March 7, 2005Benasque Neutrinos Theory Neutrinos Theory Carlos Pena Garay IAS, Princeton ~ Plan I. Neutrinos as Dirac Particles II. Neutrinos from earth and heavens III. Neutrinos as Majorana Particles Plan I. Neutrinos as Dirac Particles II. Making Neutrinos III. Neutrinos as Majorana Particles Nuclear -decay Neutrinos are left-handed All neutrinos left handed => massless Neutrinos must be massless If neutrinos are massive neutrino right-handed => Contradiction! Helicity = Chirality (massless) 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 CPT : L => R Anti-neutrinos are right-handed - 3 neutrino families (flavors) Standard Model 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 . Massive Neutrinos : Dirac (p,h) (p,-h) CPT (p,h) (p,-h) CPT Boost, if m 0 Standard Model + right handed Masses : Several distinctive features I am going to concentrate today is neutrino oscillations Parameters : Dirac Exercise Neutrino Oscillations lili W X lklk j U ij U jk Neutrino Oscillations in Vacuum Due to different masses, different phase velocities Osc. Length : distance to come back to the initial state 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 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 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 Exercise : Write P ee, P e Exercise : Minimal number of P to know all the others One example : Some real cases : Neutrino Oscillations in matter After a plane wave pass through a slab, the phase is shifted : p (x+(n-1)R) Net effect : Only the difference of potential is relevant Net effect : 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 l / l 0 sin 2 2 m sin 2 2 = 0.08 sin 2 2 = ~ 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 ~ ~ l / l 0 2m 1m e e sin 2 2 = sin 2 2 = 0.08 Large mixing Small mixing Level crossing (t) = cos a 1m + sin a 2m e -i (t) = (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 A real case: solar neutrinos The Neutrino Matrix :SM + mass ALL oscillation neutrino data BUT LSND 3 ranges