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

2. June 2005Steve Elliott, NPSS 20052 Outline Two-flavor vacuum oscillations Two-flavor matter oscillations Three-flavor oscillations –The general formalism –The “rotation” matrices

3. June 2005Steve Elliott, NPSS 20053 Consider Two Mass States  1 corresponding to m 1  2 corresponding to m 2 Think of  as a Vector

4. June 2005Steve Elliott, NPSS 20054  is a solution of H

5. June 2005Steve Elliott, NPSS 20055 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

6. June 2005Steve Elliott, NPSS 20056 Mixing Weak eigenstates are a linear superposition of mass eigenstates.

7. June 2005Steve Elliott, NPSS 20057 In Vacuum, no potential in H Denote c = cos  s = sin 

8. June 2005Steve Elliott, NPSS 20058 U H U -1

9. June 2005Steve Elliott, NPSS 20059 The energy difference (and Trig.)

10. June 2005Steve Elliott, NPSS 200510 U H U -1 becomes The algebra is going to get involved, so lets define A, B, and D such that:

11. June 2005Steve Elliott, NPSS 200511 The Diff Eq A solution to this equation should have the form:

12. June 2005Steve Elliott, NPSS 200512 Insert proposed solution

13. June 2005Steve Elliott, NPSS 200513 Two Equations

14. June 2005Steve Elliott, NPSS 200514 r + solution r - solution

15. June 2005Steve Elliott, NPSS 200515  is a superposition of these 2 solutions (D+2A) is a constant so we sweep it into a redefinition of the C’s.

16. June 2005Steve Elliott, NPSS 200516 The solutions To determine the C’s, use =1 and assume that at t=0, we have all e.

17. June 2005Steve Elliott, NPSS 200517 The time dependent solution What is the probability of finding all   at time t?

18. June 2005Steve Elliott, NPSS 200518 Transition probability

19. June 2005Steve Elliott, NPSS 200519 The Answer Complete mixing: large sin2  and long R/L would result in an “average”: that is P=1/2.

20. June 2005Steve Elliott, NPSS 200520 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.

21. June 2005Steve Elliott, NPSS 200521 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.

22. June 2005Steve Elliott, NPSS 200522 Similar algebra as before

23. June 2005Steve Elliott, NPSS 200523 Constant Density Solutions Note similar form to vacuum Oscillations. Note that sin 2 2  m can be 1 even when sin 2 2  is small. That is when: L/L 0 = cos2 

24. June 2005Steve Elliott, NPSS 200524 Variable Density Integrate over the changing density (such as in a star).

25. June 2005Steve Elliott, NPSS 200525 Three Formulism

26. June 2005Steve Elliott, NPSS 200526 Transition Probability

27. June 2005Steve Elliott, NPSS 200527 Transition Probability Real U’s

28. June 2005Steve Elliott, NPSS 200528 Complex U’s If U is complex, then we have the possibility

29. June 2005Steve Elliott, NPSS 200529 Oscillation Experiments Appearance: look for  when none are expected Disappearance: look for decrease in flux of 

30. June 2005Steve Elliott, NPSS 200530 Neutrino Sources and Oscillations Solar neutrinos –Few MeV, L~10 11 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

31. June 2005Steve Elliott, NPSS 200531 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

32. June 2005Steve Elliott, NPSS 200532 SolarReactorAtmospheric Maki, Nakagawa, Sakata, Pontecorvo

33. June 2005Steve Elliott, NPSS 200533 PDB parameterization

34. June 2005Steve Elliott, NPSS 200534 CP violation

35. June 2005Steve Elliott, NPSS 200535 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)

36. June 2005Steve Elliott, NPSS 200536 CP violation hep-ph/0306221

37. June 2005Steve Elliott, NPSS 200537 There are only 2 independent  m 2 for 3 This will be important when we discuss LSND.

38. June 2005Steve Elliott, NPSS 200538 Resources Steve Elliott - UW Phys 558 class notes Bahcall Book Many phenomenology papers