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

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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)

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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

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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 !