# Presentation: 470003

**A. Yu. Smirnov**

*International Centre for Theoretical Physics, Trieste, Italy *
*  **Institute for Nuclear Research,  RAS, Moscow, Russia*

- of the Earth

- Neutrino oscillograms

*E. Akhmedov, M. Maltoni, A.S.,  JHEP 0705:077 (2007) ; *
*arXiv:0804.1466 (hep-ph)*
*A.S. hep-ph/0610198. *

*Fermilab, April 16, 2008*

- The earth density profile

- inner
- core

- outer
- core

- upper
- mantle

- transition
- zone

- crust

- lower
-  mantle

- (phase transitions in
- silicate minerals)

- liquid

- solid

- 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. Ohlsson*
*M. Chizhov, *
*M. Maris, *
*S .Petcov*
*T. Kajita*

-  e’

- Michele
- Maltoni

- 1 - Pee

- Neutrino images

- oscillograms

## 

- core

- mantle

- flavor to flavor transitions

- Oscillations in
- multilayer medium

- - nadir angle

- core-crossing
- trajectory

- 

- zenith
-  angle

- Mass & mixing

- Oscillations

- Oscillograms

- = 33o

- 

- ``Set-up''

- accelerator
- atmospheric
- cosmic neutrinos

- Comments

- 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?

- Outline

-  from SAND to HAND

- Two effects

- 1. 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

- mass

- 1

- 2

- 3

- 3

- mass

- m2atm

- m2atm

- m2sun

- m2sun

- Inverted mass hierarchy

- Normal mass hierarchy

- |Ue3|2

- Type of the mass hierarchy

- Ue3

- ?

- Masses and mixing

- CP-violating phase

- sin13  = |Ue3|2

-  I = diag (1,  1,  ei)

- tan12 =  |Ue2|2   /  |Ue1|2

- tan23  = |U3|2   / |U3|2

- f  =  UPMNS mass

- UPMNS  = U23  I U13 IU12

- Explaining Oscillograms

- Two effects

- Oscillations in matter
- with nearly constant
- density

- Parametric
- enhancement
- of oscillations

- mantle – core - mantle

- 1 layer case: mantle

- Interference of
- different modes
- of oscillations

- In two neutrino context

- Evolution equation

- d 
- d t

- i         =  H 

- M M+
-   2E

- H =             + V(t)

- e
- 
- 

- M  is the mass matrix

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

- M M+  = U Mdiag2 U+

-  Mdiag2 =  diag (m12,  m22,  m32)

- 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

- Neutrino polarization vectors

**B** =        (sin 2m,  0,  cos2m)

- 2
-  lm

- =  ( **B** x  **P** )

- d**P**
- 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)

- Graphical  representation

**B** =        (sin 2m,  0,  cos2m)

- 2
-  lm

- =  ( B x   )

- d
- dt

- _Evolution equation_

-   = 2t/ lm

- - phase of oscillations

- P = e+e = Z + 1/2 = cos2Z/2

- probability to find   e

- Oscillations

- Constant density

- Source

- Detector

- F0(E)

- F(E)

- F (E)
- F0(E)

- E/ER

- E/ER

- thin layer

- thick layer

- k =  L/ l0

- sin2 2 = 0.824

- 

- k = 1

- k = 10

- e

- e

- Resonance enhancement

- Layer of length L

- A Yu Smirnov

- F (E)
- F0(E)

- E/ER

- E/ER

- thin layer

- thick layer

- k = 1

- k = 10

- sin2 212 = 0.08

- Small mixing angle

- sin22m =

- sin22

- ( cos2EV/m2)2    +  sin 22

- sin22m = 1

- Mixing is maximal  if

- V   =                 cos 2

- m2
-  2E

- He =  H

- Difference of  the eigenvalues

- H2 - H1 =

- m2
-  2E

- ( cos2EV/m2)2  + sin22

- Diagonalization of the Hamiltonian:

- V =  2 GF ne

- Mixing in matter

- Resonance
- condition

- Parametric enhancement of oscillations

- Enhancement associated to certain
- conditions for the phase of oscillations

- Another way of getting strong transition

- No large vacuum mixing and no matter
- enhancement of mixing or resonance
- conversion

- ``Castle wall profile’’

- V

- 

- 

-  =  = 

- V. Ermilova V. Tsarev,  V. Chechin
- E. Akhmedov
- P. Krastev, A.S., Q. Y. Liu,
- S.T. Petcov, M. Chizhov

- 1      2     3      4      5       6      7

- m

- m

- m

- m

- Parametric resonance

-  =  = 

- distance

- distance

- c1 = c2 = 0

- General case: certain correlation
- between the phases and mixing angles

*Akhmedov, A.S.*

-   sccos2m +  sccos2m= 0

- si = sini,
- ci = cosi,  (i = 1,2)

- half-phases

- Parametric enhancement in the Earth

- core

- mantle

- mantle

- mantle   core   mantle

- 1

- 2

- 3

- 4

- 1

- 2          3

- 4

- MSW-resonance
- peaks 1-2 frequency

- 1 - Pee

- Parametric peak
- 1-2 frequency

- MSW-resonance
- peaks 1-3 frequency

- Parametric
- ridges
- 1-3 frequency

- 

- Analytic vs. numeric

- Evolution

- collinearity
- condition

- (parametric resonance
-  condition)

- Graphical

- representation

- 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).

- Parametric enhancement of 1-2 mode

- core

- mantle

- mantle

- 1

- 2

- 3

- 4

- 3

- 2

- 4

- 1

- Properties of oscillograms

- Dependence on neutrino parameters and
- earth density profile (tomography)

- Sensitivity to density profile

- twist

- Sensitivity to density profile

- Shift of
- border

- Dependence on gradients

- Dependence

- on 1-3 mixing

- Flow of large probability
- toward  larger 

- Lines of flow change weakly

- Factorization of
- 13  dependence

- Position of the
- mantle MSW peak

- measurement of 13

- Other

- channels

-   normal   inverted
- neutrino  antineutrino

- For 2  system

- mass

- hierarchy

- CP-violation

- _Under CP-transformations:_

-     c

- _CP- transformations:_

-  c =  i 02  +    

- applying to the chiral components

- UPMNS   UPMNS *

-    - 

- V  - V

- usual medium is  C-asymmetric
- which leads to CP asymmetry
- of interactions

- _Under T-transformations: _

-   -

- V   V

- initialfinal

- CP-violation

-  = 60o

- Standard
- parameterization

-  = 130o

-  = 315o

- Interference of modes

- P( e  )  = |cos 23ASe i  +  sin 23AA|2

- AS  ~ i  sin212m  sin

-  L
-  l12m

- ``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, s132

- _For constant density:_

- dependence on and23is explicit

- AA  ~ i  sin213m  sin

-  L
-  l13m

- up to phase
- factors

- ``Magic lines"

- P(e  ) = c232|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’’ phase

- interference
- term

-  - ``weak’’ phase

- Dependence on 
-  disappears if

- Solar ``magic’’ lines

- does not depend on energy
- - magic baseline

- V. Barger, D. Marfatia,
-  K Whisnant
- P. Huber,  W. Winter,
- A.S.

-  AS   =  0
-  AA   =  0

- Atmospheric magic lines 

- L = k l13 m (E),    k = 1, 2, 3, ...

- = k 

- Interference terms

- AS  =  0

-   - true (experimental) value of phase

- f  - 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:

- Interference terms

- 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

- CP violation domains

- solar magic lines
- atmospheric magic lines
- relative phase lines

- Regions of different sign of P

- Interconnection
- of lines due to
- level crossing

- factorization
- is not valid

- Grid  (domains)
- does not
- change with 

- Int. phase
- line moves
- with  -change

- P

- P

- P

- Sensitivity to CP-phase

- 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

-      

- Sensitivity to CP-phase

- Contour plots for
- the probability
- difference
- P = Pmax – Pmin
- for  varying
- between  0 – 360o

- Averaging?

- Applications

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

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

- Where we are?

- 10

- 1

- 100

- 0.1

- E, GeV

- MINOS

- T2K

- CNGS

- NuFac 2800

- 0.005

- 0.03

- 0.10

- T2KK

- Degeneracy
- of parameters

- Large atmospheric
- neutrino detectors

- LAND

- LENF

- Two approaches

- Accelerators

- 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

- Atmospheric

- Neutrinos

- Small fluxes, with uncertainties

- Large effects

- Cover rich-structure regions

- No degeneracy?

- Atmospheric neutrinos

-  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 angles

- L ~ 10 – 104  km

- Several neutrino types

-  various flavors:  e and  
-  neutrinos and antineutrinos

- Cover whole
- parameter space (E, )

- Detectors

- 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 GeV

- Reducing down 20 GeV?

- TITAND

*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

- TITAND

*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)

- Number of events

- 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 ring
- MR – multi-ring

- (...) – number of events detected by  4SK years

- MC:  800 SK-years

- zenith angle

- From SAND to HAND

- 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

- Conclusions

- 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 profile
- that 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 with
- Huge atmospheric neutrino detectors

- CP and magic trajectories

- P( e  )  = |cos 23ASe i  +  sin 23AA|2

- |AS | ~ sin212m  sin

-  L
-  lm

- For high energies

- lm   l0

- for 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

- Constant rho-approximation

- Reproduces all the features of oscillograms

- Vacuum
- mimicking

- Weak matter
- effects

- Varying 1-3 mixing

- Effect of 1-2 mixing

- Oscillograms for
- tan212 = 0.45

-    e

- Effect of 1-2 mixing

-      

- Oscillograms for
- tan212 = 0.45

- Amplitude condition

- Phase condition

- 1 layer:

- _MSW resonance condition_

- S(1)11  =  S(1)22

- Im (S11 S12*) = 0

- Im S(1)11  = 0

- sin  = 0

- cos 2m = 0

- 2 layers:

- X3  = 0

- S(2)11  =  S(2)22

- Im S(2)11  = 0

- Generalized resonance condition
- valid 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

- Collinearity condition

- 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 way
- to generalize
- parametric
- resonance
- condition

- from unitarity condition

- Structures of oscillograms

- 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 maxima
- Core-enhancement
- effect

- P = sin (4m – 2c)

- Saddle points at
- low energies

- Maxima at high energies
- above resonances

- Parametric oscillations

- ``Castle wall profile’’

- 

- m

- 

- m

- parametric resonance condition

-   oscillation
-     half-phases

- -  mixing angles

- V

- 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   X

- X, Y = X, Y (ii)

- X = (X1 , X2,  X3)

-  -Pauli matrices

-  P(ea)  =sin22msin2

- L
-  lm

- Oscillations in matter

- Oscillation
- Probability
- constant density

- Amplitude of
- oscillations

- half-phase 

- oscillatory factor

-  - mixing angle in matter

- lm(E, n )

- m(E, n )

- – oscillation length in matter

- _Conditions for maximal transition probability: P = 1_

- 1. Amplitude condition:

- sin22m= 1

- 2. Phase condition:

-  =    +  k

- MSW resonance condition

- m  

- lm  l

- In vacuum:

- lm = 2 /(H2 – H1)

- Non-constant density case

- 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 realizations
- which did not contained in the  original condition;
-  more physics content

- S11    S12
- S21    S22

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

-  x
- 0

- Structure of

- oscillograms

- Interference phase condition

- AS  =  0

-   - true (experimental) value of phase

- Determining 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  =

- isthe  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

- Parameterization

- UPMNS  = U23  I U13  I U12

-  I = diag (1,  1,  ei)

## ~

- sin2 2m   = 1

- Flavor mixing  is maximal
- Level split is minimal

- In resonance:

- l  =  l0 cos 2

- Vacuum
- oscillation
- length

- Refraction
- length

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

- l / l0

- sin2 2m

- sin22  = 0.08

- sin2212  = 0.825

- 

- 

- ~ n E

- Resonance width:    nR =  2nR tan2

- Resonance layer:      n =  nR + nR

- Resonance

- Detectors

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

- they encode important  information about
- earth structure and neutrino parameters

## Oscillation length in matter

- l =

-   4
- m2

- 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) = l0cos2

- l/sin2

- l=  l0/cos2)

- (maximum at

- ~ l

-   2
- H2 - H1

- lm =

- 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:
-      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 213    = 0.025  can be measured
-       with    = 45o  accuracy  ( 99% CL)