2. A Yu Smirnov   e 2 1 mass 1 2 3 3  m 2 atm  m 2 sun Inverted mass hierarchy (ordering) Normal mass hierarchy (ordering) |U e3 | 2 Type of the mass hierarchy: Normal, Inverted Type of mass spectrum: with Hierarchy, Ordering, Degeneracy absolute mass scale U e3 = ? ?

3. Hierarchy of mass squared differences:  m 12 2  m 23 2 | = 0.02 - 0.08 m h >  m 23 2 > 0.04 eV |m 2  m 3 | > |  m 12 2  m 23 2 | = 0.19 No strong hierarchy of masses: + 0.09 - 0.05 |sin  | Bi-large or large-maximal mixing between neighboring families (1- 2) (2- 3): bi-maximal + corrections? 0 0.2 0.4 0.6 0.8 1-3 1-2 2-3 11 The heaviest mass: m h ~ (0.04 - 0.3) eV  12  C  23   33

5. m >  m 2 From oscillations: Kinematic methods: From neutrinoless double beta decay  If the effective Majorana mass m ee is measured m > m ee /3 Troitsk: m e < 2.05 eV (95%) after ``anomaly’’ subtraction Mainz: m e < 2.3 eV (95%) updated, 2004 Future: KATRIN If m e = 0.35 eV m e < 0.2 eV (90%) upper bound 5  (statistical) from 0 discovery potential

6. m ee =  k U ek 2 m k e i  x p p n n e e m ee Neutrinoless double beta decay Z  (Z + 2) + e - + e - + + 2 neutrino double beta decay: Z  (Z + 2) + e - + e - H-M, NEMO ~ 200 000 events Spectrum total energy of the electron pair F E ee Q 2 0 Mechanisms of 0   -decay Majorana mass of the electron neutrino Rate ~ |m ee | 2

7. Fifth detector Heidelberg-Moscow experiment Evidence Cosmology? If 2  0 -> mechanism? Evidence Cosmology? If 2  0 -> mechanism?

8. 76 Ge -> 76 Se + e - + e - 4.2  - evidence 5 detectors, 71.7 kg yr Q ee = 2039 keV T 1/2 = 1.19 x 10 25 y T 1/2 = (0.69 – 4.18) x 10 25 y (3  range) m ee = 0.44 eV m ee = 0.24 – 0.58 eV (3  ) Spectrum near the end point Positive claim

9.  m ee =  k m ee (k) e i  k) Assuming 3 Majorana neutrinos m L – is the lightest eigenvalue  (m ee -  m L ) - plot Vissani Klapdor-Kleingrothhous Pas, A.S m ee (k) contribution from k-eigenvalue m ee (1) = U e1 2 m L m ee (2) = U e2 2 m L 2 +  m 21 2 m ee (3) = U e3 2 m L 2 +  m 31 2 U e1 2 > U e2 2 > U e3 2 For normal mass hierarchy m 3 > m 2 > m 1 = m lightest  m 21 2  m 31 2 m ee m L Cancellation is possible m ee (3) m ee (2) m ee (1)

10.  (m ee -  m L ) - plot m ee (3) = U e3 2 m L m ee (1) = U e1 2 m L 2 +  m 13 2 m ee (2) = U e2 2 m L 2 +  m 13 2 For inverted mass hierarchy m 2 > m 1 > m 3 = m L  m 31 2 No crossing of trajectories - no cancellation m ee (2) /m ee (1) = U e2 2 /U e1 2 = tan 2  sol m ee (3) << m ee (2) m ee (3) /m ee (2) < U e3 2 /U e2 2 < 1/7 mLmL m ee m ee (3) m ee (2) m ee (1)

11. A. Strumia, F. Vissani Neutrinoless double beta decay Kinematic searches, cosmology Sensitivity limit mLmL Heidelberg- Moscow m ee < 0.05 eV - excludes degenerate spectrum m ee < 0.01 eV - excludes inverted mass hierarchy Problems: - uncertainties of nuclear matrix elements - possible other contributions apart from neutrino mass If HM result confirmed – strongly degenerate spectrum

12. A. Strumia, F. Vissani mLmL HM(3  ) Cuoricino (90%) NEMO (90%) GERDA II CUORE NEMO: 100 Mo Comments Cuoricino, CUORE: 130 Te GERDA: 76 Ge IGEX (99%)

13. m ee = sin 2  sol  m sol 2 m ee = cos2  sol  m atm 2 1. Normal mass hierarchy: m 2 >> m 1 U e3 2 << 0.04 2. Inverted mass hierarchy Among observables opposite CP phases m ee =  m atm 2 the same CP phases 3. Degenerate mass spectrum m ee = m e m ee = cos2  sol m e opposite CP phases the same CP phases Also implies:

14. Following Max Tegmark and Sasha Dolgov Relative density fluctuations:  =  1). If all components cluster:  ~ a(t)  = 3H 2 m Pl 2 /8  + c/a 2 c = 3 m Pl 2 k/8  curvature  ~  c/a 2 since in the matter dominated epoch  ~ a -3 a(t) - scale parameter Indeed, from Einstein equation: so k – parameter in Friedman-Robertson-Walker metric, k = 0 in flat Universe (matter dominated epoch)  ~  c/a 2 a -3

15. 2). If only fraction  * of matter density clusters, fluctuations grow slower:  ~ a p p ~  * 3/5 3). Neutrinos do not cluster on the small enough scales even if they are massive and non-relativistic due to high velocities determined by the free streaming scale free free ~ v t 2 Distance neutrino travel while Universe expands by factor of 2 < free neutrino clustering is suppressed (escape velocity is smaller than typical neutrino velocity) > free neutrinos cluster as cold dark matter, p = 1 On scales  Change shape of the power spectrum (in contrast to DE)

16. Fluctuation growth factor: Dark energy (DE) and photons do not cluster. (Effect of photons can be neglected) When DE dominates,  * ~ 0 and clustering stops Clustering occurs in the epoch between a MD matter start to dominate and a  D when DE starts to dominate k is the wavenumber a  D a MD p(k) a  D a MD ~  * (k) 3/5 In the epoch a MD - a  D  * (k) ~ 1 – f (k) f (k) is the the energy density in neutrinos for which 1/k > free The growth factor a  D a MD (1 – f (k)) 3/5 ~ a  D a MD (1 – 3/5 f (k)) 4700 e –4f (k) ~  ~

17. f (k)  =  i m i n i (k) –8f (k) Power spectrum: P(k) = P(k,f) P(k,0) ~ e For non-relativistic neutrinos: For very large k – (small scales), all neutrinos in spectrum satisfy 1/k < free n i = 112/cm 3 If neutrino mass spectrum is degenerate:f (k)  = 3m n f (k) and therefore suppression of power spectrum decrease with k Energy density in non-clustering component for given k

18. M. Tegmark, et al solid line m = 0 dashed line  m = 1 eV For small scales the power is suppressed by ~2 P =

19. M. Tegmark et al, 95% constraints (Galaxy clustering) mm  = - 1

20.  free  = - 1 1  22 Dependence of bound on DE equation of state S. Hannestad, astro-ph/0505551

21.  i  m i m 0 > (0.08 - 0.10) eV G. L. Fogli et al., hep-ph/0408045 m < 0.13 eV, 95% C.L. U. Seljak et al. Degenerate spectrum Heidelberg- Moscow

22. e  Large Scintillator Neutrino Detector Los Alamos Meson Physics Facility e+e+ e + p => e + + n Cherenkov cone + scintillations p         e + + e +   p         e + + e +       e t Oscillations? P = (2.64 +/- 0.67 +/- 0.45) 10 -3 L = 30 m n decay at rest  m 2 > 0.2 eV 2 200 t mineral oil scintillator Beyond ``standard’’ picture: - new sector, - new symmetry

23. Ultimate oscillation anomaly? K.Babu, S Pakvasa Disfavored by a new analysis of KARMEN collaboration Disfavored by a new analysis of KARMEN collaboration Disfavored by atmospheric neutrino data, no compatibility of LSND and all-but LSND data below 3  -level Disfavored by atmospheric neutrino data, no compatibility of LSND and all-but LSND data below 3  -level O. Peres, A.S. M. Sorel, J. Conrad, M. Shaevitz M.C. Gonzalez-Garcia, M. Maltoni, T. Schwetz G. Barenboim, L. Borissov, J. Lykken S. Palomares-Ruiz, S. Pascoli, T.Schwetz R.Fardon, A. E. Nelson, N. Weiner CPT + (3+1)

25.   e 2 1 4 mass  m 2 atm  m 2 sun 3  m 2 LSND s Generic possibility of interest even independently of the LSND result Generation of large mixing of active neutrinos due to small mixing with sterile state Produces uncertainty in interpretation of results The problem is P ~ |U e4 | 2 |U  | 2 Restricted by short baseline experiments CHOOZ, CDHS, NOMAD 2 - 3  below the observed probability 1-3 subsystem of levels is frozen

26. Compatibility of short baseline Experiments and LSND datasets 95% 90% 99% Allowed regions from combined fit of LSND and short baseline experiments hep-ex/0407027

27.   e 2 1 4 mass  m 2 atm  m 2 sun 3  m 2 LSND ’ s 5 s ’  m 2 LSND FINeSE M. Sorel, J. Conrad, M. Shaevitz x x

28. 450 t (mineral oil) 1280 PMT 12 m diameter tank L = 541 m, ~ 800 MeV Search for e appearance

29. M.H. Shaevitz

31. A Yu Smirnov For vacuum oscillations:  (t) =  k U  k e k -i  k P(      ) = | | 2 P(      ) = |  k U  k *  U  k e | 2 -i  k

32. A Yu Smirnov CP-asymmetry: A CP = P(     ) - P(    ) T-asymmetry: A T = P(     ) - P(    ) A CP = 4 J CP sin t + sin t + sin t  m 12 2 2E  m 23 2 2E  m 31 2 2E J CP = Im [U e2 U  * U e3 * U  ] = = s 12 c 12 s 13 c 13 2 s 23 c 23 sin  where is the leptonic analogue of the Jarlskog invariant L. Wolfenstein, C. Jarlskog, V. Barger, K. Whisnant, R. Phillips For vacuum oscillations:

33. A Yu Smirnov P  = |  j U  j * U  j e | 2 ijij Transition probability    CP-transformation: P  CP  = |  j U  j U  j * e | 2 P  T = |  j U  j * U  j e | 2 = P  CP ijij J CP < 0.03 Oscillating factor is small unless long baseline (2000 - 3000 km) are taken Earth matter effect is important U  j --> U  j * T-transformation:  ijij in vacuum: Usual matter is CP-asymmetric CP-violation in neutrino oscillations even for  (U  j m ) CP = ( U  j m ) * in matter: Problem is to distinguish: fundamental CP violation CP-violation due to matter effect Precise knowledge of oscillation parameters, resolve ``degeneracy’’ of parameters, ambiguity… T-violation? Global fit

34.  (  m 23 2 ) ~ 0.0001 eV 0.7 GeV T2K JPARC  SuperKamiokande accelerator, off-a 295 km 2009 start NO A Fermilab  Ash River accelerator, off-a 810 km 2.2 GeV Double CHOOZ reactor baseline L mean energy goal status 1.05 km0.004 GeV project  (sin 2 2  23 ) ~ 0.01 Hierarchy ?  m 23 2 sin 2  13 < 0.005 – 0.008 2008 start - 2011 sin 2  13 < 0.005 90% C.L.      e e  e    2008 start ? sin 2  13 < 0.006 Hierarchy

35.  axis  detector E =  p * 1 + (  ) 2     = E  /m  p * = 0.03 GeV – momentum of neutrino in the rest frame of pion E EE    narrow energy spectrum Narrow spectrum – to Reduce background from high energy NC  N     X  0     e Searches for oscillations T2K NO A

37. E ( e ) < E ( e ) < E ( x )  10 11 - 10 12  g/cc 0

38. Normal hierarchyInverted hierarchy Both resonances are in the neutrino channel 1-3 resonance is in the antineutrino channnel

39. The MSW effect can be realized in very large interval of neutrino masses  m 2 ) and mixing Very sensitive way to search for new (sterile) neutrino states The conversion effects strongly depend on Type of the mass hierarchy Strength of the 1-3 mixing (s 13 ) A way to probe the hierarchy and value of s 13  m 2 = (10 -6 - 10 7) eV 2 sin 2 2  = (10 -8 - 1) If 1-3 mixing is not too small s 13 2 > 10 - 5 strong non-oscillatory conversion is driven by 1-3 mixing In the case of normal mass hierarchy: Small mixing angle realization of the MSW effect almost completely F( e ) = F 0 (  ) No earth matter effect in e - channel but in e - channel Neutronization e - peak disappears hard e - spectrum e    

40. Beam uncertainties can be controlled if Two well separated detectors are used Properties of medium are know Comparison of signals from the two detectors: oscillation effects between them and also test properties of the original flux This is realized for oscillations of SN neutrinos inside the Earth: D1 D2 L1L1 L2L2 Fluxes arriving at the surface of the earth are the same for both detectors If sin 2  13  >  10 -4 an appearance of the Earth matter effect in e or ( e ) signal will testify for normal (inverted) mass hierarchy of neutrinos

41. A Yu Smirnov Extreme cases A. Normal hierarchy large 1-3 mixing composite, weakly (sin 2  ~ 1/3) mixed e  -spectrum Earth matter effect B. Inverted hierarchy large 1-3 mixing C. Very small 1-3 mixing unmixed, hard in antineutrino channel unmixed, hard composite, strongly (cos 2  ~ 2/3) permuted in neutrino channel composite, strongly (cos 2  ~ 2/3) permuted composite, weakly (sin 2  ~ 1/3) mixed both in neutrino and antineutrino channels Large 1-3 mixing: sin 2  13 > 10 -4

42. R.C. Schirato, G.M. Fuller, astro-ph/0205390 The shock wave can reach the region relevant for the neutrino conversion  ~ 10 4 g/cc During 3 - 5 s from the beginning of the burst Influences neutrino conversion if sin 2  13 > 10 -5 ``wave of softening of spectrum’’ The effects are in the neutrino (antineutrino) for normal (inverted) hierarchy: change the number of events delayed Earth matter effect C.Lunardini, A.S., hep-ph/0302033 R.C. Schirato, G.M. Fuller, astro-ph/0205390 K. Takahashi et al, astro-ph/0212195 Density profile with shock wave propagation at various times post-bounce h - resonance

43. G. Fuller time of propagation velocity of propagation shock wave revival time density gradient in the front size of the front Can shed some light on mechanism of explosion Studying effects of the shock wave on the properties of neutrino burst one can get (in principle) information on Steep front: breaks adiabaticity or make its violation stronger, - after passing can be restored again - influence transitions

44. F( e ) = F 0 ( e ) + p  F 0  F 0 = F 0 (  ) - F 0 ( e ) p is the permutation factor p The earth matter effect can partially explain the difference of Kamiokande and IMB: spectra of events p depends on distance traveled by neutrinos inside the earth to a given detector: 4363 km Kamioka d = 8535 km IMB 10449 km Baksan C.Lunardini, A.S. One must take into account conversion effects of supernova neutrinos Conversion in the star Earth matter effect Normal hierarchy is preferable H. Minakata, H. Nunokawa, J Bahcall, D Spergel, A.S.

45. Average energy of observed events Average energy of the original e -flux