state, some of the interactions taken into account by the differential cross section of eϩeϪ ϩϪ Ϯ ϯ 0 → Eq. ͑1͒ are forbidden. For the decay ϯ ϯ0, with a (l ¯)( ) has, after integration over the unob- → vector meson as the intermediate hadronic→ state, only vector served neutrino degrees of freedom and summation over un- and axial vector interactions are possible. Denoting the vec- observed spins, the structure ͑see, for example, Ref. ͓19͔͒ tor coupling by gV and the axial vector coupling by gA , the ϯ ϯ 0 general matrix element for the decay can be 2 ␣ 2 2 → ͉ ͉ ϭHP͓E ϩE ϩE ͔ϩh HЈC ͓EЈ ϩ␦EЈ ͔. parametrized by h ϭ2g g /(g ϩg ). The parameter h 1 2 3 ␣ 1 2 V A V A M is proportional to the neutrino helicity. In the standard ͑5͒ model, with purely left-handed neutrinos, one expects h ϭϪ1. Omitting the total strength of the interaction and The first term is the spin-averaged part of the differential other constant factors, the general matrix element is ͓5͔ cross section. The second term contains the spin correlation. As defined by Eq. ͑4͒, H is the spin-averaged part of the ϯ ϯ 0 2 ͉ ͑ ͉͒ ϰHϮh ͑sHЈ͒ matrix element for the semihadronic decay ϯ ϯ0, M → → whereas HЈ is the spin-dependent part. The symbols Ei and ϭ2͑kQ͒͑qQ͒Ϫ͑kq͒͑QQ͒Ϯm h s EiЈ are the Lorentz invariant formulations of the correspond- ing terms in Eq. ͑2͒. The spin-averaged pair production is ϫ͓2Q ͑kQ͒Ϫk ͑QQ͔͒, ͑4͒ denoted by P and the production spin correlation matrix by C␣. where the four-vectors k, q, s, and Q denote the neutrino The spin analyzer ϯ ϯ0 can resolve the ratio of momentum, the momentum, the spin vector, and longitudinal to transverse→ polarization of the intermediate pϯϪp0, respectively. Using the shorthand notations H meson, but, because of the absence of interference terms, and HЈ, the polarimeter vector of the decay is hϭHЈ /H cannot separate its transverse polarization into the left- and ͓5,6͔.Intherest frame only the three-vector hជ of the po- right-handed part. Thus our (lϮ¯)(ϯ0) events are sen- ជ sitive to , , h , and h ␦ ͓see Eq. ͑5͔͒, whereas our larimeter vector is relevant, where h is aligned to the ex- pected spin direction of the decay products. (Ϯ0¯)(ϯ0) events allow a determination of the 56 DETERMINATION OF THE MICHEL PARAMETERS AND... 5323
product h2 . The signs of and h are well known from Monte Carlo integration over the measurable quantities aជ in other experiments ͓7,8͔ and no attempt is made to remeasure the denominator of Eq. ͑6͒ is done with a full detector simu- them in this analysis. lation. This technique was used in Ref. ͓8͔. Here we have Not all of the kinematical quantities needed to evaluate applied only minor changes, such as taking the radiative cor- Eq. ͑5͒ for each detected event are well determined. For ex- rections in the semihadronic decay into account. A full de- ample, the azimuthal angle of the unmeasured momentum scription can be found in Ref. ͓20͔. around the two-pion momentum can take on a range of val- The effectiveness of this technique has been demonstrated ues, restricted by kinematical constraints. Initial-state radia- by generating events with the KORALB/TAUOLA ͓5,21͔ Monte tion, radiative corrections to the decays ϯ lϯ¯ and Carlo program and applying the fit method to these events. ϯ ϯ0, external bremsstrahlung, and uncertainties→ of The results are compatible with the input within the statisti- the→ measured momenta will also modify the evaluation of cal errors of the test, which are of order 0.01%. These tests Eq. ͑5͒. have been performed for standard model input values as well We determine the Michel parameters from a single event as for non-standard-model values. Thus, at the level of accu- likelihood fit to the selected data samples. The indetermin- racy needed here, we have demonstrated that the method is ancies described above are taken into account in the likeli- unbiased and the factorization assumption mentioned above hood function by forming a weighted sum over all possible is justified. kinematical configurations. The weights are derived by as- suming that radiative effects and the detector resolution fac- torize from the Born level matrix element and do not depend III. DATA SELECTION on the fit parameters. Formally, the per event likelihood function is taken to be The measurements presented here were performed with the CLEO II detector ͓22͔ at the Cornell Electron Storage L͑⌰͉aជ ͒ϵP͑aជ͉⌰͒ Ring. The data sample used here was collected at center of
2 mass energies around ͱsϭ10 GeV. The integrated luminos- ͦ ͑␣ជ ,ជ ͉⌰͒ͦ ͑␣ជ ͒w͑aជ ,ជ ͉␣ជ ͒dជ d␣ជ Ϫ1 6 ͵ M ity is Ϸ3.5 fb , with about 3.02ϫ10 pairs produced. ϭ , Events with exactly two oppositely charged tracks are se- ͵ ͦ ͑␣ជ ,ជ ͉⌰͒ͦ2͑␣ជ ͒w͑aជ ,ជ ͉␣ជ ͒dជ d␣ជ daជ lected. Each track must have a momentum greater than M 500 MeV/c and its distance of closest approach to the inter- ͑6͒ action point in the plane transverse to the beam must be less than 10 mm. The polar angle of each track relative to the where ͉ ͉2 is given by Eq. ͑5͒. The vector ⌰ represents the beam must satisfy the condition cos Ͻ0.71. The two set of parametersM (,,h ,h ␦) that are determined in ͉ ͉ tracks are required to be separated by an opening angle of the fit. As discussed below, we also include the ⌰ depen- more than 90°. dence of all significant sources of background in the event To suppress non- background we require that not more likelihood. The vector a contains all measured quantities, ជ than one of the two tracks has a momentum greater than 85% i.e., the momenta of the charged lepton and the two pions. of the beam energy. The total energy ͑due to photons, show- ជ The vector  contains all unmeasured quantities, such as ers associated with charged tracks, and all other sources͒ those associated with the neutrinos, the photons of the initial- measured in the electromagnetic calorimeter has to be greater state bremsstrahlung that mostly escape undetected down the than 20% and less than 85% of ͱs. Additionally, the mo- beam pipe, radiated photons in the decay, and photons from menta pជ of the two tracks have to satisfy the condition external bremsstrahlung. The vector ␣ជ represents the value i of the measured quantities before resolution and radiative ͉pជ 1ϩpជ 2͉/(͉pជ1͉ϩ͉pជ2͉)Ͼ0.05 to suppress cosmic rays. We reconstruct 0 decays using calorimeter show- ជ ␥␥ effects. The weight w(aជ ,͉␣ជ ) contains all of the resolution ers that are not matched→ to a charged track, with an energy and radiative effects, and the integrals over ␣ perform con- ជ greater than 50 MeV and a polar angle of ͉cos ͉Ͻ0.71. To volutions with the detector resolution. The acceptance func- suppress background from decays with more 0 mesons tion of the detector is denoted by and depends only on ␣. ជ than the mode under study, we reject events containing ad- The denominator in Eq. ͑6͒ ensures that the likelihood inte- ditional showers that are not associated with a reconstructed grated over aជ is normalized to unity for all values of ⌰. 0 The integration over the unmeasured quantities is done meson, that lie more than 30 cm from the closest charged analytically as far as possible. The remaining integration is track projection into the calorimeter, and that have energies most conveniently computed numerically using Monte Carlo of more than 75 MeV for polar angles of ͉cos ͉Ͻ0.71 and methods. In the Monte Carlo algorithm used here, many tri- 100 MeV for ͉cos ͉Ͼ0.71. als are executed for each observed event to generate the un- In the lepton-versus- sample exactly one 0 is required with Ϫ4Ͻ(m Ϫm 0)/ Ͻ3, where the mass resolution measured quantities ជ ͑radiated photons͒ and the ‘‘before ␥␥ m␥␥ is typically between 5 MeV/c2 and 10 MeV/c2 de- radiation and detector resolution’’ values ␣ជ of the measured m␥␥ 0 quantities. Not all trials are successful in generating ជ and ␣ជ pending on the energy. The momentum of the recon- consistent with the kinematics of pair events. We have structed 0 has to be greater than 300 MeV/c. chosen the number of trials to be 450 such that the number of The track further away in angle from the reconstructed 0 successful trials for most signal events is large enough to is required to be either an electron or a muon. Tracks are achieve adequate precision for this integration. ͑In addition, identified as electrons when their momentum and dE/dx in- the fraction of trials that are successful f hit is a measure of formation from the tracking system, as well as the energy the goodness of the hypothesis that the event is a pair.͒ The measurement in the electromagnetic calorimeter, are consis- 5324 J. P. ALEXANDER et al. 56
TABLE I. Background contribution from other decays.
Electrons, 33 531 Accepted events Muons, 21 680 Accepted events mesons, 11 177 Accepted events Estimated Estimated Estimated Event topology background ͑%͒ Event topology background ͑%͒ Event topology background ͑%͒
(eϮ¯)(ϯ00) 1.78Ϯ0.20 (Ϯ¯)(ϯ00) 1.73Ϯ0.20 (Ϯ0¯)(ϯ00) 3.95Ϯ0.45 (eϮ¯)(Kϯ0) 1.94Ϯ0.20 (Ϯ¯)(Kϯ0) 2.00Ϯ0.20 (Ϯ0¯)(Kϯ0) 4.31Ϯ0.50 (Ϯ¯)(ϯ0) 0.14Ϯ0.03 (Ϯ¯)(ϯ0) 1.29Ϯ0.18 (eϮ¯)(ϯ) 0.13Ϯ0.02 (Ϯ¯)(ϯ) 0.14Ϯ0.03 remaining sources 0.96Ϯ0.10 remaining sources 0.90Ϯ0.10 remaining sources 2.04Ϯ0.20
⌺ 4.95Ϯ0.30 ⌺ 6.06Ϯ0.35 ⌺ 10.30Ϯ0.70
tent with the electron hypothesis. Tracks with momenta (ϯ0)(Ϯ0¯) events. The ‘‘remaining sources’’ ͑Table greater than 1.5 GeV/c are identified as muons if they match I͒ are from a variety of modes. to hits in the muon counters beyond at least three absorption lengths of material. IV. DATA ANALYSIS The invariant mass of the reconstructed 0 and the 2 The spin correlation used in this analysis can be most charged pion candidate has to satisfy m ϯ 0Ͼ0.5 GeV/c . easily illustrated with the spin sensitive variable ͓24͔ of The missing mass m of the event has to satisfy the con- miss the decay ϯ ϯ0. In the rest frame, assuming the dition mmissϾ0.1ͱs. Additionally, we require the Monte standard model→VϪA interaction (h ϭϪ1), the matrix el- Carlo success rate f ͑see Sec. II͒ to be at least 6.6%. After hit ement given by Eq. ͑4͒ can be written as these three cuts the remaining background from Bhabha events, two-photon interactions, and qq¯ events is negligible, ϯ ϯ 0 2 ͉ ͑ ͉͒ ϰH͑1Ϯszhz͒, ͑7͒ as has been verified by a careful comparison of relevant ki- M → nematical distributions between data and pair Monte Carlo where the z axis has been chosen as spin quantization axis. events ͑KORALB/TAUOLA ͓5,21͔͒. For qq¯background this has The symbols sz and hz denote the z components of the spin also been confirmed by an analysis of qq¯ Monte Carlo vector and the polarimeter vector introduced in Sec. I ͓sz is events. just the longitudinal polarization and is equivalent to in P In the -versus- sample exactly two 0 mesons are re- Eq. ͑2͔͒. quired with Ϫ4Ͻ(m Ϫm 0)/ Ͻ3. A momentum cut In the case where the flight direction is chosen as the z ␥␥ m␥␥ of greater than 300 MeV/c is applied on the reconstructed axis, i.e., the spin quantization axis, left-handed leptons 0 mesons. The 0 mesons are associated with the charged correspond to szϽ0 and right-handed leptons to szϾ0. With this, Eq. ͑7͒ shows that left-handed Ϫ leptons ͑right- tracks by their nearness in angle. The invariant masses ϩ handed leptons͒ have preferentially negative-hz values, mϯ0 of the meson candidates have to be greater than Ϫ ϩ 0.5 GeV/c2. These cuts are identical to those used in the whereas right-handed leptons ͑left-handed leptons͒ l-vs- selection and suppress backgrounds from pair events have preferentially positive-hz values. Averaging hz over the with different decay modes. In addition, the missing trans- kinematically allowed rest frames yields the variable verse momentum of the event, p , and E , the total energy ͑i.e., integrating over the azimuthal angle of the momentum T tot around the two-pion momentum͒. of the observed particles, have to satisfy p /(ͱsϪE )Ͼ0.1, T tot Figure 1 shows the dependence of the measured electron p /(ͱs/2)Ͼ0.075, and E /ͱsϾ0.3. We again require T tot momentum spectrum on . One clearly sees that for f Ͼ6.6%. We verified with Monte Carlo studies, as de- hit negative- values, high-momentum leptons are preferred by scribed in the preceding paragraph, that after these cuts the the data, whereas for positive- values, low-momentum lep- contribution from non- background is negligible. tons are preferred. This correlation indicates h ϷϪ1, as This selection results in a sample of 66 388 accepted expected by the standard model. For h ϭ0, which is events, comprising 33 531 candidates in the topology (eϯ¯)(Ϯ0¯), 21 680 candidates in (ϯ¯)(Ϯ0¯), equivalent to zero spin correlation, the lepton momentum and 11 177 candidates in (ϯ0)(Ϯ0¯). These num- spectrum is independent of , as can be seen in Fig. 1. The bers of events are in good agreement with expectations based plots corresponding to Fig. 1 for the -vs- sample are very on world average branching ratios ͓23͔. similar and are not shown here. Figure 2 shows for the In all three samples, e vs , vs , and vs , the -vs- sample the spectrum of one side of the event for background from non- events is insignificant. The back- different values of from the other side ͑two entries per 2 ground from events is estimated by using the KORALB/ event͒. Again the data favor h Ϸ1, in agreement with the TAUOLA Monte Carlo program ͓5,21͔. The results are listed standard model. in Table I, where the errors reflect statistical, experimental, To take the background from other events into account, and theoretical uncertainties. The total contribution from the fit function of Eq. ͑6͒ is extended to include background: background is about 5% for (eϯ¯)(Ϯ0¯), around 6% ϯ Ϯ 0 for ( ¯)( ¯), and approximately 10% for Liϭ͓1Ϫ͑␣1ϩ•••ϩ␣n͔͒Siϩ␣1B1,iϩ•••ϩ␣nBn,i , ͑8͒ 56 DETERMINATION OF THE MICHEL PARAMETERS AND... 5325
FIG. 1. Electron momentum spectrum in different regions of . The variable is sensitive to the spin of the lepton in the decay ϯ ϯ0 ͑Ͻ0 Ϫ left handed, Ͼ0 Ϫ right handed͒. The data ͑points with errors͒ as well as the Monte Carlo expectation for → ⇒ ⇒ h ϭϪ1 ͑solid histograms͒ show clearly the spin correlation, whereas for h ϭ0 ͑dashed histograms͒ the two sides of the event are uncorrelated. The hatched histograms show the Monte Carlo predicted background ͑assuming h ϭϪ1͒.
where ␣k is the background fraction of the kth background. ϭ0.747Ϯ0.012, h ϭ0.973Ϯ0.047, e e The function Si is the likelihood of the ith signal event, given by Eq. ͑6͒. The functions B are the corresponding k,i h ␦ ϭ0.716Ϯ0.031. likelihoods of the backgrounds. For the dominant sources of e e background, listed in Table I, the functions Bk,i include their full dependence on the fit parameters ⌰ to avoid bias. The As mentioned in the Introduction, the l-vs- sample used here is correlated with the one used in Ref. ͓17͔. Because of values used in the fits for the background fractions ␣k are taken from Table I. The amount of background not included the special treatment of the low-energy muons there, the pre- in the fit is Ϸ1% in the l-vs- sample and around 2% in the cision on the Michel parameter reached in Ref. ͓17͔ is -vs- sample. The effect of this disregarded background better than the one we might obtain here. Therefore, we do ͑‘‘remaining sources’’ in Table I͒ is discussed in the system- not fit for . Instead we fixed to the value determined in atic error section. Ref. ͓17͔ of ϭ0.010Ϯ0.261 and ϭϪ0.015Ϯ0.091, The hadronic current of the spin analyzer ϯ ϯ0 is where the first result is obtained in the muon sample alone very well known. However, the q2 dependence→ of the inter- and the second one is the combined result of the muon and mediate resonance structure might be a possible source for electron sample under the assumption eϭ . With the first uncertainties, especially the contribution of the Ј meson. result we obtain, in the -vs- analysis, The Ј contribution is parametrized by  ͓25͔, and with ϩ Ϫ ϭ0.750Ϯ0.017, h ϭ1.048Ϯ0.068, e e data ͓26͔  is determined ͓25͔ to be ϭϪ0.145. In a recent CLEO measurement ͓27͔, where events were used, a h ␦ ϭ0.781Ϯ0.040. value of ϭϪ0.091Ϯ0.009 was measured. For consistency, we use the latter value together with the mass and width obtained in Ref. ͓27͔ for the and Ј mesons. Using the second result we measure, with the -vs- sample, The only fit parameter in the -vs- analysis is h2 .We ϭ0.746Ϯ0.017, h ϭ1.043Ϯ0.067, obtain
2 h ␦ ϭ0.777Ϯ0.040. h ϭ0.989Ϯ0.019. All errors shown above are statistical only. Fixing to the In the e-vs- analysis we have three fit parameters , h , standard model value of ϭ0 results and h ␦. We measure the values in shifts of Ϫ ϭϪ0.0029, (h ) ϭϪ0.015 ϭ0 ϭϪ0.015
FIG. 2. spectrum for the -vs- sample of one side of the event for different values of from the other side ͑two entries per event͒. The data are represented by the dots with error bars. The solid histograms show the Monte Carlo expectation for h2 ϭ1. The Monte Carlo expectation for h2 ϭ0 is given by the dashed histograms. Background is indicated by the hatched histograms ͑assuming h2 ϭ1͒. 5326 J. P. ALEXANDER et al. 56
FIG. 3. Contributions of single events to Ϫ2lnL for ͑a͒ (eϯ¯)(Ϯ0¯), ͑b͒ (ϯ¯)(Ϯ0¯), and ͑c͒ (ϯ0) ϫ(Ϯ0¯) events. Data ͑dots with error bars͒ and Monte Carlo estimate ͑solid histograms͒ are in good agreement. Background is represented by the hatched histograms.
Ϫ(h ) ϭϪ0.0024, and (h ␦) Ϫ(h ␦) ϭ0 ϭϪ0.015 ϭ0 ϭϪ0.0018. FIG. 4. 90% confidence limits on the reduced coupling con- ␥Ј ␥ ␥ The confidence levels of the fits are 73% in the e-vs- stants g⑀ϭg⑀/max(g⑀). analysis, 21% in the -vs- analysis, and 9% in the -vs- uncertainty in the detector resolution has been estimated by analysis. Figure 3 shows the corresponding log likelihood scaling the error matrix of the resolution by a factor of 4. The per event distributions. One sees that the data are in good systematic error due to radiation has been obtained by vary- agreement with the best-fit model. ing the amount of radiation in the fit function by Ϯ10%. The Systematic errors arise from statistical errors of the Monte uncertainty in the trigger efficiency arises from the tracking Carlo estimate of the normalization integral and from uncer- component of the trigger, whereas the uncertainty due to the tainties in the momentum dependence of the lepton identifi- neutral component of the trigger is negligible. Therefore, the cation efficiency, the acceptance function of the ϯ 0 spin systematic error due to the trigger has been evaluated with a analyzer, the background estimates, the model for the had- subsample of our data that satisfies the neutral as well as the ronic current, the detector resolution, the radiative correc- tracking component of the trigger. tions, and the trigger efficiencies. The different contributions With these systematic errors added in quadrature and us- to the systematic error are summarized in Table II. ing the sign of h determined in Refs. ͓7,8͔ we obtain the The lepton identification efficiency has been measured, as a function of momentum and polar angle, with independent results listed in Table III. The results are in agreement with lepton data samples. The systematic error given in Table II the standard VϪA interaction. arises from the statistical error of this measurement. The ac- ceptance function of the ϯ0 spin analyzer has been varied V. INTERPRETATION AND SUMMARY via its dependence on the momenta of the two pions and the The measurement of and ␦ implies that the probability angle between the two pions. The variations considered have PR ͓see Eq. ͑3͔͒ of a right-handed to participate in leptonic been determined by comparison of the Monte Carlo estimate decays is PRϽ0.044 at a 90% confidence level. Separately and data. The systematic error due to the considered back- for electrons and muons, we obtain PRϽ0.066 for the elec- ground has been evaluated by varying the fractions of the different backgrounds in the fit function over a range given tronic mode and PRϽ0.067 for the muonic mode. Both lim- by statistical, experimental, and theoretical uncertainties. The its are at a 90% confidence level. effect of the disregarded background has been studied using The 90% confidence limits on the reduced coupling con- ␥Ј ␥ ␥ the Monte Carlo estimate. The Ј contribution is measured in stants g⑀ϭg⑀/max(g⑀) obtained from the combined re- Ref. ͓27͔ with an error of ⌬ϭϮ0.009. Since  has model sults on the Michel parameters ͑Table III͒ are plotted in Fig. dependences, and to be conservative, we varied  in the 4. Without directly measuring the helicity of the neutrino in S range of Ϯ0.020. The systematic error due to has been leptonic decays, the gLL coupling cannot be distinguished V evaluated by varying within its determined range ͓17͔. The from the gLL coupling. Adding the knowledge of the param- 56 DETERMINATION OF THE MICHEL PARAMETERS AND... 5327
TABLE II. Contributions to the systematic error.
2 Source ⌬(e) ⌬( ) ⌬(h e) ⌬(h ) ⌬(h e␦e) ⌬(h ␦ ) ⌬(h ) Monte Carlo statistics Ϯ0.0025 Ϯ0.0028 Ϯ0.0122 Ϯ0.0131 Ϯ0.0090 Ϯ0.0072 Ϯ0.0014 lepton identification Ϯ0.0006 Ϯ0.0033 Ϯ0.0005 Ϯ0.0016 Ϯ0.0004 Ϯ0.0025 acceptance function of spin analyzer Ϯ0.0010 Ϯ0.0015 Ϯ0.0038 Ϯ0.0093 Ϯ0.0036 Ϯ0.0064 Ϯ0.0047 considered background Ϯ0.0009 Ϯ0.0012 Ϯ0.0018 Ϯ0.0067 Ϯ0.0008 Ϯ0.0023 Ϯ0.0022 disregarded background Ϯ0.0004 Ϯ0.0009 Ϯ0.0029 Ϯ0.0059 Ϯ0.0011 Ϯ0.0017 Ϯ0.0005 parameter  of Ј contribution Ϯ0.0002 Ϯ0.0002 Ϯ0.0012 Ϯ0.0025 Ϯ0.0002 Ϯ0.0002 Ϯ0.0010 Michel parameter a ⌬()ϭϮ0.091 Ϯ0.0170 Ϯ0.0144 Ϯ0.0125 Michel parameter a ⌬()ϭϮ0.261 Ϯ0.0448 Ϯ0.0417 Ϯ0.0302 detector resolution Ϯ0.0004 Ϯ0.0004 Ϯ0.0006 Ϯ0.0005 Ϯ0.0003 Ϯ0.0004 Ϯ0.0002 radiation Ϯ0.0013 Ϯ0.0011 Ϯ0.0018 Ϯ0.0032 Ϯ0.0021 Ϯ0.0041 Ϯ0.0007 trigger Ϯ0.0017 Ϯ0.0035 Ϯ0.0094 Ϯ0.0123 Ϯ0.0022 Ϯ0.0025 Ϯ0.0011 total a ⌬()ϭϮ0.091 Ϯ0.004 Ϯ0.018 Ϯ0.016 Ϯ0.027 Ϯ0.010 Ϯ0.017 Ϯ0.006 total a ⌬()ϭϮ0.261 Ϯ0.004 Ϯ0.045 Ϯ0.016 Ϯ0.047 Ϯ0.010 Ϯ0.032 Ϯ0.006 aReference ͓17͔.
eter does not improve the limits. The couplings with a the pure left-handed W boson WL of the standard model, ␥ right-handed , g⑀R , are mostly constrained by the determi- such a model assumes a pure right-handed W boson WR , nation of and ␦. Additional information comes from the where the mass eigenstates W1 and W2 are in general super- V measurement of , which allows one to constrain the gRL and positions of the weak eigenstates WL and WR . This model T can be parametrized by the mass ratio ϭM /M of the two gRL couplings. Compared to the situation five years ago ␣ 1 2 when only the Michel parameter was measured in decay bosons W1/2 and the mixing angle between WL/R . The and no limits on the coupling constants g␥ existed, Fig. 4 standard model is obtained in the limit ␣ 0 and 0. ⑀ → → illustrates the progress made. However, the VϪA interaction Figure 5 shows the one, two, and three contours for ␣ and obtained with the combined results on , , ␦, and h . as assumed by the standard model is still not fully experi- mentally verified for decays. For ϭ0, W2 is identical with WR and the following limit is More stringent limits can be obtained by restricting the obtained on M R : generality of the model. For example, we consider a left- 2 right symmetric model ͓28͔ for the electroweak interaction, M RϾ304 GeV/c at 90% C.L. where the parity violation has its origin in a spontaneous symmetry breaking of the left-right symmetry. In addition to The mass limit obtained for free is
TABLE III. Results. , , and ␦ denote the combined results. l , l , and l␦l are the results separated for electrons and muons.
Parameter World average ͓23͔ This analysisa Correlation coefficients
h Ϫ1.011Ϯ0.027 Ϫ0.995Ϯ0.010Ϯ0.003 0.742Ϯ0.027 0.747Ϯ0.010Ϯ0.006 (,)ϭ0.046 (,␦)ϭ0.069 1.03 Ϯ0.12 1.007Ϯ0.040Ϯ0.015 (,h )ϭ0.000 (,␦)ϭ0.158 ␦ 0.76 Ϯ0.11 0.745Ϯ0.026Ϯ0.009 (,h )ϭϪ0.241 (␦,h )ϭϪ0.276 e 0.736Ϯ0.028 0.747Ϯ0.012Ϯ0.004 (e ,e)ϭ0.046 (e ,e␦e)ϭ0.074 1.03 Ϯ0.25 0.979Ϯ0.048Ϯ0.016 ( ,h )ϭ0.000 ( , ␦ )ϭ0.216 e e e e e ␦ 1.11 Ϯ0.18 0.720Ϯ0.032Ϯ0.010 ( ,h )ϭϪ0.194 ( ␦ ,h )ϭϪ0.214 e e e e e 0.74 Ϯ0.04 0.750Ϯ0.017Ϯ0.045 ( ,)ϭ0.026 ( ,␦)ϭ0.029 1.23 Ϯ0.24 1.054Ϯ0.069Ϯ0.047 ( ,h )ϭ0.000 ( , ␦ )ϭϪ0.030 ␦ 0.71 Ϯ0.15 0.786Ϯ0.041Ϯ0.032 ( ,h )ϭϪ0.128 ( ␦ ,h )ϭϪ0.152 aTogether with the sign of h determined in ͓7,8͔. 5328 J. P. ALEXANDER et al. 56
FIG. 5. Limits on the mass ratio ␣ and the mixing angle of a left-right symmetric model. FIG. 6. 90% confidence limits on the mass ratio ␣ for ͑a͒ 2 tan ϭ0 and ͑b͒ for tan free. M 2Ͼ260 GeV/c at 90% C.L. of the neutrino helicity parameter h . The results obtained The corresponding likelihood functions are shown in Fig. 6. 2 The limit obtained in muon decay is M 2Ͼ406 GeV/c ͓23͔. are consistent with the standard model prediction. With the We have also studied the constraints given by our mea- exception of the Michel parameter , the CLEO measure- surement on extensions of the standard model with charged ments given in Ref. ͓17͔ are superseded by the results ob- Higgs bosons. The Ϫ lepton and the charged daughter lep- tained here. Despite the high statistics used, the accuracy of ton lϪ in leptonic decays mediated by charged Higgs the measurements is still dominated by statistical and not bosons are right handed ͓29,30͔. Thus, in the general ansatz systematic uncertainties. Thus there is potential to further S of Eq. ͑1͒, charged Higgs bosons are represented by the gRR improve the precision on the Michel parameters in decays coupling. From our measurement on and ␦ we obtain at the B factories soon to come into operation, as well as at future factories. 2 M HϮϾ0.91ϫtan GeV/c at 90% C.L., where  is the ratio of vacuum expectation values. The com- ACKNOWLEDGMENTS bined limit obtained from this measurement and the recent We gratefully acknowledge the effort of the CESR staff in CLEO measurement of ͓17͔ is providing us with excellent luminosity and running condi- 2 tions. J.P.A., J.R.P., and I.P.J.S. thank the NYI program of M HϮϾ1.04ϫtan GeV/c at 90% C.L. the NSF; M.S. thanks the PFF program of the NSF; G.E. All results presented here assume massless neutrinos. thanks the Heisenberg Foundation; K.K.G., M.S., H.N.N., We have studied the kinematical and dynamical effects of a T.S., and H.Y. thank the OJI program of the U.S. DOE; 24-MeV/c2 neutrino on our results. We find that for the J.R.P., K.H., M.S., and V.S. thank the A.P. Sloan Founda- Michel parameters as well as for the neutrino helicity the tion; A.W. and R.W. thank the Alexander von Humboldt existence of such a neutrino would not affect the results at Stiftung; and M.S. thanks Research Corporation for support. the level of our accuracy. This work was supported by the National Science Founda- We have presented a precision measurement of the tion, the U.S. Department of Energy, and the Natural Sci- Michel parameters , , and ␦ in leptonic decay as well as ences and Engineering Research Council of Canada.
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