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

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


Is this true? To what extent?

An example; Intrinsic degeneracy


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

Is P degeneracy necessarily two-fold?


Intrinsic degeneracy; S. Uchinami for PhD thesis

But, the answer is NO !People suspect

the answer is YES because


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

•


Sign m231

degeneracy (HM-Nunokawa 01)

23 octant degeneracy (Fogli-Lisi 96)


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


A well-defined

framework for P

degeneracy

I use Cervera et al. formula for oscillation probabilities


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


Generally, P degeneracy has simpler structure


P-degenerac

y as an invariance

of P

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


How to obtain

degeneracy solutions?

An example; intrinsic degeneracy


An example; intrinsic degeneracy2


4th-order eq. of s13!

P degeneracy as a re-parametrization invariance


Degeneracy solutions form network!


Degeneracy

solutions; how they look like?

13


II

III

V

I focus energy dependence; 13


I focus energy dependence;



How to solve P

degeneracy?


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


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


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?


Background at low E for HE beam

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


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


Two detector

method is powerful


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)


Original idea: sensitive because dynamism in 2nd oscillation maximum


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


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


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


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


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


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


Another example; sign-m2 degeneracy2


Another example; sign-m2 degeneracy3



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

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