Probing 6D operators at future e−e+ colliders

We explore the sensitivities at future e−e+ colliders to probe a set of six-dimensional operators which can modify the SM predictions on Higgs physics and electroweak precision measurements. We consider the case in which the operators are turned on simultaneously. Such an analysis yields a “conservative” interpretation on the collider sensitivities, complementary to the “optimistic” scenario where the operators are individually probed. After a detail analysis at CEPC in both “conservative” and “optimistic” scenarios, we also considered the sensitivities for FCC-ee and ILC. As an illustration of the potential of constraining new physics models, we applied sensitivity analysis to two benchmarks: holographic composite Higgs model and littlest Higgs model.


Introduction
The discovery of the Higgs Boson [1,2] is a great success of the Standard Model (SM). However, there are still many open questions left unanswered, such as hierarchy problem, dark matter, and cosmic baryon asymmetry. They hint at the existence of physics beyond the SM (BSM). One of the promising venue of probing BSM physics is through measuring the deviation of Higgs properties. This strongly motivates the construction of an electronpositron collider as a Higgs factory within next decades to study the underlying BSM physics. The proposed Higgs factories include the International Linear Collider (ILC, Japan) [3][4][5], the Circular Electron Positron Collider (CEPC, China) [6] and the Future Electron-Positron Circular Collider (FCC-ee, CERN) [7].

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In an effective field theory (EFT) of the SM, the leading effects of BSM physics above the electroweak (EW) scale can be parametrized by a set of six-dimensional (6D) operators (1.1) Here L SM describes physics in the SM. c i and Λ denote dimensionless Wilson coefficients and the cutoff scale defined by the BSM physics, respectively. Among these operators, 59 are CP-even and 17 are CP-odd. The form of the operators depends on the choice of basis [8][9][10][11][12][13].
Since the discovery of Higgs boson, the probe of the 6D operators, particularly the ones motivated by Higgs physics, at LHC and future e − e + colliders has been extensively studied [14][15][16][17][18][19][20][21][22][23][24]. There are different strategies in analyzing the sensitivities to new physics. It can be done with only a single operator tuning on at a time, which provides an "optimistic" projection of the sensitivities at the future e − e + colliders. However, new physics models tend to generate multiple such operators. Without assuming a particular model, one could go to the other extreme by turning on all operators simultaneously without assuming any correlation among them. Such an analysis, a primary effort in this paper, will result in a "conservative" interpretation on collider sensitivities due to cancellation effects among the multiple contributions. Despite this, we should keep in mind that while this approach give some information about potential degeneracies and correlations in interpreting the measurements, it is not directly applicable to specific models. New physics models typically generate a smaller set of independent operators, equivalently, predicts correlations between different operators in the complete set. For that case, one can analyze the experimental constraints or the collider sensitivities straightforwardly, utilizing the correlation matrix predicted by the specific models. It is not necessary (and also impossible) to go through all potential new physics models, for the purpose of qualitatively demonstrating the capability of a future collider. As an illustration, we pursued such analyses in two benchmark models: the holographic composite Higgs model and littlest Higgs model.
Our study partially overlaps with some recent studies on the sensitivities of probing the SM EFT at future e − e + colliders [21,[25][26][27]. The study in ref. [21] was pursued under a yet-to-be-explicitly-established assumption that the 6D EW operators can be constrained sufficiently well. Different from that, we incorporate the sensitivity analysis for these 6D EW operators, without making any first working assumption about them. This may yield a significant impact for the sensitivity discussions on the triple gauge coupling (TGC) measurement. In addition, a recently proposed operating scenario (see, e.g., [28]) is assumed for the FCC-ee analysis. Refs. [25,26] took similar strategies, with the results presented in the "κ"-scheme and in the 6D operator-scheme, respectively. Compared to these analysis, we focus more on the comparative studies on the sensitivities in the "optimistic" and "conservative" scenarios, and the sensitivities at the CEPC, ILC and FCC-ee. More than that, there exist some differences between the operator sets studied and the observables applied. We include the operator O (3)l LL (as is defined in table 1) in the analysis which was ignored in [26]. But, unlike [26] (and also [21]), our analysis does not include the Higgs decay observables, and correspondingly several operators which are sensitive to them. As for the study in [27], it mainly focused on the interpretation of the collider sensitivities in concrete benchmarks. We organize this article in the following way. We will introduce the analysis formalism and the observables applied in section 2 ad section 3, respectively. The analysis and its results will be presented in section 4. In this section, we will pursue a χ 2 fit on the sensitivities of probing the 6D operators at CEPC, in both "optimistic" and "conservative" interpretations. Then we will make a comparative study on the sensitivities at CEPC, FCCee, ILC250 (with data at 250 GeV and below) and ILC (with full data), and look into the operators O 6 in details which is difficult to probe. We will apply the analysis to study the theory of SILH in section 5, analyzing the collider sensitivities to probe its benchmarks: holographic composite Higgs model [29,30] and littlest Higgs model [31]. We conclude in section 6. More technical details and analysis results can be found in appendix.

Analysis formalism
There are 13 6D operators which are relevant to the e − e + → ZH production: 10 CP-even and 3 CP-odd ones. In this article, we focus only on the CP-even ones. We also include the triple gauge boson operator since it is often generated together with these ones in new physics scenarios. These 11 operators are summarized in table 1. This is a subset of the operators in the so called Warsaw basis [9], omitting operators with quarks.
These 11 operators can influence physics at the EW scale in four ways: (1) renormalizing wave function; (2) shifting the definition of EW parameters; (3) modifying the existing SM couplings (including the charge shifting in the gauge boson currents) and (4)

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Here g, g are the SU (2)  Three input parameters of the EW sector in the SM, typically chosen to be {α, m Z , G F }, receive shifts induced by the 6D operators Here the superscripts "sm" represents the SM definition, and "(r)" represents the reference or the measured central value used as input for the fit. Then the parameter shifts can be denoted as This formalism is independent of the definition of the field renormalization factors δZ Z and δZ A . Hence, in addition to affect the observable directly, D6 operators can also contribution to the deviation from SM prediction by shifting the definition of input parameters. From here on, we will suppress the superscript (r) for the measured observables, unless specified. Since v 2 sm /Λ 2 differs with v 2 /Λ 2 only at O( v 4 Λ −4 ) order, we also replace the former with the latter. The new physics corrections to some observables can be derived directly. One example is (2.6) Another example is Figure 1. Feynman diagrams for Higgsstrahlung process.
We have

Observables for analysis
Throughout this paper, we will consider three classes of observables: inclusive signal rates of Higgs events, angular observables in Higgs events, and electroweak precision observables (EWPOs). We will not include the total width of Higgs boson and its decay branching ratios. Correspondingly, we will not consider the operators which do not enter the inclusive production rates at tree level, but modify the Higgs decays, such as h → bb, τ τ , only. The incorporation of the Higgs decays as observables could reveal more information about a larger set of operators. We will leave such an important analysis to a future study.
Regarding theoretical predictions, we will use "δ" to denote the shift caused by wave function renormalization or by definition shift in the EW input parameters. We will use "∆" to denote the total deviation from the reference value for any given observables.

Higgs events
A. Higgs strahlung process. The first important process is e + e − → Zh, as is shown in figure 1. The signal events can be well-selected using the variable of recoiling mass. At leading order, the relevant Lagrangian is given by L hZ µēL γ µ e L + g with the coefficients In this Lagrangian, new vertices appear due to O l L , O B. W W fusion process. Another important process is the W W fusion Higgs production e + e − → ν eνe h, as shown in figure 2. Here we didn't take into account the Z associated Higgs production, with the Z boson decaying into two neutrinos. At leading order, the relevant Lagrangian is given by with the coefficients C. Z-associated di-higgs process. As the beam energy increases, di-Higgs channel switches on. An important channel is the Z association production process e − e + → Zhh.  The relevant Lagrangian for this channel is with the coefficients

Higgs production angular observables
A recent discussion on the angular observables for the process e − e + → hZ(→ l + l − ) can be found in [32,33]. Among the six independent angular observables, four are CP-even, given by Here the angular variables are defined as in

Electroweak precision tests
A. EWPOs at Z pole. The EWPOs at Z pole which are relevant to our analysis include At tree level, the Z partial decay width and the asymmetry are given by in terms of vector and axial couplings g f V,A , or by in terms of chiral couplings g L,R . Γ l,ν is defined for a single flavor, whereas Γ inv includes contribution from all possible flavors. With the 6D operators turned on, the corrections to the chiral couplings of Z boson are given by (3.14) Here δg Z and δθ w represent the effect of the EW parameter shift; δZ Z and δZ X represent the effect of field redefinition; and c (3)l L , c l L and c e R represent the effect of the charge shift in the leptonic Z current. The quark current operators are turned off in this paper, though they may contribute to some of these observables, e.g., R b , in a more general context. For more discussions on this, see, e.g., [26].
The formulae for the operator corrections to the EWPOs are presented in appendix B, with six Wilson coefficients involved: As is indicated in eq. (B.1)-(B.9), the ratio for the coefficients of c W B , c T and c , R µ , R τ and sin 2 θ lep eff , is fixed to be −1.1 : 2 : 4. This is because the three terms in these EWPOs are generated either via ∆g i L /g i L − ∆g i R /g i R , with i representing charged leptons, up quarks and down quarks, or via ∆g ν L /g ν L − ∆g l R /g l R . Both of them satisfy the relation with the combination 2δθ w − δZ X fixing this ratio. This combination also contains a c   • A b,µ FB and sin 2 θ lep eff . They have the same dependence on ξ 0 , ξ + and c e R . • R µ,τ . They have the same dependence on ξ 0 , ξ + and c e R .
These degenerate or approximately degenerate directions could be lifted by Γ Z , which is approximately proportional to ∆g i L g i L + ∆g i R g i R , and m W . Γ Z and m W have different dependences on the variables beyond ξ 0,± and c e R . Thus, we have totally six classes of non-degenerate EWPOs to probe the six Wilson coefficients. The entangled dependence of the EWPOs on the six operators also explains the relatively large magnitude for their correlation matrix entries, as are listed in appendix C.
Though sin 2 θ lep eff and s 2 w are identical in the SM, they represent different measurements. Hence they are influenced by these 6D operators in different ways. s 2 w received corrections via the EW parameter shift only (see eq. (2.6)), whereas sin 2 θ lep eff receives extra contributions caused by field redefinition (see eq. (3.15)).
B. W boson mass. The W boson mass m W = m Z c w receives contributions via the shift of the EW parameters only, resulting in C. Di-boson process. The di-boson production e − e + → W + W − can be applied to probe the TGC, and hence the operator O 3W . It is mainly influenced by the coupling Despite this, a full angular analysis might be valuable, given that the total signal rate is dominated by forward transverse W W production and hence less sensitive to anomalous couplings. We leave the latter to a future work.

Analysis of sensitivity to new physics
Before performing a full analysis on the sensitivities of probing the 6D operators at future e − e + colliders, we will start with a set of analysis using CEPC as an example. We begin with the case in which we turn on one operator at a time. This simplified approach provides an optimistic estimation on the energy scales that could be probed. It provides a basic idea on how the 6D operators individually contribute to the observables, but the potential cancellations among the contributions from different operators are ignored. The latter could dramatically change the collider sensitivities. To illustrate this point, we will consider several cases with more operators turned on. Finally, we will study the sensitivities at all
formulae for the operator corrections to the observables are summarized in appendix B. The effective operators are implemented using FeynRules and the cross sections are computed using either CalcHEP or MadGraph5 [43][44][45].

CEPC analysis: turning on operators individually
The sensitivities for probing the 6D operators at CEPC are presented in table 5 compared to other operators. This implies that more than one observables are sensitive to each of these operators, as was advertised in section 3.3.

CEPC analysis: turning on multiple operators simultaneously
Next let us turn on more 6D operators in table 1. For a comparison with the results shown in table 6, we need to project the allowed region in the space of Wilson coefficients to the relevant axis, that is, to "marginalize" the irrelevant Wilson coefficients. There is a geometric interpretation regrading this method. The χ 2 defines a 10-dimensional ellipsoid in an 11-dimensional space which is expanded by the set of Wilson coefficients. Marginalizing 10 of the 11 Wilson coefficients is equivalent to imposing the conditions LL . This is related to the fact that R b (one of the six classes of the EWPOs) is a weak observable in probing ξ 0 . With the sixth operator turned on, the lack of a sixth independent strong EWPOs yields an approximately degenerate direction in the parameter space expanded by the six operators. To break this degeneracy, extra observables (e.g., Bhabha scattering e − e + → µ − µ + ), need to be introduced.

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A full analysis for the CEPC sensitivities for probing the whole set of 6D operators is presented in figure 6, with all observables in table 4 applied. The normalized correlation matrix for this χ 2 fit is presented in table 7 of appendix C. We have the following observations on the "marginalization" results: the CEPC sensitivities are inherited from the ones presented in figure 5. The energy scale that the CEPC is able to probe decreases from dozens of TeV in the "optimistic" case to TeV or several TeV, except for O • The operator O 3W can be weakly probed only via the e − e + → W + W − production, with the energy scale accessible to the CEPC being decreased from a couple of TeV in the "optimistic" case to sub TeV (this feature is also shared by FCC-ee and ILC, as will be shown below). This is a result of the concerted action of (1) the weak dependence of the e − e + → W + W − production on O 3W due to helicity suppression at linear level [46]; and (2) the existence of approximate degeneracy for the set of EW operators to which the e − e + → W + W − production is much more sensitive (see eq. (B.19)). This effect yields a sensitivity estimation for probing O 3W several times weaker than that obtained in [21].
• The three operators {O W W , O BB , O H } contribute to the Higgs events at tree level. The energy scales that the CEPC is able to probe decrease from several TeV/TeV in the "optimistic" case to TeV/sub TeV, with potential cancellation between the operators taken into account. This is related to the fact that there is only one observable at 240 GeV which is highly sensitive to these operators, say, σ(Zh). Though σ(ννh) and the e − e + → Zh angular observables play a role in constraining the Wilson coefficients, they are too weak to completely break the remaining degeneracies. Despite this, the sensitivities for probing O W W , O BB and O W B could be improved by a couple of times by incorporating the Higgs decay measurements. For example, the decay width of the di-photon mode can be shifted by these operators, yielding As is indicated in [21,26], including the di-photon decay measurement may push the sensitivities of probing the O W W and O BB operators up to several TeVs (note, fewer or no relevant EW operators were turned on in [21,26], which may cause an uncertainty for the estimation).
• The operator O 6 contributes to the Higgs events at loop level only. The energy scales that the CEPC is able to probe decrease from sub TeV in the "optimistic" case to < O(0.1) TeV.
The χ 2 fit sensitivities can be also projected to a 2D plane expanded by two Wilson coefficients, using a marginalization method, as is shown in figures 11-13 in appendix E. . "Optimistic" (light) and "conservative" (dark) sensitivity projections for probing each of the set of 6D operators at CEPC, FCC-ee, ILC250, ILC500 and ILC. Here "ILC250" refers to a combination of the ILC data at 250 GeV and the EW precision measurements at LEP (see table 3); "ILC500" refers to a combination of "ILC250" and the ILC data at 500 GeV; and "ILC" refers to a more optimistic operating scenario, with the LEP measurements in "ILC500" replaced by the Giga-Z data.

Comparative study at future e − e + colliders
Next let us make a comparison on the sensitivities of probing the 6D operators at the future e − e + colliders. For each machine, there exist multiple possibilities for its operating scenario. For concreteness, we consider the measurement precisions at CEPC, FCC-ee and ILC with a subset of possible running scenarios, shown in table 4. The "optimistic" and "conservative" sensitivity interpretations at each machine are presented Figiure 7. Both CEPC and FCC-ee are circular e − e + colliders with non-polarized beams. Benefitting from a larger integrated luminosity at Z pole, the sensitivities at FCC-ee are mildly better than the CEPC ones, in both interpretations. The comparison with the sensitivities at ILC250, ILC500 and ILC is more involved. The ILC250 is less capable in probing these operators than both CEPC and FCC-ee, because of its relatively small luminosity at 250 GeV and the lack of data at Z-pole. However, this can be improved significantly by the data expected to be collected at a higher beam energy. 1 With the data at 500 GeV, the ILC500 performance becomes not much worse than or comparable to the CEPC and FCC-ee ones in the optimistic case. In the conservative case, the ILC500 performance becomes comparable to or even better than the CEPC and FCC-ee ones. This results in a smaller difference between the two kinds of sensitivity interpretations at ILC, compared with the ones at CEPC and FCC-ee, as is indicated in figure 7. On the other hand, the data at Giga-Z can slightly improve the sensitivities only which could be achieved at ILC500. We note that we have oversimplified the beam polarization scenario at ILC, assuming a full-time run for the polarization configuration (P e − , P e + ) = (−0.8, 0.3). Splitting time between different polarization configurations can enhance the power of breaking the operator degeneracies. This effect has been discussed in [21,25], yielding an improvement of ∼ 20-30% on the reach of the new physics scale in some of the operators.

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To get a better picture about the roles played by the observables at 500 GeV, in table 6 we present the marginalized fitting results for Λ/ √ c i (TeV) in the ILC scenarios, varying from ILC250 to ILC500 by adding one more observable at 500 GeV each time. Compared with that at CEPC and FCC-ee, the degeneracy problem for {c W B , c T , c l LL , c l L , c e R } at ILC250 is even worse, given the lack of the Z-pole data. This problem can be addressed to some extent by the e − e + → W + W − measurement at ILC500, as is indicated in table 6. σ(W + W − ) does not depend on {O W W , O BB , O H , O 6 } at tree level, but it has relatively strong sensitivities to these EW operators (see eq. (B.21)). With its help, the constraints for these operators are raised to a level compared to the ones at CEPC and FCC-ee. But this also means that the sensitivity to O 3W is still weak. A combination of the other three observables at 500 GeV, say, σ(Zh), σ(Zhh) and σ(ννh) can help constrain three of {O W W , O BB , O H , O 6 } which are weakly constrained at ILC250. Particularly, the ILC500 has a much better performance in probing O 6 , compare to CEPC and FCC-ee. This is due to the e − e + → Zhh production, an observable which is not available at CEPC and FCC-ee. Though it is less important in the "optimistic" analysis, this observable plays a crucial role in breaking the degeneracy related to O 6 in the "conservative" scenario. As for CEPC and the FCC-ee , their weakness in probing O 6 could be mitigated somewhat by combining with the LHC data for di-Higgs production, e.g., pp → hh → bbττ [47][48][49][50]. Note, the weak sensitivity to probe O 6 below the Zhh thresholds (particularly in the "conservative" scenario) may indicate that the non-linear c 6 terms, e.g., the one-loop quadratic term induced by the Higgs self-energy correction [51], need to be incorporated in the analysis. However, this term, even if being turned on, still fails to yield a bound clearly stronger than the perturbative unitarity one set by the hh → hh scattering, say, |κ 3 | < 5.5 [52]. So, we simply neglect such terms here.
Such a comparative study can be also extended to a plane expanded by two Wilson coefficients, as is shown in figure 14 in appendix E.

JHEP05(2018)081 5 Application to two benchmark composite Higgs models
In this section, we will apply our analysis to a couple of benchmark composite Higgs models. If the composite resonances are heavy, their low energy effects can be captured by a set of correlated EFT operators, named as a "SILH" parametrization [10]. The SILH parametrization contains two characteristic parameters: f , the decay constant of strong dynamics, and g ρ , the strong coupling. Its Lagrangian is given by [10] L SILH =c Here m ρ = g ρ f defines a typical composite resonance mass. To begin with, we neglect the loop-level operators listed in the second line, and rewrite the Lagrangian in the minimal operator basis using the relations [12] O Here O u,d,l y denotes the 6D Yukawa operators, say, the product of the Yukawa term and the H † H, and f runs over all fermions in the SM. These two relations can be further simplified to make connection to our analysis. While substituting O W in eq. (5.1), we omit the operator O 6 , considering its insensitivity to the observables used in the analysis. The 6D Yukawa operators O u,d,l y mainly influence the Higgs Yukawa couplings and hence are less relevant for the inclusive observables applied. The case for O B is somewhat more complicated. The quark current operators may nontrivially contribute to the Γ had . So we will exclude all EWPOs involving the Z hadronic width Γ had below, in order to safely neglect this subtlety. Then under an assumption of Λ 2 = (4πf ) 2 , the relevant Lagrangian terms are given by The SILH can have different realizations, which are characterized by the values ofc i s. Though the LHC runs are able to constrain the SILH, the experimental bounds are typically model-dependent. One LHC probe is to measure the Higgs couplings such as The current LHC runs yield a lower bound f > 600 − 700 GeV, forc H = 1 [53,54], under the assumption of no mixing effect with extra scalars. Such a bound could be pushed up to ∼ 1.5 TeV at HL-LHC. Another LHC probe is to search for the composite resonances. For example, the current searches for the fermionic top partner via its pair production set up an lower limit for the resonance mass 0.9 − 1.2 TeV [55][56][57][58], and hence yield a constraint g ρ f = m ρ > O(1) TeV. Below we will consider two benchmark models: holographic composite Higgs model and littlest Higgs model.
A. Holographic composite Higgs model. The holographic Higgs model [29,30] is based on a theory over a slice of ADS 5 space-time. This space-time, characterized by a constant radius of curvature for its internal space, is compactified with two 4D branes as boundaries. By matching the holographic Higgs model with the SILH EFT, one obtains the Wilson coefficients in the Lagrangian eq. (5.2) as [10] c T = 0c H = 1c W =c B ≈ 1 .  figure 8. According to the left panel, the parameter region with a small f or/and a weak g ρ is relatively easy to probe. This is because it yields relatively light composite resonances and hence a lower effective interacting scale. This observation is consistent with what one had in [27], where the "SILH" pattern is essentially the holographic composite Higgs model discussed here, except that several more operators were turned on in [27]. Note, as the strong coupling g ρ approaches ∼ 4π, the loop-level operators in eq.  B. Littlest Higgs model. The littlest Higgs model [31] is a composite Higgs model with collectively symmetry breaking, with a coset group SU(5)/SO (5). By matching the littlest Higgs model with the SILH EFT, one can figure out the Wilson coefficients in the Lagrangian eq. (5.2) as [10] This setup yields two vanishing coefficients in the Lagrangian eq. The sensitivities of probing the littlest Higgs model at future e − e + colliders are presented in figure 9. Similar to the holographic composite Higgs model, the parameter region with a small f or/and a weak g ρ will be probed first (left-upper panel). The sensitivity projections to the planes expanded by c T /Λ 2 , c

JHEP05(2018)081 6 Conclusions
In this article we presented a systematic study on the sensitivities of probing the UV physics at the future e − e + colliders. The effect of new physics is parametrized by a set of 6D operators at leading order in its EFT. We turned on eleven of these operators simultaneously, which can be probed by Higgs physics and EW precision measurements. The analysis provides a "conservative" projection on the collider sensitivities, complementary to the "optimistic" projection presented where these 6D operators are turned on individually. Then we made a comparative study on the sensitivities at CEPC, FCC-ee and ILC. Three running scenarios at ILC were considered: "ILC250" (ILC data at 250 GeV + EWPO measurements at LEP), "ILC500" (ILC250 + ILC data at 500 GeV) and "ILC" (ILC data at 250 and 500 GeV + GigaZ data). As an application, we analyzed two benchmark models in the composite Higgs scenario. Our results can be briefly summarized as following. e + → W W production, due to their corrections to the Higgs couplings and TGC, respectively. O 6 is difficult to probe because it contributes to e − e + → Zh at loop level only. These features are shared by FCC-ee and ILC250, ILC500, ILC (though the sensitivities to probe O 6 can be improved to some extent by measuring the e − e + → Zhh production at ILC500 and ILC). • Benefitting from a larger integrated luminosity at Z pole, the sensitivities at FCCee are mildly better than the CEPC ones, in both "optimistic" and "conservative" projections.
• An ILC run with E CM = 500 GeV (ILC500) is highly beneficial. Limited by its relatively small luminosity at 250 GeV and the lack of data at Z-pole, ILC250 is less capable in probing these operators. However, this can be adequately compensated by the data at 500 GeV. By combining with the 500 GeV data, the ILC performance is comparable to or better than the CEPC and FCC-ee ones. Moreover, compared to CEPC and FCC-ee, ILC500 performs much better in probing the O 6 operator or measuring the cubic Higgs coupling in the "conservative" analysis. This is mainly because the e − e + → Zhh production, an observable not available at CEPC and FCC-ee [26], can break the degeneracy related to O 6 . Additionally, the ILC can also benefit from time splitting among different polarization configurations [21,25].

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• As an application, the "conservative" analysis is applied to the simplified model of SILH, with the mutual dependence of the Wilson coefficients taken into account. The analysis indicates that CEPC, FCC-ee and ILC have a potential to probe its decay constant up to O(1 − 10)TeV, with the strong coupling varying between 1 − 4π.

A Feynman rules for the interaction vertices
The modified Feynman rules for the interaction vertices are listed as below

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Here the Higgs and gauge fields have been rescaled to their canonical forms. The relevant coefficients are defined as

B Observables for analysis: numerical formulae
The formulae for calculating the contributions of the 6D operators to the observables at future e − e + colliders are listed in the following. The formulae are obtained by using Mad-Graph and CalcHEP, with the model files generated by FeynRule, or by using Mathematica directly. The effect of renormalization group running from the cutoff to the Z pole or the JHEP05(2018)081 beam energy scales has been neglected for the Wilson coefficients. In the following, we use the simplified notation c i Λ 2 ≡ c i (Λ/TeV) 2 .

C Normalized correlation matrices
The normalized correlation matrix for the 6D operators is defined as Here c i and c j run over all Wilson coefficients. Obviously the correlation matrix is symmetric.  Table 7. Normalized correlation matrix in the χ 2 fit at CEPC.   Table 8. Normalized correlation matrix in the χ 2 fit at FCC-ee.  Table 9. Normalized correlation matrix in the χ 2 fit at ILC250.  Table 10. Normalized correlation matrix in the χ 2 fit at ILC500.  Table 11. Normalized correlation matrix in the χ 2 fit at ILC.

D Parameter marginalization in χ 2
The introduction on parameter marginalization in χ 2 can be found in various lecture notes (see, e.g., [59]). Below we will simply introduce this method. Let's consider n independent observables O 1 , O 2 , · · · , O n , with their measured values satisfying Gaussian distribution. These observables are all linearly dependent on m parameters or Wilson coefficients C T = {c 1 , c 2 , · · · , c m }, namely, O i = O i (c 1 , c 2 , · · · , c m ) with i = 1, · · · , n and m ≤ n. Then their probability distribution function (PDF) is given by f (δ 1 , δ 2 , · · · , δ n ) = 1 (2π) Here δ i represents a normalized deviation from the measured value. f (δ 1 , δ 2 , · · · , δ n ) can be converted to the PDF of the m parameters g(c 1 , c 2 , · · · , c m ) ∝ exp − 1 2 with dc 1 dc 2 · · · dc m g(c 1 , c 2 , · · · , c m ) = 1 (D.3) Here is a quadratic function of C T = {c 1 , c 2 , · · · , c m }. M is the correlation matrix of the m parameters which is symmetric. The marginalized distribution for a single parameter, say, c m , is defined as g M (c m ) = dc 1 dc 2 · · · dc m−1 g(c 1 , c 2 , · · · , c m ) (D.5) Separating c m from the other m − 1 paramters we have with C X = (c 1 c 2 · · · c m−1 ) T . X, Z and y are the entries of the correlation matrix M = X Z Z T y . (D.7) The (m − 1) × (m − 1) matrix X can be diagonalized by taking a unitary transformation C X → C X . In the new parameter basis, we have Here we define C X = (c 1 c 2 · · · c m−1 ) T , X = diag(x 1 x 2 · · · x m−1 ) and Z = (z 1 z 2 · · · z m−1 ) T . Integrating out C X , we obtain This relation indicates that, given a confidence level for the χ 2 analysis, the constraints for c m is completely determined by the correlation matrix M . There is a geometric interpretation regrading this. Eq. (D.8) defines a (m − 1)-dimensional ellipsoid in a m-dimensional space which is expanded by C T = (C T X , c m ). Integrating out C X is equivalent to imposing the conditions c i + cmz i x i = 0 or equivalently, the conditions ∂χ 2 ∂c i = 0, for i = 1, · · · , m − 1. Therefore, the marginalization of C X is simply a projection of the ellipsoid to the c m axis in the m-dimensional space.
The discussions above can be generalized to the case with multiple variables. In this case, χ 2 is defined as with C X = c 1 , · · · , c k representing the k parameters to marginalize. Here X and Y are k × k and (m − k) × (m − k) matrices, respectively. With this setup, we have is marginalized by integrating out the first term, yielding 16) Eq. (D.16) describes the correlation among the parameters in C Y . At a given C.L., the value of ∆χ 2 depends on the number of parameters in C Y . If C Y contains one parameter only, say, c m , eq. (D.16) is reduced to eq. (D.9), with ∆χ 2 = 1 at 1σ C.L. Again marginalizing C X is equivalent to imposing the conditions ∂χ 2 /∂c i to Eq. (D.15), with i running from 1 to k. It can be geometrically interpreted as a projection of a (m − 1)-dimensional ellipsoid in a m-dimensional space to its (m − k)-dimensional subspace, which are expanded by (C X , C Y ) and C Y , respectively. The geometric interpretation of parameter marginalization in a simple model is presented in figure 10. In this example there are two free parameters, say, c 1 and c 2 , with c 2 being marginalized. Eq. (D.8) defines an one-dimensional ellipsoid or ellipse at the c 1 − c 2 plane. Integrating out c 2 is equivalent to imposing a condition ∂χ 2 ∂c 2 = ∂χ 2 ∂c 2 = 0. The c 2 marginalization is simply a projection of the ellipse to the c 1 axis. Here the size of the ellipse is determined by the ∆χ 2 value, which is equal to one at 1σ C.L. In figure 10, the allowed range for c 1 with a marginalized c 2 is indicated by the brown line ending at the purple lines. As a comparison, if c 2 is turned off, the constraint for c 1 becomes stronger, which is denoted by the brown dashed line ending at the blue ellipse.   . "Optimistic" (left panels) and "conservative" (right panels) 2D sensitivities at CEPC, FCC-ee and ILC. The coordinate axes are in the unit of (TeV) −2 . In the "optimistic" analysis, only two 6D operators are turned on. In the "conservative" analysis, all 6D operators listed in table 1 are turned on, whereas the irrelevant Wilson coefficients are marginalized.

JHEP05(2018)081
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