Lepton Mixing and Neutrino Mass
v Werner Rodejohann T -1 (MPIK, Heidelberg) mv = m L - m D M R m D Strasbourg MANITOP Massive Neutrinos: Investigating their July 2014 Theoretical Origin and Phenomenology
1 Neutrinos!
2 Literature ArXiv: • – Bilenky, Giunti, Grimus: Phenomenology of Neutrino Oscillations, hep-ph/9812360 – Akhmedov: Neutrino Physics, hep-ph/0001264 – Grimus: Neutrino Physics – Theory, hep-ph/0307149 Textbooks: • – Fukugita, Yanagida: Physics of Neutrinos and Applications to Astrophysics – Kayser: The Physics of Massive Neutrinos – Giunti, Kim: Fundamentals of Neutrino Physics and Astrophysics – Schmitz: Neutrinophysik
3 Contents I Basics I1) Introduction I2) History of the neutrino I3) Fermion mixing, neutrinos and the Standard Model
4 Contents II Neutrino Oscillations II1) The PMNS matrix II2) Neutrino oscillations II3) Results and their interpretation – what have we learned? II4) Prospects – what do we want to know?
5 Contents III Neutrino Mass III1) Dirac vs. Majorana mass III2) Realization of Majorana masses beyond the Standard Model: 3 types of see-saw III3) Limits on neutrino mass(es) III4) Neutrinoless double beta decay
6 Contents I Basics I1) Introduction I2) History of the neutrino I3) Fermion mixing, neutrinos and the Standard Model
7 I1) Introduction Standard Model of Elementary Particle Physics: SU(3) SU(2) U(1) C × L × Y
o R + R Species # e R b R tR s R Quarks 10P 10 c R d i o u R i + R i e o L Leptons 3 13 + L e L tL Charge 3 16 c L u L b L s L d L Higgs 2 18
18 free parameters... + Gravitation + Dark Energy + Dark Matter + Baryon Asymmetry
8 Standard Model of Elementary Particle Physics: SU(3) SU(2) U(1) C × L × Y
o R + R Species # e R b R tR s R Quarks 10P 10 c R d i o u R i + R i e o L Leptons 3 13 + L e L tL Charge 3 16 c L u L b L s L d L Higgs 2 18
+ Neutrino Mass mν
9 Standard Model∗ of Particle Physics add neutrino mass matrix mν (and a new energy scale?)
Species # Quarks 10P 10 Leptons 3 13 Charge 3 16 Higgs 2 18
10 Standard Model∗ of Particle Physics add neutrino mass matrix mν (and a new energy scale?)
Species # Species # Quarks 10P 10 Quarks 10P 10 Leptons 3 13 Leptons 12 (10) 22 (20) −→ Charge 3 16 Charge 3 25 (23) Higgs 2 18 Higgs 2 27 (25)
11 SupersymmetryLHC Astrophysics ILC cosmic rays ... supernovae BBN Cosmology ... DarkBaryon Matter Asymmetry NEUTRINOS ...
Flavor physics GUT quark mixing ... SO(10) see−saw proton... decay
12 SupersymmetryLHC Astrophysics ILC cosmic rays ... supernovae BBN Cosmology ... BaryonDark Matter
NEUTRINOS Asymmetry ...
Flavor physics GUT quark mixing ... SO(10) see−saw proton... decay 13 General Remarks Neutrinos interact weakly: can probe things not testable by other means • – solar interior – geo-neutrinos – cosmic rays Neutrinos have no mass in SM • – probe scales m 1/Λ ν ∝ – happens in GUTs – connected to new concepts, e.g. new representations of SM gauge group, Lepton Number Violation particle and source physics ⇒
14 Contents I Basics I1) Introduction I2) History of the neutrino I3) Fermion mixing, neutrinos and the Standard Model
15 I2) History 1926 problem in spectrum of β-decay 1930 Pauli postulates “neutron”
16 1932 Fermi theory of β-decay
1956 discovery of ν¯e by Cowan and Reines (NP 1985) 1957 Pontecorvo suggests neutrino oscillations 1958 helicity h(ν )= 1 by Goldhaber V A e − ⇒ − 1962 discovery of νµ by Lederman, Steinberger, Schwartz (NP 1988) 1970 first discovery of solar neutrinos by Ray Davis (NP 2002); solar neutrino problem 1987 discovery of neutrinos from SN 1987A (Koshiba, NP 2002)
1991 Nν = 3 from invisible Z width 1998 SuperKamiokande shows that atmospheric neutrinos oscillate
2000 discovery of ντ 2002 SNO solves solar neutrino problem 2010 the third mixing angle
17 Future History 20ab CP shown to be violated 20cd mass ordering determined 20ef Dirac/Majorana nature shown 20gh value of neutrino mass measured 20ij new physics discovered 20kl ???
18 Contents I Basics I1) Introduction I2) History of the neutrino I3) Fermion mixing, neutrinos and the Standard Model
19 I3) Neutrinos and the Standard Model SU(3) SU(2) U(1) SU(3) U(1) with Q = I + 1 Y c × L × Y → c × em 3 2
νe Le = (2, 1) e ∼ − L e (1, 2) R ∼ − ν (1, 0) total SINGLET!! R ∼ oR +R
eR bR tR sR cR d io u R i+ R ie oL +L eL
tL cL uL bL sL dL
20 Masses in the SM:
= g L Φ e + g L Φ˜ ν + h.c. −LY e R ν R with ∗ ∗ + 0 νe ∗ φ φ L = and Φ=˜ iτ2 Φ = iτ2 = e φ0 φ+ L − after EWSB: Φ (0, v/√2)T and Φ˜ (v/√2, 0)T h i → h i → v v Y = ge eL eR + gν νL νR + h.c. me eL eR + mν νL νR + h.c. −L √2 √2 ≡ in a renormalizable, lepton number conserving model with Higgs doublets the ⇔ absence of νR means absence of mν
21 Quark Mixing
g + µ CC = W uL γ V dL −L √2 µ with u =(u,c,t)T , d =(d,s,b)T Quarks have masses they mix ↔ mass matrices need to be diagonalized ↔ flavor states not equal to mass states ↔ results in non-trivial CKM (Cabibbo-Kobayashi-Maskawa) matrix:
0.97427 0.00015 0.22534 0.00065 0.00351+0.00015 ± ± −0.00014 V = 0.22520 0.00065 0.97344+0.00016 0.0412+0.0011 | | ± −0.00016 −0.0005 0.00867+0.00029 0.0404+0.0011 0.999146+0.000021 −0.00031 −0.0005 −0.000046 small mixing related to hierarchy of masses?
22 Contents II Neutrino Oscillations II1) The PMNS matrix II2) Neutrino oscillations II3) Results and their interpretation – what have we learned? II4) Prospects – what do we want to know?
23 II1) The PMNS matrix Analogon of CKM matrix:
g µ − CC = ℓL γ U νL W −L √2 µ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
∗ να = Uαi νi connects flavor states να (α = e, µ, τ) to mass states νi (i = 1, 2, 3)
mass terms go like ℓL ℓR and νL νR (?)
(flavor = mass if m = 0 U = ½ ) ν ⇒
24 Number of parameters in U for N families: complex N N 2 N 2 2 N 2 × unitarity N 2 N 2 − rephase ν , ℓ (2 N 1) (N 1)2 i i − − − a real matrix would have 1 N (N 1) rotations around ij-axes 2 − in total: families angles phases 2 1 0 3 3 1 4 6 3 N 1 N (N 1) 1 (N 2) (N 1) 2 − 2 − − this assumes νν¯ mass term, what if Majorana mass term νT ν ?
25 Number of parameters in U for N families: complex N N 2 N 2 2 N 2 × unitarity N 2 N 2 − rephase ℓ N N(N 1) α − − a real matrix would have 1 N (N 1) rotations around ij-axes 2 − in total: families angles phases extra phases 2 1 1 1 3 3 3 2 4 6 6 3 N 1 N (N 1) 1 N (N 1) N 1 2 − 2 − − Extra N 1 “Majorana phases” because of mass term νT ν − (absent for Dirac neutrinos)
26 Majorana Phases connected to Majorana nature, hence to Lepton Number Violation • I can always write: U = UP˜ , where all Majorana phases are in • P = diag(1, eiφ1 , eiφ2 , eiφ3 ,...): 2 families: • cos θ sin θ 1 0 U = sin θ cos θ 0 eiα −
27 3 families: U = R R˜ R P • 23 13 12 −iδ 1 0 0 c13 0 s13 e c12 s12 0 = 0 c s 0 1 0 s c 0 P 23 23 − 12 12 0 s c s eiδ 0 c 0 0 1 − 23 23 − 13 13 iδ c12 c13 s12 c13 s13 e = s c c s s e−iδ c c s s s e−iδ s c P − 12 23 − 12 23 13 12 23 − 12 23 13 23 13 s s c c s e−iδ c s s c s e−iδ c c 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13 with P = diag(1, eiα, eiβ)
28 Contents II Neutrino Oscillations II1) The PMNS matrix II2) Neutrino oscillations II3) Results and their interpretation – what have we learned? II4) Prospects – what do we want to know?
29 II2) Neutrino Oscillations (in Vacuum) Neutrino produced with charged lepton α is flavor state
ν(0) = ν = U ∗ ν | i | αi αj | ji evolves with time as
ν(t) = U ∗ e−i Ej t ν | i αj | j i amplitude to find state ν = U ∗ ν : | βi βi | ii
(ν ν , t)= ν ν(t) = U U ∗ e−i Ej t ν ν A α → β h β| i βi αj h i| j i
δij
∗ −i Ei t | {z } = Uαi Uβi e
30 Probability:
P (ν ν , t) P = (ν ν , t) 2 α → β ≡ αβ |A α → β | ∗ ∗ −i (Ei−Ej ) t = Uαi Uβi Uβj Uαj e ij αβ P e−i∆ij Jij | {z } =|... = {zEXERCISE!} ∆ P = δ 4 Re αβ sin2 ij + 2 Im αβ sin∆ αβ αβ − {Jij } 2 {Jij } ij j>i j>i X X with phase
1 ∆ = 1 (E E ) t 1 p2 + m2 p2 + m2 L 2 ij 2 i − j ≃ 2 i i − j j 2 m2 m2−m2 1 mi j q i j 2 pi (1 + 2p2 ) pj (1p + 2p2 ) L 4E L ≃ i − j ≃ 2 2 2 1 m m ∆m L GeV ∆ = i − j L 1.27 ij 2 ij 4 E ≃ eV2 km E !
31 ∆ P = δ 4 Re αβ sin2 ij + 2 Im αβ sin∆ αβ αβ − {Jij } 2 {Jij } ij j>i j>i X X α = β: survival probability • α = β: transition probability • 6 2 requires U = ½ and ∆m = 0 • 6 ij 6 Pαβ = 1 conservation of probability • α ↔ Pαβ invariant under U eiφα U eiφj •Jij αj → αj Majorana phases drop out! ⇒
32 CP Violation In oscillation probabilities: U U ∗ for anti-neutrinos → Define asymmetries:
∆ = P (ν ν ) P (ν ν )= P (ν ν ) P (ν ν ) αβ α → β − α → β α → β − β → α αβ = 4 Im ij sin∆ij j>i {J } P 2 families: U is real and Im αβ = 0 α,β,i,j • {Jij } ∀ 3 families: • ∆m2 ∆m2 ∆m2 ∆ = ∆ = ∆ = sin 21 L + sin 32 L + sin 13 L J eµ − eτ µτ 2 E 2 E 2 E CP ∗ ∗ where JCP = Im Ue1 Uµ2 Ue2 Uµ1 1 = 8 sin2θ12 sin2θ23sin2θ13 cos θ13 sin δ 2 vanishes for one ∆mij = 0 or one θij = 0 or δ = 0, π
33 Two Flavor Case
Uα1 Uα2 cos θ sin θ αα 2 2 1 2 U = = 12 = Uα1 Uα2 = sin 2θ U U sin θ cos θ ⇒J | | | | 4 β1 β2 − ∆m2 and transition probability is P = sin2 2θ sin2 21 L αβ 4 E
34 35 amplitude sin2 2θ • maximal mixing for θ = π/4 ν = 1 (ν + ν ) • ⇒ α 2 1 2 q 2 2 E eV oscillation length Losc = 4π E/∆m21 = 2.48 GeV ∆m2 km • 21 2 2 L Pαβ = sin 2θ sin π ⇒ Losc is distance between two maxima (minima) e.g.: E = GeV and ∆m2 = 10−3 eV2: L 103 km osc ≃
36 37 L L : fast oscillations sin2 πL/L = 1 ≫ osc h osci 2 and P = 1 2 U 2 U 2 = U 4 + U 4 αα − | α1| | α2| | α1| | α2| sensitivity to mixing
38 L L : fast oscillations sin2 πL/L = 1 ≫ osc h osci 2 and P = 2 U 2 U 2 = U 2 U 2 + U 2 U 2 = 1 sin2 2θ αβ | α1| | α2| | α1| | β1| | α2| | β2| 2 sensitivity to mixing
39 L L : hardly oscillations and P = sin2 2θ (∆m2L/(4E))2 ≪ osc αβ sensitivity to product sin2 2θ ∆m2
40 large ∆m2: sensitivity to mixing small ∆m2: sensitivity to sin2 2θ ∆m2 maximal sensitivity when ∆m2L/E 2π ≃ 41 Characteristics of typical oscillation experiments
2 2 Source Flavor E [GeV] L [km] (∆m )min [eV ]
(−) (−) −1 2 4 −6 Atmosphere νe , νµ 10 ... 10 10 ... 10 10
−3 −2 8 −11 Sun νe 10 ... 10 10 10
−4 −2 −1 −3 Reactor SBL ν¯e 10 ... 10 10 10
−4 −2 2 −5 Reactor LBL ν¯e 10 ... 10 10 10 (−) (−) −1 2 −1 Accelerator LBL νe , νµ 10 ... 1 10 10 (−) (−) −1 Accelerator SBL νe , νµ 10 ... 1 1 1
42 Quantum Mechanics
Can’t distinguish the individual mi: coherent sum of amplitudes and interference
43 Quantum Mechanics Textbook calculation is completely wrong!! E E is not Lorentz invariant • i − j massive particles with different p and same E violates energy and/or • i momentum conservation definite p: in space this is eipx, thus no localization •
44 Quantum Mechanics consider E and p = E2 m2: j j j − j 2 q 2 ∂pj mj ∂pj pj E + mj 2 E ξ , with ξ = 2E 2 ≃ ∂mj ≡ − 2E − ∂mj mj =0 mj =0
2 2 2 ∂Ej mj mj Ej pj + mj 2 = pj + = E + (1 ξ) ≃ ∂mj 2pj 2E − mj =0
in pion decay π µν: → 2 2 2 mπ mµ mj Ej = 1 2 + 2 2 − mπ ! 2mπ thus,
2 2 1 mµ ∆mij ξ = 1+ 2 0.8 in Ei Ej (1 ξ) 2 mπ ! ≃ − ≃ − 2E
45 wave packet with size σx(> 1/σp) and group velocity vi = ∂Ei/∂pi = pi/Ei: ∼ (x v t)2 ψ exp i(E t p x) − i i ∝ − i − i − 4σ2 x
1) wave packet separation should be smaller than σx! L p L ∆v < σx < ⇒ Losc σp (loss of coherence: interference impossible)
2 2) mν should NOT be known too precisely! 2 2 2 ∂mν 2 pν Losc if known too well: ∆m δmν = δpν δxν 2 = ≫ ∂pν ⇒ ≫ ∆m 2π
(I know which state νi is exchanged, localization) In both cases: P = U 4 + U 4 (same as for L L ) αα | α1| | α2| ≫ osc
46 Quantum Mechanics total amplitude for α β should be given by → d3p A ∗ exp i(E t px) ∝ 2E Aβj Aαj {− j − } j j X Z with production and detection amplitudes (p p˜ )2 ∗ exp − j Aαj Aβj ∝ − 4σ2 p we expand around p˜j: ∂E (p) E (p) E (˜p )+ j (p p˜ )= E˜ + v (p p˜ ) j ≃ j j ∂p − j j j − j p=˜pj
and perform the integral over p: (x v t)2 A exp i(E˜ t p˜ x) − j ∝ − j − j − 4σ2 j x X
47 the probability is the integral of A 2 over t: | | v + v P = dt A 2 exp i (E˜ E˜ ) j k (˜p p˜ ) x | | ∝ − j − k v2 + v2 − j − k Z ( " j k # ) 2 2 2 (vj vk) x (E˜j E˜k) exp 2 − 2 2 2 −2 2 × (−4σx(vj + vk) − 4σp(vj + vk)) now express average momenta, energy and velocity as m2 p˜ E ξ j j ≃ − 2E
m2 m2 ˜ j p˜j j Ej E + (1 ξ) , vj = 1 2 ≃ − 2E E˜j ≃ − 2E this we insert in first exponential of P : ∆m2 L ˜ ˜ vj + vk jk (Ej Ek) 2 2 (˜pj p˜k) = " − vj + vk − − # 2E
48 the second exponential (damping term) can also be rewritten and the final probability is
2 2 2 ∆mij L 2 2 σx P exp i L coh 2π (1 ξ) osc ∝ − 2E − Ljk ! − − Ljk ! with 4√2E2 4πE Lcoh = σ and Losc = jk ∆m2 x jk ∆m2 | jk| | jk| expressing the two conditions (coherence and localization) for oscillation discussed before
49 Quantum Mechanics derivation of formula also works in QFT, when everything is a big Feynman diagram:
(Lorentz invariance, energy and momentum conservation at every vertex, etc.)
50 Contents II Neutrino Oscillations II1) The PMNS matrix II2) Neutrino oscillations II3) Results and their interpretation – what have we learned? II4) Prospects – what do we want to know?
51 II3) Results and their interpretation – what have we learned? Early main results as by-products: • – check solar fusion in Sun solar neutrino problem → – look for nucleon decay atmospheric neutrino oscillations → data for long time describable by 2-flavor formalism • genuine 3-flavor effects since 2011 • – third mixing angle! – mass ordering? – CP violation? have entered precision era •
52 3 families: U = R23 R˜13 R12 P
−iδ 1 0 0 c13 0 s13 e c12 s12 0 = 0 c s 0 1 0 s c 0 P 23 23 − 12 12 0 s c s eiδ 0 c 0 0 1 − 23 23 − 13 13 iδ c12 c13 s12 c13 s13 e = s c c s s e−iδ c c s s s e−iδ s c P − 12 23 − 12 23 13 12 23 − 12 23 13 23 13 s s c c s e−iδ c s s c s e−iδ c c 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13 with P = diag(1, eiα, eiβ)
53 Characteristics of typical oscillation experiments
2 2 Source Flavor E [GeV] L [km] (∆m )min [eV ]
(−) (−) −1 2 4 −6 Atmosphere νe , νµ 10 ... 10 10 ... 10 10
−3 −2 8 −11 Sun νe 10 ... 10 10 10
−4 −2 −1 −3 Reactor SBL ν¯e 10 ... 10 10 10
−4 −2 2 −5 Reactor LBL ν¯e 10 ... 10 10 10 (−) (−) −1 2 −1 Accelerator LBL νe , νµ 10 ... 1 10 10 (−) (−) −1 Accelerator SBL νe , νµ 10 ... 1 1 1
54 Interpretation in 3 Neutrino Framework assume ∆m2 ∆m2 ∆m2 and small θ : 21 ≪| 31|≃| 32| 13 atmospheric and accelerator neutrinos: ∆m2 L/E 1 • 21 ≪ ∆m2 P (ν ν ) sin2 2θ sin2 31 L µ → τ ≃ 23 4 E
solar and KamLAND neutrinos: ∆m2 L/E 1 • | 31| ≫ ∆m2 P (ν ν ) 1 sin2 2θ sin2 21 L e → e ≃ − 12 4 E
short baseline reactor neutrinos: ∆m2 L/E 1 • 21 ≪ ∆m2 P (ν ν ) 1 sin2 2θ sin2 31 L e → e ≃ − 13 4 E
EXERCISE!
55 Summing up
−iδ c12 c13 s12 c13 s13 e iδ iδ U = −s12 c23 − c12 s23 s13 e c12 c23 − s12 s23 s13 e s23 c13 = iδ iδ s12 s23 − c12 c23 s13 e −c12 s23 − s12 c23 s13 e c23 c13 −iδ 1 0 0 c13 0 s13 e c12 s12 0 0 c23 s23 0 1 0 −s12 c12 0 iδ 0 −s23 c23 −s13 e 0 c13 0 01
| atmospheric{z and} | SBL{z reactor} | solar{z and } LBL accelerator LBL reactor
56 −iδ c12 c13 s12 c13 s13 e iδ iδ U = −s12 c23 − c12 s23 s13 e c12 c23 − s12 s23 s13 e s23 c13 = iδ iδ s12 s23 − c12 c23 s13 e −c12 s23 − s12 c23 s13 e c23 c13 −iδ 1 0 0 c13 0 s13 e c12 s12 0 0 c23 s23 0 1 0 −s12 c12 0 iδ 0 −s23 c23 −s13 e 0 c13 0 01
| atmospheric{z and} | SBL{z reactor} | solar{z and } LBL accelerator LBL reactor 2 1 10 0 1 0 ǫ +ǫ 0 q 3 q 3 1 1 ǫ 1 2 0 q 2 −q 2 + 0 1 0 − 3 +ǫ 3 0 q q 1 1 0 q 2 +ǫ q 2 ǫ 0 1 0 01 2 1 2 2 2 1 2 (sin θ23 = 2 − ǫ) (sin θ13 = ǫ ) (sin θ12 = 3 −ǫ ) 2 2 2 ∆mA ∆mA ∆m⊙
57 Status of global Fits LBL Acc + Solar + KL + SBL Reactors + SK Atm 4 NH IH 3
σ 2 N
1
0 6.5 7.0 7.5 8.0 8.5 2.0 2.2 2.4 2.6 2.8 0.0 0.5 1.0 1.5 2.0 δ m 2 /10 -5 eV 2 ∆ m 2 /10 -3 eV 2 δ / π 4
3 σ
N 2
1
0 0.25 0.30 0.35 0.3 0.4 0.5 0.6 0.7 0.01 0.02 0.03 0.04 2 θ 2 θ 2 θ sin 12 sin 23 sin 13
Fogli, Lisi et al., March 2014
58 PMNS matrix is given by
0.779 ... 0.848 0.510 ... 0.604 0.122 ... 0.190 U = 0.183 ... 0.568 0.385 ... 0.728 0.613 ... 0.794 | | 0.200 ... 0.576 0.408 ... 0.742 0.589 ... 0.775 compare with CKM:
0.97427 0.00015 0.22534 0.00065 0.00351+0.00015 ± ± −0.00014 V = 0.22520 0.00065 0.97344+0.00016 0.0412+0.0011 | | ± −0.00016 −0.0005 0.00867+0.00029 0.0404+0.0011 0.999146+0.000021 −0.00031 −0.0005 −0.000046 WTF? large mixing?? ↔
59 Two Mass Orderings
Normal Inverted
ν3 ν2
2 ∆m21
ν1
2 ∆m32 νe
νµ
ντ 2 ∆m31
ν2
2 ∆m21
ν1 ν3
unknown sign of larger ∆m2
60 Present knowledge 1 = νT m ν with m = U diag(m , m , m ) U T L 2 ν ν 1 2 3
9 physical parameters in mν θ and m2 m2 • 12 2 − 1 θ and m2 m2 • 23 | 3 − 2| θ (or U ) • 13 | e3| m , m , m • 1 2 3 sgn(m2 m2) • 3 − 2 Dirac phase δ • Majorana phases α and β •
61 62 63 What’s that good for?
Predictions of All 63 Models 12 11 anarchy 10 texture zero SO(3) 9 A 4 S , S 8 3 4 L e-L µ-L τ 7 SRND SO(10) lopsided 6 SO(10) symmetric/asym 5 4 Number of Models 3 2 1 0 1e-05 0.0001 0.001 0.01 0.1 2 sin θ 13
Albright, Chen
64 Supersymmetrie LHC ILC Astrophysik ... kosm. Strahlung Kosmologie Supernova Dunkle Materie BBN ... NEUTRINOS Baryon− asymmetrie ...
Flavorphysik GUT Quarkmischung ... SO(10) See−Saw Protonzerfall ...
65 EXERCISE: Lepton Flavor Violation Suppose BR(µ eγ) (m m† ) 2. → ∝| ν ν eµ| In general, can it be zero? What does the experimental value U 0.15 imply | e3|≃ for the BR?
66 EXERCISE: Neutrino Telescopes
0 Φe Pee Peµ Peτ Φe 0 Φµ = Peµ Pµµ Pµτ Φµ Φ P P P Φ0 τ eτ µτ ττ τ Transition amplitude after loss of coherence:
P (ν ν )= U 2 U 2 α → β | αi| | βi| X
if θ23 = π/4 and θ13 = 0: what happens to a “pion source” 0 0 0 (Φe : Φµ : Φτ )=(1:2:0)? Obtain the first order corrections (θ = π/4 ǫ and θ = ǫ ). How large can 23 − 1 13 2 they become?
67 Contents II Neutrino Oscillations II1) The PMNS matrix II2) Neutrino oscillations II3) Results and their interpretation – what have we learned? II4) Prospects – what do we want to know?
68 Degeneracies Expand 3 flavor oscillation probabilities in terms of R = ∆m2 /∆m2 and U : ⊙ A | e3| 2 ˆ 2 ˆ 2 2 sin (1 A)∆ 2 2 2 sin A∆ P (νe νµ) sin 2θ13 sin θ23 − + R sin 2θ12 cos θ23 → ≃ (1 Aˆ)2 Aˆ2 − sin Aˆ∆ sin(1 Aˆ)∆ +sin δ sin2θ13 R sin2θ12 cos θ13 sin2θ23 sin∆ − Aˆ(1 Aˆ) − sin Aˆ∆ sin(1 Aˆ)∆ +cos δ sin2θ13 R sin2θ12 cos θ13 sin2θ23 cos ∆ − Aˆ(1 Aˆ) − 2 ˆ √ 2 ∆mA with A = 2 2 GF ne E/∆mA and ∆= 4 E L θ π/2 θ degeneracy • 23 ↔ − 23 θ -δ degeneracy • 13 δ-sgn(∆m2 ) degeneracy • A Solutions: more channels, different L/E, high precision,...
69 Typical time scale
70 3 Tasks Now that neutrino mass is introduced: Determine parameters • Explain values of parameters • Check if minimal description is correct •
71 Contents III Neutrino Mass III1) Dirac vs. Majorana mass III2) Realization of Majorana masses beyond the Standard Model: 3 types of see-saw III3) Limits on neutrino mass(es) III4) Neutrinoless double beta decay
72 III1) Dirac vs. Majorana masses Observation shows that neutrinos possess non-vanishing rest mass
Upper limits on masses imply mν < eV ∼ How can we introduce neutrino mass terms?
73 a) Dirac masses add ν (1, 0): R ∼ EWSB v D = gν L Φ˜ νR gν νL νR = mν νL νR L −→ √2
−12 But mν < eV implies gν < 10 ge ∼ ∼ ≪≪ highly unsatisfactory fine-tuning...
74 −6 actually, me = 10 mt, so WTF? point is that
u with mu md d ≃ has to be contrasted with
νe −6 with mν 10 me e ≃
75 b) Majorana masses (long version) need charge conjugation:
electron e−: [γ (i∂µ + e Aµ) m] ψ =0 (1) µ − positron e+: [γ (i∂µ e Aµ) m] ψc =0 (2) µ − − Try ψc = S ψ∗, evaluate (S∗)−1 (2)∗ and compare with (1):
S = iγ2
and thus
T T ψc = iγ ψ∗ = iγ γ ψ C ψ 2 2 0 ≡ flips all charge-like quantum numbers
76 Properties of C:
C† = CT = C−1 = C − C γ C−1 = γT µ − µ −1 T C γ5 C = γ5 −1 T C γµγ5 C =(γµγ5) properties of charged conjugate spinors:
(ψc)c = ψ ψc = ψT C c c ψ1 ψ2 = ψ2 ψ1 c c (ψL) =(ψ )R c c (ψR) =(ψ )L C flips chirality: LH becomes RH
77 b) Majorana masses (short version)
T T charge conjugation: ψc = iγ ψ∗ = iγ γ ψ C ψ 2 2 0 ≡ flips all charge-like quantum numbers connection to chirality:
c c (ψL) =(ψ )R c c (ψR) =(ψ )L
C flips chirality: LH becomes RH
78 Mass terms
= m ν ν + h.c. = m (ν ν + ν ν ) L ν L R ν L R R L both chiralities a must for mass term! Possibilities for LR:
(i) νR independent of νL: Dirac particle
c (ii) νR =(νL) : Majorana particle
νc =(ν + ν )c =(ν )c +(ν )c = ν + ν = ν : νc = ν ⇒ L R L R R L Majorana fermion is identical to its antiparticle, a truly neutral particle ⇒ all additive quantum numbers (Q, L, B,...) are zero ⇒
79 in terms of helicity states : 4 d.o.f. for Dirac particles: ±
80 in terms of helicity states : 2 d.o.f. for Majorana particles: ±
81 Mass term for Majorana particles: 1 1 = ν m ν = ν +(ν )c m (ν +(ν )c) = 1 ν m νc + h.c. LM 2 ν 2 L L ν L L 2 L ν L Majorana mass term • ν† ν∗ NOT invariant under ν eiα ν •LM ∝ ⇒ → breaks Lepton Number by 2 units ⇒
82 Dirac vs. Majorana in V A theories: observable always suppressed by (m/E)2 − suppose beam from π+ decays: π+ µ+ ν • → µ can we observe ν + p n + µ+ ? • µ → chirality is not a good quantum number: “spin flip” • emitted ν (negative helicity) is not purely left-handed: • µ (m=0) m (m=0) u↓(p)= u (p)+ u ( p) L 2E R −
+ P u↓ = 0 and µ can be produced IF u v ( Majorana!) • R 6 ∝ ↔ amplitude (m/E) probability (m/E)2 ⇒ ∝ ⇒ ∝ only NA can save the day! ⇒
83 Dirac vs. Majorana Z-decay: • Γ(Z ν ν ) m2 → D D 1 3 ν Γ(Z ν ν ) ≃ − m2 → M M Z
Meson decays • 2 m 2 BR(K+ π− e+ µ+) m 2 = U U m 10−30 | eµ| → ∝| eµ| ei µi i ∼ eV X
neutrino-antineutrino oscillations • 1 P (ν ν¯ )= U U U ∗ U ∗ m m e−i(Ej −Ei)t α → β E2 αj βj αi βi i j i,j X
84 Contents III Neutrino Mass III1) Dirac vs. Majorana mass III2) Realization of Majorana masses beyond the Standard Model: 3 types of see-saw III3) Limits on neutrino mass(es) III4) Neutrinoless double beta decay
85 III2) Realization of Majorana masses beyond the SM a) Higher dimensional operators Renormalizability: only dimension 4 terms in L SM has several problems there is a theory beyond SM, whose low energy limit → is the SM higher dimensional operators: → 1 1 = + + + ... = + + + ... L LSM L5 L6 LSM Λ O5 Λ2 O6 gauge and Lorentz invariant, only SM fields: 1 c c v2 = Lc Φ˜ ∗ Φ˜ † L EWSB (ν )c ν m (ν )c ν Λ O5 Λ −→ 2Λ L L ≡ ν L L
> 0.1 eV 14 it follows Λ c m 10 GeV Weinberg 1979 ∼ ν
86 Weinberg operator is LLΦΦ