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Neutrino Oscillations

Neutrino Oscillations. Or how we know most of what we know. Outline. Two-flavor vacuum oscillations Two-flavor matter oscillations Three-flavor oscillations The general formalism The “rotation” matrices. Consider Two Mass States.  1 corresponding to m 1  2 corresponding to m 2

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Neutrino Oscillations

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  1. Neutrino Oscillations Or how we know most of what we know

  2. Outline • Two-flavor vacuum oscillations • Two-flavor matter oscillations • Three-flavor oscillations • The general formalism • The “rotation” matrices Steve Elliott, NPSS 2005

  3. Consider Two Mass States 1 corresponding to m1 2 corresponding to m2 Think of  as a Vector Steve Elliott, NPSS 2005

  4.  is a solution of H Steve Elliott, NPSS 2005

  5. The Neutrinos Consider the weak eigenstates e, . These are not the mass eigenstates, 1, . The mass eigenstates are propagated via H. The Mixing Matrix: U Steve Elliott, NPSS 2005

  6. Mixing Weak eigenstates are a linear superposition of mass eigenstates. Steve Elliott, NPSS 2005

  7. In Vacuum, no potential in H Denote c = cos  s = sin  Steve Elliott, NPSS 2005

  8. UHU-1 Steve Elliott, NPSS 2005

  9. The energy difference (and Trig.) Steve Elliott, NPSS 2005

  10. UHU-1 becomes The algebra is going to get involved, so lets define A, B, and D such that: Steve Elliott, NPSS 2005

  11. The Diff Eq A solution to this equation should have the form: Steve Elliott, NPSS 2005

  12. Insert proposed solution Steve Elliott, NPSS 2005

  13. Two Equations Steve Elliott, NPSS 2005

  14. r+ solution r- solution Steve Elliott, NPSS 2005

  15.  is a superposition of these 2 solutions (D+2A) is a constant so we sweep it into a redefinition of the C’s. Steve Elliott, NPSS 2005

  16. The solutions To determine the C’s, use <|>=1 and assume that at t=0, we have all e. Steve Elliott, NPSS 2005

  17. The time dependent solution What is the probability of finding all  at time t? Steve Elliott, NPSS 2005

  18. Transition probability Steve Elliott, NPSS 2005

  19. The Answer Complete mixing: large sin2 and long R/L would result in an “average”: that is P=1/2. Steve Elliott, NPSS 2005

  20. What about MSW? The Sun is mostly electrons (not muons). e can forward scatter from electrons via the charged or neutral current.  can only forward scatter via the neutral current. The e picks up an effective mass term, which acts on the weak eigenstates. Steve Elliott, NPSS 2005

  21. The MSW H term. This extra term results in an oscillation probability that can have a resonance. Thus even a small mixing angle, , can have a large oscillation probability. Steve Elliott, NPSS 2005

  22. Similar algebra as before Steve Elliott, NPSS 2005

  23. Constant Density Solutions Note similar form to vacuum Oscillations. Note that sin22m can be 1 even when sin22is small. That is when: L/L0 = cos2 Steve Elliott, NPSS 2005

  24. Variable Density • Integrate over the changing density (such as in a star). Steve Elliott, NPSS 2005

  25. Three  Formulism Steve Elliott, NPSS 2005

  26. Transition Probability Steve Elliott, NPSS 2005

  27. Transition Probability Real U’s Steve Elliott, NPSS 2005

  28. Complex U’s If U is complex, then we have the possibility Steve Elliott, NPSS 2005

  29. Oscillation Experiments Appearance: look for  when none are expected Disappearance: look for decrease in flux of  Steve Elliott, NPSS 2005

  30. Neutrino Sources and Oscillations • Solar neutrinos • Few MeV, L~1011 m • Electron neutrinos • Most are disappearance expts. (Except SNO NC and SK’s slight NC sensitivity) • Reactor • Few MeV, L~10m - 300 km • Electron neutrino disappearance Steve Elliott, NPSS 2005

  31. Neutrino Sources • Accelerator • 30-50 MeV ( decay) • DIF sources can be several GeV • Various appearance and disappearance modes, various baselines • Atmospheric •  and  decay • Various energies • Baseline from 20 to 10,000 km Steve Elliott, NPSS 2005

  32. Maki, Nakagawa, Sakata, Pontecorvo Atmospheric Reactor Solar Steve Elliott, NPSS 2005

  33. PDB parameterization Steve Elliott, NPSS 2005

  34. CP violation Steve Elliott, NPSS 2005

  35. The Jarlskog Invariant Note the product of the sin of all the angles. If any angle is 0, CP violation is not observable. Note that I have seen different values of the leading constant. (taken to be 1 here) Steve Elliott, NPSS 2005

  36. CP violation hep-ph/0306221 Steve Elliott, NPSS 2005

  37. There are only 2 independent m2 for 3  This will be important when we discuss LSND. Steve Elliott, NPSS 2005

  38. Resources • Steve Elliott - UW Phys 558 class notes • Bahcall Book • Many phenomenology papers Steve Elliott, NPSS 2005

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