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. Fermilab, April 16, 2008

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

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

core

mantle

flavor to flavor transitions

Oscillations inmultilayer medium

- nadir angle

core-crossingtrajectory

-zenith angle

Mass & mixing

Oscillations

Oscillograms

= 33o

acceleratoratmosphericcosmic neutrinos

The Earth is unique

We know that neutrino masses and mixing are non-zero

Oscillograms are reality and this reality will be with us forever

First of all we need to understand their properties and physics behind

Can we observe (reconstruct) these neutrino images?

and then…

How can we use them?

With which precision?

from SAND to HAND

Two effects1. Explaining oscillograms

2. How oscillograms depend on unknown yet neutrino parameters and Earth density profile

3. How can we use them?

Interference of modes and CP-violation

e

2

1

mas

s

1

2

3

3

mas

s

m2atmm2

atm

m2sun

m2sun

Inverted mass hierarchyNormal mass hierarchy

|Ue3|2

Type of the mass hierarchyUe3

?

CP-violating phase

sin13 = |Ue3|2 I = diag (1, 1, ei)

tan12 = |Ue2|2 / |Ue1|2

tan23 = |U3|2 / |U3|2

f = UPMNS massUPMNS = U23 I U13 IU12

Oscillations in matter with nearly constant density

Parametric enhancement of oscillations

mantle – core - mantle1 layer case: mantle

Interference of different modes of oscillations

In two neutrino context

d d t i = H

M M+

2EH = + V(t)

e

M is the mass matrix

V = diag (Ve, 0, 0) effective potential

M M+ = U Mdiag2 U+

Mdiag2 = diag (m1

2, m22, m3

2)

Eigenstates and eigenvalues of Hamiltonian

Diagonalization of H

Mixing matrix

Mixing matrix in vacuum

Energy levels

Re e

+ , P = Im e

+ , e

+ e - 1/2

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

= ( B x P ) dP dt

Coincides with equation for the electron spin precession in the magnetic field

= e ,

Polarization vector:

P =+

Evolution equation:

i = H d d t

d d t i = (B )

Differentiating P and using equation of motion

= 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

Constant densitySource Detector

F0(E) F(E)

F (E)F0(E)

E/ERE/ER

thin layer thick layer

k = L/ l0 sin2 2 = 0.824

k = 1 k = 10

e e

Layer of length L

A Yu Smirnov

F (E)F0(E)

E/ERE/ER

thin layer thick layerk = 1 k = 10

sin2 212 = 0.08

Small mixing angle

sin22m =sin22

( cos2EV/m2)2 + sin 22

sin22m = 1

Mixing is maximal if

V = cos 2m2

2E He = H

Difference of the eigenvalues

H2 - H1 = m2

2E( cos2EV/m2)2 + sin22

Diagonalization of the Hamiltonian:

V = 2 GF ne

Resonance condition

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

= =

distancedistance

c1 = c2 = 0 General case: certain correlation between the phases and mixing angles

Akhmedov, A.S.

sccos2m

+ sccos2

m= 0

si = sini, ci = cosi, (i = 1,2)

half-phases

core

mantle

mantle

mantle core mantle

1

2

3

4

1

2 3

4

MSW-resonancepeaks 1-2 frequency

1 - Pee

Parametric peak1-2 frequency

MSW-resonancepeaks 1-3 frequency

Parametric ridges1-3 frequency

collinearitycondition

(parametric resonance condition)

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).

core

mantle

mantle

1

2

3

4 3

2

4

1

Dependence on neutrino parameters and earth density profile (tomography)

twist

Shift of border

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 - Vusual medium is C-asymmetricwhich leads to CP asymmetryof interactions

Under T-transformations:

- V V

initialfinal

= 60o

Standardparameterization

= 130o

= 315o

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

AS ~ i sin212m

sin L l12

m

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

AA depends mainly on m132, 13

AS depends mainly on m122,

12

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

``Factorization’’ approximation:

corrections of the order m122 /m13

, s13

2

For constant density:

dependence on and23is explicit

AA ~ i sin213m

sin L l13

m

up to phase factors

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*) ``strong’’ phaseinterference term - ``weak’’ phase

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

If all parameters but are known

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| cos 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

Contour plots for the probability difference P = Pmax – Pmin

for varying between 0 – 360o

Averaging?

Position of the mantle peak – measure of 1-3 mixing

- determination of mass hierarchy- 1-3 mixing- CP violation- Earth tomography

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

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)

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.

Tool to elaborate methods and criteria of selection of events to - enhance sensitivity to particular effects - disentangle effects (remove degeneracy )

Worthwhile to consider a possibility of measuring oscillograms withHuge 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

Reproduces all the features of oscillograms Vacuum mimicking

Weak matter effects

Oscillograms fortan212 = 0.45

e

Oscillograms fortan212 = 0.45

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

``Castle wall profile’’

m

m

parametric resonance condition

- oscillation half-phases

- mixing anglesV

d

also S. Petcov M. Chizhov

Maximal depth of oscillations

E. Kh. Akhmedov

X3 = 0

P = (1 – X3 / |X| ) sin 2 n

im

i

Probability after n periods: multiplying the evolution matrices for each layer

Evolution matrix over one period (two layers)

S = Y – i XX, Y = X, Y (ii)

X = (X1 , X2, X3) -Pauli matrices

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

AS = 0

- true (experimental) value of phaseDetermining the CP-violation phase:

f - fit value

Compare probabilities

P = P() - P(f)

P = 0(along the magic lines)

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

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

In split approximation and for constant density:

= m322 L/2E

= Pint() - Pint(f)

AA = 0

int. phase condition

depends on

c12c13 s12c13 s13e-i

- s12c23 - c12s23s13ei c12c23 - s12s23s13ei s23c13

s12s23 - c12c23 s13ei - c12s23 - s 12c23s13ei c23c13

UPMNS =

isthe Dirac CP violating phase

c12 = cos , etc.

is the ``solar’’ mixing angle is the ``atmospheric’’ mixing angle is the mixing angle restricted by CHOOZ/PaloVerde experiments

UPMNS = U23 I U13 I U12 I = diag (1, 1, ei)

sin2 2m = 1

Flavor mixing is maximalLevel split is minimal

In resonance:

l = l0 cos 2

Vacuumoscillation length

Refractionlength~~

For large mixing: cos 2 ~ 0.4the equality is broken: strongly coupled system shift of frequencies.

l / l0

sin2 2m

sin22 = 0.08 sin2212 = 0.825

~ n E

Resonance width: nR = 2nR tan2

Resonance layer: n = nR + nR

structures of oscillograms are not accidental but well determined by simple relations

they encode important information about earth structure and neutrino parameters

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)