1. Degree of polarization of  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 study more precisely oscillations of by detection of tau leptons. The small number of events with  –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  lepton in contrast to electron and muon implies that it can be partially polarized. The degree of polarization of  is one of the parameters which describe the  decay distributions. Therefore, the discussion the  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 polarization of  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. 1.The nucleons inside the nucleus do not interact with each other. 2.Their momenta are uniformly distributed in the Fermi sphere which has radius given by the Fermi momentum k F (there is a direct relation between the k F and the nuclear matter density). 3.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  +g’ (QHD). The RPA corrections are calculated by taking into account infinite sum of one particle --one hole diagrams.

6. Polarization tensor: The inclusive cross section for investigated process is the following [Horowitz at. al.]: 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:

7. The RPA Free Fermi Gas

8. Algebraic properties of Polarization tensor 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: Thus, we get decomposition of the scattering amplitude: 4x4 matrices which form basis of some closed algebra.

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. Neutrino energy E = 4 GeV FG+RPA FG

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  has high degree of polarization for almost all angles apart from forward scattering where the  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.