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

Authors: V. Beylin; M. Bezuglov; V. Kuksa; N. Volchanskiy
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We analyze an extension of the Standard Model with an additionalSU(2) 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 SU(4)→Sp(4). 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.


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## Body

## 1. Introduction

The experimental detection of the Higgs boson [1, 2] with mass MH≈125 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 extraU(1) 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 S, T, and U 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-likeSU(2)L 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),NHC=2 [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 SU(4)→Sp(4). 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 Section2, we construct vectorlike interactions for the case of SU(2) 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 withZ- and W-bosons. We construct such interactions explicitly for the case of even generations of two-color (NHC=2) H-quarks.In the simplest scenario with two generations (A=1,2) of left-handed H-quarks, the bidoublet of these quarks is presented as a matrix QL(A)aa_, where a=1,2 and a_=1,2 are indices of SU(2)L and SU(2)HC fundamental representations, respectively. (In the following all indices related to the hypercolor group are underlined.)This bidoublet transforms underU(1)Y⊗SU(2)L⊗SU(2)HC as(1)QLAaa_′=QLAaa_+igBYAθQLAaa_+i2gWθkτkabQLAba_+i2gHCφk_τk_a_b_QLAab_.Here QL(A)1a_=UL(A)a_, QL(A)2a_=DL(A)a_, and the H-quarks charges qU,D are defined by the arbitrary hypercharges YA. The right-handed singlets (with respect to electroweak SU(2)L group) have the following group transformations:(2)SRAa_′=SRAa_+igBYRAθSRAa_+i2gHCφk_τk_a_b_SRAb_,where A=1,2 and YR(A) are hypercharges of singlets. Now, the charge conjugation operation, C^, is applied to the fields of the second generation keeping the first generation of H-quarks unchanged:(3)QL2Caa_=C^QL2aa_.The transformation properties of the charge conjugated fields have the form(4)QL2Caa_′=QL2Caa_-igBY2QL2Caa_-i2gWθkτkab∗QL2Cba_-i2gHCφk_τk_a_b_∗QL2Cab_.Then, we redefine the H-quark fields (the fermion chirality is changed by the charge conjugation):(5)QR2aa_=ϵabϵa_b_QL2Cbb_,ϵab=01-10.Further, we multiply both sides of (4) by ϵabϵa_b_ and use the following properties of SU(2) group matrices:(6)ϵacϵbc=δab,ϵabτkbc∗ϵcf=τkaf.Using redefinition (5), from (4), we get(7)QR2aa_′=QR2aa_-igBY2QR2aa_+i2gWθkτkabQR2ba_+i2gHCφk_τk_a_b_QR2ab_.This transformation law coincides with the one given by formula (1) for the first generation (A=1) when Y2=-Y1.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:(8)Qaa_=QL1aa_+QR2aa_=QL1aa_+ϵabϵa_b_QL2Cbb_.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 fieldSR(2)a_ is redefined as follows:(9)SLa_=ϵa_b_C^SR2b_.This redefined field transforms as the right-handed singlet SR(1) if YR(2)=-YR(1) 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, s, which has a nonzero v.e.v., s=σ~+u, where u=s. 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, Y1=-Y2. 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 SU(2)HC group and for the case of an even number of generations.The gauge part of the model Lagrangian directly follows from (1) and (2):(10)LQ,S=-14Tμνk_Tk_μν+iQ¯γμ∂μ-igBY1Bμ-i2gWWμkτk-i2gHCTμk_τk_Q-mQQ¯Q+iS¯γμ∂μ-igBYR1Bμ-i2gHCTμk_τk_S-mSS¯S,where Tμk_ is a H-gluon field. The mass terms are formally included in (10) because they do not break SU(2)HC-symmetry of the model. The status of the SU(2)L-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 S-quark consists of the two initial chiral singlets. It should be noted that the singlet S can be useful since a composite H-meson Q¯S is a representation of the groups U(1)Y⊗SU(2)L. 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 Q and S being independent, from now on, the SU(2)L 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—σ 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 mQ→0, gW→0 has a global SU(4) symmetry corresponding to rotations in the space of the four initial chiral fermion fields. The Lagrangian with nonzero mQ can be rewritten in the form which explicitly reveals the violation of symmetry SU(4)→Sp(4) by the mass term [54, 55]. For mQ=0 the Lagrangian retains the full SU(4) symmetry but, in an analogy with QCD, one might expect the dynamical symmetry breaking by vacuum expectation value U¯U+D¯D≠0. This v.e.v. has the mass term structure and leads to the dynamical breaking of the symmetry SU(4)→Sp(4). As a result, the broken generators of SU(4) 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 withNF~ Dirac quarks in the limit of zero masses is SU(2NF~), with the chiral group being its subgroup, SU(NF~)L⊗SU(NF~)R⊂SU(2NF~) (this statement is valid for any symplectic gauge theory [56]; the group SU(2) is isomorphic to the group Sp(2)) [52, 53]. This global symmetry is often called the Pauli–Gürsey symmetry. The quark condensate breaks the Pauli–Gürsey symmetry to its subgroup Sp(2NF~) [51, 57]. In the following we will consider the simplest case of two flavors NF~=2.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 cosetSU(4)/Sp(4) is given in [56, 58], which considers different gauge groups with arbitrary numbers of flavors and colors, NF~ and NHC). These possibilities are determined by the cancellation of gauge anomalies.(i)
V-A ultraviolet completion. We can introduce a left-handed weak doublet QL=ULDL and two right-handed weak singlets UR and DR with opposite hypercharges Y(UR)=-Y(DR). 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 SU(2)L group, QL=ULDL and QR=URDR [44, 48]. The hypercharges of the doublets should be the same, Y(QL)=Y(QR). In this case the Dirac mass term, Q¯LQR+Q¯RQL, 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:(11)L=-14Tμνk_Tk_μν+iP¯La_Da_b_PLb_-12mQP¯La_M0PRa_+P¯Ra_M0†PLa_,(12)Da_b_μ=∂μδa_b_-i2gHCTk_μτa_b_k_-2igWWkμΣkδa_b_,where(13)PLa_=QL1a_QL2a_,PRa_=ϵa_b_PLb_Care left- and right-handed quartet fields (QL(1) and QL(2) are left-handed doublets introduced in the previous section). The EW term in the covariant derivative (12) involves the matrices(14)Σk=122τk00τk,k=1,2,3,that are three of ten Sp(4) generators Σα satisfying the following conditions:(15)TrΣα=0,Σα†=Σα,TrΣαΣβ=12δαβ,ΣαTM0+M0Σα=0,α,β=1,2,…,10.The mass term in Lagrangian (11) introduces the antisymmetric 4×4 matrix(16)M0=-M0T=0ϵϵ0. We have used the matrix M0 also to define the algebra of the Sp(4) generators. Although M0 has a noncanonical form, it can be brought into the form 01-10 or ϵ00ϵ 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 SU(2)HC, 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 vanishingmQ and gW the global symmetry group of Lagrangian (11) is the Pauli–Gürsey group SU(4) [52, 53], the chiral symmetry being a subgroup of the Pauli–Gürsey group:(17)PLa_⟶UPLa_,PRa_⟶U∗PRa_,U∈SU4.The mass term of the current H-quarks breaks the group SU(4) explicitly. Indeed, if we consider infinitesimal transformations U=1+iθαΣα, θα≪1, 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 Sp(4) of the Pauli–Gürsey group (see [54, 55]). H-quark condensate Q¯Q has the same spinor structure as the mass term. Thus, the dynamical breaking by the condensate Q¯Q should be also SU(4)→Sp(4) [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 SU(4); 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 groupSU(4) Lagrangian (11) possesses an additional global U(1) 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 G-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 modelSU(4)≅SO(6)→SO(5)≅Sp(4). 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 reads(18)L=iP¯LDPL-2κP¯LMPR+P¯RM†PL,(19)DμPL=∂μPL-2igWWμkΣkPL.Here κ is a H-quark–H-hadron coupling constant. The matrix M of spin-0 H-hadrons is antisymmetric. Its transformation law under the global symmetry SU(4) is(20)M⟶UMUT,U∈SU4.Being a complex antisymmetric matrix with 12 independent components, the field M can be conveniently expanded in terms of five “broken” generators βα˙ of the Pauli–Gürsey group:(21)M=122A0+iB0+Aα˙+iBα˙βα˙M0. The generators βα˙ are subjected to the conditions(22)Trβα˙=0,βα˙†=βα˙,Trβα˙βγ˙=12δα˙γ˙,TrΣαβα˙=0,βα˙TM0-M0βα˙=0,α˙,γ˙=1,2,…,5,α=1,2,…,10, and can be written explicitly as(23)βk=122τk00-τk,k=1,2,3,β4=1220110,β5=i2201-10. Now the Lagrangian of constituent H-quarks (18) can be put into the following form:(24)L=iQ¯DQ-κuQ¯Q-κσ′Q¯Q+iη~Q¯γ5Q+a~kQ¯τkQ+iπ~kQ¯γ5τkQ+12A0Q¯aa_ϵabϵa_b_Qbb_C+iB0Q¯aa_ϵabϵa_b_γ5Qbb_C+h.c.,(25)DμQ=∂μQ-i2gWWμkτkQ,where γ5=iγ0γ1γ2γ3 and(26)σ′=A0-u,η~=B0,a~k=Ak,π~k=Bk,k=1,2,3,A0=12A4+iA5,B0=12B4+iB5. From now on we use tildes to distinguish hypermesons from usual ones. The v.e.v. u=A0~Q¯Q breaks the global symmetry SU(4) spontaneously.As it is seen from the form of the covariant derivative (19), the local electroweak group is embedded into global Sp(4) and breaks it as well as its chiral subgroup explicitly. The covariant derivative of the (pseudo)scalars follows from the transformation properties of M:(27)DμM=∂μM-2igWWμkΣkM+MΣkT.Using the above derivative, the scalar sector of the model can be written as follows:(28)L=DμH†·DμH+TrDμM†·DμM-U=12Dμh·Dμh+Dμhk·Dμhk+∂μσ~·∂μσ~+Dμπ~k·Dμπ~k+∂μη~·∂μη~+Dμa~k·Dμa~k+∂μA¯0·∂μA0+∂μB¯0·∂μB0-U,where the covariant derivatives of the H-meson fields read(29)Dμπ~k=∂μπ~k+gWeklmWμlπ~m,Dμa~k=∂μa~k+gWeklmWμla~m.In (28) it is assumed that the Higgs doublet H of SM is fundamental, not composite. Its transformation properties are defined as usual in SM—the covariant derivative of H is(30)DμH=∂μH+i2gBBμH-i2gWWμkτkH, or equivalently(31)H=12h2+ih1h-ih3=12h+ihkτk01,Dμh=∂μh+12gBδ3kBμ+gWWμkhk,Dμhk=∂μhk-12gBδ3kBμ+gWWμkh-12eklmgBδ3lBμ-gWWμlhm.In Lagrangian (28) the potential term U consists of self-interactions of the scalar fields:(32)U=-∑i=03μi2Ii+∑i≤j=03λijIiIj,where I0 is the SU(2)L⊗U(1)Y invariant of the SM Higgs doublet and Ii, i=1,2,3, are three independent SU(4) invariants of the field M:(33)I0=H†H=12v+h2,I1=TrM†M-4RePfM=12u+σ′2+π~kπ~k+2B¯0B0,I2=TrM†M+4RePfM=12η~2+a~ka~k+2A¯0A0,I3=4ImPfM=-u+σ′η~+a~kπ~k+B¯0A0+A¯0B0. Here PfM=-1/4TrMM~=1/8ϵprstMprMst is the Pfaffian of M; ϵprst is the 4-dimensional Levi-Civita symbol (ϵ1234=+1); v=〈h〉 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 I3 is odd under CP conjugation. CP invariance implies thatλ03=λ13=λ23=0 and μ3=0.Tadpole equations forv,u≠0 are(34)μ02=λ00v2+12λ01u2,μ12=λ11u2+12λ01v2+ζQ¯Qu.Vacuum stability is ensured by the following inequalities:(35)Λ11=λ11-ζQ¯Q2u3>0,λ00>0,4λ00Λ11-λ012>0.Deriving (34) and (35) we have taken into account a tadpole-like source term LSB=-ζ〈Q¯Q〉(u+σ′), where ζ is a parameter proportional to the current mass mQ of the H-quarks. Such term in phenomenological fashion communicates effects of explicit breaking of the SU(4) 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 q¯q 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 σ 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 σ term (see [67, 68], for example).The masses of the (pseudo)scalar fields read(36)mσ~,H2=λ00v2+Λ11u2±λ00v2-Λ11u22+λ012v2u2,mπ~2=mB2=-ζQ¯Qu,mη~2=ma~2+2λ33u2,ma~2=mA2=-μ22+12λ02v2+12λ12u2.The physical Higgs boson becomes partially composite receiving a tiny admixture of the scalar fieldσ′:(37)h=cos⁡θsH-sin⁡θsσ~,σ′=sin⁡θsH+cos⁡θsσ~,tan⁡2θs=λ01vuλ00v2-Λ11u2,sgnsin⁡θs=-sgnλ01, where h and σ′ are the fields being mixed, while H and σ~ are physical ones.Finally, the self-interactions of scalar fields take the form(38)L=-λ00h3v+14h-14λ11Bα˙Bα˙+σ′2Bα˙Bα˙+σ′2+4uσ′-14λ01h2v+hBα˙Bα˙+σ′2+2uσ′h-14λ02h2v+hAα˙Aα˙+η~2-14λ12Bα˙Bα˙+σ′2+2uσ′Aα˙Aα˙+η~2-14λ22Aα˙Aα˙+η~22-λ33-u+σ′η~+a~kπ~k+B¯0A0+A¯0B02,where Aα˙Aα˙=2a~+a~-+a~0a~0+2A¯0A0 and Bα˙Bα˙=2π~+π~-+π~0π~0+2B¯0B0.The complete set of the lightest spin-0 H-hadrons in the model includes pNG states (pseudoscalar H-pionsπ~k and scalar complex H-diquarks/H-baryons B0), their opposite-parity chiral partners a~k and A0, and singlet H-mesons σ~ and η~. 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(39)Q′=ei/2ξQ,A0′=eiξA0,B0′=eiξB0 or equivalently the Lagrangian given by (18), (28), and (32) in terms of the quartet field PL and the antisymmetric field M is invariant under a transformation(40)PL′=ei/2ξΣ4PL,M′=ei/2ξΣ4Mei/2ξΣ4T,Σ4=122100-1,where Σ4 is a generator of Sp(4)⊂SU(4). The EW symmetry, which is spanned by the generators Σk, k=1,2,3, defined by (14), does not break the symmetry (40), since the generator Σ4 commutes with Σk. This additional global U(1)HB 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 SU(4)→Sp(4) [69].Table 1
Quantum numbers of the lightest (pseudo)scalar H-hadrons and the corresponding H-quark currents inSU(2)HC model. G~ denotes hyper-G-parity of a state (see Section 4). B~ is the H-baryon number. Qem is the electric charge. T is the weak isospin. Hyperbaryons do not carry intrinsic C- and HG-parities, since the charge conjugation reverses the sign of the H-baryon number.
State H-quark current T G ~ ( J P C ) B ~ Q e m σ ~ Q - Q 0 + ( 0 + + ) 0 0 η ~ i Q - γ 5 Q 0 + ( 0 - + ) 0 0 a ~ k Q - τ k Q 1 - ( 0 + + ) 0 ±1, 0 π ~ k i Q - γ 5 τ k Q 1 - ( 0 - + ) 0 ±1, 0 A 0 Q - a a _ C ϵ a b ϵ a _  b _ Q b b _ 0   ( 0 -   ) 1 0 B 0 i Q - a a _ C ϵ a b ϵ a _  b _ γ 5 Q b b _ 0   ( 0 +   ) 1 0


## 4. Physical Lagrangian of the Minimal Model

Now, we represent the part of physical Lagrangian which is relevant for further analysis of the most interesting case with zero hypercharge (stable H-pion scenario). The H-quark interactions with the EW bosons are vectorlike, and the corresponding Lagrangian follows from (25):(41)LQ,G=12gWU¯γμDWμ++12gWD¯γμUWμ-+12gWU¯γμU-D¯γμDcWZμ+sWAμ.Here cW and sW denote cosine and sine of the Weinberg angle. Interactions of (pseudo)scalars with photons and intermediate bosons are described by the following Lagrangians:(42)Lσ~,H,G=182gW2Wμ+W-μ+gB2+gW2ZμZμcos⁡θsH-sin⁡θsσ2,Lπ~,a~,G=igWW+μπ~0π~,μ--π~-π~,μ0+h.c.+igWcWZμ-sWAμπ~-π~,μ+-π~+π~,μ-+gW2π~+π~-cWZμ-sWAμ2-gW2π~0cWZμ-sWAμπ~+Wμ-+π~-Wμ+-12gW2π~+2Wμ-W-μ+π~-2Wμ+W+μ+gW2π~02+π~-π~+Wμ+W-μ+π~⟶a~.In the above Lagrangian L(π~/a~,G) the last term means that the interactions of the triplet scalar H-mesons a~ have the same couplings and vertices as the interactions of the H-pions.The scalar and pseudoscalar fieldsσ~, π~, and H interaction with the H-quarks is described by the Lagrangian which directly follows from (24):(43)LQ,σ~,H=-κcθσ~+sθHU¯U+D¯D+i2κπ~+U¯γ5D+i2κπ~-D¯γ5U+iκπ~0U¯γ5U-D¯γ5D,where cθ=cos⁡θs and sθ=sin⁡θs. There is a specific symmetry of the minimal hypercolor model leading to some phenomenological consequences. At the fundamental level, the Lagrangian of the current H-quarks (10) is invariant under modified charge conjugation of the H-quark fields (hyper-G-parity,HG-parity) which is defined as follows:(44)Qaa_HG=ϵabϵa_b_Qbb_C,where C is the charge conjugation, a,b are isotopic indices, and a_, b_ are hypercolor indices (it is the same notation as in the Section 2). To prove the statement, we use (6) and the properties of bilinear forms with respect to the ordinary charge conjugation(45)Q¯aa_CQbb_C=Q¯bb_Qaa_,Q¯aa_Cγ5Qbb_C=Q¯bb_γ5Qaa_,Q¯aa_CγμQbb_C=-Q¯bb_γμQaa_.By straightforward calculations one can check that Lagrangian (10) is invariant under the transformation (44), since the H-gluon Tμ and the SM fields are not transformed. To analyze transformation properties of the π~Q¯Q effective vertex in more detail, we use (44) and (45) and have(46)Q¯aa_γ5τabkπ~kQba_HG=Q¯aa_Cϵabϵa_b_γ5τbckπ~kHGϵcdϵb_c_Qdc_C=-Q¯aa_Cγ5τad∗kπ~kHGQda_C=-Q¯aa_γ5τabkπ~kHGQba_.So, the invariance condition results in the transformationπ~kHG=-π~k, that is, π~ is odd, while the SM fields are even under modified charge conjugation (44). This is a special case of the treatment of general vectorlike HC models in [70, 71]. It is observed in [70] thatHG-parity is a good quantum number of the theory and all SM particles areHG-even. Thus,HG-odd π~ has no decay modes with only SM particles in the final states. In the model under consideration decay channels of type π~±→π~0X± are allowed due toHG-parity conservation.It is important that all restrictions on the oblique corrections are fulfilled in this variant of hypercolor. If the hypercharge is zero andh-σ~ mixing is absent, then T-parameter is equal to zero. If, however, we consider a HC scenario with a nonzero hypercharge and mixing, a constraint for the T parameter value emerges (see [35, 43]). Then the h-σ~ mixing angle should be sufficiently small to avoid problems with the PT parameters and the measured properties of the SM Higgs boson.


## 5. Low-Energy Signature of the Model

In this section, we consider briefly main phenomenological consequences of the minimal model for the case of zero hypercharge. In spite of a simple structure and minimal particle content, the model can manifest a rich phenomenology and interesting signature in collider physics. Here we consider processes with the H-sigma(σ~) and H-pions (π~). It is supposed that these states are the lowest ones in the model (see, however, the results of lattice calculations in [72]). Indeed, the claim that pNG states are the lightest in the mass spectrum is based on the hypothesis of a hierarchy of H-physics scales. In other words, we suppose that other (not pNG) possible H-hadrons including vector H-mesons are heavier than the pNG bosons. Namely, the explicit SU(4) symmetry breaking is considered as a small perturbation in comparison with the dynamical symmetry breaking in analogy with the orthodox QCD, where the scale of chiral symmetry breaking is much larger than the light quark masses. From our previous analysis of the parameter space, it follows that the masses of H-mesons are of the order of 102–103 GeV. Thus, the low-energy pNG states of the minimal model can be accessed at the LHC and future linear collider.Channels of H-pion production and decay are described by the model Lagrangian (see the previous section). At the LHC these pNG states most effectively occur in two ways: in vector boson fusion (VBF) reactionpp→V∗V′∗→π~π~′, where V=W,Z,γ, or in the s-channel of qq¯′- orqq¯-fusion—Drell–Yan type (DYT) process, pp→V′∗→π~π~. Corresponding Feynman diagrams can be found in [35]. There is also an analog of usual associated production where H-pion pair is produced together with vector boson, pp→V′∗→Vπ~π~. Its cross section, is somewhat suppressed compared with the DYT reaction by extra factor gW2. The channel, however, has a specific set of final states (see below).As to VBF and DYT mechanisms, their contributions to the cross section of H-pion pair production strongly depend on the invariant H-pion mass, kinematic cuts for final states, quark pdf’s, combinatorial factors, andq→Vq′ splitting functions at high energies. Of course, NLO and NNLO corrections for these channels should be different and can be important—as is the case for Higgs production at the LHC [73–76]. A detailed analysis of LO cross sections and NLO corrections is beyond the scope of the paper.It seems that the VBF production of H-pions is suppressed, in particular, by an additionalgW4, and Drell–Yan type process dominates (see [36]). The situation is, however, more complicated due to the above-mentioned factors, and in the TeV region VV′-fusion cross section is very close to DYT or even larger (see, for example, [76]). Moreover, due to suitable pT cuts, it is possible that, as it happens for the high mass (~TeV) scalar boson production [74], s-channel qq′-fusion cross section should be comparatively small. Of course, it is not the same process; nevertheless, enhancing factors for the VBF are analogous—a lot of integrated partons with low x and pT when vector boson splits off. Namely, due to integration with quark splitting functions in the region of low partonic pT, VBF cross section can be increased by log2⁡(M/MW); M is an invariant mass of H-pion pair. Note also that large resonance s-channel contribution to VBF production with intermediate σ~ is possible if mσ~ is close to 2mπ~. This point should be considered separately.VBF cross section of H-pion pair production, as function ofqq′ center-of-mass energy and H-pion mass, was calculated in our paper [35] and σVBF(pp→qq′π~π~)≈0.01–0.02 pb when Ecm~1TeV and H-pion mass is 200–300 GeV. We also estimate the DYT cross section in this region as approximately 0.03–0.05 pb. Both of these cross sections decrease of about one order of magnitude with the mass of π~ increasing up to ~500–700 GeV.Almost the same situation is observed for the hierarchy of Higgs production mechanisms [77]—associated Higgs production dominates at s=2 TeV—but at higher energies, s≥4TeV, the situation is reversed and VBF cross section exceeds associated production by almost a half. In other words, behavior of these cross sections at high energies should be studied more carefully and it will be done in the next paper.Estimated cross sections of H-pion production are small, so, to detect a signal, large statistics and the background suppression are necessary. From this point of view, VBF reactions are more promising due to the presence of the two hard tagging jets. Adding some reasonable cuts, for the rapidity to highlight the central region of the reaction,|η|≤2.5, and for final leptons, pT≥100GeV, it is possible to separate leptons from π~± decay. These decays are also marked by large missed pT due to heavy stable neutral H-pions and neutrino.H-pion production in the process of annihilatione+e-→γ∗, Z∗→2π~;4π~ is also possible through the reactions of the type Z∗→π~+π~- and W±∗→π~±π~0. These processes have transparent signature and can be studied at future linear colliders. Note that some interesting features should be observed: production of H-pions in associated process, e+e-→Z∗→Zπ~±π~∓, e+e-→Z∗→Wπ~±π~0, or via VV′-fusion. At the ILC Higgs boson production cross sections demonstrate evolution with energy [78] which is analogous to predicted for the LHC. In the H-pion production we expect the same behavior of cross sections.To analyze a final signature in the reactions above, note that due toHG-parity conservation (see the previous section) H-pions have no tree-level decay modes having in the final states the SM particles only. The lowest order amplitudes which govern decays of the type π~→V1V2,V1V2V3, where Va=γ,Z,W, are described by triangle and box diagrams with H-quarks loops. It can be easily checked that interference contributions for the transition π~→V1V2 with U and D hyperquark loops cancel out each other. Since MU=MD, this compensation is obvious due to the opposite charges, qD=-qU, when Y1=0. Analysis of the decay π~→V1V2V3 reveals compensation of box contributions. More exactly, the diagrams with loop momenta circulating in opposite directions cancel out each other. It is easy to prove that the compensation results from generalized Furry’s theorem [79]. To this end, we should use the following properties of the Dirac and Pauli matrices:(47)Tr⁡γμ1γμ2⋯γμn=Tr⁡γμn⋯γμ2γμ1,Tr⁡τa1τa2⋯τan=-1nTr⁡τan⋯τa2τa1.The cancellation of amplitudes in the case of an even number of final bosons is inherently isotopic—it results from the zero H-quark hypercharge. If the number of final bosons is odd, such cancellation follows from the charge parity conservation along with the vectorlike structure of the H-quark EW interaction. As a result, the H-pion fields are stable in the framework of the vectorlike hypercolor model with zero hypercharge and degenerate masses in the H-quark doubletQ and triplet π~. In the previous section, it was demonstrated that this stability follows from the presence of the discrete symmetry in the model.Note that the H-quark masses remain degenerate,MU=MD, at the one-loop level. It can be easily checked that the self-energy contributions into the mass renormalization are defined by electroweak and H-pion loops. These terms are exactly the same for the U and D quarks. However, this effect does not take place in the case of the H-pion masses. The mass-splitting value of the H-pion can be calculated by summing over self-energy diagrams. Detailed analysis of the relevant amplitudes reveals that only EW diagrams contribute to the mass-splitting Δmπ~=mπ~±-mπ~0; all strong (H-quark) loops are cancelled out. As a result we get(48)Δmπ~=GFMW422π2mπ~ln⁡MZ2MW2-βZ2ln⁡μZ+βW2ln⁡μW-4βZ3μZarctan⁡2-μZ2μZβZ+arctan⁡μZ2βZ+4βW3μWarctan⁡2-μW2μWβW+arctan⁡μW2βW,where μV=MV2/mπ~2, βV=1-μV/4, and GF is Fermi’s constant. For the H-pion masses in the interval 200–800 GeV from (48) it follows that Δmπ~≈0.170–0.162GeV. Nonzero mass-splitting in the H-pion triplet violates isotopic invariance. However,HG-parity remains a conserved quantum number since it is induced by a discrete symmetry rather than a continuous transformation in the space of H-pion states. Thus, an account of higher order corrections does not lead to destabilization of the neutral H-pion.So, the analysis performed leads to the conclusion that the model involves stable weakly interacting neutral H-pion. Then, production of the H-pions at colliders manifests itself with some unique signature of the final state—charge leptons and large missing energy. The stable H-pionπ~0 can be also considered as a component of Dark Matter.For the width of the charged H-pion decay in the strong channel we get(49)Γπ~±⟶π~0π±=GF2πfπ2Uud2mπ~±Δmπ~2λ¯mπ±2,mπ~02;mπ~±2.Here fπ=132MeV, π± is a standard pion, and(50)λ¯a,b;c=1-2a+bc+a-b2c21/2.The H-pion decay width in the lepton channel is(51)Γπ~±⟶π~0l±νl=GF2mπ~±324π3∫q12q22λ¯q2,mπ~02;mπ~±23/21-3ml22q2+ml62q6dq2,where q12=ml2, q22=(Δmπ~)2, and ml is a lepton mass.Now, using (49), (51), and the value Δmπ~ from (48), we estimate decay widths, lifetimes, and track lengths in these channels as follows:(52)Γπ~±⟶π~0π±=6·10-17GeV,τπ=1.1·10-8sec,cτπ≈330cm;Γπ~±⟶π~0l±νl=3·10-15GeV,τl=2.2·10-10sec,cτl≈6.6cm.From these analysis it follows that main characteristic fingerprints of H-pions at TeV scale in the VBF, DYT, and associated production are(1)
V∗V∗→π~π~+jj—two hard tagging jets, high pT,mis from two π~0, neutrino, and a lepton (or two charged leptons) from π~±;(2)
V∗→π~π~—high pT,mis from two π~0 and νl, and final one lepton, ll¯ or π±π±, from pair of π~±;(3)
V∗→Vπ~π~—hadron jets (or ll¯ or lνl) from W or Z decays, high pT,mis from two π~0, and neutrino (from π~± and/or W±); l+l- —if there are two final charged H-pions or one charged H-pion; and W, trilepton signal from Wπ~±π~± final state.As to the production of a single scalar H-sigmaσ~ at the LHC and ILC, it is strongly suppressed reaction at the tree-level due to the small σ~-h mixing. More exactly, the tree-level σ~ production is suppressed with respect to the Higgs production by sin2⁡θs, where θs is a mixing angle.At the one-loop level both single and double H-sigma production occur in the processes of typeV∗V′∗→σ~,2σ~ and/or V∗→Δ→V′σ~,2σ~, where V∗ and V′ are vector bosons in the intermediate and final states; Δ denotes a H-quark triangle loop.Decays of the typeσ~→V1V2, where V1,2=γ,Z,W, proceed through H-quark and H-pion loops. Dominant decay channels of H-sigma are σ~→π~0π~0,π~+π~-, which take place at the tree-level and provide large decay width for mσ~⩾2mπ~. The width is mostly defined by the coupling λ11 in the limit of small mixing:(53)Γσ~⟶π~π~=3u2λ1128πmσ~1-4mπ~2mσ~2.Using the previous parametric analysis in [35] concerning the value λ11 (λHC in [35]) and u, from (53), one can get Γ(σ~→π~π~)≳10GeV when mσ~≳2mπ~.As it was noted above, the small mixingh–σ~ in conformal approximation leads to the relation mσ~≈3mπ~ and all tree-level decay widths are proportional to the square value of the σ~-h mixing angle θs. Corresponding decay widths are as follows:(54)Γσ~⟶ff¯=gW2sin2θs32πmσ~mf2MW21-4mf2mσ~23/2,Γσ~⟶ZZ=gW2sin2θs16πcW2MZ2mσ~1-4mZ2mσ~21/21+mσ~2-2MZ228MZ4,Γσ~⟶W+W-=gW2sin2θs8πMW2mσ~1-4mW2mσ~21/21+mσ~2-2MW228MW4.In (54) mf is a mass of standard fermion f and cW=cos⁡θW. In the limit of zero mixing we should consider the loop-level decay channels. Here, we consider the decay channel σ~→γγ which proceeds mainly through H-quark and H-pion loops. The width can be written in the form(55)Γσ~⟶γγ=α2mσ~16π3FQ+Fπ~+Fa~+FW+Ftop2,where the contributions of H-quarks, FQ, H-pions, Fπ~, W-bosons, FW, and top-quarks, Ftop, are defined by the following expressions:(56)FQ=-2κMQmσ~1+1-τQ-1fτQ,Fπ~=gπ~σ~mσ~1-τπ~-1fτπ~,gπ~σ~≈uλ11,Fa~=ga~σ~mσ~1-τa~-1fτa~,ga~σ~≈uλ12,FW=-gWsin⁡θsmσ~8MW2+3τW-1+3τW-12-τW-1fτW,Ftop=43gWsin⁡θsMt2mσ~MW1+1-τt-1fτt,fτ=arcsin2⁡τ,τ<1,fτ=-14ln⁡1+1-τ-11-1-τ-1-iπ2,τ>1.Nonzero σ~-h mixing influences the width viaW- andt-quark loops, their amplitudes being proportional to sin⁡θs. Using the analysis of the model parameter space in [35], we get an estimation Γ(σ~→γγ)≈5–10MeV. To calculate the σ~ production in full processes pp¯→σ~→all, a corresponding program which integrates partonic cross sections with the quark distribution functions should be used. Instead, we give an approximate evaluation of this cross section for the subprocess of vector boson fusion VV→σ~(s)→all, where V=γ,Z,W. Namely, the cross section can be calculated with sufficient accuracy using a simple formula in the framework of factorization method [80]:(57)σVV⟶σ~s=16π2Γσ~s⟶VV9sλ¯2MV2,MV2;sρσ~s,where σ~(s) is σ~ in the intermediate state with energy s and Γ(σ~(s)→VV) is a partial width. The probability density ρσ~(s) is defined by the following expression:(58)ρσ~s=1πsΓσ~ss-Mσ~22+sΓσ~2s,where Γσ~(s) is the total width of the σ~ with a mass squared equal to s. Exclusive cross section at peak energy region s=Mσ~ can be found by the change in the numerator of the expression (58) Γσ~→Γ(σ~→V′V′)=Γσ~·Br(σ~→V′V′):(59)σVV⟶σ~⟶V′V′=16π9Brσ~⟶VVBrσ~⟶V′V′mσ~21-4MV2/mσ~2≈16π9mσ~2·Brσ~⟶VVBrσ~⟶V′V′.So, the cross section at mσ~2≫MV2 is fully defined by branchings of sigma decay and mσ~. When 2mπ~>mσ~ dominant decay channels are σ~→WW,ZZ, which lead to a narrow peak (Γ≲10–100MeV). However, here we have the cross section of the subprocess and do not take into account the distribution function. Moreover, we should also average cross section over energy resolution. Both these factors reduce significantly the value of cross section. When 2mπ~<mσ~ dominant decay channel is σ~→π~π~ which leads to a wide peak (Γ~10GeV). In this case Br(σ~→VV) is small and we get very small cross section. Thus, the main signature of the H-sigma production and decay is a wide peak at 2mπ~<mσ~, mostly caused by the strong possible decay σ~→2π~ along with weak signals caused by two-photon, lepton, and quark-jet final states (from WW, ZZ, and standard π± channels). There is also specific decay mode with two stable neutral H-pions as products of σ~ decay—this manifests itself in a large missing energy together with charged leptons in the final states. As it was shown in the end of Section 3, due to the global U(1)HB symmetry, the lightest H-diquark is stable. Then, from the physical Lagrangian, it follows that the other H-diquark can decay to the stable one and something else. So, there is a possibility of constructing the Dark Matter from two types of particles: stable neutral H-pion and the lightest scalar (or pseudoscalar) H-diquark with conserved H-baryon number. Detailed consideration of the two-component scenario of the DM depends on the variety of model parameters, mass-splitting between the pNG states, and agreement with the data on the DM relic. We add that the suggested DM model does not contain (stable) H-baryon carrying the EW charge (see, for example, [81]), so there are no strong constraints for the DM relic in the case. The study is in progress now and results will be presented in the next paper.As toA0,B0 production at the colliders, these particles can be produced only by intermediate pNG states, a~a,  η~, and the Higgs boson, h, or σ~. At the tree-level these channels are suppressed by the mixing angle. They also can originate from loops with the participation of pNG.


## 6. Conclusion

The analysis performed demonstrates some unique features of the simplest minimal HC model with two generations of H-quarks andSU(2)HC as the H-confinement group. This scenario makes it possible to construct vectorlike interaction, starting from chiral nonsymmetric H-quark set of fields. In the simplest case of two-flavor scenario the set of pNG bosons, (pseudo)scalar H-mesons and H-baryons (H-diquarks), arises, which provides the rich phenomenology. The neutral H-pion π~0 is stable when Y1=0 due to hyper-G-parity conservation, so specific decay channels for π~± and σ~ with a large missing energy open. Moreover, analysis of the production and decays of (pseudo)scalar states, H-pion and H-sigma, allows distinguishing between scenarios with zero and nonzero H-quarks hypercharge [35]. At the same time, the model predicts a strong signal with large missing energy in the case 2mπ~<mσ~ or weak signal with two-vector final states in the opposite case. The presence of nonanomalous global symmetry U(1)HB in the model leads to the conservation of H-baryon charge. This, in turn, manifests itself in the presence of the stable H-baryon complex field B0. Note that the H-baryon state A0 can be stable also when MA0<MB0. This possibility will be studied separately.The minimal model under consideration has some phenomenological features which can be verified both at collider experiments and by astrophysical observations. An interesting consequence of the model structure is a possible interpretation of the stable neutral H-pions and H-baryons as particles of DM. So, the model with stable neutral fields gives the possibility of constructing two-component DM. In this work we concentrated mainly on the methodological aspects of the model. To make complete phenomenological analysis, we should consider astrophysical applications and take into account all experimental restrictions on new physics.



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