Neutrinoless Double Beta Decay: Theory

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T -1 Werner Rodejohann mv = m L - m D M R m D SSI 2015 MANITOP 17/08/15 Massive Neutrinos: Investigating their Theoretical Origin and Phenomenology

1 Neutrinos...

...still interesting...

2 What is Neutrinoless Double Beta Decay? (A, Z) (A, Z + 2) + 2 e− (0νββ) →

second order in weak interaction, Γ G4 Q5 rare! • ∝ F ⇒ not to be confused with (A, Z) (A, Z + 2) + 2 e− + 2ν ¯ (2νββ) • → e (Γ G4 Q11, but occurs more often...) ∝ F

3 Need to forbid single β decay:

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

4 How many nuclei in thisHow nuclei many condition? in How many nuclei in thisHow nuclei many condition? in ld yA ilai e etr yLs Kaufman Lisa by lecture see Giuliani, A. by Slide Q-value [MeV] 5 A 35 candidate isotopes • 10 are interesting: 48Ca, 76Ge, 82Se, 96Zr, 100Mo, 116Cd, 124Sn, 130Te, • 136Xe, 150Nd Q-value vs. natural abundance vs. reasonably priced enrichment • vs. association with a well controlled experimental technique vs... no superisotope ⇒ 0ν Natural abundance of differentνββ candidate 0 Isotopes G for νββ0 -decay of different Isotopes 35 20

18 30 16

25 14 ] -1 12

20 yrs

-14 10 15 [10

ν 8 0 G 10 6 Natural abundance [%] Natural abundance 4 5 2

0 0 48 76 82 96 100 110 116 124 130 136 150 48 76 82 96 100 110 116 124 130 136 150 Ca Ge Se Zr Mo Pd Cd Sn Te Xe Nd Ca Ge Se Zr Mo Pd Cd Sn Te Xe Nd

Isotope Isotope

T 0ν 1/a T 0ν Q−5 1/2 ∝ 1/2 ∝ 6 Neutrinoless Double Beta Decay (A, Z) (A, Z + 2) + 2 e− (0νββ) Lepton Number Violation → ⇒ Standard Interpretation (neutrino physics) • Non-Standard Interpretations (BSM = neutrino physics) • 6 Int. J. Mod. Phys. E20, 1833 (2011) [1106.1334]; J. Phys. G39, 124008 (2012) [1206.2560]; 1507.00170

7 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 baryogenesis: 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 Baryon Number Violation ⇒ 0νββ is NOT a neutrino mass experiment!!

8 Upcoming/running experiments: exciting time!! best limit was from 2001, improved 2012

Name Isotope Source = Detector; calorimetric with Source 6= Detector high ∆E low ∆E topology topology AMoRE 100Mo X – – – CANDLES 48Ca – X – – COBRA 116Cd (and 130Te) – – X – CUORE 130Te X – – – DCBA/MTD 82Se / 150Nd – – – X EXO 136Xe – – X – GERDA 76Ge X – – – CUPID 82Se / 100Mo / 116Cd / 130Te X – – – KamLAND-Zen 136Xe – X – – LUCIFER 82Se / 100Mo / 130Te X – – – LUMINEU 100Mo X – – – MAJORANA 76Ge X – – – MOON 82Se / 100Mo / 150Nd – – – X NEXT 136Xe – – X – SNO+ 130Te – X – – SuperNEMO 82Se / 150Nd – – – X XMASS 136Xe – X – –

see lecture by Lisa Kaufman

9 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): nuclear physics • Mx η : particle physics • x

10 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

11 Interpretation of Experiments Master formula:

Γ0ν = G (Q, Z) (A, Z) η 2 x |Mx x| G (Q, Z): phase space factor, typically Q5 • x (A, Z): nuclear physics; factor 2-3 uncertainty • Mx η : particle physics; many possibilities • x

12 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 u dL L

W Uei − eL i ν q νi − eL Uei W

u dL L

prediction of (100 ǫ) % of all neutrino mass mechanisms − (lecture by Andre De Gouvea)

13 Paths to Neutrino Mass

quantum number approach ingredient 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

14 dL uL

Uei − eL νi q νi − eL Uei W

uL dL

U 2 from charged current • ei m /E from spin-ﬂip and if neutrinos are Majorana particles • i i amplitude proportional to coherent sum (“eﬀective mass”)

m = U 2 m | ee| ei i P m/E eV/100 MeV is tiny: only N can save the day! ≃ A

15 Dirac vs. Majorana in V A theories: observable diﬀerence always suppressed by (m/E)2 − In terms of degrees of freedom (helicity and particle/antiparticle): • νD =(ν↑,ν↓, ν¯↑, ν¯↓) versus νM =(ν↑,ν↓) weak interactions act on chirality (left-/right-handed) • chirality is not a good quantum number (“spin ﬂip”): L = + m • ↓ E ↑ Dirac: • − − m – what we produce from a W is ℓ (¯ν↑ + E ν¯↓) − − – the ν¯↓ CANNOT interact with another W to generate another ℓ : ∆L = 0 Majorana: • − − m – what we produce from a W is ℓ (ν↑ + E ν↓) − − – the ν↓ CAN interact with another W to generate another ℓ : ∆L = 2 amplitude (m/E) probability (m/E)2 ⇒ ∝ ⇒ ∝ 16 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 2 + − + + 2 −30 meµ BR(K π e µ ) meµ = Uei Uµi mi 10 | | → ∝| | ∼ 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 Xi,j

17 The eﬀective mass Im

α (2) . 2i β |m ee | e (3) . 2i |m ee | e

m ee (1) Re |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

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

18 Fix known things, vary unknown things...

normal invertiert

2 2 m3 m2

2 ∆m21 2 m1

2 ∆m32 νe

νµ

ντ 2 ∆m31

2 m2

2 ∆m21 2 2 m1 m3

19 The usual plot

20 Alternative processes The lobster:

2 2 + − + + 2 −30 meµ BR(K π e µ ) meµ = Uei Uµi mi 10 | | → ∝| | ∼ eV X

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

Normal Inverted

32 10

30 10 (yrs) ν ]

1/2 28 10 [T

26 10

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

22 Which mass ordering with which life-time? Σ m m β | ee| 2 2 2 2 2 2 2 2i(α−β) NH ∆mA ∆m⊙ + Ue3 ∆mA ∆m⊙ + Ue3 ∆mAe q | | q | | p 0pν > 28−29 0.05 eV 0.01 eV 0.003 eV T1/2 10 yrs ≃ ≃ ∼ ⇒ ∼ IH 2 ∆m2 ∆m2 ∆m2 1 sin2 2θ sin2 α A A A − 12 p p p p 0ν > 26−27 0.1 eV 0.05 eV 0.03 eV T1/2 10 yrs ≃ ≃ ∼ ⇒ ∼ QD 3m m m 1 sin2 2θ sin2 α 0 0 0 − 12 > p 0ν > 25−26 0.1 eV T1/2 10 yrs ∼ ⇒ ∼

23 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

24 CP violation! Dirac neutrinos! something else does 0νββ!

25 Neutrino Mass m(heaviest) > m2 m2 0.05 eV | 3 − 1|≃ ∼ p 3 complementary methods to measure neutrino mass:

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

2 model-indep.; ﬁnal?; Kurie |Uei|2 m 2.3 0.2 0.1 i theo. clean worst

pP best; systemat.; mi . . . Cosmo. 0 7 0 3 0 05 NH/IH model-dep. P 2 fundament.; model-dep.; νββ Ueimi . . . 0 | | 0 3 0 1 0 05 NH/IH theo. dirty P

see lectures by Alessandro Melchiorri and Joe Formaggio

26 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

27 Current 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-A 0.31 0.26 0.23 –

QRPA-A 0.28 0.24 0.25 –

SkM-HFB-QRPA 0.29 0.24 0.28 – GERDA Bhupal Dev, Goswami, Mitra, W.R., Phys. Rev. D88

28 Generalization...

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

30 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 q 1/r 0.1 GeV ≃ ≃ NME overlap of decaying nucleons... • ↔ diﬀerent approaches (QRPA, NSM, IBM, GCM, pHFB) imply uncertainty • plus uncertainty due to model details •

31 typical model for NME: set of single particle states with a number of possible wave function configurations; obtained by solving Dirac equation in a mean background field; interaction known, treatment of fields differs

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

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

Interacting Boson Model (IBM) (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

32 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

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

33 Faessler, Fogli et al., PRD 79+87

34 NMEs are order one numbers :-)

IBM NSM best-fit: 4.06 best-fit: 1.92 6,0 6,0 IBM NSM 4 +/- 1 sqrt(4) +/- sqrt(1/10)

5,0 5,0

4,0 4,0 ν ν 0 3,0 0 3,0 M M

2,0 2,0

1,0 1,0

0,0 0,0 Ge(76) Se(82) Mo(100) Te(128) Te(130) Ge(76) Se(82) Te(128) Te(130) Xe(136) QRPA Jyvaskyla QRPA Tubingen best-fit: 2.92 best fit: 3.08 6,0 6,0 QRPA Jyvaskyla QPRA Tubingen

sqrt(9) +/. sqrt(2) sqrt(10) +/- sqrt(3) 5,0 5,0

4,0 4,0 ν ν 0 0 3,0 3,0 M M

2,0 2,0

1,0 1,0

0,0 0,0 Ge(76) Se(82) Zr(96) Mo(100)Cd(116)Te(128) Te(130) Xe(136) Ge(76) Se(82) Zr(96) Mo(100)Cd(116)Te(128) Te(130) Xe(136)

35 Experimental Aspect

aMεt without background 0ν −1  (T1/2) M t ∝  a ε with background rB ∆E   with B is background index in counts/(keV kg yr) • ∆E is energy resolution • ǫ is eﬃciency • (T 0ν )−1 (particle physics)2 • 1/2 ∝ Note: factor 2 in particle physics is combined factor of 16 in M t B ∆E × × ×

36 Inverted Ordering

Nature provides 2 scales:

IH 2 2 IH 2 2 mee max c13 ∆mA and mee min c13 ∆mA cos 2θ12 | | ≃ q | | ≃ q requires (1026 ... 1027) yrs O is the lower limit m IH ﬁxed? | ee|min

37 Inverted Hierarchy

m3 = 0.001 eV 0.1 IH, 3σ 0.25 0.125 IH, BF 1

0.5 0.25 y] 2 27 1 0.5 [eV] 4 [10 i 2 ν 2 ν

1 / 8 0 1 m h 4 T 16 2

8 32 4

0.01 16 8 0.28 0.3 0.32 0.34 0.36 0.38 2 sin θ12 Current 3σ range of sin2 θ gives factor of 2 uncertainty for m IH 12 ∼ | ee|min combined factor of 16 in M t B ∆E ⇒ ∼ × × × need precision determination of θ12! JUNO ⇒ ↔ Dueck, W.R., Zuber, PRD 83; Ge, W.R., 1507.05514

38 39

0.7

Prior

Posterior

0.6

0.5

12 0.4 θ [meV] ee

0.3 PDF dP/d min M

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 [meV] θ 3 12 40

100 100 100 48 76 82 20Ca 32Ge 34Se

Gerda Gerda+IGEX+HM

[eV] -1 [eV] -1 [eV] -1

〉 10 〉 10 〉 10 ν 〈m 〉IH ν 〈m 〉IH ν 〈m 〉IH m ν max m ν max m ν max 〈 〈 〈

〈 〉IO 〈 〉IO 〈 〉IO mν min mν min mν min

10-2 10-2 10-2 1026 1027 1028 1029 1030 1025 1026 1027 1028 1029 1030 1025 1026 1027 1028 1029 0ν 0ν 0ν T1/2 [years] T1/2 [years] T1/2 [years]

100 100 100 100 130 136 42Mo 52Te 54Xe CUORE-0 Cuoricino+CUORE-0 EXO KamLAND-Zen [eV] [eV] -1 [eV] -1 -1 〉 〉 10 NEMO-3 〉 10 10 ν ν 〈 〉IH ν 〈 〉IH 〈 〉IH m m mν max m mν max mν max 〈 〈 〈

〈 〉IO 〈 〉IO 〈 〉IO mν min mν min mν min

10-2 10-2 10-2 1024 1025 1026 1027 1028 1029 1024 1025 1026 1027 1028 1029 1025 1026 1027 1028 1029 0ν 0ν 0ν T1/2 [years] T1/2 [years] T1/2 [years] With 0νββ one can test lepton number violation • test Majorana nature of neutrinos • probe neutrino mass scale • extract Majorana phase • constrain inverted ordering • conceptually, it would increase our believe in GUTs • seesaw mechanism • leptogenesis •

41 Sterile Neutrinos?? LSND/MiniBooNE/gallium • cosmology • BBN • r-process nucleosynthesis in Supernovae • reactor anomaly • ∆m2 [eV2] U U ∆m2 [eV2] U U 41 | e4| | µ4| 51 | e5| | µ5| 3+2/2+3 0.47 0.128 0.165 0.87 0.138 0.148 1+3+1 0.47 0.129 0.154 0.87 0.142 0.163 or ∆m2 = 1.78 eV2 and U 2 = 0.151 41 | e4| Kopp, Maltoni, Schwetz, 1103.4570

see lectures by William Louis and Karsten Heeger

42 Sterile Neutrinos and 0νββ recall: m act can vanish and m act 0.03 eV cannot vanish • | ee|NH | ee|IH ∼ m = U 2 m + U 2 m e2iα + U 2 m e2iβ + U 2 m e2iΦ1 • | ee| || e1| 1 | e2| 2 | e3| 3 | e4| 4 | mact mst | {zee } | {zee } sterile contribution to 0νββ (assuming 1+3): • act st 2 2 mee NH mee ∆mst Ue4  ≫| | | | ≃ q | |  m act ≃| ee|IH  m cannot vanish and m can vanish! ⇒ | ee|NH | ee|IH usual phenomenology gets completely turned around! ⇒ Barry, W.R., Zhang, JHEP 1107; Giunti et al., PRD 87; Girardi, Meroni, Petcov, JHEP 1311; Giunti, Zavanin, 1505.00978

43 [10 Usual plot gets completely turned around!

Normal Inverted 0 10 3 ν (best-fit) 3 ν (best-fit) 1+3 ν (best-fit) 1+3 ν (best-fit)

-1 10 > (eV) ee

-3 10 0.001 0.01 0.1 0.001 0.01 0.1

mlight (eV)

44 Interpretation of Neutrino-less Double Beta Decay 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 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

45 Standard Interpretation: • dL uL

Uei − eL i ν q i ν − eL

Uei W

uL dL

Non-Standard Interpretations: uL • − d uL L dL uL c eL dL uL c d d WL − W − eL − eL ˜ eL ˜ − dR uL eL W ν χ/g˜ L uL 0 χ/g˜ χ 2 N √2 R gvL hee ∆−− χ/g˜ 0 χ

NR uL − χ/g˜ eR W − − ˜ WR eL eL uL − W eL c WL d ˜ c − dR uL d uL dL uL uL dL eL dL

46 Schechter-Valle theorem: observation of 0νββ implies Majorana neutrinos

W W d d u u

− − νe νe e e

G2 is 4 loop diagram: mBB F MeV5 10−25 eV ν ∼ (16π2)4 ∼ explicit calculation: Duerr, Lindner, Merle, 1105.0901

47 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 48 Energy Scale: 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 ⇒

49 Examples

Fermions (and no RHC): •

rate m for q2 m2 m»q m  ≫  A∝ q2 m2 →   1 µ m² − for q2 m2 µ 1m²  m ≪   mass

Note: maximum A corresponds to m ≃hqi: limits on O(mK ) Majorana neutrinos − from K+ → π µ+e+

heavy scalar/vector: • 1 1 A∝ q2 m2 → m2 −

50 Heavy neutrinos

Mitra, Senjanovic, Vissani, 1108.0004

51 T 0ν (1 eV) = T 0ν (1 TeV) RPV Supersymmetry • left-right symmetry • heavy neutrinos • color octets • leptoquarks • eﬀective operators • extra dimensions • ... • need to solve the inverse problem... ⇒

52 Distinguishing Mechanisms The inverse problem of 0νββ 1.) Other observables (LHC, LFV, KATRIN, cosmology,...) 2.) Decay products (individual e− energies, angular correlations, spectrum,...) 3.) Nuclear physics (multi-isotope, 0νECEC, 0νβ+β+,...)

53 1.) Example Left-Right Symmetry very simple extension of SM gauge group to SU(2) SU(2) U(1) L × R × B−L usual particle content:

′ νL νR LLi =   (2, 1, 1) , LRi =   (1, 2, 1) ℓL ∼ − ℓR ∼ −  i  i uL 1 uR 1 QLi =   (2, 1, 3 ) , QRi =   (1, 2, 3 ) dL ∼ dR ∼  i  i for symmetry breaking:

+ ++ + ++ δL /√2 δL δR /√2 δR ∆L   (3, 1, 2) , ∆R   (1, 3, 2) ≡ 0 + ∼ ≡ 0 + ∼ δL δL /√2 δR δR /√2  −   −  0 + φ1 φ2 φ   (2, 2, 0) ≡ − 0 ∼ φ1 φ2  

54 Left-right Symmetry very rich Higgs sector (13 extra scalars) • ′ rich gauge boson sector (Z , MW ± ) with • R 2 ′ > MZ = 1−tan2 θ MWR 1.7 MWR 4.3 TeV q W ≃ ∼ ’sterile’ neutrinos ν • R type I + type II seesaw for neutrino mass • 2 2 gR mW right-handed currents with strength GF g M • L WR m 1/M : maximal parity violation smallness of neutrino mass • ν ∝ WR ↔

(Note: in case of modiﬁed symmetry breaking g 6= g and M < M possible. . .) L R Z′ WR

55 Left-right Symmetry ′ c T 6 neutrinos with ﬂavor states nL and mass states nL =(νL, NR)

′ ′ νL KL US νL n =   =   nL =     L c c νR KR T V NR         Right-handed currents:

lep g µ − iα − µ c −iα − − = ℓLγ KLnL(W + ξe W )+ ℓRγ KRn ( ξe W + W ) LCC √2 1µ 2µ L − 1µ 2µ (K and K are 3 6 mixing matrices) L R × plus: gauge boson mixing

± iα ± W cos ξ sin ξ e W1  L  =     ± −iα ± WR sin ξ e cos ξ W2   −   

56 Connection to Neutrinos Majorana mass matrices M = f v from ∆ and M = f v from ∆ L L L h Li R R R h Ri (with fL = fR = f)

′ c ML MD ν ν 1 ′ ′ c L −1 T = ν ν     mν = ML MD M M Lmass − 2 L R T ′ ⇒ − R D MD MR νR     useful special cases

−1 T −1 T (i) type I dominance: mν = MD MR MD = MD fR /vR MD

(ii) type II dominance: mν = fL vL for case (i): mixing of light neutrinos with heavy neutrinos of order

1/2 T mν < −7 TeV Sαi Tαi 10 | |≃| |≃ r Mi ∼ Mi small (or enhanced up to 10−2 by cancellations)

57 Right-handed Currents in Double Beta Decay (A, Z) (A, Z + 2) + 2e− → 3 lep g µ c − iα − = eLγ (UeiνLi + SeiN )(W + ξe W ) LCC √2 Ri 1µ 2µ Xi=1 + e γµ(T ∗ νc + V ∗N )( ξe−iαW − + W − ) R ei Li ei Ri − 1µ 2µ ′c ′c ℓ = L iσ ∆ f L′ L iσ ∆ f L′ LY − L 2 L L L − R 2 R R R classify diagrams: mass dependent diagrams (same helicity of electrons) • triplet exchange diagrams (same helicity of electrons) • momentum dependent diagrams (diﬀerent helicity of electrons) •

58 Mass Dependent Diagrams electrons either both left- or right-handed, leading diagrams:

dL uL dR uR

W WR

Uei − − eL eR νi q NRi νi − − eL eR Uei W WR

uL uR dL dR

m 4 ∗ 2 G2 mee R G2 WL Vei ν F q2 NR F m i M A ≃ A ≃ WR i P

59 Triplet Exchange Diagrams leading diagrams:

uL uR dL dR

− − eL eR WL WR

√ 2 √ 2 2g vL hee 2g vR hee −− −− δL δR

R WL − W − eL eR

u uR dL L dR

4 2 2 heevL 2 mWL VeiMi G 2 G 2 δL F m δR F mW i m A ≃ δL A ≃ R δR P (negligible)

60 Momentum Dependent Diagrams electrons with opposite helicity

m2 1 q G2 WL + tan ξ U T ∗ S V ∗ ALR ≃ F m2 ei ei q − ei ei M 2 WR Xi i leading diagrams (long range):

uL dL dL uL

WL WL

− − eL eL νL νL

NR NR

NR NR − − eR eR WR

WR WL

dR uR dL uL 2 2 mWL ∗ 1 2 ∗ 1 λ GF m i UeiTei q η GF tan ξ i UeiTei q A ≃ WR A ≃ P P

61 Left-right symmetry dR uR

− WR eR − eR − eR dR u¯R

NRi

− eR WR NRi WR

u¯R dR

R dR u

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

62 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.

63 Type II dominance (Tello et al., 1011.3522)

2 −1 T v −1 T ∗ mν = mL mD MR mD = vL f YD f YD vL f − − vR −→ m ﬁxes M and exchange of N with W ﬁxed in terms of PMNS: ⇒ ν R R R m 4 V 2 U 2 G2 W ei ei ⇒ ANR ≃ F M M ∝ m WR X i X i

64 Constraints from Lepton Flavor Violation + dR uR eR

− eR − heµ µR WR

√ 2 2g vR hee −− −− δR δR − R eR W − eR hee

− dR uR eR

65 Constraints from Lepton Flavor Violation

R R + d u eR

− − eR µ heµ WR R

2 √2g vR hee −− −− δR δR − WR − eR eR hee uR − dR eR

mδ = 3.5 TeV Normal Inverted Normal R Inverted mδ = 3.5 TeV R 30 mδ = 2 TeV 32 m = 2 TeV 10 R 10 δ R mδ = 1 TeV R mδ = 1 TeV R

30 10 28 10 GERDA 1T (yrs) R (yrs) N ν ] ] 1/2 1/2 28 10 GERDA 1T [T [T

GERDA 40kg

26 GERDA 40kg 10 26 10 Excluded by KamLAND-Zen Excluded by KamLAND-Zen

0.0001 0.001 0.010.0001 0.1 0.001 0.01 0.1 0.0001 0.001 0.010.0001 0.1 0.001 0.01 0.1 m (eV) m (eV) light light Barry, W.R., JHEP 1309

66 Adding diagrams

d R u R d L u L

W R W − U ei − e R e L ν i N Ri q i − ν − e R e L U ei W R W

d R u R d L u L

lower bound on m(lightest) > meV ⇒ ∼ Bhupal Dev, Goswami, Mitra, W.R., Phys. Rev. D88

67 Left-right symmetry

dL uL dL uL dR uR

W W WR

Uei − − − eL eL eR i ν q NRi NRi i ν − − − eL eL eR

Uei W W WR

uL uL uR dL dL dR

uL uL dL uL dR uR dL dL

WL WL − − eL eR − − L L e WL WR e νL νL

2 2 √2 √2 N N gvL hee gvR hee R R −− −− δL δR NR NR − − W WR eR eR L − − eL eR WR

WL uL uR dL dR dL uL dR uR

68 Left-right symmetry

uL uL uR dL dL dR

W W WR Uei − − − eL eL eR νi q NRi NRi νi − − − eL eL eR Uei W W WR

uL uL uR dL dL dR S2 V 2 U 2 m ei ei ei i M M 4 M i WR i

uL uL dL uL dR uR dL dL

WL WL − − eL eR − − WL WR eL eL νL νL

√ 2 √ 2 2g vL hee 2g vR hee NR NR −− −− δL δR NR NR − − R R R WL − W − e e eL eR WR

WR WL uL uR dL dR dL uL dR uR U 2 m V 2 M U T ei i ei i U T tan ζ ei ei M 2 M 4 M 2 ei ei M 2 ∆L WR ∆R WR

69 Left-right symmetry

uL uL uR dL dL dR

W W WR Uei − − − eL eL eR νi q NRi NRi νi − − − eL eL eR Uei W W WR

uL uL uR dL dL dR 0.4 eV 2 10−8 GeV−1 4 10−16 GeV−5 × ×

uL uL dL uL dR uR dL dL

WL WL − − eL eR − − WL WR eL eL νL νL

√ 2 √ 2 2g vL hee 2g vR hee NR NR −− −− δL δR NR NR − − R R R WL − W − e e eL eR WR

WR WL u uR dL L dR dL uL dR uR — 10−15 GeV−5 6 10−9 1.4 10−10 GeV−2 × ×

70 2.) Distinguishing via decay products Consider standard plus λ-mechanism

dL uL dL uL

W − Uei − eL νL eL ν i q NR i ν − eL NR − Uei eR W WR

dL uL dR uR

dΓ dΓ 2 (1 β1 β2 cos θ) (E1 E2) (1 + β1 β2 cos θ) dE1 dE2 d cos θ ∝ − dE1 dE2 d cos θ ∝ − Arnold et al., 1005.1241

71 2.) Distinguishing via decay products Deﬁning asymmetries A =(N N )/(N + N ) and A =(N N )/(N + N ) θ + − − + − E > − < > <

300 300

250 250

D 200 D 200 meV meV \@ \@ 150 150 Ν Ν m m X X 100 100

50 50

0 0 -4 -2 0 2 4 -4 -2 0 2 4 Λ @10 -7D Λ @10 -7D

72 3.) Distinguishing via nuclear physics

Gehman, Elliott, hep-ph/0701099 3 to 4 isotopes necessary to disentangle mechanism

73 Direct versus indirect contribution 1 Example: introduce Ψi = (8, 1, 0) and Φ=(8, 2, 2 )

0 0 H H d u Φ e− Ψ i Φ Φ Ψ i e− Φ

να Ψ Ψ νβ i i d u

1-loop mν indirect contribution to 0νββ: direct contribution to 0νββ: 2 2 mee 2 yeα l GF | 2 | cud 4 A ≃ q A≃ MΨi MΦ Choubey, Dürr, Mitra, W.R., JHEP 1205

74 Direct versus indirect contribution new contribution can dominate over standard one:

75 Do Dirac neutrinos imply that there is no Lepton Number Violation? possible to construct U(1) model with χ 2 and φ 4 B−L ∼− ∼ = y L Hν + κ χ ν νc + h.c. + µ2 X 2 + λ X 4 L αβ α R,β αβ R,α R,β X | | X | | X=PH,φ,χ +λ H 2 φ 2 + λ H 2 χ 2 + λ χ 2 φ 2 µφχ2 + h.c. Hφ| | | | Hχ| | | | χφ| | | | − break it by allowing only scalar φ to obtain VEV: neutrinos are Dirac particles, and Lepton Number violated by 4 units! ⇒ neutrinoless double beta decay forbidden... ⇒ Heeck, W.R., EPL 103

76 Do Dirac neutrinos mean there is no Lepton Number Violation? Model based on gauged B L, broken by 4 units − Neutrinos are Dirac particles, ∆L = 2 forbidden, but ∆L = 4 allowed... ⇒ observable: neutrinoless quadruple beta decay (A, Z) (A, Z + 4) + 4 e− ⇒ →

0 ν4β Q −

2 β − + β 2 Mass

Z 2 Z Z+2 − Heeck, W.R., EPL 103

77 Candidates for neutrinoless quadruple beta decay

0ν2β Decay rate 2ν2β

0ν4β

Q Q0 2 0ν4β ν β Energy

Q0ν4β Other decays NA 96Zr 96Ru 0.629 τ 2ν2β 2 1019 2.8 40 → 44 1/2 ≃ × 136Xe 136Ce 0.044 τ 2ν2β 2 1021 8.9 54 → 58 1/2 ≃ × 150Nd 150Gd 2.079 τ 2ν2β 7 1018 5.6 60 → 64 1/2 ≃ ×

Heeck, W.R., EPL 103

78 Summary

79 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 ⇒

80 KATRIN and right-handed currents

u u u uL R L R

WR− WL− − W− WL R eL− eL− eR− eR− WL− WR−

d d d dL R L R Uei ν ν ν νLi L′ R′ R′

left-handed contribution • right-handed contribution • interference contribution • Neutrino masses up to m = 18.6 keV testable Warm Dark Matter! ⇔ (see lecture by Kevork Abazajian)

81 mixing down to 10−7 in reach!? ⇒ Mertens et al., 1409.0920

82 u u u i uL R L R

WR− WL− W− WL− R eL− eL− eR− eR− WL− WR−

dL dR dL dR Uei ν ν ν νLi L′ R′ R′

dΓ ′ = K (E + me)peX[1+2C tan ξ] dE LL 2 2 2 2 2 2 × |Uei| X − mi Θ(X − mi)+ |Sei| X − Mi Θ(X − Mi) h i p 4 p 2 dΓ ′ mWL 2 mWL ≃ K (E + me)peX 4 + tan ξ +2C 2 tan ξ dE RR mW mW " R R # 2 2 2 ×|Vei| X − Mi Θ(X − Mi) p 2 dΓ ′ mW = −2K mepeRe L + C tan ξ dE mW LR (" R # 2 2 2 2 × UeiTeimi X − mi Θ(X − mi)+ SeiVeiMi X − Mi Θ(X − Mi) h p with X = E E p io 0 − 83 Focus for simplicity on

u uL R

− W− WL R eL− eR−

d dL R Uei ν νLi R′ total contribution of keV neutrino with mass M to beta decay:

84 connection to 0νββ constraints:

1 m −1 −1 2 θ2 = S 2 + e 0ν −2 G0ν T 0ν S2 M /m 2 eﬀ | ej| M |Mν | 01 1/2 −| ej j e| j

Barry, Heeck, W.R., JHEP 1407

85 How the additional interactions save the day double beta decay without RHC: θ2M = 7 10−10 keV = 70 µeV • × double beta decay with RHC: (m /m )4 V 2M = 8 meV • WL WR | ei| 4 2 4 Γ (N νγ¯ ) m Sei m RHC j WL | | WL decay: → 4 2 4 • ΓSM(Nj νγ) ≃ m Tei ≃ m → WR | | WR 4 2 2 −6 2 2.5 TeV 2 beta decay: θeﬀ Sej + 1.1 10 Vej m > Sej • ≃| | × | | WR | |

86 Dirac particles: ν = νc helicity states : 4 d.o.f. 6 ±

87 Majorana particles: ν = νc helicity states : 2 d.o.f. ±

88 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 ⇒

89 Dirac vs. Majorana in V A theories: observable diﬀerence always suppressed by (m/E)2 − suppose beam from π+ decays: π+ µ+ ν (left-handed) • → µ can we observe ν + p n + µ+ ? (right-handed) • µ → chirality is not a good quantum number: “spin ﬂip” • emitted ν (negative helicity) is not purely left-handed: • µ m u (p)= u(m=0)(p)+ u(m=0)( p) ↓ L 2E R −

P u = 0 and µ+ can be produced if m = 0 and “u v” ( Majorana!) • R ↓ 6 6 ∝ ↔ amplitude (m/E) probability (m/E)2 ⇒ ∝ ⇒ ∝ only NA can save the day! ⇒

90 plus uncertainty due to model details: Short range correlations: • – Jastrow function – Unitary Correlation Operator method – Coupled Cluster method model parameters • – correlated: e.g., gA, gpp, SRC – uncorrelated: e.g., model space of single particle states – some include errors, some don’t...

91 2 2 0ν gA 0ν gV 0ν = GT 2 F M 1.25 M − gA M with

0ν = f σ σ τ − τ − H (r , E ) i MGT h | l k l k GT lk a | i Plk 0ν = f τ − τ − H (r , E ) i MF h | l k F lk a | i Plk r 1/p 1/(0.1 GeV) distance between the two decaying neutrons • lk ≃ ≃ E average energy • a sum over all multipolarities! • ’neutrino potential’ H (r , E ) integrates over the virtual neutrino • GT,F lk a momenta (if heavy particles exchanged: nucleon structure: g = g (0)/(1 q2/M 2 )2) • A A − A

92 The 2νββ matrix elements can be written as

− − hf| σa τa |nihn| σb τb |ii 2ν = Pa Pb GT En−(Mi−M )/2 M n f P − − hf| τa |nihn| τb |ii 2ν = Pa Pb MF En−(Mi−Mf )/2 Pn no direct connection to 0νββ... • sum over 1+ states (low momentum transfer) • adjust some parameters to reproduce 2νββ-rates • 2νββ observed in 48Ca, 76Ge, 82Se, 96Zr, 100Mo (plus exc. state), 116Cd, • 128Te, 130Te, 136Xe 150Nd (plus exc. state) with half-lives from 1018 to 1024 yrs can partly be tested with forward angle (= low momentum transfer) (p, n) • charge exchange reactions

93 Mixed Diagrams can dominate

L uL d dR uR dL uL WL − W R W eL νL − U ei − eR eL ν i NR Ri q N i − ν − NR − eR eL eR U ei W R W WR R dR uR dR u dL uL 2 4 2 mW UT mW V 2 mi λ NR ν U 2 A ∼ MWR q A ∼ MWR MR A ∼ q

mν −7 −2 with T M 10 (or huge enhancements up to 10 ) ≃ q R ∼ M M 2 Aλ R WR T 105 T ⇒ ≃ q m ≃ ANR W Barry, W.R., JHEP 1309

94 Mixed Diagrams can dominate

uL dL dR uR

WL − WR eL − L ν eR

NR NRi − NR − eR eR WR WR

R dR uR dR u

Normal Inverted N (L)/ν R λ/ν 1000 ν ] 1/2 1 / [T k ] 1/2 [T

0.001

1e-06 0.001 0.01 0.001 0.01 m (eV) light

Barry, W.R., 1303.6324

95 Imprint of keV neutrinos on ß-spectrum

& ' ' & e cos sin # light & # sin ' $cos ' !& !# s % ! heavy ! " " "

& light & heavy cos 2 (' ) + sin 2 (' )

Susanne Mertens 12

Mertens et al., 1409.0920

96 Imprint of keV neutrinos on ß-spectrum

sin 2 )(

keV neutrino

m heavy Susanne Mertens 1 3

Mertens et al., 1409.0920

97 Imprint of keV neutrinos on ß-spectrum

Susanne Mertens 1 5

Mertens et al., 1409.0920

98 . . . a nraesga ihadtoa neatos ..rig e.g. interactions, additional with signal increase can adfrKTI osesomething see to KATRIN for hard ⇒

Interaction strength Sin2(2θ) 10 10 10 10 iigdw to down mixing 10 10 10 -13 -12 -11 -10 -9 -8 -7

Tremaine-Gunn / Lyman-α 1 Not enough DM DM overproduction 2 Dark matter mass M Excluded by X-ray observations 99 5 10 10 DM − [keV] 7 nreach!? in 50 . . . thne currents ht-handed in total one has dΓ dΓ dΓ dΓ = + + dE dE LL dE RR dE LR

È È -4 -4 Sei = 10 , mWR = 3.5 TeV,Ξ = -5´10

108

7 10 Full D 2 106 Total RHN

eV ÈS 2 5 L@ 10 e RR p L 4 e 10 LR m

+ 3

E 10 H

K' 102 LH

dE 1

10 G d H 1

10-1

10-2 8 10 12 14 16 18 E @keVD

100 Observation of LNV at LHC implies washout eﬀects in early Universe! Example TeV-scale W : leading to washout e± e± W ±W ± and R R R → R R e± W ∓ e∓ W ±. Further, e±W ∓ e∓ W ± stays long in equilibrium R R → R R R R → R R (Frere, Hambye, Vertongen; Bhupal Dev, Lee, Mohapatra; U. Sarkar et al.)

More model-independent (Deppisch, Harz, Hirsch):

+ + ΓW (qq ℓ ℓ qq) > MX σLHC washout: log10 → 6.9 + 0.6 1 + log10 H ∼ TeV − fb

(TeV-0νββ, LFV and YB: Deppisch, Harz, Huang, Hirsch, Päs) post-Sphaleron mechanisms, τ ﬂavor eﬀects,... ↔

101 Flavor Symmetry Models: sum-rules

Normal Inverted

-1 2m 10 2 +m 3 -2 =m 10 1

-3 3σ 30% error 10 3σ exact

> (eV) TBM exact ee

-1 m 1

10 +m 2 =m -2

10 3

-3 10 0.01 0.1 0.01 0.1

mβ (eV) constraints on masses and Majorana phases Barry, W.R., Nucl. Phys. B842

102 Predictions of SO(10) theories Yukawa structure of SO(10) models depends on Higgs representations 10 ( H), 126 ( F ), 120 ( G) H ↔ H ↔ H ↔ Gives relation for mass matrices: m r(H + sF + it G) up ∝ u m H + F + iG down ∝ m r(H 3sF + it G) D ∝ − D m H 3F + it G ℓ ∝ − l M r−1F R ∝ R

Numerical ﬁt including RG, Higgs, θ13

10H + 126H : 19 free parameters

10H + 126H + 120H : 18 free parameters 20 (19) observables to be ﬁtted

103 Predictions of SO(10) theories

2 |mee| m0 M3 χ Model Fit [meV] [meV] [GeV]

12 10H + 126H NH 0.49 2.40 3.6 × 10 23.0 12 10H + 126H +SS NH 0.44 6.83 1.1 × 10 3.29

14 10H + 126H + 120H NH 2.87 1.54 9.9 × 10 11.2 13 −6 10H + 126H + 120H +SS NH 0.78 3.17 4.2 × 10 6.9 × 10

13 10H + 126H + 120H IH 35.52 30.2 1.1 × 10 13.3 13 10H + 126H + 120H + SS IH 24.22 12.0 1.2 × 10 0.6

Dueck, W.R., JHEP 1309

104