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Introduction: Massive neutrinos in the Standard Model

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:
- exactly like other fermion masses
- Majorana:
- where νc= Cν̄T= Cγ0ν*
- different chiral states need not have same mass in this case
- neutrino and “antineutrino” same particle, different chirality

Seesaw mechanism

- General mass term has both Dirac and Majorana components:
- If , we can diagonalise matrix to get eigenstateswith masses

naturally small mass 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:
- mass eigenstates propagate according toif 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:
- For two-flavour case we havegiving

therefore key variable for experiments is L/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

Neutrino oscillations

Principles of oscillation measurements

Solar neutrinos, θ12

Atmospheric neutrinos, θ23

New measurements of θ13

Principles of oscillation measurements

- Relevant physical properties are Δm2ijand 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 experimentslook for reduction in flux of original flavour
- only possibility for very low-energy neutrinos (reactor ν̄e, solar νe)
- appearance experimentslook for converted flavour
- e.g. νe events from a νμ beam

The PMNS matrix

atmosphericneutrinosνμ↔ντ

solar neutrinosνe↔νX

reactor & accel. neutrinosνμ↔νe

need all three mixing angles to be non-zero for CP violation to be possible

Solar neutrinos

- Produced as by-product of hydrogen fusion
- 4 1H →4He + 2e+ + 2νe
- reaction goes by many paths which produceneutrinos of differentenergies
- initial flavour state νeas too little energy toproduce μ, τ
- Detected by inverse βdecay, elastic scattering,or dissociation of 2H

http://www.kip.uni-heidelberg.de/tt_detektoren/neutrinos.php?lang=en

Solar neutrinos: θ12, Δm212

- requires L/E ~ 30 km/MeV in 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→ ν̅X disappearance
- requires long-baseline experiment owing to small Δm2
- expected flux from known reactor power output

Solar neutrinos: θ12, Δm212

- 3-flavour analysis of SNO+KamLAND data gives(arXiv 1109.0763)

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 Δm2region of plane
- this exclusion previouslydepended on reactordata (ν̄)

Atmospheric neutrinos: θ23, Δm223

- 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 neutrinos: θ23, Δm223

- MINOS combined fit
- 3-flavour global fit
- 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-limitedbut in qualitative agreement with this
- Therefore 3rd mixing angle θ13involves νe
- can be seen in νe appearance in νμ beam orν̅e 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 monochromaticbeam

T2K beam is 2.5° off-axis—optimised for oscillation measurement

3.22±0.43 expected

3.2σ effect

T2K analysis

- 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 decayin 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” detectorsidentical 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

- T2K :
- 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):
- measurement of Δm2 still best done bycombining 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 predictsand 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

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:
- Two experiments in pipeline should do much better
- 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
- Sensitive to
- PMNS matrix multiplied by diag(1, eiα, eiβ) introducing two additional phases
- Rate
- for SM, amplitude ∝mi/q2where m ~ 0.5 eV and q ~ 108eV
- 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
- source = detector
- calorimetric; energy signature
- target isotope fixed
- source ≠ detector
- tracker; topological signature
- target isotope variable

60Co

Experimental results

- Best results probe down to ⟨mee⟩ ~ 0.2-0.6 eV
- not yet quite in rangeof interesting limits
- Next few years: improvement of ~ ×10
- EXO-1000, CUORE,KamLAND-ZEN,GERDA/MAJORANAall hoping for ~0.02-0.06 eV

Effect on models

- Sensitivity to hierarchy: IHimplies accessible minimummass (∝ 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 oscillations
- Δm223 ~ 0.0024 eV2means ∑mν ≥ 0.05eV
- bounds are within factorof 10 and will improve soon(e.g. Planck)

Model dependence

- Quoted constraints assume
- flat geometry
- exactly 3 neutrino species with Tν= (4/11)1/3TCMB
- dark energy is a cosmological constant
- There are correlations between ∑mνand other parameters

WMAP5

Comparing different measures

W. Rodejohann, hep-ph/1206.2560

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 = 0.1eV

Conclusion: it does help to have multiple approaches.

Important things I don’t have time to discuss!

Non-standard interpretations of ββ0ν

Light sterile neutrinos

Neutrino astrophysics and cosmology

Summary and Conclusion

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!

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

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