neutrino pendulum a mechanical model for 3-flavor neutrino oscillations michael kobel (tu dresden)...
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![Page 1: Neutrino Pendulum A mechanical model for 3-flavor Neutrino Oscillations Michael Kobel (TU Dresden) Obertrubach, 5.10.2011 Schule für Astroteilchenphysik](https://reader033.vdocument.in/reader033/viewer/2022051417/5697c01b1a28abf838ccf7c6/html5/thumbnails/1.jpg)
Neutrino Pendulum
A mechanical model for3-flavor Neutrino Oscillations
Michael Kobel (TU Dresden)Obertrubach, 5.10.2011
Schule für Astroteilchenphysik
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• Free Oscillation of one pendulum: • 2 pendulums with same length ℓ, mass m
coupled by spring with strength k • 2 Eigenmodes
– Different eigenfrequencies = energies
Mode a (II + I) with
Mode b (II - I) with – Frequency (=energy) difference
increases with stronger coupling
– Coupling can be steered by varying k or d(we‘ll vary d in the following)
Model: Coupled Pendulums
2
22 2
mkd
22 a
g
2
222 b
d ℓ
+ +
a: I II
- +
b: I II
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Equations for Coupled Pendulums
k
k
K K
1
02
0
d1
ℓ
d2
ℓ
m m
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Equations of motion for l1 = l2 = l and d1 ≠ d2
22
2112221122222
2
112
2211122111112
dddddkdgmm
dddddkdgmm
k
k
ii
ijjik
ijji
mm
dd
m
dkdg :,:,:,:
2
22
2
22212
12111
222
2212
212
211
2
2
1
2
1
2
1
ij
ij
ijij
M
M
For K, B mesons damping important:
12 = 21 Damping in Coupling (K)1 , 2 Damping in Decay (B)
For Neutrinos damping negligible
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Undamped motion for l1 = l2 =: l and special case d1 = d2 =: d
12
222
2
212
112
kdgmm
kdgmm
22
22 2 :,:
mkdg
22
2
222
222
2
1
2
1
1
1
1
1
22
22
valueEwithvectorE
valueEwithvectorE
M
M
ij
ij
..
..
//
//
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Two bases in Hilbert-space
flavor-basis• eigenstates of flavor
• eigenstates of weak charge
• particles take part in weak interactions as flavor-eigenstates
• Examples:
– K0( su) or K0(s u)
– e, ,
mass-basis• eigenstates of mass
• well-defined lifetime
• Particles propagate through space-time as mass-eigenstates
• Examples:
– K0L , K0
S
– 1, 2, 3
tEtxpi eet )()(
• The coupling of flavor eigenstates leads to eigenstates with different masses e.g. for linear combination of 2 states:
a with ma2 = m2
b with mb2 = m2 + m2
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Correspondences
pendulum particlesLinear oscillation complex phase rotation
Eigenmodes fixed eigenfrequencies
Mass eigenstates fixed phase frequencies
Frequency differences different energies
Frequency differences eiEt ~ eim²t
different masses
One pendulum =
lin. combination of eigenmodes
Flavor eigenstate = lin. combination of mass eigenstates
|amplitude2| ~
total energy in oscillation
|amplitude2| ~detection probability
Beat-Frequency~ of eigenmodes
Flavor-Oscillation ~ m2 of mass eigenstates
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Three flavor Neutrino pendulumThree flavor Neutrino pendulum
coupled pendula for demonstrating
3-flavor neutrino mixing as realized in
nature
Idea: M.K.built 2004 at Uni Bonn,
extended 2006 at TU Dresden with variable mixing angles
and digital readouthttp://neutrinopendel.tu-dresden.de
Copies in: Hamburg, Münster, DESY(Zeuthen), Sussex …
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PMNS mixing matrix PMNS mixing matrix (w/o Majorana Phases) (w/o Majorana Phases) • 3 Mixing angles: 3 Mixing angles: θθ1212, , θθ2323, , θθ1313
• 1 CP-violating Dirac-Phase: 1 CP-violating Dirac-Phase: δδ (neglected in the following) (neglected in the following) • +2 mass differences +2 mass differences m2
12 , m223
3
2
1
1212
1212
1313
1313
2323
2323
100
0
0
0
010
0
0
0
001
cs
sc
ces
esc
cs
sci
ie
Θsolar, reactorθ13, δΘatmos, beam
3-flavor neutrino mixing3-flavor neutrino mixing
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flavor-oscillationsflavor-oscillations Each flavor (e.g. e) is sum of mass eigenstates (1, 2, 3)
Each mass eigenstate with fixed p has a different phase frequency i
exp(iit) = exp(iEit) = exp(i((p2+mi2)t) ~ exp(ipt+imi
2t/2p+…)
The differences ij |mi2 - mj
2| =:mij2 lead to flavor oscillations
mij2 determines the oscillation period
ij determines the oscillation amplitude
)(
)(. 2252
eVm
MeVEmL
ijij
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Current values Current values cf. global fit Th.Schwetz, M.Tortola, J.W.F Valle, arxiv 1108.1376cf. global fit Th.Schwetz, M.Tortola, J.W.F Valle, arxiv 1108.1376
Very near to tri/bi-maximal mixing (family symmetries…)23 = 45° 13 = 0° 12 = 35.3°
m223 = 2,42 x 10-3 eV2 m2
13 = 2,50 x 10-3 eV2 m212 = 0,076 x 10-5 eV2
„fast“ oscillation „slow“ oscillation
23= 46°± 3° 13= 6.5° ±1.5° (3.2)
12 = 34.0° ± 1.0°
)(MeVEkmL 3012)(MeVEkmL 123
Harrison, Perkins, Scott ’99,’02Z.Xing,’02, He, Zee, ’03, Koide ’03Chang, Kang, Kim ’04, Kang ’04
UPMNS
θsolar, reactorθ13, δθatmos, beam
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Realisation as coupled pendulaRealisation as coupled pendula
3
2e
1e
- +
+ - +
+ + +
1
2
3
1
2
3
normal inverted hierarchy
m
46/min
43/min42/min
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“Neutrino light” from the Sun (Super-Kamiokande)
Solar NeutrinosSolar Neutrinos
MeV7.2622He4 4 eep
Tcentral = 15E6 K
6.5E10 ve/cm2s
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Neutrino spectrum, uncertainties and sensitivities Neutrino spectrum, uncertainties and sensitivities (Bahcall et al., 2000)(Bahcall et al., 2000)
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Electron Neutrino Oscillation -> Electron Neutrino Oscillation ->
oscillation of e via and small m212 in
and always identical for 0
Vary modify fraction of e in and
e only eigenmode for =35°
http://neutrinopendel.tu-dresden.de (special high school thesis J. Pausch 2008)
smallersmaller
largerlarger
Possible range:20o <
< 90o
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Chlorine (Ray Davis, Homestake): Chlorine (Ray Davis, Homestake): Final Measurement resultFinal Measurement result
• Mean over 108 independent measurements: Only 32% of expected e
detected• Rdetected = 2,56 SNU
+- 0,16 (stat.) +- 0.16 (sys.)
• Solar Model Prediction
(new, 2005)R = 8,1 +- 1,2 SNU
• Significance:4.6 s.d.
37Ar Atoms / day
1.5
1.0
0.5
1 Solar Neutrino Unit (SNU) = s-1 = z.B. 1ab * cm-2 s-1
Main source of captured e: 8B
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Gallex (+ GNO): 1991-97 (+1999-2004)Gallex (+ GNO): 1991-97 (+1999-2004)
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Gallex / GNO resultsGallex / GNO results
Gallex/GNO: 69.3 ± 4.1 ± 3.6 SNU
SSM
Total: GALLEX/GNO & SAGE: 68.1 ± 3.75 SNU
Gallex,GNO
SSM prediction: 129 +8/-6 SNU* (BP98)
*) 1 SNU (solar neutrino unit) = 1 v-capture / 1036 target atoms
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50 000 t H20 Cherenkov detector
40 m high40 m
11146 Light-detektors(Photomultiplier)
50 cm
1 km deep inKamiokamine, Japan
Super Kamiokande Detektor in Japan
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Interpretation of measurements Interpretation of measurements
Bahcall:
Fraction detected:(uncertaintytheory-dominated)
Cl: (32 +- 6)%
H2O: (41 +- 7)%
Ga: (54 +- 5)%
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Solar Solar oscillations – the final proof 2002 oscillations – the final proof 2002
April 2002: SNO Experiment“Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino”http://arxiv.org/abs/nucl-ex/0204008
October 2002: Nobelprize forRaymond Davis (Homestake)
Masatoshi Koshiba (Superkamiokande)
December 2002:“First Results from KamLAND: Evidence for Reactor Anti-Neutrino Disappearance ” http://arxiv.org/abs/hep-ex/0212021
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Creighton Mine (Nickel)Sudbury, CanadaCreighton Mine (Nickel)Sudbury, Canada
Depth 2070m
1000t D2O1000t D2O
9500 PMTs9500 PMTs
SNO:SNO:Sudbury Neutrino ObservatorySudbury Neutrino Observatory
SNO:SNO:Sudbury Neutrino ObservatorySudbury Neutrino Observatory
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SNO – three independent informationsSNO – three independent informations
1000 t heavy water (D20)
CC-eppd
e
NCxx
npd
ES -- ee x x
)(14.0ES
CC
e
e
CC
NC
e
e
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They all arrive! They all arrive!
D2O data (April 2002)
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Reactor neutrinos: Do they really *oscillate*? Reactor neutrinos: Do they really *oscillate*?
Typical Energy: 2-6 MeV
Oscillation length(known today) L12 = 30km * E/MeV = 60 – 180 km
Until year 2001:Lmax = 1 km Only limits
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Ideal situation for KamLAND in KamiokaIdeal situation for KamLAND in Kamioka
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Most recent KamLAND result (2008)Most recent KamLAND result (2008)„Precision Measurement of Neutrino Oscillation Parameters with KamLAND“, Phys.Rev.Lett.100:221803,2008
L0 is the „effective“ baseline = flux-weighted average of distance = 180km
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KamLAND result (2008)KamLAND result (2008)„Precision Measurement of Neutrino Oscillation Parameters with KamLAND“, Phys.Rev.Lett.100:221803,2008
KamLAND + solar:
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ProblemsProblems
Historical Prejudice: mixing angles should be smallProblem: How to get large neutrino deficit w/ small mixing?
Today no problem: 2 mixing angles are large!
Knowing about large , but having 0
Effective 2-flavor mixing! min detection rate should be >= 50%
Problem: Observed rate of Homestake ~ 32% !
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Solution: MSW effect (1985)Solution: MSW effect (1985)Starting with e in sun via 4p 4He + 2e+ + 2 e + 27 MeV
transition to = not possible, since e not part of for =0
oscillation only to
effective 2-state oscillation: Psurv (e e) >= 50%
need additional effect for explaining Homestake (and SNO) measurement
MSW effect: oscillation enhancement in matter
+ + +
MSW Effect
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Landau-Zener Theory (1932)Landau-Zener Theory (1932)
http://pra.aps.org/pdf/PRA/v23/i6/p3107_1http://pra.aps.org/pdf/PRA/v23/i6/p3107_1
Example:
q:= Magnetic Field H2 Spin states m>0, m<0
q: = Electron density Ne(r) in sun2 Neutrino states e, (+ )
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Effect of an interaction between |1> and |2> Effect of an interaction between |1> and |2>
Example:1,2 :flavor states: e, (+ )
a,b: mass states: 1,
V: Neutrino Flavor Mixing via
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Transitions at level crossing Transitions at level crossing
Example for Neutrinos:|V12|2 m2 ~ 1/L
(oscillation length in matter)
dE/dt dm/dr ~ tan2
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Neutrino propagation in matter –MSW (Mikheyev, Smirnov, Wolfenstein) Effect
Origin: ve and vμ,τ have different interaction with mattere
e(ve can undergo CC and NC reaction, vμ,τ only NC!)
eee V
Emm
mmEdt
di
41
22
2241
22
22
:cossin
sincosVacuum:
In matter there is an additional potential in the equation of motion for ve → ve scattering (Flavor base)
eF NGE 24In matter:
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Solution can be written in terms of a mixing angle m in matter,which depends on electron density Ne, i.e. on position in sun
m
m
mm
mme
v
v
v
v
2
1
cossin
sincos
v
v
v
v e
mm
mm
m
m
cossin
sincos
2
1
For small vacuum mixing angle (1°):
For large vacuum mixing angle (32°):
Sun: surface resonance center
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Slide from Stephen Parke http://boudin.fnal.gov/AcLec/AcLecParke.html
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Simulation of MSW: Variation of Simulation of MSW: Variation of mm
smallersmaller
largerlarger
90°
45°
35°
20°
Modify m:
Sun’s center: ~ 90o , i.e. 2e“resonance” = crossing region: ~45°
Sun’s surface: ~35°, i.e. 2e
Adiabaticity:variation of Ne (i.e. mm , m)*slow* w.r.t. Lm (i.e. 1/mm
2)Hi
m = m2m + const
2m
1m
e
resonancesin2 2 = 0.825
~ NeESun’s surface
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Kamiokande
SAGE & GALLEX
Homestake
Status of Solar Oscillations ~2000Status of Solar Oscillations ~2000
LMA
LOW
SMA
Common prejudice in 2000: Small-Mixing-Angle “SMA”-MSW solution
In addition:
“Just so” observable at distance sun-earth
today’s value m2 = 8 x 10-5 eV2
L = 30 km x E/MeV
Very small m2 ~ 8 x 10-11 eV2 L = 30 x 106 km x E/MeV
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SNO mixing parameterSNO mixing parameter
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, K
e
e
(protons, He, , ,)
L=10~20 km
Primary cosmic rays
Low EnergyLimit : e = 2 : 1
E(GeV)
→
10-1 1 10 102
3D calculation
Mixture of e &
→e++e
10-1 1 10 102 E(GeV)
Flu
x ra
tio
ee
+ flux
2
Atmospheric neutrinosAtmospheric neutrinos
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Disappearance ofDisappearance of SuperKamiokande 2000:
look at e and from air showers:
• no deficit for e
• clear deficit for
• fully compatible with
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e µ
d
u
d
e-
u
u
d
W-
n p
electron event
myon event
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atmospheric neutrinosatmospheric neutrinosSuperKamiokande 2000:
described als
pendula:
e : weak coupling to
: weak coupling to e
strong coupling to
http://minos.phy.bnl.gov/nu-osc-lab/Superposition1.html
0
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Modify Modify
Non-maximal mixing of and
3 no longer eigenmode
http://neutrinopendel.tu-dresden.de(special high school thesis J. Pausch 2008)
smallersmaller
largerlarger
Possible range:30o <
< 60o
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Impact of Impact of on beam or atmospheric on beam or atmospheric
3sine
atmospheric or beam e appearance
„slow“ directly via m12 (weak coupling)
„fast“ modulation via with m23 (strong coupling)
13 = 6o
sin 13 = 0.1
sin2 213 = 0.04
0
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T2K (Tokai to Kamioka)T2K (Tokai to Kamioka)
Neutrino Super Beam
Off-Axis Detector Superkamiokande
Proton driver
First neutrinos produced on April 23rd 2009
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Takashi KobayashiJuly 14, 2011, CERN Colloquium
8 events remained
3. PID is e-like
Enhance e CC
49
7. Reconstructed neutrino energy < 1250 MeV
- Reject higher energy intrinsic beambackground from kaon decays
Signal Efficiency = 66%Background Rejection: 77% for beam ν
e
99% for NC
6 final candidate events remained!
Expected BG
1.5evts
Selection criteria & cut values are fixed before analysis. Unbiased
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A candidate
50
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Impact of Impact of on reactoron reactoree
e present in 3 sin e
e can now excite mode,
inducing fast modulation
Reactor e disappearance
Reactor neutrinos (2 MeV)
sin = 0.10= 6o
sin = 0.20= 12o
smallersmaller
largerlarger
Possible range:-6o <
< 6o
e nu mu nu
e nu mu nu
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Reactor Experiment (starting)Reactor Experiment (starting)
Double-Chooz sensitivity for (m2 = 2.0-2.5 10-3 eV2): sin2(213) < 0.03, 90% C.L.
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near far
Double CHOOZ: near and far detectorDouble CHOOZ: near and far detector
4E
Lmsin2sincos
4E
Lmsin2sin1)P(
2212
122
134
2312
132
ee
• max. sensitivity on 13: E ~ 4 MeV, Δmatm2 Losc/2 ~ 1.5 km
KamLAND
CHOOZ
sin2(212)sin2(213)
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Are neutrino pendulums a perfect model?Are neutrino pendulums a perfect model?
Few “features”Need “creative” sign convention, leading to
imperfection for understanding sequence of masses
Else perfect!
The END !