Sensitivity of Anomalous Quartic Gauge Couplings via $Z\gamma\gamma$ Production at Future hadron-hadron Colliders

Triple gauge boson production provides a promising opportunity to probe the anomalous quartic gauge couplings in understanding the details of electroweak symmetry breaking at future hadron-hadron collider facilities with increasing center of mass energy and luminosity. In this paper, we investigate the sensitivities of dimension-8 anomalous couplings related to the $ZZ\gamma\gamma$ and $Z\gamma\gamma\gamma$ quartic vertices, defined in the effective field theory framework, via $pp\to Z\gamma\gamma$ signal process with Z-boson decaying to charged leptons at the high luminosity phase of LHC (HL-LHC) and future facilities, namely the High Energy LHC (HE-LHC) and Future Circular hadron-hadron collider (FCC-hh). We analyzed the signal and relevant backgrounds via a cut based method with Monte Carlo event sampling where the detector responses of three hadron collider facilities, the center-of-mass energies of 14, 27 and 100 TeV with an integrated luminosities of 3, 15 and 30 ab$^{-1}$ are considered for the HL-LHC, HE-LHC and FCC-hh, respectively. The reconstructed 4-body invariant mass of $l^+l^-\gamma\gamma$ system is used to constrain the anomalous quartic gauge coupling parameters under the hypothesis of absence of anomalies in triple gauge couplings. Our results indicate that the sensitivity on anomalous quartic couplings $f_{T8}/\Lambda^{4}$ and $f_{T9}/\Lambda^{4}$ ($f_{T0}/\Lambda^{4}$, $f_{T1}/\Lambda^{4}$ and $f_{T2}/\Lambda^{4}$) at 95$\%$ C.L. for FCC-hh with $L_{int}$ = 30 ab$^{-1}$ without systematic errors are two (one) order better than the current experimental limits. Considering a realistic systematic uncertainty such as 10$\%$ from possible experimental sources, the sensitivity of all anomalous quartic couplings gets worsen by about 1.2$\%$, 1.7$\%$ and 1.5$\%$ compared to those without systematic uncertainty for HL-LHC, HE-LHC and FCC-hh, respectively.


I. INTRODUCTION
The Standard Model (SM) puzzle was completed with the simultaneous discovery of the scalar Higgs boson, predicted theoretically in the SM, at the CERN Large Hadron Collider (LHC) by both ATLAS and CMS collaborations [1,2]. With the discovery of this particle, the mechanism of electroweak symmetry breaking (EWSB) has become more important and still continues to be investigated. The self-interaction of the triple and quartic vector boson couplings is defined by the non-Abelian structure of the ElectroWeak (EW) sector within the framework of the SM. Any deviation in the couplings predicted by the EW sector of the SM is not observed yet with the precision measurements. While the experimental results are consistent with the couplings of W ± to Z boson, there is no experimental evidence of Z bosons coupling to photons. Therefore, studying of triple and quartic couplings can either confirm the SM and the spontaneous symmetry breaking mechanism or provide clues for the new physics Beyond Standard Model (BSM). Anomalous triple and quartic gauge boson couplings are parametrized by higher-dimensional operators in the Effective Field Theory (EFT) that can be explained in a model independent way of contribution of the new physics in the BSM. The anomalous triple gauge couplings are modified by integrating out heavy fields whereas the anomalous Quartic Gauge Couplings (aQGC) can be related to low energy limits of heavy state exchange. In this scenario, the SU (2) L U (1) Y is realized linearly and the lowest order Quartic Gauge Couplings are given by the dimension-eight operators [3,4]. These operators are so-called genuine QGC operators which generate the QGC without having TGC associated with them.
Many experimental and phenomenological studies have been carried out and revealed constraints about aQGC. Both vector-boson scattering processes (i.e. ZZjj and Zγjj process ) and triboson (i.e, Zγγ production) production are directly sensitive to to the quartic ZZγγ and Zγγγ vertices.
The new era starting with the novel machine configuration of the LHC and beyond aims to decrease the statistical error by increasing center of mass energy and luminosity in the measurements of the Higgs boson properties as well as finding clues to explain the physics beyond SM. With the configurations of beam parameters and hardware, the upgrade project HL-LHC will achieve an approximately 250 fb −1 per year to reach a target integrated luminosity of 3000 fb −1 at 7.0 TeV nominal beam energy of the LHC in a total of 12 years [67]. The other considered post-LHC hadron collider which will be installed in existing LHC tunnel is HE-LHC that is designed to operate at √ s= 27 TeV center-of-mass energy with an integrated luminosity of at least a factor of 5 larger than the HL-LHC [68]. As stated in the Update of the European Strategy for Particle Physics by the European Strategy Group, it is recommended to investigate the technical and financial feasibility of a future hadron collider at CERN with a centre-of-mass energy of at least 100 TeV.
The future project currently under consideration by CERN which comes to fore with infrastructure and technology as well as the physics opportunities is the Future Circular Collider (FCC) Study [69].
The goal of our study is to investigate the effects of anomalous quartic gauge couplings on ZZγγ and Zγγγ vertices via pp → Zγγ process where Z boson subsequently decays to e or µ pairs at HL-LHC, HE-LHC and FCC-hh. The rest of the paper is organized as follows. A brief review of theoretical framework that discusses the operators in EFT Lagrangian is introduced in Section II. The event generation tools as well as the detail of the analysis to find the optimum cuts for separating signal events from different source of backgrounds is discussed in Section III. In section IV, we give the detail of method to obtain sensitivity bounds on anomalous quartic gauge couplings, and then determine them with an integrated luminosity L int = 3 ab −1 , 15 ab −1 , 30 ab −1 for HL-LHC, HE-LHC and FCC-hh, respectively. Finally, we summarize our result and compare obtained limits to the current experimental results in Section V.

COUPLINGS
Although there is no contribution of the quartic gauge-boson couplings of the ZZγγ and Zγγγ vertices to the Zγγ production in the SM, new physics effects in the cross section of Zγγ production can be searched with high-dimensional effective operators which describe the anomalous quarticgauge boson couplings without triple gauge-boson couplings. These neutral aQGCs couplings are modeled by either linear or non-linear representations using an EFT [70][71][72]. In the non-linear representation, the electroweak symmetry breaking is due to no fundamental Higgs scalar whereas in the linear representation, it can be broken by the conventional SM Higgs mechanism. With the discovery of the Higgs boson at the LHC, it becomes important to study the anomalous quartic gauge couplings based on linear representation. In this representation, the parity conserved and charge-conjugated effective Lagrangian include the dimension-eight effective operators by assuming the SU (2)×U (1) symmetry of the EW gauge field, with a Higgs boson belongs to a SU (2) L doublet.
In this approach, the lowest dimension of operators which leads to quartic interactions but do not include two or three weak gauge boson vertices are expected to be eight. Therefore, the three where Λ is the scale of new physics, and f S,j , f M,j and f T,j represent coefficients of relevant effective operators. These coefficients are zero in the SM prediction. The expanded form of these operators and a complete list of quartic vertices modified by these operators are given in Appendix A.
Among the f M,x and f T,x operators that affect the ZZγγ and Zγγγ vertices, f M,x operators in the production of Zγγ at future of hadron-hadron colliders with high center of mass energies and luminosities were examined and their limit values were predicted in Ref [29]. Therefore, we focus on the five coefficients f T 0 /Λ 4 , f T 1 /Λ 4 , f T 2 /Λ 4 , f T 8 /Λ 4 and f T 9 /Λ 4 of the operators containing four field strength tensors for this study. Especially f T 8 /Λ 4 , and f T 9 /Λ 4 give rise to only neutral anomalous quartic gauge vertices. The effective field theory is only valid under the new physics scale in which unitarity violation does not occur. However, high-dimensional operators with nonzero aQGC can lead to a scattering amplitude that violates unitarity at sufficiently high energy values, called the unitarity bound. The value of the unitarity bound for the dimension-8 operators is determined by using a dipole form factor ensuring unitarity at high energies as: whereŝ is the maximum center-of-mass energy, Λ F F is the energy scale of the form factor. The maximal form factor scale Λ F F is calculated with a form factor tool VBFNLO 2.7.1 [73] for a given input of anomalous quartic gauge boson couplings parameters. The VBFNLO utility determines form factor using the amplitudes of on-shell VV scattering processes and computes the zeroth partial wave of the amplitude. The real part of the zeroth partial wave must be below 0.5 which is called the unitarity criterion. All channels the same electrical charge Q in V V → V V scattering (V = W/Z/γ) are combined in addition to individual check on each channel of the V V system.
The calculated Unitarity Violation (UV) bounds using the form factor tool with VBFNLO as a function of higher-dimensional operators considered in our study are given in Fig.1. The unitarity is safe in the region that is below the line for each coefficients.
The limit values with no unitarization restriction ( Λ F F = ∞) on the dimension-8 aQGC

obtained by ATLAS and CMS collaborations by
/Λ 4 and f T 9 /Λ 4 obtained from analysis of Zγγ production as well as different production channels by ATLAS and CMS collaborations.
setting all other anomalous couplings to zero are summarized in Table I. The limits on The current best limits obtained by CMS collaboration for different production channels on /Λ 4 and f T 9 /Λ 4 couplings are also presented in last column of the Table I. The best limits obtained on in association with two jets production [23] at a center-of-mass energy of 13 TeV with an integrated luminosity 35.9 fb −1 . They also reported limits on f T 8 /Λ 4 from production of two jets in association with two Z boson [24] and f T 9 /Λ 4 from electroweak production of a Z boson, a photon and two forward jets production [28] at a center-of-mass energy of 13 TeV with an integrated luminosity 137 fb −1 .

III. EVENT SELECTION AND DETAILS OF ANALYSIS
The details of the analysis are given for the effects of dimension-8 operators on anomalous quartic of cross section with respect to one of the coupling while others are kept at zero. The sensitivity of f T 8 /Λ 4 and f T 9 /Λ 4 anomalous quartic couplings for each collider options to cross sections is more significant than the other couplings as seen in the second row of Fig. 3. Therefore, we expect to obtain better limits on the f T 8 /Λ 4 and f T 9 /Λ 4 couplings with Zγγ production. The general expression for amplitude in the EFT regime for the process considered can be written as where |M SM | 2 , (M SM M * dim8 ) and |M dim8 | 2 are the SM, interference of the SM amplitude with higher dimensional operators, and the square of the new physics contributions, respectively. In order to show the effectiveness of the form factor, the cross sections at LO without and with Λ F F =1.5 and 2 TeV is presented for all three collider options in Fig.4. It can be clearly seen in Fig. 4 that the square contributions of the new physics amplitudes suppress the interference contributions of the SM amplitude with high-dimensional operators in the case where the UV limit is not applied.
However, if the new physics energy scale is heavy (i.e. Λ F F =1.5 and 2 TeV or higher), the largest new physics contribution to pp → Zγγ process is expected from the interference between the SM and the dimension-eight operators as seen from Fig. 4. For further analysis including response of the detector effects, we generate 600k events for all backgrounds and signal processes where we scan each  [80] where a cone radius is set as ∆R = 0.4 (0.2) and p j T >15 (25) GeV for HL-LHC and HE-LHC (FCC-hh) colliders.
Our main focus is to see the effects of anomalous quartic gauge boson couplings via pp → Zγγ signal process where Z boson subsequently decays to e or µ pairs. Therefore, events with two isolated photons and one pair of the same flavor and oppositely charged leptons (electrons or muons) are selected for further analysis (Cut-0). Electron and muon channels are combined to increase sensitivity even more. The signal includes nonzero effective couplings and SM contribution as well as its interference. "sm" stands for SM background process of the same final state with the signal process in our analysis. The main background processes to the selected l + l − γγ sample of events may originate from Zγj and Zjj production with hadronic jet misidentified as a photon. Such misidentifications generally arise from jets hadronizing with a neutral meson, which carries away most of the jet energy. The photons that carry a large fraction of the jet energy can exceed the reconstructed HL-LHC FCC-hh   the final-state charged leptons and photons as given Cut-1 and Cut-2 in Table II for each collider options. Since the signal event contains two photons, we can safely suppress the event contamination and avoid infrared divergences by using a minimum transverse momentum and pseudo-rapidity cuts of the leading and sub-leading photons for the other background processes. Furthermore, the normalized distributions of leading and sub-leading photons (∆R(γ 1 , γ 2 )), leading photon and leading charged lepton (∆R(γ 1 , l 1 )), leading and sub-leading charged leptons ( ∆R(l 1 , l 2 )) separation in the pseudorapidity-azimuthal angle plane as well as the invariant mass of the oppositely signed charged lepton pair are given in Fig.7 (Fig.10) for HL-LHC (FCC-hh). To have well-separated photons and charged leptons in the phase space that leads to be identified separate objects in the detector, we require separations as ∆R(γ 1 , γ 2 )> 0.4, ∆R(γ 1 , l 1 )> 0.4 and ∆R(l 1 , l 2 )< 1.4 (Cut- 3). We also impose the invariant mass window cut around the Z boson mass peak as 81 GeV < M l + l − < 101 GeV (Cut-4) to suppresses the virtual photon contribution to the di-lepton system.
Since requiring the high transverse momentum photon eliminates the fake backgrounds, we plot the transverse momentum of leading photon for HL-LHC, HE-LHC and FCC-hh in Fig.11 (left to right ) to define a region which is sensitive to aQGC. From these normalized plots we apply a cut on p γ 1 T as 160 GeV, 250 GeV and 300 GeV for each collider option, respectively (Cut-5). The flow of cuts are summarized in Table II for each hadron-hadron colliders that we analyzed. The normalized number of events after applied cuts are presented in Table III  and FCC-hh, respectively. The distribution of the reconstructed 4-body invariant mass of l + l − γγ system for HL-LHC, HE-LHC and FCC-hh options given in Fig. 12 (top to bottom, respectively). is As seen from Fig.12, the applied UV bounds impose an upper-cut in the invariant mass of the l + l − γγ system which guarantees that the unitarity constraints are always satisfied. The number of events after applying UV bounds for signals are given in the parenthesis at Table III for comparison  TABLE III: The cumulative number of events for  In order to obtain a continuous prediction for the anomalous quartic gauge couplings after Cut-5, a quadratic fit is performed to number of events for each couplings ( n bins i N N P i ) obtained by integrating the invariant mass distribution of l + l − γγ system in Fig.12. The obtaining of the 95% Confidence Level (C.L.) limit on a one-dimensional aQGC parameter is performed by χ 2 test which corresponds to 3.84 leading leptons (γ 2 ) after the event selection(Cut-0) for the signals and all relevant backgrounds processes at FCC-hh with L int = 30 ab −1 where N N P i is the total number of events in the existence of aQGC, N B i is total number of events of the corresponding SM backgrounds in ith bin, ∆ i = δ 2 sys + 1  Table IV for HL-LHC, HE-LHC and FCC-hh collider options. This table also presents limits with δ sys = 0, 3%, 5% and 10% systematic errors as well as the unitarity bounds defined as the scattering energy at which the aQGC coupling strength is set equal to the observed limit. Following the second way discussed above, we present obtained limits at 95% C.L. with UV bound applied on the invariant mass distribution of l + l − γγ system for HL-LHC, HE-LHC and FCC-hh collider options without systematic errors in Fig. 13 where the impact of the UV bound can be seen. The limits on the aQGCs with UV bounds get worsen as expected since the interference of the SM amplitude with the dimension-eight operators suppresses the square contribution of new physics amplitude.
We point out that our numerical results for the case f the same production channel used in our analysis [7,9], our obtained limits for all aQGC are one to three order of magnitude better as can be seen from the comparison of Table I and Table IV  of pp → Zγγ process with leading-order (LO) or next to leading order (NLO) predictions [85] and higher order EW corrections, the uncertainty in integrated luminosity as well as electrons and jets misidentified as photons. In our study, we focus on LO predictions but do not investigate the impact and validity of these higher-order corrections on the signal and SM background processes.
Since the main purpose of this study is not to discuss sources of the systematic uncertainty in detail but to investigate the overall effects of the systematic uncertainty on the limits values of aQGC, we consider three different scenarios of systematic uncertainty. The 95% C.L. limit values without systematic uncertainties and with three different scenarios of systematic uncertainties as δ sys = 3%, 5% and 10% for three different collider options are quoted in Table 5. The limits on the aQGC considered in this study with systematic error are weaker slightly when realistic systematic error is considered, e.g., compared with a 10% systematic error and without systematic error, sensitivity of f T 9 /Λ 4 gets worsen by about 1.2%, 1.7% and 1.5% for HL-LHC, HE-LHC and FCC-hh, respectively.
We expect two (one) order of magnitude better limits on f ZZγγ and Zγγγ vertices for 95% C.L. has performed with three systematic uncertainty scenario δ sys = 3%, 5% and 10% using the l + l − γγ invariant mass system distributions. Since O T 8 and O T 9 give rise to aQGC containing only the neutral electroweak gauge bosons among the anomalous quartic operators, we reach the remarkable sensitivity on especially f T 8 /Λ 4 and f T 9 /Λ 4 couplings for HE-LHC and FCC-hh options comparing with current experimental results as seen from Table   IV iii ) Eight operators containing field strength tensors only are as follows A complete list of corresponding quartic gauge boson vertices modified by dimension-8 operators is given in Table V.