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The role of charged Higgs boson decays in the determination of tan β-like parameters

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2013 J. Phys. G: Nucl. Part. Phys. 40 095004

(http://iopscience.iop.org/0954-3899/40/9/095004)

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IOP PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS

J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 (10pp) doi:10.1088/0954-3899/40/9/095004

The role of charged Higgs boson decays in thedetermination of tan β-like parameters

H Cardenas1, D Restrepo2 and J-Alexis Rodriguez1

1 Departamento de Fısica, Universidad Nacional de Colombia, Bogota, Colombia2 Instituto de Fısica, Universidad de Antioquia, Medellın, Colombia

E-mail: restrepo@udea.edu.co

Received 24 June 2013Published 14 August 2013Online at stacks.iop.org/JPhysG/40/095004

AbstractThe most commonly used parameterizations of the Yukawa couplings in thetwo Higgs doublet model are revisited. Some bounds on the parameter space ofthe charged Higgs sector of the two Higgs doublet model are obtained in a newparameterization, which is slightly different from that already known. Theseconstraints are obtained from flavor observables such as the measurement ofB → Xsγ and the recent measurement of Bs → μ+μ− from LHCb. The ratioRH+ = B(H+ → τ+ντ )/B(H+ → tb) is evaluated for the type-III and type-IItwo Higgs doublet models, and their differences are quantified.

(Some figures may appear in colour only in the online journal)

One simple framework beyond the standard model (SM) consists in adding a Higgs doublet,leading to a rich Higgs sector spectrum, and therefore giving a richer phenomenology thanin the SM [1, 2]. This model is usually called the two Higgs doublet model (THDM). In theTHDM there are three Goldstone bosons that give masses to the gauge bosons and there arealso five Higgs bosons: two CP-even neutral Higgs bosons, one CP-odd neutral Higgs bosonand two charged Higgs bosons.

There are multiple ways of implementing this kind of SM extension. These depend on howthe couplings between the Higgs bosons and the fermions are chosen. They should be chosencarefully in order to avoid flavor-changing neutral currents (FCNC), which are generated at thetree level in the most general version of the THDM, the so called type-III THDM. In contrast,models like the type-I and II THDM impose an additional discrete symmetry in order to avoidthe FCNC at tree level [2]. In the type-I model, the couplings between the Higgs bosons andthe fermions are chosen such that only one Higgs doublet couples to the SM fermions, whereasin the type-II model, one doublet couples with the up quark sector and the another one withthe down quark sector. In the type-III model, although both Higgs doublets couple with theSM fermions, the model is still phenomenologically possible if the FCNC at tree level aresuppressed [3].

The type-II THDM has been studied more extensively in the literature, motivated by itssimilarities in the Higgs sector at tree level with the minimal supersymmetric SM (MSSM),

0954-3899/13/095004+10$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

which requires a second Higgs doublet. However, due to quantum loop corrections, the Higgssector will actually be described by an effective potential which is similar to the most generalversion of the THDM, the so-called type-III.

From a phenomenological point of view, it is important to study the properties of themost general THDM. In this model, there are two Yukawa matrices per fermion type (up ordown type quarks and leptons) which cannot be diagonalized simultaneously. There are twodifferent ways to deal with the FCNC couplings: the first one is to adopt a particular Yukawatexture like the Cheng–Sher ansatz [4] which assumes that the couplings are proportional tothe geometric mean of the two masses (ξi j ∼ √

mimj); the second one is to impose the minimalflavor violation (MFV) hypothesis [5] as was done in the so-called aligned THDM (ATHDM)[6, 7], where the two Yukawa matrices per fermion are proportional to each other, and thenthey can be diagonalized avoiding the FCNCs at tree level. The alignment condition holdsat high energies, but it has been shown that it does not hold at low energies because of thequantum corrections concerning the quark sector [7].

Different parameterizations of the couplings have been suggested. Haber and Davidson[8] propose three different tan β-like parameters, associated to the up, down and chargedlepton sector. They also emphasize that the parameter tan β = v2/v1 of the MSSM is not aphysical parameter of the type-III THDM, because it corresponds to a specific basis of theHiggs doublets. In the framework of the ATHDM [6, 7], there are also three parameters ζ f

related to the up, down and lepton sector. In addition, there is also the version by Ibarraet al [9], who defined three different angles relative to the parameter tan β = v2/v1 of thetype-II THDM. In all cases, efforts have been made to systematically restrict the availableparameter space of the model under consideration using the known phenomenology. Our aimis to compare these different parameterizations and to try to unify all these studies in termsof the most convenient parameters. These parameters should be such that they can be easilycompared to the type-II THDM in order to take it as reference and differentiate between themodels.

The THDM is an SU (3)C ⊗ SU (2)L ⊗ U (1)Y gauge model with the particle spectrumof the SM plus two scalar doublets with hypercharge Y = 1

2 . In general, both neutralcomponents of the two scalar fields can acquire vacuum expectation values (VEV) differentfrom zero. However, using a global SU (2) transformation, it is possible to choose so thatonly one of them has non-zero VEV. Then, the three Goldstone fields are in just onedoublet,

H1 =(

G+1√2(v + S1 + iG0)

), H2 =

(H+

1√2(S2 + iS3)

). (1)

In this form, the scalar particle spectrum consists of the charged fields H± and the neutralfields φ0 = (h, H, A) which in general can be written as φ0i = Ri jS j, where the orthogonalmatrix Ri j is defined by the scalar potential. If CP invariance is assumed then A = S3

and (Hh

)=

(cos(α − β) sin(α − β)

− sin(α − β) cos(α − β)

) (S1

S2

), (2)

where the angle (α − β) appears as a physical parameter.The most general expression for the Lagrangian in the THDM is

−LY = Q0Lη

U,01 �1U

0R + Q0

LηD,01 �1D0

jR + Q0Lη

U,02 �2U

0R + Q0

LηD,02 �2D0

R

+ L0Lη

E,01 �1E0

R + L0Lη

E,02 �2E0

R + h.c., (3)

where ηU,Di are the non-diagonal mixing matrices 3×3, �i = iσ2�i, (U, D)R are right-handed

fermion singlets, QL are left-handed fermion doublets, and the index 0 indicates that the fields

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

are not mass eigenstates. The Lagrangian in the Higgs basis can be rotated in an angle β

conveniently from(η0

1, η02

)to (κ0, ρ0) and it can be written as

−LY = Q0Lκ

U,0H1U0R + Q0

LκD,0H1D0

R + Q0Lρ

U,0H2U0R + Q0

LρD,0H2D0

R

+ L0LH1κ

E,0E0R + LLH2ρ

E,0E0R + h.c.. (4)

Generally, the mass matrices in equation (3) cannot be diagonalized simultaneously. Wedefine the mass basis in the usual way using matrices V f=U,D,E

L,R and the Lagrangian turnsinto [8]

−LY = 1√2

D[κDsβ−α + ρDcβ−α]Dh + 1√2

D[κDcβ−α − ρDsβ−α]DH + i√2

Dγ5ρDDA

+ 1√2

U[κU sβ−α + ρU cβ−α]Uh + 1√2

U[κU cβ−α − ρU sβ−α]UH − i√2

Uγ5ρUUA

+ 1√2

L[κLsβ−α + ρLcβ−α]Lh + 1√2

L[κLcβ−α − ρLsβ−α]LH + i√2

Lγ5ρLLA

+ [U (VCKMρDPR − ρUVCKMPL)DH+ + νρLPRLH+ + h.c.], (5)

where K =VUL V D

L†, κ f =V f

L κ f ,0V fR

†is the corresponding diagonal matrix, and

ρ f =V fL ρ f ,0 V f †

R . It is worth noting that equation (5) exhibits tree-level FCNCs unless theρF are diagonal. A sufficient condition is that each fermion type F = {D,U, L} couples toonly one Higgs doublet [2]—this is equivalent to ηF,0

1 = 0 or ηF,02 = 0, which leads to the

relations ρF,0 = κF,0 cot β and ρF,0 = −κF,0 tan β, respectively.In the phenomenologically viable models, the third-generation Yukawa couplings

dominate, and one may define tan β-like parameters based only on the consideration of third-generation fermion couplings. One alternative is just to consider the tan β-like parameters as thethree alignment parameters to suppress FCNCs. For this Lagrangian, Haber and Davidson [8]have introduced three tan β-like parameters tan βD,E and cot βU in the case of one quark/leptongeneration. The construction of tan β-like parameters in the multi-generation THDM-III ismore complicated [8].

In the Cheng and Sher [4] parameterization, where the couplings are proportional to thegeometric mean of the two masses ξi j = λi j

√mimj/v, the third-generation Yukawa couplings

dominate, and it is possible to define tan β-like parameters from the relations

ρf33 = λ

f33

m33

v= λ

f33κ

f33, Yf = λ

f33M f , (6)

where Yf = vρf33 and M f = m f

33.3 Therefore, it is possible to establish a relationship betweenthe Haber and Davidson [8] parameterization and the one by Cheng and Sher [4].

On the other hand, the alignment hypothesis considers [6, 7]

ηU,02 = ξ ∗

U eiθηU,01 , ηF,0

2 = ξ f e−iθηF,01 , (7)

where F = D, E and on those grounds, if the alignment condition is fulfilled, the Y ′f and M′

fare proportional to each other [6, 7] and can therefore be simultaneously diagonalized. Theproportional factor is

ζ f = ξ f − tan β

1 + ξ f tan β, (8)

3 Cheng and Sher [4] parameterization can be written in general in terms of nine tan β-like parameters [10].

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

and comparing it to the notation by Haber and Davidson [8], it can be seen that the parametersζU,D,E of the alignment model [6, 7] are equivalent to the parameters tan βU,D,L since

tan βD = −ζD,

tan βL = −ζE ,

cot βU = ζU .

(9)

There is another parameterization of the alignment condition given by Ibarra et al [9], interms of phases relative to the usual parameter tan β. Thus, this parameterization is interpretedat the high energy cut-off scale, �, as:

ηf ,01 (�) = cos ψ fY

′′f η

f ,02 (�) = sin ψ fY

′′f . (10)

The above equations for the not-well-defined Yukawa Y ′′f should be solved to recover

the alignment definition. Hence, the alignment hypothesis assuming real parameters isη

f ,02 = tan ψ f η

f ,01 and after comparison with equation (7), it is obtained as tan ψ f = ξ f .

Considering that the alignment condition is satisfied (Y ′f and M′

f are proportional) then

Yf = − tan(β − ψ f )M f . (11)

Notice now that it is straightforward to recover the THDM-II (ψu = π/2)

tan(π/2 − β) = limtan ψu→∞

ζU = 1

tan β= cot β, (12)

moreover ψd,l = 0 suggests ζF = − tan β instead of the established relationship in the originalpaper [9].

The parameterization by Ibarra et al [9] is interesting because it allows one to recover atype-II THDM and preserves the parameter tan β, and is thus convenient for phenomenologicalanalyses. Moreover, in order to keep this recovery mode easier than in the previous definitions,it is convenient to re-define the alignment conditions for the up sector as

ηU,02 (�) = − cot ψUηU,0

1 (�), (13)

which produces

M′U = v√

2

(ηU,0

1 cos β + ηU,02 sin β

)= v√

2ηU,0

1

cos β

tan β(tan ψu − tan β), (14)

and

Y ′U = v√

2

(−ηU,01 sin β + ηU,0

2 cos β)

= − v√2ηU,0

1

cos β

tan ψu(tan β tan ψu + 1). (15)

Therefore, the alignment condition is

Y ′U = −1 + tan ψu tan β

tan ψu − tan βM′

U = cot(β − ψu)M′U (16)

and it allows one to establish a relationship between the parameterizations given by ζU =cot(β − ψu) and ζF = − tan(β − ψF ) where (F = D, L). Using the current parameterizationwhich is YF = − tan(β − ψF )MF and YU = cot(β − ψu)MU , the limit of the THDM-IIcorresponds to ψu = ψd = ψe = 0.

Accordingly, it is possible to establish a relationship between the Cheng and Sher [4]parameterization and the newly proposed alignment notation using equations (11) and (6),

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

Table 1. Alignment conditions defined as ηf ,02 = Xf η

f ,01 .

New Pich et al Ibarra et al

Xu − cot ψu ξ ∗U eiθ tan ξU

Xb tan ξb ξD e−iθ tan ξD

Xe tan ξe ξL e−iθ tan ξL

Table 2. tan β-like parameters defined asYf = Xf M f . The columns are obtained from the alignmentconditions in table 1.

Alignment New by [6, 7] by [9] by [8] by [4]

Xu cot(β − ψu) ζ ∗U − tan(β − ψu) cot βU λtt

Xd − tan(β − ψd ) ζD − tan(β − ψd ) − tan βD λbb

Xe − tan(β − ψe) ζL − tan(β − ψe) − tan βL λττ

assuming that at a high energy cut-off scale the two Yukawa couplings for each fermion typeare aligned. Then, they are simultaneously diagonalized and FCNCs are absent at tree level.Establishing the equalities

λF33(�) = tan(ψF − β), λu

33(�) = cot(β − ψu) (17)

and λfi j(�) = 0 for i = j. Non-zero non-diagonal couplings can be generated when evolved

from the scale � to the electroweak scale [9].To summarize, we present in table 1 three different parameterizations for the alignment

hypothesis. Table 2 contains the relationships between the meaningful parameters on themodels already presented by Haber and Davidson [8], Cheng and Sher [4], Pich et al [6, 7]and Ibarra et al [9] plus our modification.

In the second part of this work we focus on how to differentiate between the type-II andtype-III THDM, where we will use the parameterizations shown in table 2. One way of doingthis is to compare the predictions for phenomenological processes between the models. We aregoing to use processes where the charged Higgs boson is involved. The usual processes used tolook for the charged Higgs boson experimentally are the decays H+ → tb and H+ → τ+ντ .We propose the following ratio between these typical processes and compare them in theframeworks of type-II and type-III THDM:

RH+ = B(H+ → τ+ντ )

B(H+ → tb)= �(H+ → τ+ντ )

�(H+ → tb), (18)

and we evaluate it in the framework of the type-III THDM. The decay widths at tree level areexplicitly

�(H+ → tb)III = 3mH+K2tb

16πv2

[(1 − m2

t

m2H+

)(cot(β − ψu)

2m2t + tan(β − ψd )2m2

b

)− 4 cot(β − ψu) tan(β − ψd )

m2t m2

b

m2H+

](1 − m2

t

m2H+

)

�(H+ −→ τντ )III = m+Hm2

τ

16πv2

(1 − m2

τ

m2H+

)2

tan(β − ψe)2. (19)

Using table 2 it is straightforward to write down these equations in the differentparameterizations of the type-III THDM. Notice that the RH+ expression will be in terms

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

of the three independent parameters tan(β − ψ f ). The parameter space of the type-III THDMis without doubt bigger than the parameter space of the type-II THDM where only onetan β-like parameter appears. In order to present the predictions for this RH+ in the type-IITHDM we evaluate the special case where ψu = ψd = ψe = 0, using the new parameterizationintroduced in table 2.

Then, by using the available constraints from flavor physics, we will calculate the expecteddeviations of the observable RH+ from its THDM type-II value. Instead of exploring the fullparameter space of the type-III THDM, we will focus on the limiting cases where at least twoof the tan β-like parameters are equal. In this way, the type-III THDM under the alignmenthypothesis should be explored in such a way that the usual type-II THDM can be clearlyidentified to allow comparisons. Therefore, we are going to consider three different THDM-II-like models, which are based on the fact that the parameters tan βd or tan βe will be establishedexperimentally more easily than the others. This could be done considering the following threedifferent cases:

• tan(β − ψd ) = tan(β − ψe). Here tan β = tan(β − ψd ) will be used as an input in thephysical basis [11] of one specific point of the type-II THDM. Switching to the Higgsbasis [8], we will determine the range of tan(β − ψu) allowed from the constraints inflavor observables and from there we will quantify the expected deviations of RH+ fromthe THDM type-II expectations. The procedure will be repeated for each set of the threetan β-like parameters allowed from the constraints in flavor observables, to be specifiedbelow.

• cot(β −ψu) = − tan(β −ψd ). We will apply the previous procedure but with tan(β −ψe)

instead of tan(β − ψu).• cot(β − ψu) = − tan(β − ψe). The starting point to define the type-II THDM is in this

case tan β = tan(β − ψe).

Notice that the Higgs sector of the MSSM corresponds to the third case where the deviationsfrom THDM-II are in the down sector [8, 12, 13]. The Zee model of neutrino masses andmixings corresponds to the second case where the deviations from THDM-II (required toexplain the solar mixing angle) are in the lepton sector [14].

Before we deal with the ratio in equation (18), it is important to look for the availableparameter space already constrained using the low energy phenomenology. Constraints havealready been evaluated using flavor-changing processes at low energies in the THDM modelwith Yukawa alignment [7, 10] and in more general MFV constructions, where the higher-orderpowers of the Yukawas are included in MFV THDM models [15].

In order to take into account many of these observables we have used the availablesoftware 2HDM-Calculator [11] and SuperIso [16]. As in [10] we use the implementation ofthe Yukawa couplings in the code 2HDM-Calculator [11] with

[ρU ]ii = cot(β − ψu)[κU ]ii, [ρF ]ii = tan(β − ψF )[κF ]ii F = D, E, (20)

and the numerical evaluation of the flavor physics observables with SuperIso v3.2. We referthe reader to the original paper [11] for details about the flavor physics constraints used andthe set of the input parameters.

As has been pointed out in [6, 7, 10], the most restrictive processes in the ATHDMare B → Xsγ , �0(B → K∗γ ), Bu → τντ , and Z → bb. In this work we also take intoaccount the strong constraints on B → μ+μ− recently reported by the LHCb collaboration[17, 18] which help to further constrain the parameter space. The results are presented infigure 1. Two different cases are presented: tan(β − ψd ) = tan(β − ψe) (left side) andcot(β − ψu) = − tan(β − ψe) (right side) for the charged Higgs boson masses 350 and

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

Figure 1. Constraints from B → Xsγ and other processes like the decay Bs → μ+μ−.The final allowed region is the light gray (yellow) one. Two different cases are presented:tan(β − ψd ) = tan(β − ψe) (left side) and cot(β − ψu) = − tan(β − ψe) (right side) for thecharged Higgs boson masses 350 and 800 GeV, respectively, from up to down. How the constraintswork is discussed in the text.

800 GeV, respectively, from up to down. The allowed regions from B → Xsγ of the parameterspace (tan(β −ψd )− cot(β −ψu)) are shown in dark gray (green) in figure 1, and correspondto those shown in [6, 7, 10]. In this work we choose to display the other important constraints inthe same (tan(β −ψd )− cot(β −ψu)) plane in order to illustrate their effectiveness. The finalallowed region is shown in the light gray (yellow) regions. The branches with simultaneouslylarge and equal sign values of tan(β −ψd ) and cot(β −ψu) are excluded [10] from the isospinasymmetry in the exclusive decay mode B → K∗γ

�0(B → K∗γ ) ≡ �(B0 → K∗0γ ) − �(B− → K∗−γ )

�(B0 → K∗0γ ) + �(B− → K∗−γ ). (21)

In the horizontal axis, the constraint in the central region (the interval ∼[−1, 1]) is comingfrom Bs → μ+μ− with few changes for the illustrated cases, while in the vertical axis thereare strong bounds coming from Bu → τντ in the central region. As a complement to this,in figure 2 the plane tan(β − ψe) − cot(β − ψu) is shown taking the charged Higgs bosonmasses 350 and 800 GeV, the dark gray (green) region again is that allowed by B → Xsγ butprocesses like Bs → μ+μ− constrain the allowed region only to the central zone. Notice thatfor MH+ = 800 GeV the bound from Bs → Xsγ is satisfied in the whole plane (dark grey(green) background) and the final allowed region (light gray (yellow)) is obtained from the

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

Figure 2. Constraints from B → Xsγ and other processes like the decay Bs → μ+μ− in the planetan(β − ψe) − cot(β − ψu) taking the charged Higgs mass 350 and 800 GeV. The final allowedregion is the light gray (yellow) one.

Figure 3. The ratio RH+ as a function of the parameter tan(β − ψd ) for different values of thecharged Higgs boson mass mH+ = 350 800 GeV using the assumption tan(β−ψd ) = tan(β−ψe).

other already mentioned FCNC constraints. In the calculations for the process Bs → μ+μ−,it is necessary to fix the Higgs mass values in the neutral sector—we have chosen mh =125 GeV, mH = mH+ + 10 GeV and mA = mH+ + 15 GeV and we have found that theconstraints coming from Bs → μ+μ− have a mild dependence on the neutral Higgs masses.

Taking into account the whole set of constraints, the ratio RH+ (18) is plotted as afunction of the parameter tan(β − ψd ) for different values of the charged Higgs bosonmass 350 and 800 GeV. Although we have written the tree-level formulae (19), we haveevaluated RH+ using 2HDMC which includes the one loop contributions to these decays.We have considered the three already mentioned cases where tan(β − ψd ) = tan(β − ψe)

(figure 3), − tan(β − ψe) = cot(β − ψu) (figure 4) and cot(β − ψu) = − tan(β − ψd )

(figure 5). In figures 3–5, the parameter cot(β − ψu) is taking the allowed values obtainedfrom figures 1 and 2. Also, in these plots, the THDM type-II contribution is shown as ablack solid line inside the larger allowed region by the type-III THDM. The observation oflight-charged Higgs masses or RH+ > 2 × 10−1 are a clear signature of Yukawa structuresbeyond the type-II THDM. Even for small RH+ an anomalous value of tan(β − ψd ) with

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

Figure 4. The ratio RH+ as a function of the parameter tan(β−ψd ) for different values of the chargedHiggs boson mass mH+ = 350 800 GeV using the assumption cot(β − ψu) = − tan(β − ψe).

Figure 5. The ratio RH+ as a function of the parameter tan(β−ψd ) for different values of the chargedHiggs boson mass mH+ = 350 800 GeV using the assumption cot(β − ψu) = − tan(β − ψd ) and−0.5 � cot(β − ψu) � 0.5.

respect to the THDM type-II expectations could be identified. It is worth stressing that forcases cot(β − ψu) = − tan(β − ψe) and cot(β − ψu) = − tan(β − ψd ) large deviations fromthe type-II THDM are expected. However, the path from discovering the charged Higgs toactually measuring this observable may well not be possible at the LHC. Even at the futuree+e− linear colliders the attainable precision for measuring RH+ is expected to be only aroundthe level of 10% in type-II THDM [19], with some regions of the parameters where the τν

channel is not observable at all. It is worth stressing that in THDM-III, specifically in the casesillustrated in figure 4 (cot(β − ψu) = − tan(β − ψe) case), there may well be regions whereonly the τν channel is observable.

To conclude, different parameterizations of the type-III THDM have been reviewed andcompared under the alignment hypothesis. Those parameterizations are based on the workby Haber and Davidson [8], Cheng and Sher [4], Pich et al [6, 7], Ibarra et al [9] and ourmodification introduced in table 1. Equivalences between them have been found and are shownin table 2. On the other hand, different constraints on the parameter space of the ATHDM havebeen found. The relevant bounds are coming from B → Xsγ , �0(B → K∗γ ), Bu → τντ andBs → μ+μ− for the parameters involved in the charged Higgs sector. The B → Xsγ process

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J. Phys. G: Nucl. Part. Phys. 40 (2013) 095004 H Cardenas et al

is evaluated at NNLO in this framework and for the process Bs → μ+μ− the most restrictivebound by the LHCb collaboration is used. In the general type-III model it is necessary toextract three physical tan β-like parameters from collider observables, like the charged Higgsbranchings. In this work we propose the fraction RH+ = B(H+ → τ+ντ )/B(H+ → tb) inorder to quantify the differences between them. Our results show significant differencesbetween the type-II and III THDM, even when a simplified version of the type-III THDMis considered (ATHDM). The Lagrangian of these kinds of models (type-III THDM andATHDM) is similar to the effective Lagrangian coming from models with extra symmetries,either discrete symmetries or supersymmetry. By obtaining the effective Lagrangian, the extrasymmetries in those models are usually broken and their tan β parameters cease to be welldefined [8]. In this work we have illustrated the necessity to replace tan β with parametersbetter suited for general phenomenological and theoretical studies of the scalar sector. Finally,a neutral Higgs boson has been found at the LHC [20]—the next task should then be thedetermination of the underlying scalar sector.

Acknowledgments

We acknowledge C Sandoval for useful discussions and carefully reading of the paper. J-ARand HC have been supported in part by UNAL-DIB-Bogota grant 14844 and DR has beensupported in part by UdeA/2011 grant: IN1614-CE.

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