Outline • SUSY Flavor Problem • Motivation • Model • Predictions • Summary and Conclusions Steven Gabriel, OSU, Pheno ’08 – p. 1/1
SUSY Flavor Problem • Strongest constraint from K 0 − K 0 mixing: ∆ ˜ sq < 10 − 4 s /m 2 d ˜ • Solved by assuming universality: � � m 2 0 m 2 1 s = ˜ d ˜ m 2 0 1 ˜ d ˜ s d s ˜ ˜ g g ˜ ˜ s d s d Steven Gabriel, OSU, Pheno ’08 – p. 2/1
sl < 10 − 3 µ /m 2 • Constraint from µ → eγ : ∆ ˜ e ˜ • It’s similarly assumed: � � m 2 0 m 2 2 µ = e ˜ ˜ m 2 0 2 ˜ e c γ ˜ µ µ e c λ Steven Gabriel, OSU, Pheno ’08 – p. 3/1
Motivation • It has been pointed out that the form 0 A 0 B ′ − A 0 0 B C for fermion mass matrices is consistent with phenomenology (Weinberg; Wilczek and Zee; Fritzsch) • Factorizable Form (phases of A, B, B ′ , C can be absorbed into fermion fields) = ⇒ We wish to obtain this with some symmetry • Consider symmetry with 3 families belonging to 2 + 1 Steven Gabriel, OSU, Pheno ’08 – p. 4/1
• ( ˜ s ) , (˜ µ ) in same multiplets = ⇒ Explains mass d, ˜ e, ˜ degeneracy • To generate fermion masses use 2 + 1 pairs of Higgs doublets: ( H u,d 1 , H u,d 2 ) + H u,d 3 • For 2 + 1 of SU (2) , fermion mass matrices have form y ′ 0 y 1 H 3 2 H 2 − y ′ − y 1 H 3 0 2 H 1 y 2 H 2 − y 2 H 1 y 3 H 3 • If � H u 1 � / � H u 2 � = � H d 1 � / � H d 2 � , 13 and 31 entries can be rotated away Steven Gabriel, OSU, Pheno ’08 – p. 5/1
• If full (local) SU (2) , D -terms cause FCNC problems = ⇒ Discrete subgroups of SU (2) • The Higgs mass matrix ( W = H u i M ij H d j ) 0 a cb 1 M = − a 0 cb 2 b 1 b 2 0 can give large masses to all but one pair of doublets ("doublet-doublet splitting") = ⇒ MSSM at low energy 2 H u 1 H u 2 − a ∗ H u 2 H d 1 H d 2 + a ∗ H d H u = b ∗ | b 1 | 2 + | b 2 | 2 + | a | 2 , H d = c ∗ b ∗ 1 − b ∗ 1 − c ∗ b ∗ 3 3 | c | 2 | b 1 | 2 + | c | 2 | b 2 | 2 + | a | 2 � � Steven Gabriel, OSU, Pheno ’08 – p. 6/1
Fermion Mass Matrices: B ′ 0 A u a u b 1 H u B ′ , − A u a 0 u b 2 | b 1 | 2 + | b 2 | 2 + | a | 2 � B u b 1 B u b 2 C u a B ′ 0 A d a d cb 1 H d B ′ − A d a 0 d cb 2 | c | 2 | b 1 | 2 + | c | 2 | b 2 | 2 + | a | 2 � B d cb 1 B d cb 2 C d a • 13,31 entries can be rotated away • For real Yukawas, only complex c gives CP violation Steven Gabriel, OSU, Pheno ’08 – p. 7/1
The Group T ′ • T ′ is the double covering of A 4 • Only subgroup of SU (2) with doublets that are not self-conjugate • Smallest subgroup of SU (2) under which 3 does not break up Representations of T ′ : • true singlet, 1 • conjugate pair of singlets, 1 ′ , 1 ′′ • real triplet, 3 • pseudoreal doublet, 2 • conjugate pair of doublets, 2 ′ , 2 ′′ Steven Gabriel, OSU, Pheno ’08 – p. 8/1
T ′ × Z 6 Model SU (2) L Doublets: H u , H d : (2 ′ , ω ); H u 3 : (1 ′ , ω ); H ′ u , H ′ d : (2 , ω 2 ); 3 , H d H ′ u 3 , H ′ d 3 : (1 ′ , − ω 2 ); H ′′ u : (1 ′′ , − ω ); H ′′ d : (1 ′′ , − ω 2 ); 3 3 Q : (2 ′ , ω ); Q 3 : (1 ′ , ω ) SU (2) L Singlets: T : (3 , 1); D : (2 ′ , − 1); D ′ : (2 ′′ , − 1); S 1 : (1 , ω 2 ); S 2 : (1 , ω ); S 3 : (1 , − ω ); S 4 : (1 , − ω 2 ); S 5 : (1 , − 1); Q c : (2 ′ , ω ); Q c 3 : (1 ′ , ω ) • ω = e i 2 π 3 • Assignment commutes with SO (10) Grand Unification Steven Gabriel, OSU, Pheno ’08 – p. 9/1
Superpotential for SM Singlet Higgs: W = a 1 DD ′ + a 2 T 2 + b 1 T 3 + b 2 D 2 T + b 3 D ′ 2 T + b 4 DD ′ T + a 3 S 1 S 2 + a 4 S 3 S 4 + a 5 S 2 5 + b 5 S 3 1 + b 6 S 3 2 + b 7 S 2 S 2 3 + b 8 S 1 S 2 4 + b 9 S 1 S 3 S 5 + b 10 S 2 S 4 S 5 • Can generate VEV’s for all fields • No flat directions or accidental symmetries Steven Gabriel, OSU, Pheno ’08 – p. 10/1
Higgs Doublet Mass Matrix: β ( T 1 − iω 2 T 2 ) 0 0 0 0 − βωT 3 δD 2 − β ( T 1 + iω 2 T 2 ) 0 0 0 − βωT 3 − δD 1 0 0 0 0 0 0 0 ζS 3 α ( T 1 − iω 2 T 2 ) 0 0 0 0 − αωT 3 λS 1 − α ( T 1 + iω 2 T 2 ) 0 0 0 0 − αωT 3 − λS 1 0 0 0 0 γD 2 − γD 1 ξS 2 0 0 ǫS 2 0 0 m 0 Steven Gabriel, OSU, Pheno ’08 – p. 11/1
Integrating out H ′ u,d , H ′ u,d , H ′′ u,d : 3 3 0 a cb 1 − a 0 cb 2 b 1 b 2 0 � T 2 � � S 2 a = αβ � S 1 � , b 1 = γζ � S 3 D 2 � � S 2 � , b 2 = − γζ � S 3 D 1 � � S 2 � , c = δǫξ 2 � λ ξ ξ γζ � S 3 � • Light modes couple to SM singlet Higgs, H u H d Φ , with � Φ � = 0 in SUSY limit • After SUSY breaking � Φ � ∼ M SUSY Steven Gabriel, OSU, Pheno ’08 – p. 12/1
• c complex = ⇒ spontaneous CP violation • Complex B µ -parameter generated = ⇒ SUSY CP problem not fully solved • Discrete symmetries should come from broken local symmetries so that they are respected by gravity • When SU (2) reps. break up under T ′ , 1 ′ , 1 ′′ and 2 ′ , 2 ′′ always occur in pairs = ⇒ This model can be difficult to obtain from a local symmetry (e.g. 4 → 2 ′ + 2 ′′ , 5 → 3 + 1 ′ + 1 ′′ , 6 → 2 + 2 ′ + 2 ′′ ) • By extending the Abelian part of the symmetry, the model can be altered to use only complete multiplets of SU (2) Steven Gabriel, OSU, Pheno ’08 – p. 13/1
T ′ × Z 3 × Z 6 Model SU (2) L Doublets: H u , H d : (2 , ω, ω ); H u 3 : (1 , ω, ω ); H ′ u , H ′ d : (2 , 1 , ω 2 ); 3 , H d H ′ u 3 , H ′ d 3 : (1 , ω, − ω 2 ); H ′′ u : (1 , ω 2 , − ω ); H ′′ d : (1 , ω 2 , − ω 2 ); 3 3 Q : (2 , ω, ω ); Q 3 : (1 , ω, ω ) SU (2) L Singlets: T : (3 , ω, 1); T ′ : (3 , ω 2 , 1); D : (2 , ω, − 1); D ′ : (2 , ω 2 , − 1); S 1 : (1 , 1 , ω 2 ); S 2 : (1 , 1 , ω ); S 3 : (1 , 1 , − ω ); S 4 : (1 , 1 , − ω 2 ); S 5 : (1 , 1 , − 1); Q c : (2 , ω, ω ); Q c 3 : (1 , ω, ω ) • SU (2) can be broken to T ′ with a 7 Steven Gabriel, OSU, Pheno ’08 – p. 14/1
Predictions • Mass matrix forms with spontaneous CP violation: 0 A d 0 0 A u 0 B ′ B ′ u e iφ M d = , M u = − A d 0 − A u 0 d B u e iφ 0 B d C d 0 C u Small Mixing ( C u , C d >> B u , B d , B ′ u , B ′ d >> A u , A d ): � • | V ub /V cb | = m u /m c • Assuming known quark masses: 0 . 15 < θ C < 0 . 29 Large Mixing ( C u , C d ∼ B u , B d >> B ′ u , B ′ d >> A u , A d ): � � 2 � � m s /m b • θ C ≃ m d /m s 1 − 1 | V cb | 4 Steven Gabriel, OSU, Pheno ’08 – p. 15/1
Summary A Model with the following properties: • MSSM at low energy • Solves SUSY flavor problem • Ameliorates SUSY CP problem • Solves µ problem • Consistent with Grand Unification Steven Gabriel, OSU, Pheno ’08 – p. 16/1
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