Flavor Symmetry for Four Generations of Quarks and Leptons

Flavor Symmetry for Four Generations of Quarksand Leptons

Tom KephartVanderbilt University

MIAMI 2011 Conference

“An A5 Model of Four Lepton Generations,”Chian-Shu Chen, TWK and Tzu-Chiang Yuan, JHEP 1104, 015 (2011) arXiv:1011.3199 [hep-ph],

“Binary Icosahedral Flavor Symmetry for Four Generations of Quarks and Leptons,”C. S. Chen, TWK and T. C. Yuan, arXiv:1110.6233 [hep-ph].

December 17, 2011

Why Four Generations?

I Logical possibility

I Expt: LHC pheno–more of the same at higher mass? We haveseen it before.

I Theory: 4th generation provides UV completion ofSM/conformal fixed point at ∼10 TeV scale(P. Q. Hung and Chi Xiong, PLB 2011)

I Fits in AdS/CFT, Z7 orbifold model(C. M. Ho, P. Q. Hung, and TWK, arXiv:1102.3997)
















Three Generation Flavor Models

I Models with good properties exist

I A4 ≡ T (tetrahedral symmetry) three generation lepton modelE. Ma and G. Rajasekaran, PRD, 2001

I T ′ (binary tetrahedral symmetry) three generation quark andlepton model P. Frampton and TWK, IJMPA, 1995

I Many other modelsRecent reviews with extensive references:G. Altarelli, arXiv:1002.0211H. Ishimori, et al., arXiv:1003.3552











Three Generation A4 Lepton Flavor Model

I Natural tribimaximal mixings(compatible with almost all neutrino oscillation experiments)

I Three light neutrino masses3L + (1 + 1′ + 1′′)R

I Three SM charged lepton masses









T ′ three generation model

I All the attractive attributes of the A4 model

I plus

I Models quark masses and mixings

I Calculable Cabibbo angleP. Frampton, TWK, and S. Matsuzaki, PRD, 2008



I plus





I plus





I plus



Four generation A5 and I ′ flavor models

I What if we find a fourth generation at the LHC?(Suggested by conformal fixed point models, see P.Q. Hung,et al.)

I Can we preserve the good properties of three generationmodels?

I A5 ≡ I (icosahedral symmetry) four generation lepton model

I I ′ (binary icosahedral symmetry) four generation quark andlepton model

“An A5 Model of Four Lepton Generations,” C. S. Chen, TWK and T. C. Yuan, JHEP 1104, 015 (2011)arXiv:1011.3199 [hep-ph]

“Binary Icosahedral Flavor Symmetry for Four Generations of Quarks and Leptons,” C. S. Chen, TWK andT. C. Yuan, arXiv:1110.6233 [hep-ph].






















Relation between three and four generation symmetries

I Double covers

1→ Z2 → SU(2)→ SO(3)→ 1

we can restrict to the discrete cases

1→ Z2 → T ′ → A4 → 1

and

1→ Z2 → I ′ → A5 → 1

A5 as fourth generation discrete group–Lepton sectorA5 → A4

1 13 33′ 34 1 + 35 1′ + 1′′ + 3

Table: I → T (or A5 → A4) symmetry breaking.

I Need 3L + (1 + 1′ + 1′′)R at A4 level

I Choose doublets in 4LI Only the 5 contains 1′ + 1′′

I Choose 5R + 1R + 1R singlet νs

I Include 3L EW singlet


1 13 33′ 34 1 + 35 1′ + 1′′ + 3



I Choose doublets in 4L

I Only the 5 contains 1′ + 1′′




1 13 33′ 34 1 + 35 1′ + 1′′ + 3







1 13 33′ 34 1 + 35 1′ + 1′′ + 3







1 13 33′ 34 1 + 35 1′ + 1′′ + 3






A5 as fourth generation discrete group–Lepton sector

I Breaking A5 → A4 with S4, EW singlet 4 of A5, gives

I 4L → 3L + 1LI 5R + 1R + 1R → 3R + (1 + 1′ + 1′′)R + 1RI 3L → 3LI Three A4 style generations where we can have TBM

I Fourth decoupled heavier generation

I Plus other heavy νs



I 4L → 3L + 1L

I 5R + 1R + 1R → 3R + (1 + 1′ + 1′′)R + 1RI 3L → 3LI Three A4 style generations where we can have TBM





I 4L → 3L + 1LI 5R + 1R + 1R → 3R + (1 + 1′ + 1′′)R + 1R

I 3L → 3LI Three A4 style generations where we can have TBM





I 4L → 3L + 1LI 5R + 1R + 1R → 3R + (1 + 1′ + 1′′)R + 1RI 3L → 3L

I Three A4 style generations where we can have TBM


















I ′ as forth generation discrete groups–Quarks and Leptons

I ′ → T ′ I ′ → T ′

1 1 2s 23 3 2′s 23′ 3 4s 2′ + 2′′

4 1 + 3 6s 2 + 2′ + 2′′

5 1′ + 1′′ + 3

Table: I ′ → T ′ symmetry breaking.

I Choose same lepton sector as A5. (Full model has additionalZ2 ⊗ Z3 to avoid unwanted terms in Lagrangian.)

I Seek same three generation quark sector as T ′ model

I ′ as forth generation discrete groups–Quarks and Leptons

I ′ → T ′ I ′ → T ′

1 1 2s 23 3 2′s 23′ 3 4s 2′ + 2′′

4 1 + 3 6s 2 + 2′ + 2′′

5 1′ + 1′′ + 3

Table: I ′ → T ′ symmetry breaking.

I Choose same lepton sector as A5. (Full model has additionalZ2 ⊗ Z3 to avoid unwanted terms in Lagrangian.)

I Seek same three generation quark sector as T ′ model

I ′

⊗ 1 3 3′ 4 5 2s 2′s 4s 6s

1 1 3 3′ 4 5 2s 2′s 4s 6s3 3 1⊕3⊕

54⊕ 5 3′ ⊕ 4⊕ 5 3⊕3′⊕4⊕

52s ⊕4s

6s 2s⊕4s⊕6s 2′s ⊕ 4s ⊕6s ⊕ 6s

3′ 3′ 4⊕ 5 1 ⊕3′ ⊕ 5

3⊕ 4⊕ 5 3⊕3′⊕4⊕5

6s 2′s ⊕4s

2′s⊕4s⊕6s 2s ⊕ 2′s ⊕4s ⊕ 6s

4 4 3′ ⊕4⊕ 5

3⊕4⊕5

1⊕3⊕3′⊕4⊕ 5

3⊕3′⊕4⊕5⊕ 5

2′s ⊕6s

2s ⊕6s

4s⊕6s⊕6s 2s ⊕ 2′s ⊕4s ⊕ 4s ⊕6s ⊕ 6s

5 5 3 ⊕3′ ⊕4⊕ 5

3 ⊕3′ ⊕4⊕ 5

3⊕3′⊕4⊕5⊕ 5

1⊕3⊕3′⊕4⊕4⊕5⊕5

4s ⊕6s

4s ⊕6s

2s ⊕ 2′s ⊕4s⊕6s⊕6s

2s ⊕ 2′s ⊕4s ⊕ 4s ⊕6s⊕6s⊕6s

2s 2s 2s⊕4s 6s 2′s ⊕ 6s 4s ⊕ 6s 1⊕3 4 3⊕ 5 3′ ⊕ 4⊕ 5

2′s 2′s 6s 2′s ⊕4s 2s ⊕ 6s 4s ⊕ 6s 4 1 ⊕3′

3′ ⊕ 5 3⊕ 4⊕ 5

4s 4s 2s ⊕4s⊕6s

2′s ⊕4s ⊕6s

4s⊕6s⊕6s 2s ⊕ 2′s ⊕4s⊕6s⊕6s

3⊕5 3′ ⊕5

3′ ⊕ 4⊕ 5 3⊕3′⊕4⊕4⊕ 5⊕ 5

6s 6s 2′s ⊕4s ⊕6s⊕6s

2s ⊕2′s ⊕4s ⊕6s

2s ⊕ 2′s ⊕4s ⊕ 4s ⊕6s ⊕ 6s

2s ⊕ 2′s ⊕4s ⊕ 4s ⊕6s⊕6s⊕6s

3′ ⊕4⊕5

3 ⊕4⊕5

3⊕3′⊕4⊕4⊕ 5⊕ 5

1⊕3⊕3⊕3′ ⊕ 3′ ⊕4⊕4⊕5⊕5⊕ 5

Table: Multiplication rules for the binary icosahedral group I ′.

Fourth generation discrete groups–Quarks and Leptons

I Quarks assigned to “spinor” irreps of I ′

I The assignment of the quark sector under I ′ × Z2 × Z3(ud

)L

(cs

)L︸︷︷︸

U1L(2s ,+1,ω)

and

(tb

)L

(t ′

b′

)L︸︷︷︸

U2L(2s ,+1,ω2)

I

dR , sR︸︷︷︸SR

uR , cR︸︷︷︸CR︸︷︷︸

DsR(4s ,+1,+1)

, bR , b′R︸︷︷︸

DbR(2′s ,−1,ω2)

, tR , t′R︸︷︷︸

DtR(2′s ,+1,ω2)

(t ′, b′)TL , b′R and t ′R denote the chiral fields of the fourthgeneration quarks.




)L

(cs

)L︸︷︷︸

U1L(2s ,+1,ω)

and

(tb

)L

(t ′

b′

)L︸︷︷︸

U2L(2s ,+1,ω2)

I


uR , cR︸︷︷︸CR︸︷︷︸

DsR(4s ,+1,+1)

, bR , b′R︸︷︷︸

DbR(2′s ,−1,ω2)

, tR , t′R︸︷︷︸

DtR(2′s ,+1,ω2)





)L

(cs

)L︸︷︷︸

U1L(2s ,+1,ω)

and

(tb

)L

(t ′

b′

)L︸︷︷︸

U2L(2s ,+1,ω2)

I


uR , cR︸︷︷︸CR︸︷︷︸

DsR(4s ,+1,+1)

, bR , b′R︸︷︷︸

DbR(2′s ,−1,ω2)

, tR , t′R︸︷︷︸

DtR(2′s ,+1,ω2)


I ′ symmetry breaking

I As with A5 → A4, we use S4 to break I ′ → T ′

I Quark field decomposition

U1L(2s ,+1, ω) → U1L(2,+1, ω) ,

U2L(2s ,+1, ω2) → U2L(2,+1, ω2) ,

DsR(4s ,+1,+1) → SR(2′,+1,+1) + CR(2′′,+1,+1)

DbR(2′s ,−1, ω2) → DbR(2,−1, ω2) ,

DtR(2′s ,+1, ω2) → DtR(2,+1, ω2)

I ′ symmetry breaking

I As with A5 → A4, we use S4 to break I ′ → T ′

I Quark field decomposition

U1L(2s ,+1, ω) → U1L(2,+1, ω) ,

U2L(2s ,+1, ω2) → U2L(2,+1, ω2) ,

DsR(4s ,+1,+1) → SR(2′,+1,+1) + CR(2′′,+1,+1)

DbR(2′s ,−1, ω2) → DbR(2,−1, ω2) ,

DtR(2′s ,+1, ω2) → DtR(2,+1, ω2)

Full scalar sector (same as A5 model)

I S4 is an EW singlet, 4 of I ′

I H4 and H ′4 are EW doublets and 4s of I ′

I Φ3 is an EW doublet, 3 of I ′

I I ′ → T ′ breaking for scalars

S4(4,+1,+1) → S1(1,+1,+1) + S3(3,+1,+1) ,

H4(4,+1, ω2) → H1(1,+1, ω2) + H3(3,+1, ω2)

H ′4(4,−1, ω2) → H ′1(1,−1, ω2) + H ′3(3,−1, ω2)

Φ3(3,+1, ω2) → Φ3(3,+1, ω2)






S4(4,+1,+1) → S1(1,+1,+1) + S3(3,+1,+1) ,

H4(4,+1, ω2) → H1(1,+1, ω2) + H3(3,+1, ω2)

H ′4(4,−1, ω2) → H ′1(1,−1, ω2) + H ′3(3,−1, ω2)

Φ3(3,+1, ω2) → Φ3(3,+1, ω2)






S4(4,+1,+1) → S1(1,+1,+1) + S3(3,+1,+1) ,

H4(4,+1, ω2) → H1(1,+1, ω2) + H3(3,+1, ω2)

H ′4(4,−1, ω2) → H ′1(1,−1, ω2) + H ′3(3,−1, ω2)

Φ3(3,+1, ω2) → Φ3(3,+1, ω2)






S4(4,+1,+1) → S1(1,+1,+1) + S3(3,+1,+1) ,

H4(4,+1, ω2) → H1(1,+1, ω2) + H3(3,+1, ω2)

H ′4(4,−1, ω2) → H ′1(1,−1, ω2) + H ′3(3,−1, ω2)

Φ3(3,+1, ω2) → Φ3(3,+1, ω2)

I ′ results

I VeVs for H4, H ′4 and Φ3 can be chosen with a threegeneration tribimaximal mixing limit

U4gTBM =

1 0 0 00 1√

2− 1√

20

0√

13

√13

√13

0 −√

16 −

√16

√23

I and four neutrino masses

mν4 =Y 21 v

2H1

M1+

3Y 23 v

2

2M2, heavy

mν1 = mν3 =15Y 2

3 v2

2M2, light

mν2 = 0.

I ′ results

I VeVs for H4, H ′4 and Φ3 can be chosen with a threegeneration tribimaximal mixing limit

U4gTBM =

1 0 0 00 1√

2− 1√

20

0√

13

√13

√13

0 −√

16 −

√16

√23

I and four neutrino masses

mν4 =Y 21 v

2H1

M1+

3Y 23 v

2

2M2, heavy

mν1 = mν3 =15Y 2

3 v2

2M2, light

mν2 = 0.

VEVs

I S4 VEV for I ′ → T ′

〈S4〉 = (V ′S , 0, 0, 0)

I then VEVs of H ′3 and H ′1

〈H ′3〉 = (V ′31 ,V′32 ,V

′33) and 〈H ′1〉 = V ′1

I and VEV for Φ3

〈Φ3〉 = (v , v , v)

VEVs


〈S4〉 = (V ′S , 0, 0, 0)


〈H ′3〉 = (V ′31 ,V′32 ,V

′33) and 〈H ′1〉 = V ′1

I and VEV for Φ3

〈Φ3〉 = (v , v , v)

VEVs


〈S4〉 = (V ′S , 0, 0, 0)


〈H ′3〉 = (V ′31 ,V′32 ,V

′33) and 〈H ′1〉 = V ′1

I and VEV for Φ3

〈Φ3〉 = (v , v , v)

Conclusions

I Fourth generation is motivated by possibility of LHC phenoand UV completion of 3 generation models

I 3 generation A4 model of leptons generalizes naturally to 4generation A5 model of leptons

I Preserve TBM

I 3 generation T ′ model of quarks and leptons generalizesnaturally to 4 generation I ′ model of quarks and leptons

I Preserve TBM and many properties of quark sector

I Predictions of LHC pheno with extended fermion and scalarsectors, study of FCNCs, etc. –work in progress

Conclusions



I Preserve TBM




Conclusions



I Preserve TBM




Conclusions



I Preserve TBM




Conclusions



I Preserve TBM




Conclusions



I Preserve TBM




Flavor Symmetry for Four Generations of Quarks and Leptons

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