A. Yu. Smirnov

1
Neutrino oscillograms

of the Earth
A. Yu. Smirnov
International Centre for Theoretical Physics,
Trieste, Italy Institute for Nuclear Research,
RAS, Moscow, Russia
E. Akhmedov, M. Maltoni, A.S., JHEP 0705077
(2007) arXiv0804.1466 (hep-ph) A.S.
hep-ph/0610198.
Fermilab, April 16, 2008
2
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
3
P. Lipari , T. Ohlsson M. Chizhov, M. Maris, S
.Petcov T. Kajita
Neutrino images

ne ? nt
1 - Pee
Michele Maltoni
Contours of constant oscillation probability in
energy- nadir (or zenith) angle plane
oscillograms
4
Set-up''
qn

• zenith
• angle

Q p - qn
Q
- nadir angle
Mass mixing
core-crossing trajectory
Oscillations
Q 33o
Oscillograms
Oscillations in multilayer medium
core
flavor to flavor transitions
mantle
accelerator atmospheric cosmic neutrinos
5
Comments

The Earth is unique
Oscillograms are reality and this reality will
be with us forever
We know that neutrino masses and mixing are
non-zero
First of all we need to understand their
properties and physics behind
and then
Can we observe (reconstruct) these neutrino
images?
With which precision?
How can we use them?
6
Outline

1. Explaining oscillograms
Two effects
2. How oscillograms depend on unknown yet
neutrino parameters and Earth density profile
Interference of modes and CP-violation
3. How can we use them?
from SAND to HAND
7
Masses and mixing
ne
nm
nt

Ue32
?
n2
n3
Dm2sun
n1
mass
mass
Dm2atm
Dm2atm
n2
Dm2sun
n1
n3
Inverted mass hierarchy
Normal mass hierarchy
nf UPMNS nmass
UPMNS U23 Id U13 I-d U12
sin2q13 Ue32
Id diag (1, 1, eid)
Type of the mass hierarchy
Ue3
tan2q12 Ue22 / Ue12
tan2q23 Um32 / Ut32
CP-violating phase
8

Explaining Oscillograms
9
Two effects

Parametric enhancement of oscillations
Oscillations in matter with nearly constant
density
1 layer case mantle
mantle core - mantle
In two neutrino context
Interference of different modes of oscillations
10
Evolution equation

• ye
• ym
• yt

d Y d t
i H Y
Mixing matrix in vacuum
M M 2E
H V(t)
M is the mass matrix
M M U Mdiag2 U
Mdiag2 diag (m12, m22, m32)
V diag (Ve, 0, 0) effective potential
Mixing matrix
Eigenstates and eigenvalues of Hamiltonian
Diagonalization of H
Energy levels
11
Neutrino polarization vectors

Re ne nt, P Im ne nt,
ne ne - 1/2
ne nt,
Polarization vector
y
P y s/2 y
Evolution equation
d Y d t
d Y d t
i H Y
i (B s ) Y
2p lm
B (sin 2qm, 0, cos2qm)
Differentiating P and using equation of motion
d P dt
( B x P )
Coincides with equation for the electron spin
precession in the magnetic field
12
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 nene nZ 1/2 cos2qZ/2
13
Oscillations

14

15

16

17

18

19
Resonance enhancement

Source
Detector
ne
ne
Constant density
n
F(E)
F0(E)
Layer of length L
k p L/ l0
sin2 2q12 0.824
F (E) F0(E)
k 10
k 1
thick layer
thin layer
E/ER
E/ER
20

Small mixing angle
sin2 2q12 0.08
F (E) F0(E)
k 10
k 1
thick layer
thin layer
E/ER
E/ER
A Yu Smirnov
21
Mixing in matter

Diagonalization of the Hamiltonian
sin22q
sin22qm
( cos2q - 2EV/Dm2)2 sin 22q
V 2 GF ne
Mixing is maximal if
Dm2 2E
Resonance condition
V cos 2q
He Hm
sin22qm 1
Difference of the eigenvalues
Dm2 2E
( cos2q - 2EV/Dm2)2 sin22q
H2 - H1
22
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
F1
F2
q1m
q2m
1 2 3 4 5 6 7
Castle wall profile
23
Parametric resonance

si sinf i, ci cosfi, (i 1,2)
half-phases
s1c2cos2q1m s2c1cos2q2m 0
Akhmedov, A.S.
distance
distance
(f1 f2 p/2)
General case certain correlation between the
phases and mixing angles
c1 c2 0
24
Parametric enhancement in the Earth

1
mantle
2 3
core
2
1
4
mantle core mantle
3
mantle
4
25
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
26
Analytic vs. numeric

27
Evolution

collinearity condition
(parametric resonance condition)
28

29
Graphical

representation
a).
b).
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).
30
Parametric enhancement of 1-2 mode

1
mantle
core
3
4
2
2
mantle
4
3
1
31

Properties of oscillograms
Dependence on neutrino parameters and earth
density profile (tomography)
32
Sensitivity to density profile

twist
33
Sensitivity to density profile

Shift of border
34
Dependence on gradients

35
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
36
Other

channels
mass
hierarchy
For 2n system
normal ? inverted neutrino ? antineutrino
37

38
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
Under T-transformations
d ? -d
V ? V
ninitial ?? nfinal
39
CP-violation

d 60o
Standard parameterization
40

d 130o
41

d 315o
42
Interference of modes

Due to specific form of matter potential matrix
(only Vee 0)
P(n e ? nm) cos q23 ASe id sin q23AA2
atmospheric amplitude
solar amplitude
dependence on d and q23 is explicit
AS depends mainly on Dm122, q12
Factorization approximation
AA depends mainly on Dm132, q13
corrections of the order Dm122 /Dm13 2 , s132
p L l12m
AS i sin2q12m sin
For constant density
up to phase factors
p L l13m
AA i sin2q13m sin
43
Magic lines"
V. Barger, D. Marfatia, K Whisnant P. Huber, W.
Winter, A.S.

P(ne ? nm) c232AS2 s232AA2 2 s23
c23 AS AA cos(f d)
interference term
f arg (AS AA)
d - weak phase
strong phase
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
s23 sin q23
(for three layers more complicated condition)
44
Interference terms

If all parameters but d are known
d - true (experimental) value of phase
df - fit value
D P P(d) - P(df)
Interference term
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
45
Interference terms

For nm? nm 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 cosf sind
46
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
47

Int. phase line moves with d-change
Grid (domains) does not change with d
DP
48

DP
49

DP
50
Sensitivity to CP-phase

nm ? n e

• 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
51
Sensitivity to CP-phase

nm ? nm

• Contour plots for
• the probability
• difference
• P Pmax Pmin
• for d varying
• between 0 360o

Averaging?
52

Applications

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

Position of the mantle peak measure of 1-3
mixing
53
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
54
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
55
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
56
Detectors

50 kton iron calorimenter
INO Indian Neutrino observatory
HyperKamiokande
0.5 Megaton water Cherenkov detectors
UNO
Underwater detectors ANTARES, NEMO
E gt 30 50 GeV
Icecube (1000 Mton)
Reducing down 20 GeV?
TITAND (Totally Immersible Tank Assaying
Nuclear Decay)
2 Mt and more
Y. Suzuki..
57
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)
58
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)
59
Number of events

- angular resolution 3o
e-like events
- neutrino direction 10o
- energy resolution for E gt 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
60
From SAND to HAND

Huge Atmospheric Neutrino Detector
Measuring oscillograms with atmospheric neutrinos
with sensitivity to the resonance region
0.5 GeV
E gt 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?
61
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
62
CP and magic trajectories

P(n e ? nm) cos q23 ASe id sin q23AA2
solar amplitude
mainly, Dm122, q12
atmospheric amplitude
mainly, Dm132, q13
p L lm
AS sin2q12m sin
For high energies
lm ? l0
for trajectory with L l0
AS 0
P sin q23AA2
no dependence on d
For three layers more complicated condition
Contours of suppressed CP violation effects
Magic trajectories associated to
AA 0
63
Constant rho-approximation

Vacuum mimicking
Reproduces all the features of oscillograms
Weak matter effects
64

Varying 1-3 mixing
65
Effect of 1-2 mixing

nm ? n e
Oscillograms for tan2q12 0.45
66
Effect of 1-2 mixing

nm ? nm
Oscillograms for tan2q12 0.45
67
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
sin f 0
cos 2qm 0
another representation
Im (S11 S12) 0
Parametric resonance condition
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
68
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
69
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
70
Parametric oscillations

Castle wall profile
E. Kh. Akhmedov
- mixing angles
qim
f1
f2
also S. Petcov M. Chizhov
V
fi

• oscillation
• half-phases

q2m
q1m
d
Evolution matrix over one period (two layers)
s -Pauli matrices
X (X1 , X2, X3)
S Y i s X
X, Y X, Y (qi, fi)
Probability after n periods multiplying the
evolution matrices for each layer
P (1 X3 / X ) sin 2 Fn
Maximal depth of oscillations
parametric resonance condition
X3 0
71
Oscillations in matter

Oscillation Probability constant density
pL lm
P(ne -gt na) sin22qm sin2
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
sin 22qm 1
1. Amplitude condition
2. Phase condition
f p/2 pk
72
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
73
Structure of

oscillograms
74
Interference phase condition

Determining the CP-violation phase
d - true (experimental) value of phase
df - fit value
Compare probabilities
D P P(d) - P(df)
Pint(d) - Pint(df)
AS 0
(along the magic lines)
D P 0
AA 0
(f d ) - (f df) 2p k
int. phase condition
f (E, L) - ( d df)/2 p k
depends on d
In split approximation and for constant density
f Dm322 L/2E
75
Parameterization

Id diag (1, 1, eid)
UPMNS U23 Id U13 I-d U12
c12c13 s12c13
s13e-id - s12c23 -
c12s23s13eid c12c23 - s12s23s13eid
s23c13 s12s23 - c12c23 s13eid - c12s23
- s 12c23s13eid c23c13
UPMNS
c12 cos q12 , etc.
d is the Dirac CP violating phase
q12 is the solar mixing angle
q23 is the atmospheric mixing angle
q13 is the mixing angle restricted by
CHOOZ/PaloVerde experiments
76
Resonance
In resonance

sin2 2qm 1
sin2 2qm
Flavor mixing is maximal Level split is minimal
n
n
sin22q13 0.08
sin22q12 0.825
ln l0 cos 2q

Vacuum oscillation length

Refraction length
For large mixing cos 2q 0.4 the equality is
broken strongly coupled system ? shift of
frequencies.
ln / l0
n E
Resonance width DnR 2nR tan2q
Resonance layer n nR DnR
77
Detectors

structures of oscillograms are not accidental
but well determined by simple relations
they encode important information about earth
structure and neutrino parameters
78
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 l 0 /cos2q)
(maximum at
ln /sin2q
lm
2p H2 - H1
lm
l0
Resonance energy
ln
ln (ER) l 0 cos2q
E
ER
79
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 q23
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)