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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 )

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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

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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, 20000 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

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pp-chain spectrum of solar neutrino

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SNO Cherenkov light

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atmospheric neutrino Data MC FC single-ring -like (SK) 1619 2105.8 (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.

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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.

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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/0406294 SK collaboration hep-ex/0404034 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.

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Schrödinger Equation Neutrino transition probability P =S 2

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

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The step functions with c =11.85 g/cm 3 m =4.67 g/cm 3 are good approximation. Freund and Ohlsson hep-ph/9909501 If we approximate the density profile of the earth by three step functions, mantle → core → mantle, probability P S | 2 is

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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. PePe

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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

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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.

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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.

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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

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Check of the calculation. = m 21 2 / m 31 2. Akhmedov at el., JHEP 04 (2004) 078

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

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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.

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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 0.065. P is not sensitive to CP-phase.

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