CP-phase dependence of neutrino oscillation probability in matter 梅 (ume) 田 (da) 義 (yoshi) 章 (aki) with Lin Guey-Lin ( 林 貴林 ) National Chiao-Tung University in Taiwan ( 臺灣國立交通大學 )

We calculate neutrino oscillation probability in the framework of three neutrinos. The latest data is used. Matter density profile of the earth is approximated by step functions. The oscillation probability of P  e and P  are related to the angle sin 2 2  13 and sin 2 2  23. CP-phase dependence of P  e are shown as a function of baseline length L. Contents Status of neutrino oscillation Formalism of three flavor neutrino oscillation sin 2 2  13 and sin 2 2  23 dependence of transition probability (  CP =0)  cp dependence of transition probability are shown

Solar neutrino In the core of the sun, the protons will be iron after many fusion steps. Electron neutrinos made in these process, are much less than SM prediction. core  =100g/cm 3 surface  =0.01g/cm 3 many oscillations Earth Sun SK, e from 8 B → 8 B* + e + + e is 40.6% of the SM prediction. (E>5MeV, events) SNO (E>6MeV, neutrino from 8B) CC: e + D → p + p + e  NC: x + D → x + n + p ES: x + e  → x + e  Pure D 2 O phase: Nov.99 – May.01 3 He phase: n + 3 He → p + 3 H +  (0.76MeV) Salt phase: Jul.01- Sep.03, n + 35 Cl → 36 Cl+  (8.6MeV) Gallium experiments like SAGE and GALLEX /GNO (E>0.23MeV, neutrino from pp, 7 B and 8 B) neutrinos are detected by the process e + 71 Ga →e  + 71 Ge

pp-chain spectrum of solar neutrino

SNO Cherenkov light

atmospheric neutrino Data MC FC single-ring  -like (SK) (In MC, neutrino oscillation is not assumed) The energy of neutrino <25GeV L-dependence can be studied by measuring zenith angle In the vacuum, the neutrino oscillation is the function of L/E. Feb.2004, SK show the L/E plot.

K2K experiment (KEK to Kamioka, 250km) 12GeV pp collider →aluminum target → pion →  neutrino measure the number of , energy, direction at KEK → measure at Kamioka Fix the baseline length L, measure the neutrino energy dependence. Number of events are not so much. mixing angle → not so sensitive  m 31 2 → sensitive KamLAND (nuclear reactor) measure e + and n (anti- + p →e + + n ) agree with solar neutrino parameter,  m 21 2 =8.3×10 -5 eV 2, sin 2 2  12 =0.83 L/E plot are shown last summer. CHOOZ (nuclear reactor) Electron neutrino deficit was not measured. sin 2 2  13 is constrained.

There are 6 parameters  m 21,  m 31, sin 2 2  12, sin 2 2  23, sin 2 2  13,  cp 3-  allowed region of oscillation parameters 7.4×10  3 ≤  m 21 2 ≤ 9.2×10  3 (best fit 8.2×10  5, from solar + reactor) 1.9×10  3 ≤  m 31 2 ≤ 3.0×10  3 (best fit 2.4×10  3, from SK) 27.9° ≤  12 ≤ 37.3° (best fit 32.0°, from solar + reactor) sin 2 2  13 ≤ 0.18 (from atmospheric, K2K, CHOOZ) sin 2 2  23 ≥ 0.9 (from SK) J.N. Bahcall et. al, hep-ph/ SK collaboration hep-ex/ Best fit value of  m 21 2,  m 31 2,  12 sin 2 2  13 =0.1, sin 2 2  23 =1,  cp =0 will be used.

Schrödinger Equation Neutrino transition probability P  =S  2

The relation of ne, nm, nt to the mass eigenstate Feynman diagram of coherent forward elastic scattering

The step functions with  c =11.85 g/cm 3  m =4.67 g/cm 3 are good approximation. Freund and Ohlsson hep-ph/ If we approximate the density profile of the earth by three step functions, mantle → core → mantle, probability P   S  | 2 is

with density profile … calculate non-perturbatively, density is approximated by many step functions constant density … calculate non-perturbatively, use constant density perturbative result … O ((  m 21 2 /  m 31 2 ) 2 ), constant density for the lowest order, If  m 31 2 < 0, P  e ~0. PePe

 m 2 31 dependence of P  e for L=9300km (Fermi – Kamioka) sin 2 2  13 =0.1, sin 2 2  23 =1.0,  cp =0,  m 2 21 =8.2×10  5,  12 =32.0 o

sin 2 2  13 =0.1, sin 2 2  23 =1 We assume that the energy bin for the measurement is 5.5GeV-6.5GeV. We make a contour plot of P  e and P  in  13-  23 plane.

5.5GeV≤ E ≤ 6.5GeV sin 2 2  13 ≤ 0.10 sin 2 2  23 ≥ 0.92 The contours of P  e and P  are orthogonal. P  is not symmetric between  >45 o and  <45 o.

CP-phase dependence of P  e is shown. For L=9300km, CP-phase dependence → small, sin 2 2  13 and sin 2 2  23 dependence → large sin 2 2  13 and sin 2 2  23 can be fixed by L=9300km. For L = 1000km and 5000km CP-phase dependence → large sin 2 2  13 =0.1 sin 2 2  23 =1  12 =32 o  m 2 13 =2.4×10  3  m 2 13 =8.2×10  5

Check of the calculation.  =  m 21 2 /  m Akhmedov at el., JHEP 04 (2004) 078

L=8500km, Max  Min has minimum. (Max  Min = 0.01) L=5000km, Max  Min has maximum (Max  Min = 0.065)

CP-phase dependence of P  is small. L=1000km and L=9300km, CP-phase dependence is negligible. L=5000km, CP-phase dependence can be seen at around 5GeV.

Summary We relate the measurement of P  e and P  to sin 2 2  13 and sin 2 2  23. The contour graph of P  e and P  are orthogonal in  13 -  23 plane. CP-phase dependence are shown for L=1000, 5000 and 9300km. The CP-phase effects are small for L=9300km. Thus sin 2 2  13 and sin 2 2  23 dependence can be determined without the effects of CP-phase. (P  e ) max – (P  e ) min has maximum at around L=5000km. It is about P  is not sensitive to CP-phase.