Lepton Mixing and

v Werner Rodejohann T -1 (MPIK, Heidelberg) mv = m L - m D M R m D Strasbourg MANITOP Massive : 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 , hep-ph/0001264 – Grimus: Neutrino Physics – Theory, hep-ph/0307149 Textbooks: • – Fukugita, Yanagida: Physics of Neutrinos and Applications to – 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) mixing, neutrinos and the

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 beyond the Standard Model: 3 types of see-saw III3) Limits on neutrino mass(es) III4) Neutrinoless double

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 Physics: SU(3) SU(2) U(1) C × L × Y

o R + R Species # e R b R tR s R 10P 10 c R d i o u R i + R i e o L 3 13 + L e L tL 3 16 c L u L b L s L d L Higgs 2 18

18 free parameters... + Gravitation + Dark + Dark +

8 Standard Model of Elementary : 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 ... DarkBaryon Matter Asymmetry NEUTRINOS ...

Flavor physics GUT mixing ... SO(10) see−saw ... 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, 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 “”

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); 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 2010 the third

17 Future History 20ab CP shown to be violated 20cd mass ordering determined 20ef Dirac/Majorana 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, 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 ) 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 :

∆ = 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 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 with different p and same E violates energy and/or • i 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 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 with size σx(> 1/σp) and group 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 :

(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 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 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

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

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:

e−: [γ (i∂µ + e Aµ) m] ψ =0 (1) µ − 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 :

(ψc)c = ψ ψc = ψT C c c ψ1 ψ2 = ψ2 ψ1 c c (ψL) =(ψ )R c c (ψR) =(ψ )L C flips : 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 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 is identical to its , a truly ⇒ 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: “ 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



`

Ä

`



Ä

Seesaw Mechanisms are realizations of this effective operator by integrating out heavy physics

87 Master Formula: 2+2=3+1 General remarks: SU(2) U(1) with 2 2 = 3 1: L × Y ⊗ ⊕ L Φ˜ (2, +1) (2, 1) = (3, 0) (1, 0) ∼ ⊗ − ⊕ To make a singlet, couple (1, 0) or (3, 0), because 3 3 = 5 3 1 ⊗ ⊕ ⊕

Alternatively:

L Lc (2, +1) (2, +1) = (3, +2) (1, +2) ∼ ⊗ ⊕ To make a singlet, couple to (1, 2) or (3, 2). However, singlet combination is − − νℓc ℓνc, which cannot generate neutrino mass term − = (1, 0) or (3, 2) or (3, 0) ⇒ − type I type II type III

88 b) Fermion singlets (type I) introduce N (1, 0) and couple to g L Φ˜ (1, 0) R ∼ ν ∼ Hence, g L Φ˜ N is also singlet and becomes g v/√2 ν N m ν N ν R ν L R ≡ D L R in addition: Majorana mass term for NR

= ν m N + 1 N c M N + h.c. L L D R 2 R R R c 1 c 0 mD νL = 2 (νL, NR) + h.c.  mD MR   NR   1 Ψ Ψc +h.c.  ≡ 2 Mν

89 b) Fermion singlets (type I)

c 1 c 0 mD νL = 2 (νL, NR) + h.c. L  mD MR   NR  1 Ψ Ψc +h.c.  ≡ 2 Mν Diagonalization with † ∗ = D = diag(m , M) and U Mν U ν cos θ sin θ = U  sin θ cos θ  −  

90 c 1 c 0 mD νL = 2 (νL, NR) L  mD MR   NR      c 1 c † 0 mD ∗ T νL = 2 (νL, NR) U U  mD MR  U U  NR      c c T | (ν,{zN )} diag(mν , M) (ν , N) | {z } | {z } with

general formula: if m M : D ≪ R tan 2θ = 2 mD 1 MR−0 ≪ m = 1 (0 + M ) (0 M )2 + 4 m2 m2 /M ν 2 R − − R D ≃− D R M = 1 h(0 + M )+ p(0 M )2 + 4 m2 i M 2 R − R D ≃ R h p i

91 Note: mD associated with EWSB, part of SM, bounded by v/√2 = 174 GeV M is SM singlet, does whatever it wants: M m R ⇒ R ≫ D Hence, θ m /M 1 ≃ D R ≪ ν = ν cos θ N c sin θ ν with mass m m2 /M L − R ≃ L ν ≃− D R N = N cos θ + νc sin θ N with mass M M R L ≃ R ≃ R in effective mass terms 1 1 m ν νc + M N c N L≃ 2 ν L L 2 R R R compare with Weinberg operator: c v2 Λ= 2 MR −mD

also: integrate NR away with Euler-Lagrange equation

92 matrix case: block diagonalization

c 1 c 0 mD νL = (νL, N ) L 2 R  T    mD MR NR     c 1 c † 0 mD ∗ T νL = (νL, N ) 2 R U U  T  U U   mD MR NR     c c T | (ν,{zN )} diag(mν , M) (ν , N) | {z } | {z } with 6 6 diagonal matrix × 1 ρ 1 ρ 1 ρ∗ = − , † = , ∗ = − U  ρ† 1  U  ρ† 1  U  ρT 1  −       write down individual components:

93 write down individual components:

T T mν = ρ mD + mD ρ + ρ MR ρ 0 = m ρ mT ρ∗ + ρ M D − D R M = ρ† m mT ρ∗ + M − D − D R now, ρ (aka θ from before) will be of order mD/MR:

T T mν = ρ mD + mD ρ + ρ MR ρ 0 m + ρ M ρ = m M −1 ≃ D R ⇒ − D R M M ≃ R

insert ρ in mν to find:

m = m M −1 mT ν − D R D

94 2 mD mν = −MR (type I) See-Saw Mechanism Minkowski; Yanagida; Glashow; Gell-Mann, Ramond, Slansky; Mohapatra, Senjanović (77-80)

95 96 97 98 99 See-Saw Scale See-saw formula:

m = m2 /M v2/M ν D R ≃ R with m ∆m2 it follows ν ≃ A p M 1015 GeV R ≃

Note: not necessarily correct...

100 Parameter Count

m : 9=6+3 • ν m and M : 18 = 9 + 9 • D R

101 Parametrization

m = m M −1 mT = U mdiag U T ν − D R D ν is “solved” by

diag mD = iU mν R MR q where complex orthogonal R contains thep unknown parameters

c˜12c˜13 s˜12c˜13 s˜13 R =  s˜ c˜ c˜ s˜ s˜ c˜ c˜ s˜ s˜ s˜ s˜ c˜  − 12 23 − 12 23 13 12 23 − 12 23 13 23 13  s˜ s˜ c˜ c˜ s˜ c˜ s˜ s˜ c˜ s˜ c˜ c˜   12 23 − 12 23 13 − 12 23 − 12 23 13 23 13    Note: these are now complex angles, c˜12 = cos ω12 = cos(ρ12 + iσ12)

102 Higgs and Seesaw 1.) relies on Yukawa coupling with Higgs 2.) sterile neutrinos N couple to Higgs vacuum stability bound R ↔

103 Paths to Neutrino Mass

SU(2)L × U(1)Y approach ingredient quantum number L mν scale of messenger

“SM” −12 RH ν NR ∼ (1, 0) hNRΦL hv h = O(10 ) (Dirac mass)

“effective” new scale 2 – h Lc ΦΦ L h v Λ = 1014 GeV (dim 5 operator) + LNV Λ

“direct” Higgs triplet c 1 2 ∆ ∼ (3, −2) hL ∆L + µΦΦ∆ hvT Λ = M (type II seesaw) + LNV hµ ∆ 2 “indirect 1” RH ν c (hv) 1 NR ∼ (1, 0) hNRΦL + NRMRN Λ = MR (type I seesaw) + LNV R MR h 2 “indirect 2” fermion triplets (hv) 1 Σ ∼ (3, 0) hΣ LΦ + TrΣMΣΣ Λ = MΣ (type III seesaw) + LNV MΣ h

plus seesaw variants (linear, double, inverse,...) plus radiative mechanisms plus extra dimensions plusplusplus

104 c) Higgs triplet (type II)

L Lc νν∗ L∝ → has isospin I = +1 and transforms as (3, +2) 3 ∼ introduce Higgs triplet (3, 2) with (I = Q Y/2): ⇒ ∼ − 3 − H+ √2 H++ 0 0 ∆=  √2 H0 H+  →  v 0  − T     with SU(2) transformation property ∆ U ∆ U †: → = g L iτ ∆ Lc vev g v ν νc m ν νc L ν 2 −→ ν T L L ≡ ν L L

105 Constraints on vT

mν = gν vT < eV vT < eV/gν • ∼ ⇒ ∼ ρ-parameter • 2 1 2 2 M Ii (Ii + 1 Y ) v W = ρ = − 4 i i M cos θ 1 v2 Y 2  Z W  P 2 i i 1 1 I = 2 andPY = 1 = v2 + 2 v2  T I = 1 and Y = 2  2 2  v + 4 vT <  vT 8 GeV  ⇒ ∼

106 v v because T ≪ V = M 2 Tr ∆ ∆† + µ Φ† iτ ∆ Φ − ∆ 2 2 ∂V  µ v with ∂∆ = 0 one has vT = 2 M∆

coupling of SM Higgs with triplet drives minimum vT away from zero

vT can be suppressed by M∆ and/or µ compare with Weinberg operator: c M 2 Λ= ∆ gν µ Type II (or Triplet) See-Saw Mechanism Magg, Wetterich; Mohapatra, Senjanovic; Lazarides, Shafi, Wetterich; Schechter, Valle (80-82)

107 d) Fermion triplets (type III) The term

L Σc Φ˜ L∝ is a singlet if Σ (3, 0): “hyperchargeless triplet” ∼ Σ0/√2 Σ+ Σ=  Σ− Σ0/√2  −   additional terms in Lagrangian 1 L √2 Y Σc Φ+˜ Tr Σ M Σc Σ 2 Σ Σ  give a “Dirac mass term” mD = vYΣ and Majorana mass term MΣ for neutral component of Σ

108 overall mass term for neutrinos

Σ 2 (mD) mν = − MΣ

same structure as type I see-saw Type III See-Saw Mechanism Foot, Lew, He, Joshi (1989) compare with Weinberg operator: c v2 Λ= Σ 2 MΣ −(mD)

109 110 slide by T. Hambye

M Remarks Higgs and fermion triplets have SM charges coupling to gauge in • ⇒ kinetic terms: – RG effects – production at colliders – FCNC – ... naturalness in GUTs: type I type II type III • ≃ ≫ note: one, two or three of the see-saw terms may be present in m • ν

111 Remarks Higgs and fermion triplets have SM charges coupling to gauge bosons in • ⇒ kinetic terms: – RG effects – production at colliders – FCNC – ... naturalness in GUTs: type I type II type III • ≃ ≫ note: none of the see-saw terms may be present in m • ν

112 Seesaw Phenomenology:

Take advantage of (L-Higgs-N) vertex in early !

¯α † α α Γ(Ni Φ L ) Γ(Ni Φ L ) YB ε = → − → ∝ i Γ(N Φ L¯)+Γ(N Φ† L) i → i → Fukugita and Yanagida (1986)

113 Lepton Flavor Violation in Type II seesaw

114 Phenomenology of heavy singlets: Higgs Higgs potential is rather flat corrections are dangerous ⇒

−→ vacuum probably metastable: could tunnel to true vacuum replacing all fields and with new fields and forces

115 Phenomenology of heavy singlets: Higgs Higgs potential is rather flat corrections are dangerous ⇒

−→ vacuum probably metastable: could tunnel to true vacuum replacing all fields and forces with new fields and forces (not good)

116 Phenomenology of heavy singlets: Higgs Higgs coupling λ is driven to negative values by top Yukawa:

2 β 24 Tr Y †Y m4 m f(Λ) λ ∝− u u ∝ t ⇒ h ≥ vacuum stability bound

With mh = 126 GeV: vacuum probably metastable • could be λ(M ) = 0 • Pl (Holthausen, Lim, Lindner; Bezrukov et al.; Degrassi et al.; Masina)

strong dependence on top mass, αs

117 Phenomenology of heavy singlets: Higgs

Lepton-Higgs-NR vertex: Dirac Yukawa ν¯L Yν NR contribution 2 ∆β = 8 Tr Y †Y m4 λ − ν ν ∝ D Casas et al.; Strumia et al.; W.R., Zhang makes vacuum stability condition worse!

0.2 SM withoutν 14 M0 = 10 GeV, a0 = 0 11 M0 = 10 GeV, a0 = 0 4 M0 = 10 GeV, a0 = 0 0.1 ) µ

( 0 λ

−0.1

−0.2 5 10 15 10 10 10 µ [GeV]

(also effect if tricks are played to produce TeV-scale NR at colliders)

118 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

119 III3) Limits on neutrino mass(es)

Normal Inverted

ν3 ν2 2 ∆m21

ν1

2 ∆m32 νe

νµ

ν 2 τ ∆m31

ν2 2 ∆m21

ν1 ν3

∆m2 = 8 10−5 eV2 and ∆m2 ∆m2 = 2 10−3 eV2 21 × | 31|≃| 32| × normal ordering: ∆m2 > 0 • 31 inverted ordering: ∆m2 < 0 • 31 120 Neutrino masses neutrino masses (scale of their origin)−1 • ↔ neutrino mass ordering form of m • ↔ ν

m2 ∆m2 m2 ∆m2 m2: normal hierarchy (NH) • 3 ≃ A ≫ 2 ≃ ⊙ ≫ 1 m2 ∆m2 m2 m2: inverted hierarchy (IH) • 2 ≃| A|≃ 1 ≫ 3 m2 m2 m2 m2 ∆m2 : quasi-degeneracy (QD) • 3 ≃ 2 ≃ 1 ≡ 0 ≫ A

121 Neutrino masses

Neutrino mass hierarchy is moderate!

2 m2 ∆m⊙ 1 NH : 2 m3 ≥ s ∆mA ≃ 5 2 m1 > 1 ∆m⊙ IH : 1 2 0.98 m2 ∼ − 2 ∆mA ≃

122 Upper limits: direct searches (Kurie plot of 3H) • m(ν )= U 2 m2 2.3 eV at 95 % C.L. (Mainz, Troitsk) e | ei| i ≤ qX

cosmology (hot ) •

Σ= mi < eV ∼ X

neutrino-less double beta decay (0νββ) • 2 < mee = Uei mi eV | | ∼

X

123 m(ν )= U 2 m2 e | ei| i pP m(ν )NH s2 c2 ∆m2 + s2 ∆m2 m(ν )IH c2 ∆m2 e ≃ 12 13 ⊙ 13 A ≪ e ≃ 13 A q q

almost independent on mixing angles • difference of normal and inverted shows up well below KATRIN limit •

124 KATRIN

125 126 127 128 129 Σ= mi P ΣNH ∆m2 < ΣIH 2 ∆m2 ≃ A ≃ A q q

independent on mixing angles • systematics? •

130 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

131 III4) Neutrinoless double beta decay (A, Z) (A, Z + 2) + 2 e− (0νββ) →

second order in : Γ G4 rare! • ∝ F ⇒ not to be confused with (A, Z) (A, Z + 2) + 2 e− + 2ν ¯ (2νββ) • → e (which occurs more often but is still rare)

132 Need to forbid single β decay:

• even/even even/even • ⇒ → either direct (0νββ) or two simultaneous decays with virtual (energetically • forbidden) intermediate state (2νββ)

133 How many nuclei in thisHow nuclei many condition? in How many nuclei in thisHow nuclei many condition? in Q-value [MeV] ld yA Giuliani A. by Slide 134 A Upcoming/running experiments: exciting time!! best limit was from 2001...

Name Isotope source = detector; calorimetric with source 6= detector high energy res. low energy res. event topology event topology AMoRE 100Mo X – – – CANDLES 48Ca – X – – COBRA 116Cd – – X – CUORE 130Te X – – – DCBA 150Nd – – – X EXO 136Xe – – X – GERDA 76Ge X – – – KamLAND-Zen 136Xe – X – – LUCIFER 82Se X – – – MAJORANA 76Ge X – – – MOON 100Mo – – – X NEXT 136Xe – – X – SNO+ 130Te – X – – SuperNEMO 82Se – – – X XMASS 136Xe – X – –

135 Experiment Isotope Mass of Sensitivity Status Start of 0ν Isotope [kg] T1/2 [yrs] data-taking GERDA 76Ge 18 3 × 1025 running ∼ 2011 40 2 × 1026 in progress ∼ 2012 1000 6 × 1027 R&D ∼ 2015 CUORE 130Te 200 6.5 × 1026∗ in progress ∼ 2013 2.1 × 1026∗∗ MAJORANA 76Ge 30-60 (1 − 2) × 1026 in progress ∼ 2013 1000 6 × 1027 R&D ∼ 2015 EXO 136Xe 200 6.4 × 1025 in progress ∼ 2011 1000 8 × 1026 R&D ∼ 2015 SuperNEMO 82Se 100-200 (1 − 2) × 1026 R&D ∼ 2013-15 KamLAND-Zen 136Xe 400 4 × 1026 in progress ∼ 2011 1000 1027 R&D ∼ 2013-15 SNO+ 150Nd 56 4.5 × 1024 in progress ∼ 2012 500 3 × 1025 R&D ∼ 2015

136 Interpretation of Experiments Master formula:

Γ0ν = G (Q, Z) (A, Z) η 2 x |Mx x| G (Q, Z): phase space factor • x (A, Z): • Mx η : particle physics • x

137 Interpretation of Experiments Master formula:

Γ0ν = G (Q, Z) (A, Z) η 2 x |Mx x| G (Q, Z): phase space factor; calculable • x (A, Z): nuclear physics; problematic • Mx η : particle physics; interesting • x

138 3 Reasons for Multi-isotope determination 1.) credibility 2.) test NME calculation

0ν 2 T1/2(A1,Z1) G(Q ,Z ) (A ,Z ) = 2 2 |M 2 2 | T 0ν (A ,Z ) G(Q ,Z ) (A ,Z ) 2 1/2 2 2 1 1 |M 1 1 | systematic errors drop out, ratio sensitive to NME model 3.) test mechanism

0ν 2 T1/2(A1,Z1) G (Q ,Z ) (A ,Z ) = x 2 2 |Mx 2 2 | T 0ν (A ,Z ) G (Q ,Z ) (A ,Z ) 2 1/2 2 2 x 1 1 |Mx 1 1 | particle physics drops out, ratio of NMEs sensitive to mechanism

139 Why should we probe Lepton Number Violation? L and B accidentally conserved in SM • effective theory: = + 1 + 1 + ... • L LSM Λ LLNV Λ2 LLFV, BNV, LNV : B is violated • B, L often connected in GUTs • GUTs have seesaw and Majorana neutrinos • µ ˜µν B (chiral anomalies: ∂µJB,L = c Gµν G = 0 with Jµ = qi γµ qi and • L 6 Jµ = ℓi γµ ℓi) P P Lepton Number Violation as important as Violation ⇒ (0νββ is much more than a neutrino mass experiment)

140 Standard Interpretation Neutrinoless Double Beta Decay is mediated by light and massive Majorana neutrinos (the ones which oscillate) and all other mechanisms potentially leading to 0νββ give negligible or no contribution

dL uL

W ei U − eL i ν q i ν − eL Uei W

dL uL

141 Non-Standard Interpretations: There is at least one other mechanism leading to Neutrinoless Double Beta Decay and its contribution is at least of the same order as the light neutrino exchange mechanism

uL − uL dL dL uL c eL dL uL c d d W − W − eL − eL ˜ eL ˜L − dR u eL W χ/g˜ νL uL 0 χ/g˜ χ √ 2 NR 2gvL hee ∆−− χ/g˜ 0 χ L NR u − χ/g˜ eR W − − ˜ WR eL eL uL − L W e c W d ˜ c − dR uL d L uL dL uL uL dL e dL

142 Schechter-Valle theorem: no matter what process, neutrinos are Majorana:

W W d d u u

− − νe νe e e

Blackbox diagram is 4 loop: G2 mBB F MeV5 10−25 eV ν ∼ (16π2)4 ∼

143 mechanism physics parameter current limit test oscillations, 2 light neutrino exchange Uei mi 0.5 eV cosmology,

neutrino mass S2 LFV, ei −8 −1 heavy neutrino exchange 2 × 10 GeV Mi collider

2 V 16 5 flavor, heavy neutrino and RHC ei 4 × 10− GeV− M M4 i W collider R flavor, (M )ee 15 1 Higgs triplet and RHC R 10− GeV− collider m2 M4 ∆ W R R e− distribution

flavor, ˜ U S 10 2 λ-mechanism with RHC ei ei 1.4 × 10− GeV− collider, M2 W R e− distribution

flavor, ˜ −9 η-mechanism with RHC tan ζ Uei Sei 6 × 10 collider,

e− distribution ′2 λ111 collider, / Λ 5 −18 −5 short-range R SUSY 7 × 10 GeV flavor ΛSUSY = f(m˜g, m˜u , m˜ , mχ ) L dR i   b ′ ′ 1 1 −13 −2 sin 2θ λ131 λ113 2 − 2 2 × 10 GeV  m m  flavor, /  b˜ b˜  long-range R 1 2 ′ ′ collider G λ131 λ113 1 10−14 GeV−3 ∼ F m × q b Λ3 SUSY 2 −4 spectrum, Majorons |hgχi| or |hgχi| 10 . . . 1 cosmology 144 Lepton Number Violation and Majorana Neutrinos: Neutrinoless Double Beta Decay nn pp e−e− →

dL uL

W

Uei − eL

i ν q i ν − eL

Uei W

uL dL

has two requirements: ν = νc AND m = 0 ν 6

145 ∆L = 0: Neutrinoless Double Beta Decay d L 6 u L

W Im U ei − eL ν i q α i (2) . 2i ν β |m ee | e − (3) . 2i eL |m ee | e U ei W m ee u L (1) Re d L |m ee | Amplitude proportional to coherent sum (“effective mass”): m U 2 m = U 2 m + U 2 m e2iα + U 2 m e2iβ | ee|≡ ei i | e1| 1 | e2| 2 | e3| 3

P = f θ , m , U , sgn(∆m2 ),α,β 12 i | e3| A 7 out of 9 parameters of mν ! 

146 The usual plot

147 Crucial points NH: can be zero! • IH: cannot be zero! m inh = ∆m2 c2 cos 2θ • | ee|min A 13 12 gap between m inh and m pnor • | ee|min | ee|max QD: cannot be zero! m QD = m c2 cos 2θ • | ee|min 0 13 12 QD: cannot distinguish between normal and inverted ordering •

148 The usual plot (life-time instead of m ) | ee|

Normal Inverted

32 10

30 10 (yrs) ν ]

1/2 28 10 [T

GERDA 40kg 26 10

Excluded by HDM 0.0001 0.001 0.010.0001 0.1 0.001 0.01 0.1 m (eV) light

149 Plot against other observables θ 12

Normal Inverted Normal Inverted

0 0 10 10

CPV CPV CPV CPV (+,+) (+) (+,+) (+) (+,-) (-) (+,-) (-) (-,+) (-,+) (-,-) (-,-) -1 -1 10 10 > (eV) > (eV) ee ee

-3 -3 10 10 0.01 0.1 0.01 0.1 0.1 1 0.1 1 m (eV) β Σm (eV) i

Complementarity of m = U 2 m , m = U 2 m2 and Σ= m | ee| ei i β | ei| i i p P

150 CP violation! Dirac neutrinos! Something else does 0νββ!

151 Neutrino Mass > 2 2 m(heaviest) m3 m1 0.05 eV ∼ | − |≃ 3 complementary methodsp to measure neutrino mass:

Method observable now [eV] near [eV] far [eV] pro con

2 2 model-indep.; final?; Kurie |Uei| mi 2.3 0.2 0.1 pP theo. clean worst best; systemat.; Cosmo. mi 0.7 0.3 0.05 P NH/IH model-dep.

2 fundament.; model-dep.; 0νββ | Ueimi| 0.3 0.1 0.05 P NH/IH theo. dirty

152 Which mass ordering with which life-time? Σ m m β | ee| NH ∆m2 ∆m2 + U 2∆m2 ∆m2 + U 2 ∆m2 e2i(α−β) A ⊙ | e3| A ⊙ | e3| A q q 0ν > 28−29 p0.05 eV 0.01 eV 0.003 eV T1/p2 10 yrs ≃ ≃ ∼ ⇒ ∼ IH 2 ∆m2 ∆m2 ∆m2 1 sin2 2θ sin2 α A A A − 12 0ν > 26−27 p0.1 eV p0.05 eV p0.03 eVp T1/2 10 yrs ≃ ≃ ∼ ⇒ ∼ QD 3m m m 1 sin2 2θ sin2 α 0 0 0 − 12 > 0ν > 25−26 0.1 eVp T1/2 10 yrs ∼ ⇒ ∼

(for 1026 yrs you need 1026 , which are 103 mols, which are 100 kg)

153 From life-time to particle physics: Nuclear Matrix Elements

154 From life-time to particle physics: Nuclear Matrix Elements

SM vertex

e e ν ν Σ U i i ei Uei i W W

Nucl Nuclear Process Nucl

2 point-like Fermi vertices; “long-range” neutrino exchange; momentum • exchange q 1/r 0.1 GeV ≃ ≃ NME overlap of decaying ... • ↔ different approaches (QRPA, NSM, IBM, GCM, pHFB) imply uncertainty • plus uncertainty due to model details • plus convention issues (Cowell, PRC 73; Smolnikov, Grabmayr, PRC 81; • Dueck, W.R., Zuber, PRD 83)

155 typical model for NME: set of single particle states with a number of possible wave function configurations; solve in a mean background field H

Quasi-particle Random Phase Approximation (QRPA) (many single particle states, few • configurations)

Nuclear Shell Model (NSM) (many configurations, few single particle states) •

Interacting Model (IBM) (many single particle states, few configurations) (many single particle • states, few configurations)

Generating Coordinate Method (GCM) (many single particle states, few configurations) • projected Hartree-Fock-Bogoliubov model (pHFB) • tends to overestimate NMEs tends to underestimate NMEs

156 From life-time to particle physics: Nuclear Matrix Elements

8 8 (R)QRPA (Tü) 10 GCM NSM SM IBM QRPA (Tue) 7 ISM IBM-2 QRPA (Jy) PHFB IBM QRPA(J) 8 6 GCM+PNAMP IBM 6 QRPA(T) GCM ν PHFB 0 5

′ Pseudo-SU(3) 6 ν ν 0 0

4 M 4 M M 4 3

2 2 2

1

0 0 48 82 96 100 110 124 130 150 48 76 82 96 100 110 116 124 130 136 150 0 Ca Se Zr Mo Pd Sn Te Nd Ca Ge Se Zr Mo Pd Cd Sn Te Xe Nd 76 82 96 100 128 130 136 150 76 94 98 104 116 128 136 154 Ge Zr Mo Ru Cd Te Xe Sm Isotope A

Faessler, 1104.3700 Dueck, W.R., Zuber, PRD 83 Gomez-Cadenas et al., 1109.5515

to better estimate error range: correlations need to be understood

157 Faessler, Fogli et al., PRD 79 ellipse major axis: SRC (blue, red) and gA

ellipse minor axis: gpp

158 Recent Results 76Ge: • – GERDA: T > 2.1 1025 yrs 1/2 × – GERDA + IGEX + HDM: T > 3.0 1025 yrs 1/2 × 136Xe: • 25 – EXO-200: T > 1.1 10 yrs (first run with less exposure: T > 1.6 × 1025 yrs. . .) 1/2 × 1/2 – KamLAND-Zen: T > 2.6 1025 yrs 1/2 × Xe-limit is stronger than Ge-limit when:

G 2 T >T Ge MGe yrs Xe Ge G Xe Xe M

159 Xe vs. Ge: Limits on mee | | NME 76Ge 136Xe

GERDA comb KLZ comb

EDF(U) 0.32 0.27 0.13 –

ISM(U) 0.52 0.44 0.24 –

IBM-2 0.27 0.23 0.16 –

pnQRPA(U) 0.28 0.24 0.17 –

SRQRPA-B 0.25 0.21 0.15 –

SRQRPA-A 0.31 0.26 0.23 –

QRPA-A 0.28 0.24 0.25 –

SkM-HFB-QRPA 0.29 0.24 0.28 –

Bhupal Dev, Goswami, Mitra, W.R., PRD 88

160 Might not be massive neutrinos... Note: standard amplitude for light Majorana neutrino exchange: m m G2 | ee| 7 10−18 | ee| GeV−5 2.7 TeV−5 Al ≃ F q2 ≃ × 0.5 eV ≃   if new heavy particles are exchanged: c Ah ≃ M 5 for 0νββ holds: ⇒ 1 eV=1 TeV

Phenomenology in colliders, LFV ⇒

161 Left-right symmetry dR uR

− WR eR − eR − eR dR u¯R

NRi

− eR WR NRi WR

WR

u¯R dR

R dR u

Senjanovic, Keung, 1983; Tello et al.; Goswami et al.

162 dR uR

WR − eR − − eR eR

dR u¯R

NRi

− eR

WR NRi WR WR

u¯R dR

uR dR

Bhupal Dev, Goswami, Mitra, W.R., Phys. Rev. D88

163 Topics not covered Future Neutrino Experiments • Flavor Symmetries • Leptogenesis • Neutrinos and Dark Matter • Neutrinos in Cosmology • Neutrinos in Astrophysics • Neutrinos in Cosmic Rays • Sterile Neutrinos • ... •

164 At last, let me mention the quarter final :-)

165 Summary

166