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0010177580
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The term CKM matrix denotes the Cabibbo Kobayashi Maskawa mixing matrix which describes the linear superposition of the quark mass eigenstates which appears in the phenomenological weak eigenstates, see
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(1963)
Phys. Rev. Lett.
, vol.10
, pp. 531
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Cabibbo, N.1
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84927863186
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This paper is a lengthened version of C.A. Nelson, Report No. SUNY BING 7/19/92 (unpublished). Some of the results were reported by C. A. Nelson, in Proceedings of the Second Workshop on Tau Lepton Physics, edited by K. K. Gan (World Scientific, Singapore, 1993). We correct some typos in the latter: minus signs in Eqs. (15) and (16), 2sqrt 2 to sqrt 2 in Eqs. (18) and (20), and update other numbers due mainly to the larger B ( τ → ρ ν ) = 0.2460.
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5
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0002136865
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Polarization observables in the 3πν decay mode of the τ
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(1990)
Z. Phys. C
, vol.48
, pp. 75
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Rouge, A.1
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6
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84927863185
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A. Rouge, in Proceedings of the Workshop on Tau Lepton Physics, edited by M. Davier and B. Jean Marie (Editions Frontieres, Gif sur Yvette, France, 1991);
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8
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84927863184
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H. Videu, in Proceedings of the Second Workshop on Tau Lepton Physics cite1.
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20
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5544238239
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ALEPH Collaboration, has used spin correlations to measure the chiral polarization parameter for τ- → τ- ν decay and find ξ ρ=1.03 pm 0.11 pm 0.05. Our sign convention is that for mν =0 and for a pure V-A coupling, both the chirality parameter ξA =1 for τ-→ A- ν and the τ Michel parameter ξl=1 for τ- → l-ν barν. When mν not= 0, the no longer factorizing quantity (ξA SA), appearing in I(EA,EB) where the hadron helicity factor SA equiv (ΓL -ΓT)/(ΓL +ΓT), can be expressed in terms of the moduli |A(λρ, λν)|.
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(1994)
Phys. Lett. B
, vol.321
, pp. 168
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Buskulic, D.1
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31
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G. Bella, Proceedings of the Second Workshop on Tau Lepton Physics cite1;
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32
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ALEPH Collaboration, B. Gobbo, ibid.;
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34
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Most observables for CP violation in τ-τsup + production depend on the initial e- beam axis. In contrast, the tests in the text are for CP violation in τ decays and do not depend on the beam axis.
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35
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edited by, P. Drell, D. Rubin, AIP Conf. Proc. No. 302, AIP, New York
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(1994)
Lepton and Photon Interactions, Proceedings of the 16th International Conference, Ithaca, New York, 1993
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Schwarz, A.1
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W. Hollik ibid.;
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37
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M. Davier, Proceedings of the Second Workshop on Tau Lepton Physics cite1;
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38
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0026154388
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ibid. τ spin correlations at the standard model level are incorporated in the Monte Carlo simulation KORALB for γsup ast energies, see, and Z. Was
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(1991)
Comput. Phys. Commun.
, vol.64
, pp. 267
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Galik, R.S.1
Jadach, S.2
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39
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and report, 1993 (unpublished). For numerical purposes we use mτ=1.777 GeV, MZ=91.187 GeV, B(Z→ τ-τsup + ) = 0.03355, B(τ→ρν ) = 0.2460, B ( τ → { μν + eν } ) = 0.3523, and B ( τ → π ν ) = 0.1193.
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43
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in The Vancouver Meeting Particles and Fields '91, Proceedings of the Joint Meeting of the Division of Particles and Fields of the American Physical Society and the Particle Physics Division of the Canadian Association of Physicists, Vancouver, 1991, edited by D. Axen, D. Bryman, and M. Comyn (World Scientific, Singapore, 1992).
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In the simple X → ΦΦ → ( K sup + K - ) ( K sup + K - ), etc., spin correlation tests, the evidence for CP violation at the X vertex is a term(s) odd in the angle φ between the two (Ksup +K-) decay planes. For instance, sinφ or sin2φ. This was first shown in
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(1984)
Phys. Rev. D
, vol.30
, pp. 1937
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Nelson, C.A.1
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47
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For examples of useful analogies between τ lepton polarimetry tests and t quark polarimetry tests, see
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(1990)
ibid.
, vol.41
, pp. 2805
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Nelson, C.A.1
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48
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A very important practical question regarding tests for possible CP violation in τ lepton processes and in the very analogous t quark processes is: How homogeneous, and charge symmetric, are the magnetic field and other detector trigger components? At the Top Quark Collider Workshop, Madison, Wisconsin, 1992, G. L. Kane emphasized this and argued that in the case of t quark processes CP violation effects smaller than sim case 1/2 undetectable see
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(1993)
Phys. Lett. B
, vol.317
, pp. 454
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Im, C.J.C.1
Kane, G.L.2
Malde, P.J.3
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the (-Δ Nsub LR)= ε parameter of the last paper of [11]]. Similar challenges arise in the (i) design and quantification of CP symmetric detector systematic errors and from (ii) standard model backgrounds in kaon, in b quark, and in hyperon CP violation experiments. The literature on methods to test for CP violation in top quark processes includes the last paper of Ref. cite11;
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edited by, P. Zerwas, DESY Report No. 92 123A, Hamburg
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(1992)
esup +e-Collisions at 500 GeV: The Physics Potential, Proceedings of the Workshop, Munich Annecy Hamburg, 1991
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Kuhn, J.1
Bernreuther, W.2
Nachtmann, O.3
Overmann, P.4
Schröder, T.5
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65
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Jiang Liu, University of Pennsylvania Report No. UPR 0525T, 1992 (unpublished);
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69
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B406, 516(E);
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Int. J. Phys. 9, 635 (1994).
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An improved, simplified formulation of this papers' two tests would provide further checks of the CP conserving and/or violating parts of the angular distributions given in Eq. (4.20), and in Eq. (4.9) with its associated parametric transformations. We thank J. F. Donoghue for emphasizing this. The distributions in Eq. (4.20), and Eq. (4.9), are consistent with CP transformations at the helicity variable level which interchange τ- → ρ-ν→ (π-π0 ) ν variables with those for the CP conjugate decay sequence.
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We thank S. Pakvasa for emphasizing this. The argument in the text is for a pure V and A leptonic CKM coupling. For instance, a CKM phase in an S and/or P coupling contributes at the tree level only to the A(0,- case 1/2) amplitude, but not to the A(-1,- case 1/2) amplitude and so is measurable by a S2SC function. In general, T and T5 contribute unequally at the tree level to both amplitudes. Multi Higgs boson models, as well as multiloop induced leptonic CKM effects, can produce CP violating amplitudes which are observable by S2SC's.
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For more recent references and a review of multi Higgs boson models, see, in Perspectives on Higgs Physics, edited by, (World Scientific, Singapore, 1993). In theories with both CKM and non CKM sources of CP violation, the CKM phase might be measurable by S2SC's. For such a theory, see
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(1993)
Phys. Rev. D
, vol.47
, pp. 3655
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Bigi, I.I.1
Sanda, A.I.2
Uraltsev, N.G.3
Kane, G.L.4
Frampton, P.H.5
Kephart, T.W.6
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Very readable introductions to the helicity formalism are in H. Pilkuhn, Interactions to Hadrons (North Holland, Amsterdam, 1967);
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the books by Perl and by Martin and Spearman in Ref. cite18 below;
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Caltech Report No. CALT 68 1148 (unpublished). Useful symmetry transformations are in Appendix E of M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964);
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The third Euler angle locates the xa axis which is used to reference the second stage ρ- → π-π0 decay's momenta directions relative to the first stage τ- → ρ-ν decay's momenta directions. The third Euler angle γ is set equal to zero by S. M. Berman and M. Jacob, SLAC Report No. 43, 1965 (unpublished);
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compare Phys. Rev. 139, B1023 (1965)
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90
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and by, ibid.
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(1968)
, vol.169
, pp. 1342
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Chung, S.U.1
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However, in the often used Jacob Wick paper γ is set equal to -φ1τ. In the present paper, a bit awkwardly we could choose the xa coordinate axis to follow the Jacob Wick (JW) convention as we will explain in detail. Note that with the JW choice, φ1τ = - γ, the production azimuthal angle is also being used to specify the value of the third Euler rotation which is about the za axis which lies along the ρ- momentum. Essentially, the γ=-φ1τ Euler angle choice induces a rotation of the xa referencing axis backward in the ρ- rest frame by the angle φ1τ: In Eq. (2.1), in Dλ1,μ1/2ast (α,β,γ) in the JW convention we would first set γ=-φ1τ where still α=φ1τ, β= theta1τ. Since α=φ1τ, in the τ- rest frame the τ-→ρ-ν production plane is at azimuthal angle φ1τ. A boost along the ρ- momentum then gives a ρ- rest frame. But in this ρ- rest frame, the JW choice that φ1τ = -γ means that the xa referencing axis is to be chosen so that the τ-→ρ-ν production plane is also oriented with an azimuthal angle φ1τ. For the ρ-→π-π0 decay, this JW choice xa referencing axis is at tildeφa=0 where the (new) tildeφa and (same) tilde thetaa specify the π- momentum.
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93
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See also S. Orteu, Report No. UAB LFAE 89 04, 1989 (unpublished). Wigner rotations are treated clearly in Secs. 9 8 and 9 9 of M.L. Perl, High Energy Hadron Physics (Wiley, New York, 1974). See also A. D. Martin and T. D. Spearman, Elementary Particle Physics (North Holland, Amsterdam, 1970).
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At this point in the derivation one has a generalization of Eq. (4.9). This generalization is the full beam referenced state two spin correlation function. It can also be expressed in terms of observable variables by the three steps in Sec. IV of the first paper of Ref. cite10 in combination with the Wigner rotation procedure explained in the present paper in the subsection following Eq. (4.9).
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CRC Handbook of Mathematical Sciences (CRC, Boca Raton, FL, 1987).
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For instance, for a two variable correlation we would distribute N events ideally over a two dimensional ij grid according to the theoretical result, I ( x,y ) = Z0(x,y) + aZ 1 ( x,y);
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the ideal error in bin ij is σij=I(xi,yj). By chi2 minimization, the ``ideal statistical error" in the measurement of ``a" is σa= { sumij [ Z 1 ( x i , yj ) / σ ij ]2 } -1/2.
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