Neutrino microscopes

1
Neutrino microscopes
Alexei Yu. Smirnov
International Centre for Theoretical Physics

• Oscillations in low density
• medium
• - Attenuation effect
• Next order corrections
• Improved perturbation
• theory
• - Neutrino microscopes

2
Surprises?
l0 1/GFn several 103 km
refraction length

1). Probing density profile structures (of the
Earth)
r gt ln 4pE/Dm2
oscillation length
r

the smaller the energy the smaller the scale
n
r E
r ltlt l0
2). Matter effect is the lowest order effect
3). Adiabatic perturbation theory works
4). Attenuation effect the better energy
resolution
the deeper structures can be probed

5). Improved perturbation theory strange
limitations
3

In the low density medium
P.C. de Holanda, Wei Liao, A.S, Nucl. Phys.
B702307 (2004) hep-ph/0404042 A. Ioannisian,
A.S., Phys. Rev. Lett. 93241801 (2004)
hep-ph/0404060 A. Ioannisian et al., Phys.
Rev.D71033006 (2005) hep-ph/0407138
4
In the low density medium

V(x) ltlt Dm2/ 2E
Potential ltlt kinetic energy
2 E V(x) D m2
e (x) (1 -3) 10-2
e (x) ltlt 1
Small parameter
perturbation theory in e (x)
Solar neutrinos
Inside the Earth
For LMA oscillation parameters applications to
Supernova neutrinos
Oscillations appear in the first order in e (x)
Relevant channel mass-to-flavor
n2 -gt ne
P2e sin2q freg
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e
- perturbation theory
For regeneration effect in the Earth
2EVE Dm2
e (r)
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theory in terms of mass eigenstates nmass
(n1, n2 )T
Weak matter effect mixing in matter mixing in
vacuum, qm q

Mass states mix in matter
cos q sin q - sin q cos q
nmass U nm
U
nm (n1m, n2m )T eigenstates in matter
q q (V) - mixing angle of mass states in
matter
e (x) sin 2q sin 2q

e (x) sin 2qm ( cos 2q
- e (x)) 2 sin2 2q
small
id nmass /dx H(x) nmass
Evolution equation

0 0 0 Dm(x)
H(x) U(x) U(x)
D m2 2E
Dm(x) ( cos 2q - e (x)) 2 sin2
2q
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S-matrix
S-matrix in the basis of mass eigenstates nmass
(n1, n2 )T

S(x0 -gt xf) (Un Dn Un ) ... (Uj Dj Uj
) ... (U1 D1 U1 )
In j-th layer
V
Mixing matrix of mass states
cos qj sin qj - sin qj cos qj
Uj
Vj
qj q (Vj)
Dj
Evolution matrix of the eigenstates in matter
Uj
1 0 0 e
j
n
Dj
i Fmj
x
x0
xf
Phase
Following procedure of the numerical
computations . . .
Fmj D x Dm(Vj)
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Each block can be reduced to
(Uj Dj Uj ) Dj Gj
contains info about density change leads to
transitions

0 1 1 0
i Fmj
Gj 0.5 (e - 1) sin2qj
O (e 2)
Dm(Vj) D x
S(x0 -gt xf) Dn ... Dj ... D1 Sj Dn ...
Dj1 Gj Dj-1 ... D1 O(Gj Gk) ...
Dj O(1)
expansion in power of Gj
Gj O (e )
Sj Fmj Sj Dx Dm(Vj) -gt dx Dm(x)
Limit n -gt infty, D x -gt 0
Sj D x -gt dx
xf x0
Adiabatic phase
Fm(x0 -gt xf) dx Dm(x)
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S-matrix

S-matrix in the basis of mass eigenstates nmass
(n1, n2 )T
1 0 0 e
S(x0 -gt xf)
- iFm(x0 -gt xf)
- iFm(x0 -gt x)
0 e e
0
xf x0
0.5 i sin2q dx V(x)
- iFm(x -gt xf)
O(V2)
The amplitude of the oscillation transition na
-gt nb
Aa-gtb (x0 -gt xf) lt nbS(x0 -gt xf)na gt
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Integral formula

Mass-to-flavor transition
P2e sin2q freg
n2 --gt ne
xf x0
Regeneration factor
freg 0.5 sin22q dx V(x) sin Fm(x -gt xf)
xf x0
xf x
Dm2 2E
2EV(y) 2 Dm2
freg 0.5 sin22q dx V(x) sin dy
cos 2q - - sin22q
V(x)
Fm(x -gt xf)
Integration limits
x0
xf
x
The phase is integrated from a given point to the
final point
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Analytic result

2E sin22q Dm2
freg sinF0/2 Sj 0 n-1 DVj
sinFj/2
j
DVj
j1
fj 0.5(F0 - Fj)
F0
Defining
Fj
2E sin22q Dm2
x
freg Sj 0 n-1
DVjsin2F0/2 cosfj - 0.5 sinF0 sinfj

fj

If fj is large - averaging effect. This happens
for remote structures, e.g. core
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Analytic vs. numerical results

Regeneration factor as function of the zenith
angle E 10 MeV, Dm2 6 10-5 eV2, tan2q
0.4
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Adiabatic perturbation theory
Adiabatic condition
lm(x) 4ph(x)
h(x) the height of distibution
g (x) ltlt 1
At the borders of layers h(x) -gt 0
Still the theory works!
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Attenuation effect
Effects on Supernova neutrinos Features of
regeneration of the solar neutrinos Oscillation
tomography of the Earth
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Sensitivity to density profile

For mass-to-flavor transition V(x) is integrated
with sin Fm(d)
d xf - x the distance from structure to the
detector
stronger averaging effects
weaker sensitivity to structure of density
profile
larger d
larger Fm(d)
Integration with the energy resolution function
R(E, E)
freg dE R(E, E) freg(E)
xf x0
The effect of averaging
freg 0.5 sin22q dx V(x) F(xf - x) sin
Fm(x -gt xf)
averaging factor
For box-like R(E, E) with width DE
ln E p d DE
p d DE ln E
F(d) sin
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Attenuation effect

The width of the first peak
d lt ln E/DE
Attenuation factor
F
ln is the oscillation length
The sensitivity to remote structures is
suppressed
Effect of the core of the Earth is suppressed
Small structures at the surface can produce
stronger effect
d, km
The better the energy resolution, the deeper
penetration
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Attenuation length

d ln E/DE
d 4pE E/DEDm2
DE/E
ln
d, km
Core
Solar neutrinos E 10 MeV
can not be seen
10 - 20
300 km
1500 3000

Supernova neutrinos E 30 MeV
can be seen
9000
900 km
10
ln is the oscillation length
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Averaging regeneration factor

Regeneration factor averaged over the energy
intervals E (9.5 - 10.5) MeV (a), and E (8 -
10) MeV (b).
No enhancement for core crossing trajectories
in spite of larger densities
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In e order
1

e
n2 -gt ne
e
n2
ne
xf
x0
n2
ne
neglect oscillations
x
flavor oscillations
1
ne -gt n2
e
e
n2
ne
x0
xf
x
flavor oscillations
n2
ne
neglect oscillations
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Next order corrections
- Next order correction - e increases with
energy
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Second order correction

Mass-to-flavor transition
n2 --gt ne
P2e sin2q freg
xf
x
xc
In symmetric density profile
xc - is the center of trajectory
freg sin22q sinFm(xc -gt xf) I cos2q I2
. . .
Expansion in I, where
xf xc
Fm(xc -gt x) adiabatic phase from the
center to a given point x
I dx V(x) cos Fm(xc -gt x)
Estimate
O(I) - term is absent (suppressed)
for trajectories where sinFm(xc -gt
xf) k p
2EVmax Dm2
I lt e max
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Relative errors
d (fappr - fexact) / f0
f0 0.5 ef sin22q

at the surface
A. Ioannisian et al, Phys. Rev. D (2005)
Second order
First order
For the neutrino trajectory which crosses the
center of the Earth
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Improving
theory further
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Improved perturbation theory

2EV Dm2
increases with energy
Improved perturbation theory expansion with
respect to some average potential V0
2E DV 2E(V - V0) Dm2 Dm2
e
The solution in the basis of eigenstates in
matter with potential V0
( n10, n20 )
The S-matrix for n10, n20, S0, can be obtained
from the S-matrix in the basis of mass
eigenstates by substitution
Fm(xc -gt x) do not change
V -gt DV, q -gt qm0
S0 S(DV, qm0)
qm0 qm0 (V0)
- mixing in matter with the potential V0
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Regeneration factor

P2e sin2q freg
For symmetric profile
freg e0 sin22qm0 sinFm(xc -gt xf) B1(e0)
sinFm(xc -gt xf)ID B2(e0)ID2
where
B1 sin22qm0 (cos 2q0 e0 cos 2qm0)
B2 sin22qm0/2(cos 2qm0 cos 2q0 - 2sin2q -
2e0 sin22qm0) I2
xf xc
2EV0 Dm2
e0
ID dx DV(x) cos Fm(xc -gt x)
xc - is the center of trajectory
q0 qm0 - q
Fm(xc -gt x) adiabatic phase from the
center to a given point x
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Relative errors

d (10 MeV/E)2
1
5
50 MeV x 25
10 MeV x 1
20 MeV x 4
0
20
1.67
2.5
3.0
N0e
The reduced error (in second order) d/E2 as
function of 1/E for different values of the
density shift N0e
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Expansion parameter

Essentially the parameter of expansion is the
integral ID
B1, B2 lt 1
Integrating by parts
ID 2E(Vf V0)/Dm2 sin Fm(xc xf) - -
2E/Dm2 dx dV(x)/dx sin Fm(xc -gt x)
xf xc
does not depend on shift of V
The dependence on V0 due to the boundary condition
Vf - potential at the surface
The strongest suppression for V0 Vf
(also related to the attenuation effect)
xc - is the center of trajectory
Fm(xc -gt x) adiabatic phase from the
center to a given point x
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Where we are now?
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Day-Night

asymmetry
2 (N D) N D
ACC
Binned Day-Night CC event asymmetries as
functions of the electron energy (statistical
uncertainties only)
ACC - 0.021 /- 0.063 /- 0.035
Expected (0.025 - 0.030)
Previous - 0.07
Gradient with energy OK
Systematic shift of the night Spectrum?
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Day-Night

asymmetry
Day and Night CC energy spectra (statistical
uncertainties only)
ACC freg /sin2q
Gradient with energy OK
Systematic shift of the night Spectrum?
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Summary
Integral formula for oscillations in low
density medium which is valid for arbitrary
density profile

For LMA parameters it can be applied to the
solar and supernova neutrino oscillations inside
the Earth -gt oscillation tomography of the
Earth probing structures of 20 2000 km size
Attenuation effect the sensitivity to remote
structures decreases. The better the energy
resolution the weeker attenuation
With next order corrections results can be
applied up to 40 50 MeV
Expansion parameter - integral ID improvement
when the shift potential V0 V(surface)
Neutrino microscopes gt large neutrino
telescopes with low energy thresholds and good
energy resolution
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Oscillations inside the Earth

1). Incoherent fluxes of n1 and n2 arrive at
the surface of the Earth
2). In matter the mass states oscillate
3). the mass-to-flavor transitions, e.g.
n2 --gt ne are relevant
Regeneration factor
P2e sin2q freg
Pee 0.5 1 cos 2qm0 cos 2q - cos2qm0freg

4). The oscillations proceed in the weak
matter regime
2EV(x) Dm2
e (x) ltlt 1
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Oscillations inside the Earth
Oscillations in multilayer medium
Solar and supernova neutrinos mass to
flavor transitions
n2
mantle
Accelerator neutrinos LBL experiments atmospheric
neutrinos flavor to flavor transitions
core
Regeneration of the ne flux
Variety of possibilities depending on -
trajectory, - neutrino energy and - channel of
oscillations
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Solar and SN neutrinos inside the Earth

1). Incoherent fluxes of n1 and n2 arrive at
the surface of the Earth
2). In matter the mass states oscillate
3). the mass-to-flavor transitions, e.g.
n2 --gt ne are relevant
Regeneration factor
P2e sin2q freg
Pee sin2q cos 2qm0 cos 2q - cos2qm0freg
4). The oscillations proceed in the weak
matter regime
2EV(x) Dm2
e (x) ltlt 1