1. Neutrino Oscillation in Dense Matter Speaker: Law Zhiyang, National University of Singapore

2. Outline: 1.Lagrangian & Field Equations (Review) 2.Conserved Norm (In Uniform Matter Distribution) 3.Solutions to Field Equations (Uniform Matter) 4.Neutrino Oscillation in Uniform Matter 5.Conserved Norm (In Non-Uniform Matter) 6.Conclusion

3. 1.Lagrangian & Field Equations (Review):

4. Neutrino Field Equations: Eliminate right-handed field: see Halprin, PRD 34, #11, 3462(1986) * * Giunti et al PRD 43, #1, 164(1991) Assume matter is isotropic & non-relativistic:

5. Neutrino Field Equations: Notation from Kiers et al, PRD 56, # 9, 5776(1997). where,

6. 2. Conserved Norm (uniform matter): - To consider oscillation probability, a conserved norm relevant to the left-handed field is needed : For uniform, time-independent matter distribution, “A” is constant :

7. Conserved Norm (uniform matter): Consider 2 arbitrary wave functions (flavor basis) :

8. Conserved Norm (uniform matter): (1) - (2) :

9. 3. Solutions to Field Equations (Uniform Matter) : -Field equation needs to be solve to calculate probability current. For uniform matter: - Assume “ ” is traveling along, and with spin in the x-axis : Ω

10. Solutions to Field Equations (Uniform Matter) : where ;

11. Solutions to Field Equations (Uniform Matter) : Effective mass matrix (diagonal elements): This distinguishes the propagation modes (k 1/2 ) for a given “E” This quartic equation gives 4 solutions: 2 propagation modes, k 1/2, and “negative” & “positive” energy solutions * Similar dispersion relations is found by Kiers et al, PRD 56, # 9, 5776(1997).

12. Solutions to Field Equations (Uniform Matter) : Taking positive energy solution in propagation basis: Solutions in flavor basis : General solution (flavor basis) :

13. 4. Neutrino Oscillation in Uniform Matter An example: Oscillation of e : Recall : This is diagonal in flavor space; can identify e -component &  -component probability current.

14. Neutrino Oscillation in Uniform Matter e -component :  -component : From: (this is time independent)

15. Neutrino Oscillation in Uniform Matter This implies that the currents are positive when (k 1 -(  -  )), (k 2 -(  -  )) & (k 1 +  )), (k 2 +  )) are positive; Because : J(  ) = positive J(total) = constant = positive J(  ) = positive Therefore a consistent oscillation probability can be calculated.

16. Neutrino Oscillation in Uniform Matter After some algebra : Amp. =

17. Neutrino Oscillation in Uniform Matter Check vacuum oscillation in ultra-relativistic limit : This is the standard result for oscillation amplitude (vacuum).

18. Neutrino Oscillation in Uniform Matter Example 2 : Maybe in a dense astrophysical medium : Similarly for the dispersion relations :

19. Neutrino Oscillation in Uniform Matter Solving for k : left propagating solution

20. Neutrino Oscillation in Uniform Matter Oscillation length : modification to standard value

21. Neutrino Oscillation in Uniform Matter Amplitude : Resonance : But if “  ”, “E” >> “m 1 2 ”, “m 2 2 ” just vacuum amplitude

22. 5. Conserved Norm (In Non-Uniform Matter) Recall field equation : We replace : Due to the short wave length of neutrinos : Hence probability current derived is also applicable to non-uniform matter.

23. 6. Conclusion : 1.A quantum mechanical treatment of neutrino oscillation in matter is discussed. 2.A relevant conserved current is derived 3.Oscillation length & amplitude is studied for special cases together with resonance. 4.Conserved current is also applicable to non-uniform matter.

24. The End