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Susan Cartwright University of Sheffield Dirac and Majorana masses The mixing matrix and neutrino oscillations Massive neutrinos in the Standard Model In the original Standard Model, neutrinos are two- component spinors with mass exactly zero disproved by existence of neutrino oscillations—see later There are two ways to add a neutrino mass term to the SM Lagrangian Dirac: mD L R R L exactly like other fermion masses Majorana: 12 mL L L L L 12 mR R R R R where νc = Cν̄T = Cγ0ν* different chiral states need not have same mass in this case neutrino and “antineutrino” same particle, different chirality M c c M c c Seesaw mechanism General mass term has both Dirac and Majorana components: c M m mD c L L 1 Lmass 2 L R M h.c., where M M m m R D R If M mRM mD mLM , we can diagonalise matrix to get eigenstates mD mD c c c N R R L L , L L R Rc M M with masses naturally mD2 small mass mN M; m M for LH state Neutrino oscillations If neutrinos have mass, then they can be described in terms of mass eigenstates as well as weak (flavour) eigenstates no reason why these should align (and they don’t), so we have a 3×3 unitary mixing matrix (the PMNS matrix) U: Um m m mass eigenstates propagate according to m (t ) e i Emt pmL m (0) e 2 i (Mm 2E ) L m (0) if c = 1 and v ≈ c (and hence L ≈ t) therefore even if |ν(0)⟩ is a pure flavour state, |ν(L)⟩ is not Neutrino oscillations Probability of observing neutrinos of flavour ℓ' at distance L (in vacuo) from a beam of initial flavour ℓ: P ; L (0) Um 2 i (Mm / 2E )L * e U m For two-flavour case we have giving cos U sin sin cos 2 m therefore key variable for experiments is L/E 2 M L 2 2 (where M2 M22 M12 ) P ; L sin 2 sin 4 E can’t measure absolute masses Matter effects νe interact with electrons via W exchange; νμ, ντ do not This leads to an increased effective mass for a νe-dominated state in dense matter as ν propagates out through decreasing density, effective mass drops, eventually crossing another eigenvalue resulting resonant conversion can greatly enhance oscillation critical for solar neutrinos, significant for long baseline terrestrial too MSW effect—sees sign of ΔM2 Theory Summary Non-zero neutrino masses imply neutrino oscillation (change of effective flavour) hence, non-conservation of lepton family number this is experimentally established also, if 3×3 mixing, non-conservation of CP imaginary phase δ in PMNS matrix not established yet if non-zero Majorana mass, ν = ν̄ hence, non-conservation of global lepton number not established Experimental tasks determine oscillation parameters (Δm2, θ, δ) measure neutrino masses Principles of oscillation measurements Solar neutrinos, θ12 Atmospheric neutrinos, θ23 New measurements of θ13 Principles of oscillation measurements Relevant physical properties are Δm2ij and θij Experiment parameters are L, E and initial flavour e, μ but physical parameter is L/E, so result is conversion probability P(L/E), giving contour on Δm2 – sin2 2θ plane Two distinct experimental techniques disappearance experiments look for reduction in flux of original flavour only possibility for very low-energy neutrinos (reactor ν̄e, solar νe) appearance experiments look for converted flavour e.g. νe events from a νμ beam The PMNS matrix 0 0 c13 1 U 0 c23 s23 0 0 s s e i c 23 23 13 atmospheric neutrinos νμ ↔ ντ 0 s13e i c12 1 0 s12 0 c13 0 reactor & accel. neutrinos νμ ↔ νe s12 c12 0 solar neutrinos νe ↔ νX need all three mixing angles to be non-zero for CP violation to be possible 0 0 1 Solar neutrinos Produced as by-product of hydrogen fusion 4 1H → 4He + 2e+ + 2νe reaction goes by many paths which produce neutrinos of different energies initial flavour state νe as too little energy to produce μ, τ Detected by inverse β decay, elastic scattering, or dissociation of 2H http://www.kip.uni-heidelberg.de/tt_detektoren/neutrinos.php?lang=en Solar neutrinos: θ12, 2 Δm 12 2 7.5 105 eV 2 requires L/E ~ 30 km/MeV in m12 vacuum experimental approaches: solar neutrinos: νe → νX disappearance resonant conversion via MSW effect in solar interior expected flux calculated from models of solar luminosity (John Bahcall et al.) experimental normalisation via NC reactions (SNO) reactor neutrinos: ν̄e → ν̅ X disappearance requires long-baseline experiment owing to small Δm2 expected flux from known reactor power output Solar neutrinos: θ12, 2 Δm 12 NC: d + νX → p + n + νX CC: d + νe → p + p + e− ES: e− + νX → e− + νX Solar neutrinos: θ12, 3-flavour analysis of SNO+KamLAND data gives 2 21 m12 7.4100..19 105 eV 2 030 tan2 12 0.44600..029 018 sin2 13 0.02500..015 (arXiv 1109.0763) 2 Δm 12 Solar neutrinos: new results Measurement of 7Be and pep flux by Borexino line fluxes, therefore potentially more informative about energy dependence Also no day-night asymmetry excludes low Δm2 region of plane this exclusion previously depended on reactor data (ν̄) Atmospheric neutrinos: θ23, 2 Δm 23 Initially studied using neutrinos produced in cosmic-ray showers incident proton or heavier nucleus produces pions which decay to μ + νμ some of the muons then also decay (to e + νe + νμ) if they all do so then νμ:νe ratio ~ 2 Also addressed by accelerator-generated neutrino beams essentially identical process: collide proton beam from accelerator with target, collimate produced pions with magnetic horns, allow to decay in flight magnets select charge of pion, hence either νμ or ν̄μ beam Atmospheric and accelerator νs −1 0 +1 −1 cos(zenith angle) 0 +1 Atmospheric neutrinos: θ23, Δm223 MINOS combined fit 2 09 m23 2.3900..10 103 eV 2 035 sin2 223 0.95700..036 3-flavour global fit 2 07 m23 2.4200..11 103 eV 2 030 sin2 223 0.95100..024 first time that best fit θ23 ≠ 45°! Third mixing angle θ13 Absence of signal in νe shows that atmospheric mixing is νμ → ντ measurements of ντ appearance in OPERA and Super-K are statistics-limited but in qualitative agreement with this Therefore 3rd mixing angle θ13 involves νe can be seen in νe appearance in νμ beam or ν̅ e disappearance from reactors because Δm213 = Δm223 ± Δm212 and Δm212 ≪ Δm223, νμ disappearance always dominated by θ23 νe appearance: Off-axis geometry produces lower-energy, much more monochromatic beam T2K beam is 2.5° off-axis— optimised for oscillation measurement T2K analysis 11 events observed 3.22±0.43 expected 3.2σ effect Super-Kamiokande measures Cherenkov radiation from e/μ produced in interaction can distinguish the two based on ring morphology fuzzy electron ring sharp muon ring ν̅e disappearance: Multiple detectors associated with extended reactor complex ν̅e detected via inverse β decay in Gd-loaded liquid scintillator Daya Bay analysis Large difference between Δm213 and Δm212 means that L/E can be “tuned” for θ13 no ambiguity—simple 2-flavour system “near” and “far” detectors identical to minimise systematics far/near ratio R = 0.940 ± 0.011 ± 0.004 5.2σ effect ν̅e disappearance: Detector design and analysis very similar to Daya Bay Results very similar too: R = 0.920 ± 0.009 ± 0.014 4.9σ effect Results for θ13 053 063 sin2 213 0.09400..040 (NH); 0.11600..049 (IH) Daya Bay: sin2 2θ13 = 0.092 ± 0.016 ± 0.005 Reno: sin2 2θ13 = 0.113 ± 0.013 ± 0.019 Global fit (Fogli et al. arXiv 1205.5254): T2K : 009 sin2 213 0.09400..008 (NH); 0.095 0.009 (IH) measurement of Δm2 still best done by combining solar & atmospheric Δm2 Open questions We know the νe-dominated state m1 is lighter than m2 (from MSW effect), but we still don’t know if m3 > m2 (normal hierarchy) or vice versa (inverted hierarchy) longer baseline experiments, e.g. NOνA, should be able to sort this out via matter effects in Earth Constraints on phase δ are very weak can be constrained by antineutrino running and/or matter effects (NOνA again) Effect on models Tri-bimaximal mixing predicts Ue1 2 Ue2 2 Ue3 2 2 1 0 3 3 2 2 2 1 1 1 U 1 U 2 U 3 6 3 2 1 1 1 2 2 2 U 1 U 2 U 3 6 3 2 and hence θ13 = 0, which it clearly isn’t. Theorists are of course trying to rescue this with perturbations of various kinds The current hint that θ23 ≠ 45° is also inconsistent with tri-bimaximal predictions Tritium beta-decay Neutrinoless double beta decay Astrophysical constraints Neutrino mass: β decay Basic principle: observe electron spectrum of β-decay very close to endpoint presence of mc2 term for neutrino will affect this unfortunately not by much!! tritium (3H) favoured because of combination of low Q-value (18.57 keV) and shortish half-life (12.3 y) β decay status and prospects Best efforts so far by Mainz and Troitsk experiments of late 90s: 1/2 2 2 m e Uei mi 2.3 eV i 1 Two experiments in pipeline should do much better 3 KATRIN—tritium decay experiment with planned sensitivity ~0.2 eV MARE—rhenium-187 experiment, similar reach 187Re has very low Q-value but extremely long half-life MARE uses single-crystal bolometers to get good energy resolution and measure differential spectrum Very hard experiments: don’t expect results for ~5 years Neutrinoless double-β decay Even-even isobars are lighter than odd-odd (pairing term) can be energetically permitted for nucleus (Z, A) to decay to (Z±2, A) but not (Z±1, A) these decays do happen through conventional ββ2ν mode, albeit at very low rate lifetime ≫ age of universe if neutrino is Majorana particle, can also happen with no neutrino emission, ββ0ν Key features Violates lepton number by 2 units possible relevance to baryogenesis 2 mi Sensitive to mee Uei PMNS matrix multiplied by diag(1, eiα, eiβ) introducing two additional phases 2 Rate 0 Gx (Q, Z ) M x (A, Z ) x x for SM, amplitude ∝ mi/q2 where m ~ 0.5 eV and q ~ 108 eV small! nuclear matrix element M is a major systematic error theoretical calculations disagree by factors of 2 or more Relation of ⟨mee⟩ to lightest mν hep-ph/1206.2560 Most ββ isotopes are only ~10% of natural element. Enrichment is often needed. Experimental issues Signature: (A, Z) → (A, Z+2) + 2e−, so E(e−) = Q/2 —spike at energy endpoint p(e1) = −p(e2) —electrons are back to back Two experimental approaches 100 % abund source = detector Nd-150 Xe-136 Te-130 Sn-124 Cd-116 Pd-110 0.1 Mo-100 tracker; topological signature target isotope variable Zr-96 1 Se-82 source ≠ detector 10 Ge-76 calorimetric; energy signature target isotope fixed Ca-48 232Th 60Co not yet quite in range of interesting limits 10 Next few years: improvement of ~ ×10 EXO-1000, CUORE, KamLAND-ZEN, GERDA/MAJORANA all hoping for ~0.02-0.06 eV 1 0.1 Nd-150 Xe-136 Te-130 Sn-124 Cd-116 Pd-110 Mo-100 Zr-96 Se-82 ⟨mee⟩ ~ 0.2-0.6 eV Ge-76 Best results probe down to Ca-48 Experimental results 10 9 8 7 6 5 4 3 2 1 0 <mee> (eV) Effect on models Sensitivity to hierarchy: IH implies accessible minimum mass (∝ Δm213) within reach of next generation Non-observation possible even with Majorana neutrinos in NH masses and phases can conspire to cancel effect Conversely, ββ0ν decays can be driven by mechanisms other than Majorana mass (e.g. LR symmetry) such mechanisms do imply that the neutrino has a Majorana mass, but it can be very small Astrophysical constraints on mν Number density of relic neutrinos from early universe (CνB) is 112 per species per cm3 these are hot dark matter and will affect structure formation—hence leave astrophysical signatures Sensitive to ∑mν, which is bounded below by Δm223 ~ 0.0024 eV2 ∑mν (eV) oscillations 1.4 1.2 1 0.8 0.6 0.4 0.2 0 means ∑mν ≥ 0.05 eV bounds are within factor of 10 and will improve soon (e.g. Planck) Limit on ∑mν Model dependence Quoted constraints assume flat geometry exactly 3 neutrino species with Tν = (4/11)1/3 TCMB dark energy is a cosmological constant There are correlations between ∑mν and other parameters WMAP5 W. Rodejohann, hep-ph/1206.2560 Comparing different measures ββ0ν KATRIN ββ0ν Cosmology yes no yes no yes QD+M QD+D QD N-S C no N-S I low IH/NH/D m < 0.1 eV/N-S C NH yes (IH/QD)+M N-S C/I no low IH/(QD+D) NH D = Dirac; M = Majorana; QD = quasi-degenerate; NH/IH = normal/inverted; N-S C = non-standard cosmology; N-S I = non-standard interpretation of ββ0ν Assumes sensitivities of mβ = 0.2 eV, ⟨mee⟩ = 0.02 eV, ∑mν = 0.1 eV Conclusion: it does help to have multiple approaches. Non-standard interpretations of ββ0ν Light sterile neutrinos Neutrino astrophysics and cosmology All 3 neutrino mixing angles are definitely non-zero— naïve tri-bimaximal mixing ruled out Constraints on δ and determination of hierarchy should be possible with next generation of oscillation experiments Experimental limits on neutrino mass do not currently compete with cosmological constraints, but next decade should see complementarity developing Physics of massive neutrinos is a rich and interesting field! Solar neutrinos Double beta decay Hot dark matter Hannestad et al., astro-ph/1004.0695 WMAP7 + halo power spectrum WMAP7 + HPS + H0 Giusarma et al., astro-ph/1102.4774 Sterile neutrinos and axions can also contribute to hot dark matter