Neutrino Pendulum. A mechanical model for 3-flavor Neutrino Oscillations. Michael Kobel (TU Dresden) PSI,

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1 Neutrino Pendulum A mechanical model for 3-flavor Neutrino Oscillations Michael Kobel (TU Dresden) PSI,.6.016

2 Class. Mechanics: Coupled Pendulums pendulums with same length l, mass m coupled by spring with strength k Free Oscillation of one pendulum: Eigenmodes Different eigenfrequencies = energies Mode a (II + I) Mode b (II - I) with with ω a = ω ωb = ω Frequency (=energy) difference increases with stronger coupling ω + ω = g + + a: I II - + b: I II d l ω = kd m Coupling can be steered by varying k or d (we ll vary d in the following)

3 Two Coupled Neurtrino-Pendulums d 1 γ k d l k l m m ν µ ν τ γ 1 γ 0 φ 1 0 φ Neutrinos: infinite lifetime damping γ i = 0

4 m m M ij φ = m g 1 special coupling d 1 = d =: d ω + ω / = ω / 1 E.vector 1 φ d with E.value 1 E.vector with E.value 1 ==> maximal mixing 1 φ = m g φ kd m = : ω, 1 d k kd kd Uncoupled (k=0) frequency: ω / ω + ω / ( d φ d φ ) k ω ( dφ φ φ d φ ) ( 1φ 1 φ ) ( ) ω ω + Δω 1 := 1 g 1 M ij φ 1 φ kd i d m j ω + = ω = = ω11 1 M ij : ωij ω φ1 φ general coupling d 1 d ω + 1 ω ==> non maximal mixing

5 flavor-basis eigenstates of flavor eigenstates of weak charge Two bases in Hilbert-space particles take part in weak interactions as flavor-eigenstates Examples: K 0 ( s u) or K 0 ( s u) ν e, ν µ, ν τ mass-basis eigenstates of mass well-defined lifetime Particles propagate through spacetime as mass-eigenstates ν ( t) = ν e Examples: K 0 L, K 0 S i( px Et) ν 1, ν, ν 3 e Γt The coupling of flavor eigenstates leads to eigenstates with different masses e.g. for linear combination of states: ν a = (ν τ + ν µ )/ with m a = m ν b = (ν τ ν µ )/ with m b = m + m

6 Correspondences pendulum Linear oscillation Eigenmodes fixed eigenfrequencies Frequency differences ω different energies One pendulum = lin. combination of eigenmodes amplitude ~ total energy in oscillation Beat-Frequency ~ ω of eigenmodes particles complex phase rotation Mass eigenstates fixed phase frequencies Frequency differences e i Et ~ e i m²t different masses Flavor eigenstate = lin. combination of mass eigenstates amplitude ~ detection probability Flavor-Oscillation ~ m of mass eigenstates

7 Three flavor Neutrino pendulum coupled pendula for demonstrating 3-flavor neutrino mixing as realized in nature Idea: M.K. built 004 at Uni Bonn, extended 006 at TU Dresden with variable mixing angles and digital readout Copies in: Hamburg, Münster, DESY(Zeuthen), Sussex

PMNS mixing matrix (w/o Majorana Phases) 3 Mixing angles: θ 1, θ 3, θ 13 1 CP-violating Dirac-Phase: δ (neglected in the following) + mass differences m 1, m 3 = 3 1 1 1

8 PMNS mixing matrix (w/o Majorana Phases) 3 Mixing angles: θ 1, θ 3, θ 13 1 CP-violating Dirac-Phase: δ (neglected in the following) + mass differences m 1, m 3 = ν ν ν ν ν ν δ δ τ µ c s s c c e s e s c c s s c i i e Θ solar, reactor θ 13, δ Θ atmos, beam 3-flavor neutrino mixing

9 ν flavor-oscillations Each flavor (e.g. ν e ) is sum of mass eigenstates (ν 1, ν, ν 3 ) Each mass eigenstate with fixed p has a different phase frequency ω i exp(iω i t) = exp(ie i t) = exp(i( (p +m i )t) ~ exp(ipt+im i t/p+ ) The differences ω ij m i - m j =: m ij lead to flavor oscillations m ij determines the oscillation period θ ij determines the oscillation amplitude L ij =.48m = 480km E( MeV ) m m ij ( ev E( MeV ) ij ( mev ) )

Global fit 016 (www.nu-fit.org and http://arxiv.org/abs/1409.5439 ) M.C.Gonzales-Garcia, M.

10 Global fit 016 ( and ) M.C.Gonzales-Garcia, M.Maltoni, Th.Schwetz, (normal ordering) m 3 = 40 ± 4 mev m 13 = 477 ± 4 mev m 1 = 75 ± mev fast oscillation fast oscillation slow oscillation L 3 1 km E(MeV ) L 3 1 km E(MeV ) L 1 33 km E( MeV) θ 3 = 4 ± θ 13 = 8.5 ± 0. θ 1 = 33.7 ± 0.8 θ atmos, beam θ 13, δ θ solar, reactor Very near to tri / bi - maximal mixing θ 3 = 45 θ 13 = 0 θ 1 = 35.3 U PMNS Motivated by family symmetries : Harrison, Perkins, Scott 99, 0 Z.Xing, 0, He, Zee, 03, Koide 03 Chang, Kang, Kim 04, Kang 04

11 Realisation as coupled pendula ν 3 = ( ν µ + ν τ )/ - + ν = ( ν e + ν µ + ν τ )/ ν 1 = (ν e + ν µ + ν τ )/ m normal inverted hierarchy ω/π ν 3 ν ν 1 46/min ν 3 ν ν 1 ν 3 43/min ν 1 4/min ν

12 Solar Neutrinos 4 p He 4 + e + + ν e MeV T central = K Flux@Earth 6.5E10 v e /cm s Neutrino light from the Sun (Super-Kamiokande)

13 Electron Neutrino Oscillation -> θ 1 oscillation of ν e via θ 1 and small m 1 in ν µ + ν τ ν µ and ν τ always identical for θ 13 = 0 90 Vary θ 1 modify fraction of ν e in ν 1 and ν ν = ( ν e + ν µ + ν τ )/ 3 eigenmode for θ 1 =35 45 Possible range for this model: 0 o < θ 1 < 90 o (naturally large!) θ 1 smaller θ 1 larger 35 (special high school thesis J. Pausch 008) 0 N.B.!: ν e : green ν µ + ν τ : identical

14 Homestake Experiment Ray Davis Jr. 37 Cl ν + Ar + e e 37 The art of low-level physics or how to get 10 atoms out of 600 t

Chlorine (Ray Davis, Homestake): Final Measurement result Main source of captured ν e : 8 B 1 Solar Neutrino Unit (SNU) = σφ = 10 36 s -1 = z.b.

15 Chlorine (Ray Davis, Homestake): Final Measurement result Main source of captured ν e : 8 B 1 Solar Neutrino Unit (SNU) = σφ = s -1 = z.b. 1ab * 10 6 cm - s Ar Atoms / day Mean over 108 independent measurements: Only 3% of expected ν e detected R detected =,56 SNU +- 0,16 (stat.) (sys.) Solar Model Prediction (new, 005) R = 8,1 +- 1, SNU Significance: 4.6 s.d

Super Kamiokande Detektor in Japan 50 000 t H 0

16 Super Kamiokande Detektor in Japan t H 0 Cherenkov detector 40 m high 40 m Lightdetektors (Photomultiplier) 50 cm 1 km deep in Kamioka mine, Japan result: 0.41 ± 0.07 of expected ν e flux from sun

17 SNO three independent informations 1000 t heavy water (D 0) CC ν e + d p + p + - e NC ν + d p + n + x ν x + - e + e ν x ν x ES - CC ES = ν e ν e ( ν µ + ν τ ) ν e CC NC = ν e + ν µ + ν τ

18 They all arrive! D O data (April 00) Total flux 5 x 10 6 cm - s -1 ν e flux: 1.8 x 10 6 cm - s -1

Solar ν transformation the final proof 00 April 00: SNO Experiment Direct Evidence for Neutrino Flavor Transformation from Neutral- Current Interactions in the Sudbury Neutrino

19 Solar ν transformation the final proof 00 April 00: SNO Experiment Direct Evidence for Neutrino Flavor Transformation from Neutral- Current Interactions in the Sudbury Neutrino October 00: Nobelprize for Raymond Davis (Homestake) Masatoshi Koshiba (Superkamiokande) Flavor transformation established, not yet oscillation, though

21 Reactor neutrinos: Do they really *oscillate*? Typical Energy: -6 MeV Oscillation length (known today) L 1 = 33km * E/MeV = km Until year 001: L max = 1 km Only limits

KamLAND: Proof of Oscillation Long baseline θ 1 : Kamland (Japan, 00-010) Effective baseline = flux-weighted average of distance L 0 = 180km

22 KamLAND: Proof of Oscillation Long baseline θ 1 : Kamland (Japan, ) Effective baseline = flux-weighted average of distance L 0 = 180km December 00: First Results from KamLAND: Evidence for Reactor Anti-Neutrino Disappearance

23 SNO mixing parameter 003 SMA LMA LOW m²

24 KamLAND: Proof of Oscillation Kamland 008: Precision m 1 via oscillation Phys. Rev. Lett. 100, 1803 m 1 = 76 ± mev LMA

25 Neutrino spectrum, uncertainties and sensitivities (Bahcall et al., 000)

26 Theoretical Problem Starting with ν e in sun via 4p 4 He + e + + ν e + 7 MeV transition to ν 3 = (ν τ ν µ )/ not possible, since ν e not part of ν 3 for θ 13 =0 oscillation only to (ν τ + ν µ )/ effective -state oscillation: P surv (ν e ν e ) >= 50% need additional effect for explaining solar neutrino measurements Michejew-Smirnow-Wolfenstein (MSW) effect: oscillation enhancement in matter Originally introduced already 1985 small mixing angle (SMA) solution (prejudice!) θ 1 MSW Effect 1

27 Landau-Zener Theory (193) Examples: Atoms: Spin states m>0, m<0 q:= Magnetic Field B Neutrinos: states ν e, (ν µ + ν τ ) q: = Electron density N e (r) in sun

$MSW: Neutrinos in Matter ν e obtain additional diffractive potential term due to scattering with electrons (N e ) Without ν flavor-mixing (diagonal Hamiltonian υ 1 =0) With ν flavor-mixing$

28 MSW: Neutrinos in Matter ν e obtain additional diffractive potential term due to scattering with electrons (N e ) Without ν flavor-mixing (diagonal Hamiltonian υ 1 =0) With ν flavor-mixing (non-diagonal hamiltonian υ 1 0) ν e = ν (matter) ν (vacuum) avoided level crossing flavor composition of mass eigenstates changes as function of e-density N e If start as mass eigenstate: continuous change, no oscillation

29 Transitions at level crossing Example for Neutrinos: υ 1 : ~ sinθ 1 / L m (L m : oscillation length in matter) Adiabaticity (no transition at crossing) υ 1 de/dt = λλ NNNN tan θθ LL 1 1 mm de/dt: ~ cosθ 1 / λ Ne (λ Ne : scale of N e density change, ~ about 0.1x solar radius) i.e. flavor osc. fast wrt. variation of N e L m << λ Ne ~ 0.1 R and θ 1 not too small

sin cos θ θ θ θ µ = µ θ θ θ θ v v v v e m m m m m m cos sin sin cos 1 For small

30 Solution can be written in terms of a mixing angle θ m in matter, which depends on electron density N e, i.e. on position in sun = m m m m m m e v v v v 1 cos sin sin cos θ θ θ θ µ = µ θ θ θ θ v v v v e m m m m m m cos sin sin cos 1 For small vacuum mixing angle (1 ): For large vacuum mixing angle (3 ): Sun: surface resonance core

31 Modify θ m : Simulation of MSW: Variation of θ m Sun s center: ~ 90 o, i.e. ν = ν e resonance = crossing region: ~45 Sun s surface: ~35, i.e. ν = ( ν e + ν µ + ν τ ) / sin θ 1 = 0.85 H i m = m m + const resonance ν ν e m θ 1 smaller θ 1 larger 35 0 ν µ ν 1m Sun s surface ~ N e E

32 Evidence for MSW avoided level crossing Borexino 01 : No level crossing encountered (core of sun not dense enough) traverse level crossing

33 Atmospheric neutrinos Primary cosmic rays (protons, He,,,) L=10~0 km π ±, K ± µ ± ν µ +ν µ flux 3D ν µ calculation Mixture of ν e & ν µ E ν (GeV) ν µ ν µ e ± ν e Low EnergyLimit ν µ : ν e = : 1 Flux ratio π µ+ν µ e+ν µ +ν e ν µ +ν µ ν e +ν e E ν (GeV)

34 electron event ν e ν µ ν τ d u d n W - p e - µ τ u u d myon event

35 SuperKamiokande 000: described als ν µ ν τ pendula: ν e : weak coupling to ν µ, ν τ ν µ : weak coupling to ν e strong coupling to ν τ atmospheric neutrinos 0

neutrinos) Nachweis von Umwandlungen (Oszillationen)

36 nobelprize 015 Arthur B. McDonald (Queen s University, Canada) and Takaaki Kajita (University of Tokyo) Solar Neutrino Observatory (SNO) Superkamiokande (atmospheric neutrinos) Nachweis von Umwandlungen (Oszillationen) von Neutrinoarten Damit Beweis der Existenz ihrer Masse TU Dresden, WiSe 15/16 Michael Kobel 36

< 60 o (naturally ~ maximal) θ 3 smaller θ 3 larger

37 Modify θ 3 Non-maximal mixing of ν µ and ν τ ν 3 = ( ν µ + ν τ )/ no longer eigenmode Possible range for this model: 30 o < θ 3 < 60 o (naturally ~ maximal) θ 3 smaller θ 3 larger (special high school thesis J. Pausch 008)

39 Double Chooz (France) Double-Chooz sensitivity for ( m = ev ): sin (θ 13 ) < 0.03, 90% C.L.

40 Double CHOOZ: near and far detector P( ν e ν e ) 1 sin θ 13 sin m 4E 31 ν L cos 4 θ 13 sin θ 1 sin m 4E 1 ν L sin (θ 13 ) sin (θ 1 ) CHOOZ near far KamLAND max. sensitivity on θ 13 : E ν ~ 4 MeV, Δm atm L osc / ~ 1.5 km

41 Impact of θ 13 on reactor ν e ν e present in ν 3 (sin θ 13 ν e ν µ + ν τ ) ν e can now excite (ν τ ν µ ) mode, inducing fast ν τ ν µ modulation Reactor ν e ν τ ν µ disappearance Possible range for this model : -6 o < θ 13 < 6 o (naturally small!) θ 13 smaller θ 13 larger Reactor neutrinos ( MeV) sin θ 13 = 0.10 (θ 13 = 6 o ) e nu mu nu sin θ 13 = 0.0 (θ 13 = 1 o ) e nu mu nu

42 Impact of θ 13 on beam or atmospheric ν ν 3 (sinθ 13 ν e ν µ + ν τ ) atmospheric or beam ν µ ν e appearance slow directly via m 1 (weak coupling) fast modulation via ν τ ν µ with m 3 (strong coupling) θ 13 = 6 o sin θ 13 = 0.1 sin θ 13 =

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

Problems Historical Prejudice: mixing angles should be small Problem #1: How to get large neutrino deficit w/ small mixing? After SNO+ KamLAND: mixing angles θ 1 and θ 3 large!

44 Problems Historical Prejudice: mixing angles should be small Problem #1: How to get large neutrino deficit w/ small mixing? After SNO+ KamLAND: mixing angles θ 1 and θ 3 large! ( Large Mixing Angle = LMA solution) Solar neutrinos effective -flavor mixing for θ 13 0 ν 3 = (ν τ ν µ )/ not part of ν e for θ 13 =0 min solar detection rate of ν e is 50%, even with max mixing Problem #: Observed rate of Homestake ~ 3%!

45 Neutrino propagation in matter MSW (Mikheyev, Smirnov, Wolfenstein) Effect Origin: v e and v μ,τ have different interaction with matter (v e can undergo CC and NC reaction, v μ,τ only NC!) ν e,µ,τ ν e,µ,τ In matter: + 4E G F N e Vacuum: i d dt ν e 1 m cosθ m sinθ ν e 1 ν = = : ν µ 4E m sinθ m cosθ ν µ 4E ν e ( V ) µ In matter there is an additional potential in the equation of motion for ve ve scattering (Flavor base)

46 Effect of an interaction between 1> and > Example: 1, : ν flavor states: ν e, (ν µ + ν τ ) a,b: ν mass states: ν 1, ν V: Neutrino Flavor Mixing via θ 1

47 Slide from Stephen Parke

Neutrino Pendulum. A mechanical model for 3-flavor Neutrino Oscillations

Neutrino Pendulum. A mechanical model for 3-flavor Neutrino Oscillations Neutrino Pendulum A mechanical model for 3-flavor Neutrino Oscillations Michael Kobel (TU Dresden) Obertrubach, 5.0.0 Schule für Astroteilchenphysik Free Oscillation of one pendulum: pendulums with same