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

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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

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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

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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)

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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.