Parameter Degeneracy in Neutrino Oscillations (and how to solve it?)

Hisakazu Minakata

Tokyo Metropolitan University

INT Program 2010; LBL ..

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

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

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Is this true? To what extent?

An example; Intrinsic degeneracy

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P degeneracy is simplest to see by bi-P plot (HM-H.Nunokawa 01)

Is P degeneracy necessarily two-fold?

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Intrinsic degeneracy; S. Uchinami for PhD thesis

But, the answer is NO !People suspect

the answer is YES because

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ParameterDegeneracy;

definition

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)

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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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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A well-defined

framework for P

degeneracy

I use Cervera et al. formula for oscillation probabilities

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You can show 2x2x2=8

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

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Generally, P degeneracy has simpler structure

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P-degenerac

y as an invariance

of P

P-dege. from symmetry of the probability

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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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How to obtain

degeneracy solutions?

An example; intrinsic degeneracy

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An example; intrinsic degeneracy2

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4th-order eq. of s13!

P degeneracy as a re-parametrization invariance

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Degeneracy solutions form network!

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Degeneracy

solutions; how they look like?

13

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II

III

V

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I focus energy dependence; 13

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I focus energy dependence;

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July 27, 2010 INT Program LBL

How to solve P

degeneracy?

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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Project X: Off-axis NOVA --> VLBL multi-OM type approachProject X: Off-axis NOVA --> VLBL multi-OM type approach

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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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Background at low E for HE beam

Fanny Dufour@3rd T2KK WSFanny Dufour@3rd T2KK WS

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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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Two detector

method is powerful

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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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Original idea: sensitive because dynamism in 2nd oscillation maximum

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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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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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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

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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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

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

Another example; sign-m2 degeneracy

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Another example; sign-m2 degeneracy2

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Another example; sign-m2 degeneracy3

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Neutrino factory

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Nufact