#### Abstract

Resonant production of excited and quarks at the Future Circular Collider and Super Proton-Proton Collider has been researched. Dominant jet-jet decay mode has been considered. It is shown that FCC and SppC have great potential for discovery of excited () quark: up to 44.1 (36.3) and 58.4 (47.8) TeV masses, respectively. For degenerate case ( = ), these values are 45.9 and 60.9 TeV, respectively. This discovery will also afford an opportunity to determine the compositeness scale up to multi-PeV level.

#### 1. Introduction

Standard model (SM) contains plenty of elementary particles and their parameters that are not completely explained. To overcome these unsolved problems that the SM does not give answers, new models have been developed beyond the standard model (BSM) such as composite models, supersymmetry, extra dimensions, string theory, and so on. These BSM theories require higher energy level than SM energy domain to bring solutions for unanswered problems. Therefore, the SM is considered as low energy configuration of the more fundamental theory.

Numbers of particles and parameters in the SM are reduced in the frame of the composite models [1–16]. According to composite models, while SM quarks and leptons are predicted as composite particles, preons are considered as the most fundamental particles. If excited states of the SM fermions are experimentally observed, this observation will be clear proof of quarks and leptons’ compositeness.

Excited fermions are known to represent much heavier particles than the SM fermions and they could be split into two classes: excited quarks () and excited leptons (). These heavy particles could also have spin-1/2 and spin-3/2 states. From the first publication on excited leptons in 1965 [17] until today, there have been plenty of phenomenological [18–38] and experimental [39–53] studies performed on excited fermions.

Excited states of SM quarks might be shown in four possible final states with light jets, , , , and . Currently, the LHC puts experimental mass limits for all four final state cases [47, 51–54] that are (6.0), (5.5), (5.0), and (4.7) TeV for ATLAS (CMS), respectively. Like SM fermions, excited fermions also have three families and we focused on and productions which decay to dijet final states.

After the LHC physics mission is over, a new and more powerful collider will take place as an energy frontier discovery machine for the high energy physics. At CERN located in Geneva, Future Circular Collider (FCC) [55] is planned for the next step with TeV. The other project, Super Proton-Proton Collider (SppC), is planned in China at multi-TeV center of mass (CM) energies [56]; we chose TeV option in this study. Both projects promise very high luminosity. The FCC will be expected to reach 2500 integrated luminosity in ten years (Phase I) and 15000 integrated luminosity in 15 years (Phase II) [57–59]. Overall in 25 years, total integrated luminosity will be 17500 . On the other hand, the SppC will deliver collisions with 22500 integrated luminosity in 15 years (see Table 1).

In this research, we explore spin-1/2 excited and quark ( and ) decaying into dijet final states at the FCC and the SppC. In the following sections, we state spin-1/2 excited quark interaction Lagrangian, decay widths, and cross section values in Section 2, signal-background analysis to determine cuts in Section 3, and attainable mass and compositeness scale () limits and conclusions in Section 4.

#### 2. Interaction Lagrangian, Decay Widths, and Cross Sections

When left- and right-handed components of excited quarks are assigned to isodoublets, isospin structure of the first generation SM and excited quarks will be

Since interaction Lagrangian is magnetic-moment type, it contains only left-handed quark doublet and consequently right-handed excited quark doublet. For that reason, as an effective interaction Lagrangian [20, 22, 25, 54], Equation (2) was utilized for the spin-1/2 excited quarks:where compositeness scale is represented as , denotes right-handed excited quark doublet, depicts ground state left-handed quark doublet, and field strength tensors are for gluon, for SU, and for U. , , and are color parameters for gluon-quark interaction, Pauli spin matrices, and weak hyper-charge, respectively. Gauge coupling constants are , , and ; and are free parameters that are taken as equal to 1 in numerical calculations. In addition, mentioned interactions with Higgs Boson as well as mass mixing among quarks and excited quarks can be neglected since ( is vacuum expectation value of Higgs field). Indeed, 6 TeV from the LHC data, 245 GeV, and is in MeV region.

Interaction Lagrangian Equation (2) was implemented into CalcHEP [60] software by using LanHEP interface [61, 62]. In our calculations, CTEQ6L1 [63, 64] parton distribution function was used and factorizations and renormalization scale were taken equal to .

Partial decay widths of first generation excited quarks are listed in Table 2. Parameters in the last column of Table 2 are , , , , and ; here is the third component of the weak isospin of . In Figure 1, total decay widths are given for , , and TeV by scanning excited quarks mass from 6 TeV to 100 TeV. Total decay widths of and are close to each other since dominant decay modes are and . There are small differences caused by and channels. It is obviously seen that while and mass values are risen, decay widths are increased.

For the following parts of this study, we consider three cases to do analysis: (a) , (b) , and (c) (degenerate state) with , , and signal processes, respectively (here, denotes ). 6 Feynman diagrams emerge for cases (a) and (b), and 12 Feynman diagrams make contributions to signal cross section calculations for the case (c). Figure 2 presents the case (a) Feynman diagrams for illustration. Analytical expression for the cross sections at parton level corresponding to these diagrams is described by Equation (3):

In Figure 3, first generation excited quarks cross section values for three cases mentioned above are plotted for the FCC ( TeV) and the SppC ( TeV) with , , (degenerate state), and TeV. When the compositeness scale value is taken as equal to excited quark masses, cross section values are about 300 times higher at 6 TeV mass value for both collider options. Indeed, it seems that excited quark could be produced at very high mass values for both collider options. It should be noted that as the LHC experimental studies on excited quarks with dijet final states do not consider SM interference contribution to cross section [47, 51], we did not simulate interference with SM for the FCC and SppC at this stage. For the same reason, QCD corrections were disregarded in this analysis [47, 52, 65–67].

#### 3. Signal and Background Analysis

Signal processes were defined in previous section. Background process which is used in calculation is ; here denotes and for three signal cases. It is important to determine transverse momentum (), pseudo rapidity (), and invariant mass () cut values for selecting clear signal. To illustrate cut selection, only final state particles distribution originated by excited quark plots are included in Figure 4. According to these figures, cuts are applied as 2 TeV, and cuts are determined as in signal and background cross section calculations for three cases. Also, the cone angle radius is chosen as for both colliders. Additionally, invariant mass cuts are applied as mass window for again both collider options; here denotes excited quarks (, , and ) mass and is total decay widths of the excited quarks.

In order to calculate statistical significance, Equation (4) is used:

where and denote signal and background cross section values, respectively, and represents integrated luminosity. Using Equation (4), we have calculated excited quarks mass’ discovery (), observation (), and exclusion () limits on prospective frontier machines, namely, FCC and SppC.

#### 4. Results and Conclusions

Discovery, observation, and exclusion limits on the mass of excited quarks for three cases depending on integrated luminosity of the FCC and SppC with cases are plotted in Figure 5. Attainable mass limits for all three cases for FCC-Phases I and II and SppC with their final integrated luminosity values at the end of operating times are listed in Table 3. It is seen that FCC-Phase I will afford an opportunity to discover, observe, or exclude degenerate case of excited quarks up to 40.1, 43.2, and 45.6 TeV, respectively. At the end of the FCC-Phase II, these values become TeV (), TeV (), and TeV (). On the other hand, corresponding values for SppC are TeV (), TeV (), and TeV () that essentially exceed the FCC limits.

As mentioned above, we did not anticipate interference of the signal model with the SM contribution. In order to estimate this contribution, we compared discovery limits for at the FCC-Phase II. As seen from Table 3, this limit was 44.1 TeV in our case. If one takes interference terms into account, discovery limit becomes 45.0 TeV. The latter value was obtained using same discovery cuts together with corresponding statistical signification equation, namely,

where includes signal, SM, and interference contributions. Interference terms lead to slightly higher discovery limit. Therefore, presented results in this study can be considered as a bit conservative.

Concerning the role of systematic uncertainties caused by choice of PDF, factorization and renormalization scales, analysis performed at the ATLAS and CMS experiments show that their impact is less than 1% for channel [47]. As for the efficiency of jet registration, it is nearly 100% for jets with above 20 GeV [47].

In principle, compositeness scale might be quite higher than excited quark mass. If excited and quarks are not discovered at FCC or SppC, one can evaluate lower limits on compositeness scale. For illustration, we plot achievable compositeness scale depending on , , and mass for ultimate luminosity values at both colliders in Figure 6. If it is assumed that mass equals 20 TeV and it is not seen at FCC in resonant channel, according to Figure 6, this means that compositeness scale is larger than 1.2 PeV (), 2.0 PeV (), and 3.0 PeV (). Achievable scales for other values of as well as and (degenerate state) are presented in Table 4. Similar results for the SppC are given in Table 5.

In Figure 7, necessary luminosities for observation and discovery of 20 TeV mass excited u quark depending on compositeness scale are plotted for both energy-frontier colliders. It is seen that if TeV, FCC will observe with integrated luminosity and is needed for discovery, which correspond to 12 and 19.5 operation years, respectively. Concerning the SppC, it will observe with 20 TeV mass within first year and discover it in 2 years if compositeness scale is equal to 1000 TeV.

In conclusion, FCC and SppC have excellent potential for discovery of excited and quarks. If compositeness scale coincides with excited quark masses, FCC reaches = 44 TeV, = 36 TeV, and = 46 TeV (degenerate state). Corresponding values for SppC are 58 TeV, 48 TeV, and 61 TeV, respectively. If compositeness scale is higher than excited quark masses, discovery of excited quarks will afford an opportunity to determine at the same time.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study is supported by TUBITAK under the grant No. [114F337]. Authors are grateful to the Usak University, Energy, Environment and Sustainability Application and Research Center for their support.