Part III. 3-nu oscillations Passing through the Earth

Survival Probability Matter filter n source detector 1 2 1 - sin22q Non-uniform medium Matter filter n source detector Vacuum oscillations d << ln = 4 pE/Dm 2 P(averged over oscillations) 1 2 1 - sin22q Dm2/ 2E vs V Adiabatic edge Non-adiabatic conversion sin2q E Non-oscillatory adiabatic conversion Resonance at the highest density n(0) = ne = n2m n2 P = |< ne| n2 >|2 = sin2q adiabaticity

Master equation ye = ym yt d Y d t i = H Y M M+ 2E H = + V(t) p E Schroedinger M M+ 2E does not valid for equal energies of mass states ? H = + V(t) p Question to derivation 1 0 0 1 If E1 = E2 = E H = E Transition to flavor basis will also give diagonal Hamiltonian Wave packet picture resolves problem No mixing  No time evolution! Still oscillatory pattern will be observed d Y d t d Y d x x ~ t It will be oscillatory patter in space:

Evolution of 3nu system H = UPMNSMdiag2 UPMNS+ + V In the flavor basis: 1 2E H = UPMNSMdiag2 UPMNS+ + V Mdiag2 = diag (0, Dm212, Dm322) UPMNS = U23 Id U13 U12 Id = diag (1, 1, eid) ~ In the propagation basis: nf = U23Id n ~ 1 2E H = U13U12 Mdiag2 U12+U13+ + V  does not change no dependence on d and q23  no CP- violation - depends on q12 and q13

Evolution ne ne ne ne nm ~ ~ nm n2 n2 ~ ~ nt nt n3 n3 S Propagation basis ~ nf = U23Id n Id = diag (1, 1, eid ) ne ne ne ne Ae2 nm ~ ~ nm n2 n2 Ae3 ~ ~ nt nt n3 n3 projection propagation projection S A(ne  nm) = cosq23 Ae2eid + sinq23Ae3

Level crossing scheme eigenvalues Two resonances Normal hierarchy Inverted hierarchy ne nm -nt resonance Both resonances are in the neutrino channel 1-3 resonance is in the antineutrino channel

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

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

Oscillograms ne nm ,nt Contours of constant oscillation probability in energy- nadir (or zenith) angle plane P. Lipari, 1998 (unpublished), M. Chizhov, M. Maris, S .Petcov hep-ph/981050 T. Ohlsson T. Kajita Michele Maltoni

Resonance enhancement in mantle 1 mantle 1 2 mantle 2

Oscillations in multilayer media Earth matter profile Strong adiabaticity violation at the borders

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

Parametric oscillations ``Castle wall profile’’ E. Kh. Akhmedov hep-ph/9805272 qim - mixing angles f1 f2 V fi oscillation half-phases q1m q2m S. Petcov M. Chizhov hep-ph/9903399 PR D63 073003 d Evolution matrix over one period (two layers) X = (X1 , X2, X3) s -Pauli matrices S = Y – i s X X, Y = X, Y (qi, fi) Probability after n periods: multiplying the evolution matrices for each layer P = (1 – X3 / |X| ) sin 2 Fn Maximal depth of oscillations X3 = 0 parametric resonance condition

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

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

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

1 - Pee MSW-resonance peak, 1-3 mixing Parametric ridges 1-3 mixing Parametric peak 1-2 mixing MSW-resonance peaks 1-2 mixing 5p/2 3p/2 p/2

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

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

CP-violation d = 60o Standard parameterization

d = 130o

d = 315o

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

Evolution ne ne ne ne nm ~ ~ nm n2 n2 ~ ~ nt nt n3 n3 S Propagation basis ~ nf = U23Id n Id = diag (1, 1, eid ) ne ne ne ne Ae2 nm ~ ~ nm n2 n2 Ae3 ~ ~ nt nt n3 n3 projection propagation projection S A(ne  nm) = cosq23 Ae2eid + sinq23Ae3

CP-interference P(n e  nm) = |cos q23 Ae2e id + sin q23Ae3|2 Due to specific form of matter potential matrix (only Vee = 0) P(n e  nm) = |cos q23 Ae2e id + sin q23Ae3|2 ``solar’’ amplitude ``atmospheric’’ amplitude dependence on d and q23 is explicit For maximal 2-3 mixing P(ne  nm)d = |Ae2 Ae3| cos (f - d ) f = arg (Ae2* Ae3) P(nm  nm)d = - |Ae2 Ae3| cosf cos d P(nm  nt)d = - |Ae2 Ae3| sinf sind S = 0

``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 Solar ``magic’’ lines Atmospheric 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)

Interference terms How to measure the interference term? d - true 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)] AS = 0 (along the magic lines) 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

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

D P = P(d) - P(df) = const Int. phase line moves with d-change Grid (domains) does not change with d DP

DP

DP

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

Evento grams Lines of equal c2 sin2 2q13 = 0.125 No averaging

smoothing

Low energies

Low energies Dm312 /2E >> V nf = U23 Id U13 n’ 1). Matter effect on 1-3 mixing can be neglected additional 1-3 rotation: nf = U23 Id U13 n’ In the basis n ’ 1 2E H’ ~ U12 Mdiag2 U12+ + Vc132 Mdiag2 = diag (0, Dm212, Dm322) - n ’3 decouples from the rest of the system; - the problem is reduced to 2n -problem with parameters (Dm212, q12, V c13) Aee’ = Aee’(Dm212, q12, V c132) Returning to flavor basis: Aee = Aee’ c132 + A33’ s132 2). Interference of two amplitudes is averaged out (oscillations due to 1-3 mixing are averaged: ) Pee = P2 c134 + s134 P2 = |Aee’ |2

13-mixing effect 1 - 0.5sin2 2q12 sin2q12 ~ (1 – 2 sin 2 q13) sin2 q12 Survival probability LMA MSW npp nBe 1 - 0.5sin2 2q12 Earth matter effect sin2q12 III II I ln / l0 ~ E High energies Low energies ~ (1 – 2 sin 2 q13) sin2 q12 ~ (1 – 2 sin 2 q13)(1 - 0.5sin2 2q12) anticorrelate correlate

Theta 1-3 Solar neutrinos: degeneracy of 1-2 and 1-3 mixing S. Goswami, A.S. sin2q 13 = 0.017+/- 0.26

12- and 13- mixings x x x T. Schwetz et al., 0808..2016 G.L. Fogli, et al 0805.2517, v3 x x x

Hint of non-zero 1-3 mixing? Fogli et al ., 0806.2649 difference of 1-2 mixing from solar data and Kamland atmospheric: excess of sub-GeV e-like events sin2q13 = 0.016 +/- 0.010

Summary Consistent picture: interpretation of all * the results in terms of vacuum mixing of three massive neutrinos LSND ? Two effects are important for the interpretation (at the present level of accuracy): vacuum oscillations, adiabatic conversion (MSW). Oscillations in matter (multilayer medium) ~ 1s Next level: sub-leading effects related to 1-3 mixing (and CP) require more involved study New oscillation effects new matter effects nonlinear neutrino transformations Oscillograms – neutrino images of the Earth: useful tool, method of measurements? Still debates on theory of neutrinos oscillations experimental tests: Solar vs KamLAND

In the low density medium V(x) << Dm2/ 2E Potential << kinetic energy 2 E V(x) D m2 e (x) ~ (1 -3) 10-2 Small parameter: e (x) = << 1 perturbation theory in e (x) Solar neutrinos Inside the Earth For LMA oscillation parameters applications to Supernova neutrinos Oscillations appear in the first order in e (x) Relevant channel : mass-to-flavor n2 -> ne P2e = sin2q + freg

Regeneration factor Pdet = P + D Preg Total survival probability DPreg = - cos2qm0 freg determines sign of the effect Positive if suppression inside the sun is stronger than 1/2 Regeneration factor: freg = P2e - sin2q the mass-to-flavor transition n2  ne The oscillations proceed in the weak matter regime: 2EV(x) Dm2 e (x) = << 1 Can be used to develop various perturbation theories

Adiabatic perturbation theory Adiabatic condition: lm(x) 4ph(x) h(x) the height of distibution g (x) = << 1 At the borders of layers h(x) -> 0

Analytic result 2E sin22q Dm2 freg = sinF0/2 Sj = 0 …n-1 DVj sinFj/2 j Defining fj = 0.5(F0 - Fj) Fj 2E sin22q Dm2 x freg = Sj = 0 …n-1 DVj[sin2F0/2 cosfj - 0.5 sinF0 sinfj] fj If fj is large - averaging effect. This happens for remote structures, e.g. core xf x0 Fm(xc -> x) = dx Dm(x)

Analytic vs. numerical results P. de Holanda, Wei Liao, A.S. Regeneration factor as function of the zenith angle E = 10 MeV, Dm2 = 6 10-5 eV2, tan2q = 0.4

e - perturbation theory For regeneration effect in the Earth 2EVE Dm2

Precise semianalytic result For symmetric density profile using Magnus expansion (unitary in each order): P2e = sin2q + freg A.D. Supanitsky J.C.D’Olivo 2007 freg = sin2q sinFm(xc -> xf) sin 2I + cos2q sin2I A. Ioanissian, A.S. 2008 xf xc I = sin 2q dx V(x) cos Fm(xc -> x) xf x xc xf x0 Fm(xc -> x) = dx Dm(x) xc - is the center of trajectory adiabatic phase from the center of neutrino trajectory to a given point x Essentially I is the expansion parameter in the problem 2EVmax Dm2 I < = e max Estimate:

Relative errors d = (fappr - fexact) / f0 f0 = 0.5 ef sin22q at the surface A. Ioannisian et al, Phys. Rev. D (2005) Second order First order For the neutrino trajectory which crosses the center of the Earth

Earth matter effect Attenuation - genuine matter effect, - test of correctness of whole neutrino evolution Attenuation of sensitivity to remote structure related to finite energy resolution of detectors Integration with the energy resolution function R(E, E’) xf x0 freg = 0.5 sin22q dx V(x) F(xf - x) sin Fm(x -> xf) averaging factor d = xf - x the distance from structure to the detector ln E p d DE p d DE ln E F(d) = sin

Attenuation effect The width of the first peak d < ln E/DE Attenuation factor F ln is the oscillation length The sensitivity to remote structures is suppressed: Effect of the core of the Earth is suppressed Small structures at the surface can produce stronger effect d, km The better the energy resolution, the deeper penetration

Averaging regeneration factor Regeneration factor averaged over the energy intervals E = (9.5 - 10.5) MeV (a), and E = (8 - 10) MeV (b). No enhancement for core crossing trajectories in spite of larger densities

Factorization approximation Ae2 ~ AS (Dm212 , q 12) corrections of the order Dm122 /Dm13 2 , s132 Ae3 ~ AS (Dm312 , q 13) are not valid in whole energy range due to the level crossing For constant density: FS ~ H21 FS ~ Dm322 p L l12m AS ~ i sin2q12m sin p L l13m FA ~ H32 AA ~ i sin2q13m sin

Scheme of transitions ne n1 n1 n2 n2 n3 n3 Pee = S i PSei PE ie PE1e oscillations inside the Earth n3 n3 Conversion inside the star projection (if there is no earth crossing) loss of coherence Pee = S i PSei PE ie i = 1, 2, 3 PSei - probability of ne -> ni conversion inside the star PEie - probability of ni -> ne oscillations inside the Earth PEie = |Uie|2 if the burst does not cross the Earth Similarly for antineutrinos

Asymmetries Transition probability nb -> na ifj Transition probability nb -> na Pab = | Sj Uaj* Ub j e |2 ifj in vacuum: CP-transformation: Uaj --> Uaj* PabCP = | Sj Uaj Ubj*e |2 ifj T-transformation: a <-> b PabT = | Sj Ubj* Uaj e |2 = PabCP JCP < 0.03 Oscillating factor is small unless long baseline (2000 - 3000 km) are taken Earth matter effect is important Usual matter is CP-asymmetric CP-violation in neutrino oscillations even for d = 0 in matter: (Uajm)CP = ( Ua jm )* Problem is to distinguish: Precise knowledge of oscillation parameters, resolve ``degeneracy’’ of parameters, ambiguity… T-violation? fundamental CP violation CP-violation due to matter effect Global fit A Yu Smirnov

Approximations

Oscillations in matter Probability constant density pL lm P(ne -> na) = sin22qm sin2 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 sin 22qm = 1 MSW resonance condition 1. Amplitude condition: 2. Phase condition: f = p/2 + pk

Sensitivity to density profile For mass-to-flavor transition V(x) is integrated with sin Fm(d) d = xf - x the distance from structure to the detector stronger averaging effects weaker sensitivity to structure of density profile larger d larger Fm(d) Integration with the energy resolution function R(E, E’): freg = dE’ R(E, E’) freg(E’) The effect of averaging: xf x0 freg = 0.5 sin22q dx V(x) F(xf - x) sin Fm(x -> xf) averaging factor For box-like R(E, E’) with width DE: ln E p d DE p d DE ln E F(d) = sin

Integral formula P2e = sin2q + freg e - perturbation theory Regeneration factor xf x0 A. Ioannisian, A.S. freg = 0.5 sin22q dx V(x) sin Fm(x -> xf) Explicitly: xf x0 xf x Dm2 2E 2EV(y) 2 Dm2 freg = 0.5 sin22q dx V(x) sin dy cos 2q - - sin22q V(x) Fm(x -> xf) Integration limits: x0 xf x The phase is integrated from a given point to the final point

Oscillations inside the Earth 1). Incoherent fluxes of n1 and n2 arrive at the surface of the Earth 2). In matter the mass states oscillate 3). the mass-to-flavor transitions, e.g. n2 --> ne are relevant Regeneration factor: P2e = sin2q + freg Pee = 0.5[1 + cos2qm0 cos2q] - cos2qm0freg 4). The oscillations proceed in the weak matter regime: 2EV(x) Dm2 e (x) = << 1

Low density medium

Evolution equation `Physics derivation’ n1 n2 dnmass dt 1 2E m12 0 p1 = p2 = p Ei ~ p + mi2/2 In vacuum the mass states are the eigenstates of Hamiltonian n1 n2 dnmass dt 1 2E m12 0 0 m 22 nmass = i = p I + nmass Using relation nmass = U+n f find equation for the flavor states: the term pI proportional to unit matrix is omitted dnf dt M2 2E ne nm i = n f nf = where m12 0 0 m 22 mass matrix in flavor basis M2 = U U+

CP-asymmetries ACP = P(n a -> nb) - P( na -> nb ) CP-asymmetry: L. Wolfenstein, C. Jarlskog, V. Barger, K. Whisnant, R. Phillips ACP = P(n a -> nb) - P( na -> nb ) CP-asymmetry: T-asymmetry: AT = P(n a -> nb ) - P( nb -> na ) For vacuum oscillations: Dm122 2E Dm232 2E Dm312 2E ACP = 4 JCP sin t + sin t + sin t where JCP = Im [Ue2 Um2* Ue3* Um3] = = s12 c12 s13 c132 s23 c23 sind is the leptonic analogue of the Jarlskog invariant A Yu Smirnov

Regeneration factor freg = P2e - sin2q P(n2  ne) = |<ne| U(qmR) S(x0  xf) U+(qmR) U(q) |n2> |2 qmR - mixing angle at the surface of the Earth Liao Wei, P de Holanda, A.S. Adiabatic perturbation theory freg = e (R) sin22q sin2 [Fm(x0  xf)/2] + sin 2q Re{c(x0  xf)} amplitude of jump probability 2EV(R) Dm2 e (R) = If adiabaticity is realized, the regeneration depends on the potential V(R) at the surface and on the total adiabatic phase Non-adiabatic conversion appears as the interference term and therefore - linearly (in contrast to conversion in the Sun)

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 f = p/2 + p k interference phase does not depends on d P(nm  nt)d ~ - 2 s23 c23 |AS| |AA| sin f sind