Optimal determination of new physics couplings: a comparative study

We study the determination of new physics (NP) parameters using the optimal observable technique (OOT) in situations where the standard model (SM) dominates over the NP effects, and when the NP dominates over the SM contribution, using the 2-Higgs doublet model as an illustrative example; for the case of SM domination we extend our results using an effective theory parameterization of NP effects. For the case of SM dominance we concentrate on tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t\overline{t} $$\end{document} production in an e+e− collider, while for the case of NP dominance we consider both tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t\overline{t} $$\end{document} production and pair production of charged scalars, also in an e+e− collider. We discuss the effects of the efficiency of background reduction, luminosity and beam polarization, and provide a comparison of the optimal uncertainties with those obtained using a standard χ2 analysis of (Monte Carlo generated) collider data.


Introduction
Although the Standard Model (SM) of particle physics is now complete after the discovery of Higgs Boson [1,2] at the Large Hadron Collider (LHC), it leaves several questions unanswered, keeping the quest for new physics (NP) alive.However, LHC hasn't been able to pin down on any such NP yet, where searches are complicated by the huge hadronic activity and QCD backgrounds for many signals of NP.On the other hand, e + e − colliders are ideal for probing such cases.Accordingly, there are many proposals for such machines: the ILC [3-5], CLIC [6][7][8], CEPC [9,10] and FCC [11,12].Precision measurements in particular, can be performed neatly at e + e − machines due to the complete knowledge of the centre-ofmass energy, availability of the beam polarization, and lack of hadronic activitie and parton distribution function uncertainties.The top quark, being the heaviest SM particle, provides one of the most effective windows to probe NP; current measurements are still consistent with important deviations from SM prediction.
Optimal uncertainties of NP couplings depend on the relative contributions of the NP and SM for the process under observation.In this paper we will consider two complementary situations: in the first the NP generates a subdominant correction to the SM, while in the second the NP dominates.These two cases can be realized, for example, in a 2-Higgs doublet model (see, e.g.[45,46]), where, in the first, the scale of the non-SM scalars is large compared to the electroweak scale and with the collider energy; while in the second, that same scale is sufficiently low for non-SM particles to be directly produced.The specific illustrations we consider are, for the first case, t t production, and for the second, production of charged-scalar pairs.
The study of processes where the NP is not observed directly (cf. the first case above) is most conveniently done using the so-called SM effective field theory (SMEFT) approach.In our discussion below we will use this language when applicable, with the connection to a specific two-doublet model briefly spelled-out; the results thus obtained, even if motivated by a specific extension of the SM, will be more general.Within the SMEFT the Lagrangian takes the form where c i 's are the dimensionless "Wilson coefficients" and the O d i are gauge-invariant, dimensiond effective operators constructed using SM fields and their (covariant) derivatives; Λ denotes the scale of NP which, by consistency of this approach, must lie above the available energy.The lists of operators of dimension five [47], six [48,49], seven [50,51], eight [52][53][54] and nine [55][56][57] are already available.SMEFT approach has been used exhaustively to examine different properties of top-quark physics at the LHC [58][59][60][61][62][63][64][65][66][67][68][69], and also at the lepton colliders [21,[70][71][72][73][74][75][76][77][78][79].
The goals of the paper are, first, to provide an estimate of how well the coefficients can be measured for the two above cases; and second, to compare the optimal uncertainties with standard collider estimates in order to provide a measure of how far the experimental analysis can be improved (for this comparison we use Monte-Carlo generated 'data').The collider analysis we use is basic, aimed at comparing and contrasting the results obtained by both methods; it can certainly be improved, a point on which we comment in section 5.
Our paper is organized as follows: in the next section we summarize the results of the OOT; we then study the case of t t production (section 3) as an example of SM dominance over the NP, and that of charged scalars where the NP dominates over the SM (section 4); a summary is presented in section 5.

Optimal uncertainties
In this section we provide a brief summary of the statistical uncertainties of the NP physics in the OOT approach.The results here are a straightforward extension of the OOT expression obtained in [15,16,34]; the derivation is summarized in the appendix.
The theoretical cross-section for a process involving both SM and NP contributions takes the general form where g i are model-dependent coefficients (that, in general, are non-linear functions of the NP couplings), the f i are functions of the phase-space variables ϕ; for example, for a 2 → 2 scattering process such ϕ can be chosen as the scattering angle in the center-of-mass frame.The choice of the f i 's are not unique although necessarily linearly independent, but all observable results are unambiguous.
We now consider a collider experiment with integrated luminosity L int and where the events of interest follow a Poisson distribution.In this case the optimal covariance matrix for the coefficients g i is given by , If the SM contribution dominates the expression eq. ( 2.1), one can replace O → O SM = dσ SM /dϕ and the above expression reduces to the well-known result [15].
In a model where the coefficients take some specific values g (0) i , the corresponding statistical uncertainty for is determined by the χ 2 function given by where ϵ is the efficiency for the process being considered; this factor takes into account the branching ratios of the produced particles to the signal states, as well as the effect of kinematic cuts used to segregate the signal from the SM background contamination.Specifically, the region χ 2 ≤ n determines the region in parameter space around g (0) i where the optimal statistical uncertainty of the g i is below √ nσ.For cases where the NP contribution is small, i.e.O ≃ O SM , V 0 will be independent of g (0) 1 , and so will the χ 2 ≤ n region; however, in many cases of interest the SM dominates but the NP contribution to O is not negligible, and the dependence of V 0 on g (0) 1 may be important.It is worth mntioning that there are several studies applying the OOT to cases where NP effect is small, so that NP parameters can be neglected in the covariance matrix (V 0 ) [15, 19, 21-24, 27, 28].
3 Example of SM dominance: t t production at e + e − colliders As an application of the OOT we consider t t production at an e + e − future collider like the ILC assuming any NP contributions are subdominant.These conditions are realized, for example, in the so-called "flipped" two-Higgs doublet model [80][81][82][83][84][85] (for a review see [86]) where one scalar doublet couples to down-type quarks, while the second doublet couples to leptons and up-type quarks.If this second doublet is assumed to be heavy1 it will generate an effective interaction of the form ( le)(qu), where l, q are the lepton and quark left-handed doublets, and e, u the right-handed lepton and up-quark singlets, respectively.As noted in the Introduction, this situation can be studied in an effective theory approach, with the advantage that the results are not tied to a specific model.Thus, we consider first the possible low-energy manifestation of such UV complete model (i.e.below the scale of the heavy scalar) using an effective theory parameterization; within this context we will use the OOT to derive the minimum statistical uncertainty to which the Wilson couplings in the effective theory can be measured.
In the SM, pair-production of top-quark at the e + e − colliders is generated at tree-level by γ and Z mediation, as shown in the left side of fig. 1.The SMEFT contributions can be separated into two classes: those that modify the eeZ and ttZ vertices, and those that generate 4 fermion eett vertices (right side of fig.1).The relevant effective operators (in the so-called Warsaw-basis parameterization [49]) are: where g z = g2 + g ′2 and p, r, s, u are family indices.
In general, the operators in eq.(3.1) are generated by different types of NP so that they need not have a common scale Λ.The experimental constraints on the Zee coupling allow an ∼ 0.1% deviation from the the SM prediction that corresponds to Λ ∼ 7 TeV (for a unit Wilson coefficient) for group I operators [87].The constraints on the ttZ coupling (group II) are significantly weaker Λ ∼ 1 TeV [88]; the constraints on the 4-fermion operators (groups III and IV) will be of the same order.
Operators in group IV are unique in that, for the process at hand, they do not interfere with the SM, or with the operators in the other groups, and this will allow a future e + e − collider to differentiate their contributions from those generated by other types of new physics.These operators offer a convenient method to investigate a class NP effects by suppressing SM contributions in t t production for a judicious choice of beam polarization 2 .It is one of the goals of this paper to illustrate this feature using the OOT as a tool; to simplify the discussion we will then consider the effects of these operators, ignoring those that may be generated by those in groups I, II and III.Our effective Lagrangian then takes the form where c 1,2 are dimensionless (Wilson) coefficients, and Λ is the scale of new physics 3 .
The helicity amplitude 4 M (λ e − , λ e + ; λ t , λ t) for this process is given by where λ i = ±1 indicates the helicity of particle i, e 0 the U (1) em coupling constant, √ s the CM energy, s w = sin(θ w ), s 2w = sin(2θ w ) (θ w denotes the weak mixing angle), m t the top-quark mass, m z the Z-boson mass and In the above expressions, electron mass has taken to be zero; we also assumed that c 1,2 are real, if this is not the case then one must replace c a → Rec a + iλ e Imc a .We note here that, for zero electron mass, the SM contributes only to the opposite helicity amplitudes λ e − = −λ e + , whereas scalar and tensor mediated effective operators contribute only to same helicity amplitudes, so there is no EFT-SM interference; this is not the case for possible contributions from operators in groups I-III in eq.(3.1).
For s ≫ m 2 t and s 2 w ≃ 0.25 it is easy to see that the SM contribution to the total cross section has the form which vanishes when P e + = P e − = ±1 and has a maximum when P e ± = ±1; the SM polarized cross section lies above the unpolarized one when P e − ≲ a and P e + ≳ −a, or P e − ≳ a and P e + ≲ −a.
3 Within the context of the flipped two-Higgs doublet model, Λ denotes the scale of the heavy scalar and c1 the product of its Yukawa couplings to the leptons and up-type quarks. 4For the helicity amplitude calculation see [89].
Explicitly, the SM contribution is given by where and, using the notation of eq.(2.1), effective operator contribution is determined by the following choice of f i and g i : (3.9) We plot in fig. 2 this differential cross section and the associated cross section for various choices of the model parameters and beam polarizations.As expected, the SM cross section decreases as the CM energy √ s increases, and the polarized SM cross section for P e ± = −5% +80% is smaller than the unpolarized one, in accordance with eq.(3.6).We therefore will use this choice of beam polarizations for the optimal estimation of the Wilson coefficients, and compare it to that with unpolarized beams.It also evident that as Λ increases the total cross-section approaches the SM value.We identify a value Λ = Λ boundary corresponding to σ tot = 2σ SM , so that the process is SM-dominated when Λ > Λ boundary ; using fig. 2 we find Λ boundary ∼ 2 TeV and, being interested in situations where the EFT represents a correction to the SM, we will consider NP scales above this value.This also shows why t t production is such an effective process to probe NP above TeV scale.For our explicit calculations we use the following collider parameters:

Collider analysis
In this section, we provide an event level signal and background analysis of top quark pair production at e − e + machine.Our main goal here is to determine the efficiency ϵ with which we approximate the optimal observable.
As noted earlier, the SM process is mediated by Z or photon in s-channel exchange, while the NP effects we consider are generated by a contact 4-fermion interaction; as in the previous sections we will restrict ourselves to the NP effects in eq.(3.2).We will consider only the leptonic decays of the W bosons that follow from the decay of the t t pair, namely, (see fig. 3).Thus the signature will be two opposite-sign leptons of same/different flavors + two b jets+ missing energy ( / E).The leading (non-interfering) SM background contributions are generated by Zh, ZZ and W + W − Z production.We follow a standard approach, generating parton-level signal events using CalcHEP [90]; the events are then showered and analyzed using Pythia [91].For event reconstruction, and lepton and jet identification we use the following criteria:  • Lepton transverse momentum: p l T > 10 GeV.
• Exclude events where the dilepton invariant mass is in the range 75 GeV < m ll < 105 GeV to reduce the Z → ℓ + ℓ − background.
• Exclude events where the b− b invariant mass is in the range 115 GeV < m bb < 135 GeV, to reduce h → b b contamination.
where we defined ∆R = ∆η 2 + ∆ϕ 2 as the usual distance in the rapidity (η) -azimuthal angle (ϕ) plane.The invariant mass cuts are designed to exclude the SM background; the values selected are based on the event distributions plotted in fig. 4.
Using these selection criteria we find the efficiency ϵ as the ratio of observed cross section σ FS to the production cross section σ prod : For the benchmark points in table 1 we find ϵ ∼ 0.008, roughly independent of polarization.
In the following we take a conservative approach and use ϵ = 0.001 or ϵ = 0.005.

1σ surfaces of EFT parameter uncertainties
Using eq. ( 2.3), we define the optimal 1σ region of the statistical uncertainties of the EFT parameters c 1,2 for the above described reaction.We choose three different combinations of the NP coupling seed values: For P e ± = +5% −80% , both SM and BSM contribution to the top-quark pair production increase, while if we flip the polarization sign, both contributions will decrease.But we observe that P e ± = −5% +80% provides more precise ∆c i − ∆c j , which implies that the reduction of SM contribution is larger than the reduction in BSM contribution.We note that for P e ± = +5% −80% choice of polarization combination, NP uncertainties are ∼ 10 − 20% smaller compared to unpolarized beam.We also note that in all cases the uncertainties for the tensor operator coefficient c 2 are smaller than for the scalar one, as the tensor operator provides larger BSM contribution to t t production than the scalar operator.As Λ increases, BSM contribution to the production cross-section is reduced, as a result, the uncertainties of the NP couplings are also increased.This behaviour is illustrated in fig.6.
The OOT done here is primarily signal-based, where we assume that the non-interfering background effects are negligible, which is true to a great extent.However, uncertainty of NP couplings will increase when we include these remaining background effects.We take into account these effects by choosing a lower efficiency (ϵ = 0.001) than the one derived when ignoring such backgrounds (ϵ = 0.008) .An important feature of the OOT is that it provides a quantitative estimate of the degree to which a 'base hypothesis' (characterized by NP coefficients g (0)
As an example, we consider c  section) as alternative hypotheses.The degree of statistical differentiation of the alternate hypotheses from the base hypothesis are shown in fig.7 and listed in table 3, for both unpolarized and polarized beams.For unpolarized beams, hypothesis I (c 1 = 1, c2 = 0) is indistinguishable form the SM even if we choose ϵ = 0.005 and L int = 2000 fb −1 .With ϵ = 0.001(0.005)and L int = 2000 fb −1 , hypothesis II (c 1 = 0, c2 = 1) lies beyond 3σ (5σ) the exclusion (discovery) limit.Due to larger BSM contribution, hypothesis III (c 1 = 1, c2 = 1) is easier to distinguish than the other two hypotheses.Appropriate beam polarization, larger ϵ and high luminosity help the differentiation become more significant.

Optimal versus standard χ 2
In this section we provide a comparison of the optimal coefficient uncertainties with those obtained using a basic analysis of collider data.To this end we imagine a collider experiment where the data obtained is organized in a number of bins j = 1, 2, . . ., J. We consider the differential cross-section with respect to the opening angle of the outgoing particles in the CM frame as our observable for this binned analysis (as was done in the OOT approach).We denote by N obs j the number of events in the j-th bin, and N theo j (g i ) the theoretical prediction for this number; using this we define [92] (3.13)  From this we define the 1σ regions in parameter space by the condition χ 2 ≤ 1.In our case the N obs j are obtained using a numerical simulation that implements all the cuts described in section 3.1.The results for three choices of model parameters, and the comparison with the corresponding OOT results are presented in fig.8.The χ 2 statistic defined in eq.(3.13) depends on the number of bins, a dependence that is also examined in fig.8; we find that choosing 5 bins provides the most constrainted parameter space 5 and thus we use this choice in the following examples.These results give a measure of the degree to which the analysis based on eq.(3.13) can be improved.In this case, as BSM contribution is less than SM to the signal final state, the statistical fluctuation in each bin is large, as a consequence, uncertainties in NP couplings estimated from binned analysis is worse than the OOT uncertainties.

Example of NP dominance
We now consider the determination of NP parameters in cases where the NP dominates over the SM; specifically, we will consider the production of new particles at an e + e − collider.For this a specific model must be selected, and we will use the well-understood extension of the SM where an additional doublet is added to the scalar sector, the two-Higgs doublet model6 (2HDM).As noted in section 3, the so-called 'flipped' 2HDM [80][81][82][83][84][85][86] can be used to describe the type of NP effects considered in that section.
The new particles in the 2HDM are a neutral CP-even scalar, a neutral CP-odd scalar, and a pair of charged scalars H ± ; we will also assume the presence of right-handed neutrinos N R and assume that the N R and H ± are light enough to be produced at the e + e − collider.We then will study the degree to which this collider can be expected to determine some of the couplings associated with this extension of the SM.

t t production at e + e − colliders within flipped-2HDM
We first briefly revisit the process considered in section 3, assuming now that the NP can be directed produced in colliders.In this case the contact interaction on the right-hand diagram on fig. 1 is replaced by the heavy scalar exchange diagram in fig.9. We consider flipped 2HDM to elucidate this scenario.This model does not generate the effective interaction Q (3) lequ in eq.(3.2), so that c 2 = 0.The Yukawa couplings to the CP-even heavy Higgs (H) are given by [86], where and α is the angle associated with the diagonalization of the neutral CP-even scalar mass matrix.
For this model c 1 = memt v 2 k f .In order to estimate the the uncertainty of this NP coupling we will consider the resonant production of H and its subsequent decay → t t.In the numerical analysis, we consider heavy Higgs mass m H = 500 GeV, sin(β − α) ∼ 1 in the decoupling limit and small tan β (∼ 0.5).In this case we find c 1 = 5.7 × 10 −7 .Comparing uncertainty in c 1 , ∆c 1 in the flipped-2HDM is smaller than in the EFT scenario due to a greater NP contribution to t t production.

H + H − production at e + e − colliders
We consider next the pair production of H ± which provides an additional channel to probe the NP in this model in which the NP contribution dominates.We assume that the H ± have the standard minimal coupling to the photon, and parameterize the leptonic-Yukawa 7 and Z couplings as follows: where, p 3 and p 4 are the incoming momenta of H − and H + , respectively.For definiteness we will consider three different models : the flipped 2 HDM or inert doublet model (IDM) (doublet) [95], the type-II seesaw model (triplet) [96], and the scotogenic model [97]; these are characterized by 8 , e 0 tan(θ w ) = 0.17

Model
As a concrete application we will consider H + H − pair-production at a linear e + e − collider; the relevant diagrams are shown in fig.10.The corresponding helicity amplitudes, M ′ (λ e − , λ e + ), are easily obtained: M ′ (λ, λ) = 0, and 7 We assume that the masses of any other heavy neutrinos are large enough to have negligible effects at the colliders being considered. 8Here, b is a free parameter.
where θ is the scattering angle of H ± from the axis of collision, β H + = 1 − 4m 2 H + /s, and and P e ± = +10% −50% .The plots of the total cross section as a function of the CM energy are presented in fig.11.

Collider analysis of the inert doublet model
In this section we estimate the efficiency factor ϵ for H + H − production within the inert doublet model (IDM).This model is one of the simplest extension of the SM where the scalar sector is assumed to include an additional doublet H 2 , and an unbroken discrete Z 2 symmetry under which H 2 is odd whereas all other fields are even.This discrete symmetry forbids Yukawa interactions between the inert doublet H 2 and SM fermions and ensures that the lightest physical component of H 2 can serve as a Dark Matter (DM) candidate.The Lagrangian consisting the scalar dark sector can be written as where, We assume µ 2 > 0 that ensures that the vacuum expectation value of the inert doublet vanishes and guarantees that Z 2 remains unbroken.In contrast the SM scalar doublet H 1 does acquire a vacuum expectation value v.The physical modes consist of a singly-charged scalar H ± , a CP-odd neutral scalar A 0 , and CP-even neutrals h, H 0 , where h is the physical SM scalar field.The corresponding tree-level masses are given by where λ L = 1 2 (λ 1 + λ 2 + λ 3 ).The signal we will use to identify the production of an H + H − pair will be two opposite-sign e or µ leptons (OSL) plus missing energy ( / E).The decay chain can be written as where we focus on the leptonic decay modes of W .The dominant (non-interfering) SM background arises from W + W − , ZZ and W + W − Z production.The analysis of H + H − production in an e + e − collider for the IDM has been studied in detail in several publications, see, e.g., [98][99][100][101].
We use the criteria for events reconstruction as in section 3.1 in order to reduce the SM background we impose the following cuts: • C 1 : events must contain two opposite sign leptons in the final state.• C 2 : missing energy / E ≤ 370 (300) for BP1 (BP2).
It follows form fig. 12 that C 1,2 strongly reduce the (non-interfering) SM backgrounds.We also chose charged scalar masses equal to m H ± = 90 GeV and 160 GeV in view of the latest bounds [102].Using the the definition of ϵ in eq.(3.11), the efficiency factor for the two benchmark points is tabulated in table 5.These results justify the conservative choice ϵ = 0.005 we used above for all the different NP scenarios.

1σ surfaces in the a − b plane
We now apply the OOT to obtain the the optimal statistical uncertainty regions for the NP couplings (a, b); the results are presented in table 6 and the corresponding 1σ regions are shown in fig.13 with CM energy ( √ s) = 500 GeV, luminosity (L int ) = 1000 fb −1 and signal efficiency (ϵ) = 0.005.For the IDM and type II models the uncertainties in NP couplings are similar since the cross sections are almost equal.For the scotogenic model where b ̸ = 0, the t-channel diagram contributes and enhances the cross section, resulting in an increased sensitivity to b.In table cross-section, the uncertainties of NP couplings are increased by ∼30% compared to the case of m H ± = 90 GeV.We can also see that a judicious choice of polarization also enhances the cross section, leading to a reduced statistical uncertainty in the determination of the NP coefficients.For P e ± = +30% −80% the uncertainty in the parameter a (b) is reduced by ∼ 50-55% (∼25-40%).Finally, as a function of the luminosity L and the efficiency ϵ, the uncertainties scale as 1/( √ L int ϵ) with the expected result that a larger luminosity and/or efficiency also lead to reduction of the uncertainties.

Differentiation of models
We can now follow the same approach as in section 3.3 and use the OOT to estimate the extent to which different hypotheses can be distinguished.Specifically, given a "base" and alternate hypotheses, {a 0 , b 0 } and {ā, b}, respectively, we define the significance as in eq.(3.12) (where now g 0 i = g i (a 0 , b 0 ), ḡi = g i (ā, b)) and again assume that these hypotheses can be distinguished at the ≥ ℓσ level if ∆σ ≥ ℓ.
Taking a 0 = b 0 = 0 (similar to the SM) as a base hypothesis, we determine the statistical separation of the models listed above.For CM energy √ s = 500 GeV, L int = 1000 fb −1 , m H ± = 160 GeV, and ϵ = 0.005, the separation significance are listed in table 7 and corresponding plots are shown in fig.14.For unpolarized beams, we can see that for both inert doublet and type-II seesaw models are under discovery limit (that is, ∆σ < 5), but the scotogenic model is above this limit due to the enhancement of the cross-section by the t-channel N R contribution.Polarized beams (P e ± = +30% −80% ) enhances the production cross-section, which in turn provides a clear distinction (above 5σ) of the three different models from the base hypothesis.Also note that with larger luminosity, total number of events increases to provide higher significance.

Comparison between optimal and standard χ 2
The statistical uncertainties obtained using the OOT can be compared to those obtained using a basic collider analysis based on eq.(3.13), as was done in section 3.

Scotogenic 17σ 40σ
Table 7: Statistical significance of three different models with respect to the a 0 = b 0 = 0 model.
we consider the differential cross-section for H + H − pair production after subsequent decays (with the same cuts as in section 4.3) to OSL + missing energy for the binned analysis.The resulting 1σ regions in the a − b plane for three different models are shown in fig.15 for both the binned and OOT analyses (the collider parameters as in section 4.3).The 1σ contours for binned analysis (OOT) are shown by green (pink) color contours.In this scenario, with purely NP productions, the cross-section is larger, statistical fluctuation in each bin is less,  making the NP uncertainties determined through binned analysis close to OOT.

Summary and Conclusions
The OOT determines the smallest statistical uncertainty with which a NP coupling can be extracted from experiment.In this work we studied two limits of this technique, one in SMEFT (t t production in an e + e − collider) where the SM dominates over the NP contribution, and one in UV complete NP models (H + H − production in an e + e − collider) where SM contribution is subdominant.
For the first application we used an effective Lagrangian parameterization of the NP effects, including for simplicity only those operators that do not interfere with the SM contributions in this process.We find that for realistic collider parameters the 1σ statistical uncertainty of the NP parameters lie in the 20% to 100% range (depending on the values of the coefficients, the beam polarizations and the efficiency of signal background estimation).The possibility of distinguishing different NP models (defined by their values of the NP coefficients) is equally modest, with a significance below 5σ in all cases studied.
In the second application the NP particles are assumed to be light enough to be directly produced.Here we find that the sensitivity is much higher so that the NP parameters could be measured to a precision of 1% to 10% (depending on the values of the coefficients, the beam polarizations and the efficiency).Moreover, different models can be easily distinguished, provided the beams are strongly polarized, which provides a useful comparison of how efficiently the NP couplings can be extracted at the proposed future e + e − colliders.The calculations for both applications require the estimation of an efficiency factor ϵ which we obtain by performing standard collider analyses of the corresponding reactions by studying a cut based signal background analysis.
The statistical uncertainties in the NP coefficients obtained using the OOT are O(0.5) in the first case we considered (section 3), white significantly smaller, O(0.05), for the second example (section 4).This is imply due to the different values of the NP contributions to the cross sections in each case: in the first the NP effects are small, this leads to a relatively small inverse-correlation matrix M in eq.(2.2), while in the second example NP effects dominate leading ot a larger M and correspondingly smaller uncertainties.
We also compared the OOT results with those obtained a simple collider estimate of the parameter uncertainties based on the χ 2 statistic of eq.(3.13).We found that the optimal uncertainties are significantly smaller in the case where the NP effects are subdominant, but that in the case where the NP dominates the results are comparable.The collider analysis, however, can be improved by optimizing the data binning, and possibly by selecting a better suited statistic; such investigations, however, lie outside the scope of this paper.

Figure 2 :
Figure 2: Plots of the e + e − → tt cross-section.Top row: angular dependence for CM energy √ s = 500 GeV, and different values of Λ for unpolarized (left) and polarized with P e ± = −5% +80% (right) beams.Second row: total cross section as a function of √ s for c 1 = c 2 = 1 and several values of Λ, for unpolarized (left) and unpolarized (right) beams.Third row: dependence on the beam polarization for the SM (left) and the SM + EFT with c 1 = c 2 = 1 and Λ = 4 TeV (right).Bottom row: comparison of the SM total cross section (black horizontal line) with the SM + EFT with c 1 = c 2 = 1 as a function of Λ for unpolarized (left) and polarized (right) beams; the region labeled "BSM small" corresponds to σ SM+EFT > 2σ SM .

Figure 3 :
Figure 3: Production and decay of top-quark at e + e − colliders for 2l + 2b + missing energy signal.
and iii) c 0 1 = 1, c 0 2 = 1.The 1σ regions are plotted in fig. 5 for various choices of input values c (0) i and beam polarizations in ∆c i − ∆c j plane where ∆c i = c i − c 0 i .For this calculation we choose √ s= 500 GeV and L int = 1000 fb −1 .Note that, even though the SM dominates, the NP effects are significant even for Λ = 4 TeV.The 1σ uncertainties for different combination of input values are listed in table 2, while 1σ suraces are shown in fig. 5.The polarization of the initial beams in the lepton collider play a crucial role in determining the uncertainties of NP couplings.Precise extraction of NP relies on reducing the SM contribution and/or enhancing BSM contribution to the specific process.The uncertainties of NP couplings for different choices of beam polarization (within the possible ranges in accordance to collider TDR) are shown in the bottom right of fig. 5.

Figure 5 :
Figure 5: Optimal 1-σ regions in ∆c 1 − ∆c 2 plane for various choices of EFT parameters and choices of beam polarization.See figure inset and heading for details.

+80%
as the base hypothesis (that corresponds to the SM), and various other choices of the seed values of NP couplings (given in the previous model ϵ L int (fb −1 ) significance(∆σ) P e ± = 0 P e ± = −5%

Figure 7 :
Figure 7: Statistical significance (cf.eq.(3.12)) ∆σ ≤ 2, 3, 5 (respectively, red, blue and green areas) of alternate models with respect to the SM for unpolarized (left) and polarized P e ± = −5% +80% beams (right); also noted the statistical significance of 3 specific models.See figure inset and heading for details of the parameter choices.

Figure 8 :
Figure 8: Comparison of 1σ surfaces in ∆c 1 − ∆c 2 plane between OOT (light red) and cut-based analysis as in eq.(3.13) (in green) for unpolarized beams.

Figure 10 :
Figure 10: Leading pair-production mechanism for singly charged scalar pairs (H ± ) at an e ± collider.

Figure 11 :
Figure 11: The total cross-section for charged scalar pair production as a function of the CM energy ( √ s) at an e + e − collider for unpolarized beams and two different choices of beam polarization.Left: IDM; middle: type-II seesaw; right: scotogenic (see eq.(4.4)).

Figure 12 :
Figure 12: Normalized missing energy distribution of OSL + missing energy signal for IDM at the e + e − collider with √ s = 500 GeV and unpolarized beams.

Figure 15 :
Figure 15: Comparison of 1σ statistical limit for OOT (pink) and collider (green) χ 2 in a − b plane for charged scalar pair production at e + e − colliders for unpolarized beam.All the relevant parameters are written in the caption.Left: IDM; middle: type-II seesaw; right: scotogenic model (see eq. (4.4)).

Table 5 :
6, 1σ uncertainties of NP couplings for two different charged Higgs masses (m H ± = 90, 160 GeV) are listed.For m H ± =160 GeV, due to smaller production Event cross-section and efficiency factor (ϵ) for two different benchmark points of IDM

Table 6 :
Optimal 1σ statistical uncertainty in the a, b couplings for both unpolarized and polarized