Introduction ∆(27) model Conclusion
Family symmetries and fermion masses/mixings
Ivo de Medeiros Varzielas
Rudolph Peierls Centre for Theoretical Physics University of Oxford
2007/04/11
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction ∆(27) model Conclusion Outline
1 Introduction The data Family symmetries
2 ∆(27) model Overview Mass terms HPS tri-bi-maximal mixing Vacuum alignment
3 Conclusion Mixing angles predicted
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Standard model: Yukawa couplings
Yukawa Lagrangian u i c,j u i c,j ¯ LYukawa = Yij Q u H + Yij Q u H
Mass matrices
u u 0 Mij = Yij hH i
d d ¯ 0 Mij = Yij hH i
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Summary of data: masses
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Summary of data: quark mixing
Wolfenstein parameterization 1 λ λ3 2 VCKM ∼ −λ 1 λ λ3 −λ2 1 λ ≈ 0.23
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Summary of data: lepton mixing
Harrison-Perkins-Scott
q 2 q 1 − 3 3 0 q q q V = 1 1 1 PMNS 6 3 2 q 1 q 1 q 1 6 3 − 2
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion GUT scale texture fits: Quarks
Symmetric fits 3 3 0 i u u u 2 2 Y ∼ · u u ·· 1
3 −i (45)o 3 0 1.7d (0.8)e d d 2 2 Y ∼ · d (2.1)d ·· 1
d ∼ 0.13 u ∼ 0.048 Paper out soon (today?)
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion GUT scale texture fits: Charged leptons
Georgi-Jarlskog relations To good approximation: mb (M ) = 1 mτ X q → det(M ) ( ) = 11 texture zero det(Ml ) MX 1 Hints of GUT? But: ms (M ) = 1 mµ X 3
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Neutrinos?
Seesaw mechanism
ν −1 ν T mν = (MD) (MRR) (MD)
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Seesaw mechanism: masses
Seesaw formula
ν −1 ν T mν = (MD) (MRR) (MD)
1 generation example 0 m Mν = D mD mRR ν 2 ν if det (M ) = −mD; tr (M ) = mRR mD: −m2 /M 0 Mν ∼ D RR 0 MRR
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Seesaw mechanism: mixing
2 generation SD example −1 T a1 b1 M1 0 a1 b1 mν = a2 b2 0 M2 a2 b2
2 2 a1 a1a2 b1 b1b2 = 1/M1 2 + 1/M2 2 a2a1 a2 b2b1 b2
if M1 M2:
heaviest eigenstate ∼ (a1, a2) (mass ∝ 1/M1)
lightest eigenstate ∼ (b1, b2) mass ∝ 1/M2)
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Broken family symmetry
Froggatt-Nielsen Messengers hθihHi mij = Hierarchical structure suggests MX broken symmetry (hθi= 6 0) Suppresions can arise through messengers (ΨX )
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Abelian?
Simple U(1) example Field U(1) H 0 Respective mass matrix θ −1 4 3 2 d3 0 d 3 2 M = mb d c 0 3 2 1 d2 1 d c 1 hθi = 2 MX d1 2 c d1 2
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion Non-Abelian?
Reasons SM: F.S. ⊂ U(3)6; SO(10): F.S. ⊂ U(3) d 2 Specific features (e.g. M23 ∼ ) easier to explain Lepton (near) HPS mixing strongly suggests non-Abelian
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction The data ∆(27) model Family symmetries Conclusion SO(10) × SU(3)?
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Objectives
Model features Straightforward to embed into GUT / String unification Explains observed fermion data (3 generations etc.) Explains near HPS lepton mixing
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment GUT
Pati-Salam
SO(10) → SU(4) × SU(2)L × SU(2)R GUT leaves some hints d e u ν SU(4): q ↔ l: M ↔ M (d ↔ e); M ↔ MD d u e ν SU(2)R: M ↔ M (d ↔ u); M ↔ MD SU(2)R breaking associated with d > u
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Family assignments
φ¯ and ψ c The fermions ψi , ψj are triplets of the family symmetry ¯i The flavons φA are anti-triplets ¯i ¯j c Invariant mass terms: φAψi φBψj H
Desired vevs ¯ hφ3i ∝ (0, 0, 1) ¯ hφ23i ∝ (0, 1, −1) ¯ hφ123i ∝ (1, 1, 1)
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Georgi-Jarlskog field
H45
H45 (a 45 of SO(10)) acquiring a vev:
hH45i ∝ Y = T3R + (B − L)/2
Y dR = −1/2 + 1/6 = −1/3 Y eR = −1/2 − 1/2 = −1 Y νR = 1/2 − 1/2 = 0 H45 coupling to second generation → ms/mµ = 1/3
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Yukawa superpotential
Leading order terms ¯i ¯j c PY ∼ (φ3ψi )(φ3ψj )H
¯i ¯j c +(φ23ψi )(φ23ψj )HH45
¯i ¯j c +(φ23ψi )(φ123ψj )H ¯i ¯j c +(φ123ψi )(φ23ψj )H
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Yukawa superpotential
Leading order terms ¯i ¯j c PY ∼ (φ3ψi )(φ3ψj )H
¯i ¯j c +(φ23ψi )(φ23ψj )HH45
¯i ¯j c +(φ23ψi )(φ123ψj )H ¯i ¯j c +(φ123ψi )(φ23ψj )H
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Yukawa superpotential
Leading order terms ¯i ¯j c PY ∼ (φ3ψi )(φ3ψj )H
¯i ¯j c +(φ23ψi )(φ23ψj )HH45
¯i ¯j c +(φ23ψi )(φ123ψj )H ¯i ¯j c +(φ123ψi )(φ23ψj )H
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Mass matrices 1
Term by term ¯i ¯j c PY ∼ (φ3ψi )(φ3ψj )H
Respective Dirac mass matrix 0 0 0 f M = mf 0 0 0 0 0 1
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Mass matrices 2
Term by term ¯i ¯j c +(φ23ψi )(φ23ψj )HH45
Respective Dirac mass matrix 0 0 0 f f 2 f 2 M = mf 0 Y −Y 0 −Y f 2 1 + Y f 2
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Mass matrices 3
Term by term ¯i ¯j c +(φ23ψi )(φ123ψj )H
Respective Dirac mass matrix 0 0 0 f 3 f 2 3 f 2 3 M = mf Y + −Y + −3 −Y f 2 − 3 1 + Y f 2 − 3 And so on...
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Unwanted terms?
(Not) spoiling the Yukawa terms ¯i ¯j c Pspoil ∼ (φ3ψi )(φ23ψj )H
Hierarchy spoiled Terms like these must be forbidden by added symmetry
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Added symmetry (reduced)
Field U(1) ψ 0 ψc 0 H 0 H45 2 ¯ φ3 0 ¯ φ23 −1 ¯ φ123 1
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Seesaw and SD revisited
M1 < M2 M3
−1 b1 c1 . M1 0 0 b1 b2 b3 −1 mν = b2 c2 . 0 M2 0 c1 c2 c3 b c . −1 ... 3 3 0 0 M3
Wanted ν Dirac matrix b1 c1 . 0 c . ν MD = b2 c2 . ∝ b c . b3 c3 . −b c .
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Effective neutrino Lagrangian
Vevs reminder Effective terms ¯ hφ23i ∝ (0, 1, −1) ¯i ¯j Pν ∼ λ3(φ23νi )(φ23νj ) → @ hφ¯ i ∝ (1, 1, 1) ¯i ¯j 123 +λ2(φ123νi )(φ123νj ) → Enforced by effective symmetry: ¯ ¯ φ23 → −φ23 ¯ ¯ φ123 → φ123
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Getting the effective terms
Leading order terms ¯i ¯j c PY ∼ (φ23νi )(φ123νj )H → @ ¯i ¯j c +(φ123νi )(φ23νj )H → ¯i ¯j c +(φ3νi )(φ3νj )H → decouple
¯ c ¯ c PM ∼ [(φ123ν )(φ123ν )](θφ123)(θφ123) → @ ¯ c ¯ c +[(φ23ν )(φ23ν )](θφ123)(θφ3) → +(θνc)(θνc) → decouple
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment More unwanted terms
(Not) spoiling the effective terms ¯i ¯j c Pspoil ∼ (φ3νi )(φ23νj )H
¯i ¯j → Pν ∼ (φ3νi )(φ3νj ) Need added effective symmetry e.g.: ¯ ¯ φ3 → iφ3 Again the additional symmetry must be used.
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment ∆(27) invariants
Transformation properties 0 Field Z3 Z3 φ1 φ1 φ3 φ2 αφ2 φ1 2 φ3 (α) φ3 φ2
¯i Allowed: all SU(3)f invariants (e.g. φAψi ) c Disallowed: some ∆(12) invariants (e.g. ψi ψi ) ¯i ¯i Allowed: higher order invariants (e.g. φ φi φ φi )
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment ∆(27) family symmetry
Why is it interesting?
small subgroup of SU(3)f distinct "triplets" and "anti-triplets" c forbids the "quadratic" invariant ψi ψi added invariants useful for vacuum alignment discrete family symmetries don’t have associated D-terms
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Breaking the symmetry
2 generation example 2 i † V ∼ −m (ϕ ϕi ) Symmetry is continuous, continuum of vacuum states No specified direction, just magnitude
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Alignment by soft terms example
2 generation example Symmetry is discrete, breaks the continuum of vacuum states 2 i † V ∼ −m (ϕ ϕi ) 2 i † i † ±m3/2(ϕ ϕi ϕ ϕi ) 4 4 Extrema of |ϕ1| + |ϕ2| with constraint of constant magnitude: Positive: ∝ (1, 1) → V ∼ +2v 4/4 Negative: ∝ (0, 1) → V ∼ −v 4
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment ¯ ¯ φ3 and φ123
Quartic term minimisation 2 i † V ∼ −m (ϕ ϕi ) 2 i † i † ±m3/2(ϕ ϕi ϕ ϕi ) ¯ ¯ For ϕ = φ123, positive coefficient yelds hφ123i ∝ (1, 1, 1) ¯ ¯ For ϕ = φ3, negative coefficient yelds hφ3i ∝ (0, 0, 1)
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Overview Introduction Mass terms ∆(27) model HPS tri-bi-maximal mixing Conclusion Vacuum alignment Relative alignment
¯ Aligning φ23 With SU(3) invariant higher order terms: ¯i Containing φ23φ123i with positive coupling ¯ Term keeping the vanishing component away from the φ3 direction ¯ → hφ23i ∝ (0, 1, −1)
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction ∆(27) model Mixing angles predicted Conclusion The predictions
Mixing angles values PMNS angles measured experimentally 2 1 0.052 s12 ≈ 3 ±0.048 2 s12 = 0.30 ± 0.08 s2 ≈ 1 ±0.061 23 2 0.058 s2 = 0.50 ± 0.18 2 23 s13 ≈ 0.0028 2 s13 < 0.047
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction ∆(27) model Mixing angles predicted Conclusion The predictions
Mixing angles values PMNS angles measured experimentally 2 1 0.052 s12 ≈ 3 ±0.048 2 s12 = 0.30 ± 0.08 s2 ≈ 1 ±0.061 23 2 0.058 s2 = 0.50 ± 0.18 2 23 s13 ≈ 0.0028 2 s13 < 0.047
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction ∆(27) model Mixing angles predicted Conclusion Summary
∆(27) family symmetry The model is viable and unifiable. Seesaw mechanism and alignment of vevs play key roles.
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings Introduction ∆(27) model Mixing angles predicted Conclusion Summary
∆(27) family symmetry The model is viable and unifiable. Seesaw mechanism and alignment of vevs play key roles.
Ivo de Medeiros Varzielas Family symmetries and fermion masses/mixings