1. Parameter Degeneracy in Neutrino Oscillations (and how to solve it?)
INT Program 2010; LBL ..
Hisakazu Minakata
Tokyo Metropolitan University

2. Purpose of this discussion
To complete n Standard Model (SM + n mass + lepton mixing) measurement of CP phase (KM type) d and q13 is necessary
It seems that is not so easy to determine them, in particular d
If any theoretical issues involved we shall try to remove them
One of them is P degeneracy
July 27, 2010
INT Program LBL

3. P degeneracy
P degeneracy is the fact that measurement of n oscillation probability P and n-bar oscillation probability bar-P at an energy (which would determine q13 and d) actually do NOT lead to a unique solution of q13 and d
Experts may say that they know everything
Is this true? To what extent?
July 27, 2010
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4. An example; Intrinsic degeneracy
P degeneracy is simplest to see by bi-P plot (HM-H.Nunokawa 01)
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5. Is P degeneracy necessarily two-fold?
But, the answer is NO !
People suspect the answer is YES because
Intrinsic degeneracy; S. Uchinami for PhD thesis
July 27, 2010
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6. Parameter
Degeneracy;
definition
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7. P degeneracy
Let us assume that all the mixing parameters besides q13 and d are known
measurement of n oscillation probability Pme and bar-n oscillation probability bar-Pme at an energy E (which would determine q13 and d) do NOT lead to unique solution of q13 and d
Easy to solve mathematically: measurement at E=E1 and E2 (or adding more channel) solves the degeneracy
Intrinsic degeneracy (Burguet-C. et al. 01)
July 27, 2010
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8. P degeneracy (continued)
the mixing parameters besides q13 and d are not known so precisely
Mass hierarchy is not known, and may not be known either at the time of measurement of CP phase d
More solutions of q13 and d:
Sign Dm231 degeneracy (HM-Nunokawa 01)
q23 octant degeneracy (Fogli-Lisi 96)
July 27, 2010
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9. P degeneracy is doubled by unknown mass hierarchy
You can draw two ellipses from a point in P-Pbar space
Intrinsic degeneracy
Doubled by the unknown sign of m2
4-fold degeneracy
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10. A well-defined framework for P degeneracy
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11. I use Cervera et al. formula for n oscillation probabilities
You can show 2x2x2=8
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12. P degeneracy; Generalized version
Similar degeneracy occurs in, in addition to (P, PCP),
T-conjugate (P=Pme, PT=Pem)
CPT-conjugate (P, PCPT)
Golden-silver (PT, PS) channels
Generally, P degeneracy has simpler structure
July 27, 2010
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13. P-degeneracy as an invariance of P
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14. P-dege. from symmetry of the probability
are invariant under transf.
PT and PS are also invariant under the same transformation
(1) P degeneracy obvious (2) Form of the degeneracy solutions are determined by the symmetry
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15. How to obtain degeneracy solutions?
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16. An example; intrinsic degeneracy
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17. An example; intrinsic degeneracy2
4th-order eq. of s13!
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18. P degeneracy as a re-parametrization invariance
Degeneracy solutions form network!
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19. Degeneracy solutions; how they look like?
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20. q13 II
III V
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21. d
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22. I focus energy dependence; q13
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23. I focus energy dependence; d
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24. How to solve P degeneracy?
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25. Varying E at long enough baseline
atmospheric
solar
Vacuum effect comes in with L/E
Matter effect comes in with aL
Varying E implies to change relative importance between vacuum and matter effects (varying L not)
powerful for mass hierarchy
a=sqrt{2}GFNe
the best way for everything if perfect event reconstruction with 2-3 oscillation periods spanned
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26. Project X: Off-axis NOVA --> VLBL multi-OM type approach
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27. Practical issues in VLBL approach
In water background at low energies for high energy  beam highly nontrivial -> see next page
How reliable is the event reconstruction & background rejection algorithm ?
Energy resolution
Alternative way; ~100 kt scale Liquid Ar detector => feasible when?
July 27, 2010
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28. Background at low E for HE  beam
Fanny T2KK WS
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29. Varying L
If a=sqrt{2}GFNe is small
Matter effect comes in with (aL/2) = ~0.27 and relatively small even at L ~ 1000 km
By varying L, the trigonometric nature of the oscillations manifests itself (spectrum analysis helps)
Good for CPV search (w. spectrum analysis)
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30. Two detector method is powerful
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31. Kamioka-Korea 2 detector setting
Why don’t you bring one of the 2 tanks to Korea?
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32. Original idea: sensitive because dynamism in 2nd oscillation maximum
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33. Spectral information solves intrinsic degeneracy
from 1000 page Ishitsuka file
T2K
T2KK
2 detector method powerful!
SK momentum resolution ~30 MeV at 1 GeV
Ishitsuka-Kajita-HM-Nunokawa 05
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34. Two-detector setting is powerful
T2KK
Korea only
With the same input parameter and Korean detector of 0.54 Mt the sign-m2 degeneracy is NOT completely resolved
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35. T2KK vs. T2K II Comparison T2KK T2K
hep-ph/
Total mass of the detectors = Mton fid. mass
4 years neutrino beam + 4 years anti-neutrino beam
T2KK
Mass hierarchy
CP violation (sind≠0)
T2K
3 s (thick)
2 s (thin)
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36. Relative cross section error does matter
T2KK
Barger et al. 07
T2K II
Identical 2 detector setting robust to larger systematic error
It gives conservative lower bounds on sensitivity estimate of mass hierarchy and CP
July 27, 2010
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37. T2KK can solve q23 degeneracy in situ
d=0 assumed
T2KK
T2K-II + phase II reactor
sin2 2q13
T2KK 2s
(rough)
> 3s
2~3s
sin2 2q13
hep-ph/
T2KK has better sensitivity
at sin2 2q13 < 0.06~0.07 .
July 27, 2010
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sin2 q23

38. Conclusion Global overview of P degeneracy is given
In some cases, P degeneracy can be understood by the symmetry argument
More generically it is an invariance under discrete mapping of mixing parameters whose explicit form should be obtained by solving equations
Sign-Dm2 and q23 octant degeneracies are robust against spectrum analysis
Some ideas are discussed on how to solve P degeneracy
July 27, 2010
INT Program LBL

39. Another example; sign-Dm2 degeneracy
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40. Another example; sign-Dm2 degeneracy2
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41. Another example; sign-Dm2 degeneracy3
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42. Neutrino factory
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43. Nufact
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