An effective strong-coupling theory of composite particles in UV-domain

We briefly review the effective field theory of massive composite particles, their gauge couplings and characteristic energy scale in the UV-domain of UV-stable fixed point of strong four-fermion coupling, then mainly focus the discussions on the decay channels of composite particles into the final states of the SM gauge bosons, leptons and quarks. We calculate the rates of composite bosons decaying into two gauge bosons $\gamma\gamma$, $\gamma Z^0$, $W^+W^-$, $Z^0Z^0$ and give the ratios of decay rates of different channels depending on gauge couplings only. It is shown that a composite fermion decays into an elementary fermion and a composite boson, the latter being an intermediate state decays into two gauge bosons, leading to a peculiar kinematics of final states of a quark (or a lepton) and two gauge bosons. These provide experimental implications of such an effective theory of composite particles beyond the SM. We also present some speculative discussions on the channels of composite fermions decaying into $WW$, $WZ$ and $ZZ$ two boson-tagged jets with quark jets, or to four-quark jets. Moreover, at the same energy scale of composite particles produced in high-energy experiments, composite particles are also produced by high-energy sterile neutrino (dark matter) collisions, their decays lead to excesses of cosmic ray particles in space and signals of SM particles in underground laboratories.


I. INTRODUCTION
The parity-violating (chiral) gauge symmetries and spontaneous/explicit breaking of these symmetries for the hierarchy of fermion masses have been at the center of a conceptual elaboration that has played a major role in donating to mankind the beauty of the SM for particle physics.
The Nambu-Jona-Lasinio (NJL) model [1] of dimension-6 four-fermion operators at high energies and its effective counterpart, the phenomenological model [2]  A. Weak-interacting four-fermion operators and symmetry-breaking phase The dynamics of new physics at high energies may be represented by an effective theory of high-dimensional operators of fermion fields, e.g., dimension-6 four-fermion operators, preserving at least the SM gauge symmetries. The strong technicolor dynamics of extended gauge theories at the TeV scale was invoked [15,16] to have a natural scheme incorporating the relevant fourfermion operator G(ψ ia L t Ra )(t b R ψ Lib ) of the t t -condensate model [17], to generate the top-quark mass via spontaneous symmetry breaking (SSB). On the other hand, these relevant operators can be constructed on the basis of phenomenology of the SM at low-energies. In 1989, several authors [17][18][19] suggested that the SM symmetry breakdown could be a dynamical mechanism of the NJL type that intimately involves the top quark at the high-energy scale Λ. Since then, many models based on this idea have been studied [20]. In the scenes of renormalization group (scaling) invariance and resultant renormalization-group (RG) equations, there is a scaling region (IR-domain) of the infrared (IR) stable fixed point of four-fermion operators, the low-energy SM physics was supposed to be achieved by the RG equations in the IR-domain with the electroweak scale v ≈ 239.5 GeV [16,17,19]. It is in this IR-domain, we recently present the detailed study of hierarchy mass spectrum of SM fermions: from top quark to electron neutrino [7,8], on the basis of effective four-fermion operators of Einstein-Cartan type, as will be shown in Sec. II.

B. Strong-interacting four-fermion operators and gauge-symmetric phase
In addition to the IR-domain of weak-coupling four-fermion operators in the SSB phase where the low-energy SM is realized, we find a scaling region (UV-domain) of the ultraviolet (UV) stable fixed point of strong-coupling four-fermion operators in the gauge-symmetric phase. In this UVdomain at high energies, it realizes an effective theory of composite bosons and fermions composed by SM elementary fermions, these composite particles and their interactions preserving the SM gauge symmetries [9][10][11][12][13]. This is the issue that we would like to focus in this article.
In Refs. [13,14], we have already presented discussions on the possible decay channels of compos-

II. FOUR-FERMION OPERATORS BEYOND THE SM
In order for a self-contained and self-consistent article, as well as for readers' convenience, we include this section of describing effective four-fermion operators at the cutoff Λ that was similarly presented in Ref. [8] for discussing the hierarchy spectrum of SM fermions in the IR-domain. It is also necessary to present these effective four-fermion operators at the cutoff Λ to clarify its induced Lagrangian of the composite particle spectrum and interactions in the UV-domain.

A. Regularization and quantum gravity
Up to now the theoretical and experimental studies tell us the chiral gauge-field interactions to fermions in the lepton-quark family that is replicated three times and mixed. The spontaneous breaking of these chiral gauge symmetries and generating of fermion masses are made by the Higgs field sector. In the IR-fixed-point domain of weak four-fermion coupling or equivalently weak Yukawa coupling, the SM Lagrangian with all relevant operators (parametrizations) is realized and behaves an effective and renormalizable field theory in low energies. To achieve these SM relevant operators, a finite field theory of chiral-gauge interactions should be well-defined by including the quantum gravity that naturally provides a space-time regularization (UV cutoff). As an example, the finite superstring theory is proposed by postulating that instead of a simple space-time point, the fundamental space-time "constituents" is a space-time "string". The Planck scale is a plausible cut-off, at which all principle and symmetries are fully respected by gauge fields and particle spectra, fermions and bosons.
In this article, we do not discuss how a fundamental theory at the Planck scale induces highdimensional operators. Instead, as a postulation or motivation, we argue the presence of at least four-fermion operators beyond the SM from the following point view. A well-defined quantum field theory for the SM Lagrangian requires a natural regularization (UV cutoff Λ) fully preserving the SM chiral-gauge symmetry. The quantum gravity naturally provides a such regularization of discrete space-time with the minimal lengthã ≈ 1.2 a pl [21], where the Planck length a pl ∼ 10 −33 cm and scale Λ pl = π/a pl ∼ 10 19 GeV. However, the no-go theorem [22] tells us that there is no any consistent way to regularize the SM bilinear fermion Lagrangian to exactly preserve the SM chiralgauge symmetries, which must be explicitly broken at the scale of fundamental space-time cutof a. This implies that the natural quantum-gravity regularization for the SM should lead us to consider at least dimension-6 four-fermion operators originated from quantum gravity effects at short distances [53]. As a model, we adopt the four-fermion operators of the torsion-free Einstein-Cartan Lagrangian within the framework of the SM fermion content and gauge symmetries. We stress that a fundamental theory at the UV cutoff is still unknown.

B. Einstein-Cartan theory with SM gauge symmetries and fermion content
The Lagrangian of torsion-free Einstein-Cartan (EC) theory reads, where the gravitational Lagrangian L EC = L EC (e, ω), tetrad field e µ (x) = e a µ (x)γ a , spinconnection field ω µ (x) = ω ab µ (x)σ ab , the covariant derivative D µ = ∂ µ − igω µ and the axial current J d =ψγ d γ 5 ψ of massless fermion fields. The four-fermion coupling G relates to the gravitationfermion gauge coupling g and fundamental space-time cutoffã.
In the context of the SM one-family fermion content and gauge symmetries, we consider massless, two-component, left-and right-handed Weyl fermions ψ f L (doublets) and ψ f R (singlets) carrying the quantum numbers of the SM SU L (2) × U Y (1) chiral gauge symmetries, where "f " is the fermion-family index, as well as three right-handed Weyl sterile neutrinos ν f R and their lefthanded conjugated fields ν f c R = iγ 2 (ν R ) * , which do not carry any quantum number of SM gauge symmetries. Analogously to the EC theory (1), we obtain a torsion-free, diffeomorphism and local gauge-invariant Lagrangian where the SM gauge fields A µ are present in the co-variant derivative D µ to preserve the SM gauge symmetries, and axial currents read The four-fermion coupling G is unique for all four-fermion operators and high-dimensional fermion operators (d > 6) are neglected.
By using the Fierz theorem [23,24], the dimension-6 four-fermion operators in Eq. (2) can be written as [14] which preserve the SM gauge symmetries. Equations (4) and (5) where the two component Weyl fermions ψ f L and ψ f R respectively are the SU L (2) × U Y (1) gauged doublets and singlets of the SM. For the sake of compact notations, ψ f R are also used to represent ν f R , which have no any SM quantum numbers. All fermions are massless, they are four-component Dirac fermions ψ f = (ψ f L +ψ f R ), two-component left-handed Weyl neutrinos ν f L and four-component sterile Majorana neutrinos ν f M = (ν f c R + ν f R ) whose kinetic terms read In Eq. (6), f and f ′ (f, f ′ = 1, 2, 3) are fermion-family indexes summed over respectively for three lepton families (charge q = 0, −1) and three quark families (q = 2/3, −1/3). Equation (6) preserves not only the SM gauge symmetries and global fermion-family symmetries, but also the global symmetries for fermion-number conservations. We adopt the effective four-fermion operators (6) in the context of a well-defined quantum field theory at the high-energy scale Λ.
C. Fermion-family symmetry and mass eigenstate As argued in the introduction Sec. II A, the origin of effective four-fermion operators in Eqs. (1)(2)(3)(4)(5)(6) is due to the quantum gravity that couples to all fermion fields and provides a natural regularization for chiral gauge field theories, like the SM, at the UV cutoff Λ. Therefore, there is no any reason to assume different four-fermion coupling G's for different fermions that equally couple to the gravitational field in the torson-free Einstein-Cartan theory (1).
It should be further clarified that in the effective Lagrangian (6)  In order to study the gauge-boson-fermion and four-fermion interactions in terms of fermion mass-energy spectra and currents, measured as physical final states, we adopt the energy-mass eigenstates of fermions in the following discussions of entire article. The unitary chiral transfor- where U L and U R are related by a unitary matrix V, can be performed from gauge eigenstates to mass eigenstates (up-and down-quark sectors as example): and so that in Eq. (6) the fermion-family indexes f = f ′ , i.e., δ f f ′ respectively for the u-quark sector and the d-quark sector. As a result, all quark fields are mass eigenstates, the four-fermion operators (6) are "diagonal" only for each quark family without fermion-family mixing, In the following, we adopt this representation and our notations will be only for the first SM family, however, they are the same for the second and third families.
In this representation, bilinear fermion-mass operators ψ f Rψ f L + h.c., i.e., quark-mass matrices are diagonalized in the fermion-family space by the biunitary transformations where all quark masses (eigenvalues) are positive, U L and U R are related by and V u,d is a unitary matrix, see for example [25,26]. Using unitary matrices U u L,R (8) and U d L,R (9), up to a diagonal phase matrix we define the unitary quark-family mixing matrices, where the first element is the CKM matrix U = U q L ≡ U u † L U d L . The similar discussions for the lepton sector can be found in Ref. [8].
In the IR effective theory in the IR-domain, the nonzero and different expectational values of fermion-mass operators ψ f Rψ f L ∝ m f = 0 (11) and fermion-family hierarchy m f = m f ′ can be developed due to both spontaneous-symmetry and explicit-symmetry breaking of chiral symmetries [7]. In this case, apart from the breaking of the SM chiral gauge symmetries, fermion-family (flavor) U L (3) × U R (3) symmetries are broken to the U (1) symmetry for each fermion family, i.e., for the four-fermion operators (10), two-fermion operators (11) and (12).
However, the mass eigenstates of the SM elementary fermions are different from their chiral-gauge eigenstates, i.e., the interaction vertexes of the chiral-gauge-boson W ± and massive fermions are not diagonal in the fermion-family space based on the mass eigenstates of fermions, as can be seen in the kinetic terms of the effective Lagrangian (6).
In the UV effective theory in the UV-domain, instead, the chiral and flavor symmetries are preserved for m f = 0, indicating that gauge and mass eigenstates of composite particles are the same.
In Secs. (III A 3) and (IV B 3), we will come back to the discussions of the flavor-changing-neutralcurrent (FCNC) processes, considering the chiral-and flavor-symmetries breaking (preserving) of IR (UV) effective theory in the IR (UV) domain respectively.

D. SM gauge-symmetric four-fermion operators
Using Eq. (10) and we explicitly show SM gauge symmetric four-fermion operators. In the quark sector, the four-fermion operators are where a, b and i, j are the color and flavor indexes of the top and bottom quarks, the quark SU L (2) doublet ψ ia L = (t a L , b a L ) and singlet ψ a R = t a R , b a R are the eigenstates of electroweak interaction. The first and second terms in Eq. (15) are respectively the four-fermion operators of top-quark channel [17] and bottom-quark channel, whereas "terms" stands for the first and second quark families that can be obtained by substituting t → u, c and b → d, s [13,29,30].
In the lepton sector with three right-handed sterile neutrinos ν ℓ R (ℓ = e, µ, τ ), the four-fermion operators in terms of gauge eigenstates are, preserving all SM gauge symmetries, where the lepton SU L (2) doublets ℓ i L = (ν ℓ L , ℓ L ), singlets ℓ R and the conjugate fields of sterile neutrinos ν ℓc R = iγ 2 (ν ℓ R ) * . Coming from the second term in Eq. (6), the last term in Eq. (16) preserves the symmetry U lepton (1) for the lepton-number conservation, although (ν ℓ R ν ℓc R ) violates the lepton number of family "ℓ" by two units. Similarly, from the second term in Eq. (6) there are following four-fermion operators where quark fields u ℓ a,R = (u, c, t) a,R and d ℓ a,R = (d, s, b) a,R .

E. Four-fermion operators of quark-lepton interactions
Although the four-fermion operators in Eq. (6) do not have quark-lepton interactions, we consider the following SM gauge-symmetric four-fermion operators that contain quark-lepton interactions [7,8,27], where ℓ i L = (ν e L , e L ) and ψ Lia = (u La , d La ) for the first family. The (· · ·) represents for the second and third families with substitutions: e → µ, τ , ν e → ν µ , ν τ , and u → c, t and d → s, b. The four-fermion operators (18) of quark-lepton interactions are not included in Eq. (6), since leptons and quarks are in separated representations of SM gauge groups. They should be expected in the framework of Einstein-Cartan theory and SO(10) unification theory [28].

III. IR-STABLE AND UV-STABLE FIXED POINTS AND THEIR SCALING REGIONS
Apart from what is possible new physics at the UV scale Λ explaining the origin of these effective four-fermion operators, it is essential and necessary to study: (i) the phase diagram in the space of these effective four-fermion operator couplings; (ii) which dynamics of these operators undergo in terms of their couplings as functions of running energy scale µ; (iii) associating to these dynamics where infrared (IR) and/or ultraviolet (UV) stable fixed point of physical couplings locates; (iv) in the IR and/or UV domains (scaling regions) of these stable fixed points, which operators become physically relevant and renormalizable following RG equations (scaling laws), and other irrelevant operators are suppressed by the cutoff at least O(Λ −2 ).

A.
Symmetry-breaking phase and the IR-domain of the IR-stable fixed point In the NJL symmetry-breaking phase of weak coupling G > G c [54], where G c is the weak critical coupling of the NJL dynamics, the low-energy SM physics was supposed to be achieved in the IR-domain (G G c ) of IR-stable fixed point [16,17,19]. Bardeen, Hill and Lindner (BHL) proposed the effective Lagrangian of the tt-condensate model [17], and quartic λ 0 couplings are defined at an intermediate scale E (v < E < Λ), that will be clear below (30). Renormalized quantities, like Yukawa couplingḡ t (µ 2 ) and quartic couplingsλ(µ 2 ) are defined at the running energy scale µ and satisfy the RG equations in the IR-domain, where one can find A, B and RG equations for running gauge couplings g 2 1,2,3 in Eqs. (4.7), (4.8) of Ref. [17].
for G G c ≡ 8π 2 /(N c E 2 ) and intermediate energy scale E > µ v, where N c = 3 is the number of colors. The positive β-function of Eq. (22) indicates that an IR-stable fixed point G c and the IR-domain G → G c + 0 + as the running energy scale µ → v, as shown the part "I" of the β(G)function in Fig. 2. As the energy scale µ decreases, indicated by a thick arrow in Fig. 2, the RG flow is attracted to the IR-stable fixed point, and the effective SM of particle physics in low energies is realized in this IR-domain (vicinity) of IR-stable fixed point. Instead, we adopted [13,30] the experimental values of the top-quark and Higgs-boson masses, m H = 126 ± 0.5 GeV; m t = 172.9 ± 0.8 GeV, (23) and used the mass-shell conditions m t =ḡ t (m t )v/ √ 2 and m 2 H /2 =λ(m H )v 2 , as infrared boundary conditions to integrate the RG equations (20) and (21) so as to uniquely determine the functions of top-quark Yukawaḡ t (µ) and Higgs-quarticλ(µ) couplings (see Fig. 1 Our results ( Fig. 1) are radically different from the BHL results [55], not only the completely different behaviors and numerical values of the Yukawaḡ t (µ) and quarticλ(µ) couplings that will be discussed in next subsection III A 2, but also (i) the composite Higgs boson is a tightly bound state oftt pair for finite valuesZ H (µ), as if it is an elementary Higgs boson; (ii) the phase transition from the symmetry-breaking phase to the symmetric phase is indicated byλ(µ) → 0 + as µ → E + 0 − , will be discussed in Sec. III C 1; (iii) the drastically fine-tuning (hierarchy) problem can be resolved, since the intermediate scale E (v < E ≪ Λ) replaces the high-energy cutoff Λ and sets into Eq. (22), the gap-equation for more detailed discussions, see Ref. [30].
In the IR-domain (G G c ) of unique four-fermion coupling G, it seems that the four-fermion operators (15) undergo the SSB, leading to the fermion-condensation M f f ′ = −G ψ f ψ f ′ = mδ f f ′ = 0, two diagonal mass matrices of quark sectors q = 2/3 and q = −1/3 satisfying 3 + 3 mass-gap equations, see Eqs. (11) and (12). It was demonstrated [29] that as an energetically favorable solution of the SSB ground state of the SM, only top-quark is massive (m sb t = −G ψ t ψ t = 0), otherwise in addition to those become the longitudinal modes of massive gauge bosons, there would be more Goldstone modes contributing the SSB-ground-state energy. In other words, among fourfermion operators (15) and (16), the t t -condensate model (19) is the unique channel undergoing the SSB of SM gauge symmetries, for the reason that this is energetically favorable, i.e., the groundstate energy is minimal when the maximal number of Goldstone modes are three and equal to the number of the longitudinal modes of massive gauge bosons in the SM. Moreover, the four-fermion operators (16) of the lepton sector do not undergo the SSB leading to the lepton-condensation , two diagonal mass matrices of the lepton sector (q = 0 and q = −1). The reason is that the effective four-lepton coupling (GN c )/N c is N c -times smaller than the four-quark coupling (GN c ), where the color number N c = 3. In the IR-domain (G G c ) of the IR-stable fixed point G c , the effective four-quark coupling is above the critical value and the SSB occurs, whereas the effective four-lepton coupling is below the critical value and the SSB does not occur for the lepton sector.
As a result, only the top quark acquires its mass via the SSB and four-fermion operator (19) of the top-quark channel becomes the relevant operator following the RG equations in the IR domain [17]. While all other quarks and leptons do not acquire their masses via the SSB and their fourfermion operators (15,16,17,18) are irrelevant dimension-6 operators, whose tree-level amplitudes of four-fermion scatterings are highly suppressed O[(µ/Λ) 2 ], thus their deviations from the SM are nowadays experimentally inaccessible [14].

Experimental indications of tightly-bound composite Higgs boson ?
The wave-function renormalizationZ H (µ) = 1/ḡ 2 t (µ) represents the form-factor of composite Higgs boson. In the BHL case, the compositeness conditionZ H (µ) → 0, µ → Λ is adapted, this implies that the BHL loosely-bound composite Higgs boson behaves very differently from the However, the effective top-Yukawaḡ t (µ) and Higgs-quarticλ(µ) couplings monotonically and slightly decrease as the energy scale µ increases over the range (M Z < µ < E), see Fig. 1. In the range m H < µ < E, these might imply some effects on the rates or cross-sections of the following three dominate processes of Higgs-boson production and decay [3,4] or other relevant processes. Two-gluon fusion produces a Higgs boson via a top-quark loop, which is proportional to the effective Yukawa couplingḡ t (µ). Then, the produced Higgs boson decays into the twophoton state by coupling to a top-quark loop, and into the four-lepton state by coupling to two massive W -bosons or two massive Z-bosons. Due to thet t-composite nature of Higgs boson, the one-particle-irreducible (1PI) vertexes of Higgs-boson coupling to a top-quark loop, to two massive W -bosons and to two massive Z-bosons are proportional to the effective Yukawa couplingḡ t (µ).
As a result, both the Higgs-boson decaying rate Γ H→f to each of these three channels and total decay rate Γ total H = f Γ H→f are proportional toḡ 2 t (µ), however the branching ratio (Γ H→f /Γ total H ) of each Higgs-decay channel is not changed. The energy scale µ is actually the Higgs-boson energy, representing the total energy of final states, e.g., two-photon state and four-lepton states, into which the produced Higgs boson decays.
These discussions imply that the resonant amplitude (number of events) of two-photon invariant mass m γγ ≈ 126 GeV and/or four-lepton invariant mass m 4l ≈ 126 GeV is expected to become smaller as the produced Higgs-boson energy µ increases, i.e., the energy of final two-photon and/or four-lepton states increases, when the CM energy √ s of LHC p p collisions increases with a given luminosity. Suppose that the total decay rate or each channel decay rate of the SM Higgs boson is measured at the Higgs-boson energy µ = m t and the SM value of Yukawa couplingḡ 2 t (m t ) = 2m 2 t /v ≈ 1.04, see Fig. 1. In the scenario of tightly-bound composite Higgs boson, as the Higgsboson energy µ increases to µ = 2m t , the Yukawa couplingḡ 2 t (2m t ) ≈ 0.98, the variation of total decay rate or each channel decay rate is expected to be 6% for ∆ḡ 2 t ≈ 0.06. Analogously, the variation is expected to be 9% at and thus hard to be identified [14] in high-energy processes of LHC p p collisions (e.g., the Drell-Yan dilepton process, see Ref. [31]), e − e + annihilation to hadrons and deep inelastic lepton-hadron e − p scatterings at TeV scales. Nonetheless, these effects are the nonresonant new signatures of low-energy collider that show the deviations of such scenario from the SM.

Yukawa couplings and FCNC's in IR-domain
The SSB generated top-quark mass m t ∝ḡ t v breaks chiral symmetries, fermion-family mixing matrices (14) become relevant for coupling different fermion families. As a consequence, other quarks and leptons acquire their massesḡ f v, because Schwinger-Dyson (SD) equations for their self mass-energy functions acquire the explicit symmetry breaking (ESB) terms induced by the W ± -boson vector-like couplings and quark-lepton interactions at high energies, via the fermionfamily mixing matrices like the CKM (14) and PMNS matrices in the SM. References [7,8] symmetries are broken to the U (1) symmetry for each fermion family, i.e., Thus the quadrilinear four-fermion operators (10), bilinear two-fermion operators (11) and (12) are "diagonal", namely they are for each fermion family without any fermion-family mixing in the fermion-family space based on mass eigenstates of fermions. Given a definite electric charge, the U (1)-symmetry for each fermion flavor is still preserved and the quantum number of each fermion flavor is conserved. This completely prohibits any four-fermion-interacting process that violates fermion-flavor-number conservation, such as the FCNC process converting from one elementary fermion flavor ψ f to another ψ f ′ with the same electric charge.
However, such prohibition is relaxed as perturbatively turning on the interactions (6) between chiral-gauge-boson W ± and massive fermions in addition to the four-fermion interactions (10).
The reason is that these chiral-gauge interactions are not diagonal in the fermion-family space based on the mass eigenstates of fermions (11) and (12), attributed to the fact that the mass eigenstates of the SM elementary fermions are different from their chiral-gauge eigenstates. As a result, a 1PI-interacting vertex that violates the fermion-flavor-symmetry is induced via the mixing matrices of CKM type (14) and interactions among SM gauge bosons, e.g. W ± and photon, at one-loop level. Such 1PI vertex allows the FCNC processes of changing one elementary fermion ψ f flavor [mass eigenstate (8) or (9) Nevertheless, such W ± -boson contributions of SM-type to the FCNC processes are highly suppressed, since they come from the loop-level contributions of the SM W ± -boson via the fermionfamily mixing matrix of the CKM type (14) in its gauge coupling vertexes. In conclusion, apart from the SM-type suppressed contributions to FCNC, the effective Lagrangian (6) with four-fermion operators (10) in the IR-domain does not contain any additional unsuppressed 1PI vertexes that contribute to the FCNC processes.

B. Gauge-symmetric phase of strong four-fermion coupling
We turn to the brief recall on the gauge-symmetric phase of strong four-fermion coupling G, preserving SU L (2) gauge symmetry, where two-component Weyl fermion ψ L (ψ R ) is an SU L (2) doublet (singlet). The strong-coupling expansion was adopted to calculate two-point functions of composite boson and fermion fields [9], and the vertex functions of their couplings to the SU L (2)gauge bosons [10][11][12]. Detailed calculations can be found in these references and their appendixes.
In the lowest non-trivial order, we obtained the propagators of the massive composite bosons A = (ψ R ψ L ) and composite Dirac fermions: where the pole M B,D,S and residue Z S B,D,S respectively represent mass and form factor of composite particles. As long as their form-factors are finite, these composite particles behave as elementary particles. The propagators of renormalized composite particles (27) give their mass-shell conditions, The vertex functions of their SU L (2)-couplings were obtained by Ward identities from propagators (27), because the composite-particle spectra preserve the SU L (2)-gauge symmetry.
These massive composite particles formed by the such strong-coupling dynamics in the gaugesymmetric phase are completely different from the composite particles like massive Higgs boson and massless Goldstone bosons formed by the weak-coupling NJL dynamics in the symmetry-breaking phase where the SSB takes place.
C. Strong critical coupling and the UV-stable fixed point This indicates that there are two distinct phases: (i) the symmetry-breaking phase G c < G < G crit for the SM of elementray particles; (ii) the gauge-symmetric phase G > G crit for an effective theory of composite particles, and the second-order phase transition from one phase to another at the strong critical coupling G crit .

Strong critical coupling
In the symmetric phase, we indeed found [9] the existence of strong critical coupling G crit by using strong four-fermion coupling expansion to approximately calculate the two-point Green function (27)  In Refs. [13,30], we solve the full one-loop RG equations [17] for running couplingsḡ t (µ 2 ) and λ(µ 2 ) with the top-quark and Higgs-boson mass-shell conditions otherwise the effective theory would run into an instability (λ ∼ 0 − ) beyond E. These are certainly preliminary and qualitative results, since we use one-loop RG equations in low-energy and weak-coupling region and extrapolate their solutions to high-energy and strong-coupling region.
Nevertheless, the solution (30)  To close this subsection, it is worthwhile to mention that in Ref. [32] it is shown in the elementary Higgs-boson model that the quadratic term from high-order quantum corrections has a physical impact on the SSB and the phase transition to a symmetric phase occurs at the scale of order of TeV.

UV-stable fixed point
In order to show that this critical point G crit of the second-order phase transition can be a UV fixed point, we calculated the β-function in the symmetric phase of strong four-fermion coupling.
Up to the lowest non-trivial order of the strong coupling expansion, i.e., the two-fermion-loop contribution (sun-set diagrams), and we obtained the negative β-function [13] where G ≡ G × (Λ/π) 2 and the dimensionless Lorentz-scalar function Φ(p 2 /Λ 2 ) is positive and finite, monotonically decreases as the energy scale p 2 /Λ 2 increases. On the basis of the β(G)function being positive (22) and becoming negative (31), as sketched as "I", "II" and "III" in This is analogous to non-linear σ models [33], which contain a dimensionful coupling constant and are thus not perturbatively renormalizable. Nevertheless, the nonperturbative critical coupling of these σ models shows the second-order phase transition from the ordered (symmetry-breaking) phase to the disordered (symmetric) phase, and exhibits a non-trivial UV-fixed point and UVscaling domain of the renormalization group both in the lattice formulation [34] and in the 2 + ǫ formulation [35,36].
However, due to the lack of a non-perturbative method to effectively approach to the critical coupling G crit and its neighborhood, we have not been able to quantitatively determine the G crit value and properties of its scaling region i.e., the UV-domain, which will be qualitatively discussed and recalled in next section.
D. The UV-domain of UV-stable fixed point

Energy threshold of composite particles
As the running energy scale µ decreases in the vicinity of UV-stable fixed point G crit , the β(µ) function (see Fig. 2) shows that the RG flows take the effective theory of composite particles away from the UV fixed point towards the IR-domain of the IR-fixed point G c , where the low-energy SM of elementary particle physics is realized. This implies the existence of the energy threshold E thre , below which µ < E thre composite particle dissolves into its constituents of SM elementary particles.
As discussed in Sections V and VI of Ref. [11], when the energy scale µ decreases to the energy threshold E thre and G(µ) → G crit (E thre ), the phase transition occurs from the symmetric phase to the symmetry-breaking phase, all composite particles (poles) dissolve into their constituents, which are represented by three-fermion, fermion-boson and two-fermion cuts in the energy-momentum plane, as their form factor and binding energy vanish [37]. Actually, the energy threshold E thre represents the symmetry-breaking scale at the second-order phase transition G crit .  can be expanded as a series, for a/ξ ≪ 1, leading to the β-function The correlation length ξ follows the scaling law where the coefficient c 0 = (a 0 G crit ) ν and critical exponent ν need to be determined by nonperturbative numerical simulations. The physical scale E ξ ≡ ξ −1 in the UV-domain is attributed to the strong-coupling dynamics of forming composite particles. This implies the masses of composite particles (27) and (28) M and the running coupling G(µ)| µ→E thre +0 + → G crit , and the scale µ indicates the energy transfer between constituents inside composite particles.
On the basis of these discussions and observations, we advocate the following relation for (i) The values of these characteristic scales of UV-domain need some experimental knowledge in high energies, analogously to the electroweak scale (v) of IR-domain.

E. Relevant and irrelevant operators in IR-and UV-domains
In the weak-coupling IR-domain, as an energetically favorable solution for the SSB ground state [29], among four-fermion operators (6) the top-quark channel G(ψ ia L t Ra )(t b R ψ Lib ) [17] is the only physically relevant and renormalizable operators of effective dimension-4, due to the NJL-dynamics for the SSB. Namely, it becomes the effective SM Lagrangian (19) [7,13,14]. These are the properties of effective composite-particle theory in the UV-domain that we attempt to study in this article and in future.
To end this section, it is worthwhile to note that the repulsive four-fermion operators (4)  It is also worthwhile to mention that these discussions are reminiscent of the asymptotic safety

IV. COMPOSITE PARTICLES AND EFFECTIVE LAGRANGIAN IN UV-DOMAIN
In the gauge symmetric phase of an SU L (2)-chiral gauge theory containing the four-fermion operator (26), the composite particles and 1PI vertex functions, as well as their properties in the UV-domain were analyzed by using the approach of strong-coupling expansion to the dynamics of four-fermion operators at the cutoff Λ [9,[11][12][13][14]. In this section, however, based on the effective Einstein-Cartan Lagrangian (6) in the SM framework, see Sec. II D, we are going to show the composite-particle spectrum and 1PI interaction (effective Lagrangian) in terms of the quantum numbers of SM gauge symmetries. The purpose is that we shall go further to discuss composite particles decay and other processes into SM elementary gauge bosons and fermions, relevantly to possible high-energy experiments. These are the main results of this article presented in this section and following sections V, VI and VII.
A. Composite particles in UV-domain

Quark sector
Performing strong-coupling calculations similar to those detailedly presented in the Ref. [9] and their appendixes, we obtain the following results. For the u-quark channel, the massive composite boson is an SU L (2)-doublet and to form the massive composite four-component Dirac-fermion states: the SU L (2) doublet Ψ ib D and the SU L (2) singlet Ψ b S [57], The form factors [Z For the d-quark channel, the composite particles are represented by Eqs. (38)(39)(40) with the replace- (c La , s La ) or (t La , b La ) and singlet u Ra into t Ra or c Ra , as well as singlet d Ra into b Ra or s Ra .

Lepton sector
In the lepton sector, the composite boson and Weyl-fermion states formed by the first term (ℓ R -channel) of Eq. (16) are:

Quark-lepton sector
Analogously, we present for the d a R -and e R -channel of quark-lepton interactions (18), the massive composite Dirac fermions: SU L (2) doublet Ψ i D = (ℓ i L , Ψ i R ) and singlet Ψ S = (Ψ L , e R ), where the renormalized composite boson and composite Weyl-fermion states are: The composite particles from the second and third lepton families can be obtained by substitutions: e → µ, τ , ν e → ν µ , ν τ , and u → c, t and d → s, b. In general, the U Y (1)-hypercharge Y and U em -electric charge Q i of the composite particle is the sum of its constituents' hypercharges and electric charges, obeying the relation Q i = Y + t i 3L in units of e, where t i 3L is the diagonal third component of SU L (2)-isospin, t 1 3L = 1/2 for the neutrino and up quarks, and t 2 3L = −1/2 for the electron and down quarks.

Discussions and three-family replication
In the quark and lepton sectors, the massive composite boson A i , the massive composite Dirac fermions SU L (2) doublet Ψ i D and singlet Ψ S carry an electric charge Q = (2/3, −1/3, −1, 0) for u a R -and d a R -quark channels, e R -and ν R -lepton channels respectively. In the quark-lepton sector, the massive composite boson A i , the massive composite Dirac fermions SU L (2) doublet Ψ i D and singlet Ψ S carry an electric charge Q = (2/3, −1/3, −1, 0) for the d a R -e R channel and u a R -ν R channel respectively. It should be mentioned that the composite bosons Though composite particles are massive, they carry the quantum numbers of the SM chiral gauge symmetries, which are the sum of quantum numbers of their constituents of the SM elementary particles. The propagators of these composite particles have poles and residues that respectively represent their masses and form factors [9][10][11]. As long as their form factors are finite, these composite particles behave as elementary particles. It should be mentioned that the gauge symmetric masses M Π,F and form factors Z S Π,L,R of composite particles, e.g., (38)(39)(40)(41) can be different from one to another, due to some other effects that we do not study here.
To end this section, we present some discussions on the third-family composite bosons (38) for the both quark and lepton sector in one SM family, where i is the SU L (2)-isospin index, T ± is the weak isospin raising and lowering operators g i V = t i 3L − Q i sin 2 θ W and g i A = t i 3L . The weak angle θ W = tan −1 (g 1 /g 2 ) and the electron electric charge e = g 2 sin θ W . These massive fermions  gauge symmetries [58]. As a result, at tree-level we obtain the following effective Lagrangian for the first-family composite particles (38,40,42,43).
For the massive composite Dirac fermions Ψ i D and Ψ S (38-41) the u R -quark channel in the quark sector, the effective Lagrangian reads where i remains as the SU L (2)-isospin index. For the d R -quark channel, the composite particles made by u Ra → d Ra , the coupling g 1 V,A → g 2 V,A and (2/3)e → (−1/3)e. These massive composite fermions couple to the gauge bosons The Lagrangian (45) shows that the massive spectrum and interacting vertex are vector-like fully preserving the parity symmetry. Namely, each left-handed where and σ is the Pauli matrix in the isospin space. Instead, for the massive composite bosons SU L (2)doublet A i for the d R channel, i.e., the charged component A 1 ∼ (d Ra u a L ) and neutral component The last matrix-terms in Eqs. (47) and (48) respectively show that the isospin components A 1 ∼ u R u L and A 2 ∼d R d L have no electric charge, but the U Y (1) hypercharge coupling to Z 0 . The gauge couplings in the effective Lagrangian (45), (47) and (48) are consistent with SM gauge bosons couplings to the elementary fermions inside the composite particles, see Fig. 3.
The effective Lagrangian (45), (47) and (48) can be generalized to the lepton sector and leptonquark sector. Note that in the effective Lagrangian (45) and (46), we only present the spectra in the UV domain and functions of the SM gauge couplings g 1 , g 2 , g 3 at this mass scale M Π,F , rather than the electroweak scale v ≈ 239.5 GeV in the IR domain.
Since the propagators of massive composite particles are obtained from the lowest nontrivial contribution of the strong-coupling expansion [9], it cannot be precluded that there could be the interacting vertexes between massive composite particles, e.g., the Yukawa-type interactions of composite bosons and fermions, stemming from the high-order contributions of the strong-coupling expansion. In this article, we do not attempt to discuss the interacting vertexes among composite particles and their relevance in the UV-domain.

Some discussions on FCNC and anomalies in UV-domain
We attempt to have some discussions on the analogical FCNC processes of the UV effective Lagrangian (45) and (46) of composite particles in the UV-domain. The composite-fermion case is considered as an example for discussions that apply also to the composite-boson case. Recall that (i) composite fermions Ψ f are composed by the SM elementary fermions ψ f in the same family "f "; (ii) the fermion-family "diagonalized" effective Lagrangian (45,46) of massive composite particles Ψ f is obtained from the fermion-family "diagonalized" four-fermion operators (10)  considering the mixing anomaly [39].
As the running energy scale µ decreasing, and composite particle's form factor and binding energy vanishing, the composite particles become unstable and dissolve (decay) to their constitutes of SM elementary particles as final states. In the following, we discuss the decay and annihilation channels of the composite particles into the SM elementary particles, in particular two gauge bosons, as final states.

V. COMPOSITE PESUDO SCALAR BOSONS DECAY
This section V and next section VI discuss the decays and other relevant processes of composite bosons and fermions interacting with SM gauge bosons described by the effective Lagrangians (45) and (46)  To effectively study the non-perturbative nature of low-energy QCD theory, the global chiralsymmetric Lagrangian (σ-model) was adopted to describe the massless QCD-bound states of elementary u and d quarks: proton p(uud), neutron n(ddu) and pions π 0,± (uu,dd,ud). The spontaneous breaking of the SU L (2) × SU R (2) chiral symmetries leads to the massive doublet baryon fields (p, n) and massless triplet pion fields π k as three Goldstone bosons. The explicit breaking induced by the u and d quark masses leads to massive pion fields π k and the partial conservation of the axial current (PCAC) where the quark doublet ψ i = (u, d) and the axial current A j µ (x) =ψγ µ γ 5 (σ j /2)ψ. The latter couples to the pion fields π j (x) due to the spontaneous breaking of chiral symmetries, leading to the nontrivial matrix elements between the pion state |π k (p) and the vacuum The first matrix element defines the pion decay constant (form factor) f π , the second matrix element and the mass-shell condition p 2 = m 2 π defines the pion mass m π . The charged pions π ± = (π 1 ∓ iπ 2 )/2 =dγ 5 u,ūγ 5 d and neutral pion π 0 = π 3 /2 = (ūγ 5 u −dγ 5 d)/2. For the isospin j, k = 1, 2 components in Eq. (50), the first matrix element determines the rate of the decay π + (ūd) → µ + + ν µ and experimental value f π ≈ 93 MeV. In addition, for the isospin j = k = 3 component in Eq. (50) reads receiving an axial anomaly in terms of the gauge field (photon) strength F and fine-structure constant α, contributed from the triangle diagram shown in Fig. 4. This axial anomaly dominates the π 0 -decay rate, in excellent agreement with the experimental value.

B. Scalar and pesudo scalar composite bosons
We attempt to study the decays of composite bosons A i in the UV-domain, discussed in Secs. IV A and IV B. For the sake of simplicity, we adopt the first quark family ψ i L = (u, d) L and ψ R = u R , d R for illustrations. For the u-channel, the massive composite boson A i = (ū R ψ i L ) (38), which is an SU L (2) doublet and U Y (1) charged Y = −1/2, can expressed in terms of scalar and pseudo scalar fields, where the form factor [Z S Π ] −1/2 is omitted for simplifying notations. There are four components where the isospin index i = 1, 2 are relabeled as i ′ = 0, ± indicating the neutral or charged component. For the d-channel, by the substitution u → d in Eqs. (54) and (55), we have The neutral state Π 0 has the contributions from Eqs. (55) and (56)  The gauge interactions at the leading order (tree-level) of gauge couplings (47) and (48)  Instead the axial current associating to the global chiral symmetries of the effective low-energy Lagrangian of the QCD-bound states are partially conserved because of spontaneous and explicit chiral-symmetry-breaking (PCAC). However, the massive pseudo scalar fields Π 0,± (55,56) and the QCD pion fields π 0,± have the same quantum numbers of the SM symmetries. Moreover, we shall show the pseudo-scalar fields Π 0,± decay into SM elementray particles, in particular the Π 0 -decay into two SM gauge bosons, analogously to the decay π 0 → γγ, attributing to the axial anomaly (53).

C.
Composite particle Π 0 decay into two SM gauge bosons

Π 0 decay into two photons
Suppose the vacuum state is |0 in the UV domain, the composite-boson momentum state |A i (q) on the mass shell can be defined by the field operator A i (q) or A i (x) as follow with the normalization or equivalently where (38), as discussed in Sec. IV. As will be seen soon, F Π is in fact the composite-boson decay constant. From Eq. (59), the matrix element The effective Lagrangian (46) gives the equation of motion and the mass-shell condition q 2 = M 2 Π of the composite boson A i (x). From Eq. (60), we have The matrix element (59) relates to the composite-boson propagator, The matrix element of composite particle Π 0 in Eqs. (55) and (56) reads, where the g Π 0 = g Π 0 (q 2 ) is the Π 0 -coupling to its constituent fermions f = u, d quarks of mass m f and wave function u f . Actually, the coupling g Π 0 (q 2 ) is the 1PI vertex function of composite particle Π 0 and its constituent fermions and we parametrize it by the form factor F −1 Π (q 2 ) and constituent mass m f and g Π 0 is finite as m f → 0, see for example [40]. These definitions and equations apply also for scalar composite bosons S i and pseudo scalar composite bosons Π i , see Eqs. (55) and (56).
We are in the position of discussing the rate of the neutral Π 0 decay into two SM gauge bosons, i.e., diboson channels: where G and G ′ represent the SM gauge bosons W ± , Z 0 and γ. The amplitude of the decay (65) is defined by the matrix element and where k 1,2 and ǫ 1,2 are four momenta and polarizations of the SM gauge bosons G and G ′ , and J µ (z) [J ′ ν (y)] is fermion current coupling g [g ′ ] to the SM gauge boson G [G ′ ]. The amplitude T µν (k 1 , k 2 , q) has the symmetry (k 1 , µ) ↔ (k 2 , ν).
The lowest one-loop contribution to the vertex function (67) is represented by the triangle Feynman diagram, see Fig. 4. Considering Eqs. (63) and (64), the Π 0 -decay amplitude T µν (67) computed at this one-loop level is given by where the coupling vertexes gΓ µ and g ′ Γ ′ ν to the SM gauge bosons G and G ′ are given by the SM Lagrangian (45), and the trace "tr" is over the spinor space. Equation (69) is not well-defined because of the linear divergence of the momentum integral. Introducing the Pauli-Villars mass M that plays the role of the UV cutoff Λ, we adopt the Pauli-Villars regularization, to make the momentum integral to be finite and well-defined. Note that the Pauli-Villas regularization M explicitly breaks chiral gauge symmetries of effective Lagrangian.
Suppose that the pseudo scalar composite field Π 0 (x) is heavy and its mass M Π 0 is much larger than the W ± and Z 0 masses, the intermediate gauge bosons W ± and Z 0 are considered to be approximately massless, k 2 1 ≈ 0, k 2 2 ≈ 0 and 2k 1 · k 2 ≈ q 2 . In addition, the coupling-vertexes gΓ µ and g ′ Γ ′ ν in Eq. (69) are given in the SM Lagrangian (44), they contain both axial and vectorlike vertexes. We consider the vector-like vertexes so that the triangle Feynman diagram (Fig. 4) and the amplitude (67) has the AVV structure, whose nontrivial axial-anomaly amplitude can be compared with the amplitude (73) of the channel Π 0 → γ + γ. As a result, for the process Π 0 → γ + Z 0 , the couplings g = eQ i and g ′ = g 2 g i V /(2 cos θ W ), we obtain from Eq. (72) the amplitude, For the process Π 0 → W + + W − , the couplings g = g ′ = g 2 /(2 √ 2) and we obtain from Eq. (72) the amplitude, where the sum over isospin "i" gives one in this case, see Eq. (44). For the process Π 0 → Z 0 + Z 0 , the couplings g = g ′ = g 2 g i V /(2 cos θ W ) and we obtain from Eq. (72) the amplitude, Their differences from the two-photon amplitude (73) are only attributed to the different eletroweak couplings of the SM (44).

D. Ratios of different channels of Π 0 decay into two SM gauge bosons
The numerical rate of the neutral Π 0 (q) decay (65) at the mass shell (q 2 = M 2 Π 0 ) is given by integrating the squared amplitude (66) over the phase space of two gauge bosons, The diphoton channel (80)  For the reasons that the properties of composite particles, e.g., form-factors and masses (or the binding-energy depth) due to the strong-coupling dynamics, are unknown, we have not been able to calculate the total rate Γ total Π 0 and width of the composite boson Π 0 that decays into the final states of the SM elementary particles including the channel of two gauge bosons or two fermions [59]. Nevertheless, at the leading order (tree-level) of gauge interactions, we are able to calculate the rates Γ Π 0 →γγ,γZ,··· (78) of the composite boson Π 0 decaying into two gauge bosons as diboson final states, These diboson channels (84) are expected to be the most energetically favorable and largest branching ratio, Moreover, in high-energy experiments, the diboson final state and its kinematics might be more easily identified than the final state of two fermions (quarks or two jets) due to the background of the QCD dynamics.
The total decay rate and width of the composite boson, as well as the branching ratios of decay channels, are very important for the collider phenomenology of the composite boson. However, the following ratios of the branching ratios (85) for different decay channels (84), "relative branching ratio" depend only on the SM gauge couplings g and g ′ at the energy scale M Π , given by the effective Lagrangian (45). Following these discussions and using Eqs. (80-83), we can approximately estimate the decay-rate ratios (86) where the value sin 2 θ W ≈ 0.23 is approximately adopted to obtain numbers. These relations It is also possible that the composite boson Π 0 decays into the dijet final state of two quarks which was already discussed in Ref. [13,14]. Suppose that the neutral composite meson Π 0 is produced by pp collisions at the LHC, its resonance location M all channels = M Π 0 , and the rate of each decay channel depends on the values of the mass M Π 0 , decay constant F Π and SM gauge couplings at these scales. Which channel is more relevant for detections, depending not only on its theoretical branching ratio in principle, but also on its experimental measurement in practice.
The same analysis and discussion can be generalized to the decay rates of neutral composite bosons Π 0 made of charged lepton and/or neutrino pair (42) in the lepton channel. Their decay constants and masses should be approximately at the same scale of the Π 0 decay constant F Π and mass M Π 0 . Their decay rates can be obtained by Eqs. (73-79) for N c = 1, couplings Q i and g i V of the lepton sector in the SM Lagrangian (44). As an example for the two-photon final state, the rate should be (5/9) 2 N 2 c = 25/9 time smaller than the rate (80) of the quark channel. However, the ratios of branching ratios for different diboson channels are similar to Eq. (87), but modified accordingly to the leptonic Q i and g i V values. Beside, in the LHC pp collision, the production probability of leptonic composite bosons is smaller than hadronic one, due to the leptonic production rate is proportional to the small fine-structure constant α.

E.
Π ± and other composite boson decays Equations (50) and (51) of the PCAC are not applicable for the charged composite bosons Π ± decay. Therefore, we cannot use the analogy of the QCD charged pion π ± decays to obtain the rates of Π ± decay into either two quarks (jets) or two leptons, and charge-conjugated processes. In the quark case, the composite bosons Π ± are made of quarks (38), and in the lepton case the composite bosons Π ± are made of leptons (42). Nevertheless, it is expected that the Π ± -decay channel to the heaviest fermions, top quark or tau lepton, should be most favorable. Suppose that the charged composite meson Π ± is produced by pp collisions at the LHC, its resonance locates at the invariant mass M qq ≈ M Π ± of the dijet final state or other possible final states produced by two quarks (two jets). Whereas the charged lepton channel implies the final state of τ ± lepton of energy ∼ M Π ± /2 and missing energy carried away by ν τ -neutrino.
In Eqs. (55) and (56), the scalar bosons form composite quarkonium statesūu anddd that carry the same quantum numbers of QCD quarknium states, however, have much larger masses ∼ M Π 0 .
It is expected that apart from different kinematic threshold, these composite quarkonium states in principle undergo all decay channels of QCD quarkonium states. Recall that the direct decays of composite bosons Π ± and quarkonium states into dijets or dilepton without an intermediate state were preliminarily discussed [13,14]. On the basis of the SM chiral gauge symmetries and vectorlike composite fermion content, there are also other possible final decay channels, analogously to those studied in the effective theory of QCD at low energies. These will be issues in future studies.

VI. COMPOSITE FERMION DECAY AND ANNIHILATION CHANNELS
In this section VI, we turn to discussions of decay and annihilation channels of composite This implies the ratios: are similar to the ratios (87). If the bosonic resonance (84) is observed and its invariant mass is determined, the resonance of composite fermion decay (89) could possibly be identified by measuring such a peculiar final-state kinematics (90) of a single jet and two photons or other diboson states G and G ′ .

Decay into two gauge bosons and a lepton
Due to the four-fermion interaction of quark-lepton sector, see Eq. (18) and the ν e -lepton channel is given by e → ν. This indicates that a composite Dirac fermion Ψ D (43) decays into a composite meson Π 0 and a fundamental Dirac fermion, a charged lepton ℓ = [l L , ℓ R ] or a neutrino ν ℓ . The latter is an ultra-relativistic lepton. The kinematic threshold and distribution are the same as the one (89) in pure hadronic channel (90). However, the neutrino in the final states carries away mixing energy and momentum. The rate Γ Ψ D →GG ′ +a lepton of composite fermion decay (93) should be the product of Π 0 -decay rate Γ Π Π 0 →GG ′ (80-83) and the rate Γ Ψ D →Π 0 +a lepton of Ψ D decaying into a lepton and Π 0 , namely This implies the ratios: and its charge conjugate. The Weyl fields [ū L , d R ] do not form a Dirac fermion of u-or d-quark, but pick up u R andd L quarks from the vacuum, and end as u and d-quark (jets) in final states. On the other hand, the charged composite mesons Π ± , see Eqs. (55) and (56), most probably decay into two quarks (88). Therefore, the most probable final state of composite fermion (96) decay is expected to be four jets formed by four quarks, which was already discussed in Refs. [13,14].
These discussions can be generalized to the decay channels whose intermediate state is a composite quarkonium state S 0± , instead of a composite meson state Π 0± , see Eqs. (55) and (56).
Also, these discussion can be generalized to the second and third families of composite fermions.
To end this Section, we mention other possible channels of composite fermion decay.
Apart from the diphoton final state, the final states of energetic dibosons W + W − and Z 0 Z 0 are two "fat" jets in opposite directions, each of them is made by two energetic quarks, or these two bosons can decay into the final states of two lepton pairs. While in the channel Ψ D + Ψ c D → γZ 0 , the final states are an energetic photon and a "fat" jet or an energetic photon and a lepton pair.
Given by the mass-energy of two annihilating composite Dirac fermions, the kinematic massenergy of final states must be larger than 2M F . This is not an invariant energy-mass representing the resonance of an unstable composite particle. Suppose that massive composite Dirac fermions produced by the LHC pp-collision at present energies are non-relativistic particles, where the number density of the produced composite fermions n ∼ M 3 F /π is assumed. Analogously to the positronium state of electron and positron pair in the QED, it is possible that the composite fermion and its antiparticle form an intermediate unstable Coulomb bound state, then decaying into photons or other massive gauge bosons. The spin singlet of the bound state decays into two photons (even number of photons) with the probability ∼ (Q 2 α) 5 M F , the spin triplet of the bound state decays into three photons (odd number of photons) with the probability ∼ (Q 2 α) 6 M F . In these discussions, we adopt two-photon final state. As for other channels of two SM gauge bosons G and G ′ (98), in terms of their SM gauge couplings in the effective Lagrangian (45), we can obtain the ratios of different annihilation processes: depending only on the SM gauge couplings g and g ′ (45) at the mass scale M F , similarly to the ratios (87).
The following annihilation channels of two composite fermions with different charges or zero charge are also possible, and their charged conjugates. In addition, it is known that the annihilation of electron and positron, through an intermediate γ-photon or Z 0 -boson, produce a pair of particle and antiparticle in the SM. Analogously, a composite fermion and its antiparticle annihilates, through an intermediate photon or Z 0 -boson, and produce a pair of SM elementary particle and its antiparticle, where the final state is an energetic lepton pair or quark pair, the former is the dilepton channel and the latter is the dijet channel. Similarly to the channels (103) (45) and (46) Obviously, in addition to those processes discussed in this article, there are other possible experimentally relevant processes of effective four-fermion operators (6), we end this lengthy article by making some speculative considerations for experiments.

A. Some speculative considerations for LHC experiments
In the LHC pp collisions, the most probably channels of producing composite particles are the composite bosons and fermions (38)(39)(40)(41) in the first family via the four-fermion operators [13,14] G Suppose that the recent ATLAS and CMS preliminary results [43,44] of diboson resonances (dijets tagged by two bosons) with invariant masses in the energy range from 1.3 to 3.0 TeV could be further confirmed. These resonances are expected to be also seen in the channels of four quark jets, whose invariant mass and event rate should be larger, provided that these resonances are attributed to massive composite Dirac fermions at this energy range. The CMS result [45] of resonances with final states being two jets could include the event of four quark jets, two of them are geometrically close together to form a "wide jet", which should be tagged through a study of its substructure and flavor. Moreover, if composite Dirac fermions are formed by the last operator in Eq. (105), in addition to jets in final state, dark-matter particles ν R carry away missing energy-momentum [46]. Similar discussions are applied for the case of composite bosons.
Due to the W ± -and Z 0 -boson couplings g 2 to two constituent quarks (u, d) of composite fermions, in particular W ± -boson coupling to SU L (2) doublet ψ ia L = (u a L , d a L ), massive composite Dirac fermions have the following decay channels of final states: (i) dijets tagged by two highly boosted bosons WW, WZ or ZZ produced by high-energy constituent quarks (u, d) of composite fermions, together with additional quark jets; (ii) four quark jets formed by four high-energy constituent quarks (u, d) of a composite fermion with a peculiar kinematic distribution [13,14]. It is expected that the former should have smaller rate because of the SM gauge coupling g 2 , although we have not yet been able to calculate the rates of these channels. In these two aforementioned channels (i) and (ii), the final states can also be high-energy leptons, however the branching ratio of W ± and Z 0 decaying into leptons is about several times smaller than that to hadrons (jets) [41].
The composite fermion can also decay in the channel of W and Higgs (WH) bosons [42], where the Higgs boson is produced by u, d-quarks fusing into a top-quark pair via a gluon, and its production rate is then related to the QCD coupling α s = g 2 3 /4π. Similar discussions are applied for the case of composite bosons.
B. Sterile neutrinos interacting with SM particles at high energies Last but not least, all sterile neutrinos (ν i R , ν ic R ) and SM gauge-singlet (neutral) states of massive composite fermions, e.g., Ψ D ∼ [ν ℓ R , (l i L ν ℓ R )ℓ Li ], can be possible candidates of warm and cold dark matter [12,14]. They can couple or decay into the SM elementary particles in the following ways.  16) and (17) give the interactions G (l i L ν ℓ R )(ν ℓ R ℓ Li ) + (ν ℓ c R ℓ R )(l R ν ℓ c R ) + (ν ℓ c R u ℓ a,R )(ū ℓ a,R ν ℓc R ) + (ν ℓ c R d ℓ a,R )(d ℓ a,R ν ℓc R ) , among sterile neutrinos ν ℓ R , ν ℓc R (dark matter) and SM elementary particles, where the lepton SU L (2) doublets ℓ i L = (ν ℓ L , ℓ L ), singlets ℓ R and the conjugate fields of sterile neutrinos ν ℓc R = iγ 2 (ν ℓ R ) * (ℓ = e, µ, τ ), and quark fields u ℓ a,R = (u, c, t) a,R and d ℓ a,R = (d, s, b) a,R . The four-fermion coupling G in Eqs. (105) and (106) is unique. Therefore, it is expected that at the same energy scale M F > M Π ∝ E ξ (49), at which composite boson and fermion (38)(39)(40)(41) appear as resonances in the LHC pp collisions, leptonic composite boson (ē R ν ec R ) or (ν e R e L ) and composite fermion [ν ec R , (ē R ν ec R )e R ] or [ē L , (ν e R e L )ν e R ] should be formed by high-energy sterile neutrino inelastic collisions, e.g. ν e R +ν e R → e − +e + via the first or second interaction in Eq. (106). Then these leptonic composite particles decay and produce electrons and positrons. This may account for an excess of cosmic ray electrons and positrons around TeV scale [47,48] in space laboratories. In addition, recent AMS-02 results [49] show that at TeV scale the energy-dependent proton flux changes its power-law index. This implies that there would be "excess" TeV protons whose origin could be also explained by the resonance of composite bosons and fermions due to the interactions (105) and (106) of dark-matter and normal-matter particles.
We also expect that at the same energy scale M F > M Π ∝ E ξ (49), the four-fermion interactions of the last term in Eq. (105), the third and fourth terms in Eq. (106)

C. Further studies
By contrast with the low-energy effective theory of SM elementary particles in the IR-domain of weak four-fermion coupling, we have to confess that the present analysis of the effective theory of composite particles in the UV-domain of strong four-fermion coupling are completely preliminary, due to its non-perturbative nature from the theoretical point view. Adequate non-perturbative methods both analytical approaches and numerical algorithms are necessary. This is analogous to the theoretical aspects of low-energy hadron physics of non-perturbative QCD. On the other hand, from experimental point of view, it is nontrivial to proceed physically sensible final-state selection and relevant data analysis in high-energy experiments, in addition to searching for high-energy collisions and accumulating enough data for significant analysis. However, without high-energy experiments, the characteristic energy scale M F > M Π ∝ E ξ (49) of effective composite-particle theory cannot even be determined at all. It is worthwhile to mention that similar to the analogy between the Higgs mechanism and Bardeen-Cooper-Schrieffer (BCS) superconductivity, we are studying an analogy between the effective theory discussed in this article and the BCS-BEC (Bose-Einstein condensate) crossover and unitary Fermi gas of strong-interacting electrons [52], which is expected to observe in optical lattice.