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Oscillation formalism Neutrino mixing Oscillations of 2 flavors Experimental sensitivities Oscillations in 3 flavors Discovery of oscillations: atmospheric neutrinos in Super-Kamiokande

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Quark mixing in Standard Model uc t d`s` b` „From neutrinos to cosmic sources”, DK & ER States partcipating in strong interactions with well defined masses (mass matrix eigenstates): uc t ds b States participating in weak interactions: Quark mixing:

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Neutrino mixing NOT in Standard Model IF neutrinos are massive: States with well defined masses (mass matrix eigenstates): States participating in weak interactions: Lepton mixing:

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Neutrino oscillation – 2 flavors changes during propagation, hence mass states: mixing angle: are defined as different proportions of 1, 2 states states have different masses different velocities 1, 2 The ratio

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Oscillation probability – 2 flavors A state of mass m k, energy and momentum E k,p k propagates: Let’s assume an initial state: with phase:

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Oscillation probability – 2 flavors A probability that after t,x the state is still in its initial state: During propagation the contribution of 1, 2 components changes: Finally:

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Oscillation probability – 2 flavors Probability of transition from a state to a state : E – neutrino energy (in GeV) L distance from a neutrino source to detector (km) oscillation parameters experimental conditions: m mass (in eV) mixing angle Oscillation length:

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Appearance and disappearance experiments In an appearance experiment one searches for neutrinos in an initial beam of : In a disappearance experiment one counts how many of the initial neutrinos are left after passing a distance L: Note: Neutrino oscillate only if masses are non-zero and not the same

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Sensitivity to oscillations (MeV) L (m) Supernovae <100 >10 19 10 -19 - 10 -20 Solar <14 10 11 10 -10 Atmospheric >100 10 4 -10 7 10 -4 Reactor <10 <10 6 10 -5 Accelerator with short baseline >100 10 3 10 -1 Accelerator with long baseline >100 <10 6 10 -3

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Graphic illustration of neutrino oscillations For max mixing = /4 and at a distance L=L osc/2 all the initial flavor are transformed to another flavor

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Transitions between 3 mass states With 3 generations there are 3 Δ m 2 ’s but only two are independent.

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Mixing of 3 flavors For 3 flavors we need 3x3 matrix. In quark case the corresponding matrix is called CKM (Cabibo-Kobayashi-Maskava). For neutrinos MNS (Maki-Nakagava-Sakata)

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Mixing of 3 flavors (part 2) The 3x3 matrix has 4 independent real parameters: where: 4 independent parameters: Current experiments are not sensitive to φ It’s assumed

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Mixing of 3 flavors (part 3) „From neutrinos to cosmic sources”, D.Kiełczewska, E.Rondio The mixing matrix can be written: rotation by: =0

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Oscillation Probability – 3 flavors (part 1) Per analogy with 2 flavor case the amplitude for the neutrino oscillation:

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How do Neutrinos Oscillate? Amplitude

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Oscillation Probability – 3 flavors In a general case, with at least one non-zero complex phase: Note here: if = then the imaginary components disappear CP phase cannot be measured in disappearance experiments

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Oscillation Probability – 3 flavors ( =0) a 12 a 13 a 23

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Oscillation Probability – 3 flavors ( =0) Let’s assume: Then we have 2 types of experiments: Case A – „atmospheric” - small L/E: Case B – „solar” - large L/E

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Oscillation probability – 3 flavors ( =0) Case A – „atmospheric” - small L/E: Case B – „solar” - large L/E Note: for 13 =0 all formulas are the same as for 2 flavors ∆m >>δm

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More exact formula: By expanding in: one gets: L – baseline; solar term matter effects sensitivity to mass hierarchy CP violation + neutrinos - antineutrinos If LA<<1: We will introduce later: The above formula is necessary for future, more exact studies

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Let’s try to understand atmospheric neutrino puzzle

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Neutrino events in Super-K μ Upward stopping different energy scale different analysis technique different systematics Upward through-going interactions in rocks below the detector Contained events: Fully contained FC Partially contained PC e/ identification all assumed to be All have to be separated from „cosmic” muons (3Hz)

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Neutrino energy spectra Fully contained FC Partially contained PC e/μ identification all assumed to be μ Interactions in rocks Upμ stop Upμ thru

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Data MC 1ring e-like 772 707.8 -like 664 968.2 Sub-GeV (Fully Contained) E vis < 1.33 GeV, P e > 100 MeV, Pμ > 200 MeV Data MC 1-ring e-like 3266 3081.0 -like 3181 4703.9 Multi-GeV Fully Contained (E vis > 1.33 GeV ) Partially Contained (assigned as -like) Super-Kamiokande results (contained) 913 1230.0 We take ratios to cancel out errors on absolute neutrino fluxes: Too few muon neutrinos observed!

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Super-Kamiokande I results - upward going muons Up through-going , (1678days ) Data: 1.7 +- 0.04 +- 0.02 (x10 -13 cm -2 s -1 sr -1 ) MC: 1.97+-0.44 Up stopping , (1657days ) Data: 0.41+-0.02+-0.02 (x10 -13 cm -2 s -1 sr -1 ) MC: 0.73+-0.16 Again one observes a muon deficit

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Super-Kamiokande evidence for neutrino oscillations

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Interpretation of the zenith angle distributions Let’s try to find interpretation of the deficit of after passing the Earth...... Looks like disappearance... What happens to muon neutrinos? Let’s suppose an oscillation : We see that e angular distribution is as expected but what is

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Oscillations of muon neutrinos Looks like ν μ oscillates:.. Remember that we identify neutrinos by the corresponding charged lepton which they produce: But look at the masses: 106 MeV τ eV Does neutrino have enough energy to produce

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cross sections Total CC cross sections for: compared with masses: 106 MeV eV

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We don’t see the neutrinos after oscillations! The cross section for CC interaction (with ) too small NC interactions possible but then we cannot tell the neutrino flavor!

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Down, L=15 kmUp, L=12000 km For E =1 GeV For E =10 GeV For E =100 GeV Max probability of oscillation for L=L osc /2 Find corresponding Rough estimate of Oscillation length: E ν – neutrino energy (inGeV) L distance (km) The trouble is – we don’t know precisely E

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Sensitivity of angular distribution to neutrino oscillations Δm 2 = 10 -4 10 -2 10 -3 10 -1 Up-going Down-going No oscillations Expected angular distributions for oscillations with various mass parameters for the neutrino energy distribution of multi-GeV sample. Sensitivity from ~10 -4 to 10 -1 eV 2

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Zenith angle distributions e-like 1 ring -like 1 ring -like multi- ring upward going Sub-GeV Multi-GeV down Red: MC expectations Black points: Data Green: oscillations Missing are the muon neutrinos passing through the Earth! up

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Super-K up-down asymmetry expected- no oscil Data with -oscillations

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Rough estimate of the mixing angle Probability of disappearance... For large distances L oscillations happen for a variety of E so that one should take: Then: i.e. maximal mixing (at some angles we see half of neutrinos disappearing)

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A fit is performed i.e. a minimum of 2 is found. The corresponding are the best fit oscillation parameters Definition of 2 for oscillation analysis

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Results of combined fit vs m 2 flat between 0.0019 and 0.0025

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Contours for different subsamples Sub-GeV 1-ring e-like3353 Sub-GeV 1-ring -like3227 Multi-GeV 1-ring e-like 746 Multi-GeV 1-ring -like 651 PC -like 647 Multi-ring 647 Upward muons 2259 ------All 11530

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Oscillation parameters from different experiments (atmospheric)

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L/E analysis of the atmospheric neutrino data from Super-Kamiokande

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From neutrinos to cosmic sources, DK&ER Hypotheses other than oscillation Neutrino oscillation : Neutrino decoherence : Neutrino decay : Idea: use events with the best resolution in L/E

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From neutrinos to cosmic sources, DK&ER Reconstruction of E and L E observed E ν Neutrino energy is reconstructed from observed energy using relations based on MC simulation Zenith angle Flight length Neutrino flight length is estimated from zenith angle of particle direction Neutrino energy Neutrino direction

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From neutrinos to cosmic sources, DK&ER Neutrino path-length L vs angle Close to the horizon Very bad

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From neutrinos to cosmic sources, DK&ER Survival probability Null oscillation MC Best-fit expectation 1489.2 days FC+PC divide DATA/MC A dip just where oscil. min expected

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From neutrinos to cosmic sources, DK&ER Test for neutrino decay & neutrino decoherence Oscillation Decay Decoherence Alternative hypotheses excluded.

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From neutrinos to cosmic sources, DK&ER 47 Oscillation analysis – fitting L/E distribution

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Atmospheric neutrino experiments The largest statistics of atmospheric neutrino events were collected in Super-Kamiokande. The results showed: a deficit of muon neutrinos passing long distances through the Earth. first evidence of neutrino oscillatons Atmospheric neutrinos were also measured in MACRO and SOUDAN detectors. The results were consistent with neutrino oscillations.

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From neutrinos to cosmic sources, DK&ER Summary: evidence of oscillations in atmospheric neutrinos ● Missing Effect observed in different Super-K event samples and also by other experiments Stat. significance above 10 sigmas ● Angular distributions probability of disappearance depends on its path-length and energy in a way consistent with oscillation ● survival dependnce on L/E only oscillations can produce a dip ● Oscillation parameters from comparison between data and MC simulations:

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