1. 1 Neutrino Phenomenology Boris Kayser Scottish Summer School August 10, 2006 +

2. 2 What We Have Learned

3. 3 (Mass) 2 1 2 3 or 1 2 3 }  m 2 sol  m 2 atm }  m 2 sol  m 2 atm  m 2 sol = 8.0 x 10 –5 eV 2,  m 2 atm = 2.7 x 10 –3 eV 2 ~ ~ Normal Inverted Are there more mass eigenstates, as LSND suggests? The (Mass) 2 Spectrum

4. 4 This has the consequence that — | i  > =  U  i |  >. Flavor-  fraction of i = |U  i | 2. When a i interacts and produces a charged lepton, the probability that this charged lepton will be of flavor  is |U  i | 2.  Leptonic Mixing

5. 5 e [|U ei | 2 ]  [|U  i | 2 ]  [|U  i | 2 ] Normal Inverted  m 2 atm    (Mass) 2  m 2 sol }   m 2 atm    m 2 sol } or sin 2  13 The spectrum, showing its approximate flavor content, is

6. 6  m 2 atm e [|U ei | 2 ]  [|U  i | 2 ]  [|U  i | 2 ]    (Mass) 2  m 2 sol } Bounded by reactor exps. with L ~ 1 km From max. atm. mixing, From  (Up) oscillate but  (Down) don’t { { { In LMA–MSW, P sol ( e  e ) = e fraction of 2 From max. atm. mixing,    includes (  –  )/√2 From distortion of e (solar) and e (reactor) spectra

7. 7 The 3 X 3 Unitary Mixing Matrix Caution: We are assuming the mixing matrix U to be 3 x 3 and unitary. Phases in U will lead to CP violation, unless they are removable by redefining the leptons.

8. 8 U  i describes — i  W+W+ – U  i     W +  H  i  – When  i    e i  i , U  i  e i  U  i When         e i    , U  i  e –i  U  i – – Thus, one may multiply any column, or any row, of U by a complex phase factor without changing the physics. Some phases may be removed from U in this way.

9. 9 Exception : If the neutrino mass eigenstates are their own antiparticles, then — Charge conjugate i = i c = C i T One is no longer free to phase-redefine i without consequences. U can contain additional CP-violating phases.

10. 10 How Many Mixing Angles and CP Phases Does U Contain? Real parameters before constraints: 18 Unitarity constraints — Each row is a vector of length unity: – 3 Each two rows are orthogonal vectors: – 6 Rephase the three  : – 3 Rephase two i, if i  i : – 2 Total physically-significant parameters: 4 Additional (Majorana) CP phases if i = i : 2

11. 11 How Many Of The Parameters Are Mixing Angles? The mixing angles are the parameters in U when it is real. U is then a three-dimensional rotation matrix. Everyone knows such a matrix is described in terms of 3 angles. Thus, U contains 3 mixing angles. Summary Mixing angles CP phases if i  i CP phases if i = i 331

12. 12 The Mixing Matrix  12 ≈  sol ≈ 34°,  23 ≈  atm ≈ 37-53°,  13 < 10°  would lead to P(    ) ≠ P(    ). CP But note the crucial role of s 13  sin  13. c ij  cos  ij s ij  sin  ij AtmosphericCross-Mixing Solar Majorana CP phases ~

13. 13 The Majorana CP Phases The phase  i is associated with neutrino mass eigenstate i : U  i = U 0  i exp(i  i /2) for all flavors . Amp(    ) =  U  i * exp(– im i 2 L/2E) U  i is insensitive to the Majorana phases  i. Only the phase  can cause CP violation in neutrino oscillation. i

14. 14 The Open Questions

15. 15 What is the absolute scale of neutrino mass? Are neutrinos their own antiparticles? Are there “sterile” neutrinos? We must be alert to surprises !

16. 16 What is the pattern of mixing among the different types of neutrinos? What is  13 ? Is  23 maximal? Do neutrinos violate the symmetry CP? Is P(    )  P(    ) ? Is the spectrum like or ?

17. 17 What can neutrinos and the universe tell us about one another? Is CP violation by neutrinos the key to understanding the matter – antimatter asymmetry of the universe? What physics is behind neutrino mass?

18. 18 The Importance of the Questions, and How They May Be Answered

19. 19 How far above zero is the whole pattern? 3 0 (Mass) 2  m 2 atm  m 2 sol ?? 1 2 Flavor Change Tritium Decay, Double  Decay Cosmology } } What Is the Absolute Scale of Neutrino Mass?

20. 20 A Cosmic Connection Cosmological Data + Cosmological Assumptions   m i < (0.17 – 1.0) eV. Mass( i ) If there are only 3 neutrinos, 0.04 eV < Mass[Heaviest i ] < (0.07 – 0.4) eV  m 2 atm Cosmology ~ Oscillation Data   m 2 atm < Mass[Heaviest i ] Seljak, Slosar, McDonald Pastor ( )

21. 21 Are Neutrinos Majorana Particles?

22. 22 How Can the Standard Model be Modified to Include Neutrino Masses?

23. 23 Majorana Neutrinos or Dirac Neutrinos? The S(tandard) M(odel) and couplings conserve the Lepton Number L defined by— L( ) = L( – ) = – L( ) = – L ( + ) = 1. So do the Dirac charged-lepton mass terms m L R – – W – Z  + (—)(—) (—)(—) X m

24. 24 Original SM: m = 0. Why not add a Dirac mass term, m D L R Then everything conserves L, so for each mass eigenstate i, i ≠ i (Dirac neutrinos) [L( i ) = – L( i )] The SM contains no R field, only L. To add the Dirac mass term, we had to add R to the SM. X mDmD (—)(—) (—)(—) —

25. 25 Unlike L, R carries no Electroweak Isospin. Thus, no SM principle prevents the occurrence of the Majorana mass term m R R c R But this does not conserve L, so now i = i (Majorana neutrinos) [No conserved L to distinguish i from i ] We note that i = i means — i (h) = i (h) helicity X mRmR — — — — —

26. 26 The objects R and R c in m R R c R are not the mass eigenstates, but just the neutrinos in terms of which the model is constructed. m R R c R induces R R c mixing. As a result of K 0 K 0 mixing, the neutral K mass eigenstates are — K S,L  (K 0  K 0 )/  2. As a result of R R c mixing, the neutrino mass eigenstate is — i = R + R c = “ + ”.

27. 27 Many Theorists Expect Majorana Masses The Standard Model (SM) is defined by the fields it contains, its symmetries (notably Electroweak Isospin Invariance), and its renormalizability. Leaving neutrino masses aside, anything allowed by the SM symmetries occurs in nature. If this is also true for neutrino masses, then neutrinos have Majorana masses.

28. 28 The presence of Majorana masses i = i (Majorana neutrinos) L not conserved — are all equivalent Any one implies the other two.

29. 29 Electromagnetic Properties of Majorana Neutrinos Majorana neutrinos are very neutral. No charge distribution: CPT [ ] = ≠ + + + – – – But for a Majorana neutrino, [ ] CPT i [ ] i =

30. 30 No magnetic or electric dipole moment: [ ] [ ] e+e+ e–e–   = – But for a Majorana neutrino, i i = Therefore, [ i ] =   = 0

31. 31 Transition dipole moments are possible, leading to — e e 1 2  One can look for the dipole moments this way. To be visible, they would have to vastly exceed Standard Model predictions.

32. 32 How Can We Demonstrate That i = i ? We assume neutrino interactions are correctly described by the SM. Then the interactions conserve L (  – ;  + ). An Idea that Does Not Work [and illustrates why most ideas do not work] Produce a i via— ++ i ++  Spin Pion Rest Frame   (Lab) >  (  Rest Frame) ++  i ++ Lab. Frame Give the neutrino a Boost:

33. 33 The SM weak interaction causes—  i ++ Recoil Target at rest —    i    i —   i, will make  + too. i = i means that i (h) = i (h). helicity

34. 34 Minor Technical Difficulties   (Lab) >  (  Rest Frame) E  (Lab) E (  Rest Frame) m  m  E  (Lab) > 10 5 TeV if m ~ 0.05 eV Fraction of all  – decay i that get helicity flipped  ( ) 2 ~ 10 -18 if m  ~ 0.05 eV Since L-violation comes only from Majorana neutrino masses, any attempt to observe it will be at the mercy of the neutrino masses. (BK & Stodolsky) i  > ~ i i m E (  Rest Frame) i

35. 35 The Idea That Can Work — Neutrinoless Double Beta Decay [0  ] Observation would imply L and i = i. By avoiding competition, this process can cope with the small neutrino masses. i i W–W– W–W– e–e– e–e– Nuclear Process Nucl Nucl’  i

36. 36 0  e–e– e–e– uddu ( ) R L WW Whatever diagrams cause 0 , its observation would imply the existence of a Majorana mass term: Schechter and Valle ( ) R  L : A Majorana mass term

37. 37 i i W–W– W–W– e–e– e–e– Nuclear Process Nucl Nucl’ In the i is emitted [RH + O{m i /E}LH]. Thus, Amp [ i contribution]  m i Amp[0  ]   m i U ei 2  m  U ei i SM vertex  i Mixing matrix Mass ( i )

38. 38 The proportionality of 0  to mass is no surprise. 0  violates L. But the SM interactions conserve L. The L – violation in 0  comes from underlying Majorana mass terms.

39. 39 Backup Slides

40. 40 How Large is m  ? How sensitive need an experiment be? Suppose there are only 3 neutrino mass eigenstates. (More might help.) Then the spectrum looks like — sol < 2 1 3 atm 3 sol < 1 2 atm or

41. 41 m    m i U ei 2  The e (top) row of U reads —  12 ≈   ≈ 34°, but s 13 2 < 0.032

42. 42 If the spectrum looks like— then– m  ≅ m 0 [ 1 - sin 2 2   sin 2 ( ––––– )] ½. Solar mixing angle m 0 cos 2    m   m 0 At 90% CL, m 0 > 46 meV (MINOS); cos 2   > 0.28 (SNO), so m  > 13 meV. 2–122–12 Majorana CP phases { sol < atm m0m0

43. 43 If the spectrum looks like then — 0 < m  < Present Bound [(0.3–1.0) eV]. (Petcov et al.) Analyses of m  vs. Neutrino Parameters Barger, Bilenky, Farzan, Giunti, Glashow, Grimus, BK, Kim, Klapdor-Kleingrothaus, Langacker, Marfatia, Monteno, Murayama, Pascoli, Päs, Peña-Garay, Peres, Petcov, Rodejohann, Smirnov, Vissani, Whisnant, Wolfenstein, Review of  Decay: Elliott & Vogel sol < atm Evidence for 0  with m  = (0.05 – 0.84) eV? Klapdor-Kleingrothaus