A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Introduction T7 Model NLO Which Group? Conclusions

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

A Minimal Model of Neutrino Flavor

Akın Wingerter

Laboratoire de Physique Subatomique et de Cosmologie UJF Grenoble 1, CNRS/IN2P3, INPG, Grenoble, France

July 12, 2012

Work in progress w/Christoph Luhn (IPPP, Durham) & Krishna Mohan Parattu (IUCAA, Pune)

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Neutrino Mixing Matrix

¢ Neutrinos have mass and the different flavors can mix Super-Kamiokande Collaboration, Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos,” Phys. Rev. Lett. 81 (1998) 1562–1567, hep-ex/9807003 SNO Collaboration, Q. R. Ahmad et al., “Direct evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury Neutrino Observatory,” Phys. Rev. Lett. 89 (2002) 011301, nucl-ex/0204008

¢ Charged lepton and neutrino mass matrices cannot be simultaneously diagonalized ˆ † ˆ † M`+ = DLM`+ DR , Mν = ULMνUR

¢ Pontecorvo-Maki-Nakagawa-Sakata matrix B. Pontecorvo, “Mesonium and antimesonium,” Sov. Phys. JETP 6 (1957) 429 Z. Maki, M. Nakagawa, and S. Sakata, “Remarks on the uniﬁed model of elementary particles,” Prog. Theor. Phys. 28 (1962) 870–880

    νe ν1 † νµ = DLUL ν2 ντ | {z } ν3 UPMNS

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Neutrino Mixing Matrix

What we know about the mixing angles . . .

−iδ 1 0 0 c13 0 e s13 c12 s12 0 UPMNS = 0 c23 s23 0 1 0 -s12 c12 0 iδ 0 -s23 c23 -e s13 0 c13 0 0 1 −iδ c12c13 s12c13 s13e iδ iδ = -c23s12−s13s23c12e c23c12−s13s23s12e s23c13 iδ iδ s23s12−s13c23c12e -s23c12−s13c23s12e c23c13

D. Forero, M. Tortola, and J. Valle, “Global status of neutrino oscillation parameters after recent reactor measurements,” 1205.4018

Parameter Best Fit 1σ range 3σ range ◦ ◦ ◦ ◦ ◦ θ12 34.45 33.40 − 35.37 31.31 − 37.46 ◦ ◦ ◦ ◦ ◦ θ23 44.43 41.55 − 49.02 38.65 − 53.13 ◦ ◦ ◦ ◦ ◦ θ13 9.28 8.53 − 9.80 7.03 − 10.94

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Tribimaximal Mixing

¢ Until recently, our best guess was tribimaximal mixing (TBM)

P. F. Harrison, D. H. Perkins, and W. G. Scott, “Tri-bimaximal mixing and the neutrino oscillation data,” Phys. Lett. B530 (2002) 167, hep-ph/0202074

p √  ? 2√/3 1/√3 0√ UPMNS = UHPS = -1/√6 1/√3 -1/√2 -1/ 6 1/ 3 1/ 2

◦ ◦ ◦ ,→ θ12 = 35.26 , θ23 = 45 , θ13 = 0

◦ ¢ Agreement still quite good for θ12, θ23, but θ13 = 0 excluded @5σ

Forero et al, 1205.4018, DAYA-BAY Collaboration, 1203.1669, RENO Collaboration, 1204.0626 Parameter Tribimaximal Global ﬁt 1σ Daya Bay Reno ◦ ◦ ◦ θ12 35.26 33.40 − 35.37 - - ◦ ◦ ◦ θ23 45.00 41.55 − 49.02 - - ◦ ◦ ◦ ◦ ◦ θ13 0.00 8.53 − 9.80 8.8 9.8

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Tribimaximal Mixing

¢ Until recently, our best guess was tribimaximal mixing (TBM)

P. F. Harrison, D. H. Perkins, and W. G. Scott, “Tri-bimaximal mixing and the neutrino oscillation data,” Phys. Lett. B530 (2002) 167, hep-ph/0202074

p √  ? 2√/3 1/√3 0√ UPMNS = UHPS = -1/√6 1/√3 -1/√2 -1/ 6 1/ 3 1/ 2

◦ ◦ ◦ ,→ θ12 = 35.26 , θ23 = 45 , θ13 = 0

◦ ¢ Agreement still quite good for θ12, θ23, but θ13 = 0 excluded @5σ

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Post-Daya Bay Confusion

¢ Regular patter UPMNS is suggestive of a family symmetry p √    ? 2√/3 1/√3 0√ 2 1 0 UPMNS = UHPS = -1/√6 1/√3 -1/√2 ∼ 1 1 1 -1/ 6 1/ 3 1/ 2 1 1 1

¢ Daya bay and Reno rule out tribimaximal Mixing ¢ What are our options?

Give up family symmetries? Some remarks towards the end . . . ◦ Look for groups that give θ13 6= 0 R. d. A. Toorop, F. Feruglio, and C. Hagedorn, “Discrete Flavour Symmetries in Light of T2K,” Phys.Lett. B703 (2011) 447–451, 1107.3486 Keep TBM and calculate higher corrections → This talk!

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

¢ Introduce relations between families of quarks and leptons ¢ But which discrete group do we take for the family symmetry?

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

¢ Introduce relations between families of quarks and leptons ¢ But which discrete group do we take for the family symmetry?

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

¢ We scanned 76 discrete groups

¢ Many papers on A4 × Zn, n ≥ 3. Connection between A4 and TBM? ◦ ¢ ∆(96) gives θ13 6= 0 but is large ¢ We identiﬁed the smallest group that gives TBM!

105

104

103

102 Number of Models

101 (12,3) (21,1) (24,3) (27,3) (27,4) (36,3) (39,1) (42,2) (48,3) (54,8) (57,1) (60,5) (60,9) (63,1) (63,3) (72,3) (75,2) (78,2) (81,7) (81,8) (81,9) (84,2) (84,9) (93,1) (96,3) (24,12) (24,13) (36,11) (48,28) (48,29) (48,30) (48,31) (48,32) (48,33) (48,48) (48,49) (48,50) (72,15) (72,25) (72,42) (72,43) (72,44) (81,10) (84,11) (96,64) (96,65) (96,66) (96,67) (96,68) (96,69) (96,70) (96,71) (96,72) (96,74) (96,185) (96,186) (96,187) (96,188) (96,189) (96,190) (96,191) (96,192) (96,193) (96,194) (96,195) (96,197) (96,198) (96,199) (96,200) (96,201) (96,202) (96,203) (96,204) (96,226) (96,227) (96,229)

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

¢ We scanned 76 discrete groups

¢ Many papers on A4 × Zn, n ≥ 3. Connection between A4 and TBM? ◦ ¢ ∆(96) gives θ13 6= 0 but is large ¢ We identiﬁed the smallest group that gives TBM!

105

104

103

102 Number of Models

Altarelli/Feruglio and most others

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

¢ We scanned 76 discrete groups

¢ Many papers on A4 × Zn, n ≥ 3. Connection between A4 and TBM? ◦ ¢ ∆(96) gives θ13 6= 0 but is large ¢ We identiﬁed the smallest group that gives TBM!

105

104

103

102 Number of Models

Adelhart Toorop et al

Altarelli/Feruglio and most others

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

¢ We scanned 76 discrete groups

¢ Many papers on A4 × Zn, n ≥ 3. Connection between A4 and TBM? ◦ ¢ ∆(96) gives θ13 6= 0 but is large ¢ We identiﬁed the smallest group that gives TBM!

105

104

103

102 Number of Models

Our Model Adelhart Toorop et al

Altarelli/Feruglio and most others

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Discrete Flavor Symmetries

C. Luhn, K. M. Parattu, A. Wingerter, work in progress Symmetries of the model

SU(2)L × U(1)Y × T7 × U(1)R Particle content and charges

Field SU(2)L × U(1)Y T7 U(1)R L (2, -1) 3 1 e (1, 2) 1 1 µ (1, 2) 10 1 τ (1, 2) 100 1

hu (2, 1) 1 0

hd (2, -1) 1 0 ϕ (1, 0) 3 0 0 ϕe (1, 0) 3 0 Breaking the family symmetry

hϕi = (v , v , v ), hϕi = (v , 0, 0) ϕ ϕ ϕ e ϕe

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

All you need to know about T7

Tensor Products Contractions

1 × 1 = 1 x ∼ 3, y ∼ 3, z ∼ 3, 0 0 1 × 1 = 1 √ √ √ 0 1 1 1 1 00 00 3x y + 3x y + 3x y 1 × 1 = 1 √ 3 1 1 3 √2 3 3 3 2 √ B “ 1 1 ” “ 1 1 ” “ 1 1 ” C z = B x1y2 − 6 3 + 2 ı + x2y1 − 6 3 + 2 ı + x3y3 − 6 3 + 2 ı C 1 × 3 = 3 @ “ √ ” “ √ ” “ √ ”A x y − 1 3 − 1 ı + x y − 1 3 − 1 ı + x y − 1 3 − 1 ı 1 × 30 = 30 1 3 6 2 2 2 6 2 3 1 6 2 10 × 10 = 100 0 10 × 100 = 1 x ∼ 3, y ∼ 3, z ∼ 3 , 0 1 × 3 = 3 0 1 √ 1 √ 1 √ 1 6x1y1 − 6x2y3 − 6x3y2 0 0 0 3 √ 6 √ 6 √ 1 × 3 = 3 z = B− 1 6x y + 1 6x y − 1 6x y C @ 6 1 3 3 2 2 6 3 1A 00 00 0 1 √ 1 √ 1 √ 1 × 1 = 1 − 6 6x1y2 − 6 6x2y1 + 3 6x3y3 100 × 3 = 3 00 0 0 1 × 3 = 3 x ∼ 3, y ∼ 3, z ∼ 30, 0 0 0 √ √ √ 1 3 × 3 = 3 + 3 + 3 − 1 6x y + 1 6x y − 1 6x y 6 √ 1 1 3 √ 2 3 6 √ 3 2 z = B− 1 6x y − 1 6x y + 1 6x y C 3 × 30 = 1 + 10 + 100 + 3 + 30 @ 6 1 3 6 2 2 3 3 1A 1 √ 1 √ 1 √ 6x1y2 − 6x2y1 − 6x3y3 30 × 30 = 2 × 3 + 30 3 6 6

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

All you need to know about T7

Tensor Products Contractions

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

All you need to know about T7

Tensor Products Contractions

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

All you need to know about T7

Tensor Products Contractions

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

¢ Terms that are invariant, have 2 leptons and mass dimension ≤ 5 or 6: ϕ ϕ ϕ ϕ ϕ W = y eLeh + y eLµh + y eLτh + y e LLh h + y LLh h e Λ d µ Λ d τ Λ d 1 Λ2 u u 2 Λ2 u u

30⊗3⊗1⊗1 = (1+10+100+3+30)⊗1⊗1 = 1+10+100+3+30 ¢ Contract family indices (need to know Clebsch-Gordan coeﬃcients): 1√ 1√ 1√ y 3L eh ϕ + y 3L eh ϕ + y 3L eh ϕ e 3 1 d e1 e 3 2 d e2 e 3 3 d e3 1√ 1√ 1√ + y 3L µh ϕ + y 3L µh ϕ + y 3L µh ϕ µ 3 1 d e3 µ 3 2 d e1 µ 3 3 d e2 1√ 1√ 1√ + y 3L τh ϕ + y 3L τh ϕ + y 3L τh ϕ τ 3 1 d e2 τ 3 2 d e3 τ 3 3 d e1 ¢ Contract SU(2) indices and substitute vevs hϕi = (v , 0, 0), etc: L e ϕe 1√ 1√ 1√ y 3v v L(2)e + y 3v v L(2)µ + y 3v v L(2)τ e 3 d ϕe 1 µ 3 d ϕe 2 τ 3 d ϕe 3

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

¢ Terms that are invariant, have 2 leptons and mass dimension ≤ 5 or 6: ϕ ϕ ϕ ϕ ϕ W = y eLeh + y eLµh + y eLτh + y e LLh h + y LLh h e Λ d µ Λ d τ Λ d 1 Λ2 u u 2 Λ2 u u

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

¢ Terms that are invariant, have 2 leptons and mass dimension ≤ 5 or 6: ϕ ϕ ϕ ϕ ϕ W = y eLeh + y eLµh + y eLτh + y e LLh h + y LLh h e Λ d µ Λ d τ Λ d 1 Λ2 u u 2 Λ2 u u

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

¢ Terms that are invariant, have 2 leptons and mass dimension ≤ 5 or 6: ϕ ϕ ϕ ϕ ϕ W = y eLeh + y eLµh + y eLτh + y e LLh h + y LLh h e Λ d µ Λ d τ Λ d 1 Λ2 u u 2 Λ2 u u

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

¢ Mass matrices (1) (1) (1) e µ τ L1 L2 L3 (2)    √ √ √  L y 0 0 (1) 1 1 1 e L1 2y2vϕ + 2y1vϕ − 2 2y2vϕ − 2 2y2vϕ v v (2) v 2 √ e √ √ √d ϕe u (1)  1 1  M`+ = − × L  0 yµ 0 , Mν = 2 ×   6Λ 2   12Λ L2 − 2 2y2vϕ 2y2vϕ − 2 2y2vϕ + 2y1vϕ (2)  √ √ √ e  (1) 1 1 L3 0 0 yτ L − 2y v − 2y v + 2y v 2y v 3 2 2 ϕ 2 2 ϕ 1 ϕe 2 ϕ ¢ Singular value decomposition ˆ † ˆ † M`+ = DLM`+ DR , Mν = ULMν UR ¢ Neutrino mixing matrix √ √ √ √ 0 0 1 -2/ 6 -1/ 3 0  -2/ 6 -1/ 3 0  √ √ √ e,µ,τ→τ,µ,e √ √ √ U = D U† = 0 1 0 = PMNS L L    1/√6 -1/√3 -1/√2  1/√6 -1/√3 -1/√2 1 0 0 1/ 6 -1/ 3 1/ 2 1/ 6 -1/ 3 1/ 2 Form-diagonalizable! Mixing does not depend on A, B, masses do!

◦ ◦ ◦ ¢ Mixing angles: θ12 = 35.26 , θ23 = 45 , θ13 = 0 Tribimaximal

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

◦ ◦ ◦ ¢ Mixing angles: θ12 = 35.26 , θ23 = 45 , θ13 = 0 Tribimaximal

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

¢ Mass matrices (1) (1) (1) e µ τ L1 L2 L3 (2)    √ √ √  L y 0 0 (1) 1 1 1 e L1 2A + 2B − 2 2A − 2 2A v v (2) v 2 √ √ √ √d ϕe   u (1)  1 1  M`+ = − × L 0 yµ 0 , Mν = 2 × L  − 2A 2A − 2A + 2B  6Λ 2   12Λ 2  2 2  (2) (1) √ √ √ L 0 0 yτ 1 1 3 L3 − 2 2A − 2 2A + 2B 2A ¢ Singular value decomposition ˆ † ˆ † M`+ = DLM`+ DR , Mν = ULMν UR ¢ Neutrino mixing matrix √ √ √ √ 0 0 1 -2/ 6 -1/ 3 0  -2/ 6 -1/ 3 0  √ √ √ e,µ,τ→τ,µ,e √ √ √ U = D U† = 0 1 0 = PMNS L L    1/√6 -1/√3 -1/√2  1/√6 -1/√3 -1/√2 1 0 0 1/ 6 -1/ 3 1/ 2 1/ 6 -1/ 3 1/ 2 Form-diagonalizable! Mixing does not depend on A, B, masses do!

◦ ◦ ◦ ¢ Mixing angles: θ12 = 35.26 , θ23 = 45 , θ13 = 0 Tribimaximal

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions

Our T7 Model at Leading Order

◦ ◦ ◦ ¢ Mixing angles: θ12 = 35.26 , θ23 = 45 , θ13 = 0 Tribimaximal

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Next-to-Leading-Order Corrections

¢ Remember leading-order superpotential:

¢ Superpotential (now mass dimension ≤6 or 7)

C9 L e hd ϕe + C14 L µ hd ϕe + C19 L τ hd ϕe + C10 L e hd ϕ ϕ + C11 L e hd ϕ ϕe + C12 L e hd ϕe ϕe + C13 L e hu hd hd ϕe + C15 L µ hd ϕ ϕ + C16 L µ hd ϕ ϕe + C17 L µ hd ϕe ϕe + C18 L µ hu hd hd ϕe + C20 L τ hd ϕ ϕ + C21 L τ hd ϕ ϕe + C22 L τ hd ϕe ϕe + C23 L τ hu hd hd ϕe + C1 L L hu hu ϕ + C2 L L hu hu ϕe + C (LL) h h ϕ ϕ + C (LL) h h ϕ ϕ + 3 3 u u 4 30 u u C (LL) h h ϕ ϕ + C (LL) h h ϕ ϕ + 7 3 u u e e 8 30 u u e e C (LL) h h ϕ ϕ + C (LL) h h ϕ ϕ 5 3 u u e 6 30 u u e ¢ Leading-order terms, neglected terms, NLO terms

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Next-to-Leading-Order Corrections

¢ The Charged Lepton Sector

1 0 0 0 1 0 0 1 0 0 ∆M` = − √ vd C10 0 0 0 + C11 1 0 0 + C12 0 0 0 3 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 +C15 0 0 0 + C16 0 1 0 + C17 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 +C20 0 0 0 + C21 0 0 1 + C22 0 0 0 0 0 0 0 0 1 0 0 1

¢ The Neutrino Sector

2 2 v 0 0 0 √ 2 −ω −ω 1 0 0 u 2 ∆Mν = √ C3 0 0 0 + 3 2C4 −ω 2ω −1 + 4C7 0 0 1 24 3 0 0 0 −ω −1 2ω2 0 1 0 2 2 0 0 1 ω ω2 √ 2 −ω −ω 2 + C8 0 0 −1 + 4C5 ω ω2 1 + 2C6 −ω 2ω −1 0 −1 0 ω2 1 ω −ω2 −1 2ω

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Switching the constants on and oﬀ

exp(2πi) exp(2πi/3) exp(2πi/5) θ12 θ23 θ13 θ12 θ23 θ13 θ12 θ23 θ13 C3 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C4 28.52 45.00 3.47 37.73 56.24 8.90 32.28 57.91 8.50 C5 35.34 45.00 3.48 35.42 41.51 5.04 35.41 41.80 4.83 C6 31.22 45.00 2.18 36.92 37.73 5.57 33.64 36.76 5.60 C7 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C8 33.57 45.00 0.00 36.13 45.00 0.00 34.72 45.00 0.00 C10 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C11 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C12 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C15 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C16 31.68 44.95 3.58 37.08 44.83 4.10 33.85 44.90 3.79 C17 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C20 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00 C21 31.03 51.15 4.22 37.40 42.38 4.97 33.52 47.78 4.52 C22 35.26 45.00 0.00 35.26 45.00 0.00 35.26 45.00 0.00

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Switching the constants on and oﬀ

¢ The Charged Lepton Sector

¢ The Neutrino Sector

2 2 v 0 0 0 √ 2 −ω −ω 1 0 0 u 2 ∆Mν = √ C3 0 0 0 +3 2C4 −ω 2ω −1 +4 C7 0 0 1 24 3 0 0 0 −ω −1 2ω2 0 1 0 2 2 0 0 1 ω ω2 √ 2 −ω −ω 2 + C8 0 0 −1 +4 C5 ω ω2 1 + 2C6 −ω 2ω −1 0 −1 0 ω2 1 ω −ω2 −1 2ω

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Varying the Constants

¢ No constraints on the Ci

60 12 12

50 10 10 Best ﬁt

40 8 8

23 30 13 6 13 6 3σ band θ θ θ

20 4 4

10 2 2

0 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 θ12 θ12 θ23

¢ C4 = C6 = C16 = C21 = 0

60 12 12

50 10 10 Best ﬁt

40 8 8

23 30 13 6 13 6 3σ band θ θ θ

20 4 4

10 2 2

0 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 θ12 θ12 θ23

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Varying the Constants

¢ C4 = C6 = C16 = 0, C21 6= 0

60 12 12

50 10 10 Best ﬁt

40 8 8

23 30 13 6 13 6 3σ band θ θ θ

20 4 4

10 2 2

0 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 θ12 θ12 θ23

¢ C4 = C6 = C21 = 0, C16 6= 0

60 12 12

50 10 10 Best ﬁt

40 8 8

23 30 13 6 13 6 3σ band θ θ θ

20 4 4

10 2 2

0 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 θ12 θ12 θ23

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

When would we claim that a given ﬂavor symmetry is speciﬁcally well-suited to describe neutrino mixing?

¢ First of all, it should reproduce the data! TBM is excluded at 5σ. What does this mean for A4 (or T7)? ¢ It should convincingly reproduce the data!

One can (probably) always tweak TBM models to give θ13 6= 0. θ12, θ23 in agreement w/TBM. Is that enough? Angles should be insensitive to Ci → Form-diagonalizable? Or: Correlation between neutrino masses and angles ¢ Ideally, ﬂavor symmetry arises from some more fundamental theory ¢ The choice of group should be guided by above principles ¢ “Intuition”, geometric imagination and “easiness” may be misleading → Consider the case of A4

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

Arthur Cayley (1821-1895) is the ﬁrst to systematically construct groups; in 1854, he determined all groups of order 4 and 6 . . .

The Small Groups library All groups (423,164,062) of order ≤ 2000 except 1024 Hans Ulrich Besche, Bettina Eick and Eamonn O’Brien

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[1, 1] 1 r r r [2, 1] Z2 r r r [3, 1] Z3 r r r [4, 1] Z4 r r r [4, 2] Z2 × Z2 r r r [5, 1] Z5 r r r

[6, 1] S3

[6, 2] Z6 r r r [7, 1] Z7 r r r [8, 1] Z8 r r r

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[8, 2] Z4 × Z2 r r r

[8, 3] D4

[8, 4] Q8

[8, 5] Z2 × Z2 × Z2 r r r [9, 1] Z9 r r r [9, 2] Z3 × Z3 r r r

[10, 1] D5

[10, 2] Z10 r r r [11, 1] Z11 r r r [12, 1] Z3 oϕ Z4

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[12, 2] Z12 r r r

[12, 3] A4

[12, 4] D6

[12, 5] Z6 × Z2 r r r [13, 1] Z13 r r r

[14, 1] D7

[14, 2] Z14 r r r [15, 1] Z15 r r r [16, 1] Z16 r r r [16, 2] Z4 × Z4 r r r

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[16, 3] (Z4 × Z2) oϕ Z2

[16, 4] Z4 oϕ Z4 [16, 5] Z8 × Z2 r r r [16, 6] Z8 oϕ Z2

[16, 7] D8

[16, 8] QD8

[16, 9] Q16

[16, 10] Z4 × Z2 × Z2 r r r [16, 11] Z2 × D4

[16, 12] Z2 × Q8

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[16, 13] (Z4 × Z2) oϕ Z2 [16, 14] Z2 × Z2 × Z2 × Z2 r r r [17, 1] Z17 r r r

[18, 1] D9

[18, 2] Z18 r r r [18, 3] Z3 × S3

[18, 4] (Z3 × Z3) oϕ Z2 [18, 5] Z6 × Z3 r r r [19, 1] Z19 r r r [20, 1] Z5 oϕ Z4

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[20, 2] Z20 r r r [20, 3] Z5 oϕ Z4

[20, 4] D10

[20, 5] Z10 × Z2 r r r [21, 1] Z7 oϕ Z3 [21, 2] Z21 r r r

[22, 1] D11

[22, 2] Z22 r r r [23, 1] Z23 r r r [24, 1] Z3 oϕ Z8

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[24, 2] Z24 r r r [24, 3] SL(2, 3)

[24, 4] Z3 oϕ Q8

[24, 5] Z4 × S3

[24, 6] D12

[24, 7] Z2 × (Z3 oϕ Z4)

[24, 8] (Z6 × Z2) oϕ Z2 [24, 9] Z12 × Z2 r r r [24, 10] Z3 × D4

[24, 11] Z3 × Q8

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[24, 12] S4

[24, 13] Z2 × A4

[24, 14] Z2 × Z2 × S3 [24, 15] Z6 × Z2 × Z2 r r r [25, 1] Z25 r r r [25, 2] Z5 × Z5 r r r

[26, 1] D13

[26, 2] Z26 r r r [27, 1] Z27 r r r [27, 2] Z9 × Z3 r r r

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

GAP ID Group 3 U(3) U(2) U(2)×U(1) A4

[27, 3] (Z3 × Z3) oϕ Z3

[27, 4] Z9 oϕ Z3 [27, 5] Z3 × Z3 × Z3 r r r [28, 1] Z7 oϕ Z4 [28, 2] Z28 r r r

[28, 3] D14

[28, 4] Z14 × Z2 r r r [29, 1] Z29 r r r [30, 1] Z5 × S3

[30, 2] Z3 × D5

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Which Flavor Symmetry?

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

105

104

103

102 Number of Models

101 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 5 6 7 8 9 0 1 2 3 4 5 7 8 9 0 1 2 3 4 6 7 9 3 1 3 2 3 3 4 3 1 1 2 3 8 9 0 1 2 3 8 9 0 8 1 5 9 1 3 3 5 5 2 3 4 2 2 7 8 9 0 2 9 1 1 3 4 5 6 7 8 9 0 1 2 4 , , , , , , , , , , , , , , , , , , , , , , , , , 8 8 8 8 8 9 9 9 9 9 9 9 9 9 0 0 0 0 0 2 2 2 1 1 1 2 2 3 3 3 3 4 4 5 1 2 4 4 4 1 1 6 6 6 6 6 6 7 7 7 7 2 1 4 , , 7 7 6 , 9 2 8 , , , , , , , , , 4 7 0 0 3 3 2 , , , , , 5 8 1 1 1 , 4 4 , 3 6 , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , 1 2 2 4 4 2 2 3 6 3 4 4 8 8 8 8 8 8 8 8 8 5 5 6 6 6 6 7 2 2 2 2 2 7 7 8 8 8 1 8 8 4 9 9 6 6 6 6 6 6 6 6 6 6 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 2 2 3 4 4 4 4 4 4 4 4 4 7 7 7 7 7 8 8 9 9 9 9 9 9 9 9 9 9 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

¢ 50% of groups have tribimaximal models

¢ Smallest group that can produce TBM: G(21, 1) = T7

¢ Largest fraction of TBM models: G(39, 1) = T13

¢ A4 has nice geometric interpretation, but what does that mean? That humans like to think in terms of geometry?

¢ There is probably no special connection between A4 and TBM!

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Conclusions

◦ We started from TBM and NLO corrections give θ13 6= 0 Model is very economical:

Family symmetry T7 is second-smallest group w/3-dim irreps No extra (shaping) symmetries, i.e. no extra U(1) or ZN Only 2 ﬂavon ﬁelds

Vacuum stabilization kind of a headache → Either more symmetry or more ﬂavons Which ﬂavor symmetry to use? Daya Bay & Reno have shattered the TBM paradigm Probably there is no special connection between A4 and TBM Need to look beyond smallest groups and also critically review motivation for family symmetries

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions Backup

Backup Slides

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

Group is A4 × Z3

K. M. Parattu and A. Wingerter, “Tribimaximal Mixing From Small Groups,” Phys.Rev. D84 (2011) 013011, 1012.2842

(a) The 5528 bins that are ≥ 1. (b) The 1287 bins that are ≥ 1000.

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

Fix the family symmetry (we considered 76 groups)

SU(2)L × U(1)Y × A4 × Z3 × U(1)R Constant particle content; scan over all representations!

Field SU(2)L × U(1)Y A4 × Z3 U(1)R L (2, -1) 30 1 e (1, 2) 10 1 µ (1, 2) 1(8) 1 τ (1, 2) 1(5) 1

hu (2, 1) 1 0

hd (2, -1) 1 0

ϕT (1, 0) 3 0 0 ϕS (1, 0) 3 0 ξ (1, 0) 30 0

Partial scan over vevs

hϕT i = (0/1, 0/1, 0/1), hϕS i = (0/1, 0/1, 0/1), hξi = (0/1, 0/1, 0/1)

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

¢ We consider 2 models equivalent, if their Lagrangians are the same after contracting the family indices, but before the vevs are substituted ¢ In this sense, we have 39,900 inequivalent models/Lagrangians ¢ 22,932 models have non-singular charged lepton and neutrino mass matrices: ˆ † ˆ † † M`+ = DLM`+ DR , Mν = ULMνUR , UPMNS ≡ DLUL

¢ 4,481 consistent w/experiment at 3σ level (19.5%) (obsolete!) ¢ 4,233 are tribimaximal (18.5%) ¢ Probably largest set of viable neutrino models ever constructed!

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

0.25 0.18 0.20 0.16 0.20 0.14 0.15 0.12 0.15 0.10 0.10 0.08 0.10 0.06 0.04 0.05

Percentage of Vacua 0.05 Percentage of Vacua Percentage of Vacua 0.02 0.00 0.00 0.00 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 θ12 θ23 θ13

0.45 0.5 0.40 0.40 0.35 0.4 0.35 0.30 0.30 0.3 0.25 0.25 0.20 0.20 0.2 0.15 0.15 0.10 0.10

Percentage of Vacua Percentage of Vacua 0.1 Percentage of Vacua 0.05 0.05 0.00 0.0 0.00 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 θ12 θ23 θ13

(d) Number of models that give θij with the other 2 angles restricted to their 3σ interval. The histograms have 838289, 148886 and 225844 entries, respectively.

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

90 90 90

80 5.4 80 5.4 80 5.4

4.8 4.8 4.8 70 70 70 4.2 4.2 4.2 60 60 60 3.6 3.6 3.6 50 50 50 3 3 3

2 3.0 1 3.0 1 3.0 θ θ θ 40 40 40 2.4 2.4 2.4 30 30 30 1.8 1.8 1.8 20 20 20 1.2 1.2 1.2

10 0.6 10 0.6 10 0.6

0 0.0 0 0.0 0 0.0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 θ12 θ12 θ23

(e) Number of models that give θij and θmn with no constraint on the remaining angle. Each histogram has 15992118 entries.

90 90 90

80 5.4 80 5.4 80 5.4

4.8 4.8 4.8 70 70 70 4.2 4.2 4.2 60 60 60 3.6 3.6 3.6 50 50 50 3 3 3

2 3.0 1 3.0 1 3.0 θ θ θ 40 40 40 2.4 2.4 2.4 30 30 30 1.8 1.8 1.8 20 20 20 1.2 1.2 1.2

10 0.6 10 0.6 10 0.6

0 0.0 0 0.0 0 0.0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 θ12 θ12 θ23

(f) Number of models that give θij and θmn with the remaining angle restricted to its 3σ interval. The histograms have 2941000, 3675600 and 1057170 entries, respectively.

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

(g) The 5528 bins that are ≥ 1. (h) The 1287 bins that are ≥ 1000.

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor Introduction T7 Model NLO Which Group? Conclusions How did we generate this set of models?

105

104

103

102 Number of Models

¢ Histogram bars: All models,3 σ, TBM ¢ 38 groups (50%) have tribimaximal models

¢ Smallest group that can produce TBM: G(21, 1) = T7

¢ Largest fraction of TBM models: G(39, 1) = T13. Special?

¢ Group names: g ⊂ U(3), g ⊃ A4, A4 ⊂ g ⊂ U(3)

Akın Wingerter, LPSC Grenoble A Minimal Model of Neutrino Flavor