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1
**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**

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

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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**

mixing angle: mass states: ϑ are defined as different proportions of ν1 ,ν2 states ν1 ,ν2 states have different masses different velocities The ratio changes during propagation, hence

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**Oscillation probability – 2 flavors**

A state of mass mk, energy and momentum Ek,pk propagates: with phase: Let’s assume an initial state:

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**Oscillation probability – 2 flavors**

During propagation the contribution of ν1,ν2 components changes: A probability that after t,x the state α is still in its initial α state: Finally:

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**Oscillation probability – 2 flavors**

Probability of transition from a state α to a state β: oscillation parameters m - mass (in eV) ϑ - mixing angle experimental conditions: Eν – neutrino energy (in GeV) L - distance from a neutrino source to detector (km) Oscillation length:

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**Appearance and disappearance experiments**

In an appearance experiment one searches for neutrinos νβ in an initial beam of να : In a disappearance experiment one counts how many of the initial neutrinos να 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**

Eν (MeV) L (m) Supernovae <100 >1019 Solar <14 1011 10-10 Atmospheric >100 10-4 Reactor <10 <106 10-5 Accelerator with short baseline 103 10-1 long baseline 10-3

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**Graphic illustration of neutrino oscillations**

For max mixing ϑ=π/4 and at a distance L=Losc/2 all the initial flavor νa are transformed to another flavor νb

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**Transitions between 3 mass states**

With 3 generations there are 3 Δm2’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 φ=0

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**Mixing of 3 flavors (part 3)**

The mixing matrix can be written: φ=0 rotation by: rotation by: rotation by: „From neutrinos to cosmic sources”, D.Kiełczewska, E.Rondio

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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 Amplitude

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**Oscillation Probability – 3 flavors**

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

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**Oscillation Probability – 3 flavors (φ=0)**

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**Oscillation Probability – 3 flavors (φ=0)**

Let’s assume: Then we have 2 types of experiments: Case A – „atmospheric” - small L/E: Tu Ewa doszla 16/04/2008 Case B – „solar” - large L/E

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**Oscillation probability – 3 flavors (φ=0)**

∆m >>δm Case A – „atmospheric” - small L/E: Note: for ϑ13=0 all formulas are the same as for 2 flavors Case B – „solar” - large L/E

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**More exact formula: one gets: By expanding in: + neutrinos**

- antineutrinos solar term CP violation L – baseline; We will introduce later: Mena v1.pdf matter effects sensitivity to mass hierarchy If LA<<1: 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**

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

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**Neutrino energy spectra**

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

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**Super-Kamiokande results (contained)**

Sub-GeV (Fully Contained) Multi-GeV Evis < 1.33 GeV, Pe > 100 MeV, Pμ > 200 MeV Fully Contained (Evis > 1.33 GeV) Data MC 1-ring e-like μ-like Data MC 1ring e-like μ-like Partially Contained (assigned as m-like) 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 μ, (1678days) Data: (x10-13 cm-2s-1sr-1) MC: Up stopping μ, (1657days) Data: (x10-13cm-2s-1sr-1) MC: 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 νμ after passing the Earth Looks like νμ disappearance... What happens to muon neutrinos? Let’s suppose an oscillation: but what is We see that ne angular distribution is as expected

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**Oscillations of muon neutrinos**

Looks like νμ oscillates:.. Remember that we identify neutrinos by the corresponding charged lepton which they produce: But look at the masses: μ MeV τ MeV Does neutrino have enough energy to produce τ ?

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**ντ cross sections Total CC cross sections for: compared with νμ**

masses: μ MeV τ MeV

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**We don’t see the neutrinos after oscillations!**

The cross section for CC interaction (with τ ) too small NC interactions possible but then we cannot tell the neutrino flavor!

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**Rough estimate of Oscillation length: Eν – neutrino energy (inGeV)**

L - distance (km) Oscillation length: Max probability of oscillation for L=Losc/2 Find corresponding Down, L=15 km Up, L=12000 km For Eν=1 GeV For Eν=10 GeV For Eν=100 GeV The trouble is – we don’t know precisely Eν

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**Sensitivity of angular distribution to neutrino oscillations**

Expected angular distributions for oscillations with various mass parameters for the neutrino energy distribution of multi-GeV sample. No oscillations 10-2 Δm2 = 10-4 10-3 10-1 Sensitivity from ~10-4 to 10-1 eV2 Up-going Down-going

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**Zenith angle distributions**

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

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**Super-K up-down asymmetry**

expected- no oscil Data with -oscillations Fig The up-to-down asymmetry for muon (2486) and electron (2531) single ring fully contained and partially contained (665) events in SuperK, from days of live time (analyzed by 6/00), as a function of observed charged particle mo- mentum. The muon data include a point for the partially contained events (PC) with more than about 1 GeV . The hatched region indicates no-oscillation expec- tations, and the dashed line µ - oscillations with m2 = 3.2 × 10-3eV 2 and maximal mixing[21]. Learned

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**Rough estimate of the mixing angle**

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

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**Definition of 2 for oscillation analysis**

A fit is performed i.e. a minimum of χ2 is found. The corresponding are the best fit oscillation parameters

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**Results of combined fit**

c2 vs Dm2 flat between and

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**Contours for different subsamples**

Sub-GeV 1-ring e-like 3353 Sub-GeV 1-ring μ-like 3227 Multi-GeV 1-ring e-like Multi-GeV 1-ring μ-like PC μ-like Multi-ring Upward muons ------All

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**Oscillation parameters from different experiments (atmospheric)**

J. Goodman, LP2001

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

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**Hypotheses other than oscillation**

Neutrino oscillation : Neutrino decay: Neutrino decoherence : Przez oddz z jakims polem w prozni moze zostac zaburzony warunek koherencji Idea: use events with the best resolution in L/E From neutrinos to cosmic sources, DK&ER

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**Reconstruction of Eν and L**

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

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**Neutrino path-length L vs angle**

Very bad Close to the horizon From neutrinos to cosmic sources, DK&ER

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**Survival probability divide DATA/MC**

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

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**Test for neutrino decay & neutrino decoherence**

Oscillation Decay Decoherence Alternative hypotheses excluded. From neutrinos to cosmic sources, DK&ER

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**Oscillation analysis – fitting L/E distribution**

From neutrinos to cosmic sources, DK&ER

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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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**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: From neutrinos to cosmic sources, DK&ER

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