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 << 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 of solar ﬂuxes

‣ Ga experiments: pp neutrinos

1 P =1 sin2 2θ ee − 2

with sin2 2θ 0.84 Pee > 0.5

‣ Cl + Super-K: 8B neutrinos

2 Pee =sin θ

→ Pee ∼ 0.3 → stronger neutrino deﬁcit is expected The Sudbury Neutrino Observatory, SNO

SNO is sensitive to all ν ﬂavors:

3 phases:

νe ﬂux (CC): 30% total ν ﬂux (NC): 100% !! The solar neutrino problem

All neutrinos are there!!

The Sun produces νe that arrive to the Earth as 1/3 νe + 1/3 νμ + 1/3 ντ

➡ ﬂavor conversion: νe → νx Conversion mechanism ? Neutrino oscillations ?? Solar neutrino oscillations

Homestake (Eν > 0.814 MeV) 37 37 - νe + Cl → Ar + e

SAGE/GALLEX-GNO (Eν >0.233 MeV) 71 71 - νe + Ga→ Ge + e

Super-Kamiokande (Ee ≳ 5 MeV) - - νx + e → νx + e

SNO (Ee ≳ 5 MeV) - [CC] νe + d → p + p + e [NC] νx + d → νx + n + p - - [ES] νx + e → νx + e

→ only LMA allowed at 3σ → max. mixing excluded at 5σ The KamLAND reactor experiment

Kamioka Liquid scinitillator Anti-Neutrino Detector

reactor experiment: ν + p n + e+ e → 55 commercial power reactors

average distance ∼ 180 km 2 -5 2 → Eν/L sensitivity range: Δm ∼ 10 eV

→correct order of magnitude to test solar neutrino oscillations in LMA region

CPT invariance: same oscillation 2 channel as solar νe (Δm 21 , θ12) Results from KamLAND

80 no-oscillation accidentals 16 ν¯ disappearance 13C(_,n) O 2002: First evidence e 60 spallation best-fit oscillation + BG → conﬁrmation of solar LMA ν oscillations KamLAND data 40 KamLAND Coll, PRL 90 (2003) 021802

Events / 0.425 MeV 1.4 2.6 MeV prompt 20 KamLAND data analysis threshold best-fit oscillation 1.2 : spectral distortions (L/E) best-fit decay 2004 best-fit decoherence 1 0 012345678 o

i 0.8 E (MeV) prompt KamLAND Coll, PRL 94 (2005) 081801 Rat Data - BG - Geo 0.6 ie Expectation based on osci. parameters 1 0.4 determined by KamLAND 2008: 1-period oscillations observed 0.2 0.8 2 0 → high precision determination Δm 21 0.620 30 40 50 60 70 80 L /E (km/MeV) 0 ie 0.4 KamLAND Coll, PRL 100 (2008) 221803 Survival Probability 0.2

0 20 30 40 50 60 70 80 90 100 L /E (km/MeV) 0 ie Combined analysis solar + KamLAND

90%CL, 3σ KamLAND conﬁrms solar neutrino oscillations.

Best ﬁt point: 2 +0.016 sin θ12 = 0.320 -0.017 2 -5 2 Δm 21 = 7.62 ± 0.19 x 10 eV

max. mixing excluded at more than 7σ Forero, M.T., Valle, PRD86, 073012 (2012) arXiv:1205.4018 [hep-ph]

➡ Bound on θ12 dominated by solar data.

2 ➡ Bound on Δm 21 dominated by KamLAND. The atmospheric neutrino sector The atmospheric neutrino anomaly

Cosmic rays interacting with the Earth atmosphere producing pions and kaons, that decay generating neutrinos:

then, one expects:

However, this prediction is in disagreement with the experimental results: Atmospheric neutrinos

450 Sub-GeV e-like 450 Sub-GeV µ-like 400 400 350 350 1998: Evidence νμ oscillations at Super-K 300 300 250 250 200 200 150 150

oscillation channel νμ →ντ Number of Events 100 Number of Events 100 50 50 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosθ cosθ 2004: oscillatory L/E pattern Multi-GeV e-like 350 Multi-GeV µ-like + PC 140 300 120 250 100 80 200 60 150 40 100 Number of Events Number of Events 20 50 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 cosθ cosθ

Upward Through Going µ Upward Stopping µ ) 4 -1 1.4

sr 3.5 -1

s 1.2

-2 3 1 cm 2.5 -13 2 0.8 1.5 0.6

Flux(10 1 0.4 0.5 0.2 0 0 -1 -0.8 -0.6 -0.4 -0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 Super-K Coll, PRL93, 101801 (2004) cosθ cosθ

∆m2 L P =1 sin2 2θ sin2 32 Super-K Coll., PRL 8 (1998) 1562. µµ − 23 4 E ν Combined analysis atmospheric + LBL data

➡ Super-Kamiokande (I + II + III) + K2K + MINOS + T2K LBL data

NH IH

➡ Determination of 2 θ23 and Δm 31 is now dominated by LBL data

90%CL, 3σ Forero, M.T., Valle, 2012

Best ﬁt point: 2 +0.026 2 +0.022 sin θ23 = 0.600 sin θ23 = 0.613 *local bf in 0.427 -0.031 -0.040 2 +0.06 -3 2 2 +0.07 -3 2 Δ Δm 31 = -(2.43 x 10 ) eV m 31 = 2.55-0.08 x 10 eV -0.06 Short-baseline reactor experiments New generation of reactor experiments

Near Detector Far detector

First oscillation maximum

more powerful reactors (multi-core) larger detector volume 2–6 detectors at 100 m – 1 km. New generation reactor experiments

2 reactor cores + 1 FD (ND 2013) livetime: 227.9 days 2 sin 2θ13 = 0.109 ± 0.030 (stat) ±0.025 (syst) 2 → sin 2θ13 = 0 excluded at 2.9σ Double Chooz Coll, PRD 86 (2012) 052008

6 reactor cores + 6 detectors (3ND,3FD) livetime: 139 days 2 sin 2θ13 = 0.089 ± 0.010 (stat) ±0.005 (syst)

2 → sin 2θ13 = 0 excluded at 7.7σ Daya Bay Coll., Chin. Phys. C 37 (2013) 011001

6 reactor cores + 2 detectors (ND,FD) livetime: 229 days 2 sin 2θ13 = 0.113 ± 0.013 (stat) ±0.019 (syst)

→ θ13=0 excluded at 4.9σ

RENO Coll., PRL 108 (2012) 191802 3-neutrino probabilities in vacuum

νe disappearance channel

reactor exp with atm L/E reactor exp with solar L/E (CHOOZ) (KamLAND)

νμ → νe appearance channel CP violating term: only present for appearance probabilities α≠β. Genuine 3-ﬂavor effect for antineutrinos: δ →-δ 3-neutrino masses and mixings

‣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

2 Δm 31 : atmospheric + long-baseline 2 Δm 21 : solar + KamLAND 3-neutrino masses and mixings

‣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

‣From the experimental data we know:

2 2 and θ Δm 31 >> Δm 21 13 << 1 → 3-ﬂavour effects are suppressed: dominant oscillations are well described by effective 2-ﬂavour oscillations ∆m2L P (ν ν )=sin2 2θ sin2 α → β 4E Three-neutrino effects

2 ‣θ13 effects in oscillations with Δm 21 : KamLAND + solar neutrinos

2 ‣θ13 effects in oscillations with Δm 31 : atmospheric and accelerator experiments

→ νe appearance in νμ beam

2 2 ‣Δm 21 effects in oscillations with Δm 31: reactor and atmospheric experiments

‣effects of CP violation: atmospheric and accelerator experiments 3-ﬂavour oscillation parameters

10%

IH 15%

- Deviations from 2-3 maximal mixing, θ23 > 45° preferred for IH. - Poor sensitivity to δCP - No indication for correct mass ordering Forero, M.T., Valle, PRD86, 073012 (2012) arXiv:1205.4018 [hep-ph]