A. Yu. SmirnovInternational 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

innercore

outercore

upper mantle

transition zone

crustlower mantle

(phase transitions in silicate minerals)

liquidsolid

Fe

Si

PREM model A.M. Dziewonski D.L Anderson 1981

Re = 6371 km

core

mantle

flavor to flavor transitions

Oscillations inmultilayer medium

- nadir angle

core-crossingtrajectory

-zenith angle

= 33o

acceleratoratmosphericcosmic neutrinos

Applications:

Contours of constant oscillation probability in energy- nadir (or zenith) angle plane

P. Lipari ,T. OhlssonM. Chizhov, M. Maris, S .PetcovT. Kajita

e

Michele Maltoni

1 - Pee

from SAND to HAND

Explaining oscillograms

Dependence of oscillograms on neutrino parameters

Applications

CP-violation domains

What these oscillograms for?

Neutrino images of the earth6X2

Oscillations in matter with nearly constant density

Parametric enhancement of oscillations(mantle - core – - mantle)

(mantle)

Interference

constant density + corrections

Peaks due to resonance enhancement of oscillations

Low energies: adiabatic approximation

Parametric resonance parametric peaks

Smallness of 13 and m21

2/m322

in the first approximation: overlap of two 2–patternsdue to 1-2 and 1-3 mixings

interference

of modes CP-interference

interference(sub-leading effect)

MSW-resonancepeaks 1-2 frequency

1 - Pee

Parametric peak1-2 frequency

MSW-resonancepeaks 1-3 frequency

Parametric ridges1-3 frequency

mantle

mantle

1

2

1 2

core

mantle

mantle

mantle core mantle

1

2

3

4

1

2 3

4

core

mantle

mantle

1

2

3

4 3

2

4

1

determine localizations of the peaks and ridges

Generalized

resonance

condition

Intersection of the corresponding lines determines positions of the peaks

phase: = + k

depth:sin2m = 1

Generalizations of the conditions ``AMPLITUDE = 1’’ in matter with constant density to the case of 1 or 3 layers with varying densities

also determine position of other features: minima, saddle points, etc

generalized phase condition

collinearitycondition

Maximal oscillation effects:

Amplitude(resonance)condition

Phasecondition

Generalizations:

E.Kh. Akhmedov, A.S.,M Maris, S. Petcov

Dependence on neutrino parameters and earth density profile (tomography)

Flow of large probability toward larger

Lines of flow change weakly

Factorization of 13 dependence

Position of the mantle MSW peak

measurement of 13

normal inverted

neutrino antineutrino

For 2 system

Under CP-transformations:

cCP- transformations: c = i 02

+ applying to the chiral components

UPMNS UPMNS * -

V - V

usual medium is C-asymmetricwhich leads to CP asymmetryof interactions

= 60o

Standardparameterization

= 130o

= 315o

Three grids of lines:

Solar magic lines

Atmospheric magic lines

Interference phase lines

e e

fU23I I = diag (1, 1, ei )

e

e

S

~

Propagation basis~

~

~

~

projection projectionpropagation

A(e ) = cos23Ae2ei + sin23Ae3

Ae3

Ae2

P( e ) = |cos 23Ae2e i + sin 23Ae3|2

``atmospheric’’ amplitude``solar’’ amplitude

Due to specific form of matter potential matrix (only Vee = 0)

dependence on and23is explicit

P(e ) = |Ae2 Ae3| cos ( - )

P( ) = - |Ae2 Ae3| cos cos

P( ) = - |Ae2 Ae3| sin sin

For maximal 2-3 mixing

= arg (Ae2* Ae3)

= 0

Ae2 ~ AS (m212

, 12)

Ae3 ~ AS (m312

, 13)

are not valid in whole energy range due to the level crossing

S ~ H21 S ~ m322

A ~ H32

corrections of the order m122 /m13

, s13

2

AS ~ i sin212m

sin L l12

m

AA ~ i sin213m

sin L l13

m

For constant density:

P(e ) = c23

2|AS|2 + s232|AA|2 + 2 s23 c23 |AS| |AA| cos( + )

L lijm

at high energies: l12m

~ l0

L = k l0 , k = 1, 2, 3

AS = 0 for

(for three layers – more complicated condition)

s23 = sin 23 = arg (AS AA*)

Dependence on disappears if

Solar ``magic’’ lines

does not depend on energy- magic baseline

V. Barger, D. Marfatia, K WhisnantP. Huber, W. Winter, A.S.

AS = 0 AA = 0

Atmospheric magic lines

L = k l13 m

(E), k = 1, 2, 3, …

= k

AS = 0

- true (experimental) value of phasef - fit value

P = P() - P(f)

P = 0

(along the magic lines)

( + ) = - ( + f) + 2 k

(E, L) = - ( + f)/2 + k

= Pint() - Pint(f)

AA = 0

int. phase condition

depends on

How to measure the interference term?

Interference term:

P = 2 s23 c23 |AS| |AA| [ cos( + ) - cos ( + f)]

For e channel:

AS = 0

P = 0(along the magic lines)

= /2 + k

AA = 0

interference phase does not depends on

P( ) ~ - 2 s23 c23 |AS| |AA| cos cos

For channel - dependent part:

The survival probabilities is CP-even functions of No CP-violation.

P ~ 2 s23 c23 |AS| |AA| cos [cos - cos f]

P( ) ~ - 2 s23 c23 |AS| |AA| sin sin

solar magic linesatmospheric magic linesrelative phase lines

Regions of different sign of P

Interconnectionof lines due to level crossing

factorization is not valid

Grid (domains)does not change with

Int. phaseline moves with -change

P

P

P

Contour plots for the probability difference P = Pmax – Pmin

for varying between 0 – 360o

e

Emin ~ 0.57 ER

Emin 0.5 ER

when 13 0

- mass hierarchy- 1-3 mixing- CP violation

Pictures for Textbooks: neutrino images of the Earth

10

1

100

0.1

E,

GeV

MINOS

T2K

CNGS

NuFac 28000.005

0.03

0.10

T2KK

Degeneracyof parameters

Large atmospheric neutrino detectors

LAND

LENF

Intense and controlled beams

Small effect

Degeneracy of parameters

Combination of results from different experiments is in general required

Cover poor-structure regions

Systematic errors

Small fluxes, with uncertainties

Large effects

Cover rich-structure regions

No degeneracy?

E ~ 0.1 – 104 GeV

Problem:- small statistics- uncertainties in the predicted fluxes- presence of several fluxes - averaging and smoothing effects

Cost-free source

whole range of nadir anglesL ~ 10 – 104 km

Several neutrino types

- various flavors: e and

- neutrinos and antineutrinos

Cover whole parameter space (E, )

INO – Indian Neutrino observatory

HyperKamiokande

Y. Suzuki..

Icecube (1000 Mton)

50 kton iron calorimenter

0.5 Megaton water Cherenkov detectors

Underwater detectors ANTARES, NEMO

TITAND (Totally Immersible Tank Assaying Nuclear Decay)

2 Mt and more

UNO

E > 30 – 50 GeVReducing down 20 GeV?

Y. Suzuki

- Proton decay searches- Supernova neutrinos- Solar neutrinos

Totally Immersible Tank Assaying Nucleon Decay

TITAND-II: 2 modules: 4.4 Mt (200 SK)

Under sea deeper than 100 m

Cost of 1 module 420 M $

Modular structure

e-like events- angular resolution: ~ 3o

- neutrino direction: ~ 10o

- energy resolution for E > 4 GeV better than 2%

E/E = [0.6 + 2.6 E/GeV ] %

cos -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)

Fully contained events

5 – 10 GeV SR 1050 (9) 1080 (5) 1500 (10) MR 1150 (4) 1280 (3) 1690 (6)

SR – single ringMR – multi-ring

(…) – number of events detected by 4SK years

MC: 800 SK-years zenith angle

Measuring oscillograms with atmospheric neutrinos

E > 2 - 3 GeV

with sensitivity to the resonance region

Huge Atmospheric Neutrino Detector

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?

0.5 GeV

Oscillograms encode in a comprehensive way information about

the Earth matter profile and neutrino oscillation parameters.

Oscillograms have specific dependencies on 1-3 mixing angle, mass hierarchy, CP-violating phases and earth density profilethat 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

Locations of salient structures of oscillograms are determined by the collinearity and generalized phase conditions

Determination of neutrino parameters,tomography by

measuring oscillograms with Huge (multi Megaton)

atmospheric neutrino detectors?

P( e ) = |cos 23ASe i + sin 23AA|2

|AS | ~ sin212m

sin L lm

For high energies lm l0for trajectory with L = l0 AS = 0

P = |sin 23AA|2 no dependence on

For three layers – more complicated condition

Magic trajectories associated to AA = 0

``atmospheric’’ amplitude``solar’’ amplitude

mainly, m132, 13

mainly, m122, 12

Contours of suppressed CP violation effects

1 layer:

MSW resonance condition

S(1)11 = S(1)

22

Im (S11 S12*) = 0

Im S(1)11 =

0

sin = 0cos 2m = 0

2 layers:X3 = 0 S(2)

11 = S(2)22

Im S(2)11 = 0

Generalized resonance conditionvalid for both cases:

= + k

Re S(1)11 = 0

another representation:

For symmetric profile (T –invariance):

Re (S11) = 0

Im S11 = 0

Parametric resonance condition

unitarity: S(1)11 =

[S(1)22 ]*

S12- imaginary

S = a b -b* a*

Evolution matrix for one layer (2-mixing):

a, b – amplitudes of probabilities

For symmetric profile (T-invariance) b = - b*

For two layers: S(2) = S1 S2

A = S(2)12 = a2 b 1 + b2 a 1*

transition amplitude:

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

Re b = 0

Another wayto generalize parametric resonance condition

from unitarity condition

Different structures follow from different realizations of the collinearity and phase condition in the non-constant case.

Re (a1a2 b1) = 0

X3 = 0

P = 1

Re (S11) = 0

Absolute maximum (mantle, ridge A)

c1 = 0,c2 = 0

s1 = 0, c2 = 0

Local maximaCore-enhancement effect

P = sin (4m – 2c)

Saddle points at low energies

Maxima at high energiesabove resonances

P(ea) =sin22msin2

L lm

Oscillation Probabilityconstant density

Amplitude of oscillations

half-phase

oscillatory factor

- mixing angle in matterlm(E, n )

m(E, n ) – oscillation length in matter

Conditions for maximal transition probability: P = 1

1. Amplitude condition: sin22m= 12. Phase condition: = + k

MSW resonance condition

m lm l

In vacuum:

lm = 2 /(H2 – H1)

1. Take condition for constant density

Generalization of the amplitude and phase conditions to varying density case

2 . Write in terms of evolution matrix

3. Apply to varying density

This generalization leads to new realizationswhich did not contained in the original condition; more physics content

S11 S12 S21 S22

S(x) = T exp - i H dx = x 0

l = 4m2

Oscillation length in vacuum

Refraction length

l0 = 2 2 GFne

- determines the phase produced by interaction with matter

lm

E

l0

ER

Resonance energy:

l(ER) = l0cos2

l/sin2 l= l0/cos2)(maximum at

~ l

2H2 - H1

lm =

Simulations:

Monte Carlo simulations for SK 100 SK year scaled to 800 SK-years(18 Mtyr) = 4 years of TITAND-II

Assuming normal mass hierarchy:

Sensitivity to quadrant: sin2 23 = 0.45 and 0.55 can be resolved with 99% C.L. (independently of value of 1-3 mixing)

Sensitivity to CP-violation: down to sin2 213 = 0.025 can be measured with = 45o accuracy ( 99% CL)

a). Resonance in the mantle

b). Resonance in the core

c). Parametric ridge A

d). Parametric ridge B

e). Parametric ridge C

f). Saddle point

a). b).

c).

e).

d).

f).

Contour plots for the probability difference P = Pmax – Pmin

for varying between 0 – 360o

Averaging?

= P = (Re e

+ , Im e+ , e

+ e - 1/2)

B = (sin 2m, 0, cos2m) 2 lm

= ( B x ) d dt

Evolution equation

= 2t/ lm - phase of oscillations

P = e+e = Z + 1/2 = cos2Z/2 probability to find e

Enhancement associated to certain conditions for the phase of oscillations

Another way of getting strong transitionNo large vacuum mixing and no matterenhancement of mixing or resonance conversion

``Castle wall profile’’

V

= =

V. Ermilova V. Tsarev, V. ChechinE. AkhmedovP. Krastev, A.S., Q. Y. Liu, S.T. Petcov, M. Chizhov

1 2 3 4 5 6 7

m

m

m

m

of oscillograms

1. One (or zero) MSW peak in the mantle domain

2. Three parametric peaks (ridges) in the core domain

3. MSW peak in the core domain

1-3 mixing:

1-2 mixing:

Depending on value of sin213

1. Three MSW peaks in the mantle domain

2. One (or 2) parametric peak (ridges) in the core domain

1D 2D - structures regular behavior

Y. Suzuki

Totally Immersible Tank Assaying Nucleon Decay

Module: - 4 units, one unit: tank 85m X 85 m X 105 m - mass of module 3 Mt, fiducial volume 2.2 Mt - photosensors 20% coverage ( 179200 50 cm PMT)

TITAND-II: 2 modules: 4.4 Mt (200 SK)