Advances in High Energy Physics

Volume 2017 (2017), Article ID 1765340, 14 pages

https://doi.org/10.1155/2017/1765340

## An Analysis of a Minimal Vectorlike Extension of the Standard Model

^{1}Research Institute of Physics, Southern Federal University, Pr. Stachky 194, Rostov-on-Don 344090, Russia^{2}Bogoliubov Laboratory of Theoretical Physics, Joint Institute of Nuclear Research, Dubna 141980, Russia

Correspondence should be addressed to V. Kuksa

Received 17 November 2016; Accepted 4 January 2017; Published 29 January 2017

Academic Editor: Lorenzo Bianchini

Copyright © 2017 V. Beylin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We analyze an extension of the Standard Model with an additional hypercolor gauge group keeping the Higgs boson as a fundamental field. Vectorlike interactions of new hyperquarks with the intermediate vector bosons are explicitly constructed. We also consider pseudo-Nambu–Goldstone bosons caused by the symmetry breaking . A specific global symmetry of the model with zero hypercharge of the hyperquark doublets ensures the stability of a neutral pseudoscalar field. Some possible manifestations of the lightest states at colliders are also examined.

#### 1. Introduction

The experimental detection of the Higgs boson [1, 2] with mass GeV leaves unanswered many questions of the Standard Model (SM) (see [3], for example). A part of the SM puzzles can be solved by supersymmetry (SUSY) [4, 5]. Unfortunately, there are no clear indications that SUSY manifests itself in the experiments near a “naturalness” scale ~1 TeV. Obviously, SUSY is not rejected at all, but sparticles and their interactions are now expected to be observed at a much higher scale, ~5–10 TeV, because the parameter space of SUSY models is increasingly constrained by the LHC data [6–8].

Besides SUSY, a lot of ways are proposed to enlarge SM: an addition of extra groups, multi-Higgs and technicolor (TC) models, and many others (see reviews [3, 9] and references therein). However, we currently have not found any comprehensive variant of the theory of “everything” (excepting, possibly, string theory which has no phenomenological applications for now), so all problems of SM cannot be solved simultaneously. An origin of Dark Matter (DM) is also one of the known SM problems. At the moment we are skeptical of any manifestations of (sufficiently light) neutralino as the DM particle [10]. Note that there are a lot of other DM candidates which are suggested and discussed [11–19]. For example, DM can originate from the Higgs sector too (e.g., the inert Higgs model) [20, 21].

From a “technical” viewpoint, technicolor scenario [22–25] means a “duplication” of an analog of the QCD sector at a higher energy scale with confinement of the extra technifermions and technigluons. Originally, TC models were suggested to introduce dynamical electroweak (EW) symmetry breaking (EWSB) without fundamental Higgs scalars. Corresponding scalar boson arises in this case as a bound state of techniquarks—these models are Higgsless (note also the so-called “see-saw” mechanism giving a light scalar boson in TC) [26–31]. In this way both structure and interactions of the T-strong confined sector are considered as extra options to solve some SM problems (see [32–36]). It seems that the discovery of the Higgs boson closes some Higgsless technicolor scenarios and many investigations concentrate now on extra fermion sectors in confinement (the so-called hypercolor models) as a source of composite states and Dark Matter candidates.

Contributions of additional fields to the SM precision parameters are crucial for the models—variety of them is constrained [26] by the experimentally required values of Peskin–Takeuchi (PT) parameters [35, 37–41]. So, to select a realistic and reasonable extension, it is necessary to calculate EW polarization operators with an account of the model contributions. Then, the comparison of calculated values of , , and parameters with the experimental data gives some constraint on the structure of the model. As a rule, in the models with chirally nonsymmetric fermions, there appear unacceptable contributions to the PT parameters. It is the main reason why vectorlike models have been under consideration recently [35, 36, 42–44].

Thus, multiplet and chiral structure of the new fermion sector is a principal characteristic of SM extension. In the framework of technicolor models, as a rule, such multiplets have a standard-like structure, namely, left-hand doublets and right-hand singlets [45, 46]. In the hypercolor models, chirally symmetric (with respect to the weak group) set of new fermions is used [47]. However, this chirally symmetric fermion sector crucially differs from the standard one, so interpretation of the gauge fields as standard vector bosons is hypothetical.

In this work, we suggest a construction of vectorlike weak interaction which starts from standard-like chirally nonsymmetric set of new fermions doublets. This program has been carried out for zero hypercharge in the simplest model with two hyperquark (H-quark) generations and two hypercolors (HC), [44, 48]. We consider this scenario for the case of nonzero hypercharge and show that two left doublets of H-quarks can be transformed into one doublet of Dirac H-quarks with vectorlike weak interaction. This possibility can be realized if the hypercharges of generations have the same value and opposite signs. Importantly, this condition is in accordance with the absence of anomalies in the model. To form the Dirac states which correspond to constituent quarks, we have used a scalar field having nonzero vacuum expectation value (v.e.v.). This field is introduced as a scalar singlet pseudo-Nambu–Goldstone (pNG) boson in the framework of the simplest linear sigma model. We consider in detail the structure of the pNG multiplet which is defined by the global symmetry breaking . It is also shown that the Lagrangian of this minimal extension has specific global symmetries making neutral H-baryon and H-pion states stable.

The paper is organized as follows. In Section 2, we construct vectorlike interactions for the case of H-color and EW groups with even generations. The total Lagrangian together with the pNG bosons is considered in Section 3. The principal part of the physical Lagrangian of the model is presented in Section 4, where we demonstrate the presence of a specific discrete symmetry that leads to the stability of a pseudoscalar state. In Section 5, we analyze the main phenomenological consequences of the model.

#### 2. Vectorlike Interaction of the Gauge Bosons with H-Quarks

An essential point is the choice of chiral structure of the H-quark multiplets. It is known that chirally nonsymmetric interaction of the extra fermions with the SM bosons may contradict to restrictions on Peskin–Takeuchi parameters. Thus, it is reasonable to consider vectorlike (chiral-symmetric) interaction of (initially standard-like) H-quarks with - and -bosons. We construct such interactions explicitly for the case of even generations of two-color H-quarks.

In the simplest scenario with two generations () of left-handed H-quarks, the bidoublet of these quarks is presented as a matrix , where and are indices of and fundamental representations, respectively. (In the following all indices related to the hypercolor group are underlined.)

This bidoublet transforms under asHere , , and the H-quarks charges are defined by the arbitrary hypercharges . The right-handed singlets (with respect to electroweak group) have the following group transformations:where and are hypercharges of singlets. Now, the charge conjugation operation, , is applied to the fields of the second generation keeping the first generation of H-quarks unchanged:The transformation properties of the charge conjugated fields have the formThen, we redefine the H-quark fields (the fermion chirality is changed by the charge conjugation):

Further, we multiply both sides of (4) by and use the following properties of group matrices:

Using redefinition (5), from (4), we getThis transformation law coincides with the one given by formula (1) for the first generation when .

Thus, we have constructed the right-handed field partner of the first generation, using the second generation of the left-handed fields in two steps: charge conjugation and redefinition. Therefore, composing these fields we have a Dirac state:Because both parts (left- and right-handed) of the field have the same transformation properties, namely, (1), then the Dirac H-quarks interact with the EW vector bosons as chiral-symmetric fields.

Analogously, the right-handed field is redefined as follows:This redefined field transforms as the right-handed singlet if in full analogy with the previous case. This representation of the H-fields allows us to get a usual Dirac mass term after the summation of left and right parts. Both current and constituent H-quark masses can be introduced because the mass term does not violate the model symmetry. The simplest way to do this is to use a singlet real scalar, , which has a nonzero v.e.v., , where . Just interaction of the H-quarks with this scalar field provides Dirac type mass term for H-quarks. Note that, to get a Dirac state with the vectorlike interaction from two Weyl spinors, we should require the initial fields for the first and second families to have opposite hypercharges, . The same requirement follows from the condition of the absence of anomalies in the model. It should be noted that the suggested construction of vectorlike interaction is valid due to unique properties of group and for the case of an even number of generations.

The gauge part of the model Lagrangian directly follows from (1) and (2):where is a H-gluon field. The mass terms are formally included in (10) because they do not break -symmetry of the model. The status of the -singlet H-quark significantly differs from that of the standard quarks. The standard quark singlet is a right-handed part of the Dirac fermion state, while -quark consists of the two initial chiral singlets. It should be noted that the singlet can be useful since a composite H-meson is a representation of the groups . The standard Higgs doublet is the same representation, that is, the Higgs field can be considered as a composite state of the singlet and doublet H-quarks. However, due to the fields and being independent, from now on, the singlet states can not be included in the consideration.

#### 3. Fundamental Higgs Boson, Two-Color Fermions, and Pseudo-Nambu–Goldstone Bosons in the Linear Sigma Model

Here, we construct a linear sigma model involving the constituent H-quarks and lowest pseudo(scalar) H-hadrons— H-meson, pNG states, and their opposite-parity partners [45, 46, 49–51]. As it was shown in [51–53] (see also more recent papers [54, 55]), Lagrangian (10) in the limit , has a global symmetry corresponding to rotations in the space of the four initial chiral fermion fields. The Lagrangian with nonzero can be rewritten in the form which explicitly reveals the violation of symmetry by the mass term [54, 55]. For the Lagrangian retains the full symmetry but, in an analogy with QCD, one might expect the dynamical symmetry breaking by vacuum expectation value . This v.e.v. has the mass term structure and leads to the dynamical breaking of the symmetry . As a result, the broken generators of would be accompanied by a set of pNG states. The spectrum of the pNG states depends on the way of symmetry breaking.

The global symmetry of two-color QCD with Dirac quarks in the limit of zero masses is ), with the chiral group being its subgroup, (this statement is valid for any symplectic gauge theory [56]; the group is isomorphic to the group ) [52, 53]. This global symmetry is often called the Pauli–Gürsey symmetry. The quark condensate breaks the Pauli–Gürsey symmetry to its subgroup ) [51, 57]. In the following we will consider the simplest case of two flavors .

We have only two possibilities to assign EW quantum numbers to the two fundamental fermion constituents (for the general case a classification of physically relevant ultraviolet completions of composite Higgs models based on the coset is given in [56, 58], which considers different gauge groups with arbitrary numbers of flavors and colors, and ). These possibilities are determined by the cancellation of gauge anomalies.(i)*V-A ultraviolet completion*. We can introduce a left-handed weak doublet and two right-handed weak singlets and with opposite hypercharges . It is the case that is considered in most papers dealing with a new two-flavor confined sector [55, 59–63].(ii)*Vectorlike ultraviolet completion*. Both left- and right-handed fermions are grouped as fundamental representations of the weak group, and [44, 48]. The hypercharges of the doublets should be the same, . In this case the Dirac mass term, , is permitted by the EW symmetry. In this paper, we study the case of the vectorlike ultraviolet completion with zero hypercharges of the doublets.

At the fundamental level, the Lagrangian of two-flavor and two-color QCD (10) can be written in terms of a left-handed quartet field:whereare left- and right-handed quartet fields ( and are left-handed doublets introduced in the previous section). The EW term in the covariant derivative (12) involves the matricesthat are three of ten generators satisfying the following conditions:The mass term in Lagrangian (11) introduces the antisymmetric matrix We have used the matrix also to define the algebra of the generators. Although has a noncanonical form, it can be brought into the form or by a unitary transformation.

The equivalence of the Lagrangians (10) and (11) was proved in the previous section. It should be noted that the similar rearrangement of the Lagrangian in terms of the left-handed fields would be possible in any sort of techni- or hyperchromodynamics with T/H-quarks in self-contragredient representation of T/H-confinement group. The fundamental representation of , which is symplectic and pseudoreal representation, is just the simplest case. An aspect of this property is that the global symmetry group of the massless theory is larger than the chiral symmetry.

In the limit of vanishing and the global symmetry group of Lagrangian (11) is the Pauli–Gürsey group [52, 53], the chiral symmetry being a subgroup of the Pauli–Gürsey group:The mass term of the current H-quarks breaks the group explicitly. Indeed, if we consider infinitesimal transformations , , it is readily seen that the mass term in Lagrangian (11) is left invariant by the generators satisfying conditions (15); that is, the mass term is invariant under the subgroup of the Pauli–Gürsey group (see [54, 55]). H-quark condensate has the same spinor structure as the mass term. Thus, the dynamical breaking by the condensate should be also [51, 57]. If the current H-quark masses are significantly smaller than the scale of the spontaneous breaking of the Pauli–Gürsey group, we have the situation similar to the one in well-established QCD of light quarks. Putting it in terms natural to the quark-meson sigma models, there are five pNG bosons associated with the five “broken” generators of the group ; these bosons acquire small masses due to the small explicit breaking of the global symmetry of the model, while the constituent masses of the H-quarks are generated mostly by the dynamical symmetry breaking.

Before leaving our consideration of the Lagrangian of the fundamental current H-quarks, we should note that apart from the Pauli–Gürsey group Lagrangian (11) possesses an additional global symmetry as well as a new discrete symmetry. The former symmetry leads to conservation of an analog of the baryon number, while the latter one is a generalization of the -parity of QCD. The important consequences of these symmetries are discussed at the end of this section and in the next one.

Now, we proceed to construct an effective Lagrangian of a linear quark-hadron sigma model . This model describes the interactions of the constituent H-quarks and lightest (pseudo)scalar H-hadrons. The Lagrangian of the H-quark sector of the model readsHere is a H-quark–H-hadron coupling constant. The matrix of spin-0 H-hadrons is antisymmetric. Its transformation law under the global symmetry isBeing a complex antisymmetric matrix with 12 independent components, the field can be conveniently expanded in terms of five “broken” generators of the Pauli–Gürsey group: The generators are subjected to the conditions and can be written explicitly as Now the Lagrangian of constituent H-quarks (18) can be put into the following form:where and From now on we use tildes to distinguish hypermesons from usual ones. The v.e.v. breaks the global symmetry spontaneously.

As it is seen from the form of the covariant derivative (19), the local electroweak group is embedded into global and breaks it as well as its chiral subgroup explicitly. The covariant derivative of the (pseudo)scalars follows from the transformation properties of :

Using the above derivative, the scalar sector of the model can be written as follows:where the covariant derivatives of the H-meson fields readIn (28) it is assumed that the Higgs doublet of SM is fundamental, not composite. Its transformation properties are defined as usual in SM—the covariant derivative of is or equivalentlyIn Lagrangian (28) the potential term consists of self-interactions of the scalar fields:where is the invariant of the SM Higgs doublet and , , are three independent invariants of the field : Here is the Pfaffian of ; is the 4-dimensional Levi-Civita symbol (); is the Higgs-field v.e.v. We consider only renormalizable self-interactions of the scalar fields, although renormalizability in general has nothing to do with effective field theories. The invariant is odd under CP conjugation. CP invariance implies that and .

Tadpole equations for areVacuum stability is ensured by the following inequalities:Deriving (34) and (35) we have taken into account a tadpole-like source term , where is a parameter proportional to the current mass of the H-quarks. Such term in phenomenological fashion communicates effects of explicit breaking of the global symmetry to the vacuum parameters and the H-hadron spectrum. This resembles QCD—the chiral symmetry is broken both dynamically (with the quark condensate as an order parameter) and explicitly (by the quark masses). In the sigma models with linear realization of the chiral symmetry, the spontaneous breaking is induced by v.e.v. of meson field. The effects of the explicit breaking can be mimicked by different chirally noninvariant terms [64–66], but the most common one, which is sometimes referred to as “standard breaking,” is a tadpole-like term (see [67, 68], for example).

The masses of the (pseudo)scalar fields read

The physical Higgs boson becomes partially composite receiving a tiny admixture of the scalar field : where and are the fields being mixed, while and are physical ones.

Finally, the self-interactions of scalar fields take the formwhere and .

The complete set of the lightest spin-0 H-hadrons in the model includes pNG states (pseudoscalar H-pions and scalar complex H-diquarks/H-baryons ), their opposite-parity chiral partners and , and singlet H-mesons and . These H-hadrons are listed in Table 1 along with their quantum numbers and associated H-quark currents. Note that the total Lagrangian of the model given by (24), (25), (28), and (32) is invariant under a global transformation or equivalently the Lagrangian given by (18), (28), and (32) in terms of the quartet field and the antisymmetric field is invariant under a transformationwhere is a generator of . The EW symmetry, which is spanned by the generators , , defined by (14), does not break the symmetry (40), since the generator commutes with . This additional global symmetry (40) allows us to introduce a conserved H-baryon number, which makes the lightest H-diquark stable. We remind of the fact that the model contains the elementary Higgs field which is not a pNG state. There is, however, a scenario with a composite Higgs boson having also a new strongly coupled sector with the symmetry breaking pattern [69].