Krzysztof M. Graczyk Jaroslaw Nowak Institute of Theoretical Physics University of Wroclaw

1. Degree of polarization of t produced in quasielastic charge current neutrino-nucleus scattering Krzysztof M. Graczyk Jaroslaw Nowak Institute of Theoretical Physics University of Wroclaw Poland

2. Motivations • The CNGS experiments (ICARUS and OPERA) will able to studymore precisely oscillations of by detection of tau leptons. The small number of events with t –neutrinos is expected. It requires detailed analysis which should be done based on precise theoretical predictions. • Tau lepton has a short lifetime, so only its decay products can be observed. • The large mass of the t lepton in contrast to electron and muon implies that it can be partially polarized. The degree of polarization of t is one of the parameters which describe the t decay distributions. Therefore, the discussion the t polarization in the neutrino-matter scattering seems to be interesting as it can play an important role in the analysis of the experimental data. • During last year several groups have been studied the polarizationoft produced in quasielastic and inelastic neutrino -free nucleon scattering.See e.g. K. Hagiwar at. al., V. Naumov at.al. … • Can nuclear effects play role in tau neutrino-nucleus scattering? Even tau neutrino energy has to be bigger than ~3.4 GeV? • In particular, we are interested how large can be influence of nuclear effects on polarization of tau near the threshold region?

3. We consider the following process:

4. and s.

5. Model of the nucleus. • The relativistic mean field theory formalism [Walecka at. al.] is used to describe a nucleus • The ground state of the nucleus is given by the relativistic Fermi Gas (FG) model. • The nucleons inside the nucleus do not interact with each other. • Their momenta are uniformly distributed in the Fermi sphere which has radius given by the Fermi momentum kF (there is a direct relation between the kF and the nuclear matter density). • There is the Pauli Blocking effect. • To make the description of the nucleus more realistic the Ring Random Phase Approximation (RPA) is done based on the residual interaction p+r+g’ (QHD). The RPA corrections are calculated by taking into account infinite sum of one particle --one hole diagrams.

6. Polarization tensor or current-current correlation function • The polarization tensor is a chronological product of many body currents: We make simplification and use one body current instead of many body: • The inclusive cross section for investigated process is • the following [Horowitz at. al.]: Polarization tensor:

7. Free Fermi Gas The RPA

8. In the coordinate system with transfer of four-momentum the polarization tensor treated as a matrix (even with RPA corrections) can be decompose in four independent components: 4x4 matrices which form basis of some closed algebra. Thus, we get decomposition of the scattering amplitude: Algebraic properties of Polarization tensor

9. Polarization of tau Using decomposition of the scattering amplitude one can obtain formulae for the longitudinal, perpendicular and transverse components of the polarization: One can see that transverse component is zero!! For zero scattering angle also perpendicular component vanishes! More details you can find in: hep-ph/0407275

10. Discussion: Transition point, angular dependence, the sign of longitudinal polarization. FG+RPA FG Neutrino energy E = 4 GeV

11. Discussion: gap appears Neutrino energy E = 5 GeV FG+RPA FG

12. The mean value:

13. Conclusions • We discuss the nuclear effects up to 6 GeV neutrino energy. After 7 GeV the nuclear effects do not play a role[ M. Sajjad Athar Nucl Phys B 112 (Proc. Suppl.)] and one must apply resonance production and DIS formalizm. • For given scattering angle exist two kinematically allowed regions for tau energies. One is placed close to the tau mass and other is placed near the neutrino energy. • The Fermi motion widens these regions and for the 4 GeV beam energy they join each other. For higher energies (5 GeV) these two regions are separated by a large forbidden area. • One can notice that the degree of polarization strongly depends on the scattering angles. • Produced t has high degree of polarization for almost all angles apart from forward scattering where the t can be unpolarized. • One can observe that the polarization plots have minimum (sharp in the case theta=0) where the longitudinal polarization changes sign. • The RPA corrections shifts the minimum point. • The minimum of the mean value (at zero scattering angle) is increasing by RPA. • The results of this presentation are similar to those obtained for DIS. It suggest that it is the kinematics that plays the main role in the polarization effects.