1. NEUTRINO MASS, MIXING & OSCILLATION
D. P. Roy
Homi Bhabha Centre for Science Education
Tata Institute of Fundamental Research
Mumbai, India.

2. Contents
Basic Constituents of Matter and their Interactions : Matter Fermions and Gauge Bosons (Std Model)
Neutrino Chirality and Mass
Neutrino Mixing and Oscillation
Atmospheric Neutrino Oscillation
Solar Neutrino Oscillation
(Natural units: ħ & c = 1 => m = mc2 , mp  1 GeV)

3. Basic Constituents of Matter Mass (GeV)
Fermions (Spin = 1/2 ħ) e
Leptons νe νμ ντ 0
e μ 0.1 τ
Quarks u c t / d s b /3
For each Pair : Δe = 1 => Weak Int
Quarks also carry Colour Charge (C)=>Strong Int

4. Basic Int. Carriers (Gauge Bosons) mγ = 0, mg = 0, mW = 80, mZ = 90
1. E.M. Interaction Weak Interaction p
νe,μ,τ Z0 NC n p
νe,μ,τ
e,μ,τ
W - CC p e γ
ν e,μ,τ
Weak Decay n p e
W - μ νμ e
W - νe νe
GF ≈ GeV-2

5. Neutrino Chirality (Fermion Chirality)
Fermions have left & right chirality states
Strong and EM ints. conserve P => fL-fRsymmetric
Weak int breaks P maximally => only fL has weak int.
Only lefthanded quarks and leptons are isodoublets (I=1/2)
Righthanded quarks and charged leptons are singlets (I=0)
Neutrino has only weak int => no need of νR (none in SM)
All gauge ints. (Strong, EM & Weak) conserve CP => Exactly opposite case for antifermions : righthanded doublets and lefthanded singlets (no νL in SM)

6. Fermion Mass ψL is the wave function for lefthanded fermion fL
ψL is wave function for its antiparticle, i.e. righthanded antifermion fR
ψR is wave function for lefthanded antifermion fL
But fL is a doublet, while fR ( or fL) is a gauge singlet in the SM => Fermion mass explicitly breaks Isospin gauge symm of the Lagrangian, hence disallowed.
The fermion mass comes from its Yukawa interaction with scalar Higgs doublet. fL fR
‹h›
But no ν mass in SM <= No νR & νL

8. Neutrino Mass : See-Saw Model (mν ≈ 10-2 eV ≈ 10-11 GeV)
Assume presence of νR like all other fermions
νR has a unique property unlike any other fermion
=> has no gauge ch: e = C = I = 0
So νR= νL except opp. Chirality (Majorana)
Majorana Mass: ΔL=2, but breaks no gauge symm. => can be very large (M >>102 GeV)
Dirac mass (mψνRψνL): ΔI=1/2 => m ≤ 102 GeV
See-saw model =>

9. Importance of measuring the tiny neutrino mass:νL m
‹h› νR
m ~ 102 GeV
1/M
mν = m2/M
mν ~ 10-11GeV => M ~ 1015 GeV
Importance of measuring the tiny neutrino mass:
Provides indirect evidence of physics at the GUT scale ~ 1015 GeV,
which is far beyond the reach of any foreseeable accelerator.
2. Evidence for spontaneous lepton number violation (ΔL = 2) at the
GUT scale implies L being a gauge charge of the GUT sym group.
3. Lepton number violation at the GUT scale, creates an excess of
leptons over antileptons (leptogenesis), which can explain the present
baron asymmetry of the universe.
4. So it is important for understanding our very existence in the
Universe today.

10. Neutrino Mixing and Oscillation (simply consider mixing between two neutrino flavors)

11. l/λ

12. l ≥ λ / 2 ≈ 1.2E /Δm2 ═> measure Δm2 via neutrino oscillation
Solar & Reactor neutrino oscillation:
E ~ MeV, l (Sun) ~ 1011m & l (LBL reactor, KamLAND) ~ 105m
=>Δm2 ~ eV2 (Solar ν expt) & ~ 10-5 (KamLAND reactor ν expt).
Atmospheric & Accelerator neutrino Oscillation:
E ~ GeV, l (Atmospheric) ~ 104 km & l (LBL accelerator, MINOS)~ 103 km
=>Δm2 ~ 10-4 eV2 (Atmospheric ν expt) & ~ 10-3 eV2 (LBL expt: MINOS & K2K)
(These are far beyond the reach of any other method of mass measurement)
Indeed all these neutrino oscillation expts show definitive evidence of
small but nonzero Δm2 << eV2.

13. Atmospheric and Solar Neutrino Oscillations
Atmospheric neutrino oscillation expt.
Confirmation by LBL accelerator ν expts.
Solar neutrino source and expts.
Matter enhancement of solar neutrino osc.
Final soln to solar neutrino osc. (LMA soln)
Confirmation of LMA soln by LBL reactor ν expt.
Summary: present status & future prospects

15. KAMIOKA(Nucleon Decay Expt)→
KAMIOKA(Netrino Detection Expt)
KAMIOKANDE
→SUPERKAMIOKANDE (SK)
50 KT Water Cerenkov Detector
(30 m x 30 m x 60 m)
Surrounded by 12, ”PMTs
→Atmospheric, Solar, SN 1987A
& Accelerator Neutrinos (K2K)
Via Cerenkov Radiation from μ & e

16. e-like events match with expectation, i.e.
νe does not take part in oscillation =>
νμ→ ντ oscillation.
Low energy νμcorresponds
to small λ and hence shows
deficit at all zenith angles.
High energy νμ corresponds
to large λ and hence shows
deficit only for upgoing νμ
(l ~ λ).
SK, PRD (2005): Red Dashed (No Oscillation) & Green (Best Oscillation) fit

17. SK PRL ‘04
λ ∕2 = 1.2E/Δm2. So L/E ~ λ/2E ~ 500 km/GeV => Δm2 ~ 1.2/500 = 2.4x10-3 eV2.

18. νμ
Accelerator
Detector
l = 730 km (MINOS) & 250 km (K2K)

19. Solar Neutrino Source:

20. Solar Neutrino Fluxes vs Energy

21. Solar Neutrino Expts.
Chlorine (Homestake:Davis etal) & Gallium (SAGE, GALLEX) Expts:
Every few weeks the radioactive Germanium 71 and Argon 37 produced
are separated out. The number of produced atoms are counted via their
radioactive decays using GM counters => average νe flux => Pee.
Super K (Japan):
Real time expt, detecting elastic νe- scat. via Cerenkov radiation from the produced e-.
Also measures ν energy and direction from those of the produced electron.
SNO (Ontario, Canada): → Cerenkov detector (1KT D2O)

22. Matter Enhancement (Resonant Conversion): MSW effect
where s, c = sinθ, cosθ. Inserting V in RHS => M2 → M’2

23. For simplicity assume for a moment that sin θ << 1 => nondiagonal elements tiny
Approxly identify each diagonal element with an eigenvalue and the corresponding
eigenstate ν1,2 with the flavor eigenstates νe,μ.
Finite gap Γ = Δm2sin2θ => Pee = sin2θ,
Assuming adiabatic condition.

24. Adiabatic Condition (Landau-Zenner formula for Trans. Rate T)
If the solar Ne gradient is so small that the resulting λC(dλ1/dl)C << Γ
=> T is exponentially suppressed.
=> Pee = sin2θ

25. MSW Triangle
(E=1 MeV)

26. SK: PRL ‘01

27. Confirmation of LMA Soln. by LBL Reactor Neutrino Expt.(KamLAND):
1 KT liquid scintillator
(Bandyopadhyay, Choubey, Goswami, Petcov, Roy ’05)
Osc. Node at Evis ~ 5 MeV (E ~ 6 MeV) => λ = l = 1.8x105 m
=> Δm2 = 2.4E/λ = 8x10-5 eV2

28. Bandyopadhyay, Choubey, Goswami, Petcov, Roy ‘08

29. Present status of neutrino mass, mixing and oscillation:
Daya Bay=>

30. Future Prospects Third angle:
CHOOZ => sin2θ13 ≤ (→ 0.01 at Double-CHOOZ & Daya Bay Expts.)
→ SBL detectors, a few km from the Reactor, measuring νe disappearance.
2. Sign of Δm2 (atm):
May be measured via earth matter (e-) effect in atmospheric neutrino osc. Expts.
(SK & INO ?):
Problems: Effect proportional to the small νe comp in atm scale: sin2θ13 ??
Reconstruction of Eν from Eμ (without measuring π energies) ???
3. Absolute scale of neutrino mass cannot be determined via oscillation:
Tritium beta decay expts (Maintz & Moscow) => m1 ≤ 2 eV (→ 0.3 eV KATRIN)
But still much higher than the mass gaps, i.e.
4. Majorana nature of neutrino mass cannot me determined via oscillation:
NDBD expts: N (A,Z) → N’(A,Z+2) +2e- => m1 ≤ eV from H-M Ge
& CURICINO Te expts. (Upgrades of these expts planned => m1→ 0.05.)
They will show NDBD signal for the inverted mass hierarchy , but still not for
the normal hierarchy.