Physical Constraints on a Class of TwoHiggs Doublet Models with FCNC at tree level
Abstract
We analyse the constraints and some of the phenomenological implications of a class of two Higgs doublet models where there are flavourchanging neutral currents (FCNC) at tree level but the potentially dangerous FCNC couplings are suppressed by small entries of the CKM matrix . This class of models have the remarkable feature that, as a result of a discrete symmetry of the Lagrangian, the FCNC couplings are entirely fixed in the quark sector by and the ratio of the vevs of the neutral Higgs. The discrete symmetry is extended to the leptonic sector, so that there are FCNC in the leptonic sector with their flavour structure fixed by the leptonic mixing matrix. We analyse a large number of processes, including decays mediated by charged Higgs at tree level, processes involving FCNC at tree level, as well as loop induced processes. We show that in this class of models one has new physical scalars beyond the standard Higgs boson, with masses reachable at the next round of experiments.
a]F. J. Botella,
b]G. C. Branco,
c]Adrián Carmona,
b]M. Nebot,
b]Leonardo Pedro,
b]M. N. Rebelo
\affiliation[a]
Departament de Física Teòrica and IFIC,
Universitat de València  CSIC, E46100, Burjassot, Spain
\affiliation[b]Centro de Física Teórica de Partículas, and Departamento de Física
Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, P1049001 Lisboa,
Portugal
\affiliation[c]
Institute for Theoretical Physics,
ETH Zurich, 8093 Zurich, Switzerland
\emailAddfbotella@uv.es
\emailAddgbranco@tecnico.ulisboa.pt
\emailAddcarmona@itp.phys.ethz.ch
\emailAddnebot@cftp.ist.utl.pt
\emailAddleonardo@cftp.ist.utl.pt
\emailAddrebelo@tecnico.ulisboa.pt
\preprintIFIC/1403, CFTP/14004
1 Introduction
The recent discovery by both ATLAS [1] and CMS [2] of a particle at about 125 GeV, which may be consistently interpreted as a SMlike Higgs boson, has triggered an enormous interest in the scalar sector of the SM and some of its extensions. A crucial question to be probed experimentally is whether the scalar sector is more complex than the one of the SM and in particular whether there are more than one Higgs doublet. At least two Higgs doublets are present in many extensions of the SM, in particular in some models with spontaneous CP violation [3] and in supersymmetric extensions of the SM. The general two Higgs doublet models (2HDM) [4, 5, 6] without extra symmetries, have flavour changing neutral currents (FCNC) which have to be suppressed in order to avoid conflict with experiment. The simplest way of avoiding FCNC in the context of 2HDM is through the introduction of a discrete symmetry leading to natural flavour conservation (NFC) [7]. Another possibility of avoiding treelevel FCNC is through the hypothesis of aligned Yukawa couplings in flavour space [8]. Constraints arising from FCNC in the context of 2HDM have been the subject of many studies [13, 9, 10, 11, 12]. A very interesting alternative to NFC is provided by the socalled BGL models [14, 15, 16], where there are nonvanishing FCNC at tree level, but they are naturally suppressed as a result of an exact symmetry of the Lagrangian, which is spontaneously broken by the vevs of the neutral Higgs. The BGL models are highly constrained since, in the quark sector, all couplings are fixed by and the ratio of the two vevs, with no other parameters. This is to be contrasted with the situation that one encounters in the general 2HDM where there is a large number of parameters which can be expressed in terms of various unitary matrices arising from the misalignment in flavour space between pairs of Hermitian flavour matrices [17]. The search for the allowed parameter space in two Higgs doublet models has been done in the literature for a variety of scenarios [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. The extension of BGL models to the leptonic sector is essential in order to allow for the study of their phenomenological implications and, furthermore, to allow for a consistent analysis of the renormalization group evolution. The relationship between BGLtype models and the principle of Minimal Flavour Violation (MFV) [30, 31, 32, 33] has been studied and a MFV expansion was derived for the neutral Higgs couplings to fermions [15]. In this paper, we analyse the constraints on BGL type models and discuss some of their phenomenological implications. This paper is organized as follows. In the next section, we briefly review the BGL models and classify the various variants of these models while at the same time settling the notation. In the third section, we analyse the constraints on BGL models, derived from experiment. In section 4 we present our results. The explanation of the profile likelihood method used in our analysis and the input data appear in appendices. Finally, in section 5, we summarize our results and draw our conclusions.
2 Theoretical framework
We consider the extension of the SM consisting of the addition of two Higgs doublets as well as three righthanded neutrinos. In this work we only consider explicitly scenarios with Dirac type neutrinos, where no Majorana mass terms are added to the Lagrangian. However, our analysis of the flavourrelated experimental implications does not depend on the nature of the neutrinos, i.e., Majorana or Dirac. Therefore, our conclusions can be extended to the case of neutrinos being Majorana fermions provided that deviations from unitarity of the low energy leptonic mixing matrix are negligible, as it is the case in most seesaw models. The extension of BGL models to the leptonic sector, both for Dirac and Majorana neutrinos, was addressed by some of the authors in [16]. In order to fix our notation, we explicitly write the Yukawa interactions:
(1)  
where , and are matrices in flavour space.
The quark mass matrices generated after spontaneous gauge symmetry breaking are given by:
(2) 
where and denotes the relative phase of the vacuum expectation values (vevs) of the neutral components of . The matrices are diagonalized by the usual biunitary transformations:
(3)  
(4) 
The neutral and the charged Higgs interactions obtained from the quark sector of Eq. (1) are of the form:
(5)  
where , and , are orthogonal combinations of the fields , arising when one expands [3] the neutral scalar fields around their vacuum expectation values, , choosing in such a way that it has couplings to the quarks which are proportional to the mass matrices, as can be seen from Eq. (5). The required rotation is given by Eq. (27). Similarly, denotes the linear combination of orthogonal to the neutral Goldstone boson. The matrices and are given by:
(6) 
In terms of the quark mass eigenstates , the Yukawa couplings are:
(7) 
where and are the lefthanded and righthanded chirality projectors, respectively, and , , .
The flavour structure of the quark sector of two Higgs doublet models is characterized by the four matrices , , , . For the leptonic sector we have the corresponding matrices which we denote by , , , .
In order to obtain a structure for , such that there are FCNC at tree level with strength completely controlled by the Cabibbo – Kobayashi – Maskawa (CKM) mixing matrix , Branco, Grimus and Lavoura (BGL) imposed the following symmetry on the quark and scalar sector of the Lagrangian [14]:
(8) 
where , with all other quark fields transforming trivially under the symmetry. The index can be fixed as either 1, 2 or 3. Alternatively the symmetry may be chosen as:
(9) 
The symmetry given by Eq. (8) leads to Higgs FCNC in the down sector, whereas the symmetry specified by Eq. (9) leads to Higgs FCNC in the up sector. These two alternative choices of symmetry combined with the three possible ways of fixing the index give rise to six different realizations of 2HDM with the flavour structure, in the quark sector, controlled by the CKM matrix.
In the leptonic sector, with Dirac type neutrinos, there is perfect analogy with the quark sector. The requirement that FCNC at tree level have strength completely controlled by the Pontecorvo – Maki – Nakagawa – Sakata (PMNS) matrix, is enforced by one of the following symmetries. Either
(10) 
or
(11) 
where, once again, , with all other leptonic fields transforming trivially under the symmetry. The index can be fixed as either 1, 2 or 3.
These are the socalled BGL type models that we analyse in this
paper. There are thirty six different models corresponding to the
combinations of the six possible different implementations in each
sector
The symmetry given by Eq. (8) with the choice leads to the following pattern of zero textures for the Yukawa couplings:
(12)  
(13) 
where denotes an arbitrary entry. As a result of this symmetry the matrices , are of the form [14]:
(14) 
whereas
(15) 
In these equations only one new parameter not present in the SM appears, to wit, the ratio . It is the presence of the above symmetry, which prevents the appearance of additional free parameters. As a result, BGL models are very constrained but these constraints crucially depend on the variant of the BGL model considered. For example with the choice leading to Eqs. (12), (13), (14), Higgs mediated FCNC are controlled by the elements of the third row of . This leads, in a natural way, to a very strong suppression in the FCNC entering in the “dangerous” processes contributing to transitions. Indeed, in this variant of BGL models, the couplings entering in the tree level transition are proportional to leading to a suppression in the Higgs mediated transition, where denotes the Cabibbo parameter. With this strong suppression even light neutral Higgs, with masses of the order GeV are allowed. This strong natural suppression makes this variant of BGL models specially attractive. The neutral mass eigenstates are linear combinations of the fields , and with the mixing parameters determined by the Higgs potential.
Equations (12) and (13) are written in the weak basis (WB) where the symmetry is imposed. The six different BGL models can be fully defined in a covariant way under WB transformations [15] by
(16)  
(17) 
together with
(18)  
(19) 
where stands for (up) or (down) quarks, and are the projection operators defined [34] by
(20) 
and . Obviously, the zero textures written in the example given above only appear in the special WB chosen by the symmetry. A change of WB will alter these matrices without changing the physics. This fact leads to the consideration of WB invariant conditions as a powerful tool to analyse the physical implications of the flavour structure of models with two Higgs doublets [17]. The BGL example given explicitly above corresponds to .
With this notation the index refers to the sector that has no FCNC and refers to the row/column of that parametrizes the FCNC. Notice that for denoting “up” the index singles a row of , while for denoting “down” the index singles a column of . A characteristic feature of BGL models is the fact that in the WB covariant definition given by Eqs. (16) and (17) both matrices , involve the same projection operator. Different models with MFV were obtained through the generalization of BGL models [15]. Relaxing the above condition allows, for instance, to build models with Higgs mediated FCNC in both up and down sectors. It has been argued that out of the models verifying Eqs. (18) and (19) and their generalization to the leptonic sector, only BGL type models can be enforced by some symmetry [16]. Furthermore, in Ref. [35] it was shown that BGL models are the only models of this type that can be enforced by abelian symmetries.
Similarly, for the leptonic sector, the symmetries of Eqs. (10) or, in alternative (11), imply
(21)  
(22) 
where stands for neutrino () or for charged lepton () respectively. In this case
(23) 
where and are the unitary matrices that diagonalize the corresponding square mass matrices
(24) 
with and of the form
(25) 
In the leptonic sector, the PMNS mixing matrix , has large mixings, unlike the CKM matrix . Therefore, the Higgs mediated FCNC are not strongly suppressed. However, models where the Higgs mediated leptonic FCNC are present only in the neutrino sector can be easily accommodated experimentally due to the smallness of the neutrino masses.
In the next sections we label each of the thirty six different models we analyse by the pair (, ): the generation numbers refer to the projectors involved in each sector . For example, the model will have no tree level neutral flavour changing couplings in the up quark and the charged lepton sectors while the neutral flavour changing couplings in the down quark and neutrino sectors will be controlled, respectively, by and .
In BGL models the Higgs potential is constrained by the imposed symmetry to be of the form:
(26)  
the term in is a soft symmetry breaking term. Its introduction prevents the appearence of an wouldbe Goldstone boson due to an accidental continuous global symmetry of the potential, which arises when the BGL symmetry is exact. Namely, in the limit the pseudo scalar neutral field remains massless. Hermiticity would allow the coefficient to be complex, unlike the other coefficients of the scalar potential. However, freedom to rephase the scalar doublets allows to choose without loss of generality all coefficients real. As a result, does not violate CP explicitly. It can also be easily shown that it cannot violate CP spontaneously. In the absence of CP violation the scalar field does not mix with the fields and , therefore is already a physical Higgs and the mixing of and is parametrized by a single angle. There are two important rotations that define the two parameters, and , widely used in the literature:
(27) 
This rotation ensures that the field has flavour conserving couplings to the quarks with strength equal to the standard model Higgs couplings. The other rotation is:
(28) 
relating and to two of the neutral physical Higgs fields. The seven independent real parameters of the Higgs potential will fix the seven observable quantities, comprising the masses of the three neutral Higgs, the mass of the charged Higgs, the combination , , and . In our analysis we use the current limits on Higgs masses, identifying one of the Higgs with the one that was discovered by ATLAS and CMS. We make the approximation of no mixing between and identifying with the recently discovered Higgs and and with the additional physical neutral Higgs fields. This limit corresponds to and with this notation coincides with , which is the usual choice in the literature. This approximation is justified by the fact that the observed Higgs boson seems to behave as a standardlike Higgs particle. The quantity is of course already fixed by experiment. Electroweak precision tests and, in particular the and parameters, lead to constraints relating the masses of the new Higgs fields among themselves. Therefore the bounds on and , together with direct mass limits, significantly restrict the masses of the new Higgs particles, once the mass of is fixed. In our analysis we study BGL type models by combining the six possible implementations of the quark sector with the six implementations of the leptonic sector. It is illustrative to plot our results in terms of versus , since, as explained above in the context of our approximation of no mixing between and , there is not much freedom left. Therefore with these two parameters we approximately scan the whole region of parameter space. In our analysis, we impose present constraints from several relevant flavour observables, as specified in the next section.
3 Confronting experimental results
3.1 Generalities
In the class of 2HDM considered in this paper, the Yukawa interactions of the new scalars may produce new contributions, at tree and at loop level, that modify the SM predictions for many processes for which experimental information is available. As is customary, this will allow us to study the viability and interest of the different cases within this class of models. In terms of the New Physics (NP) and the SM leading contributions, one can organize the processes to be considered as follows.

Processes with tree level NP contributions mediated by and SM tree level contributions mediated, as, for example, universality in lepton decays, leptonic and semileptonic decays of mesons like , and , or decays of type .

Processes with tree level NP contributions mediated by the neutral scalars , , and

loop level SM contributions as in, for example, , , and oscillations,

highly suppressed (because of the smallness of the neutrino masses) loop level SM contributions as in, for example, or .


Processes with loop level NP contributions and

loop level SM contributions as in, for example, ,

highly suppressed (here too because of the smallness of the neutrino masses) loop level SM contributions as in, for example, or .

Besides those observables, electroweak precision information – and the oblique parameters , – are also relevant; they involve loop level contributions from the new scalars.
Table 1 summarizes this classification of the potentially relevant observables. Notice however that the table signals the possible new contributions but for each specific model type, some of them will be absent. More detailed descriptions of each type of constraint are addressed in the following subsections. Since we focus in the flavour sector, we exclude from the analysis of the experimental implications of the BGL models processes that probe additional couplings related to the scalar potential, such as , central in the Higgs discovery at the LHC, and refer the interested reader to [36].
BGL  2HDM  SM  
Charged  Neutral ,  Tree  Loop  
Tree  Loop  Tree  Loop  
✓  ✓  ✓  ✓  ✓  
Universality  ✓  ✓  ✓  ✓  ✓  
✓  ✓  ✓  ✓  
✓  ✓  ✓  ✓  
✓  ✓  ✓  ✓  
✓  ✓  ✓  
✓  ✓  ✓  
EW Precision  ✓  ✓  ✓ 
The set of observables that we consider is sufficient to obtain significant constraints for the masses of the new scalars and . Notice that, since the new contributions will be typically controlled by these masses, and the mixing matrices, with no additional parameters, we need fewer observables than would be necessary in the analysis of a more general 2HDM such as the one presented in [11].
Apart from the previous flavour related observables, direct searches at colliders may be relevant. For instance, a charged Higgs decaying to or with a mass lighter than \GeV was excluded
In the next subsections we describe in detail the different types of observables introduced above.
3.2 Processes mediated by charged scalars at tree level
Since transitions mediated within the SM by a boson may receive new mediated contributions, one has to pay attention to:

universality tests in pure leptonic decays ,

leptonic decays of pseudoscalar mesons ,

semileptonic decays of pseudoscalar mesons ,

decays of the form .
Universality
Pure leptonic decays are described by the following effective Lagrangian
(29) 
The second operator in (29) is the new contribution mediated by . The coefficient depends on the specific BGL model:
(30) 
where, for models of types and and otherwise – this concerns the lepton label of the model, the quark one is irrelevant here. Following the notation in [40, 41], we then have
(31)  
(32)  
(33) 
We consider for example universality in decays,
(34) 
where
(35) 
with and phase space functions
Semileptonic processes
Semileptonic processes may also receive tree level contributions from virtual ; the relevant effective Lagrangian for these processes is:
(36) 
where
(37) 
and, for models of types and , otherwise, while for models of types and , otherwise.
The rate of the leptonic decay of a pseudoscalar meson , with quark content , obtained from the effective Lagrangian in Eq. (36), is given by
(38) 
The scalar mediated new contribution is given by,
(39) 
Since the process is helicity suppressed and receives NP contributions proportional to , interesting channels are expected to involve heavy mesons and the lepton, as for example in , . Taking into account the different possible values of and , we must have
Therefore, for , if is negative, then the NP contribution is negligible; otherwise, if the NP contribution is enhanced by , it will typically interfere destructively with the SM contribution. An increase with respect to SM predictions, which would be interesting for example to account for some measurements, would require a NP contribution more than twice larger than the SM one, leading to tensions in other observables. The different channels considered in the analysis are collected in appendix B and radiative corrections are included according to [42].
In the case of decays of type , the analogue of Eq. (38) is
(40) 
The analysis uses experimental and results – see table 3.
While transitions are helicity suppressed two body decays, this is not the case anymore for decays. The corresponding decay amplitude is described by two form factors, and – with the momentum transfer to the pair –, associated to the P wave and the S wave components of the amplitude . The mediated amplitude can only contribute to the S wave component. Considering for example a specific case like , where the quark level weak transition is , we have
(41) 
giving then
(42) 
with coefficients and . For , we have instead
(43) 
and and . Notice that, even though BGL models still remain compatible with the present data for the decays , and , if the experimental anomalies observed in these processes, pointing towards physics beyond the SM, are confirmed no two such anomalies could be simultaneously accommodated in the BGL framework.
For decays, rather than resorting to the rate or the branching fraction to constrain the NP contributions, the CallanTreiman relation is used to relate the scalar form factor at the kinematic point to the decay constants of and :
(44) 
is a Chiral Perturbation Theory correction. The righthand side of Eq. (44), , is extracted from experiment, thus leading to a constraint on .
3.3 Processes mediated by neutral scalars at tree level
While the mediated NP contributions of the previous section compete with tree level SM amplitudes – including suppressed ones, as in decays –, the neutral scalars and produce tree level contributions that compete with loop level SM contributions. We consider three different types of processes.

Lepton flavour violating decays : in this case the SM loop contribution, proportional to neutrino masses is completely negligible and thus NP provides the only relevant one.

Mixings of neutral mesons, , where could be a downtype meson , or or the uptype meson . The distinction among down and uptype mesons is relevant since depending on the BGL model the tree level NP contributions will appear in one or the other sector.

Rare decays (including lepton flavour violating modes ): again depending on the BGL model and being one of the previous down or uptype pseudoscalar mesons, the tree level NP contributions will be present or not.
Lepton flavour violating decays
Lepton flavour violating decays of the form , such as , or are completely negligible in the SM, since the corresponding penguin and/or box amplitudes are proportional to neutrino masses. In BGL models of type