Excited muon searches at the FCC based muon-hadron colliders

We study the excited muon production at the FCC based muon-hadron colliders. We give the excited muon decay widths and production cross section. We deal with the $\mu p\rightarrow\mu^{\star}q\rightarrow\mu\gamma q$ process and we plot the transverse momentum, rapidity and invariant mass distributions of final state particles to get the discovery cuts. By using discovery cuts, we get the mass limits for excited muons. It is shown that the discovery limits on the mass of $\mu^{\star}$ are 2.2 TeV, 5.9 TeV and 7.5 TeV for $\mu63$-FCC, $\mu750$-FCC and $\mu1500$-FCC, respectively.


I. INTRODUCTION
Discovery of the Higgs boson by ATLAS and CMS collaborations in 2012 [1,2] has proved the accuracy and reliability of the Standard Model (SM) of the particle physics. But, many questions about dark matter, supersymmetric particles, extra dimensions, neutrino masses, asymmetry between matter and anti-matter, existence of new fundamental interactions, and fermion substucture are keeping their mystery and waiting to be solved. Many theories beyond the SM (BSM) have been proposed for these puzzling phenomena. Evidently, it is necessary to perform the particle physics experiments in more powerful colliders with higher energies and luminosities.
Compositeness is one of the BSM models that intend to solve the problem of fermionic families replication, by introducing more fundamental matter constituents called preons.
Excited fermions are predicted by preonic models and their existence would be a strong evidence for fermion substructure [3][4][5]. If known quarks and leptons present composite structures, reasonable explanations could be given for the still unanswered questions about the number and replication of SM families and their mass hierarchy. The appearence of excited states is an indisputable consequence of composite structure of known fermions [6][7][8][9]. In composite models, SM fermions are considered as ground states of a rich and heavier spectrum of excited states. Charged (e ⋆ , µ ⋆ , τ ⋆ ) and neutral (ν ⋆ e , ν ⋆ µ , ν ⋆ τ ) excited leptons come on the scene in the framework of composite models. Excited leptons with spin-1/2 and weak-isospin-1/2 are considered as the lowest radial and orbital excitations. Excited states with higher spins also appear in composite models [10][11][12][13][14].
Excited electrons (e ⋆ ) are extensively investigated in the field of excited leptonic state studies. To perform a main comparison it is necessary to study the other charged excited leptons (µ ⋆ and τ ⋆ ). In principle, µ ⋆ and τ ⋆ contributions would be differ from e ⋆ contribution in the mass and decay products of the SM leptons.
Enormous efforts are being made for the research and development of new particle colliders for the Large Hadron Collider (LHC) era and post-LHC era. A staged approach will be taken into consideration for the planning of these energy frontiers. The first stage is low-energy lepton colliders to make the precision measurements of the LHC discoveries.
These projects are the International Linear Collider (ILC) [52] with a center-of-mass energy of √ s = 0.5 TeV and low-energy muon collider (a µ + µ − collider, shortly µC) [53].  [54,55], and a hypothetical µp collider µ-LHC at this stage. The ILC with an increased center-of-mass energy ( √ s = 1 TeV), the Compact Linear Collider (CLIC) [56] with an optimal center-of-mass energy of 3 TeV, and the Plasma Wake-Field Accelarator-Linear Collider project (PWFA-LC) [57] are high-energy linear e + e − colliders under consideration to be built after the LHC. On the side of muon colliders, µC with √ s up to 3 TeV is planned as a high-energy muon collider [53].
The Future Circular Collider (FCC) [58] project investigates the various concepts of the circular colliders at CERN for the post-LHC era. The FCC is proposed as the future pp collider with √ s = 100 TeV and supported by European Union within the Horizon 2020 Framework Programme for research and innovation. Besides the pp option, it is also being planned to include the e + e − collider option (TLEP or FCC-ee) [59] and several ep collider options [60,61].
Building a muon collider as dedicated µ-ring tangential to the FCC will give opportunity to handle multi-TeV scale µp and µA colliders [62,63]. Assumed values for muon energy, center-of-mass energy, and average instantaneous luminosity for different FCC-based µp collider options are given in Table I. Excited muon searches would provide complementary information for the compositeness studies. This work is dedicated to search for excited muons at future FCC-based muonproton colliders. We introduce the effective Lagrangian responsible for the gauge interactions of excited muons and give their decay widths in Section II. Production cross-sections and the analysis for the µ ⋆ → µγ decay mode are presented in Section III. We summarized our results in Section IV.

II. EFFECTIVE LAGRANGIAN
A spin-1/2 excited lepton is the lowest radial and orbital excitation according to the classification by SU(2) × U(1) quantum numbers. Interactions between excited spin-1/2 leptons and ordinary leptons are of magnetic transition type [15,16,64]. The effective Lagrangian for the interaction between a spin-1/2 excited lepton, a gauge boson (V = γ, Z, W ± ), and the SM lepton is given by where Λ is the new physics scale, W µν and B µν are the field strength tensors, τ denotes the Pauli matrices, Y is the hypercharge, g and g ′ are the gauge couplings, and f and f ′ are the scaling factors for the gauge couplings of SU(2) and U(1); σ µν = i(γ µ γ ν − γ ν γ µ )/2 with γ µ being the Dirac matrices. An excited lepton has three possible decay modes: radiative decay l ⋆ → lγ, neutral weak decay l ⋆ → lZ , and charged weak decay l ⋆ → νW . Neglecting the SM lepton mass, we find the decay width of excited leptons as where f V is the new electroweak coupling parameter corresponding to the gauge boson the weak mixing angle, and m V is the mass of the gauge boson, and m * is the mass of the excited lepton. Total decay widths of excited leptons for Λ = m ⋆ and Λ = 100 TeV are given in Figure 1.

III. EXCITED MUON PRODUCTION AT µp COLLIDERS
The FCC-based µp colliders will provide the potential reach for excited muon searches through the µp → µ ⋆ X process. Feynman diagrams for the subprocesses µq(q) → µ ⋆ q(q) are shown in Figure 2. We implemented excited muon interaction vertices in high-energy physics simulation programme CALCHEP [65][66][67] and used it in our calculations. Total cross-section for the process µp → µ ⋆ X as a function of the excited muon mass is shown in Figure 3. We used the CTEQ6L parton distribution function in our calculations.  For the analysis we take into account the µγ decay mode of the µ ⋆ . We deal with the process µp → µ ⋆ X → µγX (subprocess µq(q) → µγq(q)) and impose generic cuts, p T > 20 Since pseudorapidity is defined to be η = −ln(tan(θ/2)), where θ is the polar angle, it is concluded that excited muons are produced mostly in the backward direction.            By examining these distributions we determine the discovery cuts presented in Table 2.
To determine these discovery cuts we specify the optimal regions where we cut off the most of the background but at the same time do not affect the signal so much. Since we choose the µ ⋆ → µγ decay mode of the excited muon (try to identify the excited muons through its decay products), no further cut is made on jets. Collider The invariant mass distributions following these cuts are shown in Figure 10. We define the statistical significance of the expected signal yield as where σ S denotes cross-section due to the excited muon production and σ B denotes the SM cross-section, L int is the integrated luminosity of the collider, and ǫ is the selection efficiency to detect the signal in the chosen channel (ǫ is assumed to be the same both on signal and on background). Taking into account the criteria SS > 3 (95% CL) and SS > 5 (99% CL), we derive the mass limits for excited muons. Our results are summarized in Table 3.