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

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

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 INT Program LBL

An example; Intrinsic degeneracy P degeneracy is simplest to see by bi-P plot (HM-H.Nunokawa 01) July 27, 2010 INT Program LBL

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 INT Program LBL

Parameter Degeneracy; definition July 27, 2010 INT Program LBL

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 INT Program LBL

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 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 July 27, 2010 INT Program LBL

A well-defined framework for P degeneracy July 27, 2010 INT Program LBL

I use Cervera et al. formula for n oscillation probabilities You can show 2x2x2=8 July 27, 2010 INT Program LBL

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 INT Program LBL

P-degeneracy as an invariance of P July 27, 2010 INT Program LBL

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 July 27, 2010 INT Program LBL

How to obtain degeneracy solutions? July 27, 2010 INT Program LBL

An example; intrinsic degeneracy July 27, 2010 INT Program LBL

An example; intrinsic degeneracy2 4th-order eq. of s13! July 27, 2010 INT Program LBL

P degeneracy as a re-parametrization invariance Degeneracy solutions form network! July 27, 2010 INT Program LBL

Degeneracy solutions; how they look like? July 27, 2010 INT Program LBL

q13 II III V July 27, 2010 INT Program LBL

d July 27, 2010 INT Program LBL

I focus energy dependence; q13 July 27, 2010 INT Program LBL

I focus energy dependence; d July 27, 2010 INT Program LBL

How to solve P degeneracy? July 27, 2010 INT Program LBL

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 July 27, 2010 INT Program LBL

Project X: Off-axis NOVA --> VLBL multi-OM type approach 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? July 27, 2010 INT Program LBL

Background at low E for HE  beam Fanny Dufour@3rd T2KK WS July 27, 2010 INT Program LBL

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) July 27, 2010 INT Program LBL

Two detector method is powerful July 27, 2010 INT Program LBL

Kamioka-Korea 2 detector setting Why don’t you bring one of the 2 tanks to Korea? (@EPP2010) July 27, 2010 INT Program LBL

Original idea: sensitive because dynamism in 2nd oscillation maximum July 27, 2010 INT Program LBL

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 July 27, 2010 INT Program LBL

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 July 27, 2010 INT Program LBL

T2KK vs. T2K II Comparison T2KK T2K hep-ph/0504026 Total mass of the detectors = 0.54 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) July 27, 2010 INT Program LBL

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 INT Program LBL

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/0601258 T2KK has better sensitivity at sin2 2q13 < 0.06~0.07 . July 27, 2010 INT Program LBL sin2 q23

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

Another example; sign-Dm2 degeneracy July 27, 2010 INT Program LBL

Another example; sign-Dm2 degeneracy2 July 27, 2010 INT Program LBL

Another example; sign-Dm2 degeneracy3 July 27, 2010 INT Program LBL

Neutrino factory July 27, 2010 INT Program LBL

Nufact July 27, 2010 INT Program LBL