Scalar and gauge sectors in the 3-Higgs Doublet Model under the S3-symmetry
SScalar and gauge sectors in the 3-Higgs Doublet Model under the S3-symmetry
M. Gómez-Bock ∗1 , M. Mondragón †2 , and A. Pérez-Martínez ‡21 Universidad de las Américas Puebla, UDLAP. Ex-Hacienda Sta. Catarina Mártir, Cholula,Puebla, México. Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, México01000 D.F., México.
February 19, 2021
Dedicated to the memory of Prof. Alfonso Mondragón,whose love of physics has been an inspiration to so many.
Abstract
We analyse the Higgs sector of an S model with three Higgs doublets and no explicitCP violation. After electroweak breaking there are nine physical Higgs bosons, one ofwhich corresponds to the Standard Model one. We study the scalar and gauge sectorsof this model, taking into account the conditions set by the minimisation and stabilityof the potential. We calculate the masses, trilinear and quartic Higgs self-couplings,and Higgs-gauge couplings. We consider two possible alignment scenarios, where onlyone of the three neutral scalars has couplings to the gauge bosons and corresponds tothe SM Higgs, and whose trilinear and quartic couplings reduce exactly to the SM ones.We also obtain numerically the parameter space allowed for the scalar masses in eachof the alignment scenarios. With this information we are able to analyse the structureof the one-loop scalar and gauge corrections to the neutral scalar mass matrix. Weshow that a residual Z symmetry is preserved at NLO, keeping the decoupling of oneof the neutral scalars, and a possible mixing between the other two. Small values forthe off-diagonal element favour a large mixing between the S singlet and doublet. The discovery the Higgs boson with a mass of
GeV [1, 2], and the experimental studyof its properties, will be of great relevance to understand its couplings with other particles,specifically the flavour problem and ways to address it. ∗ [email protected] † myriam@fisica.unam.mx ‡ [email protected] a r X i v : . [ h e p - ph ] F e b he organization of the fermions into generations or families may signal a possible under-lying structure in elementary particles, although its origin or nature is not yet understood.On the other hand, the Standard Model (SM) Higgs mechanism [3, 4], which is indispensableto understand the origin of the masses of gauge bosons and fermions, sheds no light on theflavor structure or the difference in the masses of the fundamental fermions.The flavour structure of fermions has been the subject of a great amount of researchthroughout the years. In view of the fact that the only difference between generations in thefermionic sector is the mass of the particles, the most direct or even natural way to proposea flavour structure is through the mass generation mechanism, the Higgs sector.One possibility to understand the flavour nature behind the SM is to construct an ex-tended scalar sector with a flavour symmetry, where the SM is embedded. Multi-Higgsextensions of the SM, with and without extra symmetries, have been extensively studied,some diverse examples are given in [5–15] (for reviews on two Higgs Doublet Models (2HDM)and multi-Higgs models see [16, 17]). Flavour symmetries can be either continuous or dis-crete. In this context discrete symmetries have been extensively studied in particle physics(for reviews of models with discrete symmetries see [18–21]). Since these models require theaddition of more Higgs fields, the phenomenological consequences in all sectors, like allowedextra processes and couplings, have to be analysed, some examples are found in [22–24].Restrictions are placed on the models by confronting their phenomenology with the exper-imental results, in this case the results of ATLAS [25] and CMS [26]. The Higgs sector isthus crucial to determine the viability and prospects of each model. Prime examples of thisprocedure are the Minimal Supersymmetric Standard Model (MSSM) and the 2HDM (seefor instance, [27–29] and [16, 30, 31] respectively).The permutation group of three objects S , with three Higgs doublets, has been proposedalready a long time ago [32–36] as a natural extension of the SM, even before all the quarkswere discovered. Since then, the S symmetry has been extensively studied in differentsettings, due to its simplicity and predictivity, see for instance [37–54]. In the leptonicsector, the S symmetry has also been previously used in different settings to calculateneutrino masses and mixing with interesting results [55–72]. Some studies of dark mattercandidates in models with S symmetry can be found in [73–77].In particular, the S model with 3 Higgs doublets (S3-3H), which we study here, hasled to very interesting results, unravelling the SM immersed in it. Our motivation in S3-3H is that the mass hierarchy between the third and the first two generations, as wellas the structure of the V CKM mixing matrix suggest that the fermions belong to a + irreducible representation (irrep) of a flavour symmetry. It is not possible to give masses toall three generations without breaking explicitly the S symmetry and/or adding extra Higgsdoublets. With one Higgs doublet, an explicit sequential breaking has been analysed [34,38].In the quark sector, it was shown that it is possible to obtain the Fritzsch textures and theNearest Neighbour Interaction (NNI) [43], thus fitting the CKM matrix. In the leptonicsector it was found that the S3-3H model can also reproduce the V P MNS matrix and predictsa non-vanishing θ reactor mixing angle, and some flavour changing neutral current andcontributions to g − were calculated [59–61, 66]. In [75], a version of the S3-3H plus anextra inert Higgs doublet was analysed, with the interesting result that it is possible to havea good dark matter (DM) candidate, coming from the inert sector and satisfying also theHiggs bounds, the indirect prospects of detection of this DM candidate have been studied2n [77].The scalar sector is also interesting, since there is an economy of parameters compared toa more generic 3HDM [41]. The vacuum stability has been studied in [78, 79], and the massstructure of the scalar bosons has been analysed in [46, 47]. In [46] it was found that thereis a residual Z symmetry after the electroweak symmetry breaking (EWSB) in the Higgspotential, and the corresponding charges for the scalars under this symmetry were given.The conditions for having spontaneous CP violation are presented in [50].In here, we keep the model as simple as possible, by not assuming an explicit breakingof the flavour symmetry or adding extra flavons. We calculate the scalar masses, and thetrilinear and quartic Higgs self-couplings and Higgs-gauge boson couplings. We considertwo possible alignment scenarios for the SM-like Higgs boson, where only one of the threeneutral scalars has couplings to the vector bosons (one is always decoupled due to the Z symmetry). We use a geometrical parameterization in spherical coordinates, which allows usto express the mixing of the vacuum expectation values ( vevs ) of the Higgs fields in the S singlet and doublet irreducible representations, in terms of one angle ( θ ) in our expressions.We scan the parameter space, taking into account the unitarity and stability conditions, andthe SM Higgs boson mass constraints, in each of the two alignment scenarios.Some of the trilinear scalar couplings have been obtained in [47], nevertheless we founddifferences with their results. Mainly in [47], the Z symmetry is not exhibited, whereaswe find it explicitly in our calculations, consistent with the results of [46]. As an additionalresult, we find that in each of the alignment limits, where only the SM-like Higgs couplesto the vector bosons, the trilinear and quartic couplings reduce exactly to the SM ones. Weshow that the one-loop neutral CP-even Higgs mass matrix is block diagonal, where the thescalar and gauge contributions derived from the calculated trilinear and quartic couplingsare taken into account. Although it reduces to a structure similar to the 2HDM mass matrix,the presence of the extra neutral scalar makes it possible to distinguish between the models.We also give two benchmark mass points in which the one-loop radiative corrections to themasses, coming from mixed scalar terms, vanish.The paper is organized as follows: in the next section we describe the model, and how the S symmetry acts on the Higgs electroweak doublets, giving the structure and characteristicsof the S3-3H Higgs potential. In Section 3, we parameterize the vacua and rotate to the Higgsbasis, to express our results in terms of physical parameters. We then calculate the tree levelmasses and explore numerically the two different alignment scenarios. Then, in Section 4, wecalculate the Higgs self-couplings and the Higgs-gauge bosons couplings; we present the oneswith CP-even neutral Higgs bosons in this section, and we complete with the CP-odd andcharged Higgs boson couplings in the Appendix. We also analyse the structure of the one-loop neutral CP-even Higgs mass matrix. Finally, we present a summary and the conclusionsof our work. The S is the smallest non-Abelian discrete group, and it corresponds to the permutationsof three objects or the symmetries (rotations and reflections) of an equilateral triangle. Itsirreducible representations (irrep) are a symmetric singlet , an anti-symmetric singlet A [40].A two-dimensional matrix representation, D i , of S can be obtained from the followingmatrices D + ( θ ) = cos θ sin θ − sin θ cos θ and D − ( θ ) = − cos θ sin θ sin θ cos θ , (1)with θ = 0 , ± π/ , where det D ± = ± .The field content of the model consists of the usual SU (2) doublets and singlets for SMquarks and leptons as well as three Higgs doublets and three right-handed neutrinos. We willassume all fields to belong to a reducible + representation of S , where all the fields inthe first two generations will be in a doublet, and the ones in the third generation will be inthe symmetric singlet. The same assignment follows for the three Higgs doublets discussedhere, and for the right-handed neutrinos, although we will only concentrate on the scalarsector in this work.We also consider here, the scalar kinetic structure of the Lagrangian through the covariantderivative of the scalar fields. It is important to analyse the covariant derivative for scalardoublets in order to verify not only the EWSB mechanism i.e. the contributions of vevs tothe gauge masses, but also the possible couplings among the Higgs and gauge bosons. Thekinetic terms are taken as usual L kin = | D µ φ | + N (cid:88) a =2 | D µ ψ a | . (2)In section 3.3, we write the kinetic Langrangian explicitly in the Higgs basis, whereas insection 4, we explicitly get the gauge-Higgs bosons (CP-even) couplings. By means of com-pleteness, we give all the couplings of gauge bosons with charged Higgs bosons in A.3. As with the Yukawa part of the Lagrangian, the terms in the potential are the ones thatpreserve the discrete S permutational symmetry, as reported in [35, 41]. The most generalHiggs potential invariant under the SU (3) c × SU (2) L × U (1) Y × S in the symmetry adaptedbasis is given as V = µ (cid:16) H † H + H † H (cid:17) + µ (cid:0) H † s H s (cid:1) + a (cid:0) H † s H s (cid:1) + b (cid:0) H † s H s (cid:1) (cid:16) H † H + H † H (cid:17) + c (cid:16) H † H + H † H (cid:17) + d (cid:16) H † H − H † H (cid:17) + ef ijk (cid:16)(cid:0) H † s H i (cid:1) (cid:16) H † j H k (cid:17) + h.c. (cid:17) + f (cid:110)(cid:0) H † s H (cid:1) (cid:16) H † H s (cid:17) + (cid:0) H † s H (cid:1) (cid:16) H † H s (cid:17)(cid:111) + g (cid:26)(cid:16) H † H − H † H (cid:17) + (cid:16) H † H + H † H (cid:17) (cid:27) + h (cid:110)(cid:0) H † s H (cid:1) (cid:0) H † s H (cid:1) + (cid:0) H † s H (cid:1) (cid:0) H † s H (cid:1) + (cid:16) H † H s (cid:17) (cid:16) H † H s (cid:17) + (cid:16) H † H s (cid:17) (cid:16) H † H s (cid:17)(cid:111) ; (3)4here f = f = f = − f = 1 . This same potential has also been analysed inRefs. [46, 49, 78, 79] without CP violation, and in Ref. [50] with spontaneous CP violation.In the following, we will analyse the complete structure of this scalar sector, in particularthe couplings of the Higgs bosons with the physical states to understand further the possiblephenomenology of different scenarios.The S Higgs potential of this model is constructed with the three SU (2) complex Higgsdoublets, each one assigned to a generation. The ones in the first two generations H , H are in the doublet S irrep and the one in the singlet irrep, H s , is assigned to the thirdgeneration. In terms of complex fields we express them as H = 1 √ φ + iφ φ + iφ , H = 1 √ φ + iφ φ + iφ , H s = 1 √ φ + iφ φ + iφ . (4)In order to simplify the calculations, we introduce the following variables as was donein [78, 79] x = H † H , x = Re ( H † H ) , x = Im ( H † H ) ,x = H † H , x = Re ( H † H s ) , x = Im ( H † H s ) ,x = H † s H s , x = Re ( H † H s ) , x = Im ( H † H s ) . (5)As an example, we give here some explicit real terms of the scalar fields in the potential,with the appropriate normalization factors x = H † H = 12 ( φ + φ + φ + φ ) ,x = Re ( H † H ) = 12 ( φ φ + φ φ + φ φ + φ φ ) ,x = Im ( H † H ) = 12 ( φ φ − φ φ + φ φ − φ φ ) . (6)Hence, using (5) into (3), the Higgs potential is expressed as: V = µ ( x + x ) + µ x + a x + b ( x + x ) x + c x + x ) − dx + 2 e [( x − x ) x + 2 x x ] + f ( x + x + x + x )+ g (cid:2) ( x − x ) + 4 x (cid:3) + h ( x + x − x − x ) . (7)From this general potential we have ten free parameters, before EWSB. Then, once the fieldsacquire vevs , we may relate them with the new variables (5) as (cid:104) x l (cid:105) = v l , for l = 1 , , , (cid:104) x (cid:105) = v v , (cid:104) x (cid:105) = v v , (cid:104) x (cid:105) = v v , and (cid:104) x (cid:105) = (cid:104) x (cid:105) = (cid:104) x (cid:105) = 0 . (8)We give more details of the EWSB process in next section.5 .2 The normal minimum In order to have a consistent Higgs potential, it is necessary to check that it is stable, i.e.bounded from below, and that it respects perturbative unitarity. These requirements imposeconstraints on the potential’s parameters. This analysis has already been done in [46], andwe use their expressions for the unitarity and stability bounds in here.A study of the stability of the different minima for a general Higgs potential of this kindcan be found in [78], they point out the existence of three types of minima or stationarypoints. In here, we will consider the EWSB using the natural choices of conservation ofelectric and CP charges, implying that only the real part of the neutral fields will acquire vevs , this is called the normal minimum . Thus, only the real parts of each one of the doubletswill acquire non-zero vacuum expectation values. Expressed in terms of the field componentsof H , H , H s , Eq.(4) we have (cid:104) φ (cid:105) = v , (cid:104) φ (cid:105) = v , (cid:104) φ (cid:105) = v , (cid:104) φ i (cid:105) = 0 , i (cid:54) = 7 , , , (9)this adds two more free parameters to the model as they should satisfy the condition (cid:113) v + v + v = v = 246 GeV . (10)The extreme point conditions for the potential are given by ∂V∂v i = 0 ←→ ∂V∂x j ∂x j ∂v i = 0 , (11)with i = 1 , , j = 1 , , ..., . These conditions express the tree level tadpole equations as µ + ( b + f + h ) v + ( c + g )( v + v )] v + 6 ev v v , (12) µ + ( b + f + h ) v + ( c + g )( v + v )] v + 3 e ( v − v ) v , (13) µ + ( b + f + h )( v + v ) + av ] v + e (3 v − v ) v . (14)These equations reduce further the original twelve free parameters relating two of them as v = 3 v . (15)Another possible solution that satisfies these equations would be e = 0 [40,79], which impliesthe presence of a Goldstone boson due to a residual SO (2) symmetry, but this scenario willnot be considered for the present work. In order to get the tree level masses of the Higgs bosons, it is necessary diagonalize the × matrix resulting from taking the second derivatives of the potential ( M H ) ij = ∂ V∂φ i ∂φ j (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) φ i (cid:105) , (16)6ith i, j = 1 , ..., . Due to the symmetry of the model, the mass matrix consists of fourdiagonal blocks, each one a × Hermitian and symmetric matrix. The Higgs mass matricesof the S − H have been reported previously in [46, 47], nevertheless our results differ from[47], by a factor of two in the Higgs couplings, because we have included the normalizationfactors / √ in the Higgs doublets. In this work, we will study the more general case with e (cid:54) = 0 , with a new parametrization which allows us to compare directly with the SM whenwe include the complete scalar couplings and scalar-gauge couplings. The expressions forthese couplings in general, as well as in the alignment limits are calculated in section 3.4.We obtain a block diagonal matrix, with four × Hermitian and symmetric submatricesfor the tree level masses, as it is expected since we do not consider CP violation. In section4.2, we also calculate the analytical expressions using the physical parameters for the Higgstrilinear couplings.The first two matrix blocks, M C , are the same for the charged scalars masses. We alsoget the matrix for the neutral scalars bosons (CP-even) M S and the matrix for the neutralpseudoscalar bosons masses M A (CP-odd), M H = diag ( M C , M C , M S , M A ) . (17)The matrix elements of the charged Higgs masses in terms of the potential parametersare given as M C = c c c c c c c c c , (18)with the elements of the symmetric mass matrix given as c = − v [2 ev + v ( f + h )] − gv , c = √ v ( ev + gv ) ,c = √ v [ ev + v ( f + h )] , c = − v [4 ev + v f + h )] − gv ,c = v [ ev + v ( f + h )] , c = − v [2 ev + v ( f + h )] v . (19)The mass matrix for neutral CP-even Higgs bosons is given by M S = s s s s s s s s s , (20)where, the elements of the scalar symmetric mass matrix are s = 3 v ( c + g ) , s = √ v [ v ( c + g ) + 3 ev ] ,s = √ v [ v ( b + f + h ) + 3 ev ] , s = v [ v ( c + g ) − ev ] ,s = v [3 ev + ( b + f + h ) v ] , s = ( av − ev ) v . (21)7or the CP-odd Higgs bosons mass matrix we find M A = a a a a a a a a a , (22)where each of the elements of the symmetric matrix are given as a = − ( v ( d + g ) + 2 ev v + hv ) , a = √ v ( v ( d + g ) + ev ) ,a = √ v ( ev + hv ) , a = − v ( d + g ) − ev v − hv ,a = v ( ev + hv ) , a = − v ( ev + hv ) v . (23) We will rewrite the vevs in spherical coordinates, as it was done in [75, 80] v = v cos ϕ sin θ, v = v sin ϕ sin θ, v = v cos θ. (24)The use of this spherical parameterization is helpful to visualize the relation among the vevs . The angle θ gives the amount of mixing between the vev of the singlet and the vevs ofthe doublets. We may obtain a relation between v , v , and v as tan ϕ = v v , tan θ = v v sin ϕ . (25)Moreover, the minimization condition of the potential (15), provides an extra constraintfor the relation between v and v i.e. it fixes also the value of ϕ . We assume all the vevs to be real and positive (otherwise we should consider a phase between two vevs ) implying ϕ = π/ , then tan ϕ = √ , thus we get tan ϕ = 1 / √ ⇒ sin ϕ = 12 , cos ϕ = √ , (26) tan θ = 2 v v ⇒ sin θ = 2 v v , cos θ = v v . (27) The usual form for the rotation matrix R i , to obtain the mass matrix and physical states isgiven as: [ M diag ] i = R Ti M I R i , i = s, a, c, I = S, A, C. (28)For the Higgs bosons we take the sub-indices i = s, a, c to refer to the neutral, pseudoscalarand charged Higgs bosons respectively. The rotation matrix is the product of two rotations,i.e., R i = A B i , where 8 = cos δ − sin δ δ cos δ
00 0 1 , B i = cos γ i γ i − sin γ i γ i , (29)then R i = cos γ i cos δ − sin δ sin γ i cos δ cos γ i sin δ cos δ sin γ i sin δ − sin γ i γ i . (30)The rotation matrix R a,c , which diagonalizes M A and M C , will transform the fieldsleading to the Goldstone states. They are given as follows: R a,c = cos γ a,c cos δ − sin δ sin γ a,c cos δ cos γ a,c sin δ cos δ sin γ i sin δ − sin γ a,c γ a,c = √ v v − − √ v vv v √ − v vv v v v . (31)Therefore, we can see that cos γ a,c = v v , sin γ a,c = − v v , sin δ = , and cos δ = √ . If wecompare with (26) and (27) we can see that δ = ϕ and γ a,c = π + θ . We will reparameterizethe matrices in terms of the angles θ and ϕ , so that the matrices take the following form R a,c = sin θ cos ϕ − sin ϕ − cos θ cos ϕ sin θ sin ϕ cos ϕ − cos θ sin ϕ cos θ θ = √ v v − − √ v vv v √ − v vv v v v . (32)Thus, we obtain the respective masses at tree level for the pseudoscalar and chargedHiggs bosons as m A = − v (cid:20) ( d + g ) sin θ + 54 e sin 2 θ + h cos θ (cid:21) , (33) m A = − v ( e θ + h ) , (34) m H ± = − v (cid:2) e sin 2 θ + 2( f + h ) cos θ + 4 g sin θ (cid:3) , (35) m H ± = − v e tan θ + ( f + h )] . (36)From these expressions, it can be seen that, when the value of tan θ is high, some masseswill be naturally heavy. 9or the diagonalization of the mass matrix of the neutral scalar bosons M S , we will havethe following rotation matrix: R s = cos γ s cos δ − sin δ sin γ s cos δ cos γ s sin δ cos δ sin γ i sin δ − sin γ s γ s . (37)In terms of the parameters of the potential, considering also the spherical parameterization,we have M a = (cid:20) ( c + g ) v sin θ + 34 ev sin 2 θ (cid:21) ,M b = (cid:2) ev sin θ + ( b + f + h ) v sin 2 θ (cid:3) ,M c = av cos θ − ev tan θ sin θ, (38)where the mixing angle α is tan 2 α = − M b M a − M c . (39)The rotation to obtain the mass matrix directly from the flavour basis is given as R s = √ ( M a − M c + Z M ) √ ( M b ) +( M a − M c + Z M ) − − √ M b √ ( M b ) +( M a − M c + Z M ) M a − M c + Z M √ ( M b ) +( M a − M c + Z M ) √ − M b √ ( M b ) +( M a − M c + Z M ) M b √ ( M b ) +( M a − M c + Z M ) M a − M c + Z M √ ( M b ) +( M a − M c + Z M ) , (40)where Z M = (cid:113) ( M b ) + ( M a − M c ) . Using again Eqs. (39) and. (26) we get tan γ s = − M b ( M a − M c + Z M ) , γ s = π + α, sin δ = 12 and cos δ = √ . (41)We will work with the angles α and ϕ , for that reason we express the rotation matrix inthe following form R s = sin α cos ϕ − sin ϕ − cos α cos ϕ sin α sin ϕ cos ϕ − cos α sin ϕ cos α α . (42)Then, we may write the scalar Higgs bosons masses as follows: m h = − ev sin 2 θ, (43)10 H ,H = 12 (cid:20) ( M a + M c ) ± (cid:113) ( M a − M c ) + ( M b ) (cid:21) , (44)we notice that e < . We see here that the structure of the masses is consistent with the onefound in Refs. [46, 79]. The expressions for m H , can be written in terms of the parametersof the model as m H = v (cid:2) ac θ c α + 2( c + g ) s α s θ + 4( b + f + h ) s α s θ c α c θ + et θ (6 s α c α s θ c θ + 3 c θ s α − s θ c α ) (cid:3) , = v (cid:8) a ( c α − θ + c α + θ ) + ( c + g )( c α − θ − c α + θ ) + ( b + f + h )( c α − θ ) − c α + θ ) )+ e t θ (cid:2) c α − θ ) − c α + θ ) + 4 s α + θ s α − θ + 2 s α + θ ) (cid:3)(cid:9) , (45) m H = v (cid:0) ac θ s α + 2( c + g ) c α s θ − b + f + h ) s α s θ c α c θ + et θ ( − s α c α s θ c θ + 3 c θ c α − s θ s α ) (cid:1) , = v (cid:8) a ( s α − θ + s α + θ ) + ( c + g )( s α + θ − s α − θ ) − ( b + f + h )( c α − θ ) − c α + θ ) )+ e t θ (cid:2) c α + θ ) − c α − θ ) + 4 c α + θ c α − θ + 2 c α + θ (cid:3)(cid:9) , (46)here we use the reduced notation for the trigonometric functions: s x ≡ sin x , c x ≡ cos x and t x ≡ tan x .The SM Higgs boson has already been measured [25, 26], and one of the neutral CP-evenHiggs of the model should correspond to it. Thus, it is important to explore the structureof these tree level masses in terms of the couplings of the Higgs potential in the interactionbasis, Eq. (7), and also in terms of the mixing angles relating the vevs , Eqs. (25). Thepossibility of a neutral Goldstone, a massless degree of freedom has been reported previouslyin [40, 79] when e = 0 is considered, and leads to v = v .From Eqs. (38), (43) and (44), we can reduce the expressions of the masses for specificcases of the parameter θ which, as we said before, gives the amount of mixing between the vev of the singlet and the vevs of the doublets. We explore a particular case, for instance θ = π/ , where we obtain the neutral CP-even Higgs masses as m h = − ev , (47) m H , = v (cid:34) a + c + e + g ± (cid:112) ( − a + c + 2 e + g ) + (3 e + 2( b + f + h )) (cid:35) , (48)where the three masses are proportional to the vev , v = 246 GeV, and combinations of theself-couplings. An extreme case is given when sin θ = 0 , then we will obtain two masslessGoldstone bosons, m h = 0 = m H , and only one neutral Higgs bosons m H = av . Anotherextreme case would be when cos θ → , from this approximation we obtain one almostmassless neutral scalar m h ≈ , for the other two neutral bosons we have m H ≈ v ( c + g ) and m H ≈ v ( c + g − e tan θ ) . In this extreme case, the dominant term is tan θ so, if we11ompare the three neutral masses we have one almost massless boson and a very heavy one.Thus, if we want a realistic scenario with no massless scalars, there must be an admixtureof the doublet and the singlet Higgs fields.In the following section, we will analyse the masses for the general cases of non-zeroparameter values. Specifically, for the numerical analysis, we explore the tree level massesfor different values of the θ parameter, in the range < θ < π/ , to keep the vevs positive. From the perspective of both EWSB and flavor physics, there is a basis that is particularlyuseful to compare with the SM or with other of its extensions, the so-called Higgs basis. Itis defined as the basis in which one Higgs field carries the full vev , φ vev , and the other Higgsfields ψ , ψ are perpendicular to it [13, 17, 81, 82]. In order to get the Goldstone bosons,which are needed for the generation of masses of the gauge bosons, we do the usual rotation.For multi-Higgs models, the Goldstone bosons are obtained with the same rotation anglefor both the CP-odd and charged Higgs bosons, and as we found in the previous section γ a = γ c = π + θ . The fields in the Higgs basis are then given by the transformation: φ vev ψ ψ = R Ta H H H s = sin θ cos ϕ sin θ sin ϕ cos θ − sin ϕ cos ϕ − cos θ cos ϕ − cos θ sin ϕ sin θ H H H s . (49)In our case, the rotation matrix takes the following form φ vev ψ ψ = √ v v v v v v − √ − √ v v − v v v v H H H s . (50)Then, the electroweak (EW) Higgs doublets in this basis are explicitly given as φ vev = G ± √ ( v + (cid:101) H + iG ) , ψ = H ± √ ( (cid:101) H a + iA ) , ψ = H ± √ ( (cid:101) H b + iA ) . (51)The rotation matrix above, (50), corresponds to the matrix built for the charged andpseudoscalars Higgs bosons mass eigenstates, denoted by H ± , H ± , G ± , A and A . Whereas,the neutral part of the Higgs doublets, denoted by (cid:101) H, (cid:101) H a y (cid:101) H b do not correspond to theirmass eigenstates, but they are in the Higgs basis. Now, in order to diagonalize the CP-evensector we rotate through the angle α , obtaining the physical states for neutral scalars as (cid:101) H (cid:101) H a (cid:101) H b = cos( α − θ ) 0 sin( α − θ )0 1 0 − sin( α − θ ) 0 cos( α − θ ) H h H . (52)12e can get the neutral CP-even Higgs physical states from either, the direct rotationEq.(40), which transforms interaction basis to physical basis; or from a two step rotation,from the interaction basis to the Higgs basis Eq.(50) and then to the physical basis Eq.(52).Either way we obtain the scalar Higgs masses given in Eqs. (43) and (44). We can seefrom expression (52), that there would be two alignment scenarios: A) Where ˜ H = H corresponds to the SM-like Higgs boson; whereas scenario B) will set ˜ H = H correspondingto the SM-like Higgs boson. These scenarios will be explored in next subsection.Neutral CP-even Neutral CP-odd Charged h odd ˜ H even ˜ H b even A odd A even H ± odd H ± evenTable 1: Z charged assignment for the intermediate states in the S3-3H model. In thealignment limit these will correspond to the physical states.Using the Higgs physical basis, we obtain the electroweak gauge bosons masses, W ± and Z , as well as their couplings with Higgs bosons, including the ones with A µ , performing thecanonical rotation with the weak angle. Using the covariant derivative for the kinetic termEq. (2) in the Higgs basis, and after expanding the Higgs field about the vacuum, we getthe Lagrangian in the physical basis. We show here some of the terms for illustration, thecomplete set of explicit couplings are given in next section and in Appendix A. L kin ≈ g v W + µ W − µ + g v sin( α − θ )2 H W + µ W − µ + g H H W + µ W − µ + g v cos( α − θ )2 H W + µ W − µ + g H H W + µ W − µ + g h h W + µ W − µ + ... ++ ( g + g (cid:48) ) v Z µ Z µ + ( g + g (cid:48) ) v sin( α − θ )4 H Z µ Z µ + ( g + g (cid:48) )8 H H Z µ Z µ + ( g + g (cid:48) ) v cos( α − θ )4 H Z µ Z µ + ( g + g (cid:48) )8 H H Z µ Z µ + ( g + g (cid:48) )8 h h Z µ Z µ + ... (53)Then, the masses for the EW gauge bosons W ± and Z are obtained in the usual form m W ± = g v , m Z = v ( g + g (cid:48) ) . (54)Furthermore, as the model has two different charged Higgs bosons H ± , H ± , we explicitlyverified that mixed charged Higgs and gauge bosons couplings do not appear (e.g. H +1 H − γ )as it should be in order to preserve the Z symmetry. We show it by calculating explicitlythe part of the Lagrangian for the photon L H + H − γ = igg (cid:48) (cid:112) g + g (cid:48) (cid:16) H +1 ∂ µ H − − H − ∂ µ H +1 + H +2 ∂ µ H − − H − ∂ µ H +2 (cid:17) A µ (55)13his coupling is similar to what appears in other multi-Higgs models with a specific Z symmetry [83], and more specifically for two Higgs doublet models as in [27]. This is exhibitedimplicitly in [46] as they calculate h SM → γγ through a loop of charged Higgs bosons.In section 4.1 we give the explicit form of the gauge-scalar couplings of the H and H neutral Higgs bosons of the model, in order to compare with SM couplings scenarios. Therest of the couplings, for the extended scalar sector, will be given in appendix A.3, where wecan see the manifestation of the Z symmetry as it allows only certain couplings. As we analysed in section 3.2, we can have extreme cases where sin θ = 0 or cos θ → , wherewe would have two and one massless bosons, respectively. From the tree level Higgs massexpressions, we can see that the masses depend on the three vevs , i.e. all three depend on θ . The extreme scenarios for θ will have severe consequences for the masses in other sectorsas well, e.g. in the Yukawa sector, we are preparing work on this direction.The neutral Higgs boson h is decoupled from the other two, due to the residual symmetry Z , as was reported already in [46]. In the next section, we explicitly calculate all possibletree level couplings among the scalars and also, between the scalars and the gauge bosons.We show that, as expected, due to the Z symmetry, h couples only in even numbers to thegauge bosons Eq.(53). Also, its trilinear scalar coupling is absent, as we will see in section 4.2,Eqs.(58) and (73), so it is immediately excluded as the SM-like Higgs boson. Nevertheless,this neutral Higgs boson could be interesting as a possible Dark Matter candidate, since itbehaves similarly to an inert Higgs boson.The above discussion leaves us with two possible scenarios for which, either H or H isaligned to have the mass and couplings of the SM Higgs boson, and the other one would bepractically decoupled from the gauge bosons. We consider both scenarios for the numericalanalysis.Scenario A is defined by setting H , which has the lower mass among H and H , as theSM Higgs. We further restrict the tree level value of its mass to be in the range m H ∈ [120 , , taking into account that it will receive radiative corrections. On the other hand,scenario B takes the heavier Higgs boson H as the SM one, with its mass restricted to thesame interval as in the previous scenario. For these two scenarios, the alignment meansthat the SM-like boson is maximally coupled to the gauge bosons, while the other one ispractically decoupled. The alignment of the CP-even scalar boson is discussed in modelswith extended scalar sector, particularly for 3HDM in [84], as can be derived from Eq.(52).A third, less natural case, would be a non-alignment scenario, where both Higgs bosonswould couple equally or similarly to the gauge bosons. This analysis would be more complex,and a way to establish the non-observation of the second neutral Higgs would be needed, wewill not consider that possibility here. As already mentioned, in scenario A we set H to be the SM-like Higgs boson with mass ∼ GeV. The alignment limit, given in Ref. [84], can be seen explicitly in our case from14q. (52) and corresponds to sin( α − θ ) = 1 and cos( α − θ ) = 0 . (56)In this scenario, H couples maximally to the gauge bosons, and H is decoupled from thegauge bosons. The third neutral scalar, h , is always decoupled from the gauge bosons dueto the Z symmetry. A study of the masses in this scenario has been performed in [46], butwith slightly different considerations as the ones taken here, as we explain below. In scenario B, we take H as the SM-like Higgs boson, coupled maximally to the gaugebosons. The alignment limit in this scenario changes, and is expressed as sin( α − θ ) = 0 and cos( α − θ ) = 1 . (57)Although H is always lighter than H , as can be seen from the expressions for the massesEqs.(45) and (46), it is decoupled from the gauge sector in this case. Scenario B assumesthat the heavier among the two Z even neutral Higgs bosons, is the SM-like one, so wehave a lighter Higgs boson that could be hidden from experimental detection. This scenariohas not been analysed in the S3-3H model before. This could be interesting in the contextof a possible Higgs decay of an exotic scalar with mass m Φ = 96 GeV, as reported byCMS [85]. This possibility, of a second lighter Higgs scalar consistent with this signal, hasbeen explored in SUSY models in [86]. There are also recent analyses along these lines in2HDM and N2HDM [87–89].
We perform a scan on the eight self-couplings and tan θ , Eqs. (3) and (27). We produce O (10 ) points with a pseudo-random generator, on these we first apply the stability andunitarity constrains as given in Ref. [46], to calculate the masses, and then we take out allpoints where the charged Higgs scalar masses are below GeV [90, 91]. On the survivingpoints, we apply the alignment constraints in both A and B scenarios. Finally, we impose arestriction on the mass of the respective SM-like Higgs boson. A similar analysis on scenarioA has been performed in [46], but in their analysis they restricted the mass of h to bealways heavier than m H (the SM-like Higgs). Another difference is that we have applied thealignment limit within an approximation, to allow for the possibility of a minimal couplingto the non-SM Higgs, and we also allow for a range of masses for the SM-like Higgs boson.On the other hand, scenario B has not been analysed before.The alignment limits, Eqs. (56) and (57) we apply with a and . uncertainty onthe ( α − θ ) values, i.e. , ± . , ± . , respectively. In the figures we show only the resultswith uncertainty. The restriction to an alignment limit with . uncertainty affectsonly the H and H ± masses in scenario A, as explained below. We take the SM-like Higgsmass in the range ∈ [120 , GeV to take into account the theoretical uncertainties.In Figure 1 we show the dependence on the neutral scalar h with tan θ , Eq. (43), forboth scenarios. The blue points correspond to the unitary and stability constraints only, the15lack points are the ones that satisfy also the respective alignment limit, A in the left paneland B in the right panel. Finally, the gray points are the ones in which the SM-like Higgsboson mass has been restricted to the range given above. The graph from the right showsthat in scenario B, where H is the SM-like Higgs, there are fewer points which comply withall restrictions as compared to scenario A. Also, the allowed upper bound for m h ∼ GeV, is lower than in scenario A, where m h (cid:46) GeV.Figure 1: Naturally decoupled neutral Higgs h tree level mass dependence on tan θ . Theblue points comply with the unitarity and stability conditions, the black points complyfurther with the alignment conditions in each scenario. Finally, the gray points show therestriction on the parameters from constraining the SM-like Higgs mass to − GeV.In Figure 2 we show the other two CP-even scalar masses dependence on tan θ , Eqs. (45,46).In the upper panel we show the mass of the SM-like Higgs boson in each scenario, the ma-genta points correspond to the points that pass the unitarity and stability constraints (alsoexcluding charged Higgs boson masses below 80 GeV), the purple ones comply further withthe alignment limit in each scenario, and the green ones have the mass restricted in the [120 , GeV range. The left graph corresponds to scenario A, with H the SM-like Higgsboson, and the right one to scenario B, with H the SM-like Higgs boson. It can be seenfrom the figures that the alignment limit restricts the higher possible values for the massesin each scenario, to ∼ and ∼ GeV for scenarios A and B, respectively. In scenario B,the points that are in the SM Higgs mass range are right at the lower limit of the alignmentconstraint. The figures show an alignment constraint with a 1% precision. The results with0.1% precision do not change significantly the allowed parameter region, basically they coverthe same area.The lower panel of Figure 2, shows the mass of the third neutral CP-even Higgs boson,the one that is decoupled from the gauge sector after the alignment limit is taken, withthe same colour code for the restrictions as in the upper panel. The left and right graphs,scenarios A and B respectively, show different mass scales to make more evident how thealignment limit cuts drastically the upper bound for the allowed values for the H mass inscenario A. As shown in the graphs, in the alignment limit at , the allowed mass for H isrestricted from m H ∼ GeV to ∼ GeV in scenario A (lower panel left). The resultswith . precision affect only H in this scenario, where they reduce the upper bound to16igure 2: Masses for the Higgs neutral bosons H , dependence on tan θ . The SM-like Higgsin each scenario is shown in the upper panel, and the non-SM Higgs boson in the lower panel.The magenta points comply with the stability and unitarity conditions, the purple ones withthe alignment limit in each case, and the green ones have the SM-like mass restricted to m H , ∈ [120 , , respectively for scenario A (left) and B (right).17 H (cid:46) GeV. In scenario B, the constraint on H to the SM Higgs mass range allows onlyvalues of m H ∼ . GeV to ∼ GeV, unaffected by the alignment precision. In this case,since this Higgs boson is basically decoupled from the gauge bosons it could have escapedexperimental detection.Figure 3: Dependence on tan θ of the CP-even neutral Higgs bosons masses, complying withthe three constraints mentioned in previous figures, for both scenarios. The upper panelshows no direct restriction on the Higgs mass, whereas in the lower panel the SM-like Higgsmass is restricted to [120 , GeV. The dark gray/black points correspond to m h , thepurple ones to m H and the orange ones to m H .In Figure 3, we show all three neutral Higgs scalar masses that comply with the unitarity,stability, and alignment constraints. The upper panel shows the allowed points before theSM Higgs mass restriction is applied, and the lower one, after it has been applied. Clearly,this restricts scenario B much more strongly, but also restricts somehow scenario A. Thenon-observation of the extra two neutral scalars would favour the region where tan θ ∼ ,which corresponds to log(tan θ ) ∼ in the figure, since it is possible to have both h and H heavy in scenario A, and h heavy in scenario B, suppressing possible loop effects. Ananalysis of a similar model with 4 Higgs doublets (S3-4H) [75], where the fourth doublet isinert and with some considerations in the Yukawa sector, shows that the region that satisfiesthe bounds on extra scalar searches [92], prefers values of tan θ (cid:46) .18otice that in scenario B, the prospect of a lighter Higgs boson to explain the possibledecay of a scalar with m φ = 96 GeV reported by CMS [85] exists, this exotic Higgs bosonrole played by H , since it is always lighter than the SM Higgs. Experimental bounds onpossible decays of this type of Higgs bosons will constrain further the parameter space.Finally in Figure 4, we show the pseudoscalar Higgs masses m A , , and charged Higgsbosons masses m H ± , dependence on tan θ , after all constraints have been applied. As alreadymentioned, points where m H ± , < GeV have already been excluded in every figure. Thefigures are shown with a precision of 1% in the alignment limit, applying it at 0.1% lowersthe upper bounds for m H ± (cid:46) GeV and m A (cid:46) GeV in scenario A, scenario B isunaffected. In Table 2, we present the range of values that comply with all constraints, fora restricted SM-like Higgs mass, for each of the nine parameters.Figure 4: Dependence of the two pseudoscalar masses m A , (upper panel) and chargedHiggses m H ± , (lower panel) on tan θ . The points shown comply with the constraints ofprevious figures plus the bounds on the SM-like Higgs boson mass for each scenario.We want to highlight here that these are tree level masses, radiative corrections willchange their actual theoretical value, which in the case of the one identified as the SMHiggs, we have taken into account with a conservative uncertainty of ∼ ± GeV. A next-to-leading order (NLO) calculation of the masses should be done in order to give more accurate19heoretical predictions, that could be tested at the LHC or future colliders. Work in thisdirection is proposed in [93]. NLO analytical expressions for the scalar contributions aregiven in 4.4. In order to perform a numerical calculation for these loop corrections, we needto establish the parameter dependence for the trilinear and quartic Higgs couplings, whichwe have calculated and whose expressions are given in section 4.2 and in the Appendix A.
Parameter Scenario A Scenario B tan θ [0.1, 100 ] [7 × − , 100] a [3 × − , 16.4] [7 × − , 14.5] b [-6.5 , 23.4 ] [0.3 ,19.9 ] c [ 0.02, 16 ] [0.07 ,14.6 ] d [ -15.5, 13 ] [ -9.1, 9.4] e [-6.2, -8 × − ] [-1.6, -9 × − ] f [-21.7 , 23.8 ] [-17.7 ,8 ] g [ -15.6, 15.9 ] [-14.2 , 13.1 ] h [ -13.6, 12.5 ] [-12.3 ,0.06 ]Table 2: Parameter range for each scenario, after all constraints are considered, including arestricted range for the SM-like Higgs in each case. In this section we calculate the trilinear and quartic couplings for the Higgs bosons, aswell as with the gauge bosons, and give the analytical expressions in terms of the physicalparameters. We analyse the contributions of these couplings to the neutral CP-even Higgsmass matrix one-loop radiative corrections. We give the explicit expressions for scenario A.
We use the Higgs basis to calculate the kinetic Lagrangian term, Eq. (2), for the EW gaugebosons. Performing the usual neutral EW rotation on the gauge fields W µ and B µ , we getthe physical gauge fields.The residual symmetry manifests itself also in the gauge-Higgs couplings. We show herethe couplings of gauge bosons with the three neutral CP-even scalar Higgs bosons, the restof the couplings are given in appendix A, in this expression we have not taken into accountthe combinatorical factor from two identical particles in the Lagrangian term. Notice that h does not couple in a single scalar coupling with the gauge bosons but it does in pairs withgauge bosons. In order to make the presence of the symmetry evident we put first the ones20hat vanish: g h W ± W ∓ = 0 , g h ZZ = 0; (58) g H W ± W ∓ = 2 M W cos( α − θ ) g µν v , g H W ± W ∓ = 2 M W sin( α − θ ) g µν v ; (59) g H ZZ = M Z cos( α − θ ) g µν v , g H ZZ = M Z sin( α − θ ) g µν v ; (60) g h h W ± W ∓ = M W g µν v , g h h ZZ = M Z g µν v ; (61) g H H W ± W ∓ = M W g µν v , g H H W ± W ∓ = M W g µν v ; (62) g H H ZZ = M Z g µν v , g H H ZZ = M Z g µν v . (63)The presence of these couplings could give rise to one-loop corrections to the gauge bosonsmasses. These corrections would be suppressed by the inverse of the square of the mass ofthe particle in the loop. The above favors scenario A, where we can have the H mass athigh enough energies for this suppression to occur. A complete renormalization procedurewould be needed to further bound the masses and the parameters of the model. As we already mentioned, the trilinear Higgs couplings will be important to estimate loopcorrections, for instance the SM Higgs mass correction, which has a quadratic dependenceon the cut-off scale energy Λ . Previously, the trilinear couplings for the neutral scalars in the3HDM with S symmetry were reported in [47], nevertheless our results differ from the onescalculated there. On the other hand, our expressions for the trilinear couplings do coincidewith the presence of a residual Z symmetry, as reported in [46]. Besides the confirmationof this residual symmetry, we additionally show that the couplings reduce to the SM onesfor the particular alignment limits.The couplings given in the Higgs potential, Eq. (3) can be obtained in terms of phys-ical parameters using the rotation matrices. The angle α given in Eq. (39) can be re-written using the relations we obtained in Eq. (44). Thus, we can write Eq. (38) in termsof the physical Higgs masses and rotation angle α . Moreover, also using Eqs. (33)-(36)and Eqs.(43)-(44), we obtain expressions for the quartic couplings in the scalar potential,21q. (3), given in terms of the physical parameters i.e. masses, vevs , and rotation angles( v, m h , m H , m H , m A , m A , m H ± , m H ± , tan α, tan θ ), as a = 1 v cos θ (cid:20) m H cos α + m H sin α − m h tan θ (cid:21) , (64) b = 1 v (cid:20) sin 2 α sin 2 θ ( m H − m H ) + m h θ + 2 m H ± (cid:21) , (65) c = 1 v sin θ (cid:20) m H sin α + m H cos α − m h − m H ± cos θ + m H ± (cid:21) , (66) d = 1 v sin θ (cid:104) ( m H ± − m A ) − ( m H ± − m A ) cos θ (cid:105) , (67) e = − m h v sin 2 θ , (68) f = 1 v (cid:20) m h θ + m A − m H ± (cid:21) , (69) g = 1 v sin θ (cid:20) m h + m H ± cos θ − m H ± (cid:21) , (70) h = 1 v (cid:20) m h θ − m A (cid:21) . (71)This parameterization of the scalar potential couplings differs slightly from the ones presentedin other works, as in [46,47], due to our normalization of the couplings in the scalar potential.From the scalar potential we can get the trilineal scalar couplings as usual, where allpossible combinations are given from the terms considered into potential. − iλ ijk = − i∂ V∂H i ∂H j ∂H k . (72)As we already mentioned, the Z residual symmetry present will imply a vanishingtrilinear h h h due to its odd charge under Z , we explicitly confirmed this and also obtainthe other trilinear and quartic scalar couplings. These couplings are essential in order todetermine experimentally the shape of the actual Higgs potential. To this end, a one-loopcalculation of the self-energy corrections and vertices should be performed. Moreover, it ispossible to restrict parameters from the SM Higgs boson mass corrections, as we are goingto consider in next section.The following analytical expressions are the scalar-scalar couplings written in the physicalbasis and in terms of the physical parameters: ∗ g h h h = 0 , (73) g H H H = − v s θ (cid:20) m h c α − θ c θ + m H (cid:0) c α c α − θ − s α s θ (cid:1)(cid:21) , (74) ∗ As we mentioned in the previous section the symmetry factor n! has to be added in front of the couplingsfor n identical particles in the vertex. H H H = 1 v s θ (cid:20) m h s α − θ c θ − m H (cid:0) c α s α − θ − s α c θ (cid:1)(cid:21) , (75) g h h H = 1 v s θ ( m h s α + θ + m H s α c θ ) , (76) g h h H = − v s θ ( m h c α + θ + m H c α c θ ) , (77) g H H H = − s α − θ vs θ (cid:32) m h (cid:18) s α − θ ) c θ (cid:19) + m H s α + m H s α (cid:33) , (78) g H H H = c α − θ vs θ (cid:32) m h (cid:18) s α − θ ) c θ (cid:19) + m H s α m H s α (cid:33) , (79)using the reduced notation s x ≡ sin x , c x ≡ cos x and t x ≡ tan x . Notice, from the couplingsto h , that there are loop corrections to the SM Higgs mass which are absent in the 2HDM,which provides a way to differentiate this model from the 2HDM in the alignment limit [30].In the following we show the analytic expressions for trilinear couplings of neutral Higgsbosons within CP-even and CP-odd states, although the couplings preserve CPg A A H = 1 vs θ ( − m h s α − θ c θ + 12 m H s α + m A s α − m A c θ s α − θ ) , (80) g A A H = 1 vs θ ( m h c α − θ c θ − m H c α − m A c α + m A c θ c α − θ ) , (81) g A A H = 1 vs θ (cid:18) m h s α − θ c θ + m H (cid:0) s α c θ − c α s θ (cid:1) + m A s θ c α − θ (cid:19) , (82) g A A H = 1 vs θ (cid:18) − m h c α − θ c θ + m H (cid:0) s α s θ − c α c θ (cid:1) + m A s θ s α − θ (cid:19) , (83) g A A h = 23 vs θ (cid:0) − m h ( c θ + c θ ) + 3 m A c θ − m A c θ (cid:1) , (84)23 H G G = m H s α − θ v , g H G G = m H c α − θ v . (85)We also encounter the residual Z symmetry in the only allowed couplings (the rest are notpresent) with the pseudoscalars given as: g H G A = c α − θ v ( m H − m A ) , g H G A = s α − θ v ( − m H + m A ) , (86) g h G A = 1 v ( m h − m A ) . (87)The quartic scalar couplings written in the physical basis and in terms of the physicalparameters are calculated from − iλ ijkl = − i∂ V∂H i ∂H j ∂H k ∂H l . (88)We also give the analytical expressions for the quartic auto-scalar couplings written in thephysical basis and in terms of the physical parameters. We are interested in particular inthe couplings with H , , in order to compare with the SM ones in the alignment limits. Thequartic couplings also necessary to calculate the one-loop corrections to the Higgs bosonsmasses. We give here some explicit four scalar couplings examples, the rest of the couplingscan be found at the Appendix A.2: g h h h h = 124 v s θ (cid:32) m h + 3 m H s α + 3 m H c α (cid:33) , (89) g H H H H = 12 v s θ (cid:32) m h s α − θ ( s α − θ + 2 s α + θ )9 c θ + m H ( s α s α − θ + c α s θ ) + m H s α s α − θ (cid:33) , (90) g H H H H = 12 v s θ (cid:32) m h c α − θ ( c α − θ + 2 c α + θ )9 c θ + m H s α c α − θ m H ( c α c α − θ − s α s θ ) (cid:33) . (91)24 .3 Couplings in scenario A We show here how the scalar couplings are reduced in the alignment limit of scenario A.Recalling that the alignment limit is given as, sin( α − θ ) = 1 , cos( α − θ ) = 0 , the trigonometricfunctions for α and θ satisfy the following relations sin α = cos θ ; cos α = − sin θ ; sin 2( α − θ ) = 0;cos(3 α − θ ) = − sin 2 α ; sin( α + θ ) = cos 2 θ ; cos( α + θ ) = − sin 2 θ. (92)In scenario A in the alignment limit, the Higgs boson H trilinear coupling coincidesexactly with the trilinear coupling of the SM Higgs boson, g H H H = 1 v s θ (cid:2) m H s α s θ (cid:3) = 12 v s α c θ m H = m H v . (93)And the H trilineal couplings reduces to g H H H = 1 v s θ (cid:20) c θ m h − s θ m H (cid:21) = 1 v s θ c θ (cid:20) m h − s θ m H (cid:21) . (94)The H quartic coupling (90) also reduces exactly to the SM one in the alignment limit, g H H H H = 12 v s θ m H ( − s θ c θ − c θ s θ ) = m H v . (95)The H − h quartic coupling reduces in this limit to g H H h h = 1 v s θ (cid:32) m h (2 s θ − s α ) − m H s α (cid:33) . (96)Some of the reduced scalar couplings for scenario A depend only on the masses involved,and are given as g H h h = 12 v ( m H + 2 m h ) , g H A A = v ( m H + 2 m A ) , g H A A = 12 v ( m H + 2 m A ) ,g H H ± H ∓ = 1 v ( m H + 2 m H ± ) , g H H ± H ∓ = v ( m H + 2 m H ± ) , g H H H H = g H H H H = 0 . (97)From these expressions, a lower bound for all the scalar masses (other than H , ), can beset at (cid:38) GeV. This is in natural agreement with the current bounds for charged scalars.The recent results for the rare three-body decay of the SM Higgs boson to photon anddileptons [94], will put extra constraints in the values of the allowed trilinear couplings.In case the alignment limit is not exact, denoting δ ≡ cos( α − θ ) , where < δ (cid:28) , weobtain g H H H ≈ m H v (1 + δt θ ) + m h v (cid:18) δ s θ c θ (cid:19) , (98)25here it can be seen that in order to have small corrections to the trilinear SM Higgs-likeboson coupling, a small value for tan θ is favoured. The same applies for the quartic coupling.Analogous expressions for the couplings can be found for scenario B. In this case, theSM-like Higgs boson would be H and the other neutral Higgs, H , would be lighter thanthe SM-like, at tree level. As we already discussed, we cannot fully discard this possibilitysince in this alignment scenario, H would not have couplings to the gauge bosons, and itcould escape experimental detection.We do not consider the most general case: without any alignment, since it implies thatboth neutral Higgses couple to the gauge bosons, which is highly restricted from the expe-rimental data. Considering CP-invariance, the renormalized neutral Higgs masses would be written as twodiagonal × block matrices, one for the CP-even neutral states of the Higgs doublets ( h , H , H ) , and the second for the CP-odd Higgs states ( A , A , G ) M φ ( s ) = M (0)2 H + ˆΠ S ( s ) 00 ˆΠ P ( s ) , (99)where M (0)2 H is the Higgs mass matrix at tree level given in section 3, the neutral part ofexpression (28), also (43) and (44). The complete renormalized neutral Higgs self-energies atone-loop level, ˆΠ S ( s ) should be taken with the usual prescription, given for example in [95],adapting it to the S3-3H model.In our model, the one-loop contributions to the unrenormalized mass corrections Π S, (˜ q ) H H ( s ) ,that come only from the scalar sector self-energy, denoted as Σ φ , would indicate correctionsdue to scalar bosons on the loop. Due to the Z residual symmetry, the only trilinear cou-pling that involves a single h , which would give rise to a one-loop mass correction, is theone with two different charged scalars h H ± H ∓ (Eq. (104) in the Appendix). This mixedcharged Higgs coupling is not present for the other two neutral Higgs bosons, avoiding themixing of h with the other neutral Higgs bosons at one-loop level, in this case via chargedHiggs loops.For the quartic couplings, there is no coupling that involves a single h with a pair ofidentical Higgs bosons (including with three identical ones), see Appendix A.2. This impliesthat there are no possible mass one-loop corrections that could mix h with the other neutralscalars, H and H . Moreover, we can see from the gauge couplings with h given in 4.1, thatthere are only corrections to the h mass but no mixing with other neutral Higgs bosons.Thus, the decoupling is kept at one-loop level, with the consequence that the one-loop scalarcorrection mass matrix will attain a block diagonal form, Σ φ ( s ) = Σ φh ( s ) 0 00 Σ φH ( s ) Σ φH H ( s )0 Σ φH H ( s ) Σ φH ( s ) . (100)26e see from the above expression, that even at one-loop the h scalar is decoupled from theother two, so the mass matrix structure of the other two neutral scalars is similar to the2HDM. Nevertheless, there will be loop corrections to the H , masses due to h , as can beseen from the couplings (76) and (77). On the other hand, h will also receive corrections toits mass via the gauge boson loop, due to the allowed couplings (61).The general scalar and vector contributions to H and H squared mass Higgs bosoncorrections are given as: Σ φH n = (cid:88) i g H n H n φ i φ i π A m φ i ) + (cid:88) i,j g H n φ i φ j π B p , m φ i , m φ j ) + (cid:88) k g H n φ ± k φ ∓ k π B p , m φ ± k , m φ ± k )+ (cid:88) i g H n H n V i V i π A m V i ) + (cid:88) i g H n V i V i π B p , m V i , m V i ) , (101)with n = 1 , . † For the mixing H we get Σ φH H = (cid:88) i g H H φ i φ i π A m φ i ) + (cid:88) i,j g H φ i φ j g H φ i φ j π B p , m φ i , m φ j )+ (cid:88) k g H φ ± k φ ∓ k g H φ ± k φ ∓ k π B p , m φ ± k , m φ ± k ) + (cid:88) i g H V i V i g H V i V i π B p , m V i , m V i )+ (cid:88) k g H φ ± k W ∓ g H φ ± l W ∓ π B p , m φ ± l , m W ) , (102)where φ i ( j ) = h , H , H , A , A , G , φ ± k = H ± , , G ± and V i = W ± , Z . In these expressions, A and B are the Passarino-Veltman functions of the masses involved [96]. The radiativecontributions to the mixing of Σ φH H ( s ) reduce when we apply the alignment limit. Forscenario A, the couplings reduce such that the one-loop corrections to the mixing term aregiven as follows Σ φH H = (cid:88) i g H H φ i φ i π A m φ i ) + (cid:88) i g H φ i φ i g H φ i φ i π B p , m φ i , m φ j )+ (cid:88) k g H φ ± k φ ∓ k g H φ ± k φ ∓ k π B p , m φ ± k , m φ ± k ) , (103)in this case we will only have φ i = h , A , A , φ ± k = H ± , , since all the terms involving gaugeand Goldstone bosons vanish. This is taking only into account scalar and gauge contributionsto the one-loop corrections. An equivalent expression can be found for scenario B.Furthermore, if we demand that the mixing correction term vanishes, i.e. Σ φH H = 0 ,taking two benchmarks for scalar masses we get tan θ ≈ O (1) , see Table 3. This value of tan θ indicates a maximal mixing between the S singlet and doublet, see Eqs.(47,48).Results in Table 3 are not conclusive, since we should take into account the fermioniccontributions to have a more accurate estimation of the radiative corrections to the Higgsboson masses. In particular, the top quark contribution is expected to be sizeble, due to itslarge Yukawa coupling. Work along these lines is in progress. † The terms where gauge bosons are involved show only the coupling contributions, the actual calculationwill have to involve the gauge fixing. tan θ light spectrum m h = m H = m A , = m H ± , = heavy spectrum m h = m H = m A , = m H ± , = . Table 3: Parameter values in scenario A that vanish the one-loop mixing parameter, Σ φH H =0 , taking into account only the scalar and gauge contributions. The S3-3H model is an interesting and promising extension of the SM, that can accommodatewell the masses and mixing of the quarks, leading to the NNI matrices [43], as well as leptons,where it naturally gives a non-zero reactor mixing angle [59, 66]. In this paper we studythe gauge and scalar sector of this model. We choose a geometrical parameterization inspherical coordinates, which allows us to express our results in terms of the mixing angle, tan θ , between fields in the doublet and the singlet irreps. From here it is clear that, in orderto have realistic physical scenarios, without massless scalars, this mixing must be alwaysdifferent from zero.We perform a numerical analysis on the allowed parameter space, taking into accountunitarity and stability bounds, as well as the current experimental bounds on the chargedmasses. As previously found [46], there is a residual Z symmetry, which decouples oneof the neutral scalars h from the gauge bosons. This rises the interesting possibility oftreating this decoupled scalar as a dark matter candidate, although we still have to probeits fermionic couplings. Work along these lines is in progress.We study two possible alignment scenarios in which one of the two Z even neutral scalars H , , is maximally coupled to the gauge bosons, and is thus taken to be as the SM Higgs.In scenario A, the lighter of the two, H is the SM-like Higgs boson. The other possibility,scenario B, where the heavier H is the SM-like Higgs boson, cannot be a priori excluded,since H could have escaped detection due to the absence of couplings to the vector bosons.We found the allowed ranges for all the self-couplings and scalar masses, in terms of tan θ in each alignment scenario, with a and . precision on the ( α − θ ) values. Ourresults show a clear restriction for all the scalar masses, which are mostly below TeV. Thiscorroborates similar analysis in this direction for scenario A [46] (although we allowed fora small deviation of the alignment limit, and for some uncertainty in the SM Higgs mass).Scenario B in this model has not been analysed before. The light H scalar in this scenarioopens the possibility that it might be regarded as the 96 GeV scalar, which was suggestedas a diphoton signal reported by CMS [85], and discussed in the literature in the context ofSUSY and 2HDM models [86, 87, 89].We calculate all trilinear and quartic couplings between the Higgs bosons, and also amongthe Higgs and gauge bosons, in terms of the physical parameters of the model. We founddiscrepancies with previously reported trilinear scalar couplings in this model [47], in whichthe Z symmetry is not present. On the contrary, our expressions do confirm the existenceof the residual Z symmetry, consistent with the results of [46]. Also, in our expressions,both the trilinear and quartic couplings for the SM-like Higgs boson reduce to the SM ones in28he exact alignment limits, but they approach this limit in a different way than the 2HDM.If the alignment limit is not exact, small corrections to the SM-like Higgs scalar couplingswill be achieved with small values of tan θ .In the exact alignment limit, some of the trilinear couplings depend only on the scalarmasses, which sets a natural lower bound for all masses, other than H , , to (cid:38) GeV. Theinclusion of radiative corrections might change these bounds.We obtained the analytical expressions for the one-loop corrections to the SM-like Higgs,due to scalar and gauge bosons in the loop, and found that the decoupling of h remains atone-loop level, as expected from a symmetry of the Lagrangian. From the reduced expressionsfor the couplings in scenario A, we calculate the value of tan θ for which the one-loop mixingof H and H vanishes, for two benchmark mass values. These results point to a value of tan θ ≈ O (1) , indicating a large mixing between the S doublet and the singlet, consistentwith what was reported in [75].The model has different one-loop couplings to the Higgs bosons through h , as comparedto the 2HDM. Thus, although it reduces to a form similar to the 2HDM due to the presenceof the residual Z symmetry, it will lead to different experimental signatures. Acknowledgements
We acknowledge useful discussions with Alexis Aguilar, Catalina Espinoza, and GenaroToledo. This work was partially supported with UNAM projects DGAPA PAPIIT IN111518and IN109321. M.G.B would like to thank UDLAP for financial support. A.P. acknowledgesfinancial support from CONACyT, through grant 332430.
A Higgs Couplings
A.1 Scalar trilinear couplings
Here we explicitly write down the trilinear couplings of neutral with charged Higgs bosonsand the rest of allowed cuartic couplings of Higgs bosons. Here we are able to see theresidual Z symmetry, the rest of the couplings are absent, as can be found from the directcalculation given in (72) and (88) g h H ± H ± = 13 vs θ (cid:18) − m h c θ + c θ c θ + 3 m H ± c θ − m H ± c θ (cid:19) , (104) g H H ± H ± = 2 vs θ (cid:18) − m h s α − θ + m H c θ s α + 2 m H ± c θ s α − m H ± c θ s α − θ (cid:19) , (105) g H H ± H ± = 2 vs θ (cid:18) m h c α − θ − m H c θ c α − m H ± c θ c α + 2 m H ± c θ c α − θ (cid:19) , (106)29 H H ± H ± = 2 vs θ c θ (cid:18) m h s α − θ + m H c θ (cid:0) s α c θ + c α s θ (cid:1) + m H ± s θ c θ c α − θ (cid:19) , (107) g H H ± H ± = 2 vs θ (cid:18) − m h c α − θ c θ − m H ( c α c θ − s α s θ ) + m H ± s α − θ s θ (cid:19) , (108) g H G ± G ± = m H s α − θ v , (109) g H G ± G ± = m H c α − θ v , (110) g h H ± G ± = 1 v ( m h − m H ± ) , (111) g H H ± G ± = c α − θ v ( m H − m H ± ) , (112) g H H ± G ± = s α − θ v ( − m H + m H ± ) , (113) g A H ± G ± = 1 v ( m A − m H ± − ( m A − m H ± ) c θ ) . (114) A.2 Quartic scalar couplings g H H H H = 116 v s θ (cid:32) m h s α − θ ) c θ (cid:0) s α + s α − θ ) (cid:1) − m H s α (3 c α s α − θ ) − s α + s θ ) + 2 m H s α (3 c α s α − θ ) + 3 s α + s θ ) (cid:33) , (115) g H H h h = 12 v s θ (cid:18) m h (cid:18) c α + θ − c α − θ (cid:19) + m H c θ s α s α c α − θ + m H c θ c α ( c α + θ + c α c α − θ ) (cid:19) , (116) g H H h h = 12 v s θ (cid:18) m h (cid:18) s α + θ − s α − θ (cid:19) + m H c θ s α ( s α + θ − c α s α − θ ) + m H c θ c α s α s α − θ (cid:19) , g H H H H = − v s θ (cid:18) m h c α c α − θ ( s α + 2 c θ s α − θ )9 c θ + m H s α s α − θ ) − m H s α c α − θ ( c α − θ c α + c α + θ )2 (cid:19) , (118) g H H H H = − v s θ (cid:18) m h s α s α − θ ( c α + 2 c θ c α − θ )9 c θ + m H s α s α − θ ) m H s α s α − θ ( s α − θ c α − s α + θ )2 (cid:19) , (119) g H H h h = − v s θ (cid:18) m h c α s θ + s α c θ ) + m H s α s α c θ s α − θ + m H s α c α c θ c α − θ (cid:19) , (120) g H H A A = 12 v s θ (cid:32) − m h c θ (3 c α + 3 c θ + 4 c α − θ ) + m H s α c α c α − θ c θ + m H c α c θ (cid:0) c α c α − θ − s α s θ (cid:1) + m A c α − m A ( c α − s θ ) (cid:33) , (121) g H H A A = 12 v s θ (cid:16) m h c θ (3 c α − c θ − s α − θ ) + m H c α s α s α − θ c θ + m H s α c θ ( s α s α − θ + c α s θ ) + m A s α + m A ( s θ − s α ) (cid:17) , (122) g H H A A = 12 v s θ (cid:32) m h c θ (2 c α + s θ s α − θ ) + 2 c α − θ ) + 2 c θ )+ m H s α (cid:0) s α + s θ + c θ s α − θ ) (cid:1) + m A s α − θ s θ + m H c α + 3 c θ + 4 c α − θ ) + c α − θ ) + 5 c α + θ ) ) (cid:33) , (123) g H H A A = 12 v s θ (cid:32) − m h c θ (2 c α + s θ s α − θ ) − s α − θ − c θ )+ m H s α s α − s θ + c θ s α − θ ) ) + m A c α − θ s θ m H c α + 3 c θ − s α − θ − c α − θ ) − c α + θ ) (cid:33) , (124) g h h A A = m h + 3 m H s α + 3 m H c α v s θ , (125) g h h A A = 1 v s θ (cid:32) − m h c θ + c θ ) + m H c θ s α ( s α + θ − s θ c θ c α − θ )+ m H c θ c α ( c α + θ + s θ c θ s α − θ ) + 2 m A c θ + 2 m A c θ c θ (cid:33) , (126) g H H A A = 1 v s θ (cid:18) m h s α + 2 s α − θ ) ) − m H s α s α c θ s α − θ − m H s α c α c θ c α − θ + 2( m A − m A ) s α c θ (cid:19) , (127) g H H A A = 12 v s θ (cid:16) − m h c θ ( s α + 2 c θ s α − θ ) ) + m H s α (2 c α + s θ s α − θ ) − c θ ) − m H s α (2 c α + s θ s α − θ ) + 2 c θ ) + m A s α − θ ) s θ (cid:17) , (128) g H h A A = 1 v s θ (cid:18) − m h c θ ( s α + θ + c θ s α )( c θ + c θ ) + m H s α c θ − m H ( c α s θ + c θ s α − θ ) (cid:19) , (129) g H h A A = 1 v s θ (cid:18) m h c θ ( c α + θ + c θ c α )( c θ + c θ ) − m H c α c θ − m H ( s θ s α − c θ c α − θ ) (cid:19) , (130) g A A A A = m h + 3 m H s α + 3 m H c α v s θ , (131) g A A A A = 12 v s θ (cid:18) m h c θ (2 c θ + c θ ) + m H ( s α c θ − s θ s α − θ ) + m H ( c α c α − θ − s α s θ ) (cid:19) , (132) g A A A A = 1 v s θ (cid:18) m h c θ − c θ s θ ) + m H c θ s α ( s α + θ − s θ c θ c α − θ ) + m H c θ c α ( c α + θ + s θ c θ s α − θ ) (cid:19) , g H H G G = 12 v (cid:18) m H s θ ( s θ + s α − c α s α − θ ) ) − m H s α s α − θ s θ + m A s α − θ (cid:19) , (134) g H H G G = 12 v (cid:18) m H s α c α − θ s θ + m H s θ ( s θ − s α − c α s α − θ ) ) + m A c α − θ (cid:19) , (135) g H H G G = − s α − θ ) v s θ (cid:0) ( m H − m H ) s α + 2 m A s θ (cid:1) , (136) g G G A A = 18 v s θ (cid:16) m H (3 c θ s α − θ ) − s α + 3 s θ ) − m H (3 c θ s α − θ ) − s α − s θ ) (cid:17) , (137) g G G A A = 12 v (cid:18) m h + m H s θ s α c α − θ − m H s θ c α s α − θ (cid:19) , (138) g H H A G = − v s θ (cid:18) m h s α − θ c θ + m H c θ − s α s α − θ ) − c α ) + m H s α s α − θ ) − m A s θ s α − θ ) (cid:19) , (139) g H H A G = − v s θ (cid:18) m h c α − θ c θ + m H s α − θ ) + m H c θ − s α s α − θ ) + 2 c α ) + m A s θ s α − θ ) (cid:19) , (140) g H H A G = 1 v s θ (cid:18) m h s α − θ ) c θ + m H s α s α − θ + m H s α c α − θ − m A c α − θ ) s θ (cid:19) , (141) g H h A G = 1 v s θ (cid:18) m h (cid:18) s α + θ c θ + s α (cid:19) − m A s α + m A s α − θ c θ (cid:19) , (142) g H h A G = 1 v s θ (cid:18) − m h (cid:18) c α + θ c θ + c α (cid:19) + m A c α − m A c α − θ c θ (cid:19) , (143) g G A A A = 1 v s θ (cid:18) − m h c θ + m H c α + s θ s α − θ ) − c θ ) − m H c α + s θ s α − θ ) + 2 c θ ) (cid:19) , (144) g G G G G = m H c α − θ + m H s α − θ v . (145)33 .3 Charged scalar-vector bosons couplings For the couplings with charged Higgs bosons we have g H ± H ± W ± W ∓ = 2 M W g µν v , (146) g H ± H ± W ± W ∓ = 2 M W g µν v , (147) g H ± H ± ZZ = g cos θ W g µν θ W , (148) g H ± H ± ZZ = g cos θ W g µν θ W , (149) g H ± H ∓ γγ = e g µν , (150) g H ± H ∓ γγ = e g µν , (151) g H ± H ∓ γZ = eg cos 2 θ W g µν cos θ W , (152) g H ± H ∓ γZ = eg cos 2 θ W g µν cos θ W . (153)And the couplings for mixed charged and neutral Higgs bosons with gauge bosons, aregiven as g H ∓ H ZW ± = g (cid:48) cos θ W sin( α − θ ) g µν , (154) g H ∓ H ZW ± = − g (cid:48) cos θ W cos( α − θ ) g µν , (155) g H ∓ H γW ± = − eg sin( α − θ ) g µν , (156) g H ∓ H γW ± = eg cos( α − θ ) g µν . (157)34he mixed charged Higgs boson and h couplings with two gauge bosons are absent. g γH +1 H +1 = e ( p + p (cid:48) ) µ , (158) g γH +2 H +2 = e ( p + p (cid:48) ) µ , (159) g W ± H ± H = ± g α − θ )( p + p (cid:48) ) µ , (160) g W ± H ± H = ± g α − θ )( p + p (cid:48) ) µ , (161) g W ± H ± h = ± g p + p (cid:48) ) µ , (162) g W ± H ± A = g p + p (cid:48) ) µ , (163) g W ± H ± A = g p + p (cid:48) ) µ , (164) g W ± G ± H = ∓ g α − θ )( p + p (cid:48) ) µ , (165) g W ± G ± H = ∓ g α − θ )( p + p (cid:48) ) µ , (166) g W ± G ± G = g p + p (cid:48) ) µ , (167) g γG + G + = e ( p + p (cid:48) ) µ . (168)35 eferences [1] Georges Aad et al. Observation of a new particle in the search for the Standard ModelHiggs boson with the ATLAS detector at the LHC. Phys. Lett. , B716:1–29, 2012.[2] Serguei Chatrchyan et al. Observation of a new boson at a mass of 125 GeV with theCMS experiment at the LHC.
Phys.Lett. , B716:30–61, 2012.[3] Peter W. Higgs. Spontaneous Symmetry Breakdown without Massless Bosons.
Phys.Rev. , 145:1156–1163, 1966.[4] Gordon Kane John F. Gunion, Howard E. Haber and Sally Dawson.
The Higgs Hunter’sGuide . Frontiers in Physics, 80. Westview Press, 2000.[5] Ricardo A. Flores and Marc Sher. Higgs Masses in the Standard, Multi-Higgs andSupersymmetric Models.
Annals Phys. , 148:95, 1983.[6] Ilya F. Ginzburg and Maria Krawczyk. Symmetries of two Higgs doublet model andCP violation.
Phys. Rev. D , 72:115013, 2005.[7] Gustavo C. Branco, M. N. Rebelo, and J. I. Silva-Marcos. CP-odd invariants in modelswith several Higgs doublets.
Phys. Lett. B , 614:187–194, 2005.[8] Celso C. Nishi. The Structure of potentials with N Higgs doublets.
Phys. Rev. ,D76:055013, 2007.[9] Per Osland, P. N. Pandita, and Levent Selbuz. Trilinear Higgs couplings in the twoHiggs doublet model with CP violation.
Phys. Rev. , D78:015003, 2008.[10] P. M. Ferreira and Joao P. Silva. Discrete and continuous symmetries in multi-Higgs-doublet models.
Phys. Rev. , D78:116007, 2008.[11] K. Olaussen, P. Osland, and M. Aa. Solberg. Symmetry and Mass Degeneration inMulti-Higgs-Doublet Models.
JHEP , 07:020, 2011.[12] Jisuke Kubo. Super Flavorsymmetry with Multiple Higgs Doublets.
Fortsch.Phys. ,61:597–621, 2013.[13] Kei Yagyu. Higgs boson couplings in multi-doublet models with natural flavour conser-vation.
Phys. Lett. , B763:102–107, 2016.[14] Miguel P. Bento, Howard E. Haber, J. C. Romão, and João P. Silva. Multi-Higgs doubletmodels: the Higgs-fermion couplings and their sum rules.
JHEP , 10:143, 2018.[15] I. de Medeiros Varzielas and Igor P. Ivanov. Recognizing symmetries in a 3HDM in abasis-independent way.
Phys. Rev. D , 100(1):015008, 2019.[16] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, Marc Sher, and Joao P. Silva.Theory and phenomenology of two-Higgs-doublet models.
Phys. Rept. , 516:1–102, 2012.3617] Igor P. Ivanov. Building and testing models with extended Higgs sectors.
Prog. Part.Nucl. Phys. , 95:160–208, 2017.[18] Hajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki, Yusuke Shimizu, Hiroshi Okada,and Morimitsu Tanimoto. Non-Abelian Discrete Symmetries in Particle Physics.
Prog.Theor. Phys. Suppl. , 183:1–163, 2010.[19] Hajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki, Hiroshi Okada, Yusuke Shimizu, andMorimitsu Tanimoto. An introduction to non-Abelian discrete symmetries for particlephysicists.
Lect. Notes Phys. , 858:1–227, 2012.[20] Guido Altarelli and Ferruccio Feruglio. Discrete Flavor Symmetries and Models ofNeutrino Mixing.
Rev. Mod. Phys. , 82:2701–2729, 2010.[21] Stephen F. King and Christoph Luhn. Neutrino Mass and Mixing with Discrete Sym-metry.
Rept. Prog. Phys. , 76:056201, 2013.[22] Stefano Moretti, Diana Rojas, and Kei Yagyu. Enhancement of the H ± W ∓ Z vertex inthe three scalar doublet model. JHEP , 08:116, 2015.[23] José Eliel Camargo-Molina, Tanumoy Mandal, Roman Pasechnik, and Jonas Wessén.Heavy charged scalars from c ¯ s fusion: A generic search strategy applied to a 3HDMwith U(1) × U(1) family symmetry.
JHEP , 03:024, 2018.[24] A. G. Akeroyd, Stefano Moretti, and Muyuan Song. Light charged Higgs boson withdominant decay to a charm quark and a bottom quark and its search at LEP2 andfuture e + e − colliders. Phys. Rev. D , 101(3):035021, 2020.[25] Georges Aad et al. Combined measurements of Higgs boson production and decay usingup to fb − of proton-proton collision data at √ s =
13 TeV collected with the ATLASexperiment.
Phys. Rev. D , 101(1):012002, 2020.[26] Albert M Sirunyan et al. Combined measurements of Higgs boson couplings in proton–proton collisions at √ s = 13 TeV.
Eur. Phys. J. C , 79(5):421, 2019.[27] J. F. Gunion and Howard E. Haber. Higgs Bosons in Supersymmetric Models. 1.
Nucl.Phys. , B272:1, 1986. [Erratum: Nucl. Phys.B402,567(1993)].[28] Marcela Carena and Howard E. Haber. Higgs boson theory and phenomenology.
Prog.Part. Nucl. Phys. , 50:63–152, 2003.[29] S. Heinemeyer. Higgs Physics. 5 2014.[30] John F. Gunion and Howard E. Haber. The CP conserving two Higgs doublet model:The Approach to the decoupling limit.
Phys. Rev. , D67:075019, 2003.[31] Miguel P. Bento, Howard E. Haber, J.C. Romão, and João P. Silva. Multi-Higgs doubletmodels: physical parametrization, sum rules and unitarity bounds.
JHEP , 11:095, 2017.3732] Sandip Pakvasa and Hirotaka Sugawara. Discrete Symmetry and Cabibbo Angle.
Phys.Lett. , B73:61–64, 1978.[33] Sandip Pakvasa and Hirotaka Sugawara. Mass of the t Quark in SU(2) x U(1).
Phys.Lett. , B82:105–107, 1979.[34] Haim Harari, Herve Haut, and Jacques Weyers. Quark Masses and Cabibbo Angles.
Phys. Lett. B , 78:459–461, 1978.[35] Emanuel Derman and Hung-Sheng Tsao. SU(2) X U(1) X S( n ) Flavor Dynamics and aBound on the Number of Flavors. Phys. Rev. , D20:1207, 1979.[36] Maurice V. Barnhill, III. Generalization of S(3) Mass Matrix Symmetry.
Phys. Lett. ,151B:257–259, 1985.[37] A. Mondragon and E. Rodriguez-Jauregui. A Parametrization of the CKM mixingmatrix from a scheme of S(3)-L x S(3)-R symmetry breaking. In , 1 1998.[38] A. Mondragon and E. Rodriguez-Jauregui. The Breaking of the flavor permutationalsymmetry: Mass textures and the CKM matrix.
Phys. Rev. , D59:093009, 1999.[39] L. Lavoura. A New model for the quark mass matrices.
Phys. Rev. , D61:077303, 2000.[40] J. Kubo, A. Mondragon, M. Mondragon, and E. Rodriguez-Jauregui. The Fla-vor symmetry.
Prog. Theor. Phys. , 109:795–807, 2003. [Erratum: Prog. Theor.Phys.114,287(2005)].[41] Jisuke Kubo, Hiroshi Okada, and Fumiaki Sakamaki. Higgs potential in minimal S(3)invariant extension of the standard model.
Phys. Rev. , D70:036007, 2004.[42] Renata Jora, Salah Nasri, and Joseph Schechter. An Approach to permutation symme-try for the electroweak theory.
Int. J. Mod. Phys. A , 21:5875–5894, 2006.[43] F. González Canales, A. Mondragón, M. Mondragón, U. J. Saldaña Salazar, andL. Velasco-Sevilla. Quark sector of S3 models: classification and comparison with ex-perimental data.
Phys. Rev. , D88:096004, 2013.[44] A. E. Cárcamo Hernández, E. Cataño Mur, and R. Martinez. Lepton masses andmixing in SU (3) C ⊗ SU (3) L ⊗ U (1) X models with a S flavor symmetry. Phys. Rev. ,D90(7):073001, 2014.[45] A. E. Cárcamo Hernández, I. de Medeiros Varzielas, and E. Schumacher. Fermion andscalar phenomenology of a two-Higgs-doublet model with S . Phys. Rev. , D93(1):016003,2016.[46] Dipankar Das and Ujjal Kumar Dey. Analysis of an extended scalar sector with S sym-metry. Phys. Rev. , D89(9):095025, 2014. [Erratum: Phys. Rev.D91,no.3,039905(2015)].3847] E. Barradas-Guevara, O. Félix-Beltrán, and E. Rodríguez-Jáuregui. Trilinear self-couplings in an S(3) flavored Higgs model.
Phys. Rev. , D90(9):095001, 2014.[48] Dipankar Das, Ujjal Kumar Dey, and Palash B. Pal. S symmetry and the quark mixingmatrix. Phys. Lett. , B753:315–318, 2016.[49] E. Barradas-Guevara, O. Félix-Beltrán, and E. Rodríguez-Jáuregui. CP breaking in S (3) flavoured Higgs model. 7 2015.[50] D. Emmanuel-Costa, O. M. Ogreid, P. Osland, and M. N. Rebelo. Spontaneous sym-metry breaking in the S -symmetric scalar sector. JHEP , 02:154, 2016. [Erratum:JHEP08,169(2016)].[51] Shao-Feng Ge, Alexander Kusenko, and Tsutomu T. Yanagida. Large Leptonic DiracCP Phase from Broken Democracy with Random Perturbations.
Phys. Lett. , B781:699–705, 2018.[52] Juan Carlos Gómez-Izquierdo and Myriam Mondragón. B–L Model with S symmetry:Nearest Neighbor Interaction Textures and Broken µ ↔ τ Symmetry.
Eur. Phys. J. C ,79(3):285, 2019.[53] Howard E. Haber, O. M. Ogreid, P. Osland, and M. N. Rebelo. Symmetries and MassDegeneracies in the Scalar Sector.
JHEP , 01:042, 2019.[54] A. Kunčinas, O. M. Ogreid, P. Osland, and M. N. Rebelo. S3 -inspired three-Higgs-doublet models: A class with a complex vacuum.
Phys. Rev. D , 101(7):075052, 2020.[55] Ernest Ma. S(3) Z(3) model of lepton mass matrices.
Phys. Rev. , D44:587–589, 1991.[56] A. Mondragon and E. Rodriguez-Jauregui. The CP violating phase delta(13) and thequark mixing angles theta(13), theta(23) and theta(12) from flavor permutational sym-metry breaking.
Phys. Rev. D , 61:113002, 2000.[57] Shao-Long Chen, Michele Frigerio, and Ernest Ma. Large neutrino mixing and normalmass hierarchy: A Discrete understanding.
Phys. Rev. , D70:073008, 2004. [Erratum:Phys. Rev.D70,079905(2004)].[58] O. Felix, A. Mondragon, M. Mondragon, and E. Peinado. Neutrino masses and mix-ings in a minimal S(3)-invariant extension of the standard model.
AIP Conf. Proc. ,917(1):383–389, 2007.[59] A. Mondragon, M. Mondragon, and E. Peinado. Lepton masses, mixings and FCNCin a minimal S(3)-invariant extension of the Standard Model.
Phys. Rev. , D76:076003,2007.[60] A. Mondragon, M. Mondragon, and E. Peinado. Nearly tri-bimaximal mixing in theS(3) flavour symmetry.
AIP Conf. Proc. , 1026:164–169, 2008.[61] A. Mondragon, M. Mondragon, and E. Peinado. S(3)-flavour symmetry as realized inlepton flavour violating processes.
J. Phys. , A41:304035, 2008.3962] Renata Jora, Joseph Schechter, and M. Naeem Shahid. Perturbed S(3) neutrinos.
Phys.Rev. D , 80:093007, 2009. [Erratum: Phys.Rev.D 82, 079902 (2010)].[63] Duane A. Dicus, Shao-Feng Ge, and Wayne W. Repko. Neutrino mixing with broken S symmetry. Phys. Rev. , D82:033005, 2010.[64] D. Meloni, S. Morisi, and E. Peinado. Fritzsch neutrino mass matrix from S symmetry. J. Phys. , G38:015003, 2011.[65] A. G. Dias, A. C. B. Machado, and C. C. Nishi. An S Model for Lepton Mass Matriceswith Nearly Minimal Texture.
Phys. Rev. , D86:093005, 2012.[66] F. Gonzalez Canales, A. Mondragon, and M. Mondragon. The S Flavour Symmetry:Neutrino Masses and Mixings.
Fortsch. Phys. , 61:546–570, 2013.[67] O. Felix-Beltran, F. González Canales, A. Mondragón, and M. Mondragón. S flavoursymmetry and the reactor mixing angle. J. Phys. Conf. Ser. , 485:012046, 2014.[68] Ernest Ma and Rahul Srivastava. Dirac or inverse seesaw neutrino masses with B − L gauge symmetry and S flavor symmetry. Phys. Lett. , B741:217–222, 2015.[69] Antonio Enrique Cárcamo Hernández, R. Martinez, and F. Ochoa. Fermion massesand mixings in the 3-3-1 model with right-handed neutrinos based on the S flavorsymmetry. Eur. Phys. J. , C76(11):634, 2016.[70] Arturo Alvarez Cruz and Myriam Mondragón. Neutrino masses, mixing, and leptogen-esis in an S3 model. 1 2017.[71] Zhi-Zhong Xing and Di Zhang. Seesaw mirroring between light and heavy Majorananeutrinos with the help of the S reflection symmetry. JHEP , 03:184, 2019.[72] J. D. García-Aguilar and Juan Carlos Gómez-Izquierdo. Flavored multiscalar S modelwith normal hierarchy neutrino mass. 10 2020.[73] Nabarun Chakrabarty. High-scale validity of a model with Three-Higgs-doublets. Phys.Rev. , D93(7):075025, 2016.[74] A.C.B. Machado and V. Pleitez. A model with two inert scalar doublets.
Annals Phys. ,364:53–67, 2016.[75] C. Espinoza, E. A. Garcés, M. Mondragón, and H. Reyes-González. The S SymmetricModel with a Dark Scalar.
Phys. Lett. , B788:185–191, 2019.[76] Subhasmita Mishra. Majorana dark matter and neutrino mass with S symmetry. Eur.Phys. J. Plus , 135(6):485, 2020.[77] C. Espinoza and M. Mondragón. Prospects of Indirect Detection for the Heavy S3 DarkDoublet. 8 2020. 4078] D. Emmanuel-Costa, O. Felix-Beltran, M. Mondragon, and E. Rodriguez-Jauregui. Sta-bility of the tree-level vacuum in a minimal S(3) extension of the standard model.
AIPConf. Proc. , 917:390–393, 2007. [390(2007)].[79] O.Felix Beltran, M. Mondragon, and E. Rodriguez-Jauregui. Conditions for vacuumstability in an S(3) extension of the standard model.
J. Phys. Conf. Ser. , 171:012028,2009.[80] C. Espinoza, E.A. Garcés, M. Mondragón, and H. Reyes-González. Unitarity and stabil-ity conditions in a 4-Higgs doublet model with an S -family symmetry. J. Phys. Conf.Ser. , 912(1):012022, 2017.[81] James D. Wells. Lectures on Higgs Boson Physics in the Standard Model and Beyond.In , 2009.[82] A. G. Akeroyd, Stefano Moretti, Kei Yagyu, and Emine Yildirim. Light charged Higgsboson scenario in 3-Higgs doublet models.
Int. J. Mod. Phys. , A32(23n24):1750145,2017.[83] Georg Keller and Daniel Wyler. The Couplings of Higgs Bosons to Two Vector Mesonsin Multi Higgs Models.
Nucl. Phys. , B274:410–428, 1986.[84] Dipankar Das and Ipsita Saha. Alignment limit in three Higgs-doublet models.
Phys.Rev. D , 100(3):035021, 2019.[85] Search for new resonances in the diphoton final state in the mass range between 80 and115 GeV in pp collisions at √ s = 8 TeV. 11 2015.[86] Sven Heinemeyer and T. Stefaniak. A Higgs Boson at 96 GeV?!
PoS ,CHARGED2018:016, 2019.[87] Ulrich Haisch and Augustinas Malinauskas. Let there be light from a second light Higgsdoublet.
JHEP , 03:135, 2018.[88] T. Biekötter, M. Chakraborti, and S. Heinemeyer. A 96 GeV Higgs boson in the N2HDM.
Eur. Phys. J. C , 80(1):2, 2020.[89] Thomas Biekötter, M. Chakraborti, and Sven Heinemeyer. An N2HDM Solution forthe possible 96 GeV Excess.
PoS , CORFU2018:015, 2019.[90] K. A. Olive et al. Review of Particle Physics.
Chin. Phys. , C38:090001, 2014.[91] P.A. Zyla et al. Review of Particle Physics.
PTEP , 2020(8):083C01, 2020.[92] Philip Bechtle, Sven Heinemeyer, Oscar Stal, Tim Stefaniak, and Georg Weiglein. Ap-plying Exclusion Likelihoods from LHC Searches to Extended Higgs Sectors.
Eur. Phys.J. C , 75(9):421, 2015. 4193] Nabarun Chakrabarty and Indrani Chakraborty. On the Higgs mass fine-tuning problemwith multi-Higgs doublet models.
Int. J. Mod. Phys. A , 34(05):1950025, 2019.[94] Evidence for Higgs boson decays to a low-mass dilepton system and a photon in pp collisions at √ s = 13 TeV with the ATLAS detector. 2 2021.[95] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein. TheHiggs Boson Masses and Mixings of the Complex MSSM in the Feynman-DiagrammaticApproach. JHEP , 02:047, 2007.[96] G. Passarino and M.J.G. Veltman. One Loop Corrections for e+ e- Annihilation Intomu+ mu- in the Weinberg Model.