Resonant and nonresonant new phenomena of four-fermion operators for experimental searches

In the fermion content and gauge symmetry of the standard model (SM), we study the four-fermion operators in the torsion-free Einstein-Cartan theory. The collider signatures of irrelevant operators are suppressed by the high-energy cutoff (torsion-field mass) $\Lambda$, and cannot be experimentally accessible at TeV scales. Whereas the dynamics of relevant operators accounts for (i) the SM symmetry-breaking in the domain of infrared-stable fixed point with the energy scale $v\approx 239.5$ GeV and (ii) composite Dirac particles restoring the SM symmetry in the domain of ultraviolet-stable fixed point with the energy scale ${\mathcal E}\gtrsim 5$ TeV. To search for the resonant phenomena of composite Dirac particles with peculiar kinematic distributions in final states, we discuss possible high-energy processes: multi-jets and dilepton Drell-Yan process in LHC $p\,p$ collisions, the resonant cross-section in $e^-e^+$ collisions annihilating to hadrons and deep inelastic lepton-hadron $e^-\,p$ scatterings. To search for the nonresonant phenomena due to the form-factor of Higgs boson, we calculate the variation of Higgs-boson production and decay rate with the CM energy in LHC. We also present the discussions on four-fermion operators in the lepton sector and the mass-squared differences for neutrino oscillations in short baseline experiments.

In order to accommodate high-dimensional operators of fermion fields in the SM-framework of a well-defined quantum field theory at the high-energy scale Λ, it is essential and necessary to study: (i) what physics beyond the SM at the scale Λ explains the origin of these operators; (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) or ultraviolet (UV) stable fixed point of physical couplings locates; (iv) in the domains (scaling regions) of these stable fixed points, which operators become physically relevant and renormalizable following renormalization group (RG) equations, and other irrelevant operators are suppressed by the cutoff at least O(Λ −2 ).
The strong technicolor dynamics of extended gauge theories at the TeV scale was invoked [5,6] to have a natural scheme incorporating the relevant four-fermion operator G(ψ ia L t Ra )(t b R ψ Lib ) of the t t -condensate model [7]. 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 [7][8][9] suggested that the symmetry breakdown of the SM 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 [10]. The low-energy SM physics was supposed to be achieved by the RG equations in the domain of the IR-stable fixed point with v ≈ 239.5 GeV [6,7,9]. In fact, the t t -condensate model was shown [11] to be energetically favorable, the top-quark and composite Higgs-boson masses are correctly obtained by solving RG equations in this IR-domain with the appropriate non-vanishing form-factor of Higgs boson in TeV scales [12,13].
Inspired by the non-vanishing form-factor of Higgs boson, the formation of composite fermions and restoration of the SM gauge symmetry in strong four-fermion coupling G [14], we preliminarily calculated the β(G)-function and showed [13] the domain of an UV-stable fixed point at TeV scales, where the particle spectrum is completely different from the SM. This is reminiscent of the asymptotic safety [15] that quantum field theories regularized at UV cutoff Λ might have a non-trivial UV-stable fixed point, RG flows are attracted into the UV-stable fixed point with a finite number of physically renormalizable operators. The weak and strong four-fermion coupling G brings us into two distinct domains. This lets us recall the QCD dynamics: asymptotically free quark states in the domain of an UV-stable fixed point and bound hadron states in the domain of a possible IR-stable fixed point.
In this Letter, we proceed a further study on this issue, distinguishing physically relevant fourfermion operators from irrelevant one in the both domains of IR-and UV-stable fixed points, and focusing on the discussion of relevant operators and their resonant and nonresonant new phenomena for experimental searches.
Four-fermion operators from quantum gravity. A well-defined quantum field theory for the SM Lagrangian requires a natural regularization (cutoff Λ) fully preserving the SM chiralgauge symmetry. The quantum gravity provides a such regularization of discrete space-time with the minimal lengthã ≈ 1.2 a pl [17], where the Planck length a pl ∼ 10 −33 cm and scale Λ pl = π/a pl ∼ 10 19 GeV. However, the no-go theorem [16] tells us that there is no any consistent way to regularize the SM bilinear fermion Lagrangian to exactly preserve the SM chiral-gauge symmetry.
This implies that the natural quantum-gravity regularization for the SM leads us to consider at least four-fermion operators.
It is known that four-fermion operators of the classical and torsion-free Einstein-Cartan (EC) theory are naturally obtained by integrating over "static" torsion fields at the Planck length, 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 basic space-time cutoffã. In the regularized and quantized EC theory [17] with a basic space-time cutoff, in addition to the leading term J d J d in Eq. (1) there are high-dimensional fermion operators (d > 6), e.g., ∂ σ J µ ∂ σ J µ , which are suppressed at least by O(ã 4 ).
We consider massless left-and right-handed Dirac fermions ψ L and ψ R carrying the SM quantum numbers, as well as right-handed Dirac sterile neutrinos ν R and their Majorana counterparts ν c R = iγ 2 (ν R ) * . Analogously to the EC theory (1), we obtain a torsion-free, diffeomorphism and local gauge-invariant Lagrangian where we omit the gauge interactions in D µ and fermion flavor indexes of axial currents J µ L,R ≡ ψ L,R γ µ γ 5 ψ L,R and j µ L ≡ν c R γ µ γ 5 ν c R . The four-fermion coupling G is unique for all four-fermion operators and high-dimensional fermion operators (d > 6) are neglected. If torsion fields that couple to fermion fields are not exactly static, propagating a short distancel > ∼ã , characterized by their large masses Λ ∝l −1 , this implies the four-fermion coupling G ∝ Λ −2 .
In this article, we only discuss the relevance of dimension-6 four-fermion operators (2), which can be written as by using the Fierz theorem [19]. Equations (3) and (4) represent repulsive and attractive operators respectively. It will be pointed out below that four-fermion operators (3) cannot be relevant and renormalizable operators of effective dimension-4 in both domains of IR and UV-stable fixed points.
We will consider only four-fermion operators (4) preserving the SM gauge symmetry without the flavor-mixing of three fermion families.
SM gauge symmetric four-fermion operators. In the quark sector, the four-fermion opera- 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. (5) are respectively the four-fermion operators of top-quark channel [7] 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.
In the lepton sector, we introduce three right-handed sterile neutrinos ν ℓ R (ℓ = e, µ, τ ) that do not carry any SM quantum number. Analogously to Eq. (5), 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. (4), the last term in Eq. (6) 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, 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 .
In addition, there are SM gauge-symmetric four-fermion operators that contain quark-lepton interactions [20], where ℓ i L = (ν e L , e L ) and ψ Lia = (u La , d La ) for the first family. The "terms" represent 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. Here we do not consider baryon-number violating operators. It would be interesting to study four-fermion operators in the framework of the SU (5) or SO(10) unification theory.  (5) undergoes the NJL-dynamics of spontaneous symmetry breaking [7]. As a result, the Λ 2 -divergence The Dirac part of lepton-sector (6) and (7) does not undergo the NJL-dynamics of spontaneous symmetry breaking because its effective four-fermion coupling (G c N c )/N c is smaller than the critical value (G c N c ) [11]. Therefore, except the top-quark channel (9), all Dirac fermions are massless and four-fermion operators (4) are irrelevant dimension-6 operators, whose tree-level amplitudes of four-fermion scatterings are suppressed O(Λ −2 ) in the IR-domain. The hierarchy of fermion masses and Yukawa-couplings is actually due to the explicit symmetry breaking and flavor-mixing, which were preliminarily studied [20,21] by using the Schwinger-Dyson equation for Dirac fermion self-energy functions and we will present some detailed discussions on this issue in the coming paper [22].
On the other hand, in the domain (UV-domain) of UV-stable fixed point G crit > G c N c , the phase transition takes place from the symmetry-breaking phase to the symmetric phase, the fourfermion operators (4) undergo the dynamics of forming composite fermions, e.g.
preserving all SM gauge symmetries, and the characteristic energy scale is E > ∼ 5 TeV [13]. In with the dynamics of forming composite fermions, due to the absence of strong coupling limit that is a necessary condition to form three-fermion bound states [14]. As a consequence, the tree-level amplitudes of four-fermion scatterings represented by these irrelevant operators (3) are suppressed In fact, using dilepton events the ALTAS collaboration [23] has carried out a searched for nonresonant new phenomena originating from contact interactions [18] (the left-left isoscalar model J µ L J L,µ , which is commonly used as a benchmark for this type of contact interaction searches [24]), and shown no significant deviations from the SM expectation up to Λ = π/ã > 10 TeV .
Moreover, it should be mentioned that in the IR domain, even taking into account the loop-level of fermion fields [14] and these loop-level contributions to the β(G)-function are negative [13].
Nonresonant new phenomena of four-fermion operators. In the IR-domain with the electroweak breaking scale v = 239.5 GeV, the dynamical symmetry breaking of four-fermion operator (5) accounts for the masses of top quark, W and Z bosons as well as a Higgs boson composed by a top-quark pair (tt) [7]. It is shown [12,13] that this mechanism consistently gives rise to the top-quark and Higgs masses of the SM (9) at low energies, provided the appropriate value of non-vanishing form-factor of composite Higgs boson at the high-energy scale E > ∼ 5 TeV.
The finite form-factor of composite Higgs boson is actually the wave-function renormalization of composite Higgs-boson field, which behaves as an elementary particle after performing the wavefunction renormalization. However, the non-vanishing form-factor of composite Higgs boson is in fact related to the effective Yukawa-coupling of Higgs boson and top quark, i.e.,Z of Eq. (9). The effective Yukawa couplingḡ t (µ) and quartic couplingλ(µ) monotonically decrease with the energy scale µ increasing in the range m H < µ < E ≈ 5 TeV (see Fig. 1). This means that the composite Higgs boson becomes more tightly bound as the the energy scale µ increases.
This should have 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 couplinḡ g t (µ). Then, the produced Higgs boson decays to the two-photon state by coupling to a topquark loop and to the four-lepton state by coupling to two massive W -bosons or two massive Zbosons. Due to thet t-composite nature of Higgs boson, the one-particle-irreducible (1PI) vertexes of Higgs-boson coupling to a top-quark loop, two massive W -bosons or two massive Z-bosons are proportional to the effective Yukawa couplingḡ t (µ). As a result, both the Higgs-boson decaying rate to each of these three channels and total decay rate are proportional toḡ 2 t (µ), which does not affect on the branching ratio of each Higgs-decay channel. The energy scale µ is actually the Higgsboson energy, representing the total energy of final states, e.g., two-photon state and four-lepton states, to 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 this scenario of composite Higgs boson, as the Higgs-boson energy µ increases to µ = 2m t , the Yukawa couplingḡ 2 t (2m t ) ≈ 0.98 (see Fig. 1), 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% or 11%, at µ = 3m t ,ḡ 2 t (3m t ) ≈ 0.95 or µ = 4m t ,ḡ 2 t (4m t ) ≈ 0.93 (see Fig. 1). This variation may be still too small to be clearly distinguished by the present LHC  identified in high-energy processes of LHC p p collisions (e.g., the Drell-Yan dilepton process, see Ref. [23]), e − e + annihilation to hadrons and deep inelastic lepton-hadron e − p scatterings at TeV scales.
Composite particles in the UV-domain. We turn to discuss the composite particle spectra where the renormalized composite three-fermion states are: The same discussions also apply for the first and second quark families by substituting the SU L (2) doublet (t La , b La ) into (u La , d La ) or (c La , s La ) and singlet t Ra into u Ra or c Ra , as well as singlet b Ra into d Ra or s Ra . Some detailed discussions can be found in Ref. [13].
Analogously, we present in this letter for the e R -channel of quark-lepton interactions (8) where the renormalized composite three-fermion states are: and the composite bosons . For the ν e R -channel, the composite particles are represented by Eq. (11) with the replacements e R → ν e R and d a R → u a R , In addition, the composite three-fermion states formed by the first term (ℓ R -channel) of Eq. (6) are: where the characteristic energy scale E ξ sets in the UV-domain When the energy scale µ decreases to the energy threshold E thre and the four-fermion coupling G(µ) and X(p n ) is expected to be affected by the missing energy-momenta carried away by neutrinos.
In addition, the polarized electron-deuteron deep inelastic (DIS) experiment [24,25] measured the right-left asymmetry, where σ R,L (q) is the cross-section for the deep-inelastic scattering of a right-or left-handed electron e R,L + N → e + X. For low energy-momentum transfer q 2 ≪ E, A = 0 for the violation of parity symmetry of the SM. As high energy-momentum transfer q approaches E and q > E, A → 0 for the restoration of parity symmetry [12]. TeV. However, this energy scale does not seems to be reached by both e + e − colliders and DIS experiments in near future.
In addition, two high-energy photons from the LHC pp collision can produce two electronpositron pairs fusing into a composite Dirac particle (13,14) with ℓ i L = e L and ℓ R = e R , which can be identified by observing the resonance in the invariant mass M e + e − of final states of two electronpositron pairs. In the CM frame, electron and positron of each pair move apart in the opposite direction and the energy-momentum of each particle is about one-fourth of the invariant mass. The cross-section for these channels can be estimated to be 4πα 2 /M 2 . The massive composite particle (13,14) that comprises electron ℓ i L = e L and sterile neutrino ℓ R = ν R is expected to be identified by observing the resonance with final states of electron and positron oppositely moving apart with energy being one-half of the invariant mass M, and the rest carried away by sterile neutrino and anti-neutrino oppositely moving apart. Compared with the multi-jets resonant phenomena due to four-quark operators (10) in LHC p p collisions [13], the probabilities of the dilepton resonant phenomena due to four-lepton operators (13) in p p collisions are smaller, because of the factor α 2 .
In general, what can be said are following. If the accessible CM energy √ s > M, the cross section for the allowed inelastic processes forming massive composite Dirac particles will be geometrical in magnitude, of order σ com ∼ 4π/M 2 in the CM frame where massive composite Dirac particles are approximately at rest. Decays of these massive composite particles to their constituents leading to unconventional events of multi-jets, jets-dilepton and multi-leptons states with peculiar kinematic distributions in the CM frame. As a result, these unconventional events will qualitatively depart from the SM, and completely dominate over the SM processes, for which cross sections go roughly as πα 2 gauge /( √ s) 2 . Thus the SM background is expected to be more or less zero. On the other hand, if the accessible CM energy √ s < M, then departures from the SM will be quantitative rather than qualitative, as described in the previous section.
In currently scheduled LHC (p p-collision) runs for next 20 years, the integrated luminosity will go from 10 fb −1 up to 10 3 fb −1 and the CM energy Neutrino sector. In the UV-domain, the four-neutrino operator (ν ℓc R -channel) of the last term of Eq. (6) forms composite three-fermion states (self-conjugate Majorana states) . These composite particles comprising only neutrinos are hard to be produced in ground laboratories, except in the early universe.
However, in the IR-domain for the SM, the last term of Eq. (6) can generate a mass term of Majorana type, because the family number N i = 3 (i = 1, 2, 3 that are different from the SU L (2) index) plays the same role as the color number N c of the top-quark in the t t -condensate. In the same way similar to the gauge and mass eigenstates of neutrinos ν ℓ L = U ℓ i ν i L , we use the 3 × 3 unitary PMNS matrix U for neutrino mixings to define the flavor eigenstates of sterile neutrinos by ν ℓ R = U ℓ i ν i R , (ℓ = e, µ, τ, i = 1, 2, 3).
Analogously to the generation of top-quark mass m t , the dynamical symmetry breaking of the U lepton (1)-symmetry generates the Majorana mass of right-handed neutrinos together with a sterile massless Goldstone boson, i.e. the bound state (ν i c R γ 5 ν i R ), and a sterile massive scalar particle, i.e. the bound state (ν i c R ν i R ), carrying two units of the lepton number of the family "i". Note that the family index "i" is summed over as the color index "a" in the t tcondensate. In the IR-domain, the sterile neutrino mass m M and sterile scalar particle mass m M H satisfy the mass-shell conditions, whereḡ t (µ 2 ) andλ(µ 2 ) obey the same RG equations (absence of gauge interactions) and boundary conditions of Eqs. (7,8,9) and (10) in Ref. [13]. However, unlike the electroweak scale v determined by the gauge-boson masses M W and M Z experimentally measured, the scale v sterile is unknown and needs to be determined by the sterile neutrino mass m M . The ratio is approximately estimated All sterile particles and gauge-singlet (neutral) states of massive composite Dirac particles (TeVscales) discussed here can be possible candidates of warm and cold dark matter [26], and they can decay into SM particles via relevant four-fermion operators in Eqs. (5)(6)(7)(8).
As a result, in the IR-domain, we write the following bilinear Dirac and Majorana mass terms of neutrinos in terms of their mass eigenstates ν i L and ν i We expect the Majorana masses to be approximately equal (degenerate), m M i ≈ m M , for three right-handed neutrino ν i R , since there is not any preferential i-th component of the condensate ν i c R ν i R . Whereas, due to the origin from the explicit symmetry breaking terms related to family flavor mixings, the Dirac masses m D i have the structure of hierarchy [20][21][22]. Moreover we assume that m M i ≫ m D i . Following the usual approach [27], diagonalizing the 2×2 mixing matrix (22) in terms of the neutrino and sterile neutrino mass eigenstate "i", we obtains the mixing angle 2θ i = tan −1 (m D i /m M i ) and two mass eigenvalues We turn to discuss the neutrino flavor oscillations in the usual framework. The mass-squared difference of neutrino mass eigenstates (i, j = 1, 2, 3),  22)]. Since ∆m 2D ij ≫ ∆m 2M ij , the neutrino masssquared difference ∆m 2M ij accounts for neutrino flavor oscillations with E ν /L ∼ ∆m 2M 12 ≃ 5 × 10 −3 eV 2 in the long-baseline experiments, where E ν and L respectively are neutrino energy and travel distance from a source to a detector. Whereas the neutrino mass-squared difference ∆m 2D ij may accounts for neutrino oscillations E ν /L ∼ ∆m 2D ij ≫ 10 −3 eV 2 in short baseline experiments [28].
Some remarks. The multitude of seemingly arbitrary parameters required to specify the SM shows the incompleteness of the SM, which is mainly manifested by our ignorance of (i) the relevant operators and dynamics that underlie the spontaneous/explicit breaking of the SM chiral-gauge symmetries, (ii) the global symmetries and mixings of puzzling replication of quark and lepton families. The relevant four-fermion operators (4) potentially give the theoretical description of the SM in the IR-domain with v ≈ 239.5 GeV, provided the UV-domain with E > ∼ 5 GeV, where the resonant and nonresonant new phenomena are distinct from the SM. We advocate that it is deserved