1. Part II.
Flavor conversion
in matter

2. Matter effects

3. Variety of matter effects
Standard
Active-sterile
Non-standard
matter
matter
mixing
Non-standard
interactions
effects
effects
due to the CC
n e-scattering
exchange of
- vector
scalar
particles
unparticle
nm - nt potential
due to loop corrections
nn - scattering
Extreme conditions
high densities
hot medium
high magnetic fields,
polarized medium
- density fluctuations

4. Matter potential for ne nm ne e Elastic forward Potentials scattering
L. Wolfenstein, 1978
Matter potential
for ne nm ne e
Elastic forward
scattering
Potentials
Ve, Vm W
V ~ eV inside the Earth for E = 10 MeV e ne
difference of potentials
Ve- Vm = 2 GFne
Refraction index:
n - 1 = V / p
~ inside the Earth
Neutrino optics
n – 1 =
< inside the Sun
~ inside the neutron star 2p V
Refraction length:
l0 =

5. Matter potential Hint(n) = < y | Hint | y > = V n n ne e W GF 2
derivation
Matter potential
At low energies: neglect the inelastic scattering and absorption
effect is reduced to the elastic forward scattering (refraction) described
by the potential V:
y is the wave function
of the medium
Hint(n) = < y | Hint | y > = V n n ne e
CC interactions with electrons W GF 2
Hint = n gm(1 - g 5) n e gm(1 - g5) e e ne
< e g0 (1 - g 5 ) e> = ne
- the electron number density
For unpolarized
medium at rest:
< e g e> = ne v
< e g g5 e > = ne le
V = 2 GFne
- averaged polarization vector of e

6. Supernova neutrinos Diffusion Collective effects Flavor conversion
inside the star
Propagation
in vacuum
Oscillations
Inside the Earth

7. nn -scattering ne ne ne nb Refraction in neutrino gases Z0 Z0 nb nb nb
A = 2 GF (1 – ve vb )
velocities ne ne
(p)
t-channel
elastic forward scattering nb nb
(q) ne
J. Pantaleone ne
(p)
u-channel
can lead to the coherent effect
Momentum exchange  flavor exchange ne
(q) nb
 flavor mixing nb
Collective flavor transformations

8. Flavor exchange ne projection If the background is in the mixed state:
J. Pantaleone
S. Samuel
V.A. Kostelecky ne
projection
If the background is in
the mixed state: ne nb
|nib = Fie |ne + Fit | nt
coherent nb
Bet ~ Si Fie*Fit
background ne
sum over particles of bg.
w.f. give projections nt
projection
Contribution to the Hamiltonian
in the flavor basis
Flavor exchange between the beam
(probe) and background neutrinos
A. Friedland
C. Lunardini
|Fie|2
Fie*Fit
Hnn =
2 GF Si (1 – ve vib )
FieFit *
|Fit|2

9. ``Soft'' neutrino mass nL nR f Exchange by very light scalar
Recently: in the context
of MaVaN scenario ln
D B Kalplan, E. Nelson,
N. Weiner , K. M. Zurek
M. Cirelli, M.C. Gonzalez-Garcia,
C. Pena-Garay
V. Barger, P Huber, D. Marfatia nL nR f
mf ~ eV lf
f = e, u, d, n fL fR
chirality flip –
true mass:
msoft = ln lf nf /mf
lf ~ f/M Pl
mvac  mvac + msoft
In the evolution equation:
generated by some short range
physics (interactions) EW scale VEV
medium dependent
mass

10. Evolution equation in matter
dnf dt
i = Htot nf
nf =
H tot = H vac + V is the total Hamiltonian M2 2E
H vac =
is the vacuum (kinetic) part Ve
matter part
Ve= 2 G Fne
V =
Htot
Dm2 2E
Dm2 4E ne nm ne nm
cos 2q + Ve sin 2q
sin 2q d dt
i =
Dm2 4E

11. Eigenstates and mixing in matter
in vacuum:
in matter:
Effective
Hamiltonian H0
H = H0 + V
Eigenstates
n1, n2
n1m, n2m
depend
on ne, E
Eigenvalues
m12/2E , m22/2E
H1m, H2m
instantaneous nf
nmass q ne n1
n2m nf nH
n1m qm q n2
Mixing angle determines flavors
(flavor composition) of the eigenstates nm qm

12. Mixing in matter sin22q sin22qm = ( cos2q - 2EV/Dm2)2 + sin 22q Dm2 2E
Diagonalization of the Hamiltonian:
sin22q
sin22qm =
( cos2q - 2EV/Dm2) 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
H2 - H1 =
( cos2q - 2EV/Dm2)2 + sin22q

13. Resonance ~ ~ ln = l0 cos 2q In resonance: sin2 2qm = 1 sin2 2qm n
Flavor mixing is maximal
Level split is minimal n
sin22q12 = 0.825
sin22q13 = 0.08
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

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

15. Oscillation length in matter
4p E
Dm2
Oscillation
length in vacuum
ln =
Refraction
length 2p
2 GFne
- determines the phase produced
by interaction with matter
l0 =
ln /sin2q
(maximum at
ln = l 0 /cos2q) lm 2p
H2 - H1
lm = l0
Resonance energy:
~ ln
ln (ER) = l 0 cos2q ER E
for small mixing

16. Oscillations in matter
Propagation of neutrinos in the matter of the Earth
- solar neutrinos
- supernova neutrinos
- accelerator neutrinos, LBL

17. Oscillations in matter
qm(E, n) = constant
In uniform matter (constant density)
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

18. Oscillations in matter
Probability
constant density pL lm
P(ne -> 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
sin 22qm = 1
MSW resonance condition
1. Amplitude condition:
2. Phase condition:
f = p/2 + pk

19. Resonance enhancement of oscillations
oscillations determined by qm and lm (D H)
Constant density n
Detector
Source ne ne
F0(E)
F(E)
k = p L/ l0
Layer of length L
F (E)
F0(E)
sin2 2qm
sin2 2qm
sin2 2q = 0.824
sin2 2q = 0.824
k = 10
k = 1
thick layer
thin layer
E/ER
E/ER
A Yu Smirnov

20. Small mixing angle sin2 2q12 = 0.08 F (E) F0(E) k = 10 k = 1
thin layer
thick layer
E/ER
E/ER
A Yu Smirnov

21. Neutrinos and antineutrinos
Continuity:
neutrino and antineutrino semiplanes
normal and inverted hierarchy P n n
resonance
layer
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 n n
With increase of mixing:
q -> p/4
ER -> 0
ln / l0
DER -> ER0
A Yu Smirnov

22. The MSW - effect Adiabatic or partially adiabatic
flavor conversion of neutrinos
in medium with varying density
Flavor of the neutrino state
follows density change

23. Adiabatic conversion Admixtures of the eigenstates
do not change (adiabaticity)
Determined by mixing qm0
in the production point
Flavors of the eigenstates
follow the density change
Flavor: qm = qm(r(t))
Phase difference of the eigenstates
changes leading to oscillations
f = (H1 - H2) t

26. Resonance density
mixing is maximal

29. Evolution of eigenstates in matter
formalism
Evolution of eigenstates in matter
In non-uniform medium the Hamiltonian depends on time:
H tot = H tot(ne(t))
Its eigenstates, nmatter, do not split the equations of motion
dnf dt ne nm
i = Htot n f
nf =
Inserting nf = U(qm) nmatter
we get evolution equation for
n1m
n2m
nmatter =
n1m
n2m
dqm dt
n1m
n2m
H2 - H1 i d dt
i =
dqm dt
qm = qm(n e(t)) -i
transitions
The Hamiltonian is non-diagonal
no split of equations
n1m n2m

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

31. Adiabatic conversion formula
Initial state:
n(0) = ne = cosqm0 n1m(0) + sinqm0 n2m(0)
Adiabatic evolution
to the surface of
the Sun (zero density):
n1m(0) --> n1
n2m(0) --> n2
Final state:
-if
n(f) = cosqm0 n1 + sinqm0 n2 e
Probability
to find ne
averaged over
oscillations
P = |< ne| n(f) >|2 = (cosq cosqm0)2 + (sinq sinqm0)2
= 0.5[ 1 + cos 2qm0 cos 2q ]
P = sin2q + cos 2q cos2qm0

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

33. Spatial picture Oscillations Adiabatic conversion distance
survival probability
distance
A Yu Smirnov

34. Adiabaticity violation
Physical
picture
Adiabaticity violation
Transitions n1m <-> n2m occur
admixtures of the eigenstates change
Flavors of the eigenstates
follow the density change
Flavor: qm = qm(r(t))
Phase difference of the eigenstates
changes leading to oscillations
f = (H1 - H2) t
A Yu Smirnov

35. A Yu Smirnov

36. A Yu Smirnov

37. Resonance density
mixing is maximal
A Yu Smirnov

38. A Yu Smirnov

39. A Yu Smirnov

40. Graphic representation
Pure adiabatic conversion
Partialy adiabatic conversion
n e nm

41. Survival Probability Matter filter n source detector 1 2 1 - sin22q
Non-uniform medium
Matter filter n
source
detector
Vacuum
oscillations
P(averged over oscillations) 1 2
sin22q
Adiabatic
edge
Non-adiabatic
conversion
sin2q E
Non-oscillatory
adiabatic conversion
Resonance
at the highest
density
n(0) = ne = n2m n2
P = |< ne| n2 >|2 = sin2q
adiabaticity

42. Oscillations versus MSW
Both require mixing,
MSW is usually accompanying
by oscillations
What is essential difference between
oscillations and the MSW effect?
Oscillations
Adiabatic conversion
Vacuum or uniform medium
with constant parameters
Non-uniform medium or/and medium
with varying in time parameters
Phase difference increase
between the eigenstates
Change of mixing in medium =
change of flavor of the eigenstates f qm
Different
degrees of
freedom
In non-uniform medium:
interplay of both processes

43. Degrees of freedom n (t) = cosqa n1m + sinqa n2m e f (t) = H dt`
Effects associated
to different
degrees of freedom
-if(t)
Arbitrary state:
n (t) = cosqa n1m + sinqa n2m e
Adiabaticity
violation
qa = qa(t) - determines the admixtures of the eigenstates
f (t) is the phase difference between the two eigenstates t
Oscillations
f (t) = H dt`
Flavors (flavor composition) of the eigenstates
are determined by the mixing angle in matter
Adiabatic
conversion
<ne |n1m> = cosqm
<nm |n1m> = - sin qm
qa(t) + f (t) -> parametric effects, etc.
Combination of effects
qm(t) + f (t) -> ad. conv. + oscillations

44. MSW: realizations conditions: Solar Neutrinos in expanding Neutrinos
slowly enough changing density
crossing the resonance
enough matter width
conditions:
Solar
Neutrinos
Neutrinos
in expanding
Supernova
Universe
Neutrinos
Active - sterile neutrino
conversion in the Early
Universe
Large mixing
realization
Both large mixing
and small mixing
realizations
High energy neutrinos ?
Large lepton asymmetry
is required

45. Solar neutrinos 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
Large mixing MSW conversion
provides the solution of the solar
neutrino problem (Homestake, Kam…)

46. 3. Oscillations in matter of the Earth
Physics of conversion
1. Adiabaticity
lm - oscillation length
in matter;
h – height of the density
profile
Adiabaticity
parameter
lm(x)
4ph(x)
g (x) = ~ 10 -4
Corrections are g2
Also relevant for oscillation in the Earth where affect ~ g
2. Loss of coherence
3. Oscillations in matter of the Earth

47. As simple as this: For small or zero 1-3 mixing
For 2n - mixing
P = sin2q + cos2q cos2qm0
Explicitly:
cos2q - y
P = sin2q + ½cos2q 1 +
(cos2q - y)2 + sin22q
2EV0
Dm2
y =
V0 = 2 GFne0
ne0 – electron number density
in the production point
For SSM density distribution is valid with accuracy 10-7

48. Energy profile of the effect
Survival
probability
LMA MSW
Adiabatic
edge
npp
nBe
sin2 2q nB
Earth matter
effect
sin2q
III II I
ln / l0 ~ E
Conversion with
small oscillation
effect
Non-oscillatory
transition
Averaged oscillations
with small matter
effect
Conversion +
oscillations

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

50. Minimal width condition
C. Lunardini, A.S.
Matter width d = n x should be large enough to produce significant effect
Phase induced by matter:
2px l0
fmatter = = 2 G nx
should be of the order p p
2 GF
d =
For a given amount of matter (width) which density profile leads to
maximal conversion?
matter with constant density equal to resonance density
For small mixing angle, strong transition appears if p
2 GF tan2q
d =

51. ``Vacuum mimicking ’’ The oscillation probability in matter: 2px lm
L. Wolfenstein
E. Akhmedov
O. Yasuda
The oscillation probability in matter:
2px lm
P(nm) =
sin 22qm sin2
If the phase of oscillation is small: 2
2px lm
P(nm) =
sin 22qm
coincides with
vacuum oscillation
result 2
2px ln
= sin 22q
Matter effect appears in the next order in small phase
With decrease of distance both oscillation effect and the matter effect
disappear but matter effect disappears first
results from the fact that non-diagonal terms of the Hamiltonian do
not have matter contributions
A Yu Smirnov

52. Adiabaticity parameter
condition:
k >> 1
H2 - H1
k =
dqm dt
most crucial in the resonance where
the mixing angle in matter changes fast
DrR lR
kR =
DrR = hn tan2q
is the width of the resonance layer n
dn/dx
hn =
is the scale of density change
lR = ln/sin2q
is the oscillation length in resonance
Dm2 sin22q hn
2E cos2q
Explicitly:
kR =

53. Adiabaticity violation
Physics derivation
If density ne(t) changes fast
dqm dt
k ~ 1
~ | H2 - H1 |
the off-diagonal terms in the Hamiltonian can not be neglected
transitions
n1m n2m
Admixtures of in a given neutrino state change
n1m n2m
``Jump probability’’
penetration under barrier: DH En
P12= e H H2
En ~ 1/hn
is the energy associated
to change of parameter DH
(density) H1
-pkR/2
P12= e
Landau-Zener n

54. The best scenario

55. Evolution equations Ensemble of neutrino polarization vectors Pw
w = D m2 /2E
(in single angle approximation)
Total polarization vectors for neutrinos and antineutrinos
inf
inf
P = dw Pw
P = dw Pw
L = (0, 0, 1)
dt Pw =(- wB + lL + mD) x Pw
l = V = 2 GF ne
dt Pw =(+ wB + lL + mD) x Pw
m = 2 GF nn
where
D = P - P
- collective vector
Eliminating usual matter effect  transition to the rotating system
around L with frequency -l

56. Physics of conversion Averaged survival probability
at the surface of the Sun
P = sin2 q + cos 2q cos2 qm0
non-oscillatory part
oscillations
qm0 is the mixing angle in matter
in the production point
The depth of oscillations:
A = sin 2q sin 2qm0
At the detector:
Pdet = P + DPreg
Earth regeneration