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

Contents :

• Introduction : T-odd asymmetry

• Top-quark decay

• Results

• Summary

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

4T-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

Afi : absorptive part

5T-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

• 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 !

72. Top-quark decay

• We consider the top-quark decay with one-gluon emission :

• Kinematics (in top rest frame)

Dalitz plot

(mb is neglected)

physical region of t → bWg is given by

8

*

• Density matrix (DM) formalism

W-decay DM :

Top-quark decay

9Top-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

10One-loop calculation

Imaginary part (absorptive part) of the scattering amplitude comes from the on-shell intermediate states.

Origin of the imaginary partin 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

• 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)

F7-9 : T-odd (P-odd)

⇔ Imaginary part

123. Numerical results

Kinematical cuts are needed to avoid the collinear decay, and to select the hard gluon jet event.

Contour plot of the F1 with kinematical cuts

• Kinematical cuts :

• Total rate F1 :

integrate over lepton angles

13T-even and T-odd angular distributions

A7 ~ 3%, A8 ~ 1%, A9 ~ ±0.1%

• T-odd asymmetries (one-loop, our results)

• T-even asymmetries (tree)

A3,4,6 : azimuthal angle distributions,

off-diagonal part of DM, interference between different polarization states

A2,5 : polar angle distributions, diagonal part of DM

large z2 limit → (t→bW decay)

A2 → f0 ~ 0.7, A5 → -2f- ~ -0.6, A3,4,6 vanish

14Up-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

154. 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.

16