Yukawa Bound States and Their LHC Phenomenology

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 vector 𝜔₈ 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.

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 of 𝑚_𝑄=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 at √𝑠≃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 the 𝑡^′ 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 subscripts 1 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 𝜋_1,8, 𝜌_1,8 and isoscalars (𝑡^′𝑡^′+𝑏^′𝑏^′)/√2 as 𝜂_1,8, 𝜔_1,8.

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 scalars 𝜋_1,8 and isosinglet vector 𝜔_1,8 states have attractive Goldstone potential, isosinglet scalar 𝜂_1,8 and isotriplet vector 𝜌_1,8 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 (𝑀∓2𝜔)𝜒^𝜋_±𝑞=±𝑑𝑞^′𝑞^′𝑞𝑉^𝜋_±𝑞,𝑞^′𝜒^𝜋₊𝑞^′+𝑉^𝜋_∓𝑞,𝑞^′𝜒^𝜋₋𝑞^′. (1) In the heavy isospin limit the possible states are classified by their definite isospins 𝐼=0,1.

In Figure 1, the Bethe-Salpeter potential for 𝜋₁ along the steepest direction 𝑞=𝑞^′ is plotted for various choices of the heavy quark mass 𝑚_𝑄. 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 𝑚_𝑄 compared to the case when the mixing is ignored [17]. Similar feature can be seen for 𝜔_1,8 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 

The 𝑉𝜋+ (solid) and 𝑉𝜋− (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 𝑉𝜋− turning on rather sharply for 𝑞≳100 GeV.

Figure 2 

The masses of color singlet isotriplet 𝜋 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 𝑚_ℎ=130 GeV and get similar result for 𝑚_ℎ=𝑚_𝑄. Around𝑚_𝑄≃400 GeV Yukawa contribution becomes dominant where binding energy changes dramatically. This “kink” behavior is due to the constraint 𝑎≥√3/𝑚_𝑄 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 mass 𝑚𝑞≡𝑚𝑄.

In this paper, we mainly review the scenario pursued in [19], where the heavy quark mass 𝑚_𝑄 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 𝑚_𝑄 up to ~TeV values.

Since 𝜋₁ 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 𝑞𝑞 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 𝑞𝑞 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 𝑀_𝜋1≲𝑀_𝜔1≲𝑀_𝜋8≲𝑀_𝜔8. In the next section, we review the production and decay properties of the 𝜔₈ meson.

3. Production and Decay of 𝜔₈

In this section, we start the collider phenomenology discussion with the review of the 𝜔₈ 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) the 𝜔₈ decay constant 𝜉≡𝑓_𝜔8/𝑚_𝜔8 defined by 0𝑉^𝜇,𝑎𝜔^𝑏₈𝑝≡1√2𝛿^𝑎𝑏𝑓_𝜔8𝑚_𝜔8𝜀^𝜇𝑝, (2)(ii) the fourth-generation quark mass 𝑚_𝑄=𝑚_𝑡′=𝑚_𝑏′ (this degeneracy led to heavy isospin), (iii) the 𝜔₈ mass 𝑚_𝜔8 (this is essentially the choice of binding energy 2𝑚_𝑄−𝑚_𝜔8), (iv) the quark mixing elements |𝑉_𝑡′𝑏|=|𝑉_𝑡𝑏′|, 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 state 𝜔₈ 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 meson 𝜔8 at hadron colliders.

At the parton level, the total cross-section is 𝜎_{𝑞𝑞→𝜔8}(𝑠)̂=32𝜋³𝛼²_𝑠9𝑚²_𝜔8𝜉²𝛿𝑚1−^2𝜔8𝑠̂. (3) Convoluting with the parton luminosity ℒ𝜏;𝜇²_𝐹=𝑑𝑥₁𝑑𝑥₂𝑓₁𝑥₁;𝜇²_𝐹𝑓₂𝑥₂;𝜇²_𝐹𝛿𝜏−𝑥₁𝑥₂, (4) we get the hadronic cross-section: 𝜎(𝑠)=𝑑𝜏𝜎(𝜏𝑠)𝐿𝜏;𝜇²_𝐹. (5) The inclusive production cross-section 𝜎_𝜔8 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 𝜇_𝑅=𝜇_𝐹=𝑚_𝜔8. The decay constant parameter defined in (2) is chosen at three different values 𝜉=𝑓_𝜔8/𝑚_𝜔8=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 𝑚_𝜔8>0.8 TeV the rate is negligibly small.

(a)

(b)

Figure 5 

Production cross-section of 𝜔8 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.

Figure 6 

Production cross-section of 𝜔8 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 of 𝑄𝑄. 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 𝜇_𝑅=𝜇_𝐹=𝑚_𝑄 to 4𝑚_𝑄 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 𝑚_𝑄 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 Γ𝜔₈→𝑞𝑞=𝜉²𝜋𝛼²_𝑠3𝑚_𝜔8𝑛_𝑓,Γ𝜔₈→𝑡𝑡=𝜉²𝜋𝛼²_𝑠3𝑚_𝜔81−4𝑚²_𝑡𝑚²_𝜔8,Γ𝜔₈→𝑡𝑡^′=𝜉²𝑉^∗_𝑡𝑏′𝑉_{𝑡′𝑏′}²𝐺²_𝐹𝑚⁵_𝜔8𝐸𝑚192𝜋^𝑡,𝑚_𝑡′,Γ𝜔₈→𝑏𝑏^′=𝜉²𝑉^∗_{𝑡′𝑏′}𝑉_𝑡′𝑏²𝐺²_𝐹𝑚⁵_𝜔8𝐸𝑚192𝜋^𝑏,𝑚_𝑏′, (6) where 𝑛_𝑓=5 is the number of light quark flavors that enhances this rate compared to 𝑡𝑡 and 𝑚≡𝑚/𝑚_𝜔8. 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 𝑉_𝑡𝑏′ 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 of 𝜔8 into light quark pair (dijets) or 𝑡𝑡.

(a)

(b)

Figure 8 

Weak decay modes: (a) exchange diagram with off-diagonal quark mixing element; (b) free quark decay.

(ii) Free-quark decay: 𝜔₈→𝑏𝑊𝑡^′, 𝑡𝑊𝑏^′; Γ_free≃122Γ_𝑡′+2Γ_𝑏′=Γ_𝑡′+Γ_𝑏′, (7) where Γ_𝑡′, Γ_𝑏′ are given at Born level as Γ_𝑡′𝑚_𝑡′=𝑉_𝑡′𝑏²𝐺_𝐹𝑚³_𝑡′8√𝐹𝑚2𝜋^𝑊,𝑚_𝑏,Γ_𝑏′𝑚_𝑏′=𝑉_𝑡𝑏′²𝐺_𝐹𝑚³_𝑏′8√𝐹𝑚2𝜋^𝑊,𝑚_𝑡, (8)(9) with 𝑚=𝑚/𝑚_𝑡′ or 𝑚/𝑚_𝑏′. These occur when one of the heavy constituent quarks decays as a free quark, which requires the flavor change via 𝑉_𝑡𝑏′. 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, Γ𝜔₈→𝜋₈𝑊=𝐺_𝐹𝑚³_𝜔832√𝑚2𝜋𝑚^𝜔8_𝜋8𝑊𝑚_𝜋8,𝑚_𝑊,Γ𝜔₈→𝜔₁𝑔=𝛼_𝑠𝑚18𝑚^2𝜔8_𝜔1𝐺𝑚_𝜔1,Γ𝜔₈→𝜋₈𝛾≃𝛼3(Δ𝑚)³𝑚²_𝑄. (10) 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 if𝑚_𝜔8−𝑚_𝜋8>𝑚_𝑊. For 𝑚_𝜔8−𝑚_𝜋8<𝑚_𝑊, 𝜔₈ can decay into 𝜋₈ℓ𝜈_ℓ 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 𝑚_𝑄 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: 𝐸𝑥,𝑥^′=𝜆1,𝑥²,𝑥^′221−2𝑥²−2𝑥^′22×2−9𝑥²+15𝑥⁴−8𝑥⁶−9𝑥^′2+18𝑥²𝑥^′2−8𝑥⁴𝑥^′2−16𝑥⁶𝑥^′2+15𝑥^′4−8𝑥²𝑥^′4+32𝑥⁴𝑥^′4−8𝑥^′6−16𝑥²𝑥^′6,𝐹𝑥,𝑦=1+𝑥²−2𝑥⁴−2𝑦²+𝑥²𝑦²+𝑦⁴𝜆1,𝑥²,𝑦²,𝑊𝑥,𝑦≃1−2𝑥²+𝑥⁴+3𝑦²+2𝑥𝑦²+3𝑥²𝑦²−4𝑦⁴×1+2𝑥+𝑥²−𝑦²𝜆1,𝑥²,𝑦²,𝐺(𝑥)=≃1−𝑥²³𝑥,𝜆(𝑎,𝑏,𝑐)=𝑎²+𝑏²+𝑐²−2(𝑎𝑏+𝑏𝑐+𝑐𝑎). (11) For more details of the derivation of the rates and approximations and simplifications taken we refer readers to [19] and references therein.

(a)

(b)

Figure 9 

Meson transition currents for 𝜔8→𝜋8 and 𝜔8→𝜔1.

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 only 𝑉_𝑡′𝑏 (=−𝑉_𝑡𝑏′), the decay constant 𝜉=𝑓_𝜔8/𝑚_𝜔8 of 𝜔₈, the mass difference between mesons Δ𝑚=𝑚_𝜔8−𝑚_𝜋8 for 𝜔₈→𝜋₈𝑊, Δ𝑚=𝑚_𝜔8−𝑚_𝜔1 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, 𝑉_𝑡′𝑏=0.1; (ii)Case 2: 𝜉=0.03, Δ𝑚=100 GeV, 𝑉_𝑡′𝑏=0.1; (iii)Case 3: 𝜉=0.1, Δ𝑚=200 GeV, 𝑉_𝑡′𝑏=0.1; (iv)Case 4: 𝜉=0.1, Δ𝑚=100 GeV, 𝑉_𝑡′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.

(a)

(b)

(c)

(d)

Figure 10 

Branching ratio of 𝜔8 as a function of 𝑚𝜔8 for given parameter sets in Cases 1 to 4.

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 are 𝜋₈𝑊, due to the large mass differences among the mesons and small mixing (suppressing the free and 𝑡^′𝑡(𝑏^′𝑏)), respectively.

In all cases, 𝑚_𝜔8−𝑚_𝜋8 is taken to be larger than 𝑀_𝑊. 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 𝑚_𝜔8≃𝑚_𝜋8>𝑚_𝜔1 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 for 𝜔₈ 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 of 𝜔8 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 of 𝜋₈

From the decay rates in Figure 10, 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 𝑚_𝜔8−𝑚_𝜋8>𝑀_𝑊. 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 that 𝜋₈→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 Γ𝜋₈→𝑊𝑔=2𝜉²_𝜋𝛼_𝑠𝛼_𝑊𝑚_𝜋81−𝑀²_𝑊/𝑚²_𝜋8³1+4𝑚²_𝑄/𝑚²_𝜋8−2𝑀²_𝑊/𝑚²_𝜋8², (12) where 𝜉_𝜋=𝑓_𝜋8/𝑚_𝜋8 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 𝑉_𝑡′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 ratio ℬ(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. The 𝜔₈ 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 for 𝜔8 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 is 𝜔₈→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 𝜔⁰₈→𝜋^±₈𝑊^∓_𝑠, 𝜋⁰₈𝑍_𝑠 to have a 2∶1 ratio, where the subscript 𝑠 signifies that 𝑊 is soft. The final state configuration is then 𝑏𝑊⁺𝑏^′𝑊⁻_𝑠, 𝑡𝑊⁻𝑡^′𝑊⁺_𝑠, and {𝑏𝑊⁺𝑡^′,𝑡𝑊⁻𝑏^′}𝑍_𝑠 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 has 𝜋₈𝑊 as dominant mode with final signal as 𝑊_𝑠𝑊𝑔. (𝑊^±_𝑠𝑊^∓𝑔 and 𝑍_𝑠𝑍𝑔). This signal is very unique and highly detectable at the LHC with 𝑝_𝑇>500 GeV gluon and boosted 𝑊 with soft 𝑊. The whole system also gives a narrow resonance.

The 𝜋₈𝛾 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 𝑝_𝑇𝑏)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 𝜋_𝑇 tends to be closer to the technirho 𝜌_𝑇 in mass such that 𝜌_𝑇→𝜋_𝑇𝜋_𝑇 is absent, while (near) degeneracy of 𝜔_𝑇 and 𝑎_𝑇 with 𝜌_𝑇 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 to 𝐻₁(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^′𝑏𝐻⁻(𝑏^′𝑡𝐻⁺): Γ^𝐻_free≃Γ𝑡^′→𝐻⁺𝑏+Γ𝑏^′→𝐻⁻𝑡,Γ𝑡^′→𝐻⁺𝑏=𝐺_𝐹𝑉_𝑡′𝑏²𝑚³_𝑡′8√𝜆𝑚2𝜋1,²_𝑏,𝑚²_𝐻^±×𝜅^𝑢_𝐴²1+𝑚²_𝑏−𝑚²_𝐻^±+𝑚²_𝑏𝜅^𝑑_𝐴²+4𝑚²_𝑏𝜅^𝑢_𝐴𝜅^𝑑_𝐴,Γ𝑏^′→𝐻⁻𝑡=Γ𝑡^′→𝐻⁺𝑏_{(𝑡′,𝑏,𝑑)↔(𝑏′,𝑡,𝑢)}, (13) where 𝜅_𝐴^𝑢(𝑑) is tan𝛽=𝑣₁/𝑣₂(1/tan𝛽) and 𝑚=𝑚/𝑚_{𝑡′(𝑏′)} for 𝑡^′(𝑏^′)→𝐻^±𝑏(𝑡). For numerics we take 𝑚_𝐻^±=200 GeV as suggested by [22]. Note that 𝑡^′𝑡𝐴(𝑏^′𝑏𝐴) 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 to 𝑡^′𝑡(𝑏^′𝑏) through Higgs exchange: Γ^𝐻±𝜔₈→𝑡^′𝑡=𝜉²𝐺²_𝐹𝑉_𝑡′𝑏²𝑚⁴_𝑄𝑚_𝜔8𝜆𝑚24𝜋1,²_𝑡,𝑚²_𝑄×1−𝑚²_𝑡−𝑚²_𝑄1+4𝑚²_𝐻±−2𝑚²_𝑡−2𝑚²_𝑄²,Γ^𝐻±𝜔₈→𝑏^′𝑏=Γ^𝐻±𝜔₈→𝑡^′𝑡_{(𝑡′𝑡)→(𝑏′𝑏)}. (14) 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 state 𝜔₈→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 Γ𝜔₈→𝑔ℎ≃𝜉²_ℎ2√2𝛼_𝑠3𝐺_𝐹𝑚²_𝑄𝑚_𝜔81−𝑚²_ℎ³1+4𝑚²_𝑄−2𝑚²_ℎ², (15) where 𝑚=𝑚/𝑚_𝜔8 and 𝜉_ℎ is the decay constant for this channel taken to be equal to 𝜉. We take 𝑚_ℎ=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 channels 𝑡^′𝑏𝐻⁻(𝑏^′𝑡𝐻⁺) 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 𝑚_𝐻^±<𝑚_𝜔8−𝑚_𝑄. 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 take 𝑚ℎ=125 GeV and 𝑚𝐻±=200 GeV.

The decay to 𝑔ℎ 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-like 𝑄𝑄 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 of 𝜔₈ 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→𝜋₈𝑊_𝑠→𝑔𝑊_𝑠𝑊 with highly boosted 𝑊 and 𝑔 with one soft 𝑊_𝑠 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=𝑚_𝜔8−𝑚_𝜋8<𝑀_𝑊. 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 heavy 𝑄𝑄 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.

Acknowledgments

The author thanks K. Ishiwata and M. B. Wise for letting him use their results in this paper. The author is also thankful for H. Yokoya’s help in updating the calculations. This work is supported by NSC 100-2811-M-002-061 and NSC 100-2119-M-002-001.