1. March 7, 2005Benasque Neutrinos Theory Neutrinos Theory Carlos Pena Garay IAS, Princeton ~

2. Plan I. Neutrinos as Dirac Particles II. Neutrinos from earth and heavens III. Neutrinos as Majorana Particles

3. Plan I. Neutrinos as Dirac Particles II. Making Neutrinos III. Neutrinos as Majorana Particles

4. Nuclear  -decay

5. Neutrinos are left-handed

6. All neutrinos left handed => massless Neutrinos must be massless If neutrinos are massive neutrino right-handed => Contradiction! Helicity = Chirality (massless)

7. Standard Model Exact Lepton number symmetry : terms never arise in perturbation theory Renormalizability : All gauge currents satisfy anomaly cancellation What about global symmetries : L is broken non perturbatively through the anomaly, but B-L is conserved to all orders : terms never arise in the Standard Model

8. CPT : L => R Anti-neutrinos are right-handed -

9. 3 neutrino families (flavors)

10. Standard Model

11. Global symmetries The results of charged-current experiments (until recently) were consistent with the existence of three different neutrino species with the separate conservation of L e, L  and L .

12. Massive Neutrinos : Dirac (p,h) (p,-h) CPT (p,h) (p,-h) CPT Boost, if m ≠ 0

13. Standard Model + right handed

15. Masses : Several distinctive features I am going to concentrate today is neutrino oscillations

16. Parameters : Dirac Exercise

17. Neutrino Oscillations lili W X lklk j U ij  U jk

18. Neutrino Oscillations in Vacuum Due to different masses, different phase velocities Osc. Length : distance to come back to the initial state

19. Production, propagation and detection as a unique process, where neutrinos are virtual particles propagating between production and detection : Neutrinos are described by propagators S (x P -x D ) Consider finite production and detection regions and finite energy and momentum resolution

20. If neutrinos can be considered real (on shell) and production, propagation, and detection can be treated separately Match initial and final conditions Wave packet formalism

21. Correct calculation of the phase Phases calculated in the same space-time point Second summand is small by either one and/or the other term => standard result Oscillation effects disappear if neutrino masses are all equal

22. Exercise : Write P ee, P e  Exercise : Minimal number of P   to know all the others

23. One example :

24. Some real cases :

25. Neutrino Oscillations in matter After a plane wave pass through a slab, the phase is shifted : p (x+(n-1)R) Net effect :

26. Only the difference of potential is relevant Net effect :

27.  cos  V e  sin 2   sin 2   m 2   m 2   m 2  i = d dt e  e  Two Neutrino Oscillations in matter sin 2 2  m = sin 2 2  ( cos2  G F N e E/  m 2 ) 2 + sin 2 2  Difference of the eigenvalues H 2 - H 1 =  m 2 2E ( cos2  G F n e E/  m 2 ) 2 + sin 2 2 

28. l / l 0 sin 2 2  m sin 2 2  = 0.08 sin 2 2  = 0.825 ~ n E Resonance width:  n R = 2n R tan2  Resonance layer: n = n R +  n R Resonance sin 2 2  m = 1 Flavor mixing is maximal Level split is minimal In resonance: l = l 0 cos 2  Vacuum oscillation length Refraction length ~ ~

29.  l  / l 0 2m 1m  e e  sin 2 2  = 0.825 sin 2 2  = 0.08 Large mixing Small mixing Level crossing  (t) =  cos  a 1m + sin  a  2m e -i  (t)

30.  = (Re e + , Im e + , e + e - 1/2) B = (sin 2  m, 0, cos2  m ) 2  l m elements of density matrix l m = 2  H oscillation length  = ( B x ) d dt Evolution equation Coincides with equation for the electron spin precession in the magnetic field  = 2  t/ l m - phase of oscillations P = e + e = Z + 1/2 = cos 2  Z /2 Graphical interpretation

31. A real case: solar neutrinos

32. The Neutrino Matrix :SM +  mass ALL oscillation neutrino data BUT LSND 3  ranges 