A. Yu. Smirnov

1. Neutrino oscillograms of the Earth 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. Fermilab, April 16, 2008

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. P. Lipari , T. Ohlsson M. Chizhov, M. Maris, S .Petcov T. Kajita Neutrino images ne nt’ 1 - Pee Michele Maltoni oscillograms Contours of constant oscillation probability in energy- nadir (or zenith) angle plane

4. ``Set-up'' qn • zenith • angle Q = p - qn Q - nadir angle Mass & mixing core-crossing trajectory Oscillations Q = 33o Oscillograms Oscillations in multilayer medium core flavor to flavor transitions mantle accelerator atmospheric cosmic neutrinos

5. Comments The Earth is unique Oscillograms are reality and this reality will be with us forever We know that neutrino masses and mixing are non-zero First of all we need to understand their properties and physics behind and then… Can we observe (reconstruct) these neutrino images? With which precision? How can we use them?

6. Outline 1. Explaining oscillograms Two effects 2. How oscillograms depend on unknown yet neutrino parameters and Earth density profile Interference of modes and CP-violation 3. How can we use them? from SAND to HAND

7. Masses and mixing ne nm nt |Ue3|2 ? n3 n2 Dm2sun n1 mass mass Dm2atm Dm2atm n2 Dm2sun n1 n3 Normal mass hierarchy Inverted mass hierarchy nf = UPMNSnmass UPMNS = U23 Id U13 I-d U12 sin2q13 = |Ue3|2 Id = diag (1, 1, eid) Type of the mass hierarchy Ue3 tan2q12 =|Ue2|2 / |Ue1|2 tan2q23 = |Um3|2 / |Ut3|2 CP-violating phase

8. Explaining Oscillograms

9. Two effects Parametric enhancement of oscillations Oscillations in matter with nearly constant density 1 layer case: mantle mantle – core - mantle In two neutrino context Interference of different modes of oscillations

10. Evolution equation ye • = ym yt d Y d t i = H Y Mixing matrix in vacuum M M+ 2E H = + V(t) M is the mass matrix M M+ = U Mdiag2 U+ Mdiag2 = diag (m12, m22, m32) V = diag (Ve, 0, 0) effective potential Mixing matrix Eigenstates and eigenvalues of Hamiltonian Diagonalization of H Energy levels

11. Neutrino polarization vectors ne nt, Re ne+ nt, P = Im ne+ nt, ne+ ne - 1/2 Polarization vector: y = P = y+s/2 y Evolution equation: d Y d t d Y d t i = H Y i = (B s ) Y 2p lm B = (sin 2qm, 0, cos2qm) Differentiating P and using equation of motion dP dt = ( B x P) Coincides with equation for the electron spin precession in the magnetic field

12. Graphical representation • = P = (Re ne+ nt, Im ne+ nt, ne+ ne - 1/2) 2p lm B = (sin 2qm, 0, cos2qm) Evolution equation dn dt = ( B x n) f= 2pt/ lm - phase of oscillations probability to find ne P = ne+ne = nZ+ 1/2 = cos2qZ/2

13. Oscillations

19. Resonance enhancement Source Detector ne ne Constant density n F(E) F0(E) Layer of length L sin2 2q12 = 0.824 k = p L/ l0 F (E) F0(E) k = 10 k = 1 thick layer thin layer E/ER E/ER

20. Small mixing angle sin2 2q12 = 0.08 F (E) F0(E) k = 10 k = 1 thick layer thin layer E/ER E/ER A Yu Smirnov

21. Mixing in matter Diagonalization of the Hamiltonian: sin22q sin22qm = ( cos2q - 2EV/Dm2)2 + sin 22q V = 2 GF ne Mixing is maximal if Dm2 2E Resonance condition V= cos 2q He = Hm sin22qm = 1 Difference of the eigenvalues Dm2 2E ( cos2q - 2EV/Dm2)2 + sin22q H2 - H1 =

22. Parametric enhancement of oscillations Enhancement associated to certain conditions for the phase of oscillations F1 = F2 = p Another way of getting strong transition No large vacuum mixing and no matter enhancement of mixing or resonance conversion 2q2m 2q1m V. Ermilova V. Tsarev, V. Chechin E. Akhmedov P. Krastev, A.S., Q. Y. Liu, S.T. Petcov, M. Chizhov V F2 F1 q1m q2m 1 2 3 4 5 6 7 ``Castle wall profile’’

23. Parametric resonance si = sinfi, ci = cosfi, (i = 1,2) half-phases s1c2cos2q1m + s2c1cos2q2m= 0 Akhmedov, A.S. distance distance (f1 = f2 = p/2) General case: certain correlation between the phases and mixing angles c1 = c2 = 0

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

25. 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

26. Analytic vs. numeric

27. Evolution collinearity condition (parametric resonance condition)

29. Graphical representation b). a). a). Resonance in the mantle b). Resonance in the core c). Parametric ridge A d). c). d). Parametric ridge B e). Parametric ridge C f). Saddle point f). e).

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

31. Properties of oscillograms Dependence on neutrino parameters and earth density profile (tomography)

32. Sensitivity to density profile twist

33. Sensitivity to density profile Shift of border

34. Dependence on gradients

35. 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

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

38. 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 Under T-transformations: d  -d V  V ninitial nfinal

39. CP-violation d = 60o Standard parameterization

40. d = 130o

41. d = 315o

42. Interference of modes Due to specific form of matter potential matrix (only Vee = 0) P(nenm) = |cos q23ASe id + sin q23AA|2 ``solar’’ amplitude ``atmospheric’’ amplitude dependence on d and q23is explicit AS depends mainly on Dm122, q12 ``Factorization’’ approximation: AA depends mainly on Dm132, q13 corrections of the order Dm122 /Dm132, s132 p L l12m AS ~ i sin2q12msin For constant density: up to phase factors p L l13m AA ~ i sin2q13msin

43. ``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) interference term f = arg (AS AA*) d - ``weak’’ phase ``strong’’ phase 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 s23 = sin q23 (for three layers – more complicated condition)

44. Interference terms If all parameters but d are known 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

45. 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| cosf sind

46. 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

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

48. DP

49. DP

50. 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