Vali Huseynov a,b , Rasmiyya Gasimova a , Nurida Akbarova a and Billura Hajiyeva a

2. Spin asymmetries arising in neutrino-lepton processes in a magnetic field and their macroscopic appearance Vali Huseynova,b, Rasmiyya Gasimovaa, Nurida Akbarovaa and Billura Hajiyevaa

3. a) Department of General and Theoretical Physics, Nakhchivan State University, AZ 7000, Nakhchivan, Azerbaijan; b) Laboratory of Physical Research, NakhchivanDivision of Azerbaijan National Academy of Sciences , AZ 7000, Nakhchivan, Azerbaijan E-mail : vgusseinov@yahoo.com

4. Considered processes (1) (2) (3) (4) (5) Motivation • The presence of a strong magnetic field leads on the one hand to the anisotropy and asymmetry in the heating of the stellar matter on the other

5. hand to the anisotropy and asymmetry of the subsequent explosion of the outer layers of the collapsing stellar core. To clarify the reasons of the anisotropy and the asymmetry arising in astrophysical phenomena connected with neutrino-lepton processes and to show the macroscopic appearance of these asymmetries it is important to investigate the dependence of the differential cross section of the considered processes in a magnetic field on the spin variables of the charged leptons.

6. Analyses of polarization effects in some neutrino-lepton processes in a magnetic field enable to use polarized electrons as a target in neutrino (antineutrino) detectors. • The polarization asymmetry in some neutrino-lepton processes provide a sensitive tool for distinguishing neutrino flavor of the incoming neutrinos (antineutrinos).

7. Some related published papers [1] Benesh C. J. and Horowitz C. J. astro-ph/ 9708033 [2] Bezchastnov V. G. and Haensel P. 1996 Phys. Rev. D54 3706. [3] Guseinov V. A. 2000 J. Phys. G26 1313. [4] Kuznetsov A. V. and Mikheev N. V. 1997 Phys. Lett. B 394 123. [5] Borisov A. V., Nanaa M. K. and Ternov I.M. 1993 Vestnik Moskovskogo universiteta.Fizika. Astronomiia,48(2) 15.

8. The main purposes to present analytic formulae for the rates (differential cross sections, differential probabilities) of the considered neutrino-lepton processes in a magnetic field with allowance for the longitudinal and transverse polarizations of the charged leptons; to analyze polarization effects; to show possible applications of the obtained results.

9. Assumptions • Neutrino (antineutrino) energies, transverse momenta and the Landau energy levels of the charged leptons, the strength of a magnetic field, the temperature of matter are arbitrary. • Polar and azimuthal angles of the incident (scattered) neutrinos and antineutrinos are arbitrary.

10. signature We use the standard Weirberg-Salam-Glashow (WSG) electroweak interaction theory.When the momentum transferred is relatively small, is the -boson mass, is the -boson mass),the four-fermion approximation of the WSG standard model can be used. The gauge of a 4-potential is -b and an external magnetic field vector Is directed along the axis

11. The amplitude and the matrix element of the processes is the matrix element of the considered processes. is the amplitude of the considered processes.

12. the Fermi constant; the normalization volume; the normalization lengths; 4-momentum of the incident (scattered) neutrino and/or antineutrino with the energy and components of the 4- momenta of the incident (scattered) neutrino and/or antineutrino; the sign of energy for the incident (scattered) neutrino and/or antineutrino;

13. component of the 4-momenta and of the initial (final) charged lepton and initial (final) charged antilepton with the energy the sign of energy for the charged lepton (the charged antilepton) and enumerate theLandau energy levels of the charged leptons (the charged antileptons)

14. -the Dirac matrices theDirac bispinors -the transition amplitude of the 4-current for theconsidered prosesses. The structure of the components of depends on the kind of polarization of the charged leptons (the charged antileptons).

15. Longitudinal polarization of charged leptons

16. for -processes, for -processes.

17. The spin coefficients corresponds to right-hand (left-hand) helicity.

18. Transverse polarization of charged leptons

20. correspondsto the charged lepton spin oriented parallel (antiparallel) to the magnetic field H. Rate of the processes

21. -the phase-spase element of a particle the Fermi-Dirac distribution function

22. the total interaction time

23. for transverse polarization case

28. the polar angle of the incident (scattered) neutrino and/or antineutrino

29. the azimuthal angle of the incident (scattered) neutrino and/or antineutrino the Laguerre function the associated Laguerre polynomial

30. for longitudinal polarization case

31. Analyses of the cross section of the process When the neutrinos scatter on the electrons with right-hand circular polarization , we have where

32. This expression shows that, if , and the neutrinos scatter on the electrons with right-hand circular polarization, the final electrons can only have right-hand circular polarization. When , and the differential cross section only depends on and it is not sensitive to the neutrino flavor. When the neutrinos scatter on the electrons with left- hand circular polarization , we have

33. The obtained expression shows that, if , and the neutrinos scatter on the electrons with left-hand circular polarization the final electrons can only have left-hand circular polarization. When , and the differential cross section depends on and therefore it is sensitive to the neutrino flavor. When the differential cross section is different for and . This result enables to come to the conclusion that initial electrons with left-hand circular polarization in neutrino-electron scattering in a magnetic

34. field can be used as a polarized target in neutrino detectors. The polarized electron targets in neutrino-electron scattering in a magnetic field can also be used for distinguishing neutrino flavor. When , and the neutrinos scatter on the electrons with right-hand left-hand circular polarization, we have the following expressions for : and

35. The last two expressions show that, if and the neutrinos scatter on the electrons with right-hand (left-hand) circular polarization, the final electrons can only have right-hand (left-hand) circular polarization. When and the differential cross sections only depends on and it is not sensitive to the neutrino flavor. When , and the differential cross section depends on and therefore it is sensitive to the neutrino flavor. In this case the differential cross section is different for and .

36. So, spin asymmetry in neutrino-electron scattering in a magnetic field enables to use electrons with left-hand circular polarization as polarized electron targets for distinguishing neutrino flavor and for detection of neutrinos. The analyses of the obtained expressions for the cross section show that electrons with right-hand circular polarization are heated by , and equally. However, the electrons with left-hand circular polarization in neutrino-electron scattering in a magnetic field are heated by and unequally. This fact leads to the asymmetry in the heating of the stellar matter.

37. In the case of the transverse polarizations of initial and final electronswhen the initial electrons are on the lowest Landau level, i.e. we have

38. Analyses of the cross section of the process When the antineutrinos scatter on the electrons with right-hand circular polarization , we have This expression shows that, if , and the antineutrinos scatter on the electrons with right-hand circular polarization, the final electrons can only have right-hand circular polarization.

39. When , and the differential cross section depends on and it is sensitive to the antineutrino flavor. When the differential cross section is different for and . This result enables to come to the conclusion that initial electrons with right-hand circular polarization in antineutrino-electron scattering in a magnetic field can be used as a polarized target in antineutrino detectors. The polarized electron targets in antineutrino-electron scattering in a magnetic field can also be used for distinguishing antineutrino flavor.

40. When the antineutrinos scatter on the electrons with left-hand circular polarization we have The obtained expression shows that, if , and the antineutrinos scatter on the electrons with left-hand circular polarization the final electrons can only have left-hand circular polarization. When , and the differential cross section depends on and therefore it is not sensitive to the antineutrino flavor.

41. When , and the antineutrinosscatter on the electrons with right-hand left-hand circular polarization, we have the following expressions for : and The last two expressions show that, if , and the antineutrinos scatter on the electrons with right-hand (left-hand) circular polarization, the final electrons can only have right-hand (left-hand) circular polarization.

42. When and the differential cross sections only depends on and it is not sensitive to the antineutrino flavor. When and and therefore the differential cross section depends on , it is sensitive to the antineutrino flavor. In this case the differential cross section is different for and So, spin asymmetry in antineutrino-electron scattering in a magnetic field enables to use electronswith right-handcircular polarization as polarized electron targets for distinguishing antineutrino flavor and for detection of antineutrinos. It is derived from the obtained expressions .

43. for the cross section that electrons with left-hand circular polarization in antineutrino-electron scattering in a magnetic field are heated by and equally. However, the of the obtained expressions for the cross section show that electrons with right-hand circularpolarization are heated byand unequally. This fact leads to the asymmetry in the heating of the stellar matter. For antineutrino-electron scattering we have

44. When the incident and scattered antineutrinos fly along the magnetic field and the initial electrons are on the lowest Landau level, the cross section of theantineutrino-electron scattering is sensitive to the incoming antineutrino flavor. Such electrons can be used as polarized electron targets for distinguishing antineutrino flavor and for detection of antineutrinos.

45. Analyses of the cross sections of the processes To obtain the general expressions for and for the cross section of these processes (via -boson exchange) we have to consider that The process is forbidden for electrons with right-hand circular polarization

46. The final neutrino and charged lepton are emitted asymmetrically. The ultra relativistic charged lepton and ultra relativistic positron in the process are emitted at small angles with respect to the direction of the high-energy-(anti) neutrino momentum. In the process the electrons and the positrons with left (right) helicity are emitted asymmetrically.

47. CONCLUSIONS • Spin asymmetry in neutrino-electron scattering in a magnetic field enables to use electrons with left-hand circular polarization as polarized electron targets for distinguishing neutrino flavor and for detection of neutrinos. • The electrons with right-hand circular polarization in neutrino-electron scattering in a magnetic field are heated by , and equally.

48. However, the electrons with left-hand circular polarization in neutrino-electron scattering in a magnetic field are heated by and unequally. This fact leads to the asymmetry in the heating of the stellar matter. • Spin asymmetry in antineutrino-electron scattering in a magnetic field enables to use electrons with right-hand circular polarization as polarized electron targets for distinguishing antineutrino flavor and for detection of antineutrinos.

49. The electrons with left-hand circular polarization in antineutrino-electron scattering in a magnetic field are heated by and equally. However, the electrons with right-hand circular polarization are heated by , and unequally that leads to the asymmetry in the heating of the stellar matter. • When the incident and scattered antineutrinos fly along the magnetic field and the initial electrons are on the lowest Landau level, the cross section of theantineutrino-electron scattering is

50. sensitive to the incoming antineutrino flavor. Such electrons can be used as polarized electron targets for distinguishing antineutrino flavor and for detection of antineutrinos. • The process is forbidden for electrons with right-hand circular polarization (=1). The final neutrino and charged lepton are emitted asymmetrically. • In the process the electrons and the positrons with left (right) helicity are emitted asymmetrically.