Two Higgs doublet models with an S 3 symmetry Diego Cogollo and Jo˜ ao Paulo Silva CFTP/UFCG 06/09/2016 Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 1 / 29
2HD Models Main feature: Two doublets, Φ 1 , Φ 2 − → many new parameters are introduced Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 2 / 29
2HD Models Main feature: Two doublets, Φ 1 , Φ 2 − → many new parameters are introduced They may be reduced by imposing extra symmetries Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 2 / 29
2HD Models Main feature: Two doublets, Φ 1 , Φ 2 − → many new parameters are introduced They may be reduced by imposing extra symmetries In the scalar sector (P. M. Ferreira, H. E. Habber and J. P. Silva, Phys. Rev. D 79 , 116004) and (I. P. Ivanov Phy. Rev. D 77 , 015017) have done the study Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 2 / 29
2HD Models Main feature: Two doublets, Φ 1 , Φ 2 − → many new parameters are introduced They may be reduced by imposing extra symmetries In the scalar sector (P. M. Ferreira, H. E. Habber and J. P. Silva, Phys. Rev. D 79 , 116004) and (I. P. Ivanov Phy. Rev. D 77 , 015017) have done the study Some attempts have been done to extend this analysis into the Yukawa sector (P.M. Ferreira and J. P. Silva Phys. Rev. D 83 , 065026; Eur. Phys. J. C 69 , 45) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 2 / 29
2HD Models Main feature: Two doublets, Φ 1 , Φ 2 − → many new parameters are introduced They may be reduced by imposing extra symmetries In the scalar sector (P. M. Ferreira, H. E. Habber and J. P. Silva, Phys. Rev. D 79 , 116004) and (I. P. Ivanov Phy. Rev. D 77 , 015017) have done the study Some attempts have been done to extend this analysis into the Yukawa sector (P.M. Ferreira and J. P. Silva Phys. Rev. D 83 , 065026; Eur. Phys. J. C 69 , 45) But there was not classification of all possible implementation on non-Abelian symmetries in both sectors Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 2 / 29
Our work! Phys. Rev D 93 , 095024(2016) We provide a complete classification of all possible implementations of S 3 in the 2hdm, consistent with: Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 3 / 29
Our work! Phys. Rev D 93 , 095024(2016) We provide a complete classification of all possible implementations of S 3 in the 2hdm, consistent with: Non-vanishing quark masses Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 3 / 29
Our work! Phys. Rev D 93 , 095024(2016) We provide a complete classification of all possible implementations of S 3 in the 2hdm, consistent with: Non-vanishing quark masses And a CKM matrix wich is not block diagonal Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 3 / 29
S 3 symmetry Consists of all permutations among three objects ( X 1 , X 2 , X 3 ) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 4 / 29
S 3 symmetry Consists of all permutations among three objects ( X 1 , X 2 , X 3 ) Its order is equal to 3!=6 Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 4 / 29
S 3 symmetry Consists of all permutations among three objects ( X 1 , X 2 , X 3 ) Its order is equal to 3!=6 All of six elements correspond to the following transformations, e : ( x 1 , x 2 , x 3 ) → ( x 1 , x 2 , x 3 ) , a 1 : ( x 1 , x 2 , x 3 ) → ( x 2 , x 1 , x 3 ) , a 2 : ( x 1 , x 2 , x 3 ) → ( x 3 , x 2 , x 1 ) , a 3 : ( x 1 , x 2 , x 3 ) → ( x 1 , x 3 , x 2 ) , (1) : ( x 1 , x 2 , x 3 ) → ( x 3 , x 1 , x 2 ) , a 4 a 5 : ( x 1 , x 2 , x 3 ) → ( x 2 , x 3 , x 1 ) . Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 4 / 29
S 3 symmetry Consists of all permutations among three objects ( X 1 , X 2 , X 3 ) Its order is equal to 3!=6 All of six elements correspond to the following transformations, e : ( x 1 , x 2 , x 3 ) → ( x 1 , x 2 , x 3 ) , a 1 : ( x 1 , x 2 , x 3 ) → ( x 2 , x 1 , x 3 ) , a 2 : ( x 1 , x 2 , x 3 ) → ( x 3 , x 2 , x 1 ) , a 3 : ( x 1 , x 2 , x 3 ) → ( x 1 , x 3 , x 2 ) , (1) : ( x 1 , x 2 , x 3 ) → ( x 3 , x 1 , x 2 ) , a 4 a 5 : ( x 1 , x 2 , x 3 ) → ( x 2 , x 3 , x 1 ) . By defining a 1 = a , a 2 = b , all of elements are written as { e , a , b , ab , ba , bab } Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 4 / 29
S 3 symmetry These elements are classified to three conjugacy classes, C 1 : { e } , C 2 : { ab , ba } , C 3 : { a , b , bab } . (2) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 5 / 29
S 3 symmetry These elements are classified to three conjugacy classes, C 1 : { e } , C 2 : { ab , ba } , C 3 : { a , b , bab } . (2) The number of irreducible representations is equal to number of conjugacy classes Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 5 / 29
S 3 symmetry These elements are classified to three conjugacy classes, C 1 : { e } , C 2 : { ab , ba } , C 3 : { a , b , bab } . (2) The number of irreducible representations is equal to number of conjugacy classes The irreducible representations of S 3 include two singlets 1 and 1 ′ , and a doublet 2 Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 5 / 29
S 3 symmetry These elements are classified to three conjugacy classes, C 1 : { e } , C 2 : { ab , ba } , C 3 : { a , b , bab } . (2) The number of irreducible representations is equal to number of conjugacy classes The irreducible representations of S 3 include two singlets 1 and 1 ′ , and a doublet 2 And the matrix form of the elements a and b in the real representation are: � 1 √ � � − 1 3 � 0 − 2 2 a = , b = . (3) √ 0 − 1 3 − 1 2 2 Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 5 / 29
S 3 symmetry And in the complex representation: � 0 � ω � � 1 0 a C = , b C = , (4) ω 2 1 0 0 Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 6 / 29
S 3 symmetry And in the complex representation: � 0 � ω � � 1 0 a C = , b C = , (4) ω 2 1 0 0 Where ω = e 2 i π/ 3 ( ω 3 = 1) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 6 / 29
S 3 symmetry And in the complex representation: � 0 � ω � � 1 0 a C = , b C = , (4) ω 2 1 0 0 Where ω = e 2 i π/ 3 ( ω 3 = 1) The two representatios are related by: a c = U . a . U † (5) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 6 / 29
S 3 symmetry And in the complex representation: � 0 � ω � � 1 0 a C = , b C = , (4) ω 2 1 0 0 Where ω = e 2 i π/ 3 ( ω 3 = 1) The two representatios are related by: a c = U . a . U † (5) Being U: � 1 � 1 i U = √ (6) 1 − i 2 Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 6 / 29
S 3 symmetry The multiplication rules for S 3 are: 1 ⊗ any = any , 1 ′ ⊗ 1 ′ = 1 , 1 ′ ⊗ 2 = 2 , (7) 1 ⊕ 1 ′ ⊕ 2 . 2 ⊗ 2 = Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 7 / 29
S 3 symmetry The multiplication rules for S 3 are: 1 ⊗ any = any , 1 ′ ⊗ 1 ′ = 1 , 1 ′ ⊗ 2 = 2 , (7) 1 ⊕ 1 ′ ⊕ 2 . 2 ⊗ 2 = In the real representation, the product of two doublets x = ( x 1 , x 2 ) ⊺ and y = ( y 1 , y 2 ) ⊺ , gives ( x ⊗ y ) 1 = x 1 y 1 + x 2 y 2 , ( x ⊗ y ) 1 ′ = x 1 y 2 − x 2 y 1 , ( x ⊗ y ) 2 = ( x 2 y 2 − x 1 y 1 , x 1 y 2 + x 2 y 1 ) ⊺ . (8) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 7 / 29
S 3 symmetry And the product of the doublet x with the singlet y ′ of 1 ′ gives ( x ⊗ y ′ ) = ( − x 2 y ′ , x 1 y ′ ) ⊺ . (9) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 8 / 29
S 3 symmetry And the product of the doublet x with the singlet y ′ of 1 ′ gives ( x ⊗ y ′ ) = ( − x 2 y ′ , x 1 y ′ ) ⊺ . (9) In the complex representation the product of two doublets, x = ( x 1 , x 2 ) ⊺ and y = ( y 1 , y 2 ) ⊺ , gives ( x ⊗ y ) 1 = x 1 y 2 + x 2 y 1 , ( x ⊗ y ) 1 ′ = x 1 y 2 − x 2 y 1 , ( x ⊗ y ) 2 = ( x 2 y 2 , x 1 y 1 ) ⊺ . (10) Diego Cogollo and Jo˜ ao Paulo Silva (CFTP/UFCG) Two Higgs doublet models with an S 3 symmetry 06/09/2016 8 / 29
Recommend
More recommend
