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Parameter Degeneracy in Neutrino Oscillations (and how to solve it?)
Hisakazu Minakata
Tokyo Metropolitan University
INT Program 2010; LBL ..
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Purpose of this discussion
• To complete Standard Model (SM + mass + lepton mixing) measurement of CP phase (KM type) and13is necessary
• It seems that is not so easy to determine them, in particular
• If any theoretical issues involved we shall try to remove them
• One of them is P degeneracy
July 27, 2010 INT Program LBL
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P degeneracy
• P degeneracy is the fact that measurement of oscillation probability P and -bar oscillation probability bar-P at an energy (which would determine 13 and actually do NOT lead to a unique solution of 13 and
• Experts may say that they know everything
July 27, 2010 INT Program LBL
Is this true? To what extent?
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An example; Intrinsic degeneracy
July 27, 2010 INT Program LBL
P degeneracy is simplest to see by bi-P plot (HM-H.Nunokawa 01)
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Is P degeneracy necessarily two-fold?
July 27, 2010 INT Program LBL
Intrinsic degeneracy; S. Uchinami for PhD thesis
But, the answer is NO !People suspect
the answer is YES because
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July 27, 2010 INT Program LBL
ParameterDegeneracy;
definition
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P degeneracy
• Let us assume that all the mixing parameters besides 13 and are known
• measurement of oscillation probability Pe and bar- oscillation probability bar-Pe at an energy E (which would determine 13 and do NOT lead to unique solution of 13 and
• Easy to solve mathematically: measurement at E=E1 and E2 (or adding more channel) solves the degeneracyJuly 27, 2010 INT Program LBL
Intrinsic degeneracy (Burguet-C. et al. 01)
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P degeneracy (continued)
• the mixing parameters besides 13 and are not known so precisely
• Mass hierarchy is not known, and may not be known either at the time of measurement of CP phase
• More solutions of 13 and
•
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Sign m231
degeneracy (HM-Nunokawa 01)
23 octant degeneracy (Fogli-Lisi 96)
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July 27, 2010 INT Program LBL
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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July 27, 2010 INT Program LBL
A well-defined
framework for P
degeneracy
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I use Cervera et al. formula for oscillation probabilities
July 27, 2010 INT Program LBL
You can show 2x2x2=8
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P degeneracy; Generalized version
• Similar degeneracy occurs in, in addition to (P, PCP),
• T-conjugate (P=Pe, PT=Pe)
• CPT-conjugate (P, PCPT)
• Golden-silver (PT, PS) channels
July 27, 2010 INT Program LBL
Generally, P degeneracy has simpler structure
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July 27, 2010 INT Program LBL
P-degenerac
y as an invariance
of P
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P-dege. from symmetry of the probability
July 27, 2010 INT Program LBL
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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July 27, 2010 INT Program LBL
How to obtain
degeneracy solutions?
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An example; intrinsic degeneracy
July 27, 2010 INT Program LBL
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An example; intrinsic degeneracy2
July 27, 2010 INT Program LBL
4th-order eq. of s13!
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P degeneracy as a re-parametrization invariance
July 27, 2010 INT Program LBL
Degeneracy solutions form network!
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July 27, 2010 INT Program LBL
Degeneracy
solutions; how they look like?
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13
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II
III
V
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July 27, 2010 INT Program LBL
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I focus energy dependence; 13
July 27, 2010 INT Program LBL
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I focus energy dependence;
July 27, 2010 INT Program LBL
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July 27, 2010 INT Program LBL
How to solve P
degeneracy?
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July 27, 2010 INT Program LBL
Varying E at long enough baseline
• 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}GFNea=sqrt{2}GFNe
atmosphericatmospheric
solarsolar
the best way for everything if perfect event reconstruction with 2-3 oscillation periods spanned
the best way for everything if perfect event reconstruction with 2-3 oscillation periods spanned
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July 27, 2010 INT Program LBL
Project X: Off-axis NOVA --> VLBL multi-OM type approachProject X: Off-axis NOVA --> VLBL multi-OM type approach
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July 27, 2010 INT Program LBL
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?
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July 27, 2010 INT Program LBL
Background at low E for HE beam
Fanny Dufour@3rd T2KK WSFanny Dufour@3rd T2KK WS
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July 27, 2010 INT Program LBL
Varying L
• 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)
If a=sqrt{2}GFNe is small If a=sqrt{2}GFNe is small
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July 27, 2010 INT Program LBL
Two detector
method is powerful
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July 27, 2010 INT Program LBL
Kamioka-Korea 2 detector setting
Why don’t you bring one of the 2 tanks to Korea? (@EPP2010)
Why don’t you bring one of the 2 tanks to Korea? (@EPP2010)
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July 27, 2010 INT Program LBL
Original idea: sensitive because dynamism in 2nd oscillation maximum
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July 27, 2010 INT Program LBL
Spectral information solves intrinsic degeneracy
from 1000 page Ishitsuka file
from 1000 page Ishitsuka file
SK momentum resolution ~30 MeV at 1 GeVSK momentum resolution ~30 MeV at 1 GeV
T2KT2K T2KKT2KK
2 detector method powerful!2 detector method powerful!
Ishitsuka-Kajita-HM-Nunokawa 05
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July 27, 2010 INT Program LBL
Two-detector setting is powerful
• With the same input parameter and Korean detector of 0.54 Mt the sign-m2 degeneracy is NOT completely resolved
T2KKT2KK Korea onlyKorea only
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July 27, 2010 INT Program LBL
T2KK vs. T2K II Comparison T2KK vs. T2K II Comparison Total mass of the detectors = 0.54 Mton fid. mass4 years neutrino beam + 4 years anti-neutrino beam
Total mass of the detectors = 0.54 Mton fid. mass4 years neutrino beam + 4 years anti-neutrino beam
3 (thick)3 (thick) 2 (thin)2 (thin)
Mass hierarchyMass hierarchy CP violation (sin≠0)CP violation (sin≠0)
hep-ph/0504026
T2K
T2KK
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July 27, 2010 INT Program LBL
Relative cross section error does matter
• Identical 2 detector setting robust to larger systematic error • It gives conservative lower bounds on sensitivity estimate
of mass hierarchy and CP
Barger et al. 07
Barger et al. 07
T2K IIT2K II
T2KKT2KK
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July 27, 2010 INT Program LBL
T2KK can solve 23 degeneracy in situ
T2K-II + phase II reactorT2KK=0 assumed
sin
2 2
13
sin2 23
sin
2 2
13
> 32~3
T2KK 2(rough)
T2KK has better sensitivityat sin2 213 < 0.06~0.07 .
hep-ph/0601258
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July 27, 2010 INT Program LBL
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-m2 and 23 octant degeneracies are robust against spectrum analysis
• Some ideas are discussed on how to solve P degeneracy
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Another example; sign-m2 degeneracy
July 27, 2010 INT Program LBL
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Another example; sign-m2 degeneracy2
July 27, 2010 INT Program LBL
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Another example; sign-m2 degeneracy3
July 27, 2010 INT Program LBL
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July 27, 2010 INT Program LBL
Neutrino factory
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July 27, 2010 INT Program LBL
Nufact