November 15th 2017



B. Machet 111Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France 222CNRS, UMR 7589, LPTHE, F-75005, Paris, France. 333Postal address: LPTHE tour 13-14, 4ème étage, UPMC Univ Paris 06, BP 126, 4 place Jussieu, F-75252 Paris Cedex 05 (France)

Abstract: The Glashow-Salam-Weinberg model for N=2 generations is extended to 8 composite Higgs multiplets by using a one-to-one correspondence between its complex Higgs doublet and very specific quadruplets of bilinear quark operators. This is the minimal number required to suitably account, simultaneously, for the pseudoscalar mesons that can be built with 4 quarks and for the masses of the gauge bosons. They are used as input, together with elementary low energy considerations, from which all other parameters, masses and couplings can be calculated. We focus in this work on the spectrum of the 8 Higgs bosons, on the mixing angles, and on the set of “horizontal” and “vertical” entangled symmetries that, within the chiral group, strongly frame this extension of the Standard Model. In particular, the () and () mixing angles satisfy the robust relation . Light scalars (below ) arise and the mass of (at least) one of the Higgs bosons grows like that of the heaviest bound state. cannot be safely tuned to zero and several parameters have no reliable expansion in terms of “small” parameters like or the mixing angles. This study does not call for extra species of fermions. The effective couplings of scalars, which depend on the non-trivial normalization of their kinetic terms, can be extremely weak. For the sake of (relative) brevity, their rich content of non-standard physics (including astrophysics), the inclusion of the 3rd generation and the taming of quantum corrections are left for a subsequent work.

PACS: 02.20.Qs 11.15.Ex 11.30.Hv 11.30.Rd 11.40.Ha 12.15.Ff 12.60.Fr 12.60.Rc


Chapter 1 Overview

1.1 Introduction

The Higgs boson of the Glashow-Salam-Weinberg (GSW) [1] model may be a fundamental scalar and the only one of this sort. There however exist in nature scalar mesons which are most probably quark-antiquark composites. It is therefore natural to wonder whether the Higgs boson could be such a particle, that is, just one member of the family of scalar mesons.

Previous tentatives led to the introduction of super-heavy quarks (techniquarks) [2]. This was thought to be the only solution to the mismatch between the electroweak scale (the mass of the or the vacuum expectation value of the Higgs boson) and the chiral scales , that unavoidably led, otherwise, to .

We shall follow here an orthogonal way and only interpret this mismatch as the need for (at least) two different scales, and thus for (at least) 2 Higgs bosons with vacuum expectation values (VEV’s) of the order of and . Instead of introducing extra fermions, we therefore prefer to introduce extra scalars, and we do it in a natural way.

In this work, the underlying definition of naturality is the simplest possible: all known particles, presently mesons and quarks 111though quarks are not particles. should be described in agreement with observations, including “the” Higgs boson (presumably the state discovered at the LHC [3]). All parameters (VEV’s, couplings) should be calculable in terms of “physical” quantities (masses of pseudoscalar mesons, mass, quark masses). Their number we shall reduce from the start as much as possible by simple and sensible physical arguments, like the absence of coupling between scalar and pseudoscalar mesons, that of flavor changing neutral currents (FCNC), the need for 3 true Goldstone bosons to provide 3 longitudinal degrees of freedom for the ’s, the role of Yukawa couplings to provide soft masses to the other “Goldstones” of the broken chiral symmetry etc. In this perspective, a model built to extend or complete the Standard Model should: first, not get in contradiction with present observations; secondly, be able to predict the properties and couplings of all states which have not been observed yet.

Along this path, one is unavoidably led to introduce several Higgs multiplets. The quarks of generations are the building blocks of pseudoscalar mesons and of scalar mesons. The total of such composite states should fit into quadruplets (or complex doublets). In particular, for 1 generation, 2 Higgs multiplets are expected. They involve 4 pseudoscalar mesons (2 neutral and 2 charged), 2 Higgs bosons (neutral scalars) and 2 charged scalars.
This would only be phraseology without the one-to-one correspondence that we demonstrate, concerning the transformations by the weak group , between the complex Higgs doublet of the GSW model and two sets of quadruplets of bilinear quark operators. The first set is made of quadruplets of the type that is, one scalar and 3 pseudoscalars, and the second set of quadruplets of the type . All quadruplets transform alike by and each one includes both parities. The two sets are parity-transformed of each other.

Each quadruplet has to be normalized. The normalization factors must in particular make the transition between bilinear quark operators of dimension and bosonic fields of dimension . Since each quadruplet includes one scalar operator with , its natural normalization is realized through the factor


In this way, the corresponding Higgs boson gets a “bosonic” VEV reminiscent of the GSW model. One normalizes all 4 elements of the same quadruplet by the same factor. Thus, to each quadruplet will be accordingly associated one “” and one “”, which makes, for 2 generations, a total of 8 “bosonic” VEV’s and 8 “fermionic” VEV’s. We shall suppose in the following that and that , though both statements can only be approximations in a theory that violates parity and also, eventually, and .

The dual nature of the components of the Higgs quadruplets will be extensively used. The simple example below shows the principle of the method. Let a given quadruplet (see (4.1)) include the charged pseudoscalar bilinear quark operator (after the normalization explained above has been implemented)


and, at the same time, a “Higgs boson” (scalar with non-vanishing VEV) which has a VEV . After the mixing of and quarks has been accounted for, can be expressed in terms of quark mass states, which yields . Now, PCAC [4] [5] for the mesons yields , in which is the charged pion (mesonic) interpolating field with dimension . This makes that, at least at low energy, one can also write , which therefore now appears as a bosonic field like the components of the Higgs doublets of the GSW model. Eventually, one can also use the Gell-Mann-Oakes-Renner (GMOR) relation [6] [7] to relate to pionic parameters, which leads finally to


In (1.3) we used the notation for when it is expressed in terms of a bosonic field. This notation we shall use throughout the paper: for any Higgs multiplet, stands for its expression in terms of bilinear quark operators, and stands for its “bosonic” form.

The calculations have been performed for N=1 and N=2 generations of quarks. It turns out that no simple argument or general principle could have anticipated the results, though they can be understood in simple terms a posteriori. It is also evident that a suitable solution cannot exist with a number of Higgs multiplets smaller that the one that we have introduced because, in particular, it could not fit observed pseudoscalar mesons. In this respect, the extension that we propose for the GSW model is minimal.

Two among its main features are the following:
* it is very fine-tuned;
* the and mixing angles and are independent parameters and, though , it cannot be turned safely to .

1.2 Main results for N=1 generation

Only 2 quarks () are present. They build up 4 pseudoscalar mesons (the 3 pions and the flavor singlet) and 4 scalars. The latter include 2 neutral states which are 2 Higgs bosons, and 2 charged scalars. These 8 states fit into 2 quadruplets.

The and masses are inputs. The 3 longitudinal ’s are built in this case from the 2 charged scalars and from the neutral pseudoscalar singlet; the three of them accordingly disappear from the physical spectrum.

The 2 Higgs bosons have respective masses and . They have the same ratio as the corresponding VEV’s, respectively and , exhibiting a large hierarchy .

PCAC provides the usual correspondence between pseudoscalar bilinear and pions and their leptonic decays through ’s are suitably described.

Very light scalars start to spring out. For one generation there is only one such particle.

While is determined by the GMOR relation, one gets 222We use the same notations as in the bulk of the paper. , which already points out at a negative quark mass parameter as will be confirmed for 2 generations. Then, .

The price to pay for this drastic truncation of the physical world is threefold:
* a very large hierarchy between the 2 bosonic VEV’s;
* a very small mass for the heaviest Higgs boson which cannot compete with the expected [3] and could naively look like the revival of the mismatch between and that led to technicolor models;
* the disappearance of the singlet pseudoscalar meson in favor of the neutral longitudinal .

These issues get on their way to a solution when one increases by 1 the number of generations. In particular, the mass of the “quasi-standard” Higgs boson becomes comparable to that of the heaviest pseudoscalar meson, instead of .

It is therefore reasonable to think that extending this approach to N=3 generations can bring a suitable agreement with the observed world. Unfortunately, the number of equations and constraints to fulfill increases so dramatically that a convenient method to solve them could not be found, yet.

1.3 Main results for N=2 generations

4 quarks () are now involved, which build up 32 and composite states. These fit into quadruplets. There are therefore in particular 8 Higgs bosons.

There are 2 mixing angles: describes the mixing between and flavor eigenstates while concerns and . Flavor and gauge symmetries are tightly entangled in this extension and the freedom to tune to by a flavor rotation no longer exists. As a consequence, the Cabibbo angle cannot describe alone correctly the physics under concern. These features are exhibited by studying successively the case when one approximates to and the one when both and .

1.3.1 The case

The Higgs bosons split into 1 triplet, 2 doublets and 1 singlet. Inside each of these, they are close to degeneracy. 3 have masses , more precisely , 2 have intermediate masses , 1 has a very small mass and the last two only get massive by quantum corrections.

The hierarchies between VEV’s stay below (instead of for 1 generation).

The situation has substantially improved with respect to 1 generation; indeed, the masses of the quasi-standard Higgs boson(s) suitably increase and depart from and the hierarchies between VEV’s decrease towards more reasonable values.

, that one identifies with the Cabibbo angle is expressed by the 2 formulæ


The first equation in (1.4), yields


off the experimental value of the Cabibbo angle


A negative sign for is needed, like for 1 generation. Since , it yields, by the GMOR relation, a negative sign for .

One however still faces problematic issues :

 the nice description of leptonic decays that we had found for 1 generation gets totally spoiled; the situation could only be improved if was very small 333 and are the vacuum expectation values of the neutral scalars of the quadruplets and respectively, which are defined in section 4.1; see also subsection 4.1.2;

 taking the masses of the charged pseudoscalar mesons as inputs, the mass of the neutral mesons is off by , unless one goes to a value larger than for , in conflict with most needed orthogonality relations 444 is the vacuum expectation value of the neutral scalar of the quadruplet , which is defined in section 4.1; see also subsection 4.1.2;

 defining the interpolating field of the meson as proportional to , the latter cannot be set orthogonal to (actually, we do not get too worried by this problem because of the mixing between neutral pseudoscalars);

 2 ratios of bosonic VEV’s, and 555 is the vacuum expectation value of the neutral scalar of the quadruplet , which is defined in section 4.1; see also subsection 4.1.2 come out too large to match intuitive arguments concerning (non-diagonal) quark condensates;

 the last problem concerns mixing, and proves later to be correlated with the previous one. On one side eqs. (1.4) give fairly good estimates of the mixing angle; the result is independent of the so-called parameters (ratios of bosonic VEV’s) and looks robust. On another side, Yukawa couplings provide diagonal and non-diagonal mass terms for the and quarks: with intuitive notations


in which depend on the parameters through the normalizing coefficients (1.1) of the 8 Higgs quadruplets. The paradox is that, at the values of the parameters which fit all other data, in particular pseudoscalar meson masses, comes very close to a pole, like if the “fermionic mixing angle” was close to maximal ().
So, either the quark mixing exhibits a dual nature (maximal mixing or close to maximal only concerns leptons at present), or one must find a way out of this paradox. It would be feasible at very small values of the parameters and , which seems excluded at .

1.3.2 The case

The first equation in (1.4) is only the approximation at of the exact formula


which shows that and cannot be dealt with independently. Using the experimental value of Cabibbo angle (1.6), (1.8) yields


The values that we find for the mixing angles correspond to a good approximation to 666We take . and 777Among first attempts to calculate the Cabibbo angle are the ones by Oakes [8] and by Weinberg [9]. Since then, it has been a most sought for goal of calculating the mixing angles from basic principle (see for example [10] and [11] in which specific hypotheses are made concerning the symmetries involved and/or the mass matrices, and the estimate for in [12] based on the sole existence of mass hierarchies among quarks)..

Despite its very small value, switching on has very important consequences, for example on the values of the parameters, and brings a very good agreement between the model and the basis of meson physics :

 leptonic decays of and are well described;

 the parameters and become very small which opens the way to a matching between bosonic and fermionic mixing;

 the masses of the neutral pion and kaon are now well accounted for, and the is only off by .

The spectrum of the Higgs bosons is changed to .

and are still expected to be very light, and so does because is expected to be of the same order of magnitude as . is preferred, though it is difficult to give yet a precise value. The positive improvement is that it does not need any longer to be larger than .

All parameters are very fine tuned. The importance of the small is just one among the symptoms of this; one often deals with rapidly varying functions which furthermore have poles, parameters that have no trustable expansions at the chiral limit, “unlucky” coincidences etc. It is probably the price to pay for naturalness as we define it: it is indeed very unlikely that some general principle or god-given symmetry miraculously tunes the values of physical observables up to many digits after the decimal dot. Nature is obviously fine tuned and a model that pretends to describe it accurately has many chances to be fine-tuned, too.

1.4 Principle of the method

One works at two levels, bosonic and fermionic.

 Bosonic considerations rely on few statements.

 The mass of the gauge bosons, which, in this framework, comes from the VEV’s of several Higgs bosons, is known.

 The masses of all charged pseudoscalar mesons is also known with high precision. One should be more careful about some neutral pseudoscalars that can mix and the definition of which in terms of quark bilinears can be unclear.

 The effective Higgs potential to be minimized is built from the genuine scalar potential, suitably chosen, to which is added the bosonised form of the Yukawa couplings. Its minima are constrained to occur at the set of bosonic VEV’s ’s.

 The VEV’s are supposed to be real and, therefore, there squares to be positive.

 Among the components of the Higgs 8 quadruplets:
 - there must exist 3 true Goldstone bosons related to the breaking of the local ;
 - all other scalar and pseudoscalar fields that do not have non-vanishing VEV’s are pseudo-Goldstone bosons that get “soft” masses via the Yukawa couplings at the same time as quarks get massive. This restricts and simplifies the scalar potential.

 The of the known pseudoscalar mesons will be calculated as the ratios of the corresponding quadratic terms in the bosonised Yukawa Lagrangian and in the kinetic terms. They depend on the VEV’s, on the mixing angle(s), and of course on the set of Yukawa couplings. Their number is reduced by a suitable and motivated choice for the Yukawa potential.

 Additional relations among Yukawa couplings arise from various sets of constraints:
 - no transition should occur between scalar and pseudoscalar states;
 - likewise, no transition should occur between charged pseudoscalar mesons;
 - similar orthogonality relations are explored among neutral pseudoscalars and, for 2 generations, most of them (but not all of them) can be satisfied.

 Fermionic considerations use the genuine (not bosonised) form of the Yukawa Lagrangian, which provides mass terms for the 4 quarks, both diagonal and non-diagonal. We mainly use them at .

 A first set of constraints comes when turning to the mixing between the and quarks; then the mixing angle becomes the Cabibbo angle ;
 A second set of constraints comes from requirements of reality for the quarks masses;
 A third set of constraints comes when studying the quark at the chiral limit .

All these constraints are checked by evaluating the masses of pseudoscalar mesons, the leptonic decays of charged pseudoscalars …and used to predict unknown quantities in particular the masses of the scalar mesons (Higgs bosons).


1.5 Changes between version 2 (this version) and version 1 of this work

Since the versions 1 and 2 have the same arXiv number, we list here the differences between the two.

The main difference concerns the extension to . It was indeed realized that, for 2 generations, assuming and led to grossly incorrect leptonic decays of and . The situation was paradoxical since no problem arose for 1 generation.

Version 2 includes accordingly the study of the 3 cases: 1 generation (it was not present in version 1 but had to be included here to show that leptonic decays were correctly described, and also to unify the notations between [Machet1][Machet2] and the case of 2 generations), 2 generations with and 2 generations with . It is shown how both “chiral scales” and weak scales can be accounted for without having to introduce extra super-heavy fermions. In relation with this, leptonic decays are investigated with special care.

is determined to be very close to , which is a very small number. Nevertheless, it has a crucial importance and cannot be safely tuned to , unlike in the Glashow-Salam-Weinberg model. The values of several parameters and the spectrum of Higgs bosons get modified, which shows that the underlying physics is very fine tuned. We give examples of this and insist on the fact that several parameters of the model lie in regions of rapid variation, eventually close to poles, that others don’t have reliable expansions in terms of small parameters like the pion mass, or

The new values of some parameters obtained at relieve tensions that arose when , in particular concerning the parameter and the masses of and .

There are now 12 figures instead of 3.

Many remarks, footnotes and 1 appendix have been added to guide the reader, such that all equations in the core of the paper can be easily reproduced.

Several references have been added.

Misprints have been corrected. Misplaced parentheses in eq.(124) of version 1 were the most important. Fortunately, this amounted to replace with for some contributions to the masses of neutral pseudoscalar mesons, which is numerically small (numbers have been corrected). Modifications to analytical expressions, for example eq. (127) of version 1 have of course been done.

1.6 Contents

 Chapter 2 is dedicated to general considerations.

* Section 2.1 establishes, in the general case of generations, the basic one-to-one relation between the complex Higgs doublet of the Glashow-Salam-Weinberg model and very specific quadruplets of bilinear and quark operators including either 1 scalar and 3 pseudoscalars, or 1 pseudoscalar and 3 scalars. To this purpose, the group of weak interactions is trivially embedded into the chiral group .

The normalization of the quadruplets is then explained, which introduces “bosonic” VEV’s of the form and “fermionic” VEV’s which are condensates.

The connection is made between parity and the 2 generators or .

* Section 2.2 presents general considerations concerning the Yukawa couplings . Arguments will be given concerning how and why they can be simplified. They are chosen as the most straightforward generalization to generations of the most general Yukawa couplings for 1 generation, in which 2 quarks are coupled to 2 Higgs doublets. Yukawa couplings are no longer passive in determining the VEV’s of the Higgs bosons. This leads to introducing their bosonised form. Subtracting it from the scalar potential yields an effective potential that can be used to find the (bosonic) VEV’s of the Higgs bosons. A first set of constraints arises from the condition that no transitions should occur between scalars and pseudoscalar mesons.

* Section 2.3 presents and motivates our simple choice for the Higgs potential. The minimization of the corresponding effective potential (see above) leads to another set of relations between its parameters, Yukawa couplings and bosonic VEV’s. Goldstones and pseudo-Goldstones are investigated, in relation with the concerned broken symmetries. The (soft) masses of the pseudo-Goldstones can be calculated from the bosonised form of the Yukawa Lagrangian.

* Section 2.4 gives general formulæ for the masses of the Higgs bosons.

 Chapter 3 deals with the simplest case of 1 generation. Using as input the masses of the ’s, pions, and quarks, bosonic and fermionic equations are solved which yield the spectrum of the 2 Higgs bosons, the values of the 4 VEV’s and all couplings. The leptonic decays of are shown to be in agreement with the usual PCAC estimate.

 Chapter 4 gives general results in the case of 2 generations.

* Section 4.1 displays the 8 Higgs quadruplets. The choice of the quadruplet that contains the 3 Goldstones of the spontaneously broken is motivated. Notations that will be used throughout the paper are given.

* Sections 4.2 and 4.3 give generalities concerning the kinetic terms, Yukawa couplings and the Higgs potential. The mass of the ’s is expressed in terms of the bosonic VEV’s.

* Section 4.7 identifies the group of transformations that moves inside the space of quadruplets. Its generators commute with the ones of the gauge group.

* Section 4.4 is devoted to charged pseudoscalar mesons. Their masses and orthogonality relations are explicitly written.

* Section 4.5 shows how the formula (1.8) relating , and the masses of charged pseudoscalar mesons is obtained very simply. It does not depend on low energy theorems like GMOR (which are badly verified for heavy mesons) and only relies on the statement that are proportional to the interpolating fields of, respectively . There is no need to know the proportionality constants, which makes this result specially robust.

* Section 4.6 gives the general formulæ for the masses and orthogonality conditions for and .

 Chapter 5 deals with the approximation , keeping of course .

* In section 5.1, the value of the Cabibbo angle is extracted from the general formula (1.8). Its value falls within of the experimental .

* In section 5.2, we write a basic set of equations that will determine the ratios of bosonic VEV’s, and we show how, from the sole spectrum of charged pseudoscalar mesons, one already gets a lower bound for the mass of the “quasi-standard” Higgs boson .

* Section 5.3 is dedicated to neutral pseudoscalar mesons and to the constraints given by their masses and orthogonality. We find a tension concerning , because its value obtained from the mesonic mass spectrum is slightly larger than 1 is in contradiction with orthogonality relations of to , and of to .

* Section 5.4 studies charged scalar mesons. Their orthogonality relations cannot be satisfied unless they align with flavor eigenstates. This is not surprising since the two of them which coincide with the charged Goldstone bosons of the broken gauge symmetry are by construction flavor eigenstates.

* Section 5.5 summarizes all the bosonic constraints.

* In section 5.11, we study the masses of and . In particular, is needed to correctly account for the mass of ; the latter is otherwise off by .

The next 3 sections deal with fermionic constraints.

* Section 5.6 lists the equations coming from the definition of quark masses in terms of Yukawa couplings and VEV’s. Additional constraints are given by using the freedom (as we already did for bosons) to turn to .

* Section 5.7 displays the constraints coming from the reality of the quark masses. Among the outcomes are: - the knowledge of ; - the expression of in terms of quark masses as given by the second line of (1.4), which requires in particular .

* In section 5.8 we calculate the “fermionic mixing angle” from Yukawa couplings and show that it tends to be maximal, in contrast with the small value of the Cabibbo angle.

* Section 5.9 studies at the chiral limit . This determines in particular the sign of and the value of the mass of the “quasi-standard” Higgs bosons .

* Section 5.10 summarizes the solution of all equations. It is shown how and , which are classically massless, are expected to get soft masses from quantum corrections. Hierarchies between VEV’s are shown to be much smaller than for 1 generation.

* Section 5.12 is dedicated to leptonic decays of and . We show that they cannot be suitably described for and . The situation is therefore, at the moment, worse than for 1 generation.

* Section 5.13 is a brief summary of the case , mainly pointing at the problems that arise.

 Chapter 6 concerns the general case and .

* In section 6.1, we use the experimental value of the Cabibbo angle to calculate , which is very close to .

* In section 6.2 we re-analyze the leptonic decays of the charged pions and kaons. Unlike at , a nice agreement can be obtained. Then, the parameters of the model are updated, a large set of them being very sensitive to . In particular, , and presumably too, are now very small.

* In section 6.3 we re-investigate the masses of and . We find that, even for , they can now be quite well accounted for. The is the worst, but its mass is only off by . One however needs a fairly large value of the condensate, which coincides with what we already suspected at namely that, in one way or another, “some” heavy quark should have a large condensate.

* In section 6.4, we update the Higgs spectrum. The masses of the 3 heaviest Higgs bosons has increased to , has risen to an intermediate mass of and the 4 others are light (they should not exceed ).

* Section 6.6 concludes the case . The tensions that occurred at have been mostly removed or on their way to be (like the paradox of the fermionic “maximal mixing”). Including the 3rd generation of quarks is of course highly wished for, but goes technically largely beyond the limits of this work. We emphasize the impressive ability of this multi-Higgs model to account for the physics of both the broken weak symmetry and that of mesons. We also largely comment on the very fined-tuned character of all the physical outputs. Their sensitivity to the small is just one example among a list of parameters which have no trustable expansions at the chiral limit, or at the limit of small mixing angles.

 Chapter 7 is a general conclusion, mostly focused on symmetries

* In section 7.1 we study how the chiral/gauge group acts inside each quadruplet, which shows that the third generator of the custodial is identical to the electric charge 888This had already been noticed in [13]..

We then study which generators of the diagonal annihilate the Higgs states, which provides the properties of invariance of the vacuum.

Next, we show that this extension of the GSW model can be a right-handed gauge theory as well as a left-handed one, and that it is in principle ready to be a left-right gauge symmetry. This requires 6 true Goldstone bosons, which cannot be achieved with only 2 generations because some states should be absent from the physical spectrum which have in reality been observed.

Then, we make some more remarks concerning parity and its breaking.

We study the “generation” chiral group of transformations and show how the 8 Higgs quadruplets fall into 4 doublets of or , and 1 triplet + 1 singlet of its diagonal subgroup.

* In section 7.2, we explain why the spectrum of the 8 Higgs bosons, 1 triplet, 2 doublets and 1 singlet can be considered to fall into representations of this generation subgroup orthogonal to the custodial .

* In section 7.3 we make miscellaneous remarks and give prospects for forthcoming works. We emphasize the very important role of the normalization of bosonic asymptotic states to determine their couplings to quarks; we outline in particular why present bounds on the masses of light scalars have to be revised. We give a list of topics to be investigated and their spreading to other domains of physics.

 in appendix .5, we collate the expressions of bilinear flavor quark operators in terms of their mass counterparts and of the mixing angles and . These formulæ are used throughout the paper.

Note : to help the reader, formulæ which are valid for the whole paper, including definitions, and final results, have been boxed.

Chapter 2 General results

Embedding the gauge group into the chiral group , we start by establishing a one-to-one correspondence between the Higgs doublet of the GSW model and quadruplets of bilinear quark operators. We then proceed to constructing our multi-Higgs extension of the standard model. We introduce Yukawa couplings, then the genuine and effective scalar potentials. The latter plays an important role because the Yukawa couplings are no longer passive in defining the vacuum of the theory. Last we give general formulæ for the masses of the Higgs bosons.

2.1 A one-to-one correspondence

2.1.1 The Higgs doublet of the GSW model

We give below the laws of transformations of the components of the Higgs doublet of the GSW model, and, by a very simple change of variables, put them in a form that matches the ones of specific bilinear fermion operators that we shall introduce later.

The generators of the group are the three hermitian matrices


where the ’s are the Pauli matrices


The Higgs doublet is generally written


in which and are considered to be real. The vacuum expectation value of arises from such that . lies in the fundamental representation of such that generators act according to


The transformed of the components are naturally defined by


such that the law of transformation (2.4) is equivalent to


It takes the form desired for later considerations


when one makes the substitutions


then rewrites


and is thus tantamount to .

Later, we shall often, instead of complex Higgs doublets, consider indifferently quadruplets, for example, in this case


keeping in mind that, to any such quadruplet is associated a complex doublet in the fundamental representation of given by (2.9).

2.1.2 Embedding the gauge group into the chiral group

For generations of quarks, that is, quarks, let us embed into the chiral group by representing its three generators as the following matrices


in which is the identity matrix. They act trivially on vectors of flavor quark eigenstates 111The superscript means “transpose”..

2.1.3 Quadruplets of bilinear quark operators

being any matrix, we now consider bilinear quark operators of the form and . and being transformations of and respectively, these bilinears transform by the chiral group according to


Writing and as


eq. (2.12) entails


in which and stand respectively for the commutator and anticommutator.

Let us now define the specific matrices


in which , and is a real matrix. Let us call the generic components of the two sets of quadruplets


of the type , made with one scalar and three pseudoscalars, and


of the type , made with one pseudoscalar and three scalars. By (2.14), the ’s transform by and according to


or, equivalently, since it is often convenient to manipulate states with given electric charge,