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Matter and Damping Effects inNeutrino Mixing and
OscillationsLicentiate Thesis
Mattias Blennow
Division of Mathematical Physics
Department of Physics
Royal Institute of Technology (KTH)
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.1/37
OutlineGeneral overview – History of neutrinos
Neutrino masses, mixing and oscillations
Neutrino oscillations in experiments
Solar neutrinos and the day-night effect
Exact solution to the two-flavor problem
Neutrino oscillations in dense matter
Damping effects in neutrino oscillations
Summary and conclusions
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.2/37
Neutrinos, what are they?Elementary particles
Spin 1/2
Chargeless
Comes in three flavors, νe, νµ, ντ , one for eachcharged lepton
Interacts only through the weak interaction
Initially believed to be massless – has a masswhich is at least a 100000 times less than theelectron mass
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.3/37
The Standard ModelTwo types of particles, quarks and leptons
Three types of interaction, strong, weak andelectromagnetic – does not include gravity
Three generations of particles, each containingtwo quarks and two leptons
Particles:
(
u νed e−
)
,
(
c νµs µ−
)
,
(
t ντb τ−
)
Force carriers:
γ, W±, Z0, Gi
Mass through Higgs mechanism and Yukawacouplings to the Higgs field
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.4/37
The birth of neutrino physicsPostulated in a letter by Wolfgang Pauli (1930)
Originally named “neutron” by Pauli
After Chadwick’s discovery of the neutron, EnricoFermi renamed Pauli’s particle to “neutrino”,meaning “small and neutral”
Initially belived to be massless
Experimentally detected (νe) by Clyde CowanJr. and Frederick Reines in 1956
The νµ detected by the Brookhaven NationalLaboratory in 1962
The ντ detected by the DONUT collaboration in2001
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.5/37
Are neutrinos oscillating?Neutrino oscillations first proposed by BrunoPontecorvo in 1957 in analogy to K0 − K0
oscillations
Theory further developed to neutrino flavoroscillations
Apparent deficiency in the measured fluxes ofsolar and atmospheric neutrinos attributed toneutrino oscillations
First oscillation evidence published by theSuper-Kamiokande collaboration in 1998
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.6/37
Massive neutrinos?Neutrinos are massless in the SM
Therefore, neutrino masses is a window forphysics beyond the SM
There are a number of different ways ofintroducing neutrino masses
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.7/37
Neutrino mixingIn general, the neutrino mass eigenfields neednot be the same as the fields participating in theweak interaction (flavor eigenstates) with thecharged lepton mass eigenfields
Mass and flavor eigenfields related by a unitarytransformation U , the leptonic mixing matrix, as
ν1ν2...
= U †
νeνµ...
Mass and flavor eigenstates related as
|να〉 =∑
i
U∗αi |νi〉
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.8/37
Standard parameterizationWith two neutrino flavors:
U =
(
c s
−s c
)
c = cos(θ), s = sin(θ)
With three neutrino flavors:
U =
c13c12 c13s12 s13e−iδ
−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13
s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13
cij = cos(θij), sij = sin(θij)
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.9/37
Neutrino oscillationsThe mass eigenstates are the eigenstates ofpropagation
The mass eigenstates obtain different phases ⇒quantum interference
The result: There is a probability of an initial αflavor neutrino to oscillate into another flavor β
The probability of finding the neutrino in the stateof flavor β after propagating the length L is thengiven by
Pαβ(L) =∑
i
∑
j
J ijαβ exp
(
−i∆m2
ij
2pL
)
,
where ∆m2ij = m2
i −m2j
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.10/37
Two-flavor oscillationsOnly one mixing parameter, θ
Only one mass squared difference, ∆m2
Oscillation formulas:
Pαα = Pββ = 1− sin2(2θ) sin2(
∆m2
4EL
)
Pαβ = Pβα = sin2(2θ) sin2(
∆m2
4EL
)
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.11/37
Three-flavor oscillationsFour mixing parameters, θ12, θ13, θ23, and δ
Two independent mass squared differences,∆m2
21 and ∆m231
No simple analytic formula for the neutrinooscillation probabilities
Formulas can be simplified in some special casesor Taylor expanded in small parameters
In general, Pαβ 6= Pβα (T-violation)
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.12/37
Current bounds on parametersThe current bounds on the neutrino oscillationparameters:
Parameter Best-fit 3σ confidence
∆m221 [10−5 eV2] 8.1 7.2-9.1
|∆m231| [10−3 eV2] 2.2 1.4-3.3
θ12 [◦] 33.2 28.6-38.1
θ23 [◦] 45.0 35.7-53.1
θ13 [◦] 0.0 ≤ 12.5M. Maltoni, et al., New J. Phys. 6, 122 (2004), hep-ph/0405172
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.13/37
Matter effectsInteraction with matter gives rise to an effectiveaddition to the Hamiltonian operator
All flavors interact via neutral-current interaction
Only νe interacts via charged-current interaction
e νe
νe e
W±
e e
να να
Z0
Affects the effective neutrino masses and mixing
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.14/37
Atmospheric neutrinosNeutrinos produced when cosmic rays hit theEarth’s atmosphere
Typical reaction:
π± −→ µ± +(−)
ν µ
followed by
µ± −→ e± +(−)
ν e +(−)
ν µ.
Typical neutrino energies are in the GeV range
At relatively low energies, φνµ/φνe ' 2 withoutoscillations
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.15/37
Super-KamiokandeSuccessor of the KamiokaNDE – the KamiokaNucleon Decay Experiment
Detects fast charged leptons from neutrinointeractions through Cherenkov radiation
Deep underground – reduces background
First experiment to present evidence for neutrinooscillations
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.16/37
The 1998 findingsAn apparent deficiency in the νµ flux for longerbaseline lengths
0
50
100
150
200
250
0
50
100
150
200
250
0
10
20
30
40
50
0
15
30
45
60
75
-1 -0.6 -0.2 0.2 0.6 10
40
80
120
160
200
-1 -0.6 -0.2 0.2 0.6 10
60
120
180
240
300
-1 -0.6 -0.2 0.2 0.6 10
20
40
60
80
100
e-likep < 0.4 GeV/c
e-likep > 0.4 GeV/c
e-likep < 2.5 GeV/c
e-likep > 2.5 GeV/c
µ-likep < 0.4 GeV/c
cosΘ
µ-likep > 0.4 GeV/c
cosΘ
µ-like
cosΘ
Partially Contained
cosΘ
sub-GeV multi-GeV
-1 -0.6 -0.2 0.2 0.6 10
25
50
75
100
125
Super-Kamiokande collaboration, Phys. Rev. Lett. 81, 1562 (1998), hep-ex/9807003
No deficiency or enhancement in the νe flux – νµoscillates mainly to ντ
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.17/37
Super-Kamiokande reanalysisIn a more recent analysis, neutrino oscillationsare compared to neutrino decay and neutrinodecoherence as descriptions of the νµdisappearance
Decay and decoherence are strongly disfavored
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 10 102
103
104 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 10 102
103
104
L/E (km/GeV)
Dat
a/P
redi
ctio
n (n
ull o
sc.)
Super-Kamiokande collaboration, Phys. Rev. Lett. 93, 101801 (2004), hep-ex/0404034
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.18/37
Solar neutrinosProduced as νe in thermonuclear reactions in thecenter of the Sun
The fluxes are predicted by the Standard SolarModel (SSM)
Homepage of John Bahcall, http://www.sns.ias.edu/˜jnb/
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.19/37
Solar neutrinos (2)Discrepancy between the detected and predictedfluxes (assuming no oscillations)
Homepage of John Bahcall, http://www.sns.ias.edu/˜jnb/
Discrepancy known as the “solar neutrinoproblem”
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.20/37
The SNOThe Sudbury Neutrino Observatory
Heavy water Cherenkov detector
Three types of detection:Elastic scatteringνα + e− → να + e−
Charged-currentνe + d→ e− + p+ p
Neutral-currentνα + d→ να + p+ n
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.21/37
Reactor neutrinosElectron anti-neutrinos produced in nuclearreactors
First neutrinos ever detected
Short-baseline – θ13 sensitivity
Long-baseline – ∆m212 sensitivity
Neutrino oscillation experiments mainly study thePee survival probability
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.22/37
CHOOZA short-baseline reactor neutrino experiment atthe Chooz power plant
Gives the current upper bound for θ13
No deficit of electron anti-neutrinos detected withany statistical significance
Analysis A
10-4
10-3
10-2
10-1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sin2(2θ)
δm2 (e
V2 )90% CL Kamiokande (multi-GeV)
90% CL Kamiokande (sub+multi-GeV)
νe → νx
90% CL
95% CL
CH
OO
Zco
llabo
ratio
n,P
hys.
Lett.
B46
6,41
5(1
999)
,hep-ex/9907037
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.23/37
KamLANDA long-baseline reactor neutrino experiment
The Japanese nuclear power plants (and somenon-Japanese power plants close to Japan) areused as a neutrino source
Average baseline of about 180 km
Detects an electron anti-neutrino disappearance
)2 (
eV2
m∆
-510
-410
θ 2tan
-110 1 10
KamLAND
95% C.L.
99% C.L.
99.73% C.L.
KamLAND best fit
Solar
95% C.L.
99% C.L.
99.73% C.L.
solar best fit
θ 2tan
0.2 0.3 0.4 0.5 0.6 0.7 0.8
)2 (
eV2
m∆KamLAND+Solar fluxes
95% C.L.
99% C.L.
99.73% C.L.
global best fit-510×4
-510×6
-510×8
-410×1
-410×1.2
KamLAND collaboration, Phys. Rev. Lett. 94, 081801 (2005), hep-ex/0406035
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.24/37
KamLAND L/E analysisAs for the Super-Kamiokande, a fit to oscillationsas well as to neutrino decay and neutrinodecoherence
The results favor neutrino oscillations
20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1
1.2
1.4
(km/MeV)eν/E0L
Rat
io2.6 MeV promptanalysis threshold
KamLAND databest-fit oscillationbest-fit decay
best-fit decoherence
KamLAND collaboration, Phys. Rev. Lett. 94, 081801 (2005), hep-ex/0406035
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.25/37
The day-night effectPaper 1 of the thesis
Earth matter effects on the solar neutrino flux
First treatment using three-flavor neutrinooscillations
Approximations:Third mass eigenstate essentially unaffectedConstant Earth matter densityNeutrinos produced at some average matterdensity inside the Sun
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.26/37
The day-night effect (2)The difference in the day and night electronneutrino survival probability scales essentially asc613
Three flavor day-night asymmetry at detectors
0.2 0.3 0.4 0.5
sin2θ
12
-5
-4.5
-4
-3.5
-3
log(
∆m2 /e
V2 )
θ13
= 0
θ13
= 9.2o
0
0.005
0.01
0.03
0.05
0.070.09
90% CL 95% CL 99% CL 99.73% CL
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.27/37
The day-night effect (3)Could be used to put constraints on θ13
Elaborated on by Akhmedov et.al. (arbitrarymatter density)
E. Kh. Akhmedov, JHEP 0405, 057 (2004), hep-ph/0412029
0.5 0.6 0.7 0.8 0.9 1
cos2θ
13
-0.01
0
0.01
0.02
0.03
0.04
0.05
AN
D
0.5 0.6 0.7 0.8 0.9 1
cos2θ
13
σ = 25% σSK
σ = 10% σSK
CHOOZ bound
CHOOZ bound
E. Kh. Akhmedov, hep-ph/0412029
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.28/37
Exact two-flavor solutionPaper 2 of the thesis
Deals with two-flavor neutrino oscillations inmatter with arbitrary density profile
The problem of neutrino oscillations in matter isrewritten as a real second order non-lineardifferential equation:
(p+Gp)2 = F (t)[G(1− p2)− p2]
Presents a series solution for p
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.29/37
Exact two-flavor solution (2)Convergence of the solution:
0 5 10 15 20E [GeV]
0
0.1
0.2
0.3
0.4
0.5
Pex
L = 3000 km
0 1000 2000 3000t [km]
0
0.2
0.4
Pex
5 10
15
20
25
30
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.30/37
Neutrinos in dense matterPaper 3 of the thesis
Three-flavor formulas → two-flavor formulasIf the mixing matrix has a zero entryIf one mass squared difference is zero
May be used as approximation in vacuum(θ13 → 0 or ∆m2
12 → 0)
In matter with high density, νe effectivelydecouples
We use degenerate perturbation theory to treatthe remaining νµ − ντ oscillations
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.31/37
Neutrinos in dense matter (2)Accuracy of the first-order approximation:
-3 -2 -1
log(VE / 1 eV2)
0
5
10
15
θ 13 [
o ]
< 0.25 % < < 0.5 % < < 1 %
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.32/37
Damping effectsPaper 4 of the thesis
Treatment of sub-leading effects on the probabilitylevel, e.g.,
Neutrino wave-packet decoherenceNeutrino quantum decoherenceNeutrino decayNeutrino absorption
Introduces “damping factors” in the neutrinooscillation formulas
Pαβ =∑
i
∑
j
DijJijαβ exp
(
−i∆m2
ij
2pL
)
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.33/37
Damping effects (2)The effect of damping on the neutrino survivalprobability Pαα:
0 2 4 6 8 10
Oscillation phase (∆)
0
0.2
0.4
0.6
0.8
1Su
rviv
al p
roba
bilit
yPure oscillationOscillation + decoherenceOscillation + decay IOscillation + decay II
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.34/37
Damping effects (3)Damping effects may alter the precisedetermination of the neutrino oscillationparameters
For example, the upper bound on θ13 fromshort-baseline reactor experiments (CHOOZ)
0 0.01 0.02 0.03 0.04 0.05Fit value of sin2 2Θ13
0
2
4
6
8
10
Fitv
alue
ofΣ
E@MeVD
sin22Θ13 -ΣE sensitivity for Reactor-I
1Σ
2Σ
3Σ
0 0.01 0.02 0.03 0.04Fit value of sin2 2Θ13
0
1
2
3
4
5
Fitv
alue
ofΣ
E@MeVD
sin22Θ13 -ΣE sensitivity for Reactor-II
1Σ
2Σ
3Σ
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.35/37
Damping effects (4)How to identify the specific effect?
Energy dependence is a better candidate:
10 20 30 40 50E @GeVD0
0.2
0.4
0.6
0.8P
ΜΜ
Decoherence
ΣE =H0,1,2,3,4,5L GeV
10 20 30 40 50E @GeVD0
0.2
0.4
0.6
0.8
PΜΜ
Decay
Α=H0,2,4,6,8,10L×10-4 GeV�����������������km
10 20 30 40 50E @GeVD0
0.2
0.4
0.6
0.8
PΜΜ
Oscillations
Ε=H0,1,2,3,4,5L×10-6 eV2
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.36/37
Summary and conclusionsThree-flavor effects in the solar day-nightasymmetry
Possible information on θ13 from day-nightdata
Non-linear differential equation and exact solutionof the two-flavor neutrino evolution in an arbitrarymatter density profile
The effective two-flavor scenario for matter withhigh density
Sub-leading effects on neutrino oscillationsentering on probability level
Implications for measurements of neutrinooscillation parametersHow to distinguish between different dampingeffects
Matter and Damping Effects in Neutrino Oscillations and Mixing – Mattias Blennow, KTH – Stockholm June 3, 2005 – p.37/37