Neutrino Pendulum A mechanical model for 3-flavor Neutrino Oscillations Michael Kobel (TU Dresden) Obertrubach, 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 d ℓ + + a: I II - + b: I II

Equations for Coupled Pendulums k kk KK 11 0 22 0 d1d1 ℓ d2d2 ℓ mm  

Equations of motion for l 1 = l 2 = l and d 1 ≠ d 2 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 l 1 = l 2 =: l and special case d 1 = d 2 =: d

Two bases in Hilbert-space flavor-basis eigenstates of flavor eigenstates of weak charge particles take part in weak interactions as flavor-eigenstates Examples: –  K 0 ( s  u) or K 0 (  s u) – e, ,  mass-basis eigenstates of mass well-defined lifetime Particles propagate through space- time as mass-eigenstates Examples: – K 0 L, K 0 S – 1, 2, 3 The coupling of flavor eigenstates leads to eigenstates with different masses e.g. for linear combination of 2 states:  a       with m a 2 = m 2 b        with m b 2 = m 2 +  m 2

Correspondences pendulumparticles Linear oscillationcomplex phase rotation Eigenmodes  fixed eigenfrequencies Mass eigenstates  fixed phase frequencies Frequency differences    different energies Frequency differences e i  Et ~ e i  m²t  different masses One pendulum = lin. combination of eigenmodes Flavor eigenstate = lin. combination of mass eigenstates |amplitude 2 | ~ total energy in oscillation |amplitude 2 | ~ detection probability Beat-Frequency ~  of eigenmodes Flavor-Oscillation ~  m 2 of mass eigenstates

Three 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 readout Copies in: Hamburg, Münster, DESY(Zeuthen), Sussex …

PMNS mixing matrix (w/o Majorana Phases) PMNS mixing matrix (w/o Majorana Phases) 3 Mixing angles: θ 12, θ 23, θ 13 3 Mixing angles: θ 12, θ 23, θ 13 1 CP-violating Dirac-Phase: δ (neglected in the following) 1 CP-violating Dirac-Phase: δ (neglected in the following) +2 mass differences +2 mass differences  m 2 12,  m 2 23 Θ solar, reactor θ 13, δΘ atmos, beam 3-flavor neutrino mixing

 flavor-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(i  i t) = exp(iE i t) = exp(i(  (p 2 +m i 2 )t) ~ exp(ipt+im i 2 t/2p+…) The differences  ij   |m i 2 - m j 2 | =:  m ij 2 lead to flavor oscillations  m ij 2 determines the oscillation period  ij determines the oscillation amplitude

Current values cf. global fit Th.Schwetz, M.Tortola, J.W.F Valle, arxiv Very near to tri/bi-maximal mixing (family symmetries…)  23 = 45°  13 = 0°  12 = 35.3°  m 2 23 = 2,42 x eV 2  m 2 13 = 2,50 x eV 2  m 2 12 = 0,076 x eV 2 „fast“ oscillation„slow“ oscillation  23  = 46°± 3°  13  = 6.5° ±1.5° (3.2  )  12 = 34.0° ± 1.0° Harrison, Perkins, Scott ’99,’02 Z.Xing,’02, He, Zee, ’03, Koide ’03 Chang, Kang, Kim ’04, Kang ’04 U PMNS  θ solar, reactor θ 13, δθ atmos, beam

Realisation as coupled pendula 3        2   e       1   e       normal inverted hierarchy m  46/min 43/min 42/min   

“Neutrino light” from the Sun (Super-Kamiokande) Solar Neutrinos T central = 15E6 K 6.5E10 v e /cm 2 s

Neutrino spectrum, uncertainties and sensitivities (Bahcall et al., 2000)

Electron Neutrino Oscillation ->   oscillation of e via   and small  m 2 12 in     and  always identical for    0 Vary    modify fraction of e in  and    e       only eigenmode for   =35° (special high school thesis J. Pausch 2008)   smaller   larger    Possible range: 20 o <    < 90 o

Chlorine (Ray Davis, Homestake): Final Measurement result Mean over 108 independent measurements: Only 32% of expected e detected R detected = 2,56 SNU +- 0,16 (stat.) (sys.) Solar Model Prediction (new, 2005) R = 8,1 +- 1,2 SNU Significance: 4.6 s.d. 37 Ar Atoms / day Solar Neutrino Unit (SNU) =   s -1 = z.B. 1ab *   cm -2 s -1 Main source of captured e  : 8 B

Gallex (+ GNO): ( )

Gallex / 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: /-6 SNU * (BP98) *) 1 SNU (solar neutrino unit) = 1 v-capture / target atoms

t H 2 0 Cherenkov detector 40 m high 40 m  Light- detektors (Photomultiplier) 50 cm  1 km deep in Kamioka mine, Japan Super Kamiokande Detektor in Japan

Interpretation of measurements Bahcall: Fraction detected: (uncertainty theory-dominated) Cl: ( )% H 2 O: ( )% Ga: ( )%

Solar oscillations – the final proof 2002 April 2002: SNO Experiment “Direct Evidence for Neutrino Flavor Transformation from Neutral- Current Interactions in the Sudbury Neutrino” October 2002: Nobelprize for Raymond Davis (Homestake) Masatoshi Koshiba (Superkamiokande) December 2002: “First Results from KamLAND: Evidence for Reactor Anti-Neutrino Disappearance ”

Creighton Mine (Nickel) Sudbury, Canada Depth 2070m 1000t D 2 O 9500 PMTs SNO: Sudbury Neutrino Observatory

SNO – three independent informations 1000 t heavy water (D 2 0) CC - eppd  e NC xx  npd ES --  ee x x

They all arrive! D 2 O data (April 2002)

Reactor neutrinos: Do they really *oscillate*? Typical Energy: 2-6 MeV Oscillation length (known today) L 12 = 30km * E/MeV = 60 – 180 km Until year 2001: L max = 1 km  Only limits

Ideal situation for KamLAND in Kamioka

Most recent KamLAND result (2008) „Precision Measurement of Neutrino Oscillation Parameters with KamLAND“, Phys.Rev.Lett.100:221803,2008 L 0 is the „effective“ baseline = flux-weighted average of distance = 180km

KamLAND result (2008) „Precision Measurement of Neutrino Oscillation Parameters with KamLAND“, Phys.Rev.Lett.100:221803,2008 KamLAND + solar:

Problems Historical Prejudice: mixing angles should be small Problem: 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) Starting with e in sun via 4p  4 He + 2e e + 27 MeV transition to  =      not possible, since e not part of  for   =0 oscillation only to      effective 2-state oscillation: P surv ( e  e ) >= 50% need additional effect for explaining Homestake (and SNO) measurement  MSW effect: oscillation enhancement in matter MSW Effect   

Landau-Zener Theory (1932) Example: q:= Magnetic Field H 2 Spin states m>0, m<0 q: = Electron density N e (r) in sun 2 Neutrino states e, (  +  )

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 Example for Neutrinos: |V 12 | 2   m 2 ~ 1/L (oscillation length in matter) dE/dt  dm/dr ~ tan2 

Neutrino propagation in matter – MSW (Mikheyev, Smirnov, Wolfenstein) Effect Origin: v e and v μ,τ have different interaction with matter e  (v e can undergo CC and NC reaction, v μ,τ only NC!) Vacuum: In matter there is an additional potential in the equation of motion for ve → ve scattering (Flavor base) In matter:

Solution can be written in terms of a mixing angle  m in matter, which depends on electron density N e, i.e. on position in sun For small vacuum mixing angle (1°): For large vacuum mixing angle (32°): Sun: surface resonance center

Slide from Stephen Parke

Simulation of MSW: Variation of  m   smaller   larger 90° 45° 35° 20° Modify  m : Sun’s center: ~ 90 o, i.e. 2   e “resonance” = crossing region: ~45° Sun’s surface: ~35°, i.e. 2   e       Adiabaticity: variation of N e (i.e. m m,  m ) *slow* w.r.t. L m (i.e. 1/  m m 2 ) H i m = m 2 m + const 2m 1m e  resonance sin 2 2   = ~ N e E Sun’s surface

Status 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  m 2 = 8 x eV 2  L = 30 km x E/MeV Very small  m 2 ~ 8 x eV 2  L = 30 x 10 6 km x E/MeV

SNO mixing parameter

 , K  ee  e   (protons, He,,,) L=10~20 km Primary cosmic rays Low EnergyLimit   : e = 2 : 1 E (GeV)   →   D calculation Mixture of e &  →e+  + e E (GeV) Flux ratio    e  e  +  flux 2 Atmospheric neutrinos

Disappearance 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 neutrinos SuperKamiokande 2000: described als     pendula: e : weak coupling to     : weak coupling to e strong coupling to  0

Modify   Non-maximal mixing of   and   3         no longer eigenmode (special high school thesis J. Pausch 2008)   smaller   larger    Possible range: 30 o <    < 60 o

Impact of   on beam or atmospheric Impact of   on beam or atmospheric 3   sin   e      atmospheric or beam   e appearance „slow“ directly via  m 12 (weak coupling) „fast“ modulation via    with  m 23 (strong coupling)  13 = 6 o sin  13 = 0.1 sin 2 2  13 =

T2K (Tokai to Kamioka) Neutrino Super Beam Off-Axis Detector Superkamiokande Proton driver First neutrinos produced on April 23rd 2009

Takashi Kobayashi July 14, 2011, CERN Colloquium 8 events remained 3. PID is e-like  Enhance e CC Reconstructed neutrino energy < 1250 MeV - Reject higher energy intrinsic beam background 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   on reactor  e e present in 3  sin   e      e can now excite      mode, inducing fast    modulation Reactor  e      disappearance Reactor neutrinos (2 MeV) sin    = 0.10    = 6 o  sin    = 0.20    = 12 o    smaller   larger    Possible range: -6 o <    < 6 o  e nu  mu nu

Reactor Experiment (starting) Double-Chooz sensitivity for (  m 2 = eV 2 ): sin 2 (2  13 ) < 0.03, 90% C.L.

nearfar Double CHOOZ: near and far detector max. sensitivity on  13 : E ~ 4 MeV, Δm atm 2  L osc /2 ~ 1.5 km KamLAND CHOOZ sin 2 (2  12 )sin 2 (2  13 )

Are neutrino pendulums a perfect model? Few “features” Need “creative” sign convention, leading to imperfection for understanding sequence of masses Else perfect! The END !