neutrino pendulum a mechanical model for 3-flavor neutrino oscillations michael kobel (tu dresden)...
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Neutrino Pendulum
A mechanical model for3-flavor Neutrino Oscillations
Michael Kobel (TU Dresden)Obertrubach, 5.10.2011
Schule für Astroteilchenphysik
• 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
Equations for Coupled Pendulums
k
k
K K
1
02
0
d1
ℓ
d2
ℓ
m m
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
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
..
..
//
//
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
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
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 …
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
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
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
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
“Neutrino light” from the Sun (Super-Kamiokande)
Solar NeutrinosSolar Neutrinos
MeV7.2622He4 4 eep
Tcentral = 15E6 K
6.5E10 ve/cm2s
Neutrino spectrum, uncertainties and sensitivities Neutrino spectrum, uncertainties and sensitivities (Bahcall et al., 2000)(Bahcall et al., 2000)
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
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
Gallex (+ GNO): 1991-97 (+1999-2004)Gallex (+ GNO): 1991-97 (+1999-2004)
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
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
Interpretation of measurements Interpretation of measurements
Bahcall:
Fraction detected:(uncertaintytheory-dominated)
Cl: (32 +- 6)%
H2O: (41 +- 7)%
Ga: (54 +- 5)%
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
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
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
They all arrive! They all arrive!
D2O data (April 2002)
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
Ideal situation for KamLAND in KamiokaIdeal situation for KamLAND in Kamioka
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
KamLAND result (2008)KamLAND result (2008)„Precision Measurement of Neutrino Oscillation Parameters with KamLAND“, Phys.Rev.Lett.100:221803,2008
KamLAND + solar:
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% !
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
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, (+ )
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
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
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:
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
Slide from Stephen Parke http://boudin.fnal.gov/AcLec/AcLecParke.html
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
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
SNO mixing parameterSNO mixing parameter
, 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
Disappearance ofDisappearance of SuperKamiokande 2000:
look at e and from air showers:
• no deficit for e
• clear deficit for
• fully compatible with
e µ
d
u
d
e-
u
u
d
W-
n p
electron event
myon event
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
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
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
T2K (Tokai to Kamioka)T2K (Tokai to Kamioka)
Neutrino Super Beam
Off-Axis Detector Superkamiokande
Proton driver
First neutrinos produced on April 23rd 2009
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
A candidate
50
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
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.
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)
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 !
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