1. Degeneracy and Correlation in Neutrino Oscillation Parameters Hisakazu Minakata Tokyo Metropolitan University Hisakazu Minakata Tokyo Metropolitan University

2. May. 2-6, 2006SNOW@Stockholm Degeneracy is common in everyday life

3. May. 2-6, 2006SNOW@Stockholm A machinery in my talk Oscillation probability draw ellipse if plotted in bi-P plane Role played by CP phase  and the matter clearly distinguished Art work by Adam Para Two solutions of S 23 2 x

4. May. 2-6, 2006SNOW@Stockholm Cause of the degeneracy; easy to understand You can draw two ellipses from a point in P-Pbar space Intrinsic degeneracy Doubled by the unknown sign of  m 2 4-fold degeneracy

5. May. 2-6, 2006SNOW@Stockholm  23 octant degeneracy P  e= sin 2 2  13 x s 2 23 Solar  m 2 on Matter effect on Solar  m 2 on Matter effect on OY Nufact03 Altogether 8-fold degeneracy

6. May. 2-6, 2006SNOW@Stockholm Structure of intrinsic & sign-  m 2 degeneracy in (matter) perturbative regime Intrinsic degeneracy;     sign(  m 2 )-  degeneracy arises because P is approx. invariant under:  m 2    m 2  T2K II MN JHEP01

7. May. 2-6, 2006SNOW@Stockholm Invitation to degeneracy solving

8. May. 2-6, 2006SNOW@Stockholm General comments Degeneracy occurs because of the lack of information, e.g., P , P  e and P  e-bar at a particular  energy Therefore, it can be resolved by adding information easy for theorists Practically, however, NOT an easy task to carry out experimentally Discussion cannot be purely “theoretical” and has to involve “feasibility of the measurement” you propose, and “practicality of the experiments” No unique answer !

9. May. 2-6, 2006SNOW@Stockholm Possible alternative strategies toward the resolution Conceptually simple Probably require multiple detectors/settings Inevitable if 3 types of degeneracy tightly couple with each other More practical and experimentalists friendly Has to show that one can do the job one by one Everything at once approach One by one resolution approach

10. May. 2-6, 2006SNOW@Stockholm General comments (continued) Lifting sign-  m 2 degeneracy requires matter effect long baseline => \gsim 1000 km required intrinsic degeneracy can be efficiently resolved by spectrum information Generally speaking,  23 octant degeneracy tends to decouple from others many ideas welcome Exception:  m 2 (ee) -  m 2 (  ) method alternative ways ?

11. May. 2-6, 2006SNOW@Stockholm factory as ultimate degeneracy solver By combining at 3 detectors at 130, 730, and 2810 km, it was claimed that neutrino factory can resolve all the 8- fold degeneracy if  13 > 1° (Donini, NuFACT03) Typical everything at once type

12. May. 2-6, 2006SNOW@Stockholm Golden + silver NuFact + superbeam 1st concrete way that proved to work for resolution of all the degeneracy Powerful, work down to 1-2 degree of  13 Note that V cb ~ 2 degree Yet, a bit expensive; conservatively ~1000 Million Euro/degeneracy (rich man’s solution) Ultimate degeneracy solver, but certainly not “Mayflower” (unless  13 ~ 1 o ) Highly nontrivial! If NuFACT reached “J-PARC stage” everything changes !

13. May. 2-6, 2006SNOW@Stockholm From “black- box”to the anatomy of degeneracy

14. May. 2-6, 2006SNOW@Stockholm Anatomy of degeneracy To make real progress, it is important to have a systematic understanding of the nature of degeneracy Need concrete framework to analyze the question 1st order in matter perturbation theory plus corresponding appropriate experimental setting; T2K II or T2KK(Kamioka-Korea) Ishitsuka et al. 05

15. May. 2-6, 2006SNOW@Stockholm Let’s start from the intrinsic degeneracy degenerate solution at VOM:  13 (2) =  13 (1)     T2K II Spectrum analysis should be powerful !

16. May. 2-6, 2006SNOW@Stockholm Spectral information IS powerful (almost) resolve intrinsic degeneracy at sin 2 2  13 =0.01 T2K II (from 1000 page Ishitsuka file) SK momentum resolution ~30 MeV at 1 GeV

17. May. 2-6, 2006SNOW@Stockholm Expected signal, BG and  2 and anti- × Kamioka and Korea and anti- × Kamioka and Korea 5 energy bins syst. Errors: 5% BG (overall) 5% BG (energy dep.) 5% signal efficiency syst. Errors: 5% BG (overall) 5% BG (energy dep.) 5% signal efficiency 4yrs + 4yrs anti- sin 2 2  13 =0.1,  =  /2 signal BG Kajita in NOVE2006

18. May. 2-6, 2006SNOW@Stockholm Sign-  m 2 degeneracy Resolution of sign-  m 2 degeneracy requires the matter effect Requires baseline ~1000 km 2nd detector seems required to go down to sin 2 2  13 =.02

19. May. 2-6, 2006SNOW@Stockholm T2K(0.54 Mt) vs. T2KK(0.27+0.27 Mt)    

20. May. 2-6, 2006SNOW@Stockholm It is not quite only the matter effect With the same input parameter and Korean detector of 0.54 Mt the sign-  m 2 degeneracy is NOT completely resolved 2 identical detector method powerful !

21. May. 2-6, 2006SNOW@Stockholm Energy dependence far more dynamic in 2nd oscillation maximum

22. May. 2-6, 2006SNOW@Stockholm How to solve  23 octant degeneracy Detect solar Dm2 term ~c 23 2 Requires baseline ~ earth radius well controlled systematic error plus enormous statistics Powerful at small  13 Combining reactor measurement of  13 Requires great precision in  13 measurement Powerful at large  13 If feasible, “Mossbauer” helps Strategy: Look for terms which depend on  23 but not through the form s 23 2 x sin 2 2  13

23. May. 2-6, 2006SNOW@Stockholm T.Kajita@NNN05 Atm ; powerful way for octant degene.

24. May. 2-6, 2006SNOW@Stockholm Reactor + accelerator method Acc-disappearance => s 2 23 =0.4 or 0.6 Acc-appearance => s 2 23 sin 2 2  13 = 0.06 Reactor => sin 2 2  13 = 0.1 Solves  23 degeneracy

25. May. 2-6, 2006SNOW@Stockholm Region of resolved  23 degeneracy T2K II (4MW+HK) + phase II type reactor Hiraide et al. (7 samurai)

26. May. 2-6, 2006SNOW@Stockholm Decoupling of the degeneracy Suppose that you succeed to solve the particular degeneracy, by forgetting about the remaining ones It does NOT necessarily mean that the problem is solved You have to verify that your treatment of degeneracy A is valid irrespective of the presence of degeneracy B One solution: decoupling between the degeneracy

27. May. 2-6, 2006SNOW@Stockholm  23 and sign-  m 2 degeneracy decouple For example, one can show, to first order in matter effect, the followings:  P(octant) = P(1st octant) - P(2nd) is invariant under the interchange of two sign-  m 2 degenerate pair  P(hierarchy) = P(  m 2 +) - P(  m 2 -) is invariant under the interchange of two  23 octant degenerate pair

28. May. 2-6, 2006SNOW@Stockholm Intrinsic degeneracy decouple ? Intrinsic degeneracy nearly decouples from the rest More importantly, in T2K II setting, it is resolved by spectrum analysis decouple from the game Altogether three degeneracies (approximately) decouple with each other in T2K (or T2KK) setting

29. May. 2-6, 2006SNOW@Stockholm Correlation rather than degeneracy?

30. May. 2-6, 2006SNOW@Stockholm Correlation ? The title of this talk contains “correlation”. What about that ? “correlation” may be incompletely resolved degeneracy No continuous degeneracy in P  e Nufact golden

31. May. 2-6, 2006SNOW@Stockholm No significant “correlation” in T2K II Spectral information important

32. May. 2-6, 2006SNOW@Stockholm Many remaining questions Does  beam help ?? Does monochromatic beam help ?? To what extent the lower energy threshold of NuFACT change the sensitivity of resolving degeneracy ? What would be a powerful supplementary measurement ? ISS NuFACT ongoing, report expected in Sep. 06 example:  12 for  m 2 ee -  m 2  method

33. May. 2-6, 2006SNOW@Stockholm Conclusion Nature of the degeneracy well understood With T2KK or similar 2 detector settings there appeared a well defined method for lifting intrinsic and sign-  m 2 degeneracy Spectral information solves intrinsic degeneracy and matter effect does sign-  m 2 degeneracy Resolving  23 degeneracy requires solar term or reactor II Consistency maintained by “decoupling”

34. May. 2-6, 2006SNOW@Stockholm  -> e appearance probability