# Yukawa Bound States and Their LHC Phenomenology

Authors: Enkhbat Tsedenbaljir
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

We present the current status on the possible bound states of extra generation quarks. These include phenomenology and search strategy at the LHC. If chiral fourth-generation quarks do exist their strong Yukawa couplings, implied by current experimental lower bound on their masses, may lead to formation of bound states. Due to nearly degenerate 4G masses suggested by Precision Electroweak Test one can employ “heavy isospin” symmetry to classify possible spectrum. Among these states, the color-octet isosinglet vectorω8 is the easiest to be produced at the LHC. The discovery potential and corresponding decay channels are covered in this paper. With possible light Higgs at ~125 GeV two-Higgs doublet version is briefly discussed.


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## Body

## 1. Introduction

The observed baryon asymmetry of the universe (BAU) cannot be accounted for within the current particle physics theory, the Standard Model (SM). An additional heavy chiral fourth generation (4G) could provide much needed enhancement of CP-violation for BAU [1, 2], while potentially providing or being part of the mechanism for electroweak symmetry breaking [3–7]. The latter phenomenon is thought to happen through the condensation of QQ- state due to strong Yukawa attractive potential. Ever since Nambu’s insight to symmetry breaking and subsequent works with Jona-Lasinio [8] this idea has been with us. Although it is quite old, in advent of the Large Hadron Collider (LHC), this is the first time we may be able to experimentally see whether this happens in Nature. Here we aim to review current status and near future prospect for phenomenological study of Yukawa bound states of the heavy 4G quarks.Current lower bound from direct searches for the 4G chiral quarks put their Yukawa couplings well beyond perturbative limit. Often cited upper limit from the perturbative unitarity violation (UV) [9–12] for 4G has now been reached at the LHC. Therefore if they exist, their Yukawa couplings are necessarily in nonperturbative regime. Phenomenologically most interesting consequence is the existence of meson-like QQ- bound states of these heavy quarks around TeV mass region. The principal binding force is that of the Yukawa potential. This fact essentially fixes what type of states one would expect once proper quantum numbers, such as heavy “isospin,” which we define shortly, and spin are given. Over the years there have been many works on collider phenomenology of extra generations, for example, by Holdom [13, 14], to which the bound state study covered here is complimentary and many more important works dealing with the different aspects of the subject are not mentioned due to lack of space.An important information which helps to classify the spectrum comes from the precision electroweak test [15, 16]. The constraint favors a small mass splitting between t′ and b′, the 4G up- and down-type quarks, respectively. This near degeneracy in the heavy mass limit, combined with the fact that their mixings to lighter generations are constrained to be small by various flavor constraints, leads to an approximate “isospin” symmetry [17–19]. This approximate symmetry helps one to classify the bound states and identify leading production and decay processes. For example, taken the exact limit, one can see that isotriplets must be produced in pairs; therefore in a realistic situation, the single production is suppressed by small mass splitting between t′ and b′ and/or small mixing to a lighter generation. From this one can identifys most interesting modes as far as the 7 and 8 TeV LHC run are concerned. The Higgs also has to be heavy for the precision test and now it is ruled out between 120 and 600 GeV. Recently announced signals at 5σ from both CMS and ATLAS [20, 21] point toward more complex situation than just the simple extension of the SM by an extra generation. In particular, the signals are thus far consistent with the SM Higgs and, if confirmed, the simple extension by 4G is ruled out. On the other hand, the question regarding BAU within the SM will remain unresolved with the confirmation of the existence of Higgs. In light of this fact the possible solutions can still include extra generation and/or multi-Higgs scenarios. A recent study of 4G SM with two-Higgs doublet remains consistent with the Precision Electroweak Constraint [22] with 125 GeV Higgs.In this report, we give a brief review of the status of the LHC phenomenology of Yukawa bound states mainly along the line of [19] by Enkhbat et al. We update these analyses with the available results from the LHC 2012 run. The structure of the review is as follows. In Section 2, we review descriptions of the Yukawa bound states. In Section 3, the expected signal types and their rates are given. Section 4 is devoted for further phenomenological discussions. In Section 5, a two-Higgs doublet model extension is discussed. The conclusion is given in Section 6.


## 2. Yukawa Bound State Descriptions

The current direct search of sequential fourth generation quarks, that is, one additional generation of left-handed weak doublet and right-handed weak singlet, has now reached the lower limit ofmQ=611 GeV at 95% C.L [23]. Their existence means that there is a new nonperturbative regime of Yukawa interaction. Such a strong regime will lead to a bound state of these quarks. Whether they are part of electroweak symmetry breaking or not, a phenomenological study of the bound states due to the strong Yukawa potential is not only interesting but very crucial for heavy quark search program.Emergence of a strong dynamics could lead to a rich spectrum and phenomenology as witnessed in the case of QCD. Although the tree-level UV is related to more asymptotically higher energy unlike confinement in the infrared regime there is a lesson from the QCD and chiral perturbation theory. Traditionally the tree-level unitarity constraint has been considered as some upper limit for the masses of possible extra heavy fermions above which some new degree of freedom or new dynamics should appear to cure the apparent unitarity violation. In reality it is a perturbative statement and should be considered with a grain of salt. One even could argue that the effective field theory should be able to cure UV if a correct frame work is supplemented. In particular, the chiral perturbation theory with only light pions has UV ats≃450MeV in the scalar-isoscalar channel, which can be treated by inverse amplitude method [24]. The justification of this approach is provided by existence of σ-like object f0(600). Regarding the UV, there is another dynamical argument based on large-N expansion, which points to much higher bound [25] around ~3 TeV for 4G quark masses.To examine what bound states are phenomenologically more interesting one needs a tool to identify the quantum numbers of these states. An important constraint on the 4G spectrum is the precision electroweak test, which requires thet′ and b′ masses to be nearly equal. This “new isospin” makes it easier to list the lowest lying states. Unlike technicolor-like theories, one expects colored objects as they are far above the QCD scale. So for every color singlet meson-like states there is a color-octet counterpart.Here we use the same QCD nomenclature for these states and add subscripts1 and 8 as the singlet and octet color representations, respectively. Thus we denote isovector states b-′t′, t-′b′, (t-′t′-b-′b′)/2 as π1,8, ρ1,8 and isoscalars (t-′t′+b-′b′)/2 as η1,8, ω1,8.At this scale the QCD coupling is subleading compared to the Yukawa couplings. The potential from the Yukawa couplings consists of contributions from the Higgs and pseudoscalar Goldstones. While the Higgs part is universally attractive the Goldstone potential crucially depends on the isospin and spin of the state. Since we are considering a heavy Higgs scenario for the electroweak precision constraints, the main binding force is that of the Goldstone exchange. One can quickly establish the following facts by examining the Goldstone exchange diagrams: while isotriplet scalarsπ1,8 and isosinglet vector ω1,8 states have attractive Goldstone potential, isosinglet scalar η1,8 and isotriplet vector ρ1,8 have repulsive one. Therefore η- and ρ-like states are most likely unbound while the lower bound states are π’s and ω’s. This is quite distinct compared to the well-studied scenario of technicolor, and its colored versions where one has ρ-like resonance as the defining prediction. Also the colored meson ω8 may be accessible since it can be produced singly at the LHC.More quantitative treatments for a strongly bound relativistic bound state remain challenging. The most reliable one is the lattice approach which is still being pursued. Thus far the best estimates have been provided by Bethe-Salpeter approach, which has many shortcomings. There is a crucial qualitative difference between the Bethe-Salpeter and simpler approaches of relativistic expansion based on variational method. The former takes into account the contribution of mixing between positive and negative frequency solutions. This has been shown to have a dramatic effect in [17] as illustrated in the case of pions. In the following we review essential results of this study and one based on relativistic expansion [18].


### 2.1. Relativistic Bethe-Salpeter Approach

Here we explain the spectrum emerging from the study of relativistic bound states based on Bethe-Salpeter equation. In the early nineties, in a series of papers Jain et al. [17] studied strong Yukawa bound states. They have shown using Bethe-Salpeter equations numerically that in the strong Yukawa coupling limit highly relativistic deeply bound states form. These are coupled set of integral equations applied for the bispinor wave function of the QQ- bound states. The bispinor amplitude χ±π(q) for exchange momentum q between Q and Q- has two components corresponding to the negative and positive frequencies. The mixing between these states, which is not taken into account in a simple relativistic expansion approach [18], plays a crucial role. In the following we give the integral equation, without the explicit form of its kernel V(q,q′) (Bethe-Salpeter potential) which can be found in [17]. Instead we plot it for the lowest lying state π1 along the steepest direction q=q′, which suffices to show the importance of this approach. The Bethe-Salpeter equations for χ±π(q) are given by
(1)(M∓2ω)χ±π(q)=±∫dq′q′q[V±π(q,q′)χ+π(q′)+V∓π(q,q′)χ-π(q′)].
In the heavy isospin limit the possible states are classified by their definite isospins I=0,1.In Figure1, the Bethe-Salpeter potential for π1 along the steepest direction q=q′ is plotted for various choices of the heavy quark mass mQ. V± are the diagonal and mixing contributions, respectively, for χ+ and χ- amplitudes. As we can see the mixing potential V- from the negative frequency solution is dramatically enhanced as we increase the mass. This contribution causes a “collapse” of the state; namely the bound state mass becomes zero, at a much lower value of mQ compared to the case when the mixing is ignored [17]. Similar feature can be seen for ω1,8 states. On the other hand this dominant contribution is repulsive for ρ- and η-like states. For this reason the lowest π and ω states are the focus of phenomenological study for the LHC. While a small amount of mass splitting is required between b′ and t′ for electroweak precision constraint, here the degenerate case is taken for simplicity. In [17], the authors did not discuss color-octet mesons which are the more interesting objects for a hadron collider like the LHC. Nevertheless, they have calculated the isosinglet and isotriplet meson masses in different formalisms. Among these, Bethe-Salpeter formalism with both positive and negative frequency solutions leads to strongest binding and one can see that it even gives a collapse at ~500 GeV. Their numerical solution as binding energy versus the 4G mass is shown in Figure 2.Figure 1
TheV+π (solid) and V-π (dashed) potentials in (1) for the π1 state, for mQ=400,500,600,700 GeV, and mH=700 GeV, plotted along q′=q. Both vanish for q=0, or zero relative motion, but go to rather large values for large q, with V-π turning on rather sharply for q≳100 GeV.Figure 2
The masses of color singlet isotripletπ and isosinglet states versus the 4G quark mass from [17] in various formalisms. Curves S1 (iso-singlet ω1) and T1 (isotriplet π1) are the results from covariant gauge formalism. S2 (isosinglet) and T2 (isotriplet) are the results from Salpeter’s equation using only positive frequency components. S3 (isosinglet) and T3 (isotriplet) are the results using both positive and negative frequency components. For more details of these formalisms see [17].Ishiwata and Wise studied the Yukawa bound states by variational method in relativistic expansion approach in [18]. The essence of their result is captured in the binding energy obtained by minimization of the QQ- potential. We show their plot here in Figure 3, where they chose mh=130 GeV and get similar result for mh=mQ. AroundmQ≃400 GeV Yukawa contribution becomes dominant where binding energy changes dramatically. This “kink” behavior is due to the constraint a≥3/mQ they put to have valid relativistic expansion. Here a is the length characterizing the size of the bound system. From this one can conclude that the relativistic expansion, although limited by the constraint on a, also indicates that the Yukawa binding becomes substantial. Although the relativistic expansion leads to qualitatively similar spectrum to the Bethe-Salpeter method, it nevertheless does not take into account the crucial contribution of the mixing from negative frequency solution.Figure 3
Variational binding energy of color singlet channels as the function of the heavy quark massmq≡mQ.In this paper, we mainly review the scenario pursued in [19], where the heavy quark mass mQ was taken to be within 500 to 700 GeV interval. Based on the numerical analysis of Jain et al. the binding energy was chosen to be up to 200 GeV. The latest bound on 4G quark mass has now ruled out the lower half of this range and with the coming new set of data the bound will be further raised probing ever stronger Yukawa coupling domain. Keeping this in mind we will also touch on the discussion for higher values of mQ up to ~TeV values.Sinceπ1 and ω1 are not produced by gg-fusion but by weak Drell-Yan processes, energywise they are not efficiently produced. π8 can be produced via strong interaction in pairs from q-q or gg scattering [26], which makes it low in luminosity at 7 and 8 TeV. Therefore, at least in the near future, ω8, which has the same quantum number as the gluon, is the most interesting object from LHC stand. It cannot be produced by gg-fusion since two massless vector particles cannot fuse to one massive vector particle according to the Landau-Yang theorem [27, 28]. This leaves q-q annihilation as the principle production mechanism for ω8. The ordering of the spectra, according to [17] (which did not actually consider octet states), would be Mπ1≲Mω1≲Mπ8≲Mω8. In the next section, we review the production and decay properties of the ω8 meson.


## 2.1. Relativistic Bethe-Salpeter Approach

Here we explain the spectrum emerging from the study of relativistic bound states based on Bethe-Salpeter equation. In the early nineties, in a series of papers Jain et al. [17] studied strong Yukawa bound states. They have shown using Bethe-Salpeter equations numerically that in the strong Yukawa coupling limit highly relativistic deeply bound states form. These are coupled set of integral equations applied for the bispinor wave function of the QQ- bound states. The bispinor amplitude χ±π(q) for exchange momentum q between Q and Q- has two components corresponding to the negative and positive frequencies. The mixing between these states, which is not taken into account in a simple relativistic expansion approach [18], plays a crucial role. In the following we give the integral equation, without the explicit form of its kernel V(q,q′) (Bethe-Salpeter potential) which can be found in [17]. Instead we plot it for the lowest lying state π1 along the steepest direction q=q′, which suffices to show the importance of this approach. The Bethe-Salpeter equations for χ±π(q) are given by
(1)(M∓2ω)χ±π(q)=±∫dq′q′q[V±π(q,q′)χ+π(q′)+V∓π(q,q′)χ-π(q′)].
In the heavy isospin limit the possible states are classified by their definite isospins I=0,1.In Figure1, the Bethe-Salpeter potential for π1 along the steepest direction q=q′ is plotted for various choices of the heavy quark mass mQ. V± are the diagonal and mixing contributions, respectively, for χ+ and χ- amplitudes. As we can see the mixing potential V- from the negative frequency solution is dramatically enhanced as we increase the mass. This contribution causes a “collapse” of the state; namely the bound state mass becomes zero, at a much lower value of mQ compared to the case when the mixing is ignored [17]. Similar feature can be seen for ω1,8 states. On the other hand this dominant contribution is repulsive for ρ- and η-like states. For this reason the lowest π and ω states are the focus of phenomenological study for the LHC. While a small amount of mass splitting is required between b′ and t′ for electroweak precision constraint, here the degenerate case is taken for simplicity. In [17], the authors did not discuss color-octet mesons which are the more interesting objects for a hadron collider like the LHC. Nevertheless, they have calculated the isosinglet and isotriplet meson masses in different formalisms. Among these, Bethe-Salpeter formalism with both positive and negative frequency solutions leads to strongest binding and one can see that it even gives a collapse at ~500 GeV. Their numerical solution as binding energy versus the 4G mass is shown in Figure 2.Figure 1
TheV+π (solid) and V-π (dashed) potentials in (1) for the π1 state, for mQ=400,500,600,700 GeV, and mH=700 GeV, plotted along q′=q. Both vanish for q=0, or zero relative motion, but go to rather large values for large q, with V-π turning on rather sharply for q≳100 GeV.Figure 2
The masses of color singlet isotripletπ and isosinglet states versus the 4G quark mass from [17] in various formalisms. Curves S1 (iso-singlet ω1) and T1 (isotriplet π1) are the results from covariant gauge formalism. S2 (isosinglet) and T2 (isotriplet) are the results from Salpeter’s equation using only positive frequency components. S3 (isosinglet) and T3 (isotriplet) are the results using both positive and negative frequency components. For more details of these formalisms see [17].Ishiwata and Wise studied the Yukawa bound states by variational method in relativistic expansion approach in [18]. The essence of their result is captured in the binding energy obtained by minimization of the QQ- potential. We show their plot here in Figure 3, where they chose mh=130 GeV and get similar result for mh=mQ. AroundmQ≃400 GeV Yukawa contribution becomes dominant where binding energy changes dramatically. This “kink” behavior is due to the constraint a≥3/mQ they put to have valid relativistic expansion. Here a is the length characterizing the size of the bound system. From this one can conclude that the relativistic expansion, although limited by the constraint on a, also indicates that the Yukawa binding becomes substantial. Although the relativistic expansion leads to qualitatively similar spectrum to the Bethe-Salpeter method, it nevertheless does not take into account the crucial contribution of the mixing from negative frequency solution.Figure 3
Variational binding energy of color singlet channels as the function of the heavy quark massmq≡mQ.In this paper, we mainly review the scenario pursued in [19], where the heavy quark mass mQ was taken to be within 500 to 700 GeV interval. Based on the numerical analysis of Jain et al. the binding energy was chosen to be up to 200 GeV. The latest bound on 4G quark mass has now ruled out the lower half of this range and with the coming new set of data the bound will be further raised probing ever stronger Yukawa coupling domain. Keeping this in mind we will also touch on the discussion for higher values of mQ up to ~TeV values.Sinceπ1 and ω1 are not produced by gg-fusion but by weak Drell-Yan processes, energywise they are not efficiently produced. π8 can be produced via strong interaction in pairs from q-q or gg scattering [26], which makes it low in luminosity at 7 and 8 TeV. Therefore, at least in the near future, ω8, which has the same quantum number as the gluon, is the most interesting object from LHC stand. It cannot be produced by gg-fusion since two massless vector particles cannot fuse to one massive vector particle according to the Landau-Yang theorem [27, 28]. This leaves q-q annihilation as the principle production mechanism for ω8. The ordering of the spectra, according to [17] (which did not actually consider octet states), would be Mπ1≲Mω1≲Mπ8≲Mω8. In the next section, we review the production and decay properties of the ω8 meson.


## 3. Production and Decay ofω8

In this section, we start the collider phenomenology discussion with the review of theω8 production and decay. We have concluded that it is the lowest lying colored object that can be singly produced. The other octet states are either pair produced due to the new isospin or heavier even potentially unbound. The following input parameters are chosen: (i)
theω8 decay constant ξ≡fω8/mω8 defined by
(2)〈0|Vμ,a|ω8b(p)〉≡12δabfω8mω8εμ(p),(ii)
the fourth-generation quark massmQ=mt′=mb′ (this degeneracy led to heavy isospin),(iii)
theω8 mass mω8 (this is essentially the choice of binding energy 2mQ-mω8),(iv)
the quark mixing elements|Vt′b|=|Vtb′|, where we assume that mixings to lighter generations are negligible. In principle if the binding mechanism is quantitatively established the decay constant and binding energy are no longer free parameters. As mentioned before, the most reliable approach is the lattice field theory and it is far from reaching the required level at this point and therefore is beyond the scope of this report. What is done thus far is the qualitative estimates of the possible spectrum in Bethe-Salpeter and relativistic expansion approaches which fail to address these quantities. Therefore these quantities are chosen as free parameters for the phenomenological study.


### 3.1. Production

As it has been stressed earlier, the color-octet isospin singlet vector stateω8 appears to be most prominent among the interesting objects for the LHC. ω8 is dominantly produced by qq- annihilation as shown in Figure 4 instead of gluon fusion. The latter is forbidden due to the Landau-Yang theorem [27, 28], which states that a massive vector particle cannot decay to two massless vector particles. One can explicitly check this fact.Figure 4
Production mechanism for the color-octet, isoscalar-vector mesonω8 at hadron colliders.At the parton level, the total cross-section is(3)σ^qq-→ω8(s^)=32π3αs29mω82ξ2δ(1-mω82s^).
Convoluting with the parton luminosity
(4)ℒ(τ;μF2)=∫∫dx1dx2f1(x1;μF2)f2(x2;μF2)δ(τ-x1x2),
we get the hadronic cross-section:
(5)σ(s)=∫dτ^σ^(τ^s)L(τ^;μF2).
The inclusive production cross-section σω8 for ω8 is given in Figures 5 and 6. The left- and right-hand sides of Figure 5 are the s=7 TeV and 14 TeV, respectively, while Figure 6 is for 8 TeV [19]. Here CTEQ6L [29] parton distribution functions (PDFs) are used with the renormalization and factorization scales to μR=μF=mω8. The decay constant parameter defined in (2) is chosen at three different values ξ=fω8/mω8=0.1, 0.03, and 0.01 for illustration. One could intuitively argue that one expects stronger binding for large decay constant ξ. Because one lacks a reliable numerical tool to probe the highly relativistic system, in [19] the authors chose these values, which one encounters in QCD mesons [30–32]. One should keep in mind that the system we are discussing here is not a confining QCD-like model. The binding energy was chosen constant to be 100 or 200 GeV treating it as a free parameter. Therefore the production cross-section ranges from pb to fb in magnitude depending only on the choice of ξ. Due to higher gluon density at the LHC one might think that the production via higher order scattering process gg→ω8g could be important. It turns out, upon explicit calculation, that for the range mω8>0.8 TeV the rate is negligibly small.Production cross-section ofω8 at the LHC running at 7 TeV (a) and 14 TeV (b) for various ξ=fω8/mω8 values. The open QQ- cross-sections at LO and NLO are also plotted for comparison.
(a)
(b)Figure 6
Production cross-section ofω8 at the LHC running at 8 TeV.Another question is whether the production of the heavy meson state is completely dominated by the open production ofQQ-. For this, the pair production cross-section is calculated at LO and NLO [33, 34] and the results are shown in the same figure. For NLO calculation CTEQ6M PDFs [29] are used. The dotted (hatched) band shows the range of uncertainty when the renormalization scale is varied from μR=μF=mQ to 4mQ at LO (NLO). From this we see that the heavy meson ω8 production can be at the same order or even bigger than the open pair production if the decay constant, therefore binding energy, is larger even though qq- annihilation is the mechanism for the former while the latter occurs dominantly through gg-fusion at the LHC. At 14 TeV the production cross-section is an order of magnitude larger than the lower energy runs and open production is increased at higher mQ values compared to ω8 production due to higher gluon luminosity.


### 3.2. Decay

Due to the lower lying states,ω8 has possible meson transition-like decays. Such channels are experimentally more distinct which will be a clear signal of the bound states. In the following, possible leading channels are given with corresponding formulae. Using these it has been shown in [19] that such a meson transition-like decay indeed is sizable and often times dominates. The decay channels of ω8 considered are as follows.(i)
Annihilation decay:ω8→qq-, tt-; tt-′, bb-′; the decay rates for these processes are
(6)Γ(ω8→qq-)=ξ2παs23mω8nf,Γ(ω8→tt-)=ξ2παs23mω81-4mt2mω82,Γ(ω8→tt-′)=ξ2|Vtb′*Vt′b′|2GF2mω85192πE(m^t,m^t′),Γ(ω8→bb-′)=ξ2|Vt′b′*Vt′b|2GF2mω85192πE(m^b,m^b′),
where nf=5 is the number of light quark flavors that enhances this rate compared to tt- and m^≡m/mω8. The first two are just inverses of the production process as shown in Figure 7, which only depend on the decay constant parameter. On the other hand, the latter two proceed with an exchange of W boson as depicted in Figures 8(a) and 8(b), where Vtb′ necessarily enters. The final configuration for this case is always bWbW. While the as same as for tt-, it is nevertheless kinematically distinct. Analogous to gg→ω8g production, the three-body ω8→ggg decay rate, following the tree-level calculation of [35], is calculated and found to be negligible.Figure 7
QCD-induced annihilation decay ofω8 into light quark pair (dijets) or tt-.Weak decay modes: (a) exchange diagram with off-diagonal quark mixing element; (b) free quark decay.
(a)
(b)(
ii
)
Free-quark decay:ω8→bWt-′, tWb-′;
(7)Γfree≃12[2Γt′+2Γb′]=Γt′+Γb′,
where Γt′, Γb′ are given at Born level as
(8)Γt′(mt′)=|Vt′b|2GFmt′382πF(m~W,m~b),(9)Γb′(mb′)=|Vtb′|2GFmb′382πF(m~W,m~t),
with m~=m/mt′ or m/mb′. These occur when one of the heavy constituent quarks decays as a free quark, which requires the flavor change via Vtb′. There are two distinct final state configurations depending on whether it decays either from the t′t-′or b′b-′ mode of the bound state. If it is via t′ decay, the final decay product would be bWbW, and bWWbWW if it is from b′ decay. Since these signals are the same as the final state of the open production, the search can be pursued along the standard 4G quark search strategy [36]. Due to the binding energy, the reconstructed heavy quark mass would be lower than the open production case [37]. For example, one of the bW pairs in the b′ decay case would have a lower invariant mass roughly by the binding energy.
(
iii
)
Meson transition:ω8→ω1g; π8W; π8γ,
(10)Γ(ω8→π8W)=GFmω83322πmω8mπ8W(m^π8,m^W),Γ(ω8→ω1g)=αs18mω82mω1G(m^ω1),Γ(ω8→π8γ)≃α3(Δm)3mQ2.
The meson transition decays shown in Figure 9 give the clearest experimental signal for the heavy bound states. One more possibility ω8→π1g is forbidden by the heavy isospin symmetry. The partial width for these decays depends on the mass difference as well as the transition amplitude of these resonances. For example, the meson transition to π8W is allowed only ifmω8-mπ8>mW. For mω8-mπ8<mW, ω8 can decay into π8ℓνℓ or π8qq-′ through the off shell W boson. However, the partial width would be negligibly small. Previous numerical studies on Yukawa bound states suggest that the binding energy is indeed large if the mQ is beyond the unitarity bound. For this reason, it is expected that the on-shell transition is to be the most likely and was pursued in the phenomenological study of [19].
The functions appearing in the rate formulae are given as follows:(11)E(x,x′)=λ(1,x2,x′2)2(1-2x2-2x′2)2×[2-9x2+15x4-8x6-9x′2+18x2x′2-8x4x′2-16x6x′2+15x′4-8x2x′4+32x4x′4-8x′6-16x2x′6],F(x,y)=(1+x2-2x4-2y2+x2y2+y4)λ(1,x2,y2),W(x,y)≃(1-2x2+x4+3y2+2xy2+3x2y2-4y4)×(1+2x+x2-y2)λ(1,x2,y2),G(x)=≃(1-x2)3x,λ(a,b,c)=a2+b2+c2-2(ab+bc+ca).
For more details of the derivation of the rates and approximations and simplifications taken we refer readers to [19] and references therein.Meson transition currents forω8→π8 and ω8→ω1.
(a)
(b)


#### 3.2.1. Numerical Estimates

For the numerical study we need to choose following parameters: the 4th and lighter generation mixings from which we take 34 mixings onlyVt′b (=-Vtb′), the decay constant ξ=fω8/mω8 of ω8, the mass difference between mesons Δm=mω8-mπ8 for ω8→π8W, Δm=mω8-mω1 for ω8→ω1g which is essentially the binding energy. Numerical studies based on Bethe-Salpeter equation and relativistic expansion of variational method suggest much higher values for the binding energy than the QCD case at few 100s of GeV. So we choose Δm to be 100 or 200 GeV. The following four cases were examined which represent distinct features in the large parameter space: (i)
Case 1:ξ=0.1, Δm=100 GeV, Vt′b=0.1;(ii)
Case 2:ξ=0.03, Δm=100 GeV, Vt′b=0.1;(iii)
Case 3:ξ=0.1, Δm=200 GeV, Vt′b=0.1;(iv)
Case 4:ξ=0.1, Δm=100 GeV, Vt′b=0.01.Case 1 can be called as a “nominal” choice for the parameters. For the upper bound of the mixing the results of [6, 7] were taken. The choices for smaller decay constant, stronger binding, and small mixing correspond to Cases 2, 3, and 4, respectively. The resulting branching fractions for the above choice of parameters are plotted in Figure 10 using expressions in (6)–(10). In Case 1, we see that the dominant decay mode in the low mass range is π8W and the constituent quark decay (free quark) in the higher values. In the latter case the 4G quark mass enhances the rate. The other decay modes are in the range of 1% (ω1g) to 10% level (qq-). We see that the qq- is 5 times larger than tt- due to the number of light generations.Branching ratio ofω8 as a function of mω8 for given parameter sets in Cases 1 to 4.
(a)
(b)
(c)
(d)In Case 2, the dominant modes are the same as the above. The smaller decay constant makes all the annihilation modes suppressed.In both Cases 3 and 4, the dominant modes areπ8W, due to the large mass differences among the mesons and small mixing (suppressing the free and t′t-(b′b-)), respectively.In all cases,mω8-mπ8 is taken to be larger than MW. If the binding is not as strong as the Bethe-Salpeter approach suggests, the decay becomes off shell therefore subdominant. In this case the dijet channel will be the dominant. Also if the spectrum turns out to have mω8≃mπ8>mω1 pattern upon a closer numerical examination ω1g could become important especially in the low mass range. In summary, the dominant channel is π8W in all the choices considered, while free quark decay can become competitive for large mass range if the mixing is close to its upper bound.The estimated total decay widths forω8 and π8 are shown in Figure 11. For ω8 it is at most 20 GeV which, considering its mass being above TeV, makes it a narrow resonance.Figure 11
Total decay width ofω8 and π8 for the four parameter sets. The upper curves are for ω8, where Case 3 is enhanced by a larger mω8-mπ8 mass difference. The lower curves, plotted 100 GeV less in mass, are for π8, where Cases 1 and 2 are barely distinguishable.


#### 3.2.2. The Decay ofπ8

From the decay rates in Figure10, we see that π8W is one of the leading channels if not the most leading one. Therefore, for complete phenomenological discussion, one has to address the decay of π8 further. In the rest of the paper we assume mω8-mπ8>MW. Such a high mass splitting is suggested by Bethe-Salpeter approach [17]. Hence one should keep in mind that its validity has to be further checked by lattice calculation [39].The first thing to note is thatπ8→π1g is forbidden by angular momentum sum rule. Further, isospin forbids the annihilation via W-exchange, while s-channel via g is forbidden by its color and isospin. This leaves only two possibilities: the free quark decay and annihilation to gW final state, where the W is transverse. The gW decay rate is given by
(12)Γ(π8→Wg)=2ξπ2αsαWmπ8(1-MW2/mπ82)3(1+4mQ2/mπ82-2MW2/mπ82)2,
where ξπ=fπ8/mπ8 is the π8 decay constant normalized by π8 mass. The decay rate is plotted in Figure 12 for the four different cases we considered earlier. In Cases 1 to 3, where Vt′b mixing is large, the free quark decay channel is the dominant. For Case 2, π8W is suppressed even more due to small decay constant. On the other hand, Case 4 gives π8W as the dominant decay, since the free quark decay is suppressed by the choice of small mixing.Figure 12
Branching ratioℬ(π8→Wg) for the four parameter sets, ranging from predominance (Case 4), to below 10-2 (Case 2).


## 3.1. Production

As it has been stressed earlier, the color-octet isospin singlet vector stateω8 appears to be most prominent among the interesting objects for the LHC. ω8 is dominantly produced by qq- annihilation as shown in Figure 4 instead of gluon fusion. The latter is forbidden due to the Landau-Yang theorem [27, 28], which states that a massive vector particle cannot decay to two massless vector particles. One can explicitly check this fact.Figure 4
Production mechanism for the color-octet, isoscalar-vector mesonω8 at hadron colliders.At the parton level, the total cross-section is(3)σ^qq-→ω8(s^)=32π3αs29mω82ξ2δ(1-mω82s^).
Convoluting with the parton luminosity
(4)ℒ(τ;μF2)=∫∫dx1dx2f1(x1;μF2)f2(x2;μF2)δ(τ-x1x2),
we get the hadronic cross-section:
(5)σ(s)=∫dτ^σ^(τ^s)L(τ^;μF2).
The inclusive production cross-section σω8 for ω8 is given in Figures 5 and 6. The left- and right-hand sides of Figure 5 are the s=7 TeV and 14 TeV, respectively, while Figure 6 is for 8 TeV [19]. Here CTEQ6L [29] parton distribution functions (PDFs) are used with the renormalization and factorization scales to μR=μF=mω8. The decay constant parameter defined in (2) is chosen at three different values ξ=fω8/mω8=0.1, 0.03, and 0.01 for illustration. One could intuitively argue that one expects stronger binding for large decay constant ξ. Because one lacks a reliable numerical tool to probe the highly relativistic system, in [19] the authors chose these values, which one encounters in QCD mesons [30–32]. One should keep in mind that the system we are discussing here is not a confining QCD-like model. The binding energy was chosen constant to be 100 or 200 GeV treating it as a free parameter. Therefore the production cross-section ranges from pb to fb in magnitude depending only on the choice of ξ. Due to higher gluon density at the LHC one might think that the production via higher order scattering process gg→ω8g could be important. It turns out, upon explicit calculation, that for the range mω8>0.8 TeV the rate is negligibly small.Production cross-section ofω8 at the LHC running at 7 TeV (a) and 14 TeV (b) for various ξ=fω8/mω8 values. The open QQ- cross-sections at LO and NLO are also plotted for comparison.
(a)
(b)Figure 6
Production cross-section ofω8 at the LHC running at 8 TeV.Another question is whether the production of the heavy meson state is completely dominated by the open production ofQQ-. For this, the pair production cross-section is calculated at LO and NLO [33, 34] and the results are shown in the same figure. For NLO calculation CTEQ6M PDFs [29] are used. The dotted (hatched) band shows the range of uncertainty when the renormalization scale is varied from μR=μF=mQ to 4mQ at LO (NLO). From this we see that the heavy meson ω8 production can be at the same order or even bigger than the open pair production if the decay constant, therefore binding energy, is larger even though qq- annihilation is the mechanism for the former while the latter occurs dominantly through gg-fusion at the LHC. At 14 TeV the production cross-section is an order of magnitude larger than the lower energy runs and open production is increased at higher mQ values compared to ω8 production due to higher gluon luminosity.


## 3.2. Decay

Due to the lower lying states,ω8 has possible meson transition-like decays. Such channels are experimentally more distinct which will be a clear signal of the bound states. In the following, possible leading channels are given with corresponding formulae. Using these it has been shown in [19] that such a meson transition-like decay indeed is sizable and often times dominates. The decay channels of ω8 considered are as follows.(i)
Annihilation decay:ω8→qq-, tt-; tt-′, bb-′; the decay rates for these processes are
(6)Γ(ω8→qq-)=ξ2παs23mω8nf,Γ(ω8→tt-)=ξ2παs23mω81-4mt2mω82,Γ(ω8→tt-′)=ξ2|Vtb′*Vt′b′|2GF2mω85192πE(m^t,m^t′),Γ(ω8→bb-′)=ξ2|Vt′b′*Vt′b|2GF2mω85192πE(m^b,m^b′),
where nf=5 is the number of light quark flavors that enhances this rate compared to tt- and m^≡m/mω8. The first two are just inverses of the production process as shown in Figure 7, which only depend on the decay constant parameter. On the other hand, the latter two proceed with an exchange of W boson as depicted in Figures 8(a) and 8(b), where Vtb′ necessarily enters. The final configuration for this case is always bWbW. While the as same as for tt-, it is nevertheless kinematically distinct. Analogous to gg→ω8g production, the three-body ω8→ggg decay rate, following the tree-level calculation of [35], is calculated and found to be negligible.Figure 7
QCD-induced annihilation decay ofω8 into light quark pair (dijets) or tt-.Weak decay modes: (a) exchange diagram with off-diagonal quark mixing element; (b) free quark decay.
(a)
(b)(
ii
)
Free-quark decay:ω8→bWt-′, tWb-′;
(7)Γfree≃12[2Γt′+2Γb′]=Γt′+Γb′,
where Γt′, Γb′ are given at Born level as
(8)Γt′(mt′)=|Vt′b|2GFmt′382πF(m~W,m~b),(9)Γb′(mb′)=|Vtb′|2GFmb′382πF(m~W,m~t),
with m~=m/mt′ or m/mb′. These occur when one of the heavy constituent quarks decays as a free quark, which requires the flavor change via Vtb′. There are two distinct final state configurations depending on whether it decays either from the t′t-′or b′b-′ mode of the bound state. If it is via t′ decay, the final decay product would be bWbW, and bWWbWW if it is from b′ decay. Since these signals are the same as the final state of the open production, the search can be pursued along the standard 4G quark search strategy [36]. Due to the binding energy, the reconstructed heavy quark mass would be lower than the open production case [37]. For example, one of the bW pairs in the b′ decay case would have a lower invariant mass roughly by the binding energy.
(
iii
)
Meson transition:ω8→ω1g; π8W; π8γ,
(10)Γ(ω8→π8W)=GFmω83322πmω8mπ8W(m^π8,m^W),Γ(ω8→ω1g)=αs18mω82mω1G(m^ω1),Γ(ω8→π8γ)≃α3(Δm)3mQ2.
The meson transition decays shown in Figure 9 give the clearest experimental signal for the heavy bound states. One more possibility ω8→π1g is forbidden by the heavy isospin symmetry. The partial width for these decays depends on the mass difference as well as the transition amplitude of these resonances. For example, the meson transition to π8W is allowed only ifmω8-mπ8>mW. For mω8-mπ8<mW, ω8 can decay into π8ℓνℓ or π8qq-′ through the off shell W boson. However, the partial width would be negligibly small. Previous numerical studies on Yukawa bound states suggest that the binding energy is indeed large if the mQ is beyond the unitarity bound. For this reason, it is expected that the on-shell transition is to be the most likely and was pursued in the phenomenological study of [19].
The functions appearing in the rate formulae are given as follows:(11)E(x,x′)=λ(1,x2,x′2)2(1-2x2-2x′2)2×[2-9x2+15x4-8x6-9x′2+18x2x′2-8x4x′2-16x6x′2+15x′4-8x2x′4+32x4x′4-8x′6-16x2x′6],F(x,y)=(1+x2-2x4-2y2+x2y2+y4)λ(1,x2,y2),W(x,y)≃(1-2x2+x4+3y2+2xy2+3x2y2-4y4)×(1+2x+x2-y2)λ(1,x2,y2),G(x)=≃(1-x2)3x,λ(a,b,c)=a2+b2+c2-2(ab+bc+ca).
For more details of the derivation of the rates and approximations and simplifications taken we refer readers to [19] and references therein.Meson transition currents forω8→π8 and ω8→ω1.
(a)
(b)


### 3.2.1. Numerical Estimates

For the numerical study we need to choose following parameters: the 4th and lighter generation mixings from which we take 34 mixings onlyVt′b (=-Vtb′), the decay constant ξ=fω8/mω8 of ω8, the mass difference between mesons Δm=mω8-mπ8 for ω8→π8W, Δm=mω8-mω1 for ω8→ω1g which is essentially the binding energy. Numerical studies based on Bethe-Salpeter equation and relativistic expansion of variational method suggest much higher values for the binding energy than the QCD case at few 100s of GeV. So we choose Δm to be 100 or 200 GeV. The following four cases were examined which represent distinct features in the large parameter space: (i)
Case 1:ξ=0.1, Δm=100 GeV, Vt′b=0.1;(ii)
Case 2:ξ=0.03, Δm=100 GeV, Vt′b=0.1;(iii)
Case 3:ξ=0.1, Δm=200 GeV, Vt′b=0.1;(iv)
Case 4:ξ=0.1, Δm=100 GeV, Vt′b=0.01.Case 1 can be called as a “nominal” choice for the parameters. For the upper bound of the mixing the results of [6, 7] were taken. The choices for smaller decay constant, stronger binding, and small mixing correspond to Cases 2, 3, and 4, respectively. The resulting branching fractions for the above choice of parameters are plotted in Figure 10 using expressions in (6)–(10). In Case 1, we see that the dominant decay mode in the low mass range is π8W and the constituent quark decay (free quark) in the higher values. In the latter case the 4G quark mass enhances the rate. The other decay modes are in the range of 1% (ω1g) to 10% level (qq-). We see that the qq- is 5 times larger than tt- due to the number of light generations.Branching ratio ofω8 as a function of mω8 for given parameter sets in Cases 1 to 4.
(a)
(b)
(c)
(d)In Case 2, the dominant modes are the same as the above. The smaller decay constant makes all the annihilation modes suppressed.In both Cases 3 and 4, the dominant modes areπ8W, due to the large mass differences among the mesons and small mixing (suppressing the free and t′t-(b′b-)), respectively.In all cases,mω8-mπ8 is taken to be larger than MW. If the binding is not as strong as the Bethe-Salpeter approach suggests, the decay becomes off shell therefore subdominant. In this case the dijet channel will be the dominant. Also if the spectrum turns out to have mω8≃mπ8>mω1 pattern upon a closer numerical examination ω1g could become important especially in the low mass range. In summary, the dominant channel is π8W in all the choices considered, while free quark decay can become competitive for large mass range if the mixing is close to its upper bound.The estimated total decay widths forω8 and π8 are shown in Figure 11. For ω8 it is at most 20 GeV which, considering its mass being above TeV, makes it a narrow resonance.Figure 11
Total decay width ofω8 and π8 for the four parameter sets. The upper curves are for ω8, where Case 3 is enhanced by a larger mω8-mπ8 mass difference. The lower curves, plotted 100 GeV less in mass, are for π8, where Cases 1 and 2 are barely distinguishable.


### 3.2.2. The Decay ofπ8

From the decay rates in Figure10, we see that π8W is one of the leading channels if not the most leading one. Therefore, for complete phenomenological discussion, one has to address the decay of π8 further. In the rest of the paper we assume mω8-mπ8>MW. Such a high mass splitting is suggested by Bethe-Salpeter approach [17]. Hence one should keep in mind that its validity has to be further checked by lattice calculation [39].The first thing to note is thatπ8→π1g is forbidden by angular momentum sum rule. Further, isospin forbids the annihilation via W-exchange, while s-channel via g is forbidden by its color and isospin. This leaves only two possibilities: the free quark decay and annihilation to gW final state, where the W is transverse. The gW decay rate is given by
(12)Γ(π8→Wg)=2ξπ2αsαWmπ8(1-MW2/mπ82)3(1+4mQ2/mπ82-2MW2/mπ82)2,
where ξπ=fπ8/mπ8 is the π8 decay constant normalized by π8 mass. The decay rate is plotted in Figure 12 for the four different cases we considered earlier. In Cases 1 to 3, where Vt′b mixing is large, the free quark decay channel is the dominant. For Case 2, π8W is suppressed even more due to small decay constant. On the other hand, Case 4 gives π8W as the dominant decay, since the free quark decay is suppressed by the choice of small mixing.Figure 12
Branching ratioℬ(π8→Wg) for the four parameter sets, ranging from predominance (Case 4), to below 10-2 (Case 2).


## 3.2.1. Numerical Estimates

For the numerical study we need to choose following parameters: the 4th and lighter generation mixings from which we take 34 mixings onlyVt′b (=-Vtb′), the decay constant ξ=fω8/mω8 of ω8, the mass difference between mesons Δm=mω8-mπ8 for ω8→π8W, Δm=mω8-mω1 for ω8→ω1g which is essentially the binding energy. Numerical studies based on Bethe-Salpeter equation and relativistic expansion of variational method suggest much higher values for the binding energy than the QCD case at few 100s of GeV. So we choose Δm to be 100 or 200 GeV. The following four cases were examined which represent distinct features in the large parameter space: (i)
Case 1:ξ=0.1, Δm=100 GeV, Vt′b=0.1;(ii)
Case 2:ξ=0.03, Δm=100 GeV, Vt′b=0.1;(iii)
Case 3:ξ=0.1, Δm=200 GeV, Vt′b=0.1;(iv)
Case 4:ξ=0.1, Δm=100 GeV, Vt′b=0.01.Case 1 can be called as a “nominal” choice for the parameters. For the upper bound of the mixing the results of [6, 7] were taken. The choices for smaller decay constant, stronger binding, and small mixing correspond to Cases 2, 3, and 4, respectively. The resulting branching fractions for the above choice of parameters are plotted in Figure 10 using expressions in (6)–(10). In Case 1, we see that the dominant decay mode in the low mass range is π8W and the constituent quark decay (free quark) in the higher values. In the latter case the 4G quark mass enhances the rate. The other decay modes are in the range of 1% (ω1g) to 10% level (qq-). We see that the qq- is 5 times larger than tt- due to the number of light generations.Branching ratio ofω8 as a function of mω8 for given parameter sets in Cases 1 to 4.
(a)
(b)
(c)
(d)In Case 2, the dominant modes are the same as the above. The smaller decay constant makes all the annihilation modes suppressed.In both Cases 3 and 4, the dominant modes areπ8W, due to the large mass differences among the mesons and small mixing (suppressing the free and t′t-(b′b-)), respectively.In all cases,mω8-mπ8 is taken to be larger than MW. If the binding is not as strong as the Bethe-Salpeter approach suggests, the decay becomes off shell therefore subdominant. In this case the dijet channel will be the dominant. Also if the spectrum turns out to have mω8≃mπ8>mω1 pattern upon a closer numerical examination ω1g could become important especially in the low mass range. In summary, the dominant channel is π8W in all the choices considered, while free quark decay can become competitive for large mass range if the mixing is close to its upper bound.The estimated total decay widths forω8 and π8 are shown in Figure 11. For ω8 it is at most 20 GeV which, considering its mass being above TeV, makes it a narrow resonance.Figure 11
Total decay width ofω8 and π8 for the four parameter sets. The upper curves are for ω8, where Case 3 is enhanced by a larger mω8-mπ8 mass difference. The lower curves, plotted 100 GeV less in mass, are for π8, where Cases 1 and 2 are barely distinguishable.


## 3.2.2. The Decay ofπ8

From the decay rates in Figure10, we see that π8W is one of the leading channels if not the most leading one. Therefore, for complete phenomenological discussion, one has to address the decay of π8 further. In the rest of the paper we assume mω8-mπ8>MW. Such a high mass splitting is suggested by Bethe-Salpeter approach [17]. Hence one should keep in mind that its validity has to be further checked by lattice calculation [39].The first thing to note is thatπ8→π1g is forbidden by angular momentum sum rule. Further, isospin forbids the annihilation via W-exchange, while s-channel via g is forbidden by its color and isospin. This leaves only two possibilities: the free quark decay and annihilation to gW final state, where the W is transverse. The gW decay rate is given by
(12)Γ(π8→Wg)=2ξπ2αsαWmπ8(1-MW2/mπ82)3(1+4mQ2/mπ82-2MW2/mπ82)2,
where ξπ=fπ8/mπ8 is the π8 decay constant normalized by π8 mass. The decay rate is plotted in Figure 12 for the four different cases we considered earlier. In Cases 1 to 3, where Vt′b mixing is large, the free quark decay channel is the dominant. For Case 2, π8W is suppressed even more due to small decay constant. On the other hand, Case 4 gives π8W as the dominant decay, since the free quark decay is suppressed by the choice of small mixing.Figure 12
Branching ratioℬ(π8→Wg) for the four parameter sets, ranging from predominance (Case 4), to below 10-2 (Case 2).


## 4. Phenomenology Discussion

The searches for a dijet resonance at the Tevatron and LHC put constraint on theories with TeV range particles. Theω8 production cross-section times dijet branching fraction at 8 TeV is plotted in Figure 13 for the four cases along with the dijet search at CMS [38] with ℒ=4 fb-1 data at s=8 TeV. This is a conservative estimate since, unlike the data, no cut was used in the dijet rate from ω8. The open QQ- pair productions via gg-fusion are also calculated at LO and NLO and are shown in orange and red bands, respectively. Although ω8 is through qq-, it is of the same order or even larger than the open production. We see that all the rates are at least an order of magnitude lower than the CMS constraint.Figure 13
Production cross-section times branching ratio forω8 into dijets, for the four parameter sets at the LHC 8 TeV. CMS upper limit [38] for the cross-section of dijet resonance production is also plotted.Since the dominant decay isω8→π8W it is necessary to examine π8 decay modes to get eventual signals at the LHC. Observe that π8→π1g is forbidden by angular momentum sum rule. The estimate for ω8→π8W rate in Figure shows that in Cases 1 to 3 it dominantly decays through free quark channel. From the isospin one expects ω80→π8±Ws∓, π80Zs to have a 2:1 ratio, where the subscript s signifies that W is soft. The final state configuration is then bW+b-′Ws-, tW-t-′Ws+, and {bW+t-′,tW-b-′}Zs with 7 and 6/8-jet systems, respectively. The associated soft W(Z) can be used as a tag. With additional soft W, the signal appears even more complicated than the open production. While they appear difficult, the boosted products can be exploited to analyze the system using their jet substructures. If the total mass can be constructed with good enough resolution the system adds up to a narrow resonance potentially giving signals for both ω8 and π8.This bring us to the discussion of Case 4, for which one hasπ8W as dominant mode with final signal as WsWg. (Ws±W∓g and ZsZg). This signal is very unique and highly detectable at the LHC with pT>500 GeV gluon and boosted W with soft W. The whole system also gives a narrow resonance.Theπ8γ channel is at most a percent level for all the Cases. Although the rate is low, due to high photon detection efficiency it can be searched and further studied in detail. Finally, the W-exchange channel to tt-′ and bb-′, while appears similar to the single t′(b′) production, and the associated boosted t (high pTb)can give distinct resonant signal.We conclude the phenomenological discussion emphasizing the difference compared to technicolor (TC) models. The first distinction is the presence of colored objects. For generic “walking" TC models [40], the technipion πT tends to be closer to the technirho ρT in mass such that ρT→πTπT is absent, while (near) degeneracy of ωT and aT with ρT is also often invoked. The signature for these technimesons are typically WZ, Wγ, and Zγ. Thus, not only the spectrum is rather different—absence of ρ and a mesons—the decay signature is also in strong contrast. The bound states due to strong Yukawa coupling, which follow simply from the existence of new heavy chiral quarks without assuming new dynamics, should be easily distinguished from technicolor. The Yukawa-bound ultraheavy mesons are also quite distinct from QCD bound states. Not only there is the absence of η (where gg fusion would be possible) and ρ type mesons, but also they have a much larger binding energy and are much smaller in size. This is brought about not only by a strong coupling constant, but also by isospin-dependent Goldstone coupling.


## 5. Extension to Two-Higgs Doublet

With the discovery of~125 GeV boson, a simple 4G extension of the SM appears unattainable if the signal is indeed confirmed as the SM Higgs. If an extra generation does exist at least the SM Higgs sector has to be extended further. Indeed a recent analysis has shown that the precision test for 4G with two Higgses is allowed by the current constraint if the lighter Higgs is indeed 125 GeV [22]. In the following, we discuss how an extra new doublet Higgs affects the phenomenology we considered so far in this paper.We take, for simplicity, the case of Type-II of two-Higgs doublet (2HD) model, where up- (down-) type fermions couple only toH1(H2). We also take the simplified limit, for illustration, wherein tanβ=v1/v2≃1 and sin(β-α)~1. Here α is h-H mixing angle. In the presence of a light Higgs, new ω8 decay channels open up (i) from free quark decay with Higgs in the final state, (ii) to gh/H, and (iii) from the annihilation via Higgs exchange analogous to the W-exchange discussed earlier. These are as follows. (i)
Free quark decay:ω8→t′b-H-(b′t-H+):
(13)ΓfreeH≃Γ(t′→H+b)+Γ(b′→H-t),Γ(t′→H+b)=GF|Vt′b|2mt′382πλ(1,m^b2,m^H±2)×{κAu2(1+m^b2-m^H±2)+m^b2κAd2+4m^b2κAuκAd},Γ(b′→H-t)=Γ(t′→H+b)|(t′,b,d)↔(b′,t,u),
where κAu(d) is tanβ=v1/v2(1/tanβ) and m^=m/mt′(b′) for t′(b′)→H±b(t). For numerics we take mH±=200 GeV as suggested by [22]. Note that t′t-A(b′b-A) is absent since the CP-odd A does not violate flavor at tree level in Type-II 2HD model. The loop induced amplitude are small at low tanβ.(ii)
Annihilation tot′t-(b′b-) through Higgs exchange:
(14)ΓH±(ω8→t′t-)=ξ2GF2|Vt′b|2mQ4mω824πλ(1,m^t2,m^Q2)×(1-m^t2-m^Q2)(1+4m^H±2-2m^t2-2m^Q2)2,ΓH±(ω8→b′b-)=ΓH±(ω8→(t′t-))|(t′t)→(b′b).
As in the previous case, there is no A-exchange contribution since A does not break flavor in Type-II case at tree level and for low tanβ the quantum effects are ignorable.(iii)
Annihilation with Higgs in the final stateω8→gh and gH. In Type-II 2HD case with tanβ≃1 and sin(β-α)~1, the rates for the light Higgs channel are given as
(15)Γ(ω8→gh)≃ξh222αs3GFmQ2mω8(1-m^h2)3(1+4m^Q2-2m^h2)2,
where m^=m/mω8 and ξh is the decay constant for this channel taken to be equal to ξ. We take mh=125 GeV. In the simplified limit we are considering that ω8→gH is suppressed. ω8 and g have the same quantum numbers except mass which makes the ω8g system CP even. Since A does not mix with h/H in Type-II case, it has definite odd CP-nature. Therefore ω8→gA is also absent. One might think that any of ω8→π8h/H/A could become important if kinematics allows. These are absent for the same reason that W from ω8→π8W is purely transverse.The rates for channelst′b-H-(b′t-H+) and gh are plotted in Figure 14. In Type-II case, there is no free quark to a final state with neutral Higgs. Since we find t′t- and b′b- via charged Higgs exchange to be very small we do not plot them here. The free quark decays with charged Higgs are present if mH±<mω8-mQ. The rates are at the same order as the free quark decay given in (9), which is expected since the heavy quark decay rates to tW are dominated by the Goldstone components. For Type-II 2HD extension, the annihilation through charged Higgs exchange is also at the same order as the W-exchange counterpart. Therefore they have no qualitative and a very minimal quantitative effect on the discussion of the leading π8W signal.Figure 14
Branching ratios for 2HD model for the parameter choices of Case 4. Here we takemh=125 GeV and mH±=200 GeV.The decay togh and gH could have large branching ratios at high ω8 mass range. For our simplified discussion, gh reaches at most ~20%. Therefore our π8W discussions remain robust. The search for this channel appears challenging. If enough ω8 is produced and its mass is high enough; boosted Higgs analysis [41] may be used.In this section we have taken Type II 2HD model as a simple illustration and have shown that the main phenomenology expected at the LHC from the bound states may not have to be drastically different. A more thorough numerical study is needed for clarifying whether this or any other variants of 2HD model are phenomenologically viable.


## 6. Conclusion and Outlook

In this short paper we revisited current phenomenological study of possible bound states of heavy 4G quarks. The direct search limits now have reached the perturbative unitarity bound making their Yukawa couplings well in the strong coupling regime, where one expects a meson-likeQ-Q bound states in ~TeV range. Existing studies in Bethe-Salpeter and relativistic expansion approaches enable one to study their phenomenology at the LHC.Electroweak Precision tests constrain the top and bottom-like 4G quark masses to be nearly degenerate leading to a “heavy isospin” symmetry, which in turn can be used to classify spectrum. Among these, one finds that color-octet isosinglet vector resonance ω8 is to be the easiest produced one at early runs of the LHC via qq--annihilation. The parameters needed for such study are the meson mass splittings, 4G quark mixing to lighter generations, the decay constant, and 4G quark mass.Possible final decay products have rich phenomenology with challenging signals. The leading decay channel ofω8 in the cases considered is π8W followed by free quark decay inside the bound state. With four different representative choices for the parameters their rates are studied. Among these, Case 4 with small mixing provides a clearest signal of possible bound state at the LHC, wherein the whole process chain is ω8→π8Ws→gWsW with highly boosted W and g with one soft Ws as the final state. Other cases, with enough total energy resolution, also may provide a resonant signal.With~125 GeV object found at the LHC being most likely a Higgs, if proven, simple 4G extension is unlikely to be viable. In light of this, we also briefly explore how this phenomenology is affected in presence of additional Higgs. The most significant change comes in the form of opening up new large annihilation decay channels gh and gH. Numerical estimates of these rates indicate that in Case 4  π8W→gWW mode remains the leading channel, leaving the conclusions with single Higgs robust. Another easily detectable signature is qq- dijet which becomes dominant if Δm=mω8-mπ8<MW. While this is unlikely considering what Bethe-Salpeter study suggests, this can be eventually resolved by Lattice effort.We have reviewed signatures for Yukawa-bound heavyQQ- mesons in the 1 to 2 TeV range. We also briefly touched upon two Higgs doublet case. What is covered in this review is qualitative in nature. As the LHC energy increases, and with higher luminosity, it could uncover new chiral quarks above the unitarity bound, with new unusual bound states. One could probe into the truly nonperturbative regime, in which these results only offer a glimpse of what might happen.



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