Probing neutral triple gauge couplings via Z\gamma ({\ell }^{+}{\ell }^{-}\gamma ) production at e^{+}{e}^{-} colliders

Abstract

Neutral triple gauge couplings (nTGCs) are absent in the Standard Model (SM) and at the dimension-6 level in the Standard Model Effective Field Theory (SMEFT), arising first from dimension-8 operators. As such, they provide a unique window for probing new physics beyond the SM. These dimension-8 operators can be mapped to nTGC form factors whose structure is consistent with the spontaneously-broken electroweak gauge symmetry of the SM. In this work, we study the probes of nTGCs in the reaction e^{+}{e}^{-}\to Z\gamma with Z\to {\ell }^{+}{\ell }^{-}(\ell =e,\mu ) at an e^{+}{e}^{-} collider. We perform a detector-level simulation and analysis of this reaction at the Circular Electron Positron Collider (CEPC) with collision energy \sqrt{s}=240 GeV and an integrated luminosity of 20 ab⁻¹. We present the sensitivity limits on probing the new physics scales of dimension-8 nTGC operators via measurements of the corresponding nTGC form factors.

Graphical abstract

Keywords

anomalous gauge coupling / future colliders / Zγ / effective field theory / neutral triple gauge coupling

1 Introduction

The Standard Model Effective Field Theory (SMEFT) [1] is a powerful framework for studying model-independently possible new physics beyond the Standard Model (SM). The SMEFT includes only the known elementary particles, which are assumed to have the SM quantum numbers and thus have the interactions with mass-dimension d\le 4 that are predicted by the SM, but the SMEFT also includes additional effective interactions with mass-dimensions d>4. Such higher-dimensional interactions could arise from new physics at energy scales beyond the electroweak scale due to possible exchanges of new massive particles and/or novel strong dynamics. The SMEFT interactions with dimension 5 may be relevant for neutrino physics, whereas collider experiments are generally sensitive to SMEFT interactions with even dimensions d\ge 6. Probing the effects of SMEFT operators may either constrain the possible high-scale new physics dynamics or provide hints to its possible nature, without assuming the ultraviolet (UV) origin or making any assumptions about its form.

There is an extensive theoretical literature classifying the SMEFT operators of dimension 6 [2, 3] and above [4, 5], and a growing number of phenomenological and experimental papers analyzing the constraints on their possible coefficients that are imposed by current data from the LHC and elsewhere. Most of these analyses have had operators with d=6 as their primary focus, often working to linear order in the SMEFT operator coefficients, i.e., quadratically in the new physics scale, an approximation that takes into account their interference with SM interactions [6-10]. To date there is no significant indication that any d=6 SMEFT operator has a non-zero coefficient, but future colliders will provide much greater precision in SMEFT probes [11-14].

A complete analysis of the phenomenology of dimension-6 operators should include their quadratic effects on event rates, which depend quartically on the new physics scale. At this level one should in general consider the effects of linear interference between dimension-8 SMEFT operators and SM amplitudes, which also depend quartically on the new physics scale, and there is a growing literature of analyses that take these into account [15-19]. Complementing these studies, it is interesting to consider processes that have no dimension-6 operator contributions, to which dimension-8 operators make the leading SMEFT contributions. These processes include quartic neutral vector-boson interactions and also neutral triple gauge couplings (nTGCs), where the nTGCs are the object of the present study.

Neutral triple gauge couplings are absent in the SM and at the level of dimension-6 operators in the SMEFT, arising first at the level of dimension-8 operators [20]. Hence nTGCs can provide a unique window for probing new physics beyond the SM [21-26]. The most direct experimental probes of nTGCs are via measurements of the corresponding form factors. A consistent formulation of nTGC form factors has recently been proposed, which matches precisely the nTGC form factors with the gauge-invariant dimension-8 effective operators of the SMEFT [23, 24]. This imposes nontrivial relations among the nTGC form factors and gives correct predictions for the contributions of the nTGC form factors to high-energy scattering amplitudes [23, 24]. These theoretical papers investigated probes of the nTGCs at both the electron−positron and hadron colliders.

In this work, we study experimental probes of the dimension-8 nTGC operators via measurements of their corresponding nTGC form factors in the reaction e^{+}{e}^{-}\to Z\gamma process with Z\to {\ell }^{+}{\ell }^{-}(\ell =e,\mu ) decays, as shown in Fig.1. For this purpose we perform detector-level simulation and analysis of nTGCs at the Circular Electron Positron Collider (CEPC) with energy \sqrt{s}= 240 GeV and an integrated luminosity of 20 ab{}^{-1}, using a model-independent approach that could also be adopted for other experiments.

This work is organized as follows. In Section 2, we first describe the theoretical framework for the nTGCs, which includes the SMEFT formulation of the dimension-8 nTGC operators and the corresponding nTGC form factors. Then, we present a detector-level simulation and analysis for the dimension-8 nTGC effective operators and the nTGC form factors via the reaction e^{+}{e}^{-}\to Z\gamma, using the CEPC detector as a benchmark. In Section 3, we analyze the uncertainties for both the signals and backgrounds. After this, we present our results in Section 4 for the sensitivities on probing the nTGCs at the CEPC. Finally, we conclude in Section 5.

2 Theoretical framework, simulation and analysis

In this section, we first present the theoretical framework for the nTGCs, including both the SMEFT formulation with dimension-8 nTGC operators and the corresponding nTGC form factors. Then, we systematically perform a detector-level simulation and analysis for the dimension-8 nTGC effective operators and the corresponding nTGC form factors via the reaction e^{+}{e}^{-}\to Z\gamma, using the CEPC as a benchmark.

2.1 Theoretical framework for the nTGCs

The dimension-8 SMEFT effective Lagrangian takes the following form:

(1){\mathcal{L}}_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{F}\mathrm{T}}^{}= {\displaystyle \sum _j{\displaystyle \frac{c_j}{{\mathrm{\Lambda }}^4}}{\mathcal{O}}_j^{}={\displaystyle \sum _j{\displaystyle \frac{\text{sign}(c_j^{})}{{\mathrm{\Lambda }}_j^4}}{\mathcal{O}}_j^{},}}

where the \{c_j^{}\} are dimensionless coefficients that may be \mathcal{O}( 1) that can have either sign. The effective cutoff \mathrm{\Lambda } for the new physics scale is connected to {\mathrm{\Lambda }}_j^{} via {\mathrm{\Lambda }}_j^{}\equiv \mathrm{\Lambda }/|c_j{|}^{1/4}.

In the present analysis we consider the following set of CP-conserving dimension-8 nTGC operators ({\mathcal{O}}_{G+},{\mathcal{O}}_{G-}, {\mathcal{O}}_{\tilde{B}W},{\mathcal{O}}_{\widetilde{BW}}) [22-24]:

(2a)g{\mathcal{O}}_{G+}={\tilde{B}}_{\mu \nu }{W}^{\alpha \mu \rho }(D_{\rho }{D}_{\lambda }{W}^{\alpha \nu \lambda }+D^{\nu }{D}^{\lambda }{W}_{\lambda \rho }^{\alpha }),

(2b)g{\mathcal{O}}_{G-}={\tilde{B}}_{\mu \nu }{W}^{a\mu \rho }(D_{\rho }{D}_{\lambda }{W}^{a\nu \lambda }-D^{\nu }{D}^{\lambda }{W}_{\lambda \rho }^a),

(2c){\mathcal{O}}_{\tilde{B}W}=\mathrm{i}{H}^{†}{\tilde{B}}_{\mu \nu }{W}^{\mu \rho }\left\{D_{\rho },D^{\nu }\right\}H+\mathrm{h}.\mathrm{c}.,

(2d){\mathcal{O}}_{\widetilde{BW}}^{}=\mathrm{i}{H}^{†}(D_{\sigma }^{}{\tilde{W}}_{\mu \nu }^a{W}^{a\mu \sigma }+D_{\sigma }^{}{\tilde{B}}_{\mu \nu }{B}^{\mu \sigma })D^{\nu }H +\text{h.c.}.

The nTGC vertex Z\gamma V^{\ast } (V=Z,\gamma) can be expressed in terms of nTGC form factors ( h_3^V,h_4^V) as follows [23, 24]:

(3)\begin{array}{rl}{\mathrm{\Gamma }}_{Z\gamma V\ast }^{\alpha \beta \mu (8)}(q_1^{},q_2^{},q_3^{})=& \frac{e(q_3^2-M_v^2)}{M_Z^2}[(h_3^V+\frac{h_4^V}{2M_Z^2}{q}_3^2)q_{2\nu }^{}{ϵ}^{\alpha \beta \mu \nu }\\ & +\frac{h_4^V}{M_Z^2}{q}_2^{\alpha }{q}_{3\nu }{q}_{2\sigma }{ϵ}^{\beta \mu \nu \alpha }].\end{array}

By matching this nTGC form factor formulation with the corresponding gauge-invariant dimension-8 nTGC operators, a nontrivial form factor relationship can be derived, h_4^Z= \frac{c_W^{}}{s_W^{}}{h}_4^{\gamma } [23, 24], and henceforth we will denote h_4^Z\equiv h_4^{} for simplicity. Thus there are three independent form-factor parameters (h_4^{},h_3^Z ,h_3^{\gamma }) [23, 24], which can be determined by matching the gauge-invariant dimension-8 nTGC operators ({\mathcal{O}}_{G+},{\mathcal{O}}_{G-}, {\mathcal{O}}_{\tilde{B}W},{\mathcal{O}}_{\widetilde{BW}}) in the broken phase of the electroweak gauge group SU(2){}_L^{}\otimesU(1){}_Y^{}. The form factors (h_4,h_3^Z,h_3^{\gamma }) are connected as follows to the cutoff scales ({\mathrm{\Lambda }}_{G+},{\mathrm{\Lambda }}_{G-}, {\mathrm{\Lambda }}_{\tilde{B}W},{\mathrm{\Lambda }}_{\widetilde{BW}}) of the corresponding dimension-8 nTGC operators [23, 24]:

(4a)h_4=-{\displaystyle \frac{1}{[{\mathrm{\Lambda }}_{G+}^4]}}{\displaystyle \frac{v^2{M}_Z^2}{s_W{c}_W}} ,

(4b)h_3^Z={\displaystyle \frac{1}{[{\mathrm{\Lambda }}_{\tilde{B}W}^4]}}{\displaystyle \frac{v^2{M}_Z^2}{2s_W{c}_W}},

(4c)h_3^{\gamma }=-{\displaystyle \frac{1}{[{\mathrm{\Lambda }}_{G-}^4]}}{\displaystyle \frac{v^2{M}_Z^2}{2c_W^2}}=-{\displaystyle \frac{1}{[{\mathrm{\Lambda }}_{\widetilde{BW}}^4]}}{\displaystyle \frac{v^2{M}_Z^2}{s_W^{}{c}_W^{}}},

where we denote [{\mathrm{\Lambda }}_j^4]=\text{sign}(c_j ){\mathrm{\Lambda }}_j^4 and [ {\mathrm{\Lambda }}_j^{-4}]=\text{sign}( c_j){\mathrm{\Lambda }}_j^{-4}.

In the following, we perform a systematic detector-level simulation and analysis of sensitivities to the nTGC dimension-8 effective operators via measurements of the nTGC form factors in the reaction e^{+}{e}^{-}\to Z\gamma at the CEPC.

2.2 CEPC detector

The Circular Electron Positron Collider (CEPC) [27, 28] is an international research facility proposed in China that is designed to meet the requirements of various physics studies, especially precision measurements. CEPC has well-defined momentum and energy, as well as a clean experimental environment in comparison with hadron colliders. The energy resolution for the electromagnetic calorimeter is at 16%\sqrt{\text{E/GeV}}\oplus1%, and for the hadronic calorimeter and muon detector is at 60%\sqrt{\text{E/GeV}}\oplus1%. The angular resolution is set to be less than 0.1 \text{mrad} [27, 29]. Thus it is possible to reconstruct angular variables in a more accurate way. Hence, CEPC is an ideal facility for probing new physics beyond the SM.

2.3 Simulation

For the purpose of this analysis, signal events are generated using MadGraph5_aMc@NLO [30] and Pythia8 [31], using the nTGC formulation described in Section 2.1. This nTGC formulation is implemented and imported to MadGraph5_aMc@NLO using FeynRules for nTGC event production at the matrix element level at leading order. Pythia8 is used for parton showering, fragmentation and describing the underlying events.

We illustrate the contributions of the three nTGC form factors ( h_4,h_3^Z, h_3^{\gamma }) with the benchmark choices shown in the second row of Tab.1, whose corresponding cross sections are shown in its third row. The dependence of the Z\gamma cross section on the nTGC form factors h_j^{} (and the corresponding cutoff scale {\mathrm{\Lambda }}_j^{}) can be expressed as follows:

(5){\sigma }_{Z\gamma }^{}={\sigma }_0^{}+{\overline{\sigma }}_1^{}{h}_j^{} +{\overline{\sigma }}_2^{}{h}_j^2={\sigma }_0^{}+{\tilde{\sigma }}_1^{}[{\mathrm{\Lambda }}_j^{-4}]+{\tilde{\sigma }}_2^{}{\mathrm{\Lambda }}_j^{-8},

where {\sigma }_0^{} is the SM contribution, {\overline{\sigma }}_1^{} or {\tilde{\sigma }}_1^{} arises from the interference term between the nTGC and SM contributions, and {\overline{\sigma }}_2^{} or {\tilde{\sigma }}_2^{} corresponds to the squared nTGC contributions. In the above we use the notation [ {\mathrm{\Lambda }}_j^{-4}]= \text{sign}(c_j) {\mathrm{\Lambda }}_j^{-4} as defined below Eq. (4). Fig.2 presents results obtained by scanning various form factors (h_4, h_3^Z, h_3^{\gamma }). The fitted curves in these plots indicate that the cross sections agree well with Eq. (5), confirming the dependence of the Z\gamma cross section with the nTGC form factors.

In Tab.1, each nTGC benchmark consists of three contributions: the SM term, the interference term between the SM and nTGC, and the squared nTGC term. We achieve accurate sample production by decomposing the Z\gamma cross section into these three terms and generating each term independently.

The package \mathtt{W}\mathtt{H}\mathtt{I}\mathtt{Z}\mathtt{A}\mathtt{R}\mathtt{D} [32] is used to simulate background events. All background samples considered in this analysis can be divided into three categories: the 2-fermion background (which is dominant), the 4-fermion backgrounds, and the resonant Higgs backgrounds. Detailed information on the background processes is given in of the Appendix.

The simulation of the detector response is handled by MokkaPlus [33], a GEANT4 [34]-based framework. We perform the full detector simulation for the signal process, whereas the background processes are simulated using Delphes [35].

2.4 Analysis strategy

The CEPC detector adopts the Particle Flow Algorithm (PFA) [36] for event reconstruction, using the dedicated toolkit Arbor [37], which collects tracks and hits from the calorimeter and composes the Particle Flow Objects (PFOs) with its clustering and matching modules. The CEPC detector acts like a “camera” that tracks every particle collision. It is not possible to observe all the particles directly in the collisions because some of them decay promptly or do not interact with the detector. However, if they decay to stable particles or interact with the apparatus, they leave signals in the subdetectors. These signals are used to reconstruct the decay products or to infer their presence as physics objects. These objects can be photons, electrons, muons, jets, missing energy, etc.

In this analysis, photons are identified in Arbor using shower shape variables obtained from the high granularity calorimeter without any matched tracks. Leptons (e^{\pm },{\mu }^{\pm }) are identified by a track-matched particle. A likelihood-based algorithm, LICH [38], is implemented in Arbor to separate electrons, muons, and hadrons. The overall lepton identification efficiencies [38] for electrons and muons are 99.7% and 99.9% respectively, where mis-identification rates are lower than 0.07%. To reconstruct fully electrons and muons, and to make sure no ambiguity exists, a lepton isolation criterion [39] is also applied by requiring , where E_{\text{cone}} is the energy within a cone with around the lepton and E_{\ell } is the energy of the lepton. Here E_{\ell } and E_{\text{cone}} are measured in GeV. The polar angle between two selected leptons systems is required to be within the range and so as to ensure that the selected leptons are isolated. Jets are also reconstructed by Arbor, after removing isolated leptons and photons so as to avoid mis-reconstruction due to lepton or photon constituents. A list of object definitions is shown in Tab.2.

This analysis is based on events with just one photon and a pair of leptons of the same flavor and opposite signs (electron and muon). Events with more than one photon or a pair of charged leptons are vetoed. The event selections summarized in Tab.3 are applied to improve the signal significance.

The event selections are optimised according to the requirements of the formulation in Ref. [21]. We first request that no selected jets be left in the signal events, so as to remove higher-order corrections appearing at Next-to-Leading Order (NLO) and beyond as much as possible, and to ensure that cross section enhancement comes from nTGC, not higher-order SM corrections or other SM jet backgrounds. This is an effective cut to remove other SM backgrounds and to improve sensitivity. In this scenario, we also require that two leptons must come from the same Z boson by requiring the invariant mass difference between the di-lepton system and the on-shell Z boson mass be smaller than 10 GeV. Events with final-state radiation photons (FSR) are suppressed by requiring that the sum of the invariant mass of the leptons and the invariant mass of leptons and photon is greater than twice the Z mass (|m_{\ell \ell }+m_{ll\gamma }|>182 GeV). We also apply the cut to suppress background contributions as shown in Fig.3. All the selections listed in Tab.3 are required so as to make the correct transformation between the SMEFT and the Effective Vertex Theory formulated in Ref. [21].

In this measurement the nTGC form factors are constrained by measurements of e^{+}{e}^{-}\to Z( \to {\ell }^{+}{\ell }^{-})\gamma where \ell =e,\mu. An event selection strategy is proposed based on the new form factor formulation and summarised in Tab.4 and Tab.5, which display the signal cut-flow results including contributions of the SM, the interference term, and the quadratic term.

3 Systematics

We have considered several sources of systematic uncertainties, which can be grouped into two types: theoretical and experimental uncertainties. Both systematic uncertainties have been assigned to the expected signal yields and then propagated to the SMEFT fits.

3.1 Signal uncertainties

Unlike hadron colliders, only a few theoretical uncertainties influence the final measurement in lepton colliders such as CEPC. There is no impact from Parton Distribution Functions or {\alpha }_s, and little dependence on higher-order QCD corrections. For completeness, a 0.5% theoretical uncertainty [40] is assumed for the signal yields.

The experimental systematic uncertainties include those in the integrated luminosity, detector acceptance, trigger efficiency, object reconstruction and identification efficiency, object energy scale, and resolution. Luminosity in the CEPC detector is monitored by the LumiCal using the high-statistics BhaBha process, and a relative accuracy of 0.1% is expected to be achieved [40]. A well-described detector geometry is used in the simulation to provide a precise model of the detector acceptance and response. These uncertainties should be negligible in our analysis. The photon identification, reconstruction, and energy calibration rely on dedicated algorithms and real data. All these photon-related uncertainties are detailed and studied in the CEPC CDR [27] and controlled at the sub-percent level. We assume conservatively a 1% uncertainty in the the photon efficiency and 0.05% uncertainties [40] in the photon energy scale (PES) and resolution (PER). The lepton uncertainties are estimated by varying the Z boson mass selection by \pm 1 GeV. The differences between the varied and nominal signal yields will be considered lepton uncertainties, which are strongly related to the lepton selection criteria.

3.2 Background uncertainties

The background yields are floated to consider background mis-modelling effects and uncertainties in cross section calculations. Fixed parameters are used to estimate uncertainties from different background processes. The event yields of the dominant 2-fermion background process are varied by \pm 5%, and the yields from other background processes (4 fermions and Higgs production) are varied by \pm 100%. These estimates are based on the recipe described in Ref. [39].

4 Results and discussion

The expected event yields for the SM Z\gamma process and backgrounds are summarized in Tab.7, as obtained after applying all event-topology based selections. The expected yields of SMEFT samples are propagated to the SMEFT fitting framework including all systematic uncertainties, and used to obtain sensitivities for the nTGCs.

A binned profile-likelihood fit is performed to set upper limits on the Wilson coefficients for dimension-8 operators at the 95% Confidence Level (C.L.). For this purpose we use the EFT fitting framework EFT-fun [41] to set 1- and 2-dimensional limits on nTGC parameters, individually and in parameter planes to exhibit their correlations. All the statistical and systematic uncertainties introduced in Section 3 are propagated to the EFT-fun [41] framework. The kinematic variables \varphi and \theta illustrated in Fig.4 are used in this measurement. The interference between SM and pure BSM contributions can be inferred directly from measuring these two variables, which enables better sensitivities for the nTGC coefficients.

Tab.8 summarizes the sensitivity reaches at the 95% CL for the new physics scales {\mathrm{\Lambda }}_i as obtained from the expected constraints on the associated form factors derived from the SMEFT dimension-8 coefficients given in the Effective Vertex Approach in Eq. (4), with all the systematic uncertainties taken into account. The constraints on the form factors derived in the Effective Vertex Approach are shown in Fig.5 and the corresponding constraints on the operator scales within the SMEFT framework are shown in Fig.6. Both figures highlight the central 95% C.L. range of the integral over the likelihood distribution, while values outside this range are excluded at this level. These depictions of the expected constraints on both the form factors and corresponding dimension-8 operator coefficients within the SMEFT framework offer a comprehensive understanding of the sensitivities to individual higher-dimensional operators.

In adition to these 1-dimensional limits, we have also studied the constraints on different pairs of form factors, so as to understand their allowed correlations. Constraints in 2-dimensional planes are displayed as contour plots in Fig.8. The solid lines in these plots represent the experimental constraints at 68% C.L., while the dashed lines indicate the 95% C.L. constraints, and areas outside the dashed (approximate) ellipses are excluded at the 95% C.L., taking into account all systematic uncertainties. We observe that the contour plots exhibit significant correlations between pairs of form factors.

As an alternative visualisation of our results, we have transformed the constraints from this form factor analysis to limits on the scales of the corresponding dimension-8 SMEFT operators in Fig.7. The aspect ratios and orientations of the (approximately) elliptical contours indicate the degrees of correlation between pairs of operator coefficients.

The expected limits obtained in this paper are slightly better than the phenomenological results estimated theoretically in Ref. [21], despite the inclusion of all sources of systematic uncertainties, such as detector acceptance, object reconstruction, identification efficiencies and resolution. In this measurement, we defined a new variable u=\cos \theta \times \cos {\theta }^{\ast } (where {\theta }^{\ast } is the decay angle as measured in the Z boson’s rest frame). By employing the BDTG method, we constructed a decision boundary in the \varphi−u plane to distinguish efficiently between events with positive and negative cross sections. This approach represents an improvement over the theoretical method that used only a single variable (\varphi), also achieves better significance and accuracy. This method is documented in Appendix B.

5 Conclusions

Since nTGC vertices Z\gamma V^{\ast } do not arise in the dimension-4 SM Lagrangian or in the SMEFT at the dimension-6 level, probing them from the contributions of dimension-8 operators provides a unique opportunity to explore new physics beyond the Standard Model (SM). We have investigated in this work the sensitivities to nTGCs through the reaction e^{+}{e}^{-}\to {\ell }^{+}{\ell }^{-}\gamma (with \ell =e,\mu), performing a detector-level analysis and simulation for an experiment at the CEPC. Experiments at other e^{+}{e}^{-} colliders with similar integrated luminosities and collision energies are expected to have similar sensitivities for probing the nTGCs.

Previous studies of the nTGC vertex Z\gamma V^{\ast } via form factors are not consistent with spontaneously-broken electroweak gauge symmetry of the SM. Recently a new formulation of the nTGC form factor framework has been proposed [23, 24], which is consistently determined by mapping to the complete set of dimension-8 nTGC operators of the SMEFT and hence is compatible with the full electroweak gauge symmetry of the SM. It was found that extra dimension-8 nTGC operators are needed to establish the consistent mapping from the dimension-8 nTGC operators to the correct nTGC form factors [23, 24]. The consistent form factor expression for the CP-conserving nTGC vertex Z\gamma V^{\ast } is shown in Eq. (3).

We have adopted the new nTGC form factor formula (3) to analyze the sensitivities to nTGCs in the Z\gamma channel with Z leptonic decays based on the benchmark luminosity 20 ab{}^{-1} and e^{+}{e}^{-} collision energy \sqrt{s} = 240 GeV at the CEPC. With these, we have obtained the nTGC sensitivity limits (95% C.L.) that take into account a single nonzero nTGC parameter at a time (as shown in Tab.8), as well as the sensitivity contours (95% C.L.) for each pair of nTGC form factors or for each pair of cutoff scales of dimension-8 nTGC operators (as shown in Fig.7 and Fig.8).

Our results were obtained by a dedicated simulation with a realistic detector configuration and a full treatment of the systematic experimental uncertainties as well as statistical uncertainties. A cut-based method is employed for the entire analysis, providing significantly stronger sensitivities compared to previous theoretical analyses [21]. Additionally, we optimized the extraction of the interference term using the BDTG method, which effectively separates positive and negative events, thereby enhancing the overall sensitivity.

Tab.8 shows that measurements of nTGCs at CEPC and other e^{+}{e}^{-} Higgs factories have the potential to probe energy scales well beyond their center-of-mass energies, even exceeding a TeV in the most sensitive case of the nTGC operator {\mathcal{O}}_{G+}^{}. These results are encouraging and confirm that nTGC measurements provide an interesting window to the dimension-8 new physics, extending the utility of the SMEFT beyond the dimension-6 level.

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