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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)