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A. Yu. Smirnov ICTP, Trieste & INR, Moscow
I. MSW effect: main notions, physics of effect II. Realizations and observational status III. Matter effects on neutrino oscillations
- resonance enhancement - parametric effects - oscillations in low density medium ...
Adiabatic or partially adiabatic flavor conversion of neutrinos in medium with varying density
Flavor of the neutrino state follows density change
1. Main notions: - refraction- mixing in matter- resonance- adiabaticity
2. Dynamics - physical picture of the effect- graphic representation
Difference of potentials is important for νe νµ :
νe
νee
e
W
Ve- Vµ = 2 GFne
Elastic forward scattering
PotentialsVe, Vµ
L. Wolfenstein, 1978
Refraction index:
n - 1 = V / p
~ 10-20 inside the Earth< 10-18 inside the Sun~ 10-6 inside the neutron star
V ~ 10-13 eV inside the Earth for E = 10 MeV
n - 1
Neutrino optics focusing of neutrinos fluxes by stars complete internal reflection, etc
Refraction length:
l0 = 2π / (Ve - Vµ)= 2 π/GFne
θ
Mixing angle determines flavors (flavor composition) of the eigenstates
ν1m, ν2m
H = H0 + VEffectiveHamiltonian
Eigenstates depend on ne, E
Eigenvalues
H0
ν1, ν2
m12/2E , m2
2/2E H1m, H2m
νµ
νe
ν2m
ν1m
ν2
ν1
θm
νfνf
νfνf
νmassνmass
νHνH
θ
θm
sin2 2θm = 1
Flavor mixing is maximalLevel split is minimal
In resonance:
lν = l0 cos 2θ
Vacuumoscillation length
Refractionlength~~
For large mixing: cos 2θ ~ 0.4the equality is broken: strongly coupled systemshift of frequencieslν / l0
sin2 2θm
sin2 2θ = 0.08 sin2 2θ = 0.825
νν
~ n E
Resonance width: ∆nR = 2nR tan2θ
Resonance layer: n = nR + ∆nR Determines scale of ρ and E of strong flavor transition occurs
Manifestations depend on density profile
dθmdt
Adiabaticitycondition
H2 - H1
Crucial in the resonance layer: - the mixing changes fast - level splitting is minimal
∆rR > lR
lR = lν/sin2θ
∆rR = nR / (dn/dx)R tan2θ
External conditions (density) change slowly the system has time to adjust them
<< 1
Essence: transitions between the neutrino eigenstates can be neglected
ν1m <−−> ν2mThe eigenstatespropagate independently
if vacuum mixing is small
If vacuum mixing is large, the point of maximal adiabaticity violation is shifted to larger densities
n(a.v.) −> nR0 > nR
nR0 = ∆m2/ 2 2 GF E
Oscillation length in resonanceWidth of the res. layer
A Yu Smirnov
Admixtures of the eigenstatesdo not change (adiabaticity)
Flavors of the eigenstatesfollow the density change
Phase difference of the eigenstateschanges leading to oscillations
Determined by mixing θm0
in the production point
Flavor: θm = θm(ρ(t))
φ = (H1 - H2) t
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Resonance densitymixing is maximal
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The picture is universal in terms of variable y = (nR - n ) / ∆nR no explicit dependence on oscillation parameters, density distribution, etc.only initial value y0 matters
(nR - n) / ∆nR
surv
ival
pro
babi
lity
resonance
productionpointy0 = - 5
averagedprobability
oscillationband
(distance)
resonance layer
A Yu Smirnov
Flavors of the eigenstatesfollow the density change
Phase difference of the eigenstateschanges leading to oscillations
Flavor: θm = θm(ρ(t))
φ = (H1 - H2) t
Transitions ν1m <-> ν2m occur admixtures of the eigenstates change
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Resonance densitymixing is maximal
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Pure adiabatic conversion Partialy adiabatic conversion
νµ
ν e
Large mixingrealization Both large mixing
and small mixing realizations
slowly enough changing densitycrossing the resonanceenough matter width
Active - sterile neutrinoconversion in the Early Universe
High energy neutrinos ?Large lepton asymmetryis required
Resonance layer:nR Ye = 20 g/ccRR= 0.24 Rsun
In the production point: sin2θm
0 = 0.94 cos2 θm
0 = 0.06
E = 10 MeV
ν2mν1m
Large mixing MSW conversion provides the solution of the solarneutrino problem (Homestake, Kam…)
Adiabatic solution at large mixing
∆m2 = 6.3 10-5 eV2
tan2θ = 0.39∆m2 = 8.2 10-5eV2
tan2θ = 0.40
sin2θ13 = 0.0
KL
The KamLAND collaboration, hep-ex/0406035
solar vs. KamLAND solar and KamLAND
solar
A Yu Smirnov
lν / l0 ~ E
Surv
ival
pro
babi
lity
sin2θ
νpp
νBe
νB
IIIIII
Non-oscillatory transition
Conversion + oscillations
Oscillations with small matter effect
Conversion with small oscillation effect
Earth mattereffect
tan2θ = 0.41, ∆m2 = 7.3 10-5 eV2
y
surv
ival
pro
babi
lity
surv
ival
pro
babi
lity
surv
ival
pro
babi
lity
surv
ival
pro
babi
lity
core
E = 14 MeV
E = 6 MeV
surface
E = 2 MeV
E = 0.86 MeV
distance distancey
y y
resonance
- Very good global fit ofthe solar neutrino data;
- there is no statistically significant deviations from SSM + LMA description
- Very good global fit ofthe solar neutrino data;
- there is no statistically significant deviations from SSM + LMA description
- Homestake rate -smaller difference
- boron neutrino flux vs SNO
Agreement with KamLAND result in assumption of CPTNotice: KL signal waspredicted on the basis
of LMA
Agreement with KamLAND result in assumption of CPTNotice: KL signal waspredicted on the basis
of LMA Test of the mattereffectTest of the mattereffect
No viable alternative
Other possible effects beyond LMA are restrictedby 10% ???
Effect of the non standard interactions still can be large (at least in terms of the shift of allowed region of oscillationparameters)
Signatures:- upturn of spectrum- day-night effect
V -> aMSW VV -> aMSW V
SK+SNO:
< Pee > = 0.31 (3σ)+ 0.12- 0.08
<Pee> < 0.43 (3σ)<Pee> < 0.43 (3σ)
2ν - vacuum oscillations: <Pee> > 0.5
Global analysis of solar and reactor (KamLAND+CHOOZ)data with ∆m12
2 , sin2θ12 , aMSWunconstrained:G. L. Fogli and E. Lisisubm. to New J. of Physics
Notice: in the best fit point aMSW > 1reflects mismatch of the b.f. valuesfrom solar and KamLAND analysis
free parameter
precise description of the LMAconversion both in the Sun and in the Earth taking into account various corrections
precise description of the LMAconversion both in the Sun and in the Earth taking into account various corrections
estimation of accuracyof approximations madeestimation of accuracyof approximations made
accurate analyticexpressions for probabilities and observables as functions of oscillation parameters
accurate analyticexpressions for probabilities and observables as functions of oscillation parameters
Identification of the LMA solution opens new possibilities in
γ (x) = ~ 10 -7lm(x) 4πh(x)
ε (r) = ~ 0.02 2EVE∆m2
Adiabaticity violation-> adiabatic perturbation theory
∆rprod / h Integration over production region
Regeneration effect in the EarthSmall
par
amet
ers
A Yu Smirnov
E (νe) < E (νe) < E ( νx)
n ~ (1011 - 10 12) g/cc -> 0
New aspects of the MSW dynamicsNew aspects of the MSW dynamics
2). Small mixing MSW conversion 2). Small mixing MSW conversion due to 1 due to 1 -- 3 mixing3 mixing
1). Probe of whole level crossing scheme1). Probe of whole level crossing scheme
3).3). AdiabaticityAdiabaticity violation effectviolation effect
4). Time dependent 4). Time dependent adiabaticity adiabaticity violation violation -- shock wave effectshock wave effect
5). Earth matter effect and its interplay5). Earth matter effect and its interplaywith conversion inside the starwith conversion inside the star
The relative width of the resonance: 2tan 2θ13 < 1
Can be realized due to 1 - 3 mixing and ∆m213
Shift of the resonance is small: cos 2θ13 > 0.9
sin2 θ13 > 10 -3Adiabaticconversion
For the progenitor profile:
sin2 θ13 < 10 -5
Strong adiabaticityviolation
sin2 θ1310-6 10-5 10-4 10-3 10-2 10-1 100
excluded
Partial adiabaticity
Shock wave effectsDistortion of spectrum
New interestingeffects associated
to 1-3 mixing
Observable effects dependon type of mass hierarchy: normal - inverted Mass hierarchyMass hierarchy
sin2 θ13sin2 θ13
A Yu Smirnov
sin2 θ13 > 10 -3In the adiabatic case
The density in the production point is very large: n(0) >> nR
Mixing is strongly suppressed: admixture of ν1m is negligible ν(t) = ν2m interference is negligible
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Resonance densitymixing is maximal
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R.C. Schirato, G.M. Fuller, astro-ph/0205390 The shock wave can reach the regionrelevant for the neutrino conversion
ρ ~ 104 g/ccDuring 3 - 5 s from the beginningof the burstInfluences neutrino conversion ifsin 2θ13 > 10-5
``wave of softening of spectrum’’
The effects are in the neutrino (antineutrino) for normal (inverted)hierarchy:
change the number of events
delayed Earth matter effectC.Lunardini, A.S., hep-ph/0302033
R.C. Schirato, G.M. Fuller, astro-ph/0205390
K. Takahashi et al, astro-ph/0212195
Density profile with shock wave propagationat various times post-bounce
h - resonance
G. Fuller
time of propagationvelocity of propagationshock wave revival timedensity gradient in the frontsize of the front
Can shed some light onmechanism of explosion
Studying effects of the shock wave on the properties of neutrino burstone can get (in principle) information on
Steep front: breaks adiabaticityor make its violation stronger,
- after passing can be restored again- influence transitions
F(νe) = F0(νe) + p ∆F0
∆F0 = F0(νµ) - F0(νe)
p is the permutation factorp
The earth matter effect can partially explain the difference of Kamiokandeand IMB: spectra of events
p depends on distance traveled by neutrinos inside the earth toa given detector:
4363 km Kamioka d = 8535 km IMB
10449 km Baksan C.Lunardini, A.S.
One must take into account conversion effects of supernova neutrinos Conversion in the star
Earth matter effect
Normal hierarchy is preferableH. Minakata, H. Nunokawa, J Bahcall, D Spergel, A.S.
type of neutrinostates involved:
flavor-flavor, flavor-sterilemass-flavor
Density profileDensity profile
Properties of mediumProperties of medium
Channel of oscillationsChannel of oscillations
Polarized, moving,chemical composition,thermal bath
Presence of newinteractionPresence of newinteraction
Neutrino gas
NSI, flavor changing
Constant, monotonousperiodic,fluctuating
- Resonance enhancement of oscilations- Parametric effects- Oscillations in low density medium- Effect of density fluctuations
Regeneration of the νe flux
core
mantle
ν2
Solar and supernovaneutrinosmass to flavortransitions
Solar and supernovaneutrinosmass to flavortransitions
Variety of possibilitiesdepending on- trajectory, - neutrino energy and - channel of oscillations
Variety of possibilitiesdepending on- trajectory, - neutrino energy and - channel of oscillations
Accelerator neutrinosLBL experimentsatmospheric neutrinos:flavor to flavor transitions
Accelerator neutrinosLBL experimentsatmospheric neutrinos:flavor to flavor transitions
Oscillations inmultilayer mediumOscillations inmultilayer medium
A Yu Smirnov
Constant densityConstant density
Source DetectorF0(E) F(E)
F (E)F0(E)
E/ER E/ER
thin layer thick layer
k = π L/ l0 sin2 2θ = 0.824
ν
k = 1 k = 10
νe νe
sin2 2θ = 0.824
Layer of length L
oscillations determined by θm and lm (∆ H)
A Yu Smirnov
F (E)F0(E)
E/ERE/ER
thin layer thick layerk = 1 k = 10
sin2 2θ = 0.08
High energy neutrinos in the mantle of the Earth (constant density is a good first approximation)
AtmosphericneutrinosAtmosphericneutrinos
Accelerator neutrinos, LBL experiments
Accelerator neutrinos, LBL experiments
H. Shellman, T. Ohlsson,M. Lindner …
Enhancement associated to certain conditions for the phase of oscillations
Another way to get strong transitionNo large vacuum mixing and no matterenhancement of mixing or resonance conversion
``Castle wall profile’’
V Φ1 Φ2
Φ1 = Φ2 = π
VR
V. Ermilova V. Tsarev, V. ChechinE. AkhmedovP. Krastev, A.S., Q. Y. Liu, S.T. Petcov, M. Chizhov
``Castle wall profile’’
Φ1
θ1m
Φ2
distance
θ2m
Resonance condition:
s1c2cos2θ1m + s2c1cos2θ2
m = 0
oscillation phases
mixing angles
si = sinΦι/2, ci = cosΦι/2, (i = 1,2)
V
d
also S. Petcov M. Chizhov
Simplest realization:
= maximal depth of oscillations
c1 = c2 = 0 Φ1 = Φ2 = πΦ1 = Φ2 = πIn general, certain correlation betweenthe phases and mixing angles
general
Φ1 = Φ2 = π
E. Kh. Akhmedov
Contours of constant up - down asymmetry of the e -like events
A U/D = 2 U - DU + D
U: cos ΘZ = (-1.0 - -0.6)
s232 = 0.5
5.0 5.8 6.77.5
8.3 8.8
CHOOZ
AU/D %
Akhmedov, A. Dighe,P. Lipari, A.S. Nucl. Phys.
Multi-GeV e-like events
For ∆m2 = 2 10 -3 eV2
D: cos ΘZ = (0.6 - 1.0)
maximal effect (~9%) is expected for the sample with 2 times larger energies than the present Multi-GeV sample
core
mantle
mantle
mantle core mantle
A. N. Ioannisian, A. Shep-ph/0404 060
ε (x) = << 1
V(x) << ∆ m2/ 2E V(x) << ∆ m2/ 2E
Potential << kinetic energy
For LMA oscillation parametersapplications to
2 E V(x)∆ m2
Small parameter:
perturbation theory in ε (x)ε (x) ~ (1 -3) 10-2
Relevant channel :mass-to-flavor ν2 −> νe
Oscillations appear in the firstorder in ε (x)
E. Akhmedov, et al
freg = 0.5 sin22θ dx V(x) sin Φm(x -> xf) xf
x0
P2e = sin2θ + fregP2e = sin2θ + fregMass-to-flavortransition:
ν2 --> νeν2 --> νe
xf
x0
xf
x
x
V(x)Φm(x -> xf)
x0 xfIntegrationlimits:
The phase is integrated from a given point to the final point
∆m2
2E 2EV(y) 2∆m2freg = 0.5 sin22θ dx V(x) sin dy cos 2θ - - sin22θ
Regeneration factor
I = dx V(x) cos Φm(xc -> x) xf
xc
P2e = sin2θ + fregP2e = sin2θ + fregMass-to-flavor transition:
ν2 --> νeν2 --> νe
Φm(xc -> x) adiabatic phase from the center to a given point x
2EVmax∆m2
freg = sin22θ sinΦm(xc -> xf) I + cos2θ I2 + . . .
In symmetric density profilexc - is the centerof trajectory
xf
xcx
Expansion in I, where
O(I) - term is absent (suppressed) for trajectories where sinΦm(xc -> xf) = k π
I < = ε max
Estimate:
Improved perturbation theory: expansion with respect to someaverage potential V0
freg = 0.5 sin22θ dx V(x) F(xf - x) sin Φm(x -> xf) xf
x0
For mass-to-flavor transition V(x) is integrated with sin Φm(d) d = xf - x the distance from structure to the detector
larger dlarger d larger Φm(d)larger Φm(d)stronger averaging effects
stronger averaging effects
weaker sensitivityto structure of density profile
weaker sensitivityto structure of density profile
Integration with the energy resolution function R(E, E’): freg = dE’ R(E, E’) freg(E’)
The effect of averaging:
For box-like R(E, E’) with width ∆E:
F(d) = sin lν E π d ∆E
π d ∆E lν E
averaging factor
F
d, km
The width of the first peak
d < lν E/∆Ed < lν E/∆E
lν 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
The better the energy resolution, the deeper penetration
Attenuation factor
dθm(x’)dx’
x
x0
c(x) - the amplitude of transition between the eigenstates in matter (adiabaticity violation effect)
c(x) = - dx’ exp i dx’’ ∆m(x’’)x
x’
phase Φ(x’ -> x)
freg = ε (R) sin22θ sin2 [Φm(x0-> xf)/2] + sin 2θ Re{c(x0 -> xf)}
Regeneration factor: freg = P2e - sin2θ
ε (R) = is the parameter at the surface of the Earth2EV(R) ∆m2
If adiabaticity is conserved the regeneration depends on the potential V(R) at the surface and total adiabatic phase
∆m(x) = ( cos 2θ − ε (x)) 2 + sin2 2θ∆ m2
2E
P. De Holanda, Wei Liao, A.S
φj = φj =
freg = sinΦ0/2 Σj = 0 …n-1 ∆Vj sinΦj/2
φj = 0.5(Φ0 - Φj)
2E sin22θ∆m2
freg =
Σj = 0 …n-1 ∆Vj[sin2Φ0/2 cosφj - 0.5 sinΦ0 sinφj]
Defining
2E sin22θ∆m2 x
jj+1
∆Vj
Φ0
Φj
φjIf φj is large - averaging effect. This happens for remote structures, e.g. core
Effect of shells at small depth (~ 10 km ) is important.For small cos θZ - interference of contributions from
different shells - oscillatory behaviour of freg For large cos θZ - the distance is small and they can be accounted as one layer.
Regeneration factor as function of the zenith angleE = 10 MeV, ∆m2 = 6 10-5 eV2, tan2θ = 0.4
H = HMSW + δV(r) /2 σ3H = HMSW + δV(r) /2 σ3
randomly fluctuatingterm
Successive neutrinos see different density profiles. Conversion probabilities should be averaged over noise (different profiles)
Equation for density matrix ρaveraged over the noise
Delta correlated white noise:
δV(r) = ξ V(r)
< δV(r1) δV(r2)> = ξ2 <V >2 2 τ δ(r1- r2) < δV(r1) δV(r2)> = ξ2 <V >2 2 τ δ(r1- r2)
Decoherence: P -> 1/2
correlation length
dρdt = -i [HMSW, ρ] + ξ2<V >2 τ [σ3 ,[σ3 , ρ] ]
8% 4% 2%0% ξ
M. Guzzo, P. De Holanda, N Reggiani
A.B. Balantekin, F.N. Loreti; C Burges
∆matm2
Solar vs KamLAND Solar & KamLAND
M Guzzo, P.C. De Holanda, N Reggiani
Density fluctuation effect: shift of the allowed region to smaller ∆m2
Relevant correlations length~ 10 - 100 km
Fluctuations in the resonance layer R ~ (0.2 - 0.3) RSun
For ξ = 5 - 8 %new solution at small ∆m2 ?
Bound: ξ < 5 %, (1σ)Bound: ξ < 5 %, (1σ)
A.B. Balantekin, H. Yuksel
A Yu Smirnov
Large mixing MSW provides the solution of the solar neutrino problem; - it leads to determination of ∆m12
2 and θ12We have now detailed physical picture of the effect and itsvery precise analytical description both in the Sun and in the Earth
Large mixing MSW provides the solution of the solar neutrino problem; - it leads to determination of ∆m12
2 and θ12We have now detailed physical picture of the effect and itsvery precise analytical description both in the Sun and in the Earth
Small mixing MSW driven by 1-3 mixing can be realized for SN neutrinos - gives information on 1-3 mixing and type of mass hierarchy. - opens a unique possibility to perform monitoring of shock wave
Small mixing MSW driven by 1-3 mixing can be realized for SN neutrinos - gives information on 1-3 mixing and type of mass hierarchy. - opens a unique possibility to perform monitoring of shock wave
A number of matter effects can be associated to neutrino propagation inside the Earth:
- resonance enhancement of oscillations - parametric effects in multi-layer medium- effects in low density medium, attenuation of the removed structures, etc. (solar and atmospheric neutrinos)
A number of matter effects can be associated to neutrino propagation inside the Earth:
- resonance enhancement of oscillations - parametric effects in multi-layer medium- effects in low density medium, attenuation of the removed structures, etc. (solar and atmospheric neutrinos)
MSW - adiabatic (or partially-adiabatic) conversion in medium with varying density
MSW - adiabatic (or partially-adiabatic) conversion in medium with varying density
θ12 + θC = 45οθ12 + θC = 45ο
- physics of sub-leading effectscomplex dynamics of severalsub-leading effects
- matter in extreme conditions- beyond refraction- collective effects - matter effects due to new
interactions
Further studies of the effects in solar atmospheric neutrinos and in accelerator experiments
Very rich program: -tests of MSW and matter effects - detection of sub-leading effects- searches for new neutrino states - new neutrino interactions etc.