1. 1 T-odd asymmetries in top-quark decay Hiroshi Yokoya (Niigata U) KEKPH2007 3/1-3 (2007), KEK in collaboration with Kaoru Hagiwara (KEK) and Kentarou Mawatari (KIAS)

2. 2 Contents : Introduction : T-odd asymmetry Top-quark decay Results Summary

3. 3 T-transformation is defined as reversing spatial momenta and spins without interchanging initial and final states. What is T-odd asymmetry ? ~ T-odd means that change sign under T-transformation. Non-zero T-odd asymmetry needs P-violating interaction (Weak), or polarization measurement. T : P : ~ ~ e.g. naïve-T parity T : ~ time-reversal : triple product of three momenta : triple product of two momenta and spin

4. 4 T-odd asymmetry and Unitarity Time-reversal violating term → T-odd quantity comes from the absorptive part of the scattering amplitude in CP conserving theory. Unitarity of S-matrix : T-odd quantity (non-forward amplitude) : forward amplitude (i=f=k) → optical theorem A fi : absorptive part

5. 5 T-odd asymmetry In perturbation theory, the absorptive part can be predicted by the imaginary part of loop-diagrams * = Im T-odd asymmetry in hard process ⇔ test for the absorptive part of non-forward amplitude sign ? size ? shape ?

6. 6 T-odd asymmetries in hard processes have been calculated in the e + e - annihilation, Semi-Inclusive DIS, and DY processes in one-loop level. absorptive parts of these processes are related with each other by crossing and analyticity so far, no experimental measurements for these processes Korner,Malic,Merebashvili (’00) T-odd asymmetry Korner,Kramer,Shcierholz,Fabricius, Schmitt (’80) Brandenburg,Dixon,Shadmi (’96) Hagiwara,Hikasa,Kai (’83) (W-jet) Hagiwara,Hikasa,Kai ( ’ 84) (Z-jet) Hagiwara,Kuruma,Yamada ( ’ 92) e + e - → 3-jets Semi-Inclusive DIS Drell-Yan Top-quark decay Our new calculation !

7. 7 2. Top-quark decay We consider the top-quark decay with one-gluon emission : Kinematics (in top rest frame) Dalitz plot (m b is neglected) physical region of t → bWg is given by

8. 8 * Density matrix (DM) formalism  W-decay DM : Top-quark decay

9. 9 Top-decay density matrix Real part of DM → tree diagrams : Imaginary part of DM → interference of the tree and imaginary part of the one-loop diagrams : Couture (’89), Barger et al. (’90)  Top-decay DM : expanding the amplitude as In leading-order

10. 10 One-loop calculation Imaginary part (absorptive part) of the scattering amplitude comes from the on-shell intermediate states. Origin of the imaginary part in the loop integrals; Passarino,Veltman (’79), Oldenborgh (’91) IR divergences are regulated by using gluon mass scheme We calculate these diagrams by two different methods; 1. analytic calculation by standard Feynman parameter integrals 2. express by loop scalar functions and use the fortran code “FF” and check results by the gauge invariance. in the integrand

11. 11 Now, combining Top-decay and W-decay DM’s, the decay rate is written as the lepton angular distribution. Lepton angular distribution : nine structure functions reflecting the W’s polarization (3x3=9) F 7-9 : T-odd (P-odd) ⇔ Imaginary part

12. 12 3. Numerical results Kinematical cuts are needed to avoid the collinear decay, and to select the hard gluon jet event. Contour plot of the F 1 with kinematical cuts Kinematical cuts : Total rate F 1 : integrate over lepton angles

13. 13 T-even and T-odd angular distributions A 7 ~ 3%, A 8 ~ 1%, A 9 ~ ±0.1% T-odd asymmetries (one-loop, our results) T-even asymmetries (tree) A 3,4,6 : azimuthal angle distributions, off-diagonal part of DM, interference between different polarization states A 2,5 : polar angle distributions, diagonal part of DM large z 2 limit → (t → bW decay) A 2 → f 0 ~ 0.7, A 5 → -2f - ~ -0.6, A 3,4,6 vanish

14. 14 Up-down asymmetry At the LHC experiment, about 800000 of the top-quark events are expected for the “single lepton plus jets” channel (10fb -1 ) By the branching fraction of top-decay in SM, about 70000 of events may be identified as an event with a hard gluon jet. Up-down asymmetry : Lepton direction w.r.t. the top-decay plane

15. 15 4. Summary T-odd asymmetry emerges from the absorptive part of the scattering amplitudes. In hard process, it can be predicted, and comparison with experiments would be an interesting test. We calculate the T-odd asymmetry for the top-quark decay with one-gluon emission, in one-loop level. T-odd asymmetries are predicted (~3% at most), which may be observed at the LHC or in future the ILC.

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