On the Higgs potential in the Minimal S 3 -Invariant Extension of - PowerPoint PPT Presentation

Stage I: A quick overview... The Higgs Mechanism It is known that: Only those local gauge symmetries which are broken will imply their generators to acquire mass. W + , W − , and Z are the only massive gauge bosons... Gluons and photons are massless... U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 19 / 61

Stage I: A quick overview... The Higgs Mechanism It is known that: Only those local gauge symmetries which are broken will imply their generators to acquire mass. W + , W − , and Z are the only massive gauge bosons... Gluons and photons are massless... An isospin weak doublet is introduced, and with it the Higgs lagrangian is constructed as a G SM invariant. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 19 / 61

Stage I: A quick overview... The Higgs Mechanism L H = ( D µ φ ) † ( D µ φ ) − µ 2 φ † φ − λ ( φ † φ ) 2 www . nature . com / nphys / journal / v 7 / n 1 / fig tab / nphys 1874 F 1 . html U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 20 / 61

Stage I: A quick overview... The Higgs Mechanism G SM = SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ U (1) b − l (before spontaneous symmetry breaking) U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 21 / 61

Stage I: A quick overview... The Higgs Mechanism G SM = SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ U (1) b − l (before spontaneous symmetry breaking) G Phys . World = SU (3) C ⊗ U (1) EM ⊗ U (1) b − l (after spontaneous symmetry breaking) U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 21 / 61

Stage I: A quick overview... The Higgs Mechanism www . quantumdiaries . org / 2011 / 10 / 10 / who − ate − the − higgs / U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 22 / 61

Stage I: A quick overview... The Higgs Mechanism Yukawa interactions U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 23 / 61

Stage I: A quick overview... The Higgs Mechanism Y f L φ f R + h . c . U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 24 / 61

Stage I: A quick overview... The Higgs Mechanism Y f L φ f R + h . c . vY f L f R + h . c . U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 24 / 61

Stage I: A quick overview... The flavour problem Houston, we have a flavour problem... U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 25 / 61

Stage I: A quick overview... The flavour problem Matter in the universe is found to be constituted by a single family of fermions ( ν e ,e,u,d), nevertheless, U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 26 / 61

Stage I: A quick overview... The flavour problem Matter in the universe is found to be constituted by a single family of fermions ( ν e ,e,u,d), nevertheless, in the “quantum world” there are additionally two more replicas. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 26 / 61

Stage I: A quick overview... The flavour problem Matter in the universe is found to be constituted by a single family of fermions ( ν e ,e,u,d), nevertheless, in the “quantum world” there are additionally two more replicas. Why 3 (at least) generations of matter? U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 26 / 61

Stage I: A quick overview... The flavour problem Let’s take a quick deeper look into this mystery. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 27 / 61

Stage I: A quick overview... The flavour problem Some aspects of the flavour problem: Quark weak mixing angles (PDG 2010): θ 12 ≈ 13 . 0 o θ 23 ≈ 2 . 4 o θ 13 ≈ 0 . 2 o Lepton weak mixing angles (Shwetz, Tortola, & Valle 2011): Θ 12 ≈ 33 . 9 o Θ 23 ≈ 46 . 1 o Θ 13 ≈ 6 . 5 o ( IH 7 . 2 o ) U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 28 / 61

Stage I: A quick overview... The flavour problem Some aspects of the flavour problem: Quark weak mixing angles (PDG 2010): θ 12 ≈ 13 . 0 o θ 23 ≈ 2 . 4 o θ 13 ≈ 0 . 2 o Lepton weak mixing angles (Shwetz, Tortola, & Valle 2011): Θ 12 ≈ 33 . 9 o Θ 23 ≈ 46 . 1 o Θ 13 ≈ 6 . 5 o ( IH 7 . 2 o ) CP-violation just occurs in the weak sector. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 28 / 61

Stage I: A quick overview... The flavour problem Some aspects of the flavour problem: Quark weak mixing angles (PDG 2010): θ 12 ≈ 13 . 0 o θ 23 ≈ 2 . 4 o θ 13 ≈ 0 . 2 o Lepton weak mixing angles (Shwetz, Tortola, & Valle 2011): Θ 12 ≈ 33 . 9 o Θ 23 ≈ 46 . 1 o Θ 13 ≈ 6 . 5 o ( IH 7 . 2 o ) CP-violation just occurs in the weak sector. The mass hierarchy. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 28 / 61

Stage I: A quick overview... The flavour problem Mass-Hierarchy Plot U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 29 / 61

Stage II: A quick glance into the M S 3 IESM Stage II: A quick glance into the M S 3 IESM U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 30 / 61

Stage II: A quick glance into the M S 3 IESM Some references of works with an S 3 symmetry... S. Pakvasa et al, Phys. Lett. 73B, 61 (1978) E. Derman, Phys. Rev. D19, 317 (1979) D. Wyler, Phys. Rev. D19, 330 (1979) R. Yahalom, Phys. Rev. D29, 536 (1984) A. Mondrag´ on et al, Phys. Rev. D59, 093009, (1999) J. Kubo et al, Prog. Theor. Phys. 109, 795 (2003) J. Kubo et al, Phys. Rev. D70, 036007 (2004) A. Mondragon et al, Phys. Rev. D76, 076003, (2007) D. Meloni et al, Nucl. Part. Phys. 38 015003, (2011) T. Teshima et al, arXiv:1103.6127 (2011) G. Bhattacharyya et al, Phys. Rev. D83, 011701 (2011) And many more... I apologize for those references I don’t include. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 31 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 32 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 32 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 32 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . A group theory study reveals: U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 33 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . A group theory study reveals: S 3 irreps: 1 S , 1 A , and 2 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 33 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . A group theory study reveals: S 3 irreps: 1 S , 1 A , and 2 In a 3d-real representation: 1 S and 2 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 33 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . A group theory study reveals: S 3 irreps: 1 S , 1 A , and 2 In a 3d-real representation: 1 S and 2 After fermions gain mass, U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 33 / 61

Stage II: A quick glance into the M S 3 IESM families become distinguishable ... U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 34 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . A group theory study reveals: S 3 irreps: 1 S , 1 A , and 2 In a 3d-real representation: 1 S and 2 After fermions gain mass, families become distinguishable . U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 35 / 61

Stage II: A quick glance into the M S 3 IESM Why S 3 as the flavour symmetry? Before the introduction of the Yukawa interactions, 3 families are indistinguishable . A group theory study reveals: S 3 irreps: 1 S , 1 A , and 2 In a 3d-real representation: 1 S and 2 After fermions gain mass, families become distinguishable . Was the mass distributed following the irreps of S 3 in the 3d-real representation? U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 35 / 61

Stage II: A quick glance into the M S 3 IESM The Log-Mass Plot U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 36 / 61

Stage II: A quick glance into the M S 3 IESM The Log-Mass Plot U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 37 / 61

Stage II: A quick glance into the M S 3 IESM Smile! U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 38 / 61

Stage II: A quick glance into the M S 3 IESM Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 39 / 61

Stage II: A quick glance into the M S 3 IESM Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). But any age (mass) comparison that we want to do on this global picture will be among all the families. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 39 / 61

Stage II: A quick glance into the M S 3 IESM Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). But any age (mass) comparison that we want to do on this global picture will be among all the families. Nevertheless, families are independent from each other, they don’t share both last names (different quantum numbers). U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 39 / 61

Stage II: A quick glance into the M S 3 IESM Three generations (flavours): Grandparents Parents Children are distributed in different families (Sectors: neutrinos, charged leptons, u quarks, d quarks). But any age (mass) comparison that we want to do on this global picture will be among all the families. Nevertheless, families are independent from each other, they don’t share both last names (different quantum numbers). Any conclusion about the mass hierarchy (flavour structure) can not be done from this global picture . U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 39 / 61

Stage II: A quick glance into the M S 3 IESM To see the flavour structure we need to interrelate families in each independent sector . U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 40 / 61

Stage II: A quick glance into the M S 3 IESM Say Tequila! U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 41 / 61

Stage II: A quick glance into the M S 3 IESM Mass-Percent Plot U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 42 / 61

Stage II: A quick glance into the M S 3 IESM The Flavour Spectrum U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 43 / 61

Stage II: A quick glance into the M S 3 IESM The Flavour Spectrum U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 44 / 61

Stage II: A quick glance into the M S 3 IESM The Flavour Spectrum S 3 it’s conserved. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 44 / 61

Stage II: A quick glance into the M S 3 IESM The Flavour Spectrum S 3 it’s conserved. A single Higgs weak doublet:  0 0  µ 1 M f = 0 µ 1 0   0 0 µ 3 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 44 / 61

Stage II: A quick glance into the M S 3 IESM The Flavour Spectrum S 3 it’s conserved. We need at least to introduce additionally two Higgs weak-doublets more to the SM. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 45 / 61

Stage II: A quick glance into the M S 3 IESM The Flavour Spectrum S 3 it’s conserved. We need at least to introduce additionally two Higgs weak-doublets more to the SM.  µ 1 + µ 2  µ 4 µ 5 M f = µ 4 µ 1 − µ 2 µ 6   µ 7 µ 8 µ 3 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 45 / 61

Stage II: A quick glance into the M S 3 IESM A special feature arises from the theory: U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 46 / 61

Stage II: A quick glance into the M S 3 IESM A special feature arises from the theory: The concepts of flavours and generations are taken to a more fundamental level. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 46 / 61

Stage III: Flavouring the Higgs potential Stage III: Flavouring the Higgs potential U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 47 / 61

Stage III: Flavouring the Higgs potential Some references of works with an S 3 invariant Higgs potential... S. Pakvasa and H. Sugawara, Phys. Lett. 73B, 61 (1978) E. Derman, Phys. Rev. D19, 317 (1979) D. Wyler, Phys. Rev. D19, 330 (1979) R. Yahalom, Phys. Rev. D29, 536 (1984) Y. Koide, Phys. Rev. D60, 077301 (1999) J. Kubo et al, Phys. Rev. D70, 036007 (2004) S. Chen et al, Phys. Rev. D70, 073008 (2004) O. F´ elix-Beltr´ an et al, J. Phys.: Conf. Ser. 171 012028 (2009) D. Meloni et al, Nucl. Part. Phys. 38 015003, (2011) G. Bhattacharyya et al, Phys. Rev. D83, 011701 (2011) Again, there are many more, I apologize for those not included. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 48 / 61

Stage III: Flavouring the Higgs potential Cooking recipe for a Higgs Potential with S 3 flavour U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 49 / 61

Stage III: Flavouring the Higgs potential The essential ingredients to work it out were: U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 50 / 61

Stage III: Flavouring the Higgs potential The essential ingredients to work it out were: The tensorial products between irreps: 1 S ⊗ 1 S = 1 S 1 A ⊗ 1 A = 1 S 1 A ⊗ 1 S = 1 A 1 S ⊗ 2 = 2 1 A ⊗ 2 = 2 2 ⊗ 2 = 1 S + 1 A + 2 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 50 / 61

Stage III: Flavouring the Higgs potential The essential ingredients to work it out were: The tensorial products between irreps: 1 S ⊗ 1 S = 1 S 1 A ⊗ 1 A = 1 S 1 A ⊗ 1 S = 1 A 1 S ⊗ 2 = 2 1 A ⊗ 2 = 2 2 ⊗ 2 = 1 S + 1 A + 2 To carefully carry the weak ( SU (2) L ) index. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 50 / 61

Stage III: Flavouring the Higgs potential 1. Find out all the l.i. S 3 -invariant terms for 2 and 4 scalar fields. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 51 / 61

Stage III: Flavouring the Higgs potential 1. Find out all the l.i. S 3 -invariant terms for 2 and 4 scalar fields. n = 2: 1 S ⊗ 1 S [ 2 ⊗ 2 ] S U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 51 / 61

Stage III: Flavouring the Higgs potential 1. Find out all the l.i. S 3 -invariant terms for 2 and 4 scalar fields. n = 2: 1 S ⊗ 1 S [ 2 ⊗ 2 ] S n = 4: 1 S ⊗ 1 S ⊗ 1 S ⊗ 1 S [( 1 S ⊗ 2 ) ⊗ ( 1 S ⊗ 2 )] S [( 1 S ⊗ 2 ) ⊗ ( 2 ⊗ 2 ) 2 ] S ( 2 ⊗ 2 ) A ⊗ ( 2 ⊗ 2 ) A ( 2 ⊗ 2 ) S ⊗ ( 2 ⊗ 2 ) S [( 2 ⊗ 2 ) 2 ⊗ ( 2 ⊗ 2 ) 2 ] S U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 51 / 61

Stage III: Flavouring the Higgs potential 2. Take an explicit convention for the whole theory (Yukawa lagrangian and Higgs potential) of where to place the symmetric and antisymmetric doublet components. � H DA � H D = H DS 1 ( f DA , f DS ) T ⊗ ( g DA , g DS ) T = √ ( f DA g DA + f DS g DS ) 1 S 2 ⊕ 1 ( f DA g DS − f DS g DA ) 1 A √ 2 � f DA g DS + f DS g DA � ⊕ 1 √ f DA g DA − f DS g DS 2 2 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 52 / 61

Stage III: Flavouring the Higgs potential 3. For each S 3 -invariant term make all the different independent weak indexes contractions. For example: U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 53 / 61

Stage III: Flavouring the Higgs potential 3. For each S 3 -invariant term make all the different independent weak indexes contractions. For example: ( 2 ⊗ 2 ) S ⊗ ( 2 ⊗ 2 ) S = 1 S 3 U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 53 / 61

Stage III: Flavouring the Higgs potential 3. For each S 3 -invariant term make all the different independent weak indexes contractions. For example: ( 2 ⊗ 2 ) S ⊗ ( 2 ⊗ 2 ) S = 1 S 3 ( 2 w ⊗ 2 w ) S ⊗ ( 2 w ′ ⊗ 2 w ′ ) S = 1 S 3 ⊗ G SM ( 2 w ⊗ 2 w ′ ) S ⊗ ( 2 w ⊗ 2 w ′ ) S = 1 S 3 ⊗ G SM ( 2 w ⊗ 2 w ′ ) S ⊗ ( 2 w ′ ⊗ 2 w ) S = 1 S 3 ⊗ G SM U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 53 / 61

Stage III: Flavouring the Higgs potential 3. For each S 3 -invariant term make all the different independent weak indexes contractions. For example: ( 2 ⊗ 2 ) S ⊗ ( 2 ⊗ 2 ) S = 1 S 3 ( 2 w ⊗ 2 w ) S ⊗ ( 2 w ′ ⊗ 2 w ′ ) S = 1 S 3 ⊗ G SM ( 2 w ⊗ 2 w ′ ) S ⊗ ( 2 w ⊗ 2 w ′ ) S = 1 S 3 ⊗ G SM ( 2 w ⊗ 2 w ′ ) S ⊗ ( 2 w ′ ⊗ 2 w ) S = 1 S 3 ⊗ G SM In terms of the Higgs fields: U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 53 / 61

Stage III: Flavouring the Higgs potential 3. For each S 3 -invariant term make all the different independent weak indexes contractions. For example: ( 2 ⊗ 2 ) S ⊗ ( 2 ⊗ 2 ) S = 1 S 3 ( 2 w ⊗ 2 w ) S ⊗ ( 2 w ′ ⊗ 2 w ′ ) S = 1 S 3 ⊗ G SM ( 2 w ⊗ 2 w ′ ) S ⊗ ( 2 w ⊗ 2 w ′ ) S = 1 S 3 ⊗ G SM ( 2 w ⊗ 2 w ′ ) S ⊗ ( 2 w ′ ⊗ 2 w ) S = 1 S 3 ⊗ G SM In terms of the Higgs fields: 2 ( H † 1 w H 1 w + H † 1 2 w H 2 w ) 2 1 w H 1 w ) 2 + ( H † 2 w H 2 w ) 2 + ( H † 1 w H 2 w ) 2 + ( H † 2 [( H † 1 2 w H 1 w ) 2 ] 1 w H 1 w ) 2 + ( H † 2 w H 2 w ) 2 + ( H † 1 w H 2 w ) 2 + ( H † 2 [( H † 1 2 w H 1 w ) 2 ] U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 53 / 61

Stage III: Flavouring the Higgs potential 4. Assign the same self-coupling parameter for each different contraction of the same S 3 -invariant term. U. J. Salda˜ na Salazar (IF-UNAM) On the Higgs potential in the M S 3 IESM 25 October, 2011. 54 / 61