Neutrino Pendulum. A mechanical model for 3-flavor Neutrino Oscillations
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1 Neutrino Pendulum A mechanical model for 3-flavor Neutrino Oscillations Michael Kobel (TU Dresden) Obertrubach, Schule für Astroteilchenphysik
2 Free Oscillation of one pendulum: pendulums with same length l, mass m coupled by spring with strength k Eigenmodes Different eigenfrequencies = energies Mode a (II + I) Mode b (II - I) Model: Coupled Pendulums with with w a = w wb = w Frequency (=energy) difference increases with stronger coupling w + Dw g = l + + a: I II - + b: I II d l Dw = kd ml Coupling can be steered by varying k or d (we ll vary d in the following)
3 Equations for Coupled Pendulums d g k d l k l m m K K g g 0 f 0 f
4 Equations of motion for l = l = l and d d ( ) ( ) ( ) ( ) f g f f g f f f f f g f f g f f f f & l & & l && l & l & & l && l = = - d d d d d k d m g m d d d d d k d m g m k k i i ij j i k ij j i m m d d m d kd g G = G = D = = :, :, :, : g g w w l l l ł Ł G + G - G - G + G G = G ł Ł D + - D - D + D = ł Ł - G ł Ł = - ł Ł ij ij ij ij M M w w w w w w f f f f f f & & && && For K, B mesons damping important: G = G Damping in Coupling (K) G, G Damping in Decay (B) For Neutrinos damping negligible
5 Undamped motion for l = l =: l and special case d = d =: d ml ml M ij && f = -mlg && f = -mlg f - kd f - kd g kd = : w, = : Dw l ml && f = -M && Łf ł Łf ł ( f -f ) ( f -f ) w + Dw / - Dw / = Ł - Dw / w + Dw / ł E. vector with E. value w Łł E. vector with E. value w Ł-ł ij f + Dw
6 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) n e, n m, n t mass-basis eigenstates of mass well-defined lifetime Particles propagate through spacetime as mass-eigenstates u( t) = u e Examples: K 0 L, K 0 S n, n, n 3 r r i ( px -Et ) e -Gt The coupling of flavor eigenstates leads to eigenstates with different masses e.g. for linear combination of states: n a = (n t + n m )/ with m a = m n b = (n t - n m )/ with m b = m + Dm
7 Correspondences pendulum Linear oscillation Eigenmodes fixed eigenfrequencies Frequency differences Dw different energies One pendulum = lin. combination of eigenmodes amplitude ~ total energy in oscillation Beat-Frequency ~Dw of eigenmodes particles complex phase rotation Mass eigenstates fixed phase frequencies Frequency differences e idet ~ e idm²t different masses Flavor eigenstate = lin. combination of mass eigenstates amplitude ~ detection probability Flavor-Oscillation ~ Dm of mass eigenstates
8 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
9 3-flavor neutrino mixing n n Łn e m t ł = 0 c 0 3 Ł0 - s 3 s c łł - c s e id 0 0 s 3 c e 0 -id 3 łł - c s 0 s c n n łłn 3 ł Θ atmos, beam θ 3, δ Θ solar, reactor PMNS mixing matrix (w/o Majorana Phases) 3 Mixing angles: θ, θ 3, θ 3 CP-violating Dirac-Phase: δ (neglected in the following) + mass differences Dm, Dm 3
10 n flavor-oscillations Each flavor (e.g. n e ) is sum of mass eigenstates (n, n, n 3 ) Each mass eigenstate with fixed p has a different phase frequency w i exp(iw i t) = exp(ie i t) = exp(i( (p +m i )t) ~ exp(ipt+im i t/p+ ) The differences Dw ij ~ m i - m j =: Dm ij lead to flavor oscillations Dm ij determines the oscillation period q ij determines the oscillation amplitude L ij = 5 E( MeV Dm ( ev ) ). m ij
11 Current values cf. global fit Th.Schwetz, M.Tortola, J.W.F Valle, arxiv Dm 3 =,4 x 0-3 ev Dm 3 =,50 x 0-3 ev Dm = 0,076 x 0-3 ev fast oscillation L» 3 km E(MeV ) slow oscillation L» 30 km E(MeV ) q 3 = 46 ± 3 q 3 = 6.5 ±.5 (3.s) q = 34.0 ±.0 θ atmos, beam θ 3, δ θ solar, reactor Very near to tri/bi-maximal mixing (family symmetries ) q 3 = 45 q 3 = 0 q = 35.3 U PMNS» Ł ł Harrison, Perkins, Scott 99, 0 Z.Xing, 0, He, Zee, 03, Koide 03 Chang, Kang, Kim 04, Kang 04
12 Realisation as coupled pendula n 3 = ( - n m + n t )/ - + n = (-n e + n m + n t )/ n = (n e + n m + n t )/ m normal inverted hierarchy w/p n 3 n 46/min n 3 n n n n 3 43/min n 4/min n
13 4 p fi He 4 + e + + n e MeV T central = 5E6 K 6.5E0 v e /cm s Neutrino light from the Sun (Super-Kamiokande)
14 Neutrino spectrum, uncertainties and sensitivities (Bahcall et al., 000)
15 Electron Neutrino Oscillation -> q oscillation of n e via q and small Dm in n m + n t n m and n t always identical for q 3 = 0 Vary q modify fraction of n e in n and n n = (-n e + n m + n t )/ 3 only eigenmode for q =35 Possible range: 0 o < q < 90 o q smaller q larger (special high school thesis J. Pausch 008)
16 Chlorine (Ray Davis, Homestake): Final Measurement result Main source of captured n e : 8 B Solar Neutrino Unit (SNU) = sf = 0-36 s - = z.b. ab * 0 6 cm - s - 37 Ar Atoms / day Mean over 08 independent measurements: Only 3% of expected n e detected R detected =,56 SNU +- 0,6 (stat.) (sys.) Solar Model Prediction (new, 005) R = 8, +-, SNU Significance: 4.6 s.d
17 Gallex (+ GNO): ( )
18 Gallex / GNO results SSM prediction: 9 +8/-6 SNU * (BP98) SSM Gallex, GNO Gallex/GNO: 69.3 ± 4. ± 3.6 SNU Total: GALLEX/GNO & SAGE: 68. ± 3.75 SNU *) SNU (solar neutrino unit) = v-capture / 0 36 target atoms
19 Super Kamiokande Detektor in Japan t H 0 Cherenkov detector 40 m high 40 m 46 Lightdetektors (Photomultiplier) 50 cm km deep in Kamioka mine, Japan
20 Interpretation of measurements Fraction detected: (uncertainty theory-dominated) Cl: (3 +- 6)% H O: (4 +- 7)% Ga: ( )% Bahcall:
21 Solar n oscillations 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) December 00: First Results from KamLAND: Evidence for Reactor Anti-Neutrino Disappearance
22 SNO: Sudbury Neutrino Observatory Creighton Mine (Nickel) Sudbury, Canada Depth 070m 000t D O 9500 PMTs
23 SNO three independent informations 000 t heavy water (D 0) CC n e + d p + p + - e NC n + d p + n + x n x n + - e + e ES - x n x CC ES = n e n e + 0.4( n m + n t ) CC NC = n e n e + n m + n t
24 D O data (April 00) They all arrive!
25 Reactor neutrinos: Do they really *oscillate*? Typical Energy: -6 MeV Oscillation length (known today) L = 30km * E/MeV = km Until year 00: L max = km Only limits
26
27 Ideal situation for KamLAND in Kamioka
28 Most recent KamLAND result (008) Precision Measurement of Neutrino Oscillation Parameters with KamLAND, Phys.Rev.Lett.00:803,008 L 0 is the effective baseline = flux-weighted average of distance = 80km
29 KamLAND result (008) Precision Measurement of Neutrino Oscillation Parameters with KamLAND, Phys.Rev.Lett.00:803,008 KamLAND + solar:
30 Problems Historical Prejudice: mixing angles should be small Problem: How to get large neutrino deficit w/ small mixing? Today no problem: mixing angles are large! Knowing about large q, but having q 3» 0 Effective -flavor mixing! min detection rate should be >= 50% Problem: Observed rate of Homestake ~ 3%!
31 Solution: MSW effect (985) Starting with n e in sun via 4p 4 He + e + + n e + 7 MeV transition to n 3 = (n t - n m )/ not possible, since n e not part of n 3 for q 3 =0 oscillation only to (n t + n m )/ effective -state oscillation: P surv (n e n e ) >= 50% need additional effect for explaining Homestake (and SNO) measurement MSW effect: oscillation enhancement in matter q MSW Effect
32 Landau-Zener Theory (93) Example: q:= Magnetic Field H Spin states m>0, m<0 q: = Electron density N e (r) in sun Neutrino states n e, (n m + n t )
33 Effect of an interaction between > and > Example:, : n flavor states: n e, (n m + n t ) a,b: n mass states: n, n V: Neutrino Flavor Mixing via q
34 Transitions at level crossing Example for Neutrinos: V Dm ~ /L (oscillation length in matter) de/dt dm/dr ~ tanq
35 n e,m,t n (v e,m,t e can undergo CC and NC reaction, v μ,τ only NC!) In matter: + 4E G F N e Vacuum: i d dt n Łn e m ł = 4E Ł - Dm Dm cosq sinq Dm Dm sinq n e cosq łłn m ł = : 4E ( V ) n Łn e m ł In matter there is an additional potential in the equation of motion for ve ve scattering (Flavor base)
36 q v q - q v m cos m sin m e = Łv ł Ł q q łłv m sin m cos m m ł v Łv e m ł = cosq m Ł - sinqm sinq m v cosqm łłv m m ł Sun: surface resonance center For small vacuum mixing angle ( ): For large vacuum mixing angle (3 ):
37
38 Modify q m : Simulation of MSW: Variation of q m Sun s center: ~ 90 o, i.e. n = -n e resonance = crossing region: ~45 Sun s surface: ~35, i.e. n = (-n e + n m + n t ) / 3 Adiabaticity: variation of N e (i.e. m m, q m ) *slow* w.r.t. L m (i.e. /Dm m ) sin q = 0.85 H i m = m m + const resonance n m n e q smaller q larger n m n m Sun s surface ~ N e E
39 Status of Solar Oscillations ~000 Common prejudice in 000: Small-Mixing-Angle SMA -MSW solution In addition: Just so observable at distance sun-earth today s value Dm = 8 x 0-5 ev L = 30 km x E/MeV Very small Dm ~ 8 x 0 - ev L = 30 x 0 6 km x E/MeV SMA LMA LOW SAGE & GALLEX Kamiokande Homestake
40 SNO mixing parameter
41 Atmospheric neutrinos Primary cosmic rays (protons, He,,,) L=0~0 km p, K m n m +n m flux 3D n m calculation Mixture of n e & n m E n (GeV) n m n m e n e Low EnergyLimit n m : n e = : Flux ratio p m+n m e+n m +n e n m +n m n e +n e E n (GeV)
42 Disappearance of n m SuperKamiokande 000: look at n e and n m from air showers: no deficit for n e clear deficit for n m fully compatible with n m n t
43 electron event n e n µ n t d u d n W - p e - m - t - u u d myon event
44 SuperKamiokande 000: described als n m n t pendula: n e : weak coupling to n m, n t n m : weak coupling to n e strong coupling to n t atmospheric neutrinos 0
45 Modify q 3 Non-maximal mixing of n m and n t n 3 = ( - n m + n t )/ no longer eigenmode Possible range: 30 o < q 3 < 60 o q 3 smaller q 3 larger (special high school thesis J. Pausch 008)
46
47 Impact of q 3 on beam or atmospheric n n 3 ~ (sinq 3 n e - n m + n t ) atmospheric or beam n m n e appearance slow directly via Dm (weak coupling) fast modulation via n t - n m with Dm 3 (strong coupling) q 3 = 6 o sin q 3 = 0. sin q 3 =
48 TK (Tokai to Kamioka) First neutrinos produced on April 3rd 009 Off-Axis Detector Superkamiokande Neutrino Super Beam Proton driver
49 Takashi Kobayashi July 4, 0, CERN Colloquium 3. PID is e-like ŁEnhance n e CC 8 events remained 7. Reconstructed neutrino energy < 50 MeV - Reject higher energy intrinsic beam background from kaon decays 6 final candidate events remained! Signal Efficiency = 66% Background Rejection: 77% for beam ν e 99% for NC Selection criteria & cut values are fixed before analysis. Unbiased Expected BG.5evts 49
50 A candidate 50
51 Impact of q 3 on reactor n e n e present in n 3 ~ (sin q 3 n e - n m + n t ) n e can now excite (n t - n m ) mode, inducing fast n t - n m modulation Reactor n e n t - n m disappearance Possible range: -6 o < q 3 < 6 o q 3 smaller q 3 larger Reactor neutrinos ( MeV) sin q 3 = 0.0 (q 3 = 6 o ) - e nu - mu nu sin q 3 = 0.0 (q 3 = o ) - e nu - mu nu
52 Reactor Experiment (starting) Double-Chooz sensitivity for (Dm = ev ): sin (q 3 ) < 0.03, 90% C.L.
53 Double CHOOZ: near and far detector P( n e fi n e )» - sin q 3 sin Dm 4E 3 n L - cos 4 q 3 sin q sin Dm 4E n L sin (q 3 ) sin (q ) CHOOZ near far KamLAND max. sensitivity on q 3 : E n ~ 4 MeV, Δm atm Ł L osc / ~.5 km
54 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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