Signals of $W'$ and $Z'$ bosons at the LHC in the $SU(3) \times SO(5) \times U(1)$ gauge-Higgs unification

The $pp\to \{W, W'\} \to l\nu$ and $pp\to \{\gamma, Z, Z'\} \to l^+l^-$ ($l=e,\mu$) processes in the $SU(3)_C\times SO(5)\times U(1)_X$ gauge-Higgs unification (GHU) models are studied, where $W'$ and $Z'$ bosons are Kaluza-Klein (KK) exited states of the electroweak gauge bosons. From the experimental data collected at the Large Hadron Collider, constraints on the KK mass scale and the Aharonov-Bohm phase are obtained. One can explore the KK mass scale in the GUT inspired GHU model up to 18 TeV for the luminosity 300 fb$^{-1}$ and 22 TeV for the luminosity 3000 fb$^{-1}$ at $\sqrt{s}=14$ TeV.


Introduction
Experimental results at the Large Hadron Collider (LHC) collected at a center-of-mass energy √ s = 13 TeV during the year 2015-2018 have been presented by ATLAS [1][2][3][4] and CMS [5,6] groups. No direct signals of new physics beyond the standard model (SM) have been observed at the LHC so far. In many models such as the sequential standard model, left-right symmetric model, grand unified theories (GUT) and models with an extra dimension, there appear W or Z bosons [7][8][9][10]. Physics of W and Z bosons at the LHC has been an important subject .
The effects of Z bosons in the GHU B-model at the ILC have been studied in Ref. [53]. The deviation of the forward-backward asymmetry from the SM prediction is about −1% in the e + e − → µ + µ − process at the 250 GeV ILC with polarized left-handed electron beams for θ H 0.10, where the KK mass scale is 13 TeV. The deviation of the differential left-right asymmetry reaches to about −20% in the forward region with the same parameters. The A-model and B-model can be distinguished by the dependence in the forward-backward asymmetry and left-right asymmetry on the polarization of electron and positron beams, as the two models exhibit opposite dependence on the polarization.
Another specific feature of the GHU B-model is the appearance of the two step phase transitions at T ∼ 2.6 TeV and T = 163 GeV [54]. At sufficiently high temperature, the effective potential has a minimum at θ H = π. The two phases θ H = 0 and θ H = π become degenerate at T ∼ m KK , where the two phases have SU (2) L ×U (1) Y and SU (2) R ×U (1) Y symmetry, respectively. As the temperature becomes lower, the θ H = 0 state becomes the true vacuum and a first-order phase transition from θ H = π to θ H = 0 takes place at T ∼ 2.6 TeV. This transition is called the left-right phase transition. At T = 163 GeV, the electroweak symmetry breaking (EWSB) occurs and the Higgs boson acquires a vacuum expectation value (VEV). This electroweak phase transition is found to be weakly first order.
In this paper, the pp → lν and pp → l + l − (l = e, µ) processes in the GHU A-model and B-model at the LHC are studied. Because significant differences between observables in the SM and those in the GHU models at the ILC are predicted [48,49,53]

Model
The SU (3) C × SO(5) × U (1) X GHU A-model and B-model have been given in Ref. [46] and Ref. [50], in which the action, orbifold boundary conditions (BCs), wave functions and formulae to determine the mass spectrum of each field are explained. The details of the models are not repeated here. In this section, we briefly introduce the models and explain definitions of relevant parameters.
There are matter fields having no zero modes, which is referred as the dark fermions.
With specified parameters, masses and couplings are all determined. It has been shown that one can introduce the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the GHU Bmodel while flavor changing neutral currents (FCNCs) are naturally suppressed [51]. The tree-level FCNCs exist only in the down-type quark sector and the magnitude of their couplings is of the order of O(10 −6 ). In this paper, the CKM matrix is not introduced for simplicity.
As benchmark points, eight parameter sets are taken as shown in Table 2. These benchmark points are chosen for the following reason. In the A-model, there are two free parameters z L (or m KK ) and n F (the number of dark fermions). The effects of n F on physics of the gauge bosons, Higgs bosons, quarks and leptons turn out very small. Once a set of parameters (n F , z L ) is set, θ H is determined and couplings among these fields are determined by θ H . Hence a relevant free parameter is effectively θ H only. The value of n F affects a lower limit of θ H and a dark fermion mass. For a larger n F , the lower limit of θ H is smaller and the lowest mode of the dark fermion has a lower mass. We take n F = 4, where the lower limit of θ H is θ H 0.08.     Table 2, where the decay widths are calculated by the couplings shown in Tables 3, 4 A3    Table 2, where sin 2 θ 0 W = 0.2306. The value less than 10 −4 is written as 0.  The W (1) and W R couplings with right-handed fermions are tiny and negligible except for W (1) Rt R b R coupling. Because of the large W (1) couplings with left-handed fermions,   Table 2, where sin 2 θ 0 W = 0.2306. Other information is the same as in Table 4.   Table 2, where sin 2 θ 0 W = 0.2306. Other information is the same as in Table 4.

as shown in
The contribution from the higher KK modes are calculated in Ref. [53] and found to be small. Numerically, the deviation of the e + e − → µ + µ − cross section with the first and second KK modes from the deviation with only the first KK modes is O(1)% for √ s < 3 TeV. Thus we take only the first KK modes into account in the following.

Differential Cross Sections
In this section, formulae for the cross sections of the pp → l + l − and pp → lν processes are summarized [76,77].
The cross section of the pp → l + l − process, σ pp→l + l − , is written in terms of the partonlevel cross section σ ff →l + l − (s) as   Table 2, where sin 2 θ 0 W = 0.2305. Other information is the same as in Table 4.   Table 2, where sin 2 θ 0 W = 0.2307. Other information is the same as in Table 4.
where F f (x 1,2 , Q) are the parton distribution function (PDF) at an energy scale Q and we take Q = √ s in this paper. By introducing the invariant mass of the lepton pair denoted as m ll ≡ √ s, the differential cross section with respect to the invariant mass is written as Similarly, the cross section of the pp → lν process, σ pp→lν , is written as where σ ff →lν (s) is the parton-level cross section. The cross section of the pp → lν process is explored by measuring the transverse mass (m T ) or the transverse momentum of the charged-lepton (p T ). These two parameters are related as m T = 2p T E miss T (1 − cos φ), where E miss T is the missing energy and φ is the angle between the charged-lepton and missing transverse momentum in the transverse plane. For pp collisions at high energies, the transverse momentum of the parton may be ignored. φ π and the masses of the leptons are ignored, and one finds m T 2 p T . The transverse momentum is defined as p T ≡ ( √ s/2)| sin θ|. The total cross section is written by using the transverse mass as The differential cross section with respect to the transverse mass is

Constraints from LHC Experiments
Constraints on the GHU A-model from the early stage of LHC experiment are estimated in Refs [46,47]. In this section, we update the constraints by using the LHC run 2 data.
Constraints on the GHU B-model are obtained as well. We use CT10 [78] for the PDF and ManeParse [79] to numerically evaluate the cross sections.
First, we recall the constraints on the Z bosons from the pp → e + e − and µ + µ − processes. To see the θ H and m KK dependence, the differential cross sections dσ(pp →  Table 9. The observed (expected) upper limits at 95% confidence level (CL) on N sig are 4.4 (5.0) and 3.8 (4.0) for pp → e + e − and pp → µ + µ − , which disfavors the parameter sets B + and B L at 95% CL. The parameter sets A1 and A2 are disfavored by the expected upper limits at 95% CL for pp → e + e − and pp → µ + µ − .
Next, we consider the constraints on the W bosons from the pp → eν and µν processes.
We consider the constraint from the ATLAS group [3]. The CRs and SRs are not specified there. The differential cross sections in the GHU model are larger than that in the SM above M T ∼ 2300 GeV for the parameter sets (B L , B, B H ) and (B + , B    The discovery significance is given by [10,80] Although the formula (5.3) is derived in the large number limit, we use this formula independent of the numbers of events for simplicity. For a given Z, the corresponding p-value, which is defined as the probability of obtaining a larger excess, is the same as where F is the Gaussian cumulative distribution function, µ is a mean and σ is a standard deviation. Usually, an excess larger than 5σ is qualified as a discovery. Therefore, we estimate the parameter set which gives Z 5. The p = 0.05 corresponds to the discovery significance Z = 1.64, and a model is allowed at a 95% CL for Z < 1.64 [80]. We calculate the p-value by assuming that events follow the Poisson distribution.
In the A-model, the differential cross section of the pp → lν process is almost equal to that in the SM for m KK > 8 TeV at √ s = 14 TeV. For the parameter set A3, the differential cross section of pp → e + e − process is larger above m min = 2.287 TeV. With the luminosity 300 fb −1 , N GHU = 70.1, N SM = 28.8 and the corresponding discovery significance is 6.49. The numbers of events and discovery significance of pp → µ + µ − process are smaller than those of the pp → e + e − process.
For the B-model, the discovery significance of the pp → eν process is larger than that of the pp → µν and pp → l + l − processes for the same parameter sets considered bellow.
We choose parameter sets z L O(10 12 ) with integral m KK /TeV. The results for the pp → eν and pp → e + e − processes are summarized in Tables 11 and 12. The masses and decay widths of the W and Z for those parameters are shown in Tables 13, 14       We add that backgrounds coming from other processes, acceptance and efficiency have not been taken into account in the evaluation in this section.

Summary and Discussions
In this paper we studied the pp → {W, W } → lν and pp → {γ, Z, Z } → l + l − (l = e, µ) processes in the SU (3) C × SO(5) × U (1) X GHU models. Due to the behavior of the wave functions of various fields in the fifth dimension, the Z couplings of right-and left-  The differential cross sections of the pp → l + l − processes in the GHU models are smaller than those in the SM for the invariant mass m ll 2 TeV. From the searches for events in the dilepton final states at √ s = 13 TeV with up to 140 fb −1 of data [2], the A-model is constrained as θ H 0.08 and m KK 9.5 TeV, and the B-model is constrained as θ H 0.10 and m KK 13 TeV.
The differential cross sections of the pp → lν processes in the GHU B-model are also smaller than those in the SM for the transverse mass m T 2 TeV. The constraint on the       Table 13, where sin 2 θ 0 W = 0.2309.
with a right-handed electron beam at √ s = 250 GeV, where the statistical uncertainty with the 250 fb −1 luminosity data is about 0.1% [53]. To reduce theoretical uncertainties, further studies beyond the tree-level are necessary.
Collider physics of radions, KK gravitons and KK gluons are also important subjects in models defined on a higher dimensional spacetime [81,82]. For instance, KK gluons mediate dijet and tt production processes at hadron colliders. In the tt production processes, the forward-backward asymmetry [83,84] and the charge asymmetry [85,86] have been measured, which so far have been consistent with the SM predictions. Effects of KK gluons in the pp → tt process need be studied in the GHU models as well.