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Neutrino Oscillations Neutrino Mixing

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Neutrino Oscillations Neutrino Mixing Neutrino oscillations Neutrino mixing ‣ Mixing is described by the Maki-Nakagawa-Sakata (MNS) matrix: ναL = UαiνkL k U = Ul†Uν ‣ leptonic weak charged current: 3 CC jρ † =2 αLγρναL =2 αLγρUαkνkL α=e,µ,τ α=e,µ,τ k=1 ‣NxN unitary matrix: NxN mixing parameters → N(N-1)/2 mixing angles + N(N+1)/2 phases ‣Lagrangian invariant under global phase transformations of Dirac fields: α eiθα α, ν eiφk ν → k → k CC i(θe φ1) i(θα θe) i(φk φ1) jρ † 2 αLe− − e− − γρUαke − νkL → 1 N-1 N-1 α,k → 2N-1 phases can be eliminated: (N-1)(N-2)/2 physical phases Neutrino mixing ‣For Majorana neutrinos, the lagrangian is NOT invariant under global phase transformations of the Majorana fields: iφk T 2iφk T ν e ν νkL †νkL e νkL †νkL k → k C → C → only N phases can be eliminated by rephasing charged lepton fields: CC iθα jρ † 2 αLe− γρUαkνkL → N α,k → N(N-1)/2 physical phases: (N-1)(N-2)/2 Dirac phases → effect in ν oscil. (N-1) Majorana phases → relevant for 0νββ Exercises: 1)Check number angles/phases NxN matrix 2) how many angles/phases N=2,3 ? Neutrino mixing ‣2-neutrino mixing depends on 1 angle only (+1 Majorana phase) ‣3-neutrino mixing is described by 3 angles and 1 Dirac (+2 Majorana) CP violating phases. atmospheric + LBL reactor disapp + LBL solar + KamLAND measurements appearance searches measurements Neutrino oscillations ‣flavour states are admixtures of flavor eigenstates: ναL = UαiνkL k d ‣Neutrino evolution equation: i ν = H ν − dt| | E1 0 ν H = in the 2-neutrino mass eigenstates basis j : 0 E 2 iEj t νj e− νj | → m2 | m2 since neutrinos are relativistic: t = L and E p + j p + j j 2p 2E (equal momentum approach) ‣Hamiltonian in the flavour basis: ∆m2 cos 2θ sin 2θ Hflavour = UHmassU † = − 4E sin 2θ cos 2θ with ∆m2 = m2 m2 2 − 1 Exercise: derive expression for Hflavour Neutrino oscillations picture d i ν = H ν − dt| | Production Propagation Detection ν = U ∗ ν iEj t ν = ν U | α αj | j νj : e− β| j| βj j j different propagation coherent superposition projection over flavour phases change νj of massive states eigenstates composition Neutrino oscillations Neutrino oscillation amplitude: ν ν = νβ(t) να(0) = νβ νj(t) νj(t) νj(0) νj(0) να A α→ β | | | | j 2 mj i 2E = Uβj e− Uα∗j j 2 2 mj i 2E Pνα νβ = Uα∗jUβj e− Neutrino oscillation probability: → j 2 2 ∆mijL P = δ 4 Re(U ∗ U U U ∗ )sin + αβ αβ − αi αj βi βj 4E i>j 2 ∆mijL +2 Im(U ∗ U U U ∗ )sin αi αj βi βj 2E i>j Exercise: derive general formula for oscillation probability General properties of neutrino oscillations P (ν ν )=1 ‣Conservation of probability: α → β β ‣For antineutrinos: U →U* ‣Neutrino oscillations violate flavour lepton number conservation (expected from mixing) but conserve total lepton number ‣Complex phases in the mixing matrix induce CP violation: P (ν ν ) = P (ν ν ) α → β α → β ‣Neutrino oscillations do not depend on the absolute neutrino mass scale and Majorana phases. ‣Neutrino oscillations are sensitive only to mass squared differences: ∆m2 = m2 m2 kj k − j 2-neutrino oscillations ‣2-neutrino mixing matrix: cos θ sin θ sin θ cos θ − ‣2-neutrino oscillation probability (α≠β): 2 ∆m2 L i 21 P (ν ν )= U U ∗ + U U ∗ e− 2E α → β α1 β1 α2 β2 ∆m2L =sin 2(2θ)sin2 Exercise 4E ‣The oscillation phase: ∆m2 L ∆m2 [eV 2]L[km] φ = 21 =1.27 21 Exercise 4E E[GeV ] → short distances, ϕ << 1: oscillations do not develop, Pαβ = 0 → long distance, ϕ ∼ 1: oscillations are observable → very long distances, ϕ >> 1: oscillations are averaged out: 1 P sin2(2θ) αβ 2 2-neutrino oscillation probability ∆m2L P =sin2(2θ)sin2 αβ 4E oscillation amplitude averaged oscillations oscillation length: 4πE first oscillation maximum: L = osc ∆m2 Appearance vs disappearance experiments ∆m2L ‣appearance experiments: P =sin2(2θ)sin2 αβ 4E α = β → appearance of a neutrino of a new flavour β in a beam of να ∆m2L ‣disappearance experiments: P =1 sin2(2θ)sin2 αα − 4E → measurement of the survival probability of a neutrino of given flavour Example for 2-neutrino oscillations in vacuum: reactor experiment Production and detection of reactor neutrinos The CHOOZ reactor experiment Chooz B Nuclear Power Station 2 x 4200 MWth 2 ‣ 2ν approx (Δm , θ): 2 2 2 ∆m L Pee =1 sin (2θ)sin − 4E i distance = 1.0 km ‣ L = 1 km, E∼MeV Depth 300 mwe → sensitive to Δm2∼ 10-3 eV2 neutrino target ‣ non–observation of νe disappearance: Chooz Underground Neutrino Laboratory Ardennes, France ratio between measured and expected number of events Matter effects on neutrino oscillations ‣When neutrinos pass trough matter, the interactions with the particles in the medium induce an effective potential for the neutrinos. [→ the coherent forward scattering amplitude leads to an index of refraction neutrinos. L. Wolfenstein, 1978] → modifies the mixing between flavor states and propagation states as well as the eigenvalues of the Hamiltonian, leading to a different oscillation probability wrt vacuum. Effective matter potential ‣ Effective four-fermion interaction Hamiltonian (CC+NC) G Hνα = F ν γ (1 γ )ν fγµ(gα,f gα,f γ )f int √ α µ − 5 α V − A 5 2 j in ordinary matter: f=e-,p,n µα Jmatt To obtain the matter-induced potential we integrate over f-variables: 1 non-relativistic fγµf = N δ 2 f µ,0 for a unpolarised medium: fγ γµf =0 5 neutral Ne = Np 1 J µα = [N (gα,e + gα,p)+N gα,n] matt 2 e V V n V Effective matter potential 1 J µα = [N (gα,e + gα,p)+N gα,n] matt 2 e V V n V 1 1 1 J µα =(N N , N , N ) matt e − 2 n −2 n −2 n 1 1 1 V = √2G diag(N N , N , N ) matt F e − 2 n −2 n −2 n ‣only νe are sensitive to CC (no μ,τ in ordinary matter) ‣NC has the same effect for all flavours → it has no effect on evolution (however it can be important in presence of sterile neutrinos) ‣for antineutrinos the potential has opposite sign 2-neutrino oscillations in matter E 0 ‣ Hvac = 1 2-neutrino Hamiltonian in vacuum (mass basis): 0 E 2 ‣In the flavour basis, where effective matter potential is diagonal: ∆m2 ∆m2 matt vac 4E cos 2θ + VCC 4E sin 2θ Hf = Hf + Veff = − ∆m2 ∆m2 4E sin 2θ 4E cos 2θ VCC = √2GF Ne Diagonalizing the Hamiltonian, we identify the mixing angle and mass splitting in matter: 2 matt ∆M cos 2θM sin 2θM Exercise: calculate Hf = − 2 4E sin 2θM cos 2θM θM and ΔM 2 In general: Ne=Ne(x), so θM and ΔM will be function of x as well → however, in some cases analytical solutions can be obtained 2-ν oscillations in constant matter ‣If Ne is constant (good approximation for oscillations in the Earth crust): 2 → θM and ΔM are constant as well → we can use vacuum expression for oscillation probability, replacing “vacuum” parameters by “matter” parameters: ∆M 2L P =sin2(2θ )sin2 αβ M 4E sin2 2θ sin2 2θ = M 2 2 2EV sin 2θ + (cos 2θ A) A = − ∆m2 ∆M 2 = ∆m2 sin2 2θ + (cos 2θ A)2 − There is a resonance effect for A = cos2θ → MSW effect Wolfenstein, 1978 Mikheyev & Smirnov, 1986 2-ν oscillations in constant matter mixing angle in matter: 2 2 sin 2θ sin 2θM = sin2 2θ + (cos 2θ A)2 − 2EV A = ∆m2 A ‣A << cos2θ, small matter effect → vacuum oscillations: θM = θ ‣A >> cos2θ, matter effects dominate → oscillations are suppressed: θM ≈ 0 ‣A = cos2θ,resonance takes place→ maximal mixing θM ≈ π/4 → resonance condition is satisfied for neutrinos for Δm2 > 0 for antineutrinos for Δm2 < 0 Solar neutrinos: the MSW effect ‣ neutrino oscillations in matter were first discussed by Wolfenstein, Mikheyev and Smirnov (MSW effect) ‣ electron neutrino is born at the center of the Sun as: ν = cos θ νm +sinθ νm | e M | 1 M | 2 m m → ν1 and ν2 evolve adiabatically until the solar surface and propagate in vacuum from the Sun to the Earth: P (ν ν )=P prodP det + P prodP det e → e e1 1e e2 2e prod 2 det 2 Pe1 = cos θM ,P1e = cos θ prod 2 det 2 Pe2 =sin θM ,P2e =sin θ 2 2 2 2 Pee = cos θM cos θ +sin θM sin θ Solar neutrinos: the MSW effect 2 2 2 2 Pee = cos θM cos θ +sin θM sin θ ‣In the center of the Sun: 5 2 2EV E 8 10− eV A = 0.2 ν × ∆m2 MeV ∆m2 c 3 Exercise: check this formula: ne ∼100 mol/cm and resonance occurs for A = cos(2θ) = 0.4 → Eres ≈ 2 MeV ‣For E < 2 MeV → vacuum osc: θM = θ 1 P =1 sin2 2θ ee − 2 2 ‣For E > 2 MeV → strong matter effect: θM = π/2 Pee =sin θ → Pee (E) will be crucial to understand solar neutrino data Example for 2-neutrino oscillations in matter: solar neutrinos 1 Solar neutrinos produced in nuclear reactions in the Reaction source Flux (cm−2s−1) core of the Sun: pp→ de+ ν pp 5.97(1 ± 0.006)×1010 pp cycle pe− p → d ν pep 1.41(1 ± 0.011)×108 3He p → 4He e+ ν hep 7.90(1± 0.15)×103 4 + 4p→ He + 2e + 2νe + γ 7Be e− → 7Li νγ 7Be 5.07(1 ± 0.06)×109 8B → 8Be∗ e+ ν 8B 5.94(1 ± 0.11)×106 CNO 13N → 13Ce+ ν 13N 2.88(1 ± 0.15)×108 15 15 + 15 0.17 8 O → Ne ν O 2.15(1 ± 0.16)× 10 17 17 + 17 0.19 6 F → Oe ν F 5.82(1 ± 0.17)×10 ν fluxes SSM predictions ν energy spectra Radiochemical solar experiments Homestake (Cl) experiment: 1967-2002 ‣ gold mine in Homestake (South Dakota) ‣ 615 tons of perchloro-ethylene (C2Cl4) ‣ detection process (radiochemical) ‣ only 1/3 of SSM prediction detected: Gallium radiochemical experiments: 50% deficit Solar neutrinos in Super-Kamiokande - water cherenkov detector - sensitive to all neutrino flavors: - - νx e → νx e - threshold energy ∼ 4-5 MeV - real-time detector: (E, t) ➡ Super-Kamiokande detects less neutrinos than expected according to the SSM (40%) The solar neutrino problem ➡ All the experiments detect less neutrinos than expected (30-50%) Why the deficit observed is different? ‣different type of neutrinos observed → radiochemical: νe while Super-K: να ‣different E-range sensitivity: → Cl: E > 0.814 MeV → Ga: E > 0.233 MeV → Super-K: E > 5 MeV ∼30% ∼50% ~40% Different energy suppression
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