1. Neutrino Astrophysics
The MSW effect
and
Neutrino Astrophysics
A. Yu. Smirnov
ICTP, Trieste & INR, Moscow
* Context
* Refraction, Resonance, Adiabaticity
* MSW: physical picture of the effect
* Large mixing MSW solution of the solar neutrino problem
* Supernova neutrinos and MSW effect

2. Context: Neutrino mass Neutrino mixing and Oscillations Matter effect
W. Pauli
E. Fermi
Neutrino mixing
and Oscillations
Matter effect
B. Pontecorvo,
Z. Maki, M. Nakagawa, S. Sakata
Neutrino
Refraction
L.Wolfenstein
Spectroscopy of
Solar Neutrinos
Homestake
J.N.Bahcall
G.T.Zatsepin, V.A. Kuzmin
Experiment
R. Davis Jr.,
D.S. Hammer,
K.S. Hoffman
A Yu Smirnov

3. References
[1] L. Wolfenstein, ``Neutrino oscillations in matter’’, Phys. Rev. D17, (1978)
[2] L. Wolfenstein, ``Effect of matter on neutrino oscillations’’, In Proc. of ``Neutrino -78’’,
Purdue Univ. C3 - C6.
[3] L. Wolfenstein, ``Neutrino oscillations and stellar collapse’’,
Phys. Rev. D20, (1979)
[4] S. P. Mikheyev and A. Yu. Smirnov, ``Resonance enhancement of oscillations in matter
and solar neutrino spectroscopy’’, Sov. J. Nucl. Phys. 42 (1985)
[5] S. P. Mikheyev and A. Yu. Smirnov, ``Resonance amplifications of n- oscillations
in matter and solar neutrino spectroscopy’’, Nuovo Cimento C9 (1986) 24.
[6] S. P. Mikheyev and A. Yu. Smirnov, ``Neutrino oscillations in variable-density
medium and n-bursts due to gravitational collapse of stars’’,
Sov. Phys. JETP, 64 (1986)
[7] S. P. Mikheyev and A. Yu. Smirnov, Proc. of the 6th Moriond workshop on
``Massive neutrinos in astrophysics and particle physics, Tignes, France,
eds. O Fackler and J. Tran Thanh Van, (1986) p.355.

4. Flavors, masses, mixing Mixing = ne nm nt n1 n2 n3 e m ns
Flavor neutrino states:
Mass eigenstates ne nm nt n1 n2 n3 m1 m2 m3 e m t
correspond to certain
charged leptons
Mixing
interact in pairs
Eigenstates of the
CC weak interactions
Flavor
states
Mass
eigenstates = ns
Sterile
neutrinos?
A Yu Smirnov

5. Mass spectrum and mixingne nm nt
|Ue3|2 n3 n2
Dm2sun n1
mass
Dm2atm
Dm2atm
mass
|Ue3|2 n2 n1 n3
Dm2sun
Normal mass hierarchy
(ordering)
Inverted mass hierarchy
(ordering)
Type of mass spectrum: with Hierarchy, Ordering, Degeneracy absolute mass scale
Type of the mass hierarchy: Normal, Inverted
Ue3 = ?
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6. Two aspects of mixing ne = cosq n1 + sinq n2 n2 = sinq ne + cosq nm
vacuum
mixing
angle
ne = cosq n1 + sinq n2
n2 = sinq ne + cosq nm
nm = - sinq n1 + cosq n2
inversely
n1 = cosq ne - sinq nm
coherent mixtures
of mass eigenstates
flavor composition of
the mass eigenstates n2 ne n2 n1
wave
packets n2 n1 nm n1
Flavors of eigenstates n2 ne n1
Interference of the parts of
wave packets with the same
flavor depends on the
phase difference Df
between n1 and n2
The relative phases
of the mass states
in ne and nm
are opposite n2 nm n1
A. Yu. Smirnov

7. Vacuum oscillations oscillations: n2 ne n1 Df = Dvphase t Df = 0 Dm2
Propagation in vacuum:
Flavors of mass eigenstates do not change
Determined by q
Admixtures of mass eigenstates
do not change: no n1 <-> n2 transitions n2 ne n1
Df = Dvphase t
Df = 0
Dm2 2E
Due to difference of masses n1 and n2
have different phase velocities:
Dvphase =
Dm2 = m22 - m12
Oscillation length:
oscillations:
ln = 2p/Dvphase = 4pE/Dm2
effects of the phase difference
increase which changes
the interference pattern
Amplitude (depth) of oscillations:
A = sin22q
A. Yu. Smirnov

8. Matter Effect: Refraction
L. Wolfenstein, 1978 ne e
Elastic forward
scattering
Potentials
Ve, Vm W
V ~ eV inside the Earth for E = 10 MeV ne
Difference of potentials is important
for ne nm : e
Ve- Vm = 2 GFne
Refraction index:
n - 1 = V / p
Refraction length:
~ inside the Earth
l0 = 2p / (Ve - Vm)
n - 1
< inside the Sun
= 2 p/GFne
~ inside the neutron star
focusing of neutrinos fluxes by stars
complete internal reflection, etc
Neutrino optics

9. Neutrino eigenstates in matter
in vacuum:
in matter:
Effective
Hamiltonean H0
H = H0 + V
V = Ve - Vm
Eigenstates
n1, n2
n1m, n2m
depend
on ne, E
m1, m2
m1m, m2m
Eigenvalues
m12/2E , m22/2E
H1m, H2m
Mixing in matter ne n1
n2m
is determined with respect
to eigenstates in matter
n1m q n2 nm qm qm
is the mixing angle in matter

10. Resonance ~ ~ In resonance: sin2 2qm sin2 2qm = 1 n n
Mixing in matter is maximal
Level split is minimal
sin2 2q = 0.08
sin2 2q = 0.825
ln = l0 cos 2q ~
Vacuum
oscillation
length ~
Refraction
length
For large mixing: cos 2q =
the equality is broken
the case of strongly coupled system
shift of frequencies
ln / l0
~ n E
Resonance width: DnR = 2nR tan2q
Resonance layer: n = nR + DnR
A Yu Smirnov

11. Level crossing n resonance Dependence of the neutrino eigenvalues
sin2 2q = 0.825 ne
n2m
Dependence of the neutrino eigenvalues
on the matter potential (density) nm
Large
mixing ln l0
2E V
Dm2 =
n1m
ln/ l0
V. Rubakov, private comm.
N. Cabibbo, Savonlinna 1985
H. Bethe, PRL 57 (1986) 1271
sin2 2q = 0.08 ne ln l0
= cos 2q
Small
mixing
n2m nm
ln/ l0
Crossing point - resonance
the level split in minimal
the oscillation length is maximal
n1m
For maximal mixing: at zero density
A Yu Smirnov

12. Two effects MSW Adiabatic Resonance enhancement (partially adiabatic)
neutrino conversion
Resonance enhancement
of neutrino oscillations
Density
profiles:
Variable density
Constant density
Change of the phase
difference between
neutrino eigenstates
Change of mixing, or
flavor of the neutrino
eigenstates
Degrees of
freedom:
In general:
Interplay of oscillations
and adiabatic conversion
MSW
A Yu Smirnov

13. Oscillations in matter
In uniform matter (constant density)
qm(E, n) = constant
mixing is constant
Flavors of the eigenstates do not change
Admixtures of matter eigenstates
do not change: no n1m <-> n2m transitions
Oscillations
Monotonous increase of the phase difference
between the eigenstates Dfm
as in vacuum
n2m ne
n1m
Dfm = 0
Dfm = (H2 - H1) L
Parameters of oscillations (depth and length)
are determined by mixing in matter
and by effective energy split in matter
sin22q, ln
sin22qm, lm

14. Resonance enhancement of oscillationsne ne
F0(E)
F(E)
Source
Layer of matter with constant density, length L
Detector
k = p L/ l0
thin layer
thick layer
F (E)
F0(E)
k = 1
k = 10
sin2 2q = 0.824
sin2 2q = 0.824
E/ER
E/ER
A Yu Smirnov

15. Resonance enhancement of oscillationsne ne
F0(E)
F(E)
Source
Layer of matter with constant density, length L
Detector
k = p L/ l0
thin layer
thick layer
F (E)
F0(E)
k = 1
k = 10
sin2 2q = 0.08
sin2 2q = 0.08
E/ER
E/ER
A Yu Smirnov

16. Resonance enhancement
layer
of oscillations
Continuity:
neutrino and antineutrino semiplanes
normal and inverted hierarchy P
Oscillations (amplitude of oscillations)
are enhanced in the resonance layer
ln / l0
E = (ER - DER) -- (ER + DER)
DER = ERtan 2q = ER0sin 2q
ER0 = Dm2 / 2V P
With increase of mixing:
q -> p/4
ER -> 0
ln / l0
DER -> ER0
A Yu Smirnov

17. MSW: adiabatic conversion
H = H(t) depends on time
Non-uniform matter
density changes on
the way of neutrinos:
qm = qm(n e(t))
mixing changes in the
course of propagation
n1m n2m are no more the
eigenstates of propagation
-> n1m <-> n2m transitions
ne = n e(t)
However
if the density changes slowly enough (adiabaticity condition)
n1m <-> n2m transitions can be neglected
Flavors of eigenstates change
according to the density change
determined by qm
Admixtures of the eigenstates,
n1m n2m, do not change
fixed by mixing in
the production point
Phase difference increases according to the level split
which changes with density
MSW
Effect is related to the change of flavors
of the neutrino eigenstates in matter with varying density

18. Adiabaticity DrR > lR External conditions (density) dqm
change slowly
so the system has time to
adjust itself
dqm dt
Adiabaticity condition
<< 1
H2 - H1
transitions between
the neutrino eigenstates
can be neglected
The eigenstates
propagate
independently
n1m <--> n2m
Crucial in the resonance layer:
- the mixing angle changes fast
- level splitting is minimal
if vacuum mixing
is small
DrR > lR
lR = ln/sin2q is the oscillation width in resonance
DrR = nR / (dn/dx)R tan2q is the width of the resonance layer
If vacuum mixing is large
the point of maximal adiabaticity violation
is shifted to larger dencities
n(a.v.) -> nR0 > nR
nR0 = Dm2/ 2 2 GF E

19. Adiabatic conversion and initial condition n0 < nR n0 > nR
The picture of conversion depends on how far from the resonance layer in the density
scale the neutrino is produced
n0 < nR
n0 > nR
n0 ~ nR
nR - n0 >> DnR
n0 - nR >> DnR
Interplay of
conversion and
oscillations
Oscillations with
small matter effect
Non-oscillatory
conversion
nR ~ 1/E
All three possibilities are realized for the solar neutrinos
in different energy ranges
A Yu Smirnov

20. Adiabatic conversion n1m <--> n2m n0 >> nR P = sin2 q
Non-oscillatory transition
n2m
n1m n2 n1
interference suppressed
Mixing suppressed
Resonance
n0 > nR
Adiabatic conversion + oscillations
n2m
n1m n2 n1
n0 < nR
Small matter corrections
n2m
n1m n2 n1 ne
A. Yu. Smirnov

21. The MSW effect
The picture of adiabatic conversion is universal in terms of variable y = (nR - n ) / DnR
(no explicit dependence on oscillation parameters density distribution, etc.)
Only initial value y0 matters.
production
point
y0 = - 5
resonance
survival probability
oscillation
band
averaged
probability
(nR - n) / DnR
(distance)
A Yu Smirnov

22. Adiabaticity violation
Fast density change
n2m n1m
n0 >> nR
n2m
n1m n2 n1
Resonance ne
Admixture of n1m increases

23. Solar Neutrinos n 4p + 2e- 4He + 2ne + 26.73 MeV Adiabatic conversion
in matter of the Sun
electron neutrinos are produced
F = cm-2 c-1
r : ( ) g/cc
total flux at the Earth
Oscillations
in vacuum n
Oscillations
in matter
of the Earth
J.N. Bahcall

24. Large mixing MSW solution
solar data
solar data + KamLAND
P. de Holanda, A.S.
sin2q13 = 0.0
Dm2 = eV2
Dm2 = eV2
tan2q = 0.40
tan2q = 0.41

25. Profile of the effect Adiabatic solution npp nBe nB
Survival probability
Earth matter
effect
sin2q I
III II
ln / l0 ~ E
Conversion with
small oscillation
effect
Non-oscillatory
transition
Oscillations with
small matter effect
Conversion +
oscillations
A Yu Smirnov

26. Conversion inside the Sun
tan2q = 0.41,
Dm2 = eV2
surface
core
survival probability
survival probability
resonance
E = 14 MeV
E = 2 MeV y y
distance
distance
survival probability
survival probability
E = 6 MeV
E = 0.86 MeV y y

27. LMA MSW solution An example: E = 10 MeV Resonance layer:
nR Ye = 20 g/cc
RR= 0.24 Rsun
In the production point:
sin2qm0 = 0.94
cos2 qm0 = 0.06
n2m
n1m
Evolution of the eigenstate n2m
Flavor of neutrino state follows density change

28. Inside the Earth. Regeneration
Oscillations
in the matter
of the Earth n2 n2
Regeneration of
the ne flux
core
Day - Night asymmetry
Variations of signal
during nights (zenith
angle dependence),
Seasonal variations
mantle
Spectrum distortion
Parametric effects for the
core crossing trajectories

29. Inside the Earth freg Averaging of oscillations,
divergency of the wave packets
incoherent fluxes of n1 and n2
arrive at the surface of the Earth
n1 and n2 oscillate inside the Earth
ln /l0
Regeneration of the ne flux
ln /l0 ~ 0.03
E = 10 MeV
P ~ sin2 q + freg
freg ~ 0.5 sin 22q ln /l0
The Day -Night asymmetry:
AND = freg/P ~ %
Oscillations
+ adiabatic
conversion
distance
A Yu Smirnov

30. Supernova neutrinos r ~ (1011 - 10 12 ) g/cc 0
E (ne) < E (ne) < E ( nx )
A Yu Smirnov

31. SN neutrinos and MSW effect
The MSW effect can be realized
in very large interval of
neutrino masses ( Dm2 ) and mixing
Dm2 = ( ) eV2
sin2 2q = ( )
Very sensitive way to search for new
(sterile) neutrino states
A way to probe
the hierarchy and
value of s13
Type of the mass hierarchy
The conversion effects
strongly depend on
Strength of the 1-3 mixing (s13)
Small mixing angle realization
of the MSW effect
In the case of normal mass hierarchy:
ne <-> nm /nt
almost completely
If 1-3 mixing is not too small
s132 >
F(ne) = F0( nm)
hard ne- spectrum
strong non-oscillatory conversion
is driven by 1-3 mixing
No earth matter effect in ne - channel
but in ne - channel
Neutronization ne - peak disappears

32. SN87A and the Earth matter effect
C.Lunardini
A.S. p
F(ne) = F0(ne) + p DF0
p = (1 - P1e) is the permutation factor
P1e is the probability of n1-> ne transition
inside the Earth
DF0 = F0(nm) - F0(ne)
p depends on distance traveled
by neutrinos inside the earth to a given
detector:
4363 km Kamioka
d = km IMB
10449 km Baksan p
Can partially explain the difference
of energy distributions of events
detected by Kamiokande and IMB:
at E ~ 40 MeV the signal is suppressed
at Kamikande and enhanced at IMB

33. Shock Wave Effect The shock wave can reach the region
relevant for the neutrino conversion
R.C. Schirato, G.M. Fuller, astro-ph/
r ~ 104 g/cc
During s from the beginning
of the burst
Influences neutrino conversion if
sin 2q13 > 10-5
The effects are in the neutrino
(antineutrino) for normal (inverted)
hierarchy:
h - resonance
change the number of events
R.C. Schirato, G.M. Fuller, astro-ph/
``wave of softening of spectrum’’
K. Takahashi et al, astro-ph/
delayed Earth matter effect
Density profile with shock wave propagation
at various times post-bounce
C.Lunardini, A.S., hep-ph/

34. Monitoring shock wave with neutrinos
G. Fuller
Studying effects of the shock wave
on the properties of neutrino burst
one can get (in principle) information on
time of propagation
velocity of propagation
shock wave revival time
density gradient in the front
size of the front
Can shed some light on
mechanism of explosion

35. Summary Two matter effects: Resonance enhancement of oscillations:
I. Resonance enhancement of oscillation
in matter with constant density
Two matter effects:
2. Adiabatic (quasi-adiabatic) conversion
in medium with varying density (MSW)
(a number of other matter effects exist)
Can be realized for neutrinos propagating
in the matter of the Earth (atmospheric
neutrinos, accelerator LBL experiments,
SN neutrinos ...)
Resonance enhancement
of oscillations:
Provides the solution of the solar neutrino
problem
Large mixing MSW effect:
Determination of oscillation parameters
Dm q12
Can be realized in supernova for 1-3 mixing
probe of 1-3 mixing, type of mass hierarchy
astrophysics, monitoring of a shock wave
Small mixing MSW effect:
A Yu Smirnov