Fermion and scalar phenomenology of a 2-Higgs doublet model with S 3
A. E. Cárcamo Hernández, I. de Medeiros Varzielas, E. Schumacher
DDO-TH 15/12
Fermion and scalar phenomenology of a 2-Higgs doublet model with S A. E. C´arcamo Hern´andez, ∗ I. de Medeiros Varzielas, † and E. Schumacher ‡ Universidad T´ecnica Federico Santa Mar´ıa and Centro Cient´ıfico-Tecnol´ogico de Valpara´ısoCasilla 110-V, Valpara´ıso, Chile School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, U.K. Fakult¨at f¨ur Physik, Technische Universit¨at DortmundD-44221 Dortmund, Germany
We propose a 2-Higgs doublet model where the symmetry is extended by S ⊗ Z ⊗ Z (cid:48) ⊗ Z andthe field content is enlarged by extra SU (2) L singlet scalar fields. S makes the model predictiveand leads to viable fermion masses and mixing. The observed hierarchy of the quark masses arisesfrom the Z (cid:48) and Z symmetries. The light neutrino masses are generated through a type I seesawmechanism with two heavy Majorana neutrinos. In the lepton sector we obtain mixing angles thatare nearly tri-bi-maximal, in an excellent agreement with the observed lepton parameters. Thevacuum expectation values required for the model are naturally obtained from the scalar potential,and we analyze the scalar sector properties further constraining the model through rare top decays(like t → ch ), the h → γγ decay channel and the T and S parameters. I. INTRODUCTION
The flavor puzzle is not understood in the context of the Standard Model (SM), which does not specify the Yukawastructures and has no justification for the number of generations. As such, extensions addressing the fermion massesand mixing are particularly appealing. With neutrino experiments increasingly constraining the mixing angles in theleptonic sector many models focus only on this sector, aiming to explain the near tri-bi-maximal structure of thePMNS matrix through some non-Abelian symmetry.Discrete flavor symmetries have shown a lot of promise and S , as the smallest non-Abelian group has been considerablystudied in the literature since [1], with interesting results for quarks, leptons or both, and remains a popular group[2–15]. Other popular groups are the smallest groups with triplet representations, particularly A which has only atriplet and three distinct singlets. A was used in [16–20] and more recently in [21–35]. With just triplets and singletrepresentations the groups T [36–43] and ∆(27) [44–52] are also promising as flavor symmetries. For recent reviewson the use of discrete flavor groups, see Refs. [53, 54].In this work we make use of the S group to formulate a 2-Higgs doublet model (2HDM) with an extra S ⊗ Z ⊗ Z (cid:48) ⊗ Z symmetry. Assigning the SM fermions under this symmetry and using scalars transforming under the differentirreducible representations of S , we provide an existence proof of models leading to the viable mixing inspired quarktextures presented in [55], by building a minimal realization. We then consider the model in the lepton sector wherewe obtain viable masses and mixing angles by using assignments that lead to a charged lepton texture similar to thatof the down-type quarks, with the neutrino sector being completed through a type I seesaw. We discuss the scalarpotential in some detail, showing it leads to the Vacuum Expectation Values (VEVs) used to obtain the fermionmasses, and analyzing phenomenological processes that constrain the parameters of the model such as t → ch and h → γγ .The paper is outlined as follows. In Section II we describe the field and symmetry content of the model, includinga brief revision of the quark mass and mixing angles presented in [55] (Section II A) and the equivalent analysis forthe lepton sector (Section II B). Section III contains the analysis of the phenomenology associated with the extendedscalar sector, presenting the Yukawa couplings, an analysis of rare top decays, then considering the h → γγ rate(Section III A) and the T and S parameters (Section III B). We present our conclusions in Section IV. We relegatesome technical discussions that are relevant for the paper to the Appendix. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - ph ] J u l Field q L q L q L U R u R d R d R d R l L l L l L l R l R l R ν R ν R S Z Z (cid:48) Z -3 -2 0 1 0 4 3 3 -3 0 0 4 5 3 0 0Table I: Assignments of the SM fermions under the flavor symmetries.Field φ φ ξ χ ζSU (2) L S Z Z (cid:48) Z SU (2) L and the flavor symmetries. II. THE MODEL
We consider an extension of the SM with extra scalar fields and discrete symmetries, which reproduces the predictivemixing inspired textures proposed in Ref. [55], i.e. the Cabbibo mixing arises from the down-type quark sectorwhereas the up-type quark sector contributes to the remaining mixing angles. These textures describe the chargedfermion masses and quark mixing pattern in terms of different powers of the Wolfenstein parameter λ = 0 .
225 andorder one parameters. Because of the required mismatch between the down-type quark and up-type quark textures,to obtain these textures in a model we use two Higgs doublets distinguished by a symmetry (in our model, a Z ). Inthe following, we describe our 2HDM with the inclusion of the S ⊗ Z ⊗ Z (cid:48) ⊗ Z discrete symmetry and four singletscalar fields, assigned in a S doublet, one S trivial singlet and one S non trivial singlet. We use the S discretegroup since it is the smallest non-Abelian group, having a doublet and two singlets as irreducible representations.The full symmetry G of the model is broken spontaneously in two steps: G = SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ S ⊗ Z ⊗ Z (cid:48) ⊗ Z (1) ⇓ Λ SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ Z ⇓ Λ EW SU (3) C ⊗ U (1) em , where the different symmetry breaking scales satisfy the following hierarchy Λ (cid:29) Λ EW , where Λ EW = 246 GeV isthe electroweak symmetry breaking scale.The content of the model, which includes the particle assignments under the different symmetries, is shown in TablesI and II. The S symmetry reduces the number of parameters in the Yukawa sector of this 2HDM making it morepredictive. The Z symmetry allows to completely decouple the bottom quark from the remaining down and strangequarks. As can be seen from the scalar field assignments, the two scalar SU (2) L doublets have different Z charges( φ being neutral). The Z (cid:48) and Z symmetries shape the hierarchical structure of the quark mass matrices necessaryto get a realistic pattern of quark masses and mixing.The Higgs doublets φ l ( l = 1 ,
2) acquire VEVs that break SU (2) L φ l = (cid:32) v l √ (cid:33) , l = 1 , . (2)We decompose the Higgs fields around this minimum as φ l = (cid:32) ϕ + l √ ( v l + ρ l + iη l ) (cid:33) = (cid:32) √ ( ω l + iτ l ) √ ( v l + ρ l + iη l ) (cid:33) , (3)where (cid:104) ρ l (cid:105) = (cid:104) η l (cid:105) = (cid:104) ω l (cid:105) = (cid:104) τ l (cid:105) = 0 , l = 1 , . (4)From an analysis of the scalar potential (see Appendix B), we obtain the following VEVs for the SM singlet scalars: (cid:104) ξ (cid:105) = v ξ (1 , , (cid:104) χ (cid:105) = v χ , (cid:104) ζ (cid:105) = v ζ , (5)i.e., the VEV of ξ is aligned as (1 ,
0) in the S direction.For the up and down-type quarks, the Yukawa terms invariant under the symmetries are L UY = ε ( u )33 q L (cid:101) φ u R + ε ( u )23 q L (cid:101) φ u R χ Λ + ε ( u )13 q L (cid:101) φ u R χ Λ + ε ( u )22 q L (cid:101) φ U R ξχ Λ + ε ( u )11 q L (cid:101) φ U R ξχ ζ Λ + h.c. (6) L DY = ε ( d )33 q L φ d R χ Λ + ε ( d )22 q L φ d R χ Λ + ε ( d )12 q L φ d R χ Λ + ε ( d )21 q L φ d R χ Λ + ε ( d )11 q L φ d R χ Λ + h.c. (7)The invariant Yukawa terms for charged leptons and neutrinos are L lY = ε ( l )33 l L φ l R χ Λ + ε ( l )23 l L φ l R χ Λ + ε ( l )22 l L φ l R χ Λ + ε ( l )32 l L φ l R χ Λ + ε ( l )11 l L φ l R χ ζ Λ + h.c. (8) L νY = ε ( ν )11 l L (cid:101) φ ν R χ Λ + ε ( ν )12 l L (cid:101) φ ν R χ Λ + ε ( ν )21 l L (cid:101) φ ν R + ε ( ν )22 l L (cid:101) φ ν R + ε ( ν )31 l L (cid:101) φ ν R + ε ( ν )32 l L (cid:101) φ ν R + M ν R ν c R + M ν R ν c R + M ν R ν c R + h.c. (9)The Z symmetry is the smallest cyclic symmetry that allows χ Λ in the Yukawa terms responsible for the downquark and electron masses, which we want to suppress by λ ( λ = 0 .
225 is one of the Wolfenstein parameters) withoutrequiring small dimensionless Yukawa couplings. Furthermore, the Z (cid:48) symmetry is responsible for coupling the scalar ζ with U R as well as with l R , which helps to explain the smallness of the up quark and electron mass in this model.The hierarchy of charged fermion masses and quark mixing matrix elements is therefore explained by both the Z (cid:48) and Z symmetries. Given that in this scenario the quark masses are related with the quark mixing parameters, weset the VEVs of the SU (2) L singlet scalars with respect to the Wolfenstein parameter λ and the new physics scale Λ: v ξ ∼ v ζ ∼ v χ = λ Λ . (10)These scalars therefore acquire VEVs at a scale unrelated with Λ EW . We have checked numerically that this regimeis a valid minimum of the global potential for a suitable region of the parameter space (see Appendix B). As wewill see in the following sections, in order to obtain realistic fermion masses and mixing without requiring a stronghierarchy among the Yukawa couplings, the VEVs of the SU (2) L doublets ( v and v ) should be of the same order ofmagnitude. A. Quark masses and mixing
Using Eqs. (6) and (7) we find the mass matrices for up and down-type quarks in the form: M U = v √ c λ a λ b λ a λ a , M D = v √ e λ f λ e λ f λ
00 0 g λ , (11)where a k ( k = 1 , , b , c , g , f , f , e and e are O (1) parameters. Here we assume that all dimensionlessparameters given in Eq. (11) are real excepting a , which we assume to be complex. These are the viable quarktextures presented in [55], which we briefly review here.The hermitian combinations M U M † U and M D M TD are M U M † U = v | a | λ + c λ a a λ a a λ a ∗ a λ a λ + b λ a a λ a ∗ a λ a a λ a , (12) M D M TD = v λ e + λ f e e λ + f f λ e e λ + f f λ λ e + λ f
00 0 λ g , (13)and are approximately diagonalized by unitary rotation matrices R U and R D : R † U M U M † U R U = m u m c
00 0 m t , R U (cid:39) c s s e iδ − c s e iδ c s s e − iδ − c s c c , (14) R TD M D M TD R D = m d m s
00 0 m b , R D = c s − s c
00 0 1 , (15)where c ij = cos θ ij , s ij = sin θ ij (with i (cid:54) = j and i, j = 1 , , θ ij and δ are the quark mixing angles and the CPviolating phase, respectively, in the usual parametrization. They are given bytan θ (cid:39) f f λ, tan θ (cid:39) a a λ , (16)tan θ (cid:39) | a | a λ , δ = − arg ( a ) . Therefore, the up and down-type quark masses are approximately given by m u (cid:39) c λ v √ , m c (cid:39) b λ v √ , m t (cid:39) a v √ , (17) m d (cid:39) | e f − e f | λ √ v, m s (cid:39) f λ v √ , m b (cid:39) g λ v √ . (18)We also find that the CKM quark mixing matrix is approximately V CKM = R † U R D (cid:39) c c c s e iδ s e − iδ c s s − c s c c + e − iδ s s s − c s − s s − e − iδ c c s c s − e − iδ c s s c c . (19)It is noteworthy that Eq. (11) provides an elegant understanding of all SM fermion masses and mixing angles throughtheir scalings by powers of the Wolfenstein parameter λ = 0 .
225 with O (1) coefficients.The Wolfenstein parametrization [56] of the CKM matrix is: V W (cid:39) − λ λ Aλ ( ρ − iη ) − λ − λ Aλ Aλ (1 − ρ − iη ) − Aλ , (20)with λ = 0 . ± . , A = 0 . +0 . − . , (21) ρ = 0 . ± . , η = 0 . ± . , (22) ρ (cid:39) ρ (cid:18) − λ (cid:19) , η (cid:39) η (cid:18) − λ (cid:19) . (23)From the comparison with (20), we find: a (cid:39) , a (cid:39) A (cid:39) . , a (cid:39) − A (cid:112) ρ + η e iδ (cid:39) − . e iδ , (24) δ = 67 ◦ , b (cid:39) m c λ m t (cid:39) . , c (cid:39) m u λ m t (cid:39) . . (25)Note that a is required to be complex, as previously assumed, and its magnitude is a bit smaller than the remaining O (1) coefficients.Since the charged fermion masses and quark mixing hierarchy arises from the Z (cid:48) ⊗ Z symmetry breaking, and inorder to have the right value of the Cabbibo mixing, we need e ≈ f . We fit the parameters e , f , f and g inEq. (11) to reproduce the down-type quark masses and quark mixing parameters. As can be seen from the aboveformulas, the quark sector of our model contains ten effective free parameters, i.e., | a | , a , a , b , c , e , f , f , g andthe phase arg( a ), to describe the quark mass and mixing pattern, which is characterized by ten physical observables,i.e., the six quark masses, the three mixing angles and the CP violating phase. Furthermore, in our model theseparameters are of the same order of magnitude. The results for the down-type quark masses, the three quark mixingangles and the CP violating phase δ in Tables III and IV correspond to the best fit values: e (cid:39) . , f (cid:39) . , f (cid:39) . , g (cid:39) . . (26)As pointed out in [55], the CKM matrix in our model is consistent with the experimental data. The agreement of ourmodel with the experimental data is as good as in the models of Refs. [9, 11, 29, 33, 47, 57, 58] and better than, forexample, those in Refs. [59–66]. The obtained and experimental values of the magnitudes of the CKM parameters,i.e., three quark mixing parameters and the CP violating phase δ are shown in Table III. The experimental valuesof the CKM magnitudes and the Jarlskog invariant are taken from Ref. [67], whereas the experimental values of thequark masses, which are given at the M Z scale, have been taken from Ref. [68]. Observable Model value Experimental value m u ( MeV ) 1 .
47 1 . +0 . − . m c ( MeV ) 641 635 ± m t ( GeV ) 172 . . ± . ± . m d ( MeV ) 3 .
00 2 . +0 . − . m s ( MeV ) 59 . . +16 . − . m b ( GeV ) 2 .
82 2 . +0 . − . Table III: Model and experimental values of the quark masses.Observable Model value Experimental valuesin θ . . θ . . θ . . δ ◦ ◦ Table IV: Model and experimental values of CKM parameters.
B. Lepton masses and mixing
This S flavor model obtains the viable quark textures proposed in [55] as shown in section II A. We now proceed toanalyze the lepton sector of the model. From the charged lepton Yukawa terms of Eq. (8) it follows that the chargedlepton mass matrix takes the following form: M l = v √ x λ y λ z λ y λ z λ . (27)where x , y , y , z , z , are O (1) parameters, assumed to be real, for simplicity.Then, the charged lepton mass matrix satisfies the following relations: M l M Tl = v x λ z λ + y λ z z λ + y y λ z z λ + y y λ z λ + y λ , (28) M Tl M l = v x λ (cid:0) y + y (cid:1) λ ( y z + y z ) λ y z + y z ) λ (cid:0) z + z (cid:1) λ . (29)Therefore, the matrix M l M Tl can be diagonalized by rotation matrix R l according to: R Tl M l M Tl R l = m e m µ
00 0 m τ , R l = θ l − sin θ l θ l cos θ l , tan θ l (cid:39) − z z . (30)The charged lepton masses are approximately given by m e = x λ v √ , m µ (cid:39) | y z − y z | (cid:112) z + z λ v √ , m τ (cid:39) (cid:113) z + z λ v √ . (31)From the neutrino Yukawa terms it follows that the full 5 × M ν = (cid:32) × M Dν (cid:0) M Dν (cid:1) T M R (cid:33) , (32)where: M Dν = λ ε ( ν )11 v √ λ ε ( ν )12 v √ ε ( ν )21 v √ ε ( ν )22 v √ ε ( ν )31 v √ ε ( ν )33 v √ = A FB EC D , M R = (cid:32) M M M M (cid:33) . (33)Since ( M R ) ii >> v , the light neutrino mass matrix is generated through a type I seesaw mechanism and is given by M L = M Dν M − R (cid:0) M Dν (cid:1) T = A FB EC D (cid:32) − M M − M M M M − M M M M − M M − M M − M M (cid:33) (cid:32) A B CF E D (cid:33) = − ( M A − M AF + M F ) M − M M BF M − ABM − F EM + AEM ) M − M M CF M − ACM − F DM + ADM ) M − M M BF M − ABM − F EM + AEM ) M − M M − ( M B − M BE + M E ) M − M M BDM − BCM + CEM − DEM ) M − M M CF M − ACM − F DM + ADM ) M − M M BDM − BCM + CEM − DEM ) M − M M − ( M C − M CD + M D ) M − M M = W W X cos ϕ W Y cos ( ϕ − (cid:37) ) W X cos ϕ X XY cos (cid:37)W Y cos ( ϕ − (cid:37) ) XY cos (cid:37) Y . (34)In order to demonstrate these structures can be fit to the data, we set ϕ = (cid:37) for simplicity, to obtain M L = W κW X W YκW X X κXYW Y κXY Y , κ = cos ϕ. (35)Assuming that the neutrino Yukawa couplings are real, we find that for the normal (NH) and inverted (IH) masshierarchies, the light neutrino mass matrix is diagonalized by a rotation matrix R ν , according to R Tν M L R ν = m ν
00 0 m ν , R ν = − Y √ W + Y W √ W + Y sin θ ν W √ W + Y cos θ ν θ ν − sin θ νW √ W + Y Y √ W + Y sin θ ν Y √ W + Y cos θ ν , for NH (36)tan θ ν = − (cid:115) m − X X − m , m ν = 0 , m ν , = W + X + Y ∓ (cid:113) ( W − X + Y ) − κ X ( W + Y )2 .R Tν M L R ν = m ν m ν
00 0 0 , R ν = W √ W + Y − Y √ W + Y sin θ ν − Y √ W + Y cos θ ν θ ν − sin θ νY √ W + Y W √ W + Y sin θ ν W √ W + Y cos θ ν , for IH (37)tan θ ν = − (cid:115) m − X X − m , m ν , = W + X + Y ∓ (cid:113) ( W − X + Y ) − κ X ( W + Y ) , m ν = 0 . The smallness of the active neutrinos masses is a consequence of their scaling with the inverse of the large Majorananeutrino masses, as expected from the type I seesaw mechanism implemented in our model.With the rotation matrices in the charged lepton sector R l , Eq. (30), and the neutrino sector R ν , Eqs. (36) and (37)for NH and IH, respectively, we obtain the PMNS mixing matrix U = R Tl R ν = − Y √ W + Y W √ W + Y sin θ ν W √ W + Y cos θ νW √ W + Y sin θ l cos θ l cos θ ν + Y √ W + Y sin θ l sin θ ν Y √ W + Y cos θ ν sin θ l − cos θ l sin θ νW √ W + Y cos θ l Y √ W + Y cos θ l sin θ ν − cos θ ν sin θ l sin θ l sin θ ν + Y √ W + Y cos θ l cos θ ν for NH , W √ W + Y − Y √ W + Y sin θ ν − Y √ W + Y cos θ νY √ W + Y sin θ l cos θ l cos θ ν + W √ W + Y sin θ ν sin θ l W √ X + Y sin θ l cos θ ν − cos θ l sin θ νY √ W + Y cos θ l W √ W + Y sin θ ν cos θ l − cos θ ν sin θ l sin θ l sin θ ν + W √ W + Y cos θ l cos θ ν for IH . (38)By comparing with the standard parametrization we derive the mixing angles for NH and IHsin θ = W sin θ ν Y + (1 − cos θ ν ) W , sin θ = W cos θ ν W + Y , sin θ = (cid:0) √ W + Y sin θ ν cos θ l − Y cos θ ν sin θ l (cid:1) (1 − cos θ ν ) W + Y , for NH (39)sin θ = Y sin θ ν W + (1 − cos θ ν ) Y , sin θ = Y cos θ ν W + Y , sin θ = (cid:0) √ W + Y sin θ ν cos θ l − W cos θ ν sin θ l (cid:1) (1 − cos θ ν ) Y + W , for IH. (40)We further simplify the analysis by considering x = y = z , (41)so that the charged lepton masses will be determined by three dimensionless effective parameters, i.e, x , y and z ,whereas the neutrino mass squared splittings and neutrino mixing parameters will be controlled by four dimensionlesseffective parameters, i.e, κ , W , X and Y . Varying the parameters x , y , z , κ , W , X and Y , we fit the chargedlepton masses, the neutrino mass squared splittings ∆ m , ∆ m (defined as ∆ m ij = m i − m j ) and the leptonicmixing angles sin θ , sin θ and sin θ to their experimental values for NH and IH. Therefore the lepton sectorof our model contains seven effective free parameters, i.e., x , y , z , κ , W , X and Y , and describes the leptonmasses and mixing pattern, characterized by eight physical observables, i.e., the three charged lepton masses, the twoneutrino mass squared splittings and the three leptonic mixing angles. The results shown in Table V correspond tothe following best-fit values: κ (cid:39) . , W (cid:39) . eV , X (cid:39) . eV , Y (cid:39) . eV ,x (cid:39) . , y (cid:39) . , z (cid:39) . , for NH, (42) κ (cid:39) . × − , W (cid:39) . eV , X (cid:39) . eV , Y (cid:39) . eV ,x (cid:39) . , y (cid:39) . , z (cid:39) . , for IH. (43)Using the best-fit values given above, we obtain the following neutrino masses for NH and IH m = 0 , m ≈ , m ≈ , for NH, (44) m ≈ , m ≈ , m = 0 , for IH. (45)The obtained and experimental values of the observables in the lepton sector are shown in Table V. Given thatthe lightest neutrino is predicted to be massless in our model, the neutrino masses are hierarchical, which putsthe overall neutrino mass scale below the current experimental reach (the same applies to the cosmological bound (cid:80) k =1 m ν k < .
23 eV on the sum of the neutrino masses [69, 70]). Therefore, our model fulfills the cosmologicalcontraints on neutrino masses for both normal and inverted hierarchies.The experimental values of the charged lepton masses, which are given at the M Z scale, have been taken from Ref.[68] , whereas the experimental values of the neutrino mass squared splittings and leptonic mixing angles for bothNH and IH, are taken from Ref. [71]. The obtained charged lepton masses, neutrino mass squared splittings andlepton mixing angles are in excellent agreement with the experimental data, showing that the model can perfectlyaccount for all the observables in the lepton sector. We recall that for the sake of simplicity, we assumed all leptonicparameters to be real and further restricted the set of parameters, but a non-vanishing CP violating phase in thePMNS mixing matrix can be generated by allowing one or several parameters in the neutrino mass matrix of Eq. (32)to be complex. Observable Model value Experimental value m e ( MeV ) 0 .
487 0 . m µ ( MeV ) 102 . . ± . m τ ( GeV ) 1 .
75 1 . ± . m (10 − eV ) (NH) 7 .
60 7 . +0 . − . ∆ m (10 − eV ) (NH) 2 .
48 2 . +0 . − . sin θ (NH) 0 .
323 0 . ± . θ (NH) 0 .
567 0 . +0 . − . sin θ (NH) 0 . . ± . m (10 − eV ) (IH) 7 .
60 7 . +0 . − . ∆ m (10 − eV ) (IH) 2 .
48 2 . +0 . − . sin θ (IH) 0 .
323 0 . ± . θ (IH) 0 .
573 0 . +0 . − . sin θ (IH) 0 . . ± . We can now predict the amplitude for neutrinoless double beta (0 νββ ) decay in our model, which is proportional tothe effective Majorana neutrino mass m ββ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:88) k U ek m ν k (cid:12)(cid:12)(cid:12)(cid:12) , (46)where U ek and m ν k are the PMNS mixing matrix elements and the Majorana neutrino masses, respectively.Then, from Eqs. (38) and (42)-(45), we predict the following effective neutrino mass for both hierarchies: m ββ = (cid:40) νββ decay experiments. The present best upper limit on thisparameter m ββ ≤
160 meV comes from the recently quoted EXO-200 experiment [72, 73] T νββ / ( Xe) ≥ . × yr at the 90 % CL. This limit will be improved within the not too distant future. The GERDA experiment [74, 75] iscurrently moving to “phase-II”, at the end of which it is expected to reach T νββ / ( Ge) ≥ × yr, correspondingto m ββ ≤
100 MeV. A bolometric CUORE experiment, using
Te [76], is currently under construction. Its estimatedsensitivity is around T νββ / ( Te) ∼ yr corresponding to m ββ ≤
50 meV. There are also proposals for ton-scalenext-to-next generation 0 νββ experiments with
Xe [77, 78] and Ge [74, 79] claiming sensitivities over T νββ / ∼ yr, corresponding to m ββ ∼ −
30 meV. For recent experimental reviews, see for example Ref. [80] and referencestherein. Thus, according to Eq. (47) our model predicts T νββ / at the level of sensitivities of the next generation ornext-to-next generation 0 νββ experiments. III. SCALAR PHENOMENOLOGY
The renormalizable scalar potential involving only the SU (2) doublets φ i is V ( φ i ) = − (cid:88) i =1 µ i ( φ † i φ i ) + (cid:88) i =1 κ i ( φ † i φ i ) ,V ( φ , φ ) = γ ( φ † φ )( φ † φ ) + κ ( φ † φ )( φ † φ ) ,V ( ξ, χ, ζ, φ i ) = (cid:16) λ ξ ( ξξ ) + λ χ ( χ † χ ) + λ ζ ( ζ † ζ ) (cid:17) (cid:88) i =1 λ i ( φ † i φ i ) , whereas the remaining terms are V ( ξ ) = − µ ξ ( ξξ ) + γ ξ, ( ξξ ) ξ + κ ξ, ( ξξ ) ( ξξ ) + κ ξ, ( ξξ ) ( ξξ ) ,V ( χ ) = − µ χ ( χ † χ ) + κ χ ( χ † χ ) ,V ( ζ ) = − µ ζ ( ζ † ζ ) + κ ζ ( ζ † ζ ) ,V ( ξ, χ, ζ ) = λ ( ξξ ) ( χ † χ ) + λ ( ξξ ) ( ζ † ζ ) + λ ( ζ † ζ )( χ † χ ) . To obtain a viable low-energy model with one CP-odd and one charged Goldstone boson, we consider the followingsoft breaking terms: V soft ( ζ, χ ) = − µ χζ ( ζχ + ζ † χ † ) , (48) V soft ( φ i , φ j ) = − µ (cid:104)(cid:16) φ † φ (cid:17) + (cid:16) φ † φ (cid:17)(cid:105) . (49)The mass matrices of the low-energy CP-even neutral scalars ρ , , CP-odd neutral scalars η , and charged scalars ϕ ± , can be written as M = 12 (cid:32) κ v + v v µ γv v − µ γv v − µ κ v + v v µ (cid:33) ,M = µ (cid:32) v v − − v v (cid:33) ,M = µ + κ v v (cid:32) v v − − v v (cid:33) . (50)0The physical low-energy scalar mass eigenstates are connected with the weak scalar states by the following relations[81, 82] (cid:32) hH (cid:33) = (cid:32) sin α − cos α − cos α − sin α (cid:33) (cid:32) ρ ρ (cid:33) , tan 2 α = 2 (cid:0) γv v − µ (cid:1) κ v − κ v ) + µ (cid:16) v v − v v (cid:17) , (51) (cid:32) π A (cid:33) = (cid:32) cos β sin β sin β − cos β (cid:33) (cid:32) η η (cid:33) , (cid:32) π ± H ± (cid:33) = (cid:32) cos β sin β sin β − cos β (cid:33) (cid:32) ϕ ± ϕ ± (cid:33) , tan β = v v with the low-energy physical scalar masses given by m h = 12 v (cid:18) κ v + κ v v + µ v − v (cid:113) γ v v − γµ v v + κ v − κ κ v v + κ v + µ (cid:19) , (52) m H = 12 v (cid:18) κ v + κ v v + µ v + v (cid:113) γ v v − γµ v v + κ v − κ κ v v + κ v + µ (cid:19) , (53) m A = µ (cid:18) v v + v v (cid:19) , m H ± = µ + κ v v (cid:18) v v + v v (cid:19) . (54)The physical low-energy scalar spectrum of our model includes two massive charged Higgses ( H ± ), one CP-odd Higgs( A ) and two neutral CP-even Higgs ( h, H ) bosons. The scalar h is identified as the SM-like 126 GeV Higgs bosonfound at the LHC. It it noteworthy that the neutral π and charged π ± Goldstone bosons are associated with thelongitudinal components of the Z and W ± gauge bosons, respectively.Thanks to the specific shape of the Yukawa couplings dictated by the discrete symmetries, the present model is flavorconserving in the down-type and charged lepton sectors because for those sectors we have a special case of Yukawaalignment [83–85]. φ generates the masses of the first two down-type quark generations, whereas φ is responsibleonly for the bottom Yukawa, conversely, φ is associated only with the electron Yukawa, while φ generates the massesof the remaining charged leptons. The Yukawa couplings of both doublets are therefore aligned in these sectors. Dueto the lack of Flavor Changing Neutral Currents (FCNCs) in the down-type sector, tightly constrained Kaon andB-meson mixings are protected against neutral scalar contributions. Mixing occurs exclusively in the up-type sector,where both φ and φ couple to the third generation of up-type quarks. Consequently, top quark FCNCs arise thatcan be exploited as a probe of new physics since associated processes are strongly suppressed in the SM. Explicitly,we obtain the following structures for the up and down-type Yukawas in the scalar and fermion mass bases using therotation matrices (14), (30), (51) and the corresponding transformations of the right handed fields. Y dh = y hdd y hds y hdb y hsd y hss y hsb y hbd y hbs y hbb = √ − c α m d vs β − c α m s vs β
00 0 m b s α vc β , (55) Y dH = y Hdd y Hds y Hdb y Hsd y Hss y Hsb y Hbd y Hbs y Hbb = √ − m d s α vs β − m s s α vs β
00 0 − c α m b vc β , (56) Y uh = y huu y huc y hut y hcu y hcc y hct y htu y htc y htt (cid:39) √ m u s α vc β m t v V tb V ub (cid:16) c α s β + s α c β (cid:17) m c s α vc β m t v V tb V cb (cid:16) c α s β + s α c β (cid:17) m t v (cid:16) V tb s α c β − c α s β O ( λ ) (cid:17) , (57) Y uH = y Huu y Huc y Hut y Hcu y Hcc y Hct y Htu y Htc y Htt (cid:39) √ − c α m u vc β m t v V tb V ub (cid:16) s α s β − c α c β (cid:17) − c α m c vc β m t v V tb V cb (cid:16) s α s β − c α c β (cid:17) − m t v (cid:16) V tb c α c β + s α s β O ( λ ) (cid:17) , (58)with the notations sin( x ) ≡ s x , cos( x ) ≡ c x and tan( x ) ≡ t x and V ij denote the CKM matrix elements. Furthermore,the mixing angles α and β are defined in Eq. (51). As in other 2HDMs the couplings depend crucially on theparameters α and β , but should comply with the current bounds if tan β is neither unnaturally large or small, in1 (a) (b) Figure 1: (a)Br( t → hc ) [%] in the α − β plane. (b)Br( t → hc ) [%] as a function of α for β = π/
10 (blue, solid), β = π/ β = π/ y h,Hct couplings are enhanced for small β values leading to apotentially large Br( t → hc ) observable at future experiments. which cases deviations from the bottom and top Yukawa couplings with respect to the SM will become very large.This agrees with our previous statement that the fermion mass hierarchies and mixing are best explained by tan β values of O (1). As explained above, FCNCs are absent in the down-type quark sector since the matrices Y dh,H donot have off-diagonal entries. The up-type Yukawa couplings Y h,Hut,ct , however, allow for the tree-level decays t → hq ( q = u, c ), whose branching ratios are currently limited by ATLAS to Br( t → hq ) < .
79% @ 95% C.L. [86] and byCMS to Br( t → cq ) < .
56% @ 95% C.L (observed limit) and Br( t → cq ) < . +0 . − . % (expected limit) [87]. Since y ut is negligibly small compared to y ct , we consider only the stronger CMS constraint that can be interpreted as anupper bound on the off-diagonal top Yukawas to (cid:113) | y hct | + | y hct | = √ m t v (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) V tb V cb (cid:18) s α c β + c α s β (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < . , (59)which translates to (cid:12)(cid:12)(cid:12)(cid:12) c α − β c β s β (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) . . (60)The t → ch channel is particularly interesting since its branching ratio Br( t → hc ) SM (cid:39) − [86] is extremelysuppressed in the SM, but can be potentially large in our model allowing it to be probed at future collider experiments.As shown in Fig. 1 our model predictions can reach branching ratios of O (0 . α − β plane,allowing to further constrain our model parameter space with experimental searches for rare top decays.Recently an analysis of up-type FCNCs in the 2HDM type III has been performed [88] parametrizing the flavorviolating y hct coupling as y hct = v λ ct √ m t m c according to the Cheng–Sher Ansatz [89] (this type of FCNC was shownto be remarkably stable under radiative corrections [90]). Focusing on the cc → tt as well as the t → cg channels, theyfind that λ ct can still take values of up to 10 −
20 depending on the neutral heavy Higgs mass. With y hct ∝ v V cb V tb √ m t our model corresponds to λ ct ≈ and is therefore well below the critical region. Indeed, following the analysis of [91]we find numerically that the loop induced decays t → cg , t → cγ and t → cZ are several orders of magnitude belowthe current LHC sensitivity. Explicitly, varying the free model parameters α, β and the scalar masses m H , m A and m H ± , we expect the branching ratios to be approximatelyBr( t → cg ) ∼ O (10 − ) , Br( t → cγ ) ∼ O (10 − ) , Br( t → cZ ) ∼ O (10 − ) , (61)as opposed to the current upper limits from ATLAS and CMS [92, 93]Br( t → cg ) < . × − , Br( t → cγ, cZ ) < × − . (62)2 (a) (b) Figure 2: (a)Br( t → hg ) in the α − β plane with m H = m A = 500 GeV. (b)Br( t → hg ) as a function of m H and m A for α = π/ β = π/
4. The decay rate is to a large extent independent of the charged Higgs mass m H ± . The largest branching ratio of the three channels, Br( t → cg ), is shown in Fig. 2 as a function of α and β for fixed m H and m A (a), as well as for variable m H and m A with fixed α and β (b). As it turns out, the charged Higgscontribution is tiny and does not affect the prediction for any values of m H ± .In the charged lepton sector we obtain Y lh = √ y hee y heµ y heτ y hµe y hµµ y hµτ y hτe y hτµ y hττ = √ − c α m e vs β m µ s α vc β
00 0 m τ s α vc β , (63) Y lH = √ y Hee y Heµ y Heτ y Hµe y Hµµ y Hµτ y Hτe y Hτµ y Hττ = √ − m e s α vs β − c α m µ vc β
00 0 − c α m τ vc β . (64)The charged leptons are also free of FCNCs due to the lack of off-diagonal Yukawa couplings. Consequently, therecently reported anomaly in h → µτ decays cannot be explained in our present model, even though it was possibleto account for this in other multi-Higgs models with S or other discrete symmetries [94–97].The charged Higgs couplings that are relevant, e.g., for B s,d − B s,d mixing and the radiative decays b → qγ ( q = s, d ),are given by Y LH ± = √ y du y dc y dt y su y sc y st y bu y bc y bt = √ V ud V tb + V cb t β m u v − V us V tb t β m c v − V ∗ td m t vt β V us V tb + V cb t β m u v V ud V tb t β m c v − V ∗ ts m t vt β V tb t β m t v , (65) Y RH ± = √ y ud y us y ub y cd y cs y cb y td y ts y tb = √ V ud m d vt β V us m s vt β V ub t β m b v V cd m d vt β V cs m s vt β V cb t β m b v V td m d vt β V ts m s vt β V tb t β m b v , (66) Y eνH ± = √ m e vt β , Y µνH ± = √ m µ v t β ( c θ l − s θ l ) , Y τνH ± = √ m τ v t β ( c θ l + s θ l ) , (67)3 h γγWWW h WW γγ h γγttth γγH ± H ± H ± h H ± H ± γγ Figure 3: One-loop Feynman diagrams in the Unitary Gauge contributing to the h → γγ decay. where in the last equation we summed over the neutrino mass eigenstates as they are usually undetected in typicalflavor experiments. Here, the couplings y bu and y bc that could be used to explain the outstanding anomaly in B → D ( ∗ ) τ ν decays [98] are zero, hence no difference from 2HDMs of type II is to be expected in these channels.On the other hand, the charged scalar sector is tightly constrained by b → sγ measurements, where the charged scalar H ± leads to an additional loop diagram replacing the W ± . Recently a lower bound of 480 GeV was placed on thecharged Higgs in the 2HDM type II [99]. Following the analysis of [100] we estimate a lower bound on the chargedHiggs mass imposed on our model by constraints on the Wilson coefficients involved in Br( b → sγ ). Since tan β dropsout in the product of the corresponding Yukawa couplings y tb ( y bt ) and y ts ( y st ), the prediction is independent of tan β and the lower limit is roughly m H ± (cid:38)
500 GeV.
A. Constraints from h → γγ In our 2HDM the h → γγ decay receives additional contributions from loops with charged scalars H ± , as shown inFig. 3, and therefore sets bounds on the masses of these scalars as well as on the angles α and β .The explicit form of the h → γγ decay rate is [101–108]Γ ( h → γγ ) = α em m h π v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) f a hff N c Q f F / (cid:0) (cid:37) f (cid:1) + a hW W F ( (cid:37) W ) + λ hH ± H ∓ v m H ± F ( (cid:37) H ± ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (68)Here (cid:37) i are the mass ratios (cid:37) i = m h M i , with M i = m f , M W , and m H ± , α em is the fine structure constant, N C is thecolor factor ( N C = 1 for leptons, N C = 3 for quarks), and Q f is the electric charge of the fermion in the loop. Fromthe fermion-loop contributions we consider only the dominant top quark term. Furthermore, λ hH ± H ∓ is the trilinearcoupling between the SM-like Higgs and a pair of charged Higgses, which is given by λ hH ± H ∓ = − γ + κ v sin 2 β cos ( α + β ) . (69)Besides that a htt and a hW W are the deviation factors from the SM Higgs-top quark coupling and the SM Higgs- W a htt (cid:39) sin α cos β , (70) a hW W = sin ( α − β ) , (71)where in a htt we neglected the contribution suppressed by small CKM entries.The dimensionless loop factors F / ( (cid:37) ) and F ( (cid:37) ) (for spin-1 / F / ( (cid:37) ) = 2 [ (cid:37) + ( (cid:37) − f ( (cid:37) )] (cid:37) − , (72) F ( (cid:37) ) = − (cid:2) (cid:37) + 3 (cid:37) + 3 (2 (cid:37) − f ( (cid:37) ) (cid:3) (cid:37) − , (73) F ( (cid:37) ) = − [ (cid:37) − f ( (cid:37) )] (cid:37) − , (74)with f ( (cid:37) ) = arcsin √ (cid:37), for (cid:37) ≤ − (cid:20) ln (cid:18) √ − (cid:37) − − √ − (cid:37) − (cid:19) − iπ (cid:21) , for (cid:37) > . (75)In what follows we determine the constraints that the Higgs diphoton signal strength imposes on our model. To thisend, we introduce the ratio R γγ , which normalizes the γγ signal predicted by our model relative to that of the SM: R γγ = σ ( pp → h ) Γ ( h → γγ ) σ ( pp → h ) SM Γ ( h → γγ ) SM (cid:39) a htt Γ ( h → γγ )Γ ( h → γγ ) SM . (76)The normalization given by Eq. (76) for h → γγ was also used in Refs. [94, 108–113].The ratio R γγ has been measured by CMS and ATLAS with the best-fit signals [114, 115] R CMS γγ = 1 . +0 . − . and R ATLAS γγ = 1 . ± . . Figure (4(a)) shows the sensitivity of the ratio R γγ under variations of the mixing angle α for m H ± = 500 GeV, γ + κ = 1 and different values of the mixing angle β . It follows that as the mixing angle β is increased, the rangeof α consistent with LHC observations of h → γγ moves away from π/
2. On the other hand, the decay rate is largelyindependent of the charged Higgs mass or the sum of the couplings γ + κ , which is consistent with the contributionmediated by charged scalars to the h → γγ process being a small correction. In fact we checked numerically it staysalmost constant when m H ± is varied from 500 GeV to 1 TeV for fixed values of α, β , and the quartic couplings ofthe scalar potential. For the same values of the charged Higgs mass and quartic couplings, we show in Figure (4(b))the Z-shaped allowed region in the α - β plane that is consistent with the Higgs diphoton decay rate constraints at theLHC, and overlay it with the relatively weak bound in Eq. (60) that arises from top quark FCNCs. B. T and S parameters The extra scalars affect the oblique corrections of the SM, and these values are measured in high precision experiments.Consequently, they act as a further constraint on the validity of our model. The oblique corrections are parametrizedin terms of the two well-known quantities T and S . In this section we calculate one-loop contributions to the obliqueparameters T and S defined as [116–118] T = Π (cid:0) q (cid:1) − Π (cid:0) q (cid:1) α EM ( M Z ) M W (cid:12)(cid:12)(cid:12)(cid:12) q =0 , S = 2 sin 2 θ W α EM ( M Z ) d Π (cid:0) q (cid:1) dq (cid:12)(cid:12)(cid:12)(cid:12) q =0 . (77)Π (0), Π (0), and Π (cid:0) q (cid:1) are the vacuum polarization amplitudes with { W µ , W µ } , { W µ , W µ } and { W µ , B µ } external gauge bosons, respectively, where q is their momentum. We note that in the definitions of the T and S parameters, the new physics is assumed to be heavy when compared to M W and M Z .The Feynman diagrams contributing to the T and S parameters are shown in Figs. 5 and 6.5 (a) (b) Figure 4: The constraints on the model imposed by keeping R γγ inside the experimentally allowed 1 σ range determined byCMS and ATLAS to be 1 . +0 . − . and 1 . ± .
27, respectively [114, 115]. (4.(a)) shows the ratio R γγ as a function of themixing angle α of the CP-even neutral scalars h and H for m H ± = 500 GeV, γ + κ = 1 and different values of the mixingangle β ; the blue, red and green curves correspond β set to 0, π and π , respectively, and the horizontal lines are the minimumand maximum values of the ratio R γγ . (4.(b)) shows the allowed region in the α - β plane consistent with the Higgs diphotondecay rate constraint at the LHC, superimposed with the constraint imposed by Eq.(60). We split the T and S emphasizing the contributions arising from new physics as T = T SM + ∆ T and S = S SM + ∆ S ,where T SM and S SM are the SM contributions given by T SM = − π cos θ W ln (cid:18) m h m W (cid:19) , (78) S SM = 112 π ln (cid:18) m h m W (cid:19) , (79)while ∆ T and ∆ S contain all the contributions involving in our model the heavy scalars∆ T (cid:39) − ( α − β )16 π cos θ W ln (cid:18) m H m h (cid:19) + 116 π v α EM ( M Z ) (cid:2) m H ± − F (cid:0) m A , m H ± (cid:1)(cid:3) + sin ( α − β )16 π v α EM ( M Z ) (cid:2) F (cid:0) m h , m A (cid:1) − F (cid:0) m h , m H ± (cid:1)(cid:3) + cos ( α − β )16 π v α EM ( M Z ) (cid:2) F (cid:0) m H , m A (cid:1) − F (cid:0) m H , m H ± (cid:1)(cid:3) , (80)∆ S (cid:39) π (cid:20) cos ( α − β ) ln (cid:18) m H m h (cid:19) + sin ( α − β ) K (cid:0) m h , m A , m H ± (cid:1) + cos ( α − β ) K (cid:0) m H , m A , m H ± (cid:1)(cid:21) , (81)where we introduced the functions [104, 119–125] F (cid:0) m , m (cid:1) = m m m − m ln (cid:18) m m (cid:19) , lim m → m F (cid:0) m , m (cid:1) = m , (82) K (cid:0) m , m , m (cid:1) = 1( m − m ) (cid:26) m (cid:0) m − m (cid:1) ln (cid:18) m m (cid:19) − m (cid:0) m − m (cid:1) ln (cid:18) m m (cid:19) − (cid:2) m m (cid:0) m − m (cid:1) + 5 (cid:0) m − m (cid:1)(cid:3)(cid:27) , (83)6 W W π B W W H BW W H A W W H H W W H H W W H A Figure 5: One-loop Feynman diagrams contributing to the T parameter. The fields H and H are linear combinations of thecharged Higgses H ± , similarly to how W ± gauge bosons are defined in terms of W and W . Likewise, the fields π and π are linear combinations of the charged Goldstone bosons π ± . with the propertieslim m → m K ( m , m , m ) = K ( m , m ) = ln (cid:18) m m (cid:19) , lim m → m K ( m , m , m ) = K ( m , m ) = − m + 27 m m − m m + 6 (cid:0) m − m m (cid:1) ln (cid:16) m m (cid:17) + 5 m m − m ) , lim m → m K ( m , m , m ) = K ( m , m ) . (84)The experimental results on T and S restrict ∆ T and ∆ S to lie inside a region in the ∆ S − ∆ T plane. At the 95%confidence level, these are the elliptic contours shown in Fig. 7. The origin ∆ S = ∆ T = 0 is the SM value with m h = 125 . m t = 176 GeV. We analyze the T and S parameter constraints on our model by consideringtwo benchmark scenarios, in both keeping α − β = π . In the first scenario we assume that the CP-even and CP-oddneutral Higgs bosons have degenerate masses of 500 GeV, below which the LHC has not detected any scalars beyondthe SM-like state. In this first scenario, we find that the T and S parameters constrain the charged Higgs masses tothe range 550 GeV ≤ m H ± ≤
580 GeV, which is consistent with the lower bound m H ± (cid:38)
500 GeV obtained from b → sγ constraints [99]. In the second scenario, we assume that the charged Higgses and CP-even neutral Higgseshave degenerate masses of 500 GeV. In this second scenario, the T and S parameter constraints are fulfilled if theCP-odd neutral Higgs boson mass is in the range 375 GeV ≤ m A ≤
495 GeV.7 W Bπ π W BH H W Bπ H W BH A Figure 6: One-loop Feynman diagrams contributing to the S parameter. The fields H and H are linear combinations of thecharged Higgses H ± , similarly to how W ± gauge bosons are defined in terms of W and W . IV. CONCLUSIONS
We have constructed a viable 2-Higgs doublet extension of the Standard Model which features additionally an S flavor symmetry and extra scalars that break S . This leads to textures for fermion masses, and consists in anexistence proof of models leading to the quark texture in [55]. Overall, the model can fit the observed masses, CKMand PMNS mixing angles very well. The model has in total seventeen effective free parameters, which are fitted toreproduce the experimental values of eighteen observables in the quark and lepton sectors, i.e., nine charged fermionmasses, two neutrino mass squared splittings, three lepton mixing parameters, three quark mixing angles and one CPviolating phase of the CKM quark mixing matrix. The model predicts one massless neutrino for both normal andinverted hierarchies in the active neutrino mass spectrum as well as an effective Majorana neutrino mass, relevantfor neutrinoless double beta decay, with values m ββ = 4 meV and 50 meV, for the normal and the inverted neutrinospectrum, respectively. In the latter case our prediction is within the declared reach of the next generation bolometricCUORE experiment [76] or, more realistically, of the next-to-next generation tonne-scale 0 νββ -decay experiments.The sum of the light active neutrino masses in our model is 59 meV and 0 . (cid:80) k =1 m ν k < .
23 eV. The additional scalarsmediate flavor changing neutral current processes, but due to the specific shape of the Yukawa couplings dictated bythe flavor symmetry these processes occur only in the up-type quark sector. In the scalar sector the enlarged fieldcontent of the model leads to constraints from both rare top decays and from a h → γγ rate that can be distinguishedfrom the SM prediction. Among rare top decays, t → ch is particularly promising as its branching ratio can reach O (0 . h → γγ , we find that it depends only slightly on the mass of the chargedHiggs and the dependence on the quartic scalar couplings is negligible, but the dominant top quark and vector bosoncontributions are modified in our model and allow us to place constraints on the hierarchy of the SU (2) doubletVEVs ( β ) and the mixing of their CP-even mass eigenstates ( α ) that are much stronger than those obtained from theup-type quark flavor changing processes. We also showed for a few benchmark scenarios that our model is compatiblewith the present bounds for the oblique parameters T and S . Acknowledgments
This project has received funding from the European Union’s Seventh Framework Programme for research, tech-nological development and demonstration under grant agreement no PIEF-GA-2012-327195 SIFT. A.E.C.H thanksSouthampton University for hospitality where part of this work was done. A.E.C.H was supported by Fondecyt8 (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) S (cid:68) T (cid:45) (cid:45) (cid:45) (cid:45) (cid:68) S (cid:68) T Figure 7: The ∆ S − ∆ T plane, where the ellipses contain the experimentally allowed region at 95% confidence level taken from[126–128]. We set α − β = π . Figures (a) and (b) correspond to m A = m H = 500 GeV and m H = m H ± = 500 GeV,respectively. The charged Higgs and CP-odd neutral Higgs boson masses vary between 550 GeV ≤ m H ± ≤
580 GeV (Fig.7(a), 375 GeV ≤ m A ≤
495 GeV (Fig. 7(b). The nearly vertical lines going up towards the ellipses correspond to ∆ T and ∆ S parameters in our model as masses are varied in the aforementioned ranges. (Chile), Grant No. 11130115 and by DGIP internal Grant No. 111458. Appendix A: The product rules for S . The S group has three irreducible representations: , (cid:48) and . Denoting the basis vectors for two S doublets as( x , x ) T and ( y , y ) T and y (cid:48) a non trivial S singlet, the S multiplication rules are [129]: (cid:32) x x (cid:33) ⊗ (cid:32) y y (cid:33) = ( x y + x y ) + ( x y − x y ) (cid:48) + (cid:32) x y − x y x y + x y (cid:33) , (A1) (cid:32) x x (cid:33) ⊗ ( y (cid:48) ) (cid:48) = (cid:32) − x y (cid:48) x y (cid:48) (cid:33) , ( x (cid:48) ) (cid:48) ⊗ ( y (cid:48) ) (cid:48) = ( x (cid:48) y (cid:48) ) . (A2) Appendix B: Decoupling and S VEVs
We assume that all SM singlet scalars acquire VEVs much larger than the electroweak symmetry breaking scale. Thisimplies that the mixing angle between the scalar singlets and the SU (2) doublet scalars is strongly suppressed sinceit is of the order of v , λ Λ , as follows from the method of recursive expansion of Refs. [130–132]. Consequently, themixing between these scalar singlets and the SM Higgs doublets can be neglected. We also checked numerically thatthe masses of the low-energy scalars are nearly unaffected by SM singlet VEVs of O (500 GeV) and higher.For simplicity we assume a CP invariant scalar potential with only real couplings as done in Refs. [10, 11, 42, 94].In the regime where the VEVs decouple, and also because the 1 (cid:48) scalar ζ is charged under Z (cid:48) , the relevant terms fordetermining the direction of the ξ VEV in S are V ( ξ ) = − µ ξ ( ξξ ) + γ ξ, ( ξξ ) ξ + κ ξ, ( ξξ ) ( ξξ ) + κ ξ, ( ξξ ) ( ξξ ) + κ ξ, [( ξξ ) ξ ] ξ, (B1)9From the minimization conditions of the high-energy scalar potential, we find the following relations: ∂ (cid:104) V (cid:105) ∂v ξ = 2 v ξ (cid:104) µ ξ + 2 ( κ ξ, + κ ξ, + κ ξ, ) (cid:16) v ξ + v ξ (cid:17)(cid:105) + 3 γ ξ, (cid:16) v ξ − v ξ (cid:17) = 0 ∂ (cid:104) V (cid:105) ∂v ξ = 2 v ξ (cid:110)(cid:104) µ ξ + 2 ( κ ξ, + κ ξ, + κ ξ, ) (cid:16) v ξ + v ξ (cid:17)(cid:105) + 3 γ ξ, v ξ (cid:111) = 0 , (B2)Then, from an analysis of the minimization equations given by Eq. (B2), we obtain for a large range of the parameterspace the following VEV direction for ξ : (cid:104) ξ (cid:105) = v ξ (1 , . (B3)From the expressions given in Eq. (B2), and using the vacuum configuration for the S scalar doublets given in Eq.(5), we find the relation between the parameters and the magnitude of the VEV: µ ξ = − v ξ (cid:2) γ ξ, + 4 ( κ ξ, + κ ξ, + κ ξ, ) v ξ (cid:3) , (B4)These results show that the VEV direction for the S doublet ξ in Eq. (5) is consistent with a global minimum of thescalar potential of our model. [1] S. Pakvasa and H. Sugawara, Phys. Lett. , 61 (1978). doi:10.1016/0370-2693(78)90172-7[2] H. Cardenas, A. C. B. Machado, V. Pleitez and J.-A. Rodriguez, Phys. Rev. D , no. 3, 035028 (2013)doi:10.1103/PhysRevD.87.035028 [arXiv:1212.1665 [hep-ph]].[3] A. G. Dias, A. C. B. Machado and C. C. Nishi, Phys. Rev. D , 093005 (2012) doi:10.1103/PhysRevD.86.093005[arXiv:1206.6362 [hep-ph]].[4] S. Dev, R. R. Gautam and L. Singh, Phys. Lett. B , 284 (2012) doi:10.1016/j.physletb.2012.01.051 [arXiv:1201.3755[hep-ph]].[5] D. Meloni, JHEP , 124 (2012) doi:10.1007/JHEP05(2012)124 [arXiv:1203.3126 [hep-ph]].[6] F. Gonz´alez Canales, A. Mondrag´on, M. Mondrag´on, U. J. Salda˜na Salazar and L. Velasco-Sevilla, Phys. Rev. D ,096004 (2013) doi:10.1103/PhysRevD.88.096004 [arXiv:1304.6644 [hep-ph]].[7] E. Ma and B. Melic, Phys. Lett. B , 402 (2013) doi:10.1016/j.physletb.2013.07.015 [arXiv:1303.6928 [hep-ph]].[8] Y. Kajiyama, H. Okada and K. Yagyu, Nucl. Phys. B , 358 (2014) doi:10.1016/j.nuclphysb.2014.08.009[arXiv:1309.6234 [hep-ph]].[9] A. E. C´arcamo Hern´andez, R. Martinez and F. Ochoa, Eur. Phys. J. C , no. 11, 634 (2016) doi:10.1140/epjc/s10052-016-4480-3 [arXiv:1309.6567 [hep-ph]].[10] A. E. C´arcamo Hern´andez, E. Cata˜no Mur and R. Martinez, Phys. Rev. D , no. 7, 073001 (2014)doi:10.1103/PhysRevD.90.073001 [arXiv:1407.5217 [hep-ph]].[11] A. E. C´arcamo Hern´andez, R. Martinez and J. Nisperuza, Eur. Phys. J. C , no. 2, 72 (2015) doi:10.1140/epjc/s10052-015-3278-z [arXiv:1401.0937 [hep-ph]].[12] V. V. Vien and H. N. Long, Zh. Eksp. Teor. Fiz. , 991 (2014) [J. Exp. Theor. Phys. , no. 6, 869 (2014)]doi:10.7868/S0044451014060044, 10.1134/S1063776114050173 [arXiv:1404.6119 [hep-ph]].[13] E. Ma and R. Srivastava, Phys. Lett. B , 217 (2015) doi:10.1016/j.physletb.2014.12.049 [arXiv:1411.5042 [hep-ph]].[14] D. Das and U. K. Dey, Phys. Rev. D , no. 9, 095025 (2014) Erratum: [Phys. Rev. D , no. 3, 039905 (2015)]doi:10.1103/PhysRevD.91.039905, 10.1103/PhysRevD.89.095025 [arXiv:1404.2491 [hep-ph]].[15] D. Das, U. K. Dey and P. B. Pal, Phys. Lett. B , 315 (2016) doi:10.1016/j.physletb.2015.12.038 [arXiv:1507.06509[hep-ph]].[16] E. Ma and G. Rajasekaran, Phys. Rev. D , 113012 (2001) doi:10.1103/PhysRevD.64.113012 [hep-ph/0106291].[17] K. S. Babu, E. Ma and J. W. F. Valle, Phys. Lett. B , 207 (2003) doi:10.1016/S0370-2693(02)03153-2 [hep-ph/0206292].[18] G. Altarelli and F. Feruglio, Nucl. Phys. B , 64 (2005) doi:10.1016/j.nuclphysb.2005.05.005 [hep-ph/0504165].[19] G. Altarelli and F. Feruglio, Nucl. Phys. B , 215 (2006) doi:10.1016/j.nuclphysb.2006.02.015 [hep-ph/0512103].[20] I. de Medeiros Varzielas, S. F. King and G. G. Ross, Phys. Lett. B , 153 (2007) doi:10.1016/j.physletb.2006.11.015[hep-ph/0512313].[21] I. de Medeiros Varzielas and D. Pidt, JHEP , 065 (2013) doi:10.1007/JHEP03(2013)065 [arXiv:1211.5370 [hep-ph]].[22] H. Ishimori and E. Ma, Phys. Rev. D , 045030 (2012) doi:10.1103/PhysRevD.86.045030 [arXiv:1205.0075 [hep-ph]].[23] Y. H. Ahn, S. K. Kang and C. S. Kim, Phys. Rev. D , no. 11, 113012 (2013) doi:10.1103/PhysRevD.87.113012[arXiv:1304.0921 [hep-ph]]. [24] N. Memenga, W. Rodejohann and H. Zhang, Phys. Rev. D , no. 5, 053021 (2013) doi:10.1103/PhysRevD.87.053021[arXiv:1301.2963 [hep-ph]].[25] S. Bhattacharya, E. Ma, A. Natale and A. Rashed, Phys. Rev. D , 097301 (2013) doi:10.1103/PhysRevD.87.097301[arXiv:1302.6266 [hep-ph]].[26] P. M. Ferreira, L. Lavoura and P. O. Ludl, Phys. Lett. B , 767 (2013) doi:10.1016/j.physletb.2013.09.058[arXiv:1306.1500 [hep-ph]].[27] R. Gonzalez Felipe, H. Serodio and J. P. Silva, Phys. Rev. D , no. 1, 015015 (2013) doi:10.1103/PhysRevD.88.015015[arXiv:1304.3468 [hep-ph]].[28] A. E. Carcamo Hernandez, I. de Medeiros Varzielas, S. G. Kovalenko, H. P¨as and I. Schmidt, Phys. Rev. D , no. 7,076014 (2013) doi:10.1103/PhysRevD.88.076014 [arXiv:1307.6499 [hep-ph]].[29] S. F. King, S. Morisi, E. Peinado and J. W. F. Valle, Phys. Lett. B , 68 (2013) doi:10.1016/j.physletb.2013.05.067[arXiv:1301.7065 [hep-ph]].[30] S. Morisi, D. V. Forero, J. C. Rom˜ao and J. W. F. Valle, Phys. Rev. D , no. 1, 016003 (2013)doi:10.1103/PhysRevD.88.016003 [arXiv:1305.6774 [hep-ph]].[31] S. Morisi, M. Nebot, K. M. Patel, E. Peinado and J. W. F. Valle, Phys. Rev. D , 036001 (2013)doi:10.1103/PhysRevD.88.036001 [arXiv:1303.4394 [hep-ph]].[32] R. Gonz´alez Felipe, H. Serˆodio and J. P. Silva, Phys. Rev. D , no. 5, 055010 (2013) doi:10.1103/PhysRevD.87.055010[arXiv:1302.0861 [hep-ph]].[33] M. D. Campos, A. E. C´arcamo Hern´andez, S. Kovalenko, I. Schmidt and E. Schumacher, Phys. Rev. D , no. 1, 016006(2014) doi:10.1103/PhysRevD.90.016006 [arXiv:1403.2525 [hep-ph]].[34] A. E. C´arcamo Hern´andez and R. Martinez, Nucl. Phys. B , 337 (2016) doi:10.1016/j.nuclphysb.2016.02.025[arXiv:1501.05937 [hep-ph]].[35] S. Pramanick and A. Raychaudhuri, Phys. Rev. D , no. 3, 033007 (2016) doi:10.1103/PhysRevD.93.033007[arXiv:1508.02330 [hep-ph]].[36] C. Luhn, S. Nasri and P. Ramond, Phys. Lett. B , 27 (2007) doi:10.1016/j.physletb.2007.06.059 [arXiv:0706.2341[hep-ph]].[37] C. Hagedorn, M. A. Schmidt and A. Y. Smirnov, Phys. Rev. D , 036002 (2009) doi:10.1103/PhysRevD.79.036002[arXiv:0811.2955 [hep-ph]].[38] Q. H. Cao, S. Khalil, E. Ma and H. Okada, Phys. Rev. Lett. , 131801 (2011) doi:10.1103/PhysRevLett.106.131801[arXiv:1009.5415 [hep-ph]].[39] C. Luhn, K. M. Parattu and A. Wingerter, JHEP , 096 (2012) doi:10.1007/JHEP12(2012)096 [arXiv:1210.1197[hep-ph]].[40] Y. Kajiyama, H. Okada and K. Yagyu, JHEP , 196 (2013) doi:10.1007/JHEP10(2013)196 [arXiv:1307.0480 [hep-ph]].[41] C. Bonilla, S. Morisi, E. Peinado and J. W. F. Valle, Phys. Lett. B , 99 (2015) doi:10.1016/j.physletb.2015.01.017[arXiv:1411.4883 [hep-ph]].[42] A. E. C´arcamo Hern´andez and R. Martinez, J. Phys. G , no. 4, 045003 (2016) doi:10.1088/0954-3899/43/4/045003[arXiv:1501.07261 [hep-ph]].[43] C. Arbel´aez, A. E. C´arcamo Hern´andez, S. Kovalenko and I. Schmidt, Phys. Rev. D , no. 11, 115015 (2015)doi:10.1103/PhysRevD.92.115015 [arXiv:1507.03852 [hep-ph]].[44] I. de Medeiros Varzielas, S. F. King and G. G. Ross, Phys. Lett. B , 201 (2007) doi:10.1016/j.physletb.2007.03.009[hep-ph/0607045].[45] E. Ma, Mod. Phys. Lett. A , 1917 (2006) doi:10.1142/S0217732306021190 [hep-ph/0607056].[46] I. de Medeiros Varzielas, D. Emmanuel-Costa and P. Leser, Phys. Lett. B , 193 (2012)doi:10.1016/j.physletb.2012.08.008 [arXiv:1204.3633 [hep-ph]].[47] G. Bhattacharyya, I. de Medeiros Varzielas and P. Leser, Phys. Rev. Lett. , 241603 (2012)doi:10.1103/PhysRevLett.109.241603 [arXiv:1210.0545 [hep-ph]].[48] E. Ma, Phys. Lett. B , 161 (2013) doi:10.1016/j.physletb.2013.05.011 [arXiv:1304.1603 [hep-ph]].[49] I. de Medeiros Varzielas and D. Pidt, J. Phys. G , 025004 (2014) doi:10.1088/0954-3899/41/2/025004 [arXiv:1307.0711[hep-ph]].[50] A. Aranda, C. Bonilla, S. Morisi, E. Peinado and J. W. F. Valle, Phys. Rev. D , no. 3, 033001 (2014)doi:10.1103/PhysRevD.89.033001 [arXiv:1307.3553 [hep-ph]].[51] I. de Medeiros Varzielas and D. Pidt, JHEP , 206 (2013) doi:10.1007/JHEP11(2013)206 [arXiv:1307.6545 [hep-ph]].[52] I. de Medeiros Varzielas, JHEP , 157 (2015) doi:10.1007/JHEP08(2015)157 [arXiv:1507.00338 [hep-ph]].[53] S. F. King and C. Luhn, Rept. Prog. Phys. , 056201 (2013) doi:10.1088/0034-4885/76/5/056201 [arXiv:1301.1340[hep-ph]].[54] S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, New J. Phys. , 045018 (2014) doi:10.1088/1367-2630/16/4/045018 [arXiv:1402.4271 [hep-ph]].[55] A. E. C´arcamo Hern´andez and I. de Medeiros Varzielas, J. Phys. G , no. 6, 065002 (2015) doi:10.1088/0954-3899/42/6/065002 [arXiv:1410.2481 [hep-ph]].[56] L. Wolfenstein, Phys. Rev. Lett. , 1945 (1983). doi:10.1103/PhysRevLett.51.1945[57] G. C. Branco, D. Emmanuel-Costa and C. Simoes, Phys. Lett. B , 62 (2010) doi:10.1016/j.physletb.2010.05.009[arXiv:1001.5065 [hep-ph]].[58] A. E. Carcamo Hernandez and R. Rahman, Rev. Mex. Fis. , no. 2, 100 (2016) [arXiv:1007.0447 [hep-ph]].[59] M. C. Chen and K. T. Mahanthappa, Phys. Lett. B , 34 (2007) doi:10.1016/j.physletb.2007.06.064 [arXiv:0705.0714 [hep-ph]].[60] Z. z. Xing, D. Yang and S. Zhou, Phys. Lett. B , 304 (2010) doi:10.1016/j.physletb.2010.05.045 [arXiv:1004.4234[hep-ph]].[61] G. C. Branco, H. R. C. Ferreira, A. G. Hessler and J. I. Silva-Marcos, JHEP , 001 (2012)doi:10.1007/JHEP05(2012)001 [arXiv:1101.5808 [hep-ph]].[62] A. E. Carcamo Hernandez, C. O. Dib, N. Neill H and A. R. Zerwekh, JHEP , 132 (2012)doi:10.1007/JHEP02(2012)132 [arXiv:1201.0878 [hep-ph]].[63] V. V. Vien and H. N. Long, J. Korean Phys. Soc. , no. 12, 1809 (2015) doi:10.3938/jkps.66.1809 [arXiv:1408.4333[hep-ph]].[64] M. Abbas and S. Khalil, Phys. Rev. D , no. 5, 053003 (2015) doi:10.1103/PhysRevD.91.053003 [arXiv:1406.6716 [hep-ph]].[65] H. Ishimori and S. F. King, Phys. Lett. B , 33 (2014) doi:10.1016/j.physletb.2014.06.003 [arXiv:1403.4395 [hep-ph]].[66] H. Ishimori, S. F. King, H. Okada and M. Tanimoto, Phys. Lett. B , 172 (2015) doi:10.1016/j.physletb.2015.02.027[arXiv:1411.5845 [hep-ph]].[67] K. A. Olive et al. [Particle Data Group], Chin. Phys. C , 090001 (2014). doi:10.1088/1674-1137/38/9/090001[68] K. Bora, Horizon (2013) [arXiv:1206.5909 [hep-ph]].[69] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. , A16 (2014) doi:10.1051/0004-6361/201321591[arXiv:1303.5076 [astro-ph.CO]].[70] S. M. Bilenky, Phys. Part. Nucl. , no. 4, 475 (2015) doi:10.1134/S1063779615040024 [arXiv:1501.00232 [hep-ph]].[71] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D , no. 9, 093006 (2014) doi:10.1103/PhysRevD.90.093006[arXiv:1405.7540 [hep-ph]].[72] M. Auger et al. [EXO-200 Collaboration], Phys. Rev. Lett. , 032505 (2012) doi:10.1103/PhysRevLett.109.032505[arXiv:1205.5608 [hep-ex]].[73] D. J. Auty [EXO-200 Collaboration].[74] I. Abt et al. , hep-ex/0404039.[75] K. H. Ackermann et al. [GERDA Collaboration], Eur. Phys. J. C , no. 3, 2330 (2013) doi:10.1140/epjc/s10052-013-2330-0 [arXiv:1212.4067 [physics.ins-det]].[76] F. Alessandria et al. [CUORE Collaboration], arXiv:1109.0494 [nucl-ex].[77] A. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev. C , 045504 (2012) doi:10.1103/PhysRevC.85.045504[arXiv:1201.4664 [hep-ex]].[78] J. B. Albert et al. [EXO-200 Collaboration], Phys. Rev. D , no. 9, 092004 (2014) doi:10.1103/PhysRevD.90.092004[arXiv:1409.6829 [hep-ex]].[79] C. E. Aalseth et al. [Majorana Collaboration], Nucl. Phys. Proc. Suppl. , 44 (2011)doi:10.1016/j.nuclphysbps.2011.04.063 [arXiv:1101.0119 [nucl-ex]].[80] S. M. Bilenky and C. Giunti, Int. J. Mod. Phys. A , no. 04n05, 1530001 (2015) doi:10.1142/S0217751X1530001X[arXiv:1411.4791 [hep-ph]].[81] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. , 1 (2012)doi:10.1016/j.physrep.2012.02.002 [arXiv:1106.0034 [hep-ph]].[82] G. Bhattacharyya and D. Das, Pramana , no. 3, 40 (2016) doi:10.1007/s12043-016-1252-4 [arXiv:1507.06424 [hep-ph]].[83] A. Pich and P. Tuzon, Phys. Rev. D , 091702 (2009) doi:10.1103/PhysRevD.80.091702 [arXiv:0908.1554 [hep-ph]].[84] H. Serodio, Phys. Lett. B , 133 (2011) doi:10.1016/j.physletb.2011.04.069 [arXiv:1104.2545 [hep-ph]].[85] I. de Medeiros Varzielas, Phys. Lett. B , 597 (2011) doi:10.1016/j.physletb.2011.06.042 [arXiv:1104.2601 [hep-ph]].[86] G. Aad et al. [ATLAS Collaboration], JHEP , 008 (2014) doi:10.1007/JHEP06(2014)008 [arXiv:1403.6293 [hep-ex]].[87] CMS Collaboration [CMS Collaboration], CMS-PAS-HIG-13-034.[88] C. S. Kim, Y. W. Yoon and X. B. Yuan, JHEP , 038 (2015) doi:10.1007/JHEP12(2015)038 [arXiv:1509.00491[hep-ph]].[89] T. P. Cheng and M. Sher, Phys. Rev. D , 3484 (1987). doi:10.1103/PhysRevD.35.3484[90] G. Cvetic, C. S. Kim and S. S. Hwang, Phys. Rev. D , 116003 (1998) doi:10.1103/PhysRevD.58.116003 [hep-ph/9806282].[91] D. Atwood, L. Reina and A. Soni, Phys. Rev. D , 3156 (1997) doi:10.1103/PhysRevD.55.3156 [hep-ph/9609279].[92] S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. , no. 17, 171802 (2014)doi:10.1103/PhysRevLett.112.171802 [arXiv:1312.4194 [hep-ex]].[93] G. Aad et al. [ATLAS Collaboration], Eur. Phys. J. C , no. 1, 12 (2016) doi:10.1140/epjc/s10052-015-3851-5[arXiv:1508.05796 [hep-ex]].[94] M. D. Campos, A. E. C´arcamo Hern´andez, H. P¨as and E. Schumacher, Phys. Rev. D , no. 11, 116011 (2015)doi:10.1103/PhysRevD.91.116011 [arXiv:1408.1652 [hep-ph]].[95] D. Aristizabal Sierra and A. Vicente, Phys. Rev. D , no. 11, 115004 (2014) doi:10.1103/PhysRevD.90.115004[arXiv:1409.7690 [hep-ph]].[96] J. Heeck, M. Holthausen, W. Rodejohann and Y. Shimizu, Nucl. Phys. B , 281 (2015)doi:10.1016/j.nuclphysb.2015.04.025 [arXiv:1412.3671 [hep-ph]].[97] I. de Medeiros Varzielas, O. Fischer and V. Maurer, JHEP , 080 (2015) doi:10.1007/JHEP08(2015)080[arXiv:1504.03955 [hep-ph]].[98] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D , no. 7, 072012 (2013) doi:10.1103/PhysRevD.88.072012[arXiv:1303.0571 [hep-ex]]. [99] M. Misiak et al. , Phys. Rev. Lett. , no. 22, 221801 (2015) doi:10.1103/PhysRevLett.114.221801 [arXiv:1503.01789[hep-ph]].[100] M. Trott and M. B. Wise, JHEP , 157 (2010) doi:10.1007/JHEP11(2010)157 [arXiv:1009.2813 [hep-ph]].[101] M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Sov. J. Nucl. Phys. , 711 (1979) [Yad. Fiz. ,1368 (1979)].[102] M. B. Gavela, G. Girardi, C. Malleville and P. Sorba, Nucl. Phys. B , 257 (1981). doi:10.1016/0550-3213(81)90529-0[103] P. Kalyniak, R. Bates and J. N. Ng, Phys. Rev. D , 755 (1986). doi:10.1103/PhysRevD.33.755[104] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, Front. Phys. , 1 (2000).[105] M. Spira, Fortsch. Phys. , 203 (1998) doi:10.1002/(SICI)1521-3978(199804)46:3¡203::AID-PROP203¿3.0.CO;2-4 [hep-ph/9705337].[106] A. Djouadi, Phys. Rept. , 1 (2008) doi:10.1016/j.physrep.2007.10.005 [hep-ph/0503173].[107] W. J. Marciano, C. Zhang and S. Willenbrock, Phys. Rev. D , 013002 (2012) doi:10.1103/PhysRevD.85.013002[arXiv:1109.5304 [hep-ph]].[108] L. Wang and X. F. Han, Phys. Rev. D , 095007 (2012) doi:10.1103/PhysRevD.86.095007 [arXiv:1206.1673 [hep-ph]].[109] A. E. Carcamo Hernandez, C. O. Dib and A. R. Zerwekh, Eur. Phys. J. C , 2822 (2014) doi:10.1140/epjc/s10052-014-2822-6 [arXiv:1304.0286 [hep-ph]].[110] G. Bhattacharyya and D. Das, Phys. Rev. D , 015005 (2015) doi:10.1103/PhysRevD.91.015005 [arXiv:1408.6133 [hep-ph]].[111] E. C. F. S. Fortes, A. C. B. Machado, J. Monta˜no and V. Pleitez, J. Phys. G , no. 11, 115001 (2015) doi:10.1088/0954-3899/42/11/115001 [arXiv:1408.0780 [hep-ph]].[112] A. E. Carcamo Hernandez, C. O. Dib and A. R. Zerwekh, Nucl. Part. Phys. Proc. , 35 (2015)doi:10.1016/j.nuclphysbps.2015.10.079 [arXiv:1503.08472 [hep-ph]].[113] A. E. C´arcamo Hern´andez, C. O. Dib and A. R. Zerwekh, arXiv:1506.03631 [hep-ph].[114] V. Khachatryan et al. [CMS Collaboration], Eur. Phys. J. C , no. 10, 3076 (2014) doi:10.1140/epjc/s10052-014-3076-z[arXiv:1407.0558 [hep-ex]].[115] G. Aad et al. [ATLAS Collaboration], Phys. Rev. D , no. 11, 112015 (2014) doi:10.1103/PhysRevD.90.112015[arXiv:1408.7084 [hep-ex]].[116] M. E. Peskin and T. Takeuchi, Phys. Rev. D , 381 (1992). doi:10.1103/PhysRevD.46.381[117] G. Altarelli and R. Barbieri, Phys. Lett. B , 161 (1991). doi:10.1016/0370-2693(91)91378-9[118] R. Barbieri, A. Pomarol, R. Rattazzi and A. Strumia, Nucl. Phys. B , 127 (2004) doi:10.1016/j.nuclphysb.2004.10.014[hep-ph/0405040].[119] S. Bertolini, Nucl. Phys. B , 77 (1986). doi:10.1016/0550-3213(86)90341-X[120] W. Hollik, Z. Phys. C , 291 (1986). doi:10.1007/BF01552507[121] W. Hollik, Z. Phys. C , 569 (1988). doi:10.1007/BF01549716[122] C. D. Froggatt, R. G. Moorhouse and I. G. Knowles, Phys. Rev. D , 2471 (1992). doi:10.1103/PhysRevD.45.2471[123] W. Grimus, L. Lavoura, O. M. Ogreid and P. Osland, Nucl. Phys. B , 81 (2008) doi:10.1016/j.nuclphysb.2008.04.019[arXiv:0802.4353 [hep-ph]].[124] H. E. Haber and D. O’Neil, Phys. Rev. D , 055017 (2011) doi:10.1103/PhysRevD.83.055017 [arXiv:1011.6188 [hep-ph]].[125] A. E. C´arcamo Hern´andez, S. Kovalenko and I. Schmidt, Phys. Rev. D , 095014 (2015) doi:10.1103/PhysRevD.91.095014[arXiv:1503.03026 [hep-ph]].[126] M. Baak and R. Kogler, arXiv:1306.0571 [hep-ph].[127] M. Baak et al. , Eur. Phys. J. C , 2205 (2012) doi:10.1140/epjc/s10052-012-2205-9 [arXiv:1209.2716 [hep-ph]].[128] M. Baak, M. Goebel, J. Haller, A. Hoecker, D. Ludwig, K. Moenig, M. Schott and J. Stelzer, Eur. Phys. J. C , 2003(2012) doi:10.1140/epjc/s10052-012-2003-4 [arXiv:1107.0975 [hep-ph]].[129] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Prog. Theor. Phys. Suppl. , 1 (2010)doi:10.1143/PTPS.183.1 [arXiv:1003.3552 [hep-th]].[130] W. Grimus and L. Lavoura, JHEP , 042 (2000) doi:10.1088/1126-6708/2000/11/042 [hep-ph/0008179].[131] C. Alvarado, R. Martinez and F. Ochoa, Phys. Rev. D , 025027 (2012) doi:10.1103/PhysRevD.86.025027[arXiv:1207.0014 [hep-ph]].[132] A. E. Carcamo Hernandez, R. Martinez and F. Ochoa, Phys. Rev. D87