A. Yu. Smirnov

1. Neutrino oscillograms of the Earth and CP-violation A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia E. Akhmedov, M. Maltoni, A.S., JHEP 0705:077 (2007) ; arXiv:0804.1466 (hep-ph) A.S. hep-ph/0610198. Melbourne neutrino theory workshop

2. The earth density profile A.M. Dziewonski D.L Anderson 1981 PREM model Fe inner core Si outer core (phase transitions in silicate minerals) transition zone lower mantle crust upper mantle Re = 6371 km liquid solid

3. ``Set-up'' qn • zenith • angle Q = p - qn Q - nadir angle Oscillations in multilayer medium core-crossing trajectory Q = 33o flavor to flavor transitions Applications: core accelerator atmospheric cosmic neutrinos mantle

4. Neutrino images of the Earth 1 - Pee ne nm ,nt P. Lipari , T. Ohlsson M. Chizhov, M. Maris, S .Petcov T. Kajita Michele Maltoni oscillograms Contours of constant oscillation probability in energy- nadir (or zenith) angle plane

6. Outline Now Tomorrow Forever Neutrino images of the earth 6X2 Explaining oscillograms Dependence of oscillograms on neutrino parameters CP-violation domains What these oscillograms for? Applications from SAND to HAND

7. Explaining Oscillograms

8. Physics: Oscillations in matter with nearly constant density Interference Parametric enhancement of oscillations Smallness of q13and Dm212/Dm322 (mantle) (mantle - core – - mantle) constant density + corrections in the first approximation: overlap of two 2n–patterns due to 1-2 and 1-3 mixings Parametric resonance  parametric peaks Low energies: adiabatic approximation interference (sub-leading effect) Peaks due to resonance enhancement of oscillations interference of modes CP-interference

9. 1 - Pee MSW-resonance peaks 1-3 frequency Parametric ridges 1-3 frequency Parametric peak 1-2 frequency MSW-resonance peaks 1-2 frequency 5p/2 3p/2 p/2

10. Resonance enhancement in mantle 1 mantle 1 2 mantle 2

11. Parametric enhancement in the Earth 1 mantle 2 3 1 4 core 2 mantle core mantle 3 mantle 4

12. Parametric enhancement of 1-2 mode 1 mantle core 3 4 2 2 mantle 4 3 1

13. Two conditions determine localizations of the peaks and ridges Collinearity Generalized condition phase condition Generalizations of the conditions ``AMPLITUDE = 1’’ in matter with constant density to the case of 1 or 3 layers with varying densities Generalized resonance condition phase: f = p/2 + pk depth: sin2qm = 1 Intersection of the corresponding lines determines positions of the peaks also determine position of other features: minima, saddle points, etc

14. Structure of oscillograms Maximal oscillation effects: Amplitude (resonance) condition Phase condition Generalizations: collinearity condition generalized phase condition E.Kh. Akhmedov, A.S., M Maris, S. Petcov

15. Analytic vs. numeric

16. Dependence on parameters Dependence on neutrino parameters and earth density profile (tomography)

17. Dependence on 1-3 mixing Flow of large probability toward larger Qn Lines of flow change weakly Factorization of q13 dependence Position of the mantle MSW peak measurement of q13

18. Other channels mass hierarchy For 2n system normal  inverted neutrino  antineutrino

20. CP-violation n  nc nc = i g0 g2 n + CP- transformations: applying to the chiral components Under CP-transformations: UPMNS UPMNS * d  - d V  - V usual medium is C-asymmetric which leads to CP asymmetry of interactions

21. CP-violation d = 60o Standard parameterization

22. d = 130o

23. d = 315o

24. CP-violation domains Solar magic lines Three grids of lines: Atmospheric magic lines Interference phase lines

25. Evolution Propagation basis ~ nf = U23Idn Id = diag (1, 1, eid ) ne ne ne ne Ae2 nm ~ ~ nm n2 n2 Ae3 ~ ~ nt n3 nt n3 propagation projection projection S A(nenm) = cosq23Ae2eid + sinq23Ae3

26. CP-interference Due to specific form of matter potential matrix (only Vee = 0) P(nenm) = |cos q23Ae2e id + sin q23Ae3|2 ``solar’’ amplitude ``atmospheric’’ amplitude dependence on d and q23is explicit For maximal 2-3 mixing f = arg (Ae2* Ae3) P(ne nm)d = |Ae2 Ae3| cos (f - d ) P(nm nm)d = - |Ae2 Ae3| cosf cos d P(nm nt)d = - |Ae2 Ae3| sinf sind S = 0

27. Factorization approximation Ae2 ~ AS (Dm212,q 12) corrections of the order Dm122 /Dm132, s132 Ae3 ~ AS (Dm312,q 13) are not valid in whole energy range due to the level crossing For constant density: FS ~ Dm322 p L l12m FS ~ H21 AS ~ i sin2q12msin p L l13m FA ~ H32 AA ~ i sin2q13msin

28. ``Magic lines" V. Barger, D. Marfatia, K Whisnant P. Huber, W. Winter, A.S. P(ne nm) = c232|AS|2 + s232|AA|2 + 2 s23 c23 |AS| |AA| cos(f + d) s23 = sin q23 f = arg (AS AA*) p L lijm Dependence on d disappears if AS = 0 AA = 0 = k p Atmospheric magic lines Solar ``magic’’ lines at high energies: l12m~ l0 AS = 0 for L = k l13 m(E), k = 1, 2, 3, … L = k l0 , k = 1, 2, 3 does not depend on energy - magic baseline (for three layers – more complicated condition)

29. Interference terms How to measure the interference term? d - true (experimental) value of phase df - fit value Interference term: D P = P(d) - P(df) = Pint(d) - Pint(df) For ne nm channel: DP = 2 s23 c23 |AS| |AA| [ cos(f + d) - cos (f + df)] (along the magic lines) AS = 0 AA = 0 D P = 0 (f+d ) = - (f + df) + 2p k int. phase condition f(E, L) = - ( d + df)/2 + p k depends on d

30. Interference terms For nmnm channel d - dependent part: P(nm nm)d ~ - 2 s23 c23 |AS| |AA| cosf cosd The survival probabilities is CP-even functions of d No CP-violation. DP ~ 2 s23 c23 |AS| |AA| cosf [cosd - cos df] AS = 0 (along the magic lines) D P = 0 AA = 0 interference phase does not depends on d f= p/2 + p k P(nm nt)d ~ - 2 s23 c23 |AS| |AA| sin f sind

31. CP violation domains Interconnection of lines due to level crossing factorization is not valid solar magic lines atmospheric magic lines relative phase lines Regions of different sign of DP

32. Int. phase line moves with d-change Grid (domains) does not change with d DP

33. DP

34. DP

35. Sensitivity to CP-phase nm ne • Contour plots for • the probability • difference • P = Pmax – Pmin for d varying between 0 – 360o Emin ~ 0.57 ER when q13 0 Emin 0.5 ER

36. Applications

37. What for? Tomography General Measurements knowledge of neutrino Searches for parameters Pictures for Textbooks: neutrino images of the Earth new physics • mass hierarchy • 1-3 mixing • CP violation

38. Where we are? Large atmospheric neutrino detectors 100 LAND NuFac 2800 0.005 CNGS 0.03 0.10 10 LENF E, GeV MINOS T2KK T2K 1 Degeneracy of parameters 0.1

39. Two approaches Atmospheric Accelerators Neutrinos Intense and controlled beams Small fluxes, with uncertainties Small effect Large effects Cover rich-structure regions Cover poor-structure regions No degeneracy? Degeneracy of parameters Combination of results from different experiments is in general required Systematic errors

40. Atmospheric neutrinos Cost-free source • various flavors: ne and nm • neutrinos and antineutrinos Several neutrino types E ~ 0.1 – 104 GeV Cover whole parameter space (E, Q) whole range of nadir angles L ~ 10 – 104 km • small statistics • uncertainties in the predicted fluxes • presence of several fluxes • averaging and smoothing effects Problem:

41. Detectors 50 kton iron calorimenter INO – Indian Neutrino observatory HyperKamiokande 0.5 Megaton water Cherenkov detectors UNO Underwater detectors ANTARES, NEMO E > 30 – 50 GeV Icecube (1000 Mton) Reducing down 20 GeV? TITAND (Totally Immersible Tank Assaying Nuclear Decay) 2 Mt and more Y. Suzuki..

42. TITAND - Proton decay searches - Supernova neutrinos - Solar neutrinos Totally Immersible Tank Assaying Nucleon Decay Y. Suzuki Modular structure Under sea deeper than 100 m Cost of 1 module 420 M $ TITAND-II: 2 modules: 4.4 Mt (200 SK)

43. Number of events - angular resolution: ~ 3o e-like events - neutrino direction: ~ 10o - energy resolution for E > 4 GeV better than 2% DE/E = [0.6 + 2.6 E/GeV ] % zenith angle MC: 800 SK-years Fully contained events cos Q -1 / -0.8 -0.8 / -0.6 -0.6 / -0.4 2.5 – 5 GeV SR 2760 (10) 3320 (20) 3680 (15) MR 2680 (9) 2980 (12) 3780 (13) 5 – 10 GeV SR 1050 (9) 1080 (5) 1500 (10) MR 1150 (4) 1280 (3) 1690 (6) SR – single ring MR – multi-ring (…) – number of events detected by 4SK years

44. From SAND to HAND Huge Atmospheric Neutrino Detector Measuring oscillograms with atmospheric neutrinos with sensitivity to the resonance region 0.5 GeV E > 2 - 3 GeV Better angular and energy resolution Spacing of PMT ? V = 5 - 10 MGt Should we reconsider a possibility to use atmospheric neutrinos? develop new techniques to detect atmospheric neutrinos with low threshold in huge volumes?

45. Summary Oscillograms encode in a comprehensive way information about the Earth matter profile and neutrino oscillation parameters. Locations of salient structures of oscillograms are determined by the collinearity and generalized phase conditions Oscillograms have specific dependencies on 1-3 mixing angle, mass hierarchy, CP-violating phases and earth density profile that allows us to disentangle their effects. CP-effect has a domain structure. The borders of domains are determined by three grids of lines: the solar magic lines, the atmospheric magic lines and the lines of interference phase condition Determination of neutrino parameters,tomography by measuring oscillograms with Huge (multi Megaton) atmospheric neutrino detectors?

46. CP and magic trajectories P(nenm) = |cos q23ASe id + sin q23AA|2 ``solar’’ amplitude mainly, Dm122, q12 ``atmospheric’’ amplitude mainly, Dm132, q13 p L lm |AS | ~ sin2q12msin For high energies lm l0 for trajectory with L = l0 AS = 0 P = |sin q23AA|2 no dependence on d For three layers – more complicated condition Contours of suppressed CP violation effects Magic trajectories associated to AA = 0

47. Amplitude condition Phase condition MSW resonance condition 1 layer: S(1)11 = S(1)22 f = p/2 + pk unitarity: S(1)11 = [S(1)22 ]* Re S(1)11 = 0 Im S(1)11 = 0 cos 2qm = 0 sin f = 0 another representation: Parametric resonance condition Im (S11 S12*) = 0 2 layers: X3 = 0 S(2)11 = S(2)22 For symmetric profile (T –invariance): Im S(2)11 = 0 S12- imaginary Generalized resonance condition valid for both cases: Re (S11) = 0 Im S11 = 0

48. Another way to generalize parametric resonance condition Collinearity condition Evolution matrix for one layer (2n-mixing): a b -b* a* from unitarity condition S = a, b – amplitudes of probabilities For symmetric profile (T-invariance) b = - b*  Re b = 0 For two layers: S(2) = S1 S2 transition amplitude: A = S(2)12 = a2 b 1 + b2 a 1* The amplitude is potentially maximal if both terms have the same phase (collinear in the complex space): arg (a1a2 b1) = arg (b2) Due to symmetry of the core Re b2 = 0  Re (a1a2 b1) = 0 Due to symmetry of whole profile it gives extrema condition for 3 layers

49. Structures of oscillograms Different structures follow from different realizations of the collinearity and phase condition in the non-constant case. Re (a1a2 b1) = 0 Re (S11) = 0 s1 = 0, c2 = 0 X3 = 0 c1 = 0, c2 = 0 P = 1 Local maxima Core-enhancement effect Absolute maximum (mantle, ridge A) P = sin (4qm – 2qc) Saddle points at low energies Maxima at high energies above resonances

50. Oscillations in matter Oscillation Probability constant density pL lm P(ne -> na) =sin22qmsin2 half-phase f Amplitude of oscillations oscillatory factor qm(E, n ) - mixing angle in matter qm q lm(E, n ) – oscillation length in matter In vacuum: lm ln lm = 2 p/(H2 – H1) Conditions for maximal transition probability: P = 1 MSW resonance condition sin22qm= 1 1. Amplitude condition: 2. Phase condition: f = p/2 + pk