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A. Yu. Smirnov

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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

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

``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

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

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

Explaining Oscillograms

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

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

Resonance enhancement in mantle

1

mantle

1

2

mantle

2

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

1

mantle

core

3

4

2

2

mantle

4

3

1

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

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

Analytic vs. numeric

Dependence on parameters

Dependence on neutrino parameters and

earth density profile (tomography)

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

Other

channels

mass

hierarchy

For 2n system

normal inverted

neutrino antineutrino

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

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

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(nenm) = cosq23Ae2eid + sinq23Ae3

CP-interference

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

P(nenm) = |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

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

``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)

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

Interference terms

For nmnm 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

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

Int. phase

line moves

with d-change

Grid (domains)

does not

change with d

DP

DP

DP

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

Applications

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

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

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

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:

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..

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)

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

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?

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?

CP and magic trajectories

P(nenm) = |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

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

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

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

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

Non-constant density case

Generalization of the amplitude and phase conditions to varying density case

- Take condition for
- constant density

x

0

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

S11 S12

S21 S22

2 . Write in terms of

evolution matrix

3. Apply to varying density

This generalization leads to new realizations

which did not contained in the original condition;

more physics content

Oscillation length in matter

4p E

Dm2

Oscillation

length in vacuum

ln=

2p

2 GFne

Refraction

length

- determines the phase produced

by interaction with matter

l0 =

ln = l0/cos2q)

(maximum at

ln /sin2q

lm

2p

H2 - H1

lm =

l0

Resonance energy:

~ ln

ln (ER) = l0cos2q

E

ER

Sensitivity

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:

sin2q23 = 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 2q13 = 0.025 can be measured

with Dd = 45o accuracy ( 99% CL)

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).

Sensitivity to CP-phase

nm nm

- Contour plots for
- the probability
- difference
- P = Pmax – Pmin
for d varying

between 0 – 360o

Averaging?

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

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’’

Salient features

of oscillograms

Depending on value of sin2q13

1-3 mixing:

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

1-2 mixing:

1. Three MSW peaks in the mantle domain

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

1D 2D - structures

regular behavior

TITAND

Totally Immersible Tank

Assaying Nucleon Decay

Y. Suzuki

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)