Flavor Pendula Part 0: General Introduction Part I: Neutral Meson Mixing Part II: 3-flavor Neutrino Mixing Michael Kobel (TU Dresden) 22.9.09 BND school Rathen
Coupled Pendula Free Oscillation of one pendulum: 2 pendula with same length l, mass m coupled by spring with strength k 2 Eigenmodes Different eigenfrequencies = energies 2 2 Mode a (II + I) with ω a = ω 0 Mode b (II - I) with 2 2 ω b = ω 0 + Δ Frequency (=energy) difference increases with stronger coupling 2 Δω = kd ml Coupling can be steered by varying k or d (we ll vary d in the following) 2 2 2 ω 0 2 ω = g l d + + a: I II - + b: I II l
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) mass-basis eigenstates of mass well-defined lifetime Particles propagate through space-time as mass-eigenstates υ( t) = υ e Examples: K 0 L, K 0 S r r i( px Et) e Γt ν e, ν μ, ν τ ν 1, ν 2, ν 3 Like coupled pendula, the coupling of particles leads to eigenstates with different masses and lifetimes, e.g. for linear combination of 2 states: ν a = (ν τ + ν μ )/ 2 with m 2 a = m 2 0 ν b = (ν τ ν μ )/ 2 with m 2 b = m 02 + Δm 2
Correspondences pendulum Linear oscillation Eigenmodes fixed eigenfrequencies Frequency differences Δω different energies One pendulum = lin. combination of eigenmodes amplitude 2 ~ total energy in oscillation Beat-Frequency ~Δω of eigenmodes particles complex phase rotation Mass eigenstates fixed phase frequencies Frequency differences e iδet ~ e iδm²t different masses Flavor eigenstate = lin. combination of mass eigenstates amplitude 2 ~ detection probability Flavor-Oscillation ~ Δm (2) of mass eigenstates
Part I: Pendula for Neutral Meson Mixing coupled pendula for demonstrating KK-, DD- and BB-mixing Idea: Klaus Schubert built 1987 in Institut für Hochenergiephysik Universität Heidelberg at the occasion of the discovery of BB-mixing by the ARGUS-Collaboration at DESY
Part I: Some sheets stolen from Gerhard Raven s
m1 m2 Γ x = y = ( Γ + Γ ) / 2 Γ 1 B 0 0.75 ~0.01 B s 26 ~0.10 K 0 0.95 0.996 2 1 1 Γ2 + Γ 2 D 0 0.009 0.007
How can pendula model all this? Just start with one pendulum (e.g. B 0 or B 0 )! B 0 ( b d) B 0 ( b d) oscillations: very few decay channels common to both Each flavor is damped separately Just few flavor-oscillation periods observable B s0 ( b s) B s0 ( b s) oscillations: mixing via V ts2 = 0.04 2 > V td2 = 0.009 2 much faster compared to B 0 well, model it by slower decay relative to mixing many flavor-oscillation (beat) periods observable D 0 ( u c) D 0 ( u c) oscillations: mixing via (m b, m s ) << m t much slower compared to B 0 again, model it by faster decay relative to mixing no flavor-oscillation period observable K 0 ( s d) K 0 ( s d) oscillations: most decay channels in common ( ππ, πππ ) The damping is in the coupling! Large Phase space difference for decays of mass eigenstates (CP-cons.!) The damping has to be very different for the eigenmodes After a while only the K L0 eigenmode remains
What about CP-violation?
CP-violation with pendula Realisation: differences between the K 0 and K 0 pendula Different moments of inertia ml 2 (m and/or l different) Different coupling strengths (lever arm d different) corresponds to differences of V td and V td * in mixing diagram K L has unequal amounts of K 0 and K 0 wait for K L and have a look! Equal amounts of K 0 and K 0 evolve differently former eigenmodes now develop! (cf. CPLear Experiment)
Is CP violation with pendula a perfect model? No! all mechanical devices are T invariant! if CP is violated and T is conserved: CPT is violated! In fact, we have modelled CPT violation!
Part II: Neutrino flavor pendula coupled pendula for demonstrating 3-flavor neutrino mixing as realized in nature Idea: Michael Kobel built 2004 at Uni Bonn, extended 2006 at TU Dresden with variable mixing angles and digital readout http://neutrinopendel.tu-dresden.de
3-flavor neutrino mixing νe ν μ ντ = 1 0 0 c 0 23 s 23 s c 0 23 23 c13 0 s13e iδ 0 1 0 s 13 e c 0 iδ 13 c s 0 12 12 s c 12 12 0 0 ν1 0 ν 2 1 ν 3 Θ atmos, beam θ 13, δ Θ solar, reactor PMNS mixing matrix (w/o Majorana Phases) 3 Mixing angles: θ 12, θ 23, θ 13 1 CP-violating Dirac-Phase: δ
ν flavor-oscillations oscillations Each flavor (e.g. ν e ) is sum of mass eigenstates (ν 1, ν 2, 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 2 +m i2 )t) ~ exp(ipt+im i2 t/2p+ ) The differences Δω ij m i 2 -m j2 =: Δm ij 2 lead to flavor oscillations Δm ij2 determines the oscillation period θ ij determines the oscillation amplitude L ij = 2 5m E( MeV 2 Δm ( ev ) ). 2 ij
Current values cf.. global fit Th.Schwetz et al., NJP 10 (2008) Δm 2 23 = 2,4 x 10-3 ev 2 Δm 2 13 = 2,5 x 10-3 ev 2 Δm 2 12 = 0,08 x 10-3 ev 2 fast oscillation slow oscillation L 1 ( ) L 12 = 30 km E( MeV ) 23= km E MeV θ 23 = 45 ± 3 θ 13 < 11 (90% CL) θ 12 = 33.5 ± 1.5 Θ atmos, beam θ 13, δ Θ solar, reactor consistent with so-called tri/bi-maximal mixing θ 23 = 45 θ 13 = 0 θ 12 = 35.3 2 1 0 6 3 U 1 1 1 PMNS 6 3 2 1 1 1 6 3 2 Harrison, Perkins, Scott 99, 02 Z.Xing, 02, He, Zee, 03, Koide 03 Chang, Kang, Kim 04, Kang 04
Realisation as coupled pendula ν 3 = ( ν μ + ν τ )/ 2 - + ν 2 = ( ν e + ν μ + ν τ )/ 3 + - + ν 1 = (2ν e + ν μ + ν τ )/ 6 + + + m normal inverted hierarchy ν 3 ν 2 ν 1 ω/2π 46/min P 3 ν 2 ν 1 ν 3 43/min P 1 42/min P 2
The solar neutrino deficit Davis: only sensitiv to ν e Result: Only 30% of expected ν e detected
need for enhancement (MSW effect) nuclear fusion: 100% ν e leavethesun(w/o MSW effect) 4p 4 He + 2e + + 2ν e + 27 MeV slow oscillation via Δm 2 12 and θ 12, pendula: weak coupling T/2 + + + transition to (ν τ ν μ )/ 2 not possible, since ν e not in ν 3 oscillation only to (ν τ + ν μ )/ 2 P(ν e ν e ) > 50% since just ν 1 and ν 2 count need for enhancement (MSW effect) 2θ 12 MSW Effekt 12
atmospheric neutrinos SuperKamiokande 2000: look at ν e and ν μ from air showers: no deficit for ν e cleardeficitforν μ fully compatible with ν μ ν τ
SuperKamiokande 2000: 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 http://minos.phy.bnl.gov/nu-osc-lab/superposition1.html
Modify θ 23 Non-maximal mixing of ν μ and ν τ ν 3 = ( ν μ + ν τ )/ 2 no longer eigenmode Possible range: 30 o < θ 23 < 60 o θ 23 θ 23 23 smaller 23 larger http://neutrinopendel.tu-dresden.de (special high school thesis J. Pausch 2008)
Modify θ 12 Modify fraction of ν e in ν 1 and ν 2 ν 2 = ( ν e + ν μ + ν τ )/ 3 no longer eigenmode Possible range: 20 o < θ 12 < 55 o θ 12 θ 12 12 smaller 12 larger http://neutrinopendel.tu-dresden.de (J. Pausch)
Modify θ 13 ν e present in ν 3 (sinθ 13 ν e ν μ + ν τ ) ν e can now excite (ν τ ν μ ) mode, inducing fast ν τ ν μ modulation Possible range: -6 o < θ 13 < 6 o Reactor neutrinos (2 MeV) sin θ 13 = 0.10 (θ 13 = 6 o ) θ 13 θ 13 13 smaller 13 larger e nu mu nu sin θ 13 = 0.20 (θ 13 = 12 o ) e nu mu nu
Impact of θ 13 on atmospheric ν ν 3 = (sinθ 13 ν e ν μ + ν τ )/ 2.01 reactor ν e ν τ + ν μ disappearance and atmospheric ν μ ν e appearance slow directly via Δm 12 (weak coupling) fast modulation via ν τ ν μ with Δm 23 (strong coupling) θ 13 = 6 o sin θ 13 = 0.1 sin 2 2θ 13 = 0.04 0
Are neutrino pendula a perfect model? Few features Need creative sign convention, leading to imperfection for understanding sequence of masses imperfection for θ 23 45 o some (ν τ ν μ ) present in ν 1 and ν 2 but ν e (ν τ ν μ ) still not possible! Else perfect! The END!