2.1. SM gauge and fermion sector

The weak and electromagnetic interactions are partly unified into the Electroweak theory, which has the gauge symmetry SU(2) _L x U(1) _Y . The Lagrangian for the gauge and fermion sector (leptons) is given by:

where

Here L denotes the left-handed lepton doublet, while e _R corresponds to the right-handed electron, σ ⁱ are the Pauli matrices and g y g ^’ denote the gauge coupling constants. The tensors Wμνi and B _μv are given by:

In addition we need to include the quarks, with the left-handed components forming a doublet of weak isospin, while the right-handed ones transform as singlets. Quarks are also triplets of the color interactions, which is described by the gauge symmetry SU(3) _c . Thus, the full SM gauge symmetry is: SU(3) _c x SU(2) _L x U(1) _Y .

2.2. SSB and the Higgs

Insights from the role of the vacuum in condensed matter, helped to identify how a symmetry is realized in QFT, namely:

SSB happens for instance in strong interactions, where the Chiral symmetry is not manifest, and the pions are identified as pseudo Nambu-Goldstone bosons (NGB) (pseudo means they are not exactly NGB, due to the small mass of the light quarks (u, d, s)).

In the case of the weak interactions, the problem was how to include the mass for the charged W bosons, the mediators of the weak interactions that manifest in neutron beta decay. The solution came with the BEH mechanism [⁶-⁸], which proved that within the context of a local symmetry, the degrees of freedom associated with the Goldstone bosons become the longitudinal modes of the gauge bosons, which then acquire a mass.

Let us discuss this mechanism for a simple Abelian theory, with Lagrangian:

where λ > 0.

This Lagrangian (6) is invariant under the parity (P) transformation φ → -φ. However, when one minimizes the potential, we find that:

• (a) For μ ² > 0, the minimum is invariant under P, and it occurs for:

• (b) For μ ² < 0, the minimum is displaced from the origin, and does not respect P, i.e.

Now, there is a degeneracy between v y -v. Then, one can study the fluctuations around the new minimum, which are interpreted as the particles, i.e. ξ(x) ≡ φ(x)-φ0 = φ(x)-v.

Then, the Lagrangian for the field ξ (the fluctuation) becomes:

Thanks to SSB, this Lagrangian contains now a scalar field ξ with mass mξ=2λv=-2μ2. When one uses SSB within the context of a gauge theory, it is possible to generate masses for the gauge bosons.

Within the SM the gauge symmetry for the electroweak interactions is SU(2) × U(1) _Y , with gauge bosons Wμ±,Wμ3 and B _μ . The minimal Higgs sector includes one Higgs doublet, which can be written as follows:

The mass terms for the W; Z is obtained from the the Higgs Lagrangian, which includes the Higgs kinetic term, its gauge interactions, and the Higgs potential, i.e.

The potential V (Φ) is written as:

For μ ² < 0, the gauge symmetry SU(2) _L × U(1) _Y is broken to U(1) _em . This is illustrated in Fig. 1, where we can see the different options for the vev (red points), we also display the 3-dimensional image of the potential, i.e. the mexican hat potential. In this case, part of the scalar degrees of freedom from the doublet, the so-called pseudo Goldstone bosons (pGB), become the longitudinal modes of the W ^± and Z, which then become massive. The charged pGB is identified as: G±=(1/2)(ϕ1±iϕ2), while the neutral pGB correspond to: G ⁰ = φ ₄. The remaining degree of freedom (φ ₃) includes a physical d.o.f., which by itself “forms an incomplete scalar multiplet”, as Higgs pointed out in his classic paper [⁶].

Figure 1 The Higgs potential when SSB occurs (left), and the 3-dimensional image, i.e. the mexican hat potential (right). 

However, the gauge symmetry leaves us some freedom to remove non-physical degrees of freedom. This is done in the so called Unitary gauge, where one can take: φ ₁ = φ ₂ = φ ₄ = 0 y φ ₃ = v + h; and h corresponds to the excitations from the vev. Thus, the Higgs doublet Φ can be written in the unitary gauge as:

Then, the potential V (φ) becomes:

The second term in the potential (quadratic in h) indicates that the Higgs (h) mass is m ² _h = 2v ² ‚. The cubic and quartic terms (h ³ ; h ⁴) describe the 3- and 4-point Higgs self-couplings.

On the other hand, the interaction of the Higgs with the gauge bosons, is contained in the kinetic term, which can be written as:

We define new vector fields:

Thus, the Lagrangian (15) contains mass terms for these fields, i.e.MW2Wμ+W-μ+(1/2)MZ2ZμZμ. The corresponding_p masses given by M _W = (vg=2), M _Z = (v=2) g2+g'2, while the photon remains massless, i.e. M _A =0.

We can also express the electric charge (e) in terms of the gauge couplings g y g ^’ , as follows: e = (gg ^’ /g2+g'2) = g sin θ _W , where θ _W denotes the weak mixing angle. Within the SM, there holds a relationship between the gauge boson masses, which turns out to be of crucial importance, i.e. MW=MZcos⁡θW, with cos⁡θW=(g/g2+g'2) and sin⁡θW=(g'/g2+g'2). Finally, one can also expres the Fermi constant as: GF=2(g2/8MW2)=(2/2v2).

The numerical value of the fermi constant is: GF=1.16637×10-5 GeV ^-2 [³⁴], which implies: v ~ 246 GeV, this value is known as the Electroweak scale. One can check that the interactions of the type hVV and hhVV are contained in this Lagrangian, namely:

In the SM, the fermions are chiral fields, i.e. the left and right-handed fermions transform differently under the gauge symmetry, and therefore they should be massless. However, a miracle happens here: the Higgs mechanism with a minimal Higgs doublet induces the mass for the SM fermions too. Fermion masses are obtained from the Yukawa Lagrangian that includes the coupling of the fermion doublet F _L with the Higgs doublet (Φ) and the right-handed fermion singlet f _R , which for the leptons and quarks take the form:

After SSB, all the SM fermions acquire masses (except the neutrinos), which is given by: m _f = y _f (v=2). The Yukawa Lagrangian includes the Higgs-fermion interactions too, which turn out to be proportional to the fermion mass, i.e. (hf f) = m _f /v.

2.3. SM Higgs parameters: LEP and indirect constraints

Within the SM, the Higgs sector includes only two parameters: the (dimensional) quadratic mass term μ ² and the dimensionless quartic coupling λ. Once SSB is implemented, these parameters can be traded by the Higgs vev (v) and the Higgs mass (m _h ). But what to expect for λ? Over the years, a cocktail of arguments was developed that helped to constrain the Higgs mass, namely:

A graphic summary of these constraints was presented in Ref. [³⁹], which we show in Fig. 2 (left). Thus, already by 2000, it was known that the favored Higgs mass range was 110 < m _h < 180 GeV. Fig. 2 (right) , from Ref. [⁴⁰], shows that the probability distribution for the Higgs mass, which is constrained by radiative corrections to lay in the range 110 < m _h < 130 GeV.

Figure 2 Higgs mass limts from theoretical considerations (left) and EWPT (right). Figure 2 (left) is from [³⁹], and Fig. 2 (right) is from [⁴⁰]. 

3.1. General Couplings and Feynman rules

As mentioned in the introduction, in order to study the Higgs properties at the LHC, it is both instructive and useful to present a parametrization of the Higgs couplings that is general enough, while at the same time it makes easy to reduce to the expressions to the SM limit, or some other popular SM extension, such as the 2HDM.

Thus, we shall start by describing the coupling of the scalar boson (h) with Vector bosons W ^± , Z. In this case we can write the interaction Lagrangian with terms of dimension-4, consistent with Lorentz symmetry and derivable from a renormalizable model, as follows:

When the Higgs particle corresponds to the minimal SM, one has κ _W = κ _Z = 1, while values κ _W = κ _Z ≠ 1 arise in models with several Higgs doubles, which respect the so called custodial symmetry.

On the other hand, the interaction of the Higgs with fermion f, with f = q or l for quarks and leptons, respectively, can also be written in terms of a dimension-4 Lagrangian that respects Lorentz invariance, namely:

The CP-conserving and CP-violating factors S _ij , P _ij , which include the flavor physics we are interested in, are written as:

Within the SM c _f = 1 and d _f = e _f = g _f = 0, which signals the fact that within the SM the Higgs-fermion couplings are CP-conserving and flavor diagonal, i.e. no FCNC scalar interactions. However, as we shall see next, the LHC analysis has been reported by considering c _f ≠ 0 (but with d _f = e _f = g _f = 0). These factors take into account the possibility that the Higgs boson is part of an enlarged Higgs sector, and the fermion masses coming from more than one Higgs doublet. As we shall discuss in the next sections, for specific multi-Higgs models, the explicit dependence on the Yukawa matrix structure, i.e. flavor physics, is contained in the factor η _ij , ηij'. On the other hand, when we have e _f ≠ 0, but d _f = g _f = 0, it indicates that the Higgs-fermion couplings are CP violating, but still flavor diagonal. In models with two or more Higgs doublets, it is possible to have both flavor changing scalar interactions (d _f ; g _f ≠ 0) and CP violation.

3.2. Higgs decays

For the study of Higgs phenomenology, we need to know first the Higgs branching ratios, including its decay into the most relevant modes (BR(h → XX)). For the mass range left by the analysis of EWPT, i.e. m _h ≈ 105 → 130, we know that its main decays modes must be: h→bb¯,cc¯,τ+τ-,γγ,ZZ,WW+,gg, whose decay widths are presented in the literature (See for instance ref. [¹⁵]). The Branching ratio for the mode h → XX is defined as:

The total width Γ _total is given by the sum over all the partial widths, and its value is about 10 ^-3 GeV. These decays have been evaluated in the literature, including QCD and EW corrections. Results for the corresponding branching ratios are shown in Fig. 3, from Ref. [⁴¹]. We can see that the dominant mode is h→bb¯, while h → ZZ, WW ⁺ come next, and h →τ ⁺ τ ^- is also relevant. Although the loop decay into a photon pair reaches a branching ratio of only about 2 ×10 ^-3 for m _h = 120 - 130 GeV, it has a great relevance because it provides a very clean signature. On the other hand, the loop decay into gluon pair is important, not because it could be detected as a Higgs decay, but because it serves for the Higgs production mechanism through gluon fusion.

Figure 3 Higgs couplings derived at LHC. Figure from Ref. [⁴¹]. 

3.3. Higgs search at LEP

The clean environment of electron-positron colliders was used first to search for the Higgs boson. At the first stage of LEP, which worked on the Z-pole, the Higgs was searched through the Z decay: Z → f f + h; from the non-observation of such decay it was possible to put the bound: m _h ≥ 80 Gev. Then, the second phase of LEP worked with a cm energy of 200 GeV, and then the search for the Higgs used the reaction (Bjorken mechanism): e ⁺ e ^- → Z +h. Again, the absence of a signal resulted in the bound: m _h ≥ 200 - m _Z ≃ 110 GeV.

In the late stages of LEP, with a cm energy of about 205 GeV, this bound was slightly extended, up to about 114 GeV. Just when the CERN plans marked that LEP must be closed, in order to start the LHC project, some debate arose because there was a claim for the presence of a Higgs signal with m _h = 115 Gev, which produced lots of enthusiasm among some experimentalist (who pledged for more time and energy for LEP) and also among some theorists, who quickly cooked models that could explain such mass value. The CERN director decided to close LEP and keep the plans for LHC, which a posteriori seems it was the best decision that could be made.

3.4. Higgs production at Hadron colliders

Although at a proton collider it is also possible to produce Higgs bosons with mechanism similar to the one used at LEP, the Bjorken mechanism, namely with quarks and antiquarks colliding to produce the Higgs boson in association with a massive gauge boson (W or Z), it turns out that the dominant production mechanism is gluon fusion, despite the fact that it occurs at loop level (with top quarks circulating in the loop). This is so, because gluons carry the largest fraction of momentum from the colliding protons, and also because its parton distribution function provides with the largest luminosities. The production of the Higgs boson with a pair of heavy quarks (top mainly) also reaches a significant cross section. The resulting cross sections for these mechanisms in proton proton colliders, times the branching ratios of some relevant modes, as a function of the cm energy, is shown in Fig. 4 [⁴¹].

Figure 4 Higgs cross section times BR for different modes, at LHC with Ecm = 8 TeV. Figure from [⁴¹] 

3.5. Current Higgs profile from LHC

LHC started collecting data from pp collisions with cm energy of S=7 TeV. After some initial claims for a Higgs signal around 140 GeV, which was again quickly explained by some theory models, the Higgs became real when both collaborations (ATLAS and CMS) announced on July 4th, 2012 that some events on the γγ and ZZ* channels were observed, which would correspond to a SM-like Higgs particle with mass m _h = 125 ¡ 126 GeV. The observation of a resonance decaying into γγ indicated that its spin must be either 0 or 2, in accordance with Yang theorem. An scalar, with spin-0, seemed the most natural explanation, which was reinforced by the second decay mode, namely the four-lepton signal, which could be interpreted as coming from the decay h → ZZ ^* → llll. Habemus Higgs!

As LHC continued to accumulate luminosity, more Higgs decay modes were observed, which confirmed that the particle observed by ATLAS and CMS was indeed a Higgs-like particle. The interpretation of the Higgs signal was further reinforced after LHC started taking data with higher energy (S=13 TeV). Now the signal can be appreciated even by the public eye, for instance the data from CMS on the four-lepton signal shown in Fig. 5 (from ^{Ref. [ 42]}) shows clearly a bump in the invariant mass at 125 GeV (pink), which stands clearly above the SM backgrounds. There are now a variety of signals that have been measured, which seem all consistent with the SM, as it is shown in Fig. 6 (left), from ATLAS (^{Ref. [43]}). The mass has been measured with better precision, as it can be seen in Fig. 6 (right) .

Figure 5 Higgs signal from the decay h → ZZ ^* from CMS. Figure from Ref. [⁴²] 

Figure 6 Higgs signals (left) and mass summary (right) from ATLAS. Figure from ref. [⁴³] 

The current LHC data on Higgs production has been used to derive bounds on deviation from the SM predictions for the Higgs couplings. For instance, Fig. 7 from Atlas Collaboration (from ^{Ref. [44]}) shows the allowed deviations for the Higgs couplings with gauge bosons and fermions, as parametrized by the constants κ _V and κ _F . Thus, one can see that best fit lays quite close to the point κ _V = κ _F = 1, which corresponds to the SM limit. However, small deviations are allowed within the precision reached at LHC.

Figure 7 Higgs couplings with vectors and fermions. Figure from Ref. [⁴⁴] 

In essence, the Higgs particle couples to a pair of massive gauge bosons or fermions with a strength proportional to their masses. So far, the LHC has tested only a few of them, namely the Higgs couplings with the gauge bosons and the heaviest fermions of the third family. This is shown in Fig. 8, from both ATLAS [⁴⁴] and CMS [⁴⁵] collaborations, where one can appreciate that the Higgs couplings lay on a straight line, modulo some small deviations for the Higgs coupling with bb¯ (which is still consistent at one sigma); the mode μμ has not been detected yet. These results, allow some regions of parameter space for the so-called “private Higgs” hypothesis, where each fermion type gets its mass from a different ent Higgs doublet, as it will be discussed briefly in our conclusions. Moreover, these constraints are obtained assuming SM-like pattern for the Higgs couplings, however when one considers new physics, i.e. models beyond the SM [⁴⁶], it is possible to have non-standard Higgs couplings, including the flavor and CP-violating ones. These will be discussed in the coming sections.

Figure 8 Higgs couplings as function of the mass from CMS (from [⁴⁵]) and ATLAS (from [⁴⁴]). 

4.1. Higgs and Supersymmetry

One of the most widely studied extensions of the SM is Supersymmetry. This beautiful idea implies that for each fermion there is a bosonic superpartner, as they are related by a SUSY transformation. The minimal implementation of such idea is the so called minimal SUSY extension of the SM (MSSM) (for a review see [⁴⁸, ⁴⁹]). The MSSM contains 3 families of chiral superfields, which include the quarks and leptons as well as their scalar superpartners, the squarks and sleptons. The gauge sector is described with the vector superfields associated with SU(3) _c × SU(2) _L × U(1) _Y , and includes the usual gauge bosons, but also their fermionic superpartners, the gauginos. In order to give masses to both up-and down-type fermions, one must have two Higgs doublets, described by chiral superfields, and thus one also has the Higgsinos, as the fermionic superpartners. A discrete symmetry, R- parity, must be imposed in order to respect lepton and baryon number conservation. This R- parity implies that superpertners are produced by pairs, and the lightest superpartner must be stable, which provides the seeds for a dark matter candidate, which was considered a great virtue of SUSY models. As the experimental evidence shows, SUSY must be broken, and a realistic scheme is provided by the Soft-breaking terms. Namely, SUSY is assumed to be broken in a Hidden sector, which is then transmitted to the MSSM superpartners by a suitable mediator, which may involve gravity or gauge interactions, or anomaly mediation.

The search for the signals of the superparners has been one of the central goals of current and future colliders [⁵⁰]. However, the LHC has provided bounds on the new physics scale (Λ), either from the search for new particles or from the effects of new interactions, with values that are now entering into the multi-TeV range. This result is casting some doubts about the theoretical motivations for new physics scenarios that were promoted assuming a mass scale of order TeV. This is particularly disturbing for the concept of naturalness, and its supersymmetric implementation, since the bounds on the mass of superpartners are passing the TeV limit too [⁵⁰]. However, SUSY is such a beautiful theory that it certainly desserves further work on its foundations and possible realization in nature. Thus, one has to wait for the next LHC stages, with higher energy and luminosity, in order to obtain stronger limits, both on the search for new particles, such as heavier Higgs bosons [⁵¹-⁵³] or others, and for precision tests of the SM properties.

4.2. Higgs and flavor: A more flavored Higgs sector

Although the Higgs boson has diagonal couplings to the SM fermions at tree-level, it could mix with new particles that have a non-aligned flavor structure, and this permits to induce corrections to the diagonal Yukawa couplings and/or new flavor-violating (FV) Higgs interactions. Non-standard Higgs couplings are predicted in many models of physics beyond the SM. This could happen, for instance, in extensions of the SM that contain additional scalar fields, which have non-aligned couplings to the SM fermions. When these fields mix with the Higgs boson, it is possible to induce new FV Higgs interactions. This transmission of the flavor structure to the Higgs bosons, was discussed in our earlier work [⁵⁴], where we called it more flavored Higgs boson.

This occurs, for instance, in Froggatt-Nielsen type models, where one includes a SM singlet (Flavon) that participates in the generation of the Yukawa hierarchies. This singlet mixes with the Higgs doublets, and induces FV Higgs couplings at tree-level. The phenomenology for the mixing of the SM Higgs doublet with a flavon singlet has been studied in Refs. [⁵⁵, ⁵⁶]. Mixing of SM fermions with exotic ones, could also induce FV Higgs couplings at tree-level [⁵⁷]. On the other hand, within supersymmetric models, like the MSSM, the sfermion/gauginos can have non-diagonal couplings to Higgs bosons and SM fermions. Then, FV couplings of the Higgs with SM fermions could be induced at loop-level [⁵⁴].

In the next section, we shall discuss what is probably the simplest model that contains a more flavored Higgs sector, namely the general two-Higgs doublet model [⁴⁶].

4.3. Higgs and Dark Matter

Understanding the nature of dark matter is another aspect of physics beyond the SM, which motivates extensions of the SM. The dark matter could interact with ordinary mater through the Higgs boson, an scenario called the Higgs portal. It could also happen that dark matter is contained within an extension of the Higgs sector, which could include a singlet, an extra (Inert) Higgs doublet, a mixture of them or even a higher-dimensional multiplet [⁵⁸]. DM candidates can also arise in composite Higgs models [⁵⁹].

The case when DM candidate is contained in an extra Higgs doublet is known as the IDM [⁶⁰], which has been studied extensively in the literature [⁶¹-⁶³]. Including dark matter candidate with extra sources of CP violation, has motivated the constructions of models with inert doublet and singlet [⁶⁴].

Within those models, one can accommodate a neutral and stable scalar particle, which has the right mass and couplings to satisfy the constraints from DM relic density and direct/indirect searches. It also implies modifications of the SM-like Higgs couplings, which must mass the LHC Higgs constraints. The heavy Higgs spectrum provides additional signatures of these models, which could also be searched at future colliders.

5.1. Yukawa textures: parallel and complementary

The initial construction of the 2HDM with textures [⁸²], considered first the specific form with six-zeros, and some variations with cyclic textures. In that case a specific pattern of FCNC Higgs-fermion couplings of size mimj/v was found, which is known nowadays as the Cheng-Sher ansatz. For this type of Higgs-fermion vertex it is possible to satisfy the FCNC bounds with Higgs masses lighter than O(TeV). The extension of the 2HDM-Tx with a four-zero texture was presented in [⁸³, ⁸⁴]. The phenomenological consequences of these textures for Higgs physics (Hermitian 4-textures or non-hermitian 6-textures) were considered in [⁸⁵], while further phenomenological studies were presented in [⁸⁶-⁸⁹].

When the fermions (f = U, D, L) couples to both Higgs doublets, after SSB the mass matrix is given by:

Although the texture pattern is defined by considering the zeros appearing in the fermion mass matrix M _f , we can have several options for the textures of the Yukawa matrices, Y ₁ an Y ₂. In our work on the 2HDM-Tx [⁸¹], we considered several possibilities, namely:

For instance, a four-zero texture mass matrix is of the form:

In the parallel case we have that both Yukawa matrices (Y _i , i = 1, 2) have the same form, namely:

An example of complementary case, where Y ₁ ^f and Y ₂ ^f (which only has a non-zero 33 entry) combine to produce a mass matrix with four-zero texture, is the following:

Finally, an example of semi-parallel textures is the following:

in this example, Y ₁ ^f has a four-zero texture, while Y ₂ ^f has only a non-zero 33 entry.

The detailed study of these Yukawa matrices, including its diagonalization, and the phenomenological consequences, was presented in our Ref. [⁸¹]. Next, we shall derive the Yukawa lagrangian of the general 2HDM-III, and we shall employ those textures when considering the 2HDM-Tx.

5.2. The Yukawa lagrangian

Within the general Two-Higgs doublet model, each Higgs doublet couples to a given fermion of type f through the Yukawa matrices Y ₁ ^f and Y ₂ ^f , which combine after SSB to produce a fermion mass matrix with some texture. We shall write the Yukawa Lagrangian in the 2HDM-III for the quark sector, following the notation from [⁸⁵], namely:

where the quark doublets, quark singlets and Higgs doublets are written as:

And similarly for the leptons. We have defined the conjugate doublets as: Φ~j=iσ2Φj*=ϕj0*,-ϕj-T with ϕj0=(φi+iχi)/2.

On the other hand, the CP-properties of the Higgs spectrum depend on the parameters of the Higgs potential. For the 2HDM, this has been studied extensively [²⁴], using either explicit constructions [⁶⁹], or employing the basis-independent conditions that classify whether the vacuum respects CP, it violates explicitly CP or CP is spontaneously broken [⁹⁰-⁹³]. Here we shall follow the methods and notation from Ref. [⁹⁴]. Thus, the Higgs mass eigenstates (H _i ) are obtained by the orthogonal rotations, as follows:

We identify H ₄ = G ⁰ as the neutral pseudo-Goldstone boson.

For the case with a CP-conserving Higgs potential, one has that H4=cos⁡βχ1+sin⁡βχ2 is the Goldstone boson, where tan β = v ₂ =v ₁. In this case the couplings of the CP-even Higgs bosons, h and H, with W W or ZZ bosons are given by ghVV=sin⁡(β-α)ghVVsm and gHVV=cos⁡(β-α)ghVVsm, respectively.

The matrix R can be used to express the Higgs doublets in terms of the physical Higgs mass eigenstate, as follows:

and

The values of q _ra are shown in Table II; they are written as combination of the θ _ij , which are the mixing angles appearing in the rotation matrix that diagonalize the mass matrix for neutral Higgs.

After substituting these expressions in the Yukawa Lagrangian one obtains the Higgs-fermion interactions for the general 2HDM-III. They are written in terms of the quark mass matrices (q = U, D), which receive contributions from both vev’s, namely: Mq=(v1/2)Y1q+(v2/2)Y2q . As we mentioned in the previous subsection, the diagonal form of the quark mass matrices (M¯q) is obtained by applying the bi-unitary transformations, namely: M¯q=VLqMqVRq†. Explicit expressions for these matrices are shown in [⁸¹]. Thus, the couplings of the neutral Higgs boson (H _r , a = 1, 2, 3) with fermions of type f can be expressed in the form of Eq. 23, as follows:

For the up-type quarks, the factors Sijur and Pijur are written as:

and

Similarly, for the down-type quarks we find:

and

This Lagrangian is the most general one, which includes the possibility to have: i) deviations from the SM (diagonal) Higgs couplings, ii) new flavor violating Higgs couplings and iii) new sources of CP violation, coming either from the Higgs potential or the Yukawa matrices. Further simplifications can be obtained when one assumes that the Yukawa matrices are hermitic, i.e.Y~2Q=Y~2Q† (with Q = U, D) or when the Higgs potential is CP conserving.

Next we shall present some limiting cases for the couplings of Higgs H ^r (r = 1, 2, 3) with fermions of type f, expressed in terms of the factors Sijfa,Pijfa, which in turn can be written as:

As it is discussed in detail in ref. [⁸³], when one assumes parallel Yukawa matrices of the four-textur type, one can express those factors in terms of the quark masses, the mixing angles and some flavor-dependent parameters. In fact, in that case the ηijf parameters satisfy the Cheng-Sher ansazt and are universal (the same for all Higgs bosons), then it is possible, and convenient, to express them as follows:

The parameters χ~ijf can be constrained by considering all types of low energy FCNC transitions, which produce the viable regions of parameter space.

5.3. The 2HDM-III with non-Hermitian Yukawa matrices and CPC Higgs potential

In this case we shall consider that the Higgs sector is CP conserving (CPC), while the Yukawa matrices are non-hermitian. Then, without loss of generality, we can assume that H ₃ = A is CP odd, with H ₁ = h and H ₂ = H being CP even; then: cos θ ₁₂= sin (β - α), sin θ ₁₂ = cos (β - α), sin θ ₁₃ = 0, and e-iθ13=1. The expressions for the neutral Higgs masses eigenstates can be written now in terms of the angles α (which diagonalizes the CP-even Higgs bosons) and β (tan β = v ₂/v ₁):

where a = 1, 2, and for the CP-conserving limit v^a and b ŵ _a have a vanishing phase ξ = 0. Here, the components of v^a, ŵ _a are given as:

and

Additionally, when one assumes a 4-texture for the Yukawa matrices, the Higgs-fermion couplings further simplify as Y~2ijU=χijmimj/v . Then, coefficients (37) and (38) for up sector for r = 1,2,3, and with H ₁ = h being identified as the light SM-like Higgs boson, are shown in Table III.

The η parameters are given as follows:

Similar expressions are obtained for d-type quarks and leptons.

5.4. The 2HDM-III with Hermitian Yukawa matrices and CPV Higgs

In this case we assume the hermiticity condition for the Yukawa matrices, but the Higgs sector could be CP violating. For simplicity we shall consider that the Hermitic Yukawa matrices that obey a four-texture form, but now CP is violated in the Higgs sector.

Then, one obtains the following expressions for the couplings of the neutral Higgs bosons with the up-type quarks, namely:

and

Similar expressions can be obtained for the down-type quarks and leptons, as well as for the charged Higgs couplings.

5.5. The 2HDM-III with Hermitic textures and CP-conservation (2HDM-Tx)

In this case we assume the hermiticity condition for the Yukawa matrices, and the Higgs sector is CP conserving. For simplicity we shall consider that the Yukawa matrices obey a four-texture form. Then, the Higgs-fermion couplings take the following form. For r = 1, 2 one gets:

For both Higgs bosons H ₁ and H ₂ one has: Pij1u=Pij2u=0. On the other hand, for r = 3, one has: Sij3u=0 and

5.6. The 2HDM of type I, II, X, Y: CPC case

In all these cases each fermion type (U,D,L) couple only with one Higgs doublet, as shown in Table I. Now, we have that: d _fr = g _fr = 0, while the non-zero coupling constants (c _f for h, H or e _f for A), are shown in Table IV for models I and II, while Table V shows the corresponding couplings for models X and Y.

Then, the Higgs couplings with fermions and gauge bosons in these models are determined by the mixing angles α (which diagonalizes the neutral CP-even Higgs mass matrix) and tan β = v ₂ /v ₁. For a quick test of Higgs couplings, one can rely on the Universal Higgs fit [⁹⁵], where bounds on the parameters ϵ _X are derived, which are defined as the (small) deviations of the Higgs couplings from the SM values, i.e. ghXX=ghXXsm(1+ϵX). We find very convenient, in order to use these results and get a quick estimate of the bounds, to write our parameters as: η _X = 1 + ϵ _X . For fermions, the allowed values are: ϵ _t = -0.21 ± 0.23, ϵ _b = -0.19 ± 0.3, ϵ _τ = 0 ± 0.18. However, specific tests of the mixing angles α and β, or combinations of them, have been presented by the LHC collaborations; for instance the CMS constraints on the mixing angles α and β, for 2HDM-I and 2HDM-II, are shown in Fig. 9 [⁹⁶].

Figure 9 CMS constraints on the mixing angles α and β, for 2HDM-I and 2HDM-II. Figures from Ref. [⁹⁶]. 

On the other hand, the Higgs coupling with the fermions (bb¯,cc¯,τ+τ-), which could be measured at next-linear collider (NLC) with a precision of a few percent. This is particularly interesting for the large tan β region, where the corrections to the coupling hb¯b could change the sign [⁹⁸], and modify the dominant decay of the light Higgs, as well as the associated production of the Higgs with b-quark pairs [⁹⁹, ¹⁰⁰].

6.1. LFV Higgs decays

Within the SM, LFV processes vanish at any order of perturbation theory, which motivates the study of SM extensions that predict sizable LFV effects that could be at the reach of detection. In particular, the observation of neutrino oscillations, which is associated with massive neutrinos, motivates the occurrence of Lepton Flavor Violation (LFV) in nature [¹⁰¹,¹⁰²], which could be tested with the charged lepton decays, l _i → l _j γ and li→ljl¯klk. Another interesting possibility, which became even more relevant after the Higgs discovery, is the decay h → τμ, which was studied first in Refs. [¹⁰³, ¹⁰⁴], with subsequent analyses on the detectability of the signal appearing soon after [¹⁰⁵-¹⁰⁸]. This motivated a plethora of calculations in the framework of several SM extensions, such as theories with massive neutrinos, supersymmetric theories, etc. [⁵⁴,¹⁰⁹-¹¹³]. (For more on LFV Higgs decays see also [¹¹⁴-¹¹⁸]).

Nowadays, the decay h → τµ is included in the Higgs boson studies performed at LHC, which offers a great opportunity to search for new physics at the LHC. Along this line, a slight excess of h → τµ signal was reported at the LHC run I, with a significance of 2.4 standard deviations [¹¹⁹], but subsequent studies [¹²⁰, ¹²¹] ruled out such an excess and instead put the limit BR(h→μ¯τ)<1.2×10-2 with 95% C.L. Current bounds on LFV Higgs decays from CMS are shown in Fig. 11, for the plane of LFV Higgs couplings |Y _τµ |−|Y _µτ |, which also shows the constraints obtained from LFV lepton decays; we can see that the LFV Higgs decay provides the strongest constraints on these parameters.

Figure 11 Bounds on LFV Higgs couplings. Figure from [¹²¹]. 

Now we can use those strongest constraints on the LFV Higgs decay, in order to test the 2HDM-Tx, for the choices of textures that me mentioned before: Parallel, Complementary and semi-Parallel. This is shown in Fig. 12, for the cases studied in Ref. [⁸¹], and we can see that all of them satisfy the current LHC limits, but the predicted rates could tested at the coming LHC runs.

Figure 12 Constraints on LFV Higgs couplings. Figure from [⁸¹]. 

6.2. FCNC top decays

Top quark rare decays has been studied for several years as a channel to search of new physics [¹²²-¹²⁷], which included a variety of theoretical calculation for BR(t → ch) [¹²⁸-¹³⁰]. We use the following expression for the branching ratio: BR(t → ch) = (Γ(t→ch)/Γ_tot), where the total top width is given by: Γ_tot ≈1.55 GeV, and the width for the FCNC top decay (in the CPC case) is:

The resulting rates for BR(t → ch), are shown in Fig. 13, for the cases considered in Ref. [⁸¹].

Figure 13 Constraints on Higgs couplings and implications BR(t→ch). Figure from [⁸¹]. 

So far, LHC has provided the limit B.R.(t → ch) < 2 ×10⁻³ [¹³¹]. On the other hand, Ref. [¹³²-¹³⁴], provides some estimates for the branching ratios for t → ch that could be proved at the different phases of LHC. For instance, it is claimed there that top decay processes provide the best channel to discover top FCNC interactions, while only in some cases it is surpassed by single top production, when up and charm quark FCNC interactions are involved. In some of the examples discussed in Ref. [¹³³], the maximum rates predicted to be observable, with a 3 σ statistical significance or more, is about is BR < 5.8 ×10⁻⁵, for one LHC year with a luminosity of 6000 fb ⁻¹. This implies that the top FCNC branching ratio that arise within the 2HDM-Tx can be proved at LHC.