consider t -channel W − W − → e − e − from L = − g eγ µ (1 − γ 5 ) ν M 2 W µ ¯ √ 2 M ∝ ¯ e 3 γ µ (1 − γ 5 ) ν ¯ e 4 γ ν (1 − γ 5 ) ν ν c γ ν (1 + γ 5 ) e c = ¯ e 3 γ µ (1 − γ 5 ) ν ¯ 4 ( − 1) time-ordered product ν ¯ ν is propagator ∝ ( k / + m ) f † v , ¯ e = f † ¯ v , e c = f † v + ¯ use e = f u + ¯ u + ¯ f u ⇒ e c f ¯ 4 gets a v 4 24
how to calculate identities: e γ µ (1 − γ 5 ) ν ¯ | start = ( e c ) c γ µ (1 − γ 5 ) ( ν c ) c ψ = ( ψ c ) c | T = − ( e c ) T C − 1 γ µ (1 − γ 5 ) C ν cT ψ c = − ψ T C − 1 and ψ c = C ψ | = − ( e c ) T � µ − ( γ µ γ 5 ) T � Cγ µ C − 1 = − γ T µ , Cγ µ γ 5 C − 1 = ( γ µ γ 5 ) T ν cT − γ T | = ( e c ) T [ γ µ (1 + γ 5 )] T ν cT | collecting signs = − ν c γ µ (1 + γ 5 ) e c | is scalar and fermion exchange 25
Contents I) Dirac vs. Majorana neutrinos I1) Basics I2) Seesaw mechanisms I3) Summary neutrino physics II) Neutrinoless double beta decay: standard interpretation II1) Basics II2) Implications for and from neutrino physics II3) Nuclear physics aspects II4) Exotic physics 26
I2) Seesaw Mechanisms how to generate Majorana neutrino masses . . . a) Higher dimensional operators include d ≥ 5 operators, gauge and Lorentz invariant, only SM fields: L = L SM + L 5 + L 6 + . . . = L SM + 1 Λ O 5 + 1 Λ 2 O 6 + . . . there is only one dimension 5 term! “leading order new physics” → c v 2 Λ O 5 = c 1 Φ T L c SSB Λ L ˜ Φ ˜ 2 Λ ν L ν c L ≡ m ν ν L ν c − L Majorana mass term! ∆ L = 2 � � 10 14 GeV Λ > 0 . 1 eV it follows ∼ c Weinberg 1979 m ν 27
g � ` d L f ` � a L b Weinberg operator is LL ΦΦ seesaw mechanisms are “UV-completions” of this effective operator by integrating out heavy physics 28
Master formula: 2 × 2 = 3 + 1 SU (2) L × U (1) Y with 2 ⊗ 2 = 3 ⊕ 1 : L ˜ Φ ∼ (2 , +1) ⊗ (2 , − 1) = (3 , 0) ⊕ (1 , 0) to make a singlet, couple to (1 , 0) or (3 , 0) , because 3 ⊗ 3 = 5 ⊕ 3 ⊕ 1 Alternatively: L c L ∼ (2 , − 1) ⊗ (2 , − 1) = (3 , − 2) ⊕ (1 , − 2) to make a singlet, couple to (1 , +2) or (3 , +2) . However, singlet combination (1 , − 2) is ν ℓ c − ℓ ν c , which cannot generate neutrino mass term = ⇒ (1 , 0) or (3 , +2) or (3 , 0) type I type II type III 29
g � ` d L f ` � a L b Origin of small masses has only 3 tree-level realizations L = c • N R ∼ (1 , 0) type I seesaw Φ ∗ ˜ Φ † L Λ L c ˜ • ∆ ∼ (3 , 2) type II seesaw • Σ ∼ (3 , 0) type III seesaw seesaws include new representations, new energy scales, new concepts 30
Type I Seesaw introduce N R ∼ (1 , 0) and couple to g ν L ˜ Φ ∼ (1 , 0) √ becomes g ν v/ 2 ν L N R ≡ m D ν L N R in addition: Majorana mass term for N R : 1 2 M R N c R N R using ν L m D N R = N c R m T D ν c L : ν c 0 m D L + h.c. L = 1 2 ( ν L , N c R ) m D M R N R 2 Ψ M ν Ψ c + h.c. ≡ 1 Dirac + Majorana mass term is a Majorana mass term! 31
Diagonalization: ν c 0 m D L L = 1 2 ( ν L , N c R ) m D M R N R ν c 0 m D L U ∗ = 1 U † U T ( ν L , N c R ) U 2 m D M R N R � �� � � �� � � �� � ( ν c , N ) T ( ν, N c ) diag( m ν , M ) with general formula: if m D ≪ M R : 2 m D tan 2 θ = ≪ 1 M R − 0 � � � (0 − M R ) 2 + 4 m 2 m ν = 1 ≃ − m 2 (0 + M R ) − D /M R D 2 � � � (0 − M R ) 2 + 4 m 2 M = 1 (0 + M R ) + ≃ M R D 2 32
√ Note: m D associated with EWSB, part of SM, bounded by v/ 2 = 174 GeV M R is SM singlet, does whatever it wants: ⇒ M R ≫ m D Hence, θ ≃ m D /M R ≪ 1 ν = ν L cos θ − N c with mass m ν ≃ − m 2 R sin θ ≃ ν L D /M R N = N R cos θ + ν c L sin θ ≃ N R with mass M ≃ M R in effective mass terms L ≃ 1 L + 1 2 m ν ν L ν c 2 M R N c R N R compare with Weinberg operator: Λ = − c v 2 M R m 2 D also: integrate N R away with Euler-Lagrange equation 33
matrix case: block diagonalization ν c 0 m D L L = 1 2 ( ν L , N c R ) m T M R N R D ν c 0 m D L = 1 U † U ∗ U T ( ν L , N c R ) U 2 m T M R N R � �� � D � �� � � �� � ( ν c , N ) T ( ν, N c ) diag( m ν , M ) with 6 × 6 diagonal matrix − ρ ∗ 1 − ρ 1 ρ 1 U † = U ∗ = , , U = ρ † − ρ † ρ T 1 1 1 write down individual components: 34
write down individual components: ρ m T D + m D ρ + ρ M R ρ T m ν = D ρ ∗ + ρ M R m D − ρ m T 0 = − ρ † m D − m T D ρ ∗ + M R M = now, ρ (aka θ from before) will be of order m D /M R : ρ m T D + m D ρ + ρ M R ρ T m ν = m D + ρ M R ⇒ ρ = − m D M − 1 0 ≃ R M ≃ M R insert ρ in m ν to find: m ν = − m D M − 1 R m T D 35
m ν = m 2 = m 2 D SM = m SM ǫ M R M R (type I) Seesaw Mechanism Minkowski; Yanagida; Glashow; Gell-Mann, Ramond, Slansky; Mohapatra, Senjanović (77-80) we found the requested suppression mechanism! 36
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Seesaw Formalism ν c 0 m D L = 1 M R ≫ m D ν L , ¯ L N c m ν = m T D M − 1 2(¯ R ) = ⇒ R m D m T M R N R D arbitrary 6 × 6 (?) matrix with new aspects: • fermionic singlets N R ∼ (1 , 0) • new energy scale M R ( ∝ 1 /m ν ) • lepton number violation 41
See-Saw Phenomenology Full mass matrix: m diag 0 m D 0 N S U T with U = ν = U M = M diag m T M R 0 T V D R • N is the PMNS matrix: non-unitary R ) − 1 describes mixing of heavy neutrinos with SM leptons • S = m † D ( M ∗ 42
Type II Seesaw L ∝ L c L → ν T ν has isospin I 3 = 1 and transforms as ∼ (3 , − 2) ⇒ introduce Higgs triplet ∼ (3 , +2) with ( I 3 = Q − Y/ 2 ): √ ∆ + 2 ∆ ++ − 0 0 → ∆ = √ 2 ∆ 0 − ∆ + v T 0 gives mass matrix: L = 1 → 1 L ν L ≡ 1 vev 2 y ν L c iσ 2 ∆ L 2 y ν v T ν c 2 m ν ν c − L ν L ↔ neutrino mass without right-handed neutrinos! 43
v T ≪ v because � ∆ † ∆ � + µ Φ † ∆ ˜ V = − M 2 ∆ Tr Φ v T = µ v 2 with ∂V ∂ ∆ = 0 one has M 2 ∆ v T can be suppressed by M ∆ and/or µ compare with Weinberg operator: Λ = c M 2 ∆ g ν µ Type II (or Triplet) Seesaw Mechanism Magg, Wetterich; Mohapatra, Senjanovic; Lazarides, Shafi, Wetterich; Schechter, Valle (80-82) 44
Seesaw Summary g � ` d L f ` � a L b 45
Paths to Neutrino Mass quantum number approach ingredient scale mν L of messenger “SM” h = O (10 − 12) RH ν NR ∼ (1 , 0) hNR Φ L hv (Dirac mass) h v 2 “effective” new scale Λ = 1014 GeV h Lc Φ Φ L – Λ + LNV (dim 5 operator) “direct” Higgs triplet hµ M 2 1 hLc ∆ L + µ ΦΦ∆ ∆ ∼ (3 , 2) hvT Λ = ∆ + LNV (type II seesaw) ( hv )2 “indirect 1” RH ν hNR Φ L + NRMRNc Λ = 1 NR ∼ (1 , 0) h MR R MR + LNV (type I seesaw) ( hv )2 “indirect 2” fermion triplets Λ = 1 Σ ∼ (3 , 0) h Σ L Φ + TrΣ M ΣΣ h M Σ M Σ + LNV (type III seesaw) plus seesaw variants (linear, double, inverse, . . . ) plus radiative mechanisms plus extra dimensions plusplusplus 46
Contents I) Dirac vs. Majorana neutrinos I1) Basics I2) Seesaw mechanisms I3) Summary neutrino physics II) Neutrinoless double beta decay: standard interpretation II1) Basics II2) Implications for and from neutrino physics II3) Nuclear physics aspects II4) Exotic physics 47
I3) Summary neutrino physics common prediction of all mechanisms: in basis with diagonal charged leptons L = 1 2 ν T m ν ν with m ν = U diag( m 1 , m 2 , m 3 ) U T with PMNS matrix s 13 e − iδ c 12 c 13 s 12 c 13 − s 12 c 23 − c 12 s 23 s 13 e iδ c 12 c 23 − s 12 s 23 s 13 e iδ U = P s 23 c 13 s 12 s 23 − c 12 c 23 s 13 e iδ − c 12 s 23 − s 12 c 23 s 13 e iδ c 23 c 13 with P = diag( e iα , e iβ , 1) ( ↔ Majorana, lepton number violation) ⇒ 3 angles, 3 phases, 3 masses “three Majorana neutrino paradigm” 48
Status 2016 9 physical parameters in m ν • θ 12 and m 2 2 − m 2 1 • θ 23 and | m 2 3 − m 2 2 | • θ 13 • m 1 , m 2 , m 3 • sgn( m 2 3 − m 2 2 ) • Dirac phase δ • Majorana phases α and β 49
Lisi et al. , 1601.07777 (3-flavor, matter effects in solar, atm., LBL expts) 50
m 2 normal inverted t m 2 m 2 3 2 ∆ m 2 21 m 2 1 d ∆ m 2 s ν e 32 ν µ b ν τ ∆ m 2 31 m 2 m 2 c 2 ∆ m 2 21 m 2 m 2 m 2 3 u 1 Why so different? ↔ Flavor symmetries! 51
PMNS-matrix: 0 . 801 . . . 0 . 845 0 . 514 . . . 0 . 580 0 . 137 . . . 0 . 158 | U | = 0 . 225 . . . 0 . 517 0 . 441 . . . 0 . 699 0 . 614 . . . 0 . 793 0 . 246 . . . 0 . 529 0 . 464 . . . 0 . 713 0 . 590 . . . 0 . 776 CKM-matrix: 0 . 00355 +0 . 00015 0 . 97427 ± 0 . 00014 0 . 22536 ± 0 . 00061 − 0 . 00014 | V | = 0 . 22522 ± 0 . 00061 0 . 97341 ± 0 . 00015 0 . 0414 ± 0 . 0012 0 . 00886 +0 . 00033 0 . 0405 +0 . 0011 0 . 99914 ± 0 . 00005 − 0 . 00032 − 0 . 0012 large mixing in PMNS, small mixing in CKM Why so different? ↔ Flavor symmetries! 52
Neutrino masses • neutrino masses ↔ scale of their origin • neutrino mass ordering ↔ form of m ν • m 2 3 ≃ ∆ m 2 A ≫ m 2 2 ≃ ∆ m 2 ⊙ ≫ m 2 1 : normal hierarchy (NH) • m 2 2 ≃ | ∆ m 2 A | ≃ m 2 1 ≫ m 2 3 : inverted hierarchy (IH) • m 2 3 ≃ m 2 2 ≃ m 2 1 ≡ m 2 0 ≫ ∆ m 2 A : quasi-degeneracy (QD) 53
Contents I) Dirac vs. Majorana neutrinos I1) Basics I2) Seesaw mechanisms I3) Summary neutrino physics II) Neutrinoless double beta decay: standard interpretation II1) Basics II2) Implications for and from neutrino physics II3) Nuclear physics aspects II4) Exotic physics 54
II) Neutrinoless Double Beta Decay: Standard Interpretation ( A, Z ) → ( A, Z + 2) + 2 e − (0 νββ ) • second order in weak interaction: Γ ∝ G 4 F ⇒ rare! • not to be confused with ( A, Z ) → ( A, Z + 2) + 2 e − + 2 ¯ ν e (2 νββ ) (which occurs more often but is still rare) 55
Contents I) Dirac vs. Majorana neutrinos I1) Basics I2) Seesaw mechanisms I3) Summary neutrino physics II) Neutrinoless double beta decay: standard interpretation II1) Basics II2) Implications for and from neutrino physics II3) Nuclear physics aspects II4) Exotic physics 56
II1) Basics Need to forbid single β decay: • ⇒ even/even → even/even • either direct (0 νββ ) or two simultaneous decays with virtual (energetically forbidden) intermediate state (2 νββ ) 57
��,������������������������������- ��,������������������������������- '(������)#�*+ � Slide by A. Giuliani 58
• 35 candidate isotopes • 9 are interesting: 48 Ca, 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 130 Te, 136 Xe, 150 Nd • Q -value vs. natural abundance vs. reasonably priced enrichment vs. association with a well controlled experimental technique vs. . . . ⇒ no superisotope G 0 ν for 0 νββ -decay of different Isotopes Natural abundance of different 0 νββ candidate Isotopes 35 20 18 30 16 Natural abundance [%] 25 14 G 0 ν [10 -14 yrs -1 ] 12 20 10 15 8 6 10 4 5 2 0 0 48 Ca 76 Ge 82 Se 96 Zr 100 Mo 110 Pd 116 Cd 124 Sn 130 Te 136 Xe 150 Nd 48 Ca 76 Ge 82 Se 96 Zr 100 Mo 110 Pd 116 Cd 124 Sn 130 Te 136 Xe 150 Nd Isotope Isotope T 0 ν T 0 ν 1 / 2 ∝ Q − 5 1 / 2 ∝ 1 /a 59
G [10 − 14 yrs − 1 ] Isotope Q [keV] nat. abund. [%] 48 Ca 6.35 4273.7 0.187 76 Ge 0.623 2039.1 7.8 82 Se 2.70 2995.5 9.2 96 Zr 5.63 3347.7 2.8 100 Mo 4.36 3035.0 9.6 110 Pd 1.40 2004.0 11.8 116 Cd 4.62 2809.1 7.6 124 Sn 2.55 2287.7 5.6 130 Te 4.09 2530.3 34.5 136 Xe 4.31 2461.9 8.9 150 Nd 19.2 3367.3 5.6 Most mechanisms: G x ∝ Q 5 60
Experimental Aspects g • experimental signature: sum of electron energies = Q (plus: 2 electrons and daughter isotope) • background of 2 νββ ↔ resolution 61
if claimed, typical spectrum will look like: ⇒ first reason for multi-isotope determination 62
Experimental Aspects number of events (measuring time t ≪ T 0 ν 1 / 2 life-time): N = ln 2 a M t N A ( T 0 ν 1 / 2 ) − 1 with • a is abundance of isotope • M is used mass • t is time of measurement • N A is Avogadro’s number 63
Experimental Aspects Number of events (measuring time t ≪ T 0 ν 1 / 2 life-time): N = ln 2 a M t N A ( T 0 ν 1 / 2 ) − 1 suppose there is no background: • if you want 10 26 yrs you need 10 26 atoms • 10 26 atoms are 10 3 mols • 10 3 mols are 100 kg From now on you can only loose: efficiency, background, natural abundance, . . . 64
Experimental Aspects without background a M ε t 1 / 2 ) − 1 ∝ ( T 0 ν � M t with background a ε B ∆ E with • B is background index in counts/(keV kg yr) • ∆ E is energy resolution • ǫ is efficiency 1 / 2 ) − 1 ∝ (particle physics) 2 • ( T 0 ν Note: factor 2 in particle physics is combined factor of 16 in M × t × B × ∆ E 65
Sensitivity and backgrounds 1-tonne 76 Ge Example T " 0 " = ln(2) N $ t/ UL( % ) Neutrinoless Double Beta Decay Goldhaber-Grodzins-Sunyar Celebration slide by J.F. Wilkerson 66
Interpretation of Experiments Master formula: Γ 0 ν = G x ( Q, Z ) |M x ( A, Z ) η x | 2 • G x ( Q, Z ) : phase space factor • M x ( A, Z ) : nuclear physics • η x : particle physics 67
Interpretation of Experiments Master formula: Γ 0 ν = G x ( Q, Z ) |M x ( A, Z ) η x | 2 • G x ( Q, Z ) : phase space factor; calculable • M x ( A, Z ) : nuclear physics; problematic • η x : particle physics; interesting 68
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 0 νββ give negligible or no contribution d L u L W U ei e − L ν i q ν i e − L U ei W u L d L 69
d L u L W U ei e − L ν i q ν i e − L U ei W u L d L • U 2 ei from charged current • m i /E i from spin-flip and if neutrinos are Majorana particles amplitude proportional to coherent sum (“effective mass”) �� U 2 � � | m ee | = ei m i � m/E ≃ eV/100 MeV is tiny: only N A can save the day! 70
The effective mass Im 2i α | | . (2) m e 2i β ee | | e . (3) m ee m ee Re | | (1) m ee amplitude proportional to coherent sum (“effective mass”): �� U 2 � � � � | U e 1 | 2 m 1 + | U e 2 | 2 m 2 e 2 iα + | U e 3 | 2 m 3 e 2 iβ � � = | m ee | ≡ ei m i � � � θ 12 , | U e 3 | , m i , sgn(∆ m 2 = f A ) , α, β 7 out of 9 parameters of neutrino physics! 71
Contents I) Dirac vs. Majorana neutrinos I1) Basics I2) Seesaw mechanisms I3) Summary neutrino physics II) Neutrinoless double beta decay: standard interpretation II1) Basics II2) Implications for and from neutrino physics II3) Nuclear physics aspects II4) Exotic physics 72
II2) Implications for and from neutrino physics �� U 2 � � | m ee | = ei m i � : fix known things, vary known unknown things, assume no unknown unknowns are present: normal invertiert m 2 m 2 3 2 ∆ m 2 21 m 2 1 ∆ m 2 ν e 32 ν µ ν τ ∆ m 2 31 m 2 2 ∆ m 2 21 m 2 m 2 3 1 73
The usual plot 74
The usual plot: the other way around (life-time instead of | m ee | ) Normal Inverted 32 10 30 10 [T 1/2 ] ν (yrs) 28 10 26 10 Excluded by HDM 0.0001 0.001 0.01 0.1 0.0001 0.001 0.01 0.1 m light (eV) 75
Which mass ordering with which life-time? Σ m β | m ee | � � � A e 2 i ( α − β ) � � ⊙ + | U e 3 | 2 � ∆ m 2 ∆ m 2 ⊙ + | U e 3 | 2 ∆ m 2 � ∆ m 2 ∆ m 2 � NH A A � � ∼ 10 28 − 29 yrs 1 / 2 > ∼ 0 . 003 eV ⇒ T 0 ν ≃ 0 . 05 eV ≃ 0 . 01 eV � � � � 1 − sin 2 2 θ 12 sin 2 α ∆ m 2 ∆ m 2 ∆ m 2 IH 2 A A A ∼ 10 26 − 27 yrs 1 / 2 > ∼ 0 . 03 eV ⇒ T 0 ν ≃ 0 . 1 eV ≃ 0 . 05 eV � 1 − sin 2 2 θ 12 sin 2 α QD 3 m 0 m 0 m 0 ∼ 10 25 − 26 yrs > 1 / 2 > ∼ 0 . 1 eV ⇒ T 0 ν 76
The usual plot: include other neutrino mass approaches Normal Inverted 0 10 <m ee > = 0.4 eV <m ee > = 0.4 eV CPV CPV (+,+) (+,+) -1 10 (+,-) (+,-) (-,+) (-,+) <m ee > (eV) (-,-) (-,-) m β = 0.35 eV m β = 0.35 eV m β = 0.2 eV m β = 0.2 eV -2 10 -3 10 Σ = 0.1 eV Σ = 0.2 eV Σ = 0.5 eV Σ = 0.1 eV Σ = 0.2 eV Σ = 0.5 eV Σ = 1 eV Σ = 1 eV -4 10 0.001 0.01 0.1 0.001 0.01 0.1 m light (eV) 77
Plot against other observables θ 1 Normal Inverted Normal Inverted 0 0 10 10 CPV CPV CPV CPV (+,+) (+) (+,+) (+) (+,-) (-) (+,-) (-) (-,+) (-,+) (-,-) (-,-) -1 -1 10 10 <m ee > (eV) <m ee > (eV) -2 -2 10 10 -3 -3 10 10 0.01 0.1 0.01 0.1 0.1 1 0.1 1 m β (eV) Σ m i (eV) i and Σ = � m i � | U ei | 2 m 2 Complementarity of | m ee | = U 2 ei m i , m β = 78
CP violation! Dirac neutrinos! something else does 0 νββ ! 79
Neutrino Mass � m (heaviest) > | m 2 3 − m 2 1 | ≃ 0 . 05 eV ∼ 3 complementary methods to measure neutrino mass: Method observable now [eV] near [eV] far [eV] pro con �� | U ei | 2 m 2 model-indep.; final?; Kurie 2 . 3 0 . 2 0 . 1 i theo. clean worst � m i best; systemat.; Cosmo. 0 . 5 0 . 2 0 . 05 NH/IH model-dep. | � U 2 fundament.; model-dep.; 0 νββ ei m i | 0 . 3 0 . 1 0 . 05 NH/IH theo. dirty 80
Cosmological Limits stacking more and more data sets on top of each other, limits become stronger including more and more parameters, limits become weaker needed to break degeneracies, but induces systematic issues Planck15+BAO+SN+H 0 4.0 Planck15+BAO+SN 3.5 N eff 3.0 2.5 62 64 66 68 70 72 74 H 0 (km s -1 Mpc -1 ) Riess et al. , Palanque-Delabrouille et al. , Hannestad, 1604.01424 1506.05976 PRL 95 most extreme (1410.7244) : at 1 σ : Σ m ν ≤ 0 . 08 eV, disfavors inverted ordering . . . 81
2 kinds of neutrino masses 1) ee -element of mass matrix � ∝ | ( m ν ) ee | with ( m ν ) ee = h ee v 2 1 in L eff = 1 h αβ Φ T L c Λ L α ˜ Φ ˜ β T 0 ν Λ 2 1 / 2 direct probe of fundamental object in low energy Lagrangian! 2) neutrino mass scale: QD neutrinos | m ee | QD = m 0 � 13 e 2 iα + s 2 13 e 2 iβ � � c 2 12 c 2 13 + s 2 12 c 2 � 1 + tan 2 θ 12 ⇒ m 0 ≤ | m ee | exp 1 − tan 2 θ 12 − 2 | U e 3 | 2 ≃ 3 | m ee | exp min ≃ eV min same order as Mainz/Troitsk! 82
Alternative processes ( A, Z ) → ( A, Z + 2) ∗ + 2 e − (0 νββ ) ∗ ( A, Z ) → ( A, Z − 2) + 2 e + (0 νβ + β + ) e − b + ( A, Z ) → ( A, Z − 2) + e + (0 νβ + EC) 2 e − b + ( A, Z ) → ( A, Z − 2) ∗ (0 ν ECEC) depend on same particle physics parameters, but more difficult to realize/test BUT: ratio to 0 νββ is test of NME calculation and mechanism 83
Alternative processes the lobster: � | m eµ | � 2 � � 2 � BR( K + → π − e + µ + ) ∝ | m eµ | 2 = ∼ 10 − 30 � � U ei U µi m i � � eV 84
Reconstructing m ν m ν = 85
Recent Results • 76 Ge: – GERDA: T 1 / 2 > 2 . 1 × 10 25 yrs – GERDA + IGEX + HDM: T 1 / 2 > 3 . 0 × 10 25 yrs • 136 Xe: – EXO-200: T 1 / 2 > 1 . 1 × 10 25 yrs (first run with less exposure: T 1 / 2 > 1 . 6 × 1025 yrs . . . ) – KamLAND-Zen: T 1 / 2 > 2 . 6 × 10 25 yrs Xe-limit is stronger than Ge-limit when: � � 2 G Ge M Ge � � T Xe > T Ge yrs � � G Xe M Xe � � 86
Current Limits on | m ee | 76 Ge 136 Xe NME 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-A 0.31 0.26 0.23 – QRPA-A 0.28 0.24 0.25 – SkM-HFB-QRPA 0.29 0.24 0.28 – 87
Inverted Ordering Nature provides 2 scales: � � | m ee | IH max ≃ c 2 | m ee | IH min ≃ c 2 ∆ m 2 ∆ m 2 and A cos 2 θ 12 13 13 A requires O (10 26 . . . 10 27 ) yrs is the lower limit | m ee | IH min fixed? 88
Inverted Hierarchy m 3 = 0.001 eV 0.1 IH, 3 σ 0.25 0.125 1 IH, BF 1 / 2 [10 27 y] 0.5 0.25 2 � m ν � [eV] 1 0.5 4 2 T 0 ν 1 8 4 2 16 8 4 32 16 0.01 8 0.28 0.3 0.32 0.34 0.36 0.38 sin 2 θ 12 Current 3 σ range of sin 2 θ 12 gives factor of ∼ 2 uncertainty for | m ee | IH min ⇒ combined factor of ∼ 16 in M × t × B × ∆ E ⇒ need precision determination of θ 12 ! ↔ JUNO 89
90 Ge, W.R., PRD 92 ; http://nupro.hepforge.org 0.7 Prior Posterior 0.6 0.5 min M ee [meV] PDF dP/d θ 12 0.4 0.3 0.2 0.1 10 0 o o o o o o o o o 1 10 100 30 31 32 33 34 35 36 37 38 m 3 [meV] θ 12
CP Violation? Majorana phases: consider IH spectrum � � � cos 2 θ 12 + e 2 iα sin 2 θ 12 � 1 − sin 2 2 θ 12 sin 2 α � = | m ee | ∝ α can be probed if • uncertainties on | m ee | from NME smaller than 2 • σ ( | m ee | ) < ∼ 15% A ) < ∼ 10% (IH) or σ ( m 0 ) < • σ (∆ m 2 ∼ 10% (QD) • sin 2 θ 12 > ∼ 0 . 29 • 2 α ∈ [ π/ 4 , 3 π/ 4] or [5 π/ 4 , 7 π/ 4] Pascoli, Petcov, W.R., PLB 549 No to “no-go” from Barger et al. , PLB 540 91
Majorana phases Normal Inverted Normal Inverted 0 0 10 10 CPV CPV CPV CPV (+,+) (+) (+,+) (+) (+,-) (-) (+,-) (-) (-,+) (-,+) (-,-) (-,-) -1 -1 10 10 <m ee > (eV) <m ee > (eV) -2 -2 10 10 -3 -3 10 10 0.01 0.1 0.01 0.1 0.1 1 0.1 1 m β (eV) Σ m i (eV) 92
Majorana phases � � � cos 2 θ 12 + e 2 iα sin 2 θ 12 � 1 − sin 2 2 θ 12 sin 2 α � = | m ee | ∝ sin 2 θ 12 = 0.25 + sin 2 θ 12 = 0.31 + sin 2 θ 12 = 0.38 + − 3% − 3% − 3% 10 1 σ ββ = 0.03 eV σ Σ = 0.1 eV 10 0 observed Σ [eV] 10 -1 σ ββ = 0.01 eV σ Σ = 0.05 eV 10 0 10 -1 1 2 3 4 1 2 3 4 1 2 3 4 uncertainty in |< m >| from NME data consistent with α 21 = π data consistent with α 21 = 0 |< m >| and Σ inconsistent at 2 σ CP violation established at 2 σ 21 = 8x10 -5 + 31 = 2.2x10 -3 + sin 2 θ 13 = 0 + − 0.002, ∆ m 2 − 2%, ∆ m 2 observed |< m >| = 0.3 eV − 3% Pascoli, Petcov, Schwetz, hep-ph/0505226 93
Vanishing | m ee | Im 2i α | | . (2) m e 2i β ee | | e . (3) m ee m ee Re | | (1) m ee only for NH ⇒ rule out possibility by ruling out NH unnatural? texture zero!?! 94
Vanishing | m ee | does it stay zero? • seesaw RG effects • NLO see-saw terms: m ν = m 2 D /M R + O ( m 4 D /M 3 R ) • actually: A ∝ U 2 ei m i ≃ | m ee | + O ( m 3 i /q 4 ) q 2 − m 2 q 2 i • Planck scale (Weinberg operator with Planck mass) v 2 ≃ 10 − 6 eV | m ee | = M Pl 95
Renormalization ( m 0 ν ) ee I 2 ( m 0 ( m 0 ν ) eµ I e I µ ν ) eτ I e I τ e ( m 0 ν ) µµ I 2 ( m 0 m ν = I α ν · ν ) µτ I µ I τ µ ( m 0 ν ) ττ I 2 · · τ where C α ln λ 16 π 2 α ν ln λ 1 16 π 2 y 2 I α ≃ 1 + Λ and I α ν ≃ 1 + Λ and � � α SM = − 3 g 2 2 + 2( y 2 τ + y 2 µ + y 2 y 2 t + y 2 b + y 2 c + y 2 s + y 2 d + y 2 e ) + 6 + λ H ν u � � α MSSM = − 6 5 g 2 1 − 6 g 2 y 2 t + y 2 c + y 2 2 + 6 ν u ⇒ main effect: rescaling of | m ee | , typically increases from low to high scale 96
Renormalization Antusch et al. , hep-ph/0305273 97
Flavor Symmetry Models: sum-rules Normal Inverted -1 10 2m 2 + m 3 = m 1 -2 10 <m ee > (eV) -3 3 σ 30% error 10 3 σ exact TBM exact -1 10 m 1 + m 2 = m 3 -2 10 -3 10 0.01 0.1 0.01 0.1 m β (eV) constraints on masses and Majorana phases Barry, W.R., Nucl. Phys. B842 98
Predictions of SO (10) theories Yukawa structure of SO (10) models depends on Higgs representations 10 H ( ↔ H ) , 126 H ( ↔ F ) , 120 H ( ↔ G ) Gives relation for mass matrices: m up ∝ r ( H + sF + it u G ) m down ∝ H + F + iG m D ∝ r ( H − 3 sF + it D G ) m ℓ ∝ H − 3 F + it l G M R ∝ r − 1 R F Numerical fit including RG, Higgs, θ 13 10 H + 126 H : 19 free parameters 10 H + 126 H + 120 H : 18 free parameters 20 (19) observables to be fitted 99
Predictions of SO (10) theories χ 2 | m ee | m 0 M 3 Model Fit [meV] [meV] [GeV] 3 . 6 × 10 12 NH 10 H + 126 H 0 . 49 2 . 40 23 . 0 1 . 1 × 10 12 10 H + 126 H + SS NH 0 . 44 6 . 83 3 . 29 9 . 9 × 10 14 NH 10 H + 126 H + 120 H 2 . 87 1 . 54 11 . 2 4 . 2 × 10 13 6 . 9 × 10 − 6 10 H + 126 H + 120 H + SS NH 0 . 78 3 . 17 1 . 1 × 10 13 10 H + 126 H + 120 H IH 35 . 52 30 . 2 13 . 3 1 . 2 × 10 13 10 H + 126 H + 120 H + SS IH 24 . 22 12 . 0 0 . 6 Dueck, W.R., JHEP 1309 100
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