aainstitutetext: Department of Physics, Indian Institute of Technology
Guwahati, Assam 781039, Indiabbinstitutetext:  Department of Physics and
Astrophysics, University of California, Riverside, California 92521, USA

# Optimal determination of New Physics couplings: A comparative study

Subhaditya Bhattacharya a Sahabub Jahedi b Jose Wudka<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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
$t\bar{t}$ production in an $e^{+}e^{-}$ collider, while for the case of NP
dominance we consider both $t\bar{t}$ 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 $\chi^{2}$ analysis of (Monte Carlo generated) collider data.

###### Keywords:

Specific BSM Phenomenology, Multi-Higgs Models, SMEFT, $e^{+}e^{-}$
experiments

##  1 Introduction

Although the Standard Model (SM) of particle physics is now complete after the
discovery of Higgs Boson ATLAS:2012yve ; CMS:2012qbp  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 Behnke:2013xla ; Baer:2013cma ; ILC:2019gyn , CLIC
CLICdp:2018cto ; Roloff:2018dqu ; Aicheler:2018arh , CEPC CEPC-
SPPCStudyGroup:2015csa ; CEPC-SPPCStudyGroup:2015esa  and FCC
TLEPDesignStudyWorkingGroup:2013myl ; FCC:2018evy . Precision measurements in
particular, can be performed neatly at $e^{+}e^{-}$ machines due to the
complete knowledge of the centre-of-mass 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.

The Optimal Observable Technique (OOT) Atwood:1991ka ; Davier:1992nw ;
Diehl:1993br ; Gunion:1996vv  is a powerful tool for determining the smallest
statistical uncertainty to which any NP couplings can be determined in a given
experimental environment; the degree to which any two models can be
differentiated can also be predicted. The OOT has been applied to a variety of
cases such as Higgs physics Hagiwara:2000tk ; Dutta:2008bh ; Hioki:2007jc ,
top-quark physics at $e^{+}\,e^{-}$ colliders Grzadkowski:1996pc ;
Grzadkowski:1997cj ; Grzadkowski:1998bh ; Grzadkowski:1999kx ;
Grzadkowski:2000nx , $\gamma\gamma$ colliders Grzadkowski:2004iw ;
Grzadkowski:2005ye , and $e\gamma$ colliders Cao:2006pu . A few works have
appeared using this approach in top-quark physics at the LHC Gunion:1998hm ;
Hioki:2012vn ; Hioki:2014eca  and various aspects of flavour physics
Bhattacharya:2015ida ; Calcuttawala:2017usw ; Calcuttawala:2018wgo . Recently,
the OOT has been used in studying heavy charged fermion Bhattacharya:2021ltd
production and and di-boson production Jahedi:2022duc ; Jahedi:2023myu  at the
$e^{+}\,e^{-}$ collider. We also note that there exists some other statistical
techniques, which uses optimisation for NP extraction, such as multivariate
analysis Holmstrom:1995bt ; TMVA:2007ngy ; Voss:2007jxm ; Bhat:2010zz , matrix
method D0:2004rvt ; D0:2004lvh ; Fiedler:2010sg ; Artoisenet:2010cn , etc.

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.
Gunion:1989we ; Branco:2011iw ), 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, $t\bar{t}$ 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

$\mathcal{L}_{\tt eff}=\mathcal{L}_{\tt
SM}+\frac{1}{\Lambda^{d-4}}\sum_{i}c_{i}O^{d}_{i}\,,$ (1)

where $c_{i}$’s are the dimensionless “Wilson coefficients” and the
$O^{d}_{i}$ are gauge-invariant, dimension-$d$ effective operators constructed
using SM fields and their (covariant) derivatives; $\Lambda$ denotes the scale
of NP which, by consistency of this approach, must lie above the available
energy. The lists of operators of dimension five Weinberg:1979sa , six
Buchmuller:1985jz ; Grzadkowski:2010es , seven Lehman:2014jma ;
Bhattacharya:2015vja , eight Li:2020gnx ; Murphy:2020rsh ; Murphy:2020cly  and
nine Li:2020xlh ; Liao:2020jmn ; Li:2020tsi  are already available. SMEFT
approach has been used exhaustively to examine different properties of top-
quark physics at the LHC Aguilar-Saavedra:2010uur ; Durieux:2014xla ;
Buckley:2015nca ; Buckley:2015lku ; Degrande:2018fog ; Chala:2018agk ;
DHondt:2018cww ; Aguilar-Saavedra:2018ksv ; Hartland:2019bjb ; Neumann:2019kvk
; Maltoni:2019aot ; Brivio:2019ius , and also at the lepton colliders
Kane:1991bg ; Grzadkowski:1997cj ; Brzezinski:1997av ; Boos:1999ca ;
Rontsch:2015una ; Janot:2015yza ; Englert:2017dev ; Durieux:2018tev ;
Durieux:2018ggn ; Jafari:2019seq ; Bissmann:2020mfi .

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 $t\bar{t}$ 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.

##  2 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 Diehl:1993br ; Gunion:1996vv ;
Bhattacharya:2021ltd ; the derivation is summarized in the appendix.

The theoretical cross-section for a process involving both SM and NP
contributions takes the general form

${\cal O}(\phi)=\frac{d\sigma_{\tt theo}}{d\phi}=\frac{d\sigma_{\tt
SM}}{d\phi}+\sum_{i}g_{i}f_{i}(\phi)\,,$ (2)

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 $\phi$; for example, for a $2\rightarrow 2$ scattering process
such $\phi$ 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
${\mathfrak{L}}_{\tt int}$ 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 ,

$V=\frac{1}{{\mathfrak{L}}_{\tt int}}M^{-1}\,,\qquad
M_{ij}=\int\frac{f_{i}(\phi)f_{j}(\phi)}{{\cal O}(\phi)}d\phi\,.$ (3)

If the SM contribution dominates the expression eq. 2, one can replace ${\cal
O}\rightarrow{\cal O}_{\tt SM}=d\sigma_{\tt SM}/d\phi$ and the above
expression reduces to the well-known result Diehl:1993br .

In a model where the coefficients take some specific values
$g_{i}^{\left(0\right)}$, the corresponding statistical uncertainty for is
determined by the $\chi^{2}$ function given by

$\chi^{2}=\epsilon\sum_{i}\delta g_{i}\delta
g_{j}(V_{0}^{-1})_{ij},\quad\delta g_{i}=g_{i}-g_{i}^{\left(0\right)},\quad
V_{0}=V|_{g=g^{\left(0\right)}};$ (4)

where $\epsilon$ 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
$\chi^{2}\leq n$ determines the region in parameter space around
$g_{i}^{\left(0\right)}$ where the optimal statistical uncertainty of the
$g_{i}$ is below $\sqrt{n}\sigma$. For cases where the NP contribution is
small, i.e. ${\cal O}\simeq{\cal O}_{\tt SM}$, $V_{0}$ will be independent of
$g_{1}^{\left(0\right)}$, and so will the $\chi^{2}\leq n$ region; however, in
many cases of interest the SM dominates but the NP contribution to ${\cal O}$
is not negligible, and the dependence of $V_{0}$ on $g_{1}^{\left(0\right)}$
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 ($V_{0}$) Diehl:1993br ;
Grzadkowski:1997cj ; Grzadkowski:1998bh ; Grzadkowski:1999kx ;
Grzadkowski:2000nx ; Cao:2006pu ; Gunion:1998hm ; Hioki:2007jc .

##  3 Example of SM dominance: $t\,\bar{t}$ production at $e^{+}e^{-}$
colliders

As an application of the OOT we consider $t\bar{t}$ 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 Barger:1989fj ; Grossman:1994jb ;
Akeroyd:1994ga ; Akeroyd:1996he ; Akeroyd:1998ui ; Aoki:2009ha  (for a review
see Wang:2022yhm ) 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 heavy 111More specifically, we assume the second
doublet has a small expectation value, large mass, and small mixing with the
other doublet. it will generate an effective interaction of the form
$(\bar{l}e)(\bar{q}u)$, 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 $\gamma$ 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).

Figure 1: Top-quark pair production at $e^{+}\,e^{-}$ collider. Left: SM
contribution; Right: SMEFT contribution.

The relevant effective operators (in the so-called Warsaw-basis
parameterization Grzadkowski:2010es ) are:

$\begin{array}[]{c|lll}\text{group}&\lx@intercol\hfil\text{Operator}\hfil\lx@intercol&\text{vertex}\cr\hline\cr\hline\cr\hbox{\multirowsetup
I}&Q_{\varphi
l}^{\left(1\right)}&=(i\varphi^{\dagger}D_{\mu}\varphi+\text{H.c.})(\overline{l}_{p}\gamma^{\mu}l_{r})&\phantom{-}\frac{1}{2}{\tt
v}^{2}g_{\tt z}\bar{e}\not{Z}P_{L}e\cr&Q_{\varphi
l}^{\left(3\right)}&=(i\varphi^{\dagger}\tau^{I}D_{\mu}\varphi+\text{H.c.})(\overline{l}_{p}\tau^{I}\gamma^{\mu}l_{r})&-\frac{1}{2}{\tt
v}^{2}g_{\tt z}\bar{e}\not{Z}P_{L}e\cr&Q_{\varphi
e}&=(i\varphi^{\dagger}D_{\mu}\varphi+\text{H.c.})(\overline{e}_{p}\gamma^{\mu}e_{r})&\phantom{-}\frac{1}{2}{\tt
v}^{2}g_{\tt z}\bar{e}\not{Z}P_{R}e\cr\hline\cr\hbox{\multirowsetup
II}&Q_{\varphi
q}^{\left(1\right)}&=(i\varphi^{\dagger}D_{\mu}\varphi+\text{H.c.})(\overline{q}_{p}\gamma^{\mu}q_{r})&\phantom{-}\frac{1}{2}{\tt
v}^{2}g_{\tt z}\bar{t}\not{Z}P_{L}t\cr&Q_{\varphi
q}^{\left(3\right)}&=(i\varphi^{\dagger}\tau^{I}D_{\mu}\varphi+\text{H.c.})(\overline{q}_{p}\tau^{I}\gamma^{\mu}q_{r})&-\frac{1}{2}{\tt
v}^{2}g_{\tt z}\bar{t}\not{Z}P_{L}t\cr&Q_{\varphi
u}&=(i\varphi^{\dagger}D_{\mu}\varphi+\text{H.c.})(\overline{u}_{p}\gamma^{\mu}u_{r})&\phantom{-}\frac{1}{2}{\tt
v}^{2}g_{\tt z}\bar{t}\not{Z}P_{R}t\cr\hline\cr\hbox{\multirowsetup
III}&Q_{lq}^{\left(1\right)}&=(\overline{l}_{p}\gamma_{\mu}l_{r})(\overline{q}_{s}\gamma^{\mu}q_{u})&\phantom{-}(\overline{e}\gamma_{\mu}P_{L}e)(\overline{t}\gamma^{\mu}P_{L}t)\cr&Q_{lq}^{\left(3\right)}&=(\overline{l}_{p}\gamma_{\mu}\tau^{I}l_{r})(\overline{q}_{s}\gamma^{\mu}\tau^{I}q_{u})&-(\overline{e}\gamma_{\mu}P_{L}e)(\overline{t}\gamma^{\mu}P_{L}t)\cr&Q_{lu}&=(\overline{l}_{p}\gamma_{\mu}l_{r})(\overline{u}_{s}\gamma^{\mu}u_{u})&\phantom{-}(\overline{e}\gamma_{\mu}P_{L}e)(\overline{t}\gamma^{\mu}P_{R}t)\cr&Q_{qe}&=(\overline{q}_{p}\gamma_{\mu}q_{r})(\overline{e}_{s}\gamma^{\mu}e_{u})&\phantom{-}(\overline{e}\gamma_{\mu}P_{R}e)(\overline{t}\gamma^{\mu}P_{L}t)\cr&Q_{eu}&=(\overline{e}_{p}\gamma_{\mu}e_{r})(\overline{u}_{s}\gamma^{\mu}u_{u})&\phantom{-}(\overline{e}\gamma_{\mu}P_{R}e)(\overline{t}\gamma^{\mu}P_{R}t)\cr\hline\cr\hbox{\multirowsetup
IV}&Q_{lequ}^{\left(1\right)}&=(\overline{l}_{p}^{j}e_{r})\epsilon_{jk}(\overline{q}_{s}^{k}u_{u})&-(\overline{e}P_{R}e)(\overline{t}P_{R}t)\cr&Q_{lequ}^{\left(3\right)}&=(\overline{l}_{p}^{j}\sigma_{\mu\nu}e_{r})\epsilon_{jk}(\overline{q}_{s}^{k}\sigma^{\mu\nu}u_{u})&-(\overline{e}\sigma_{\mu\nu}P_{R}e)(\overline{t}\sigma^{\mu\nu}P_{R}t)\end{array}$
(5)

where $g_{\tt z}=\sqrt{g^{2}+g^{\prime}{}^{2}}$ and $p,\,r,\,s,\,u$ are family
indices.

In general, the operators in eq. 5 are generated by different types of NP so
that they need not have a common scale $\Lambda$. The experimental constraints
on the $Zee$ coupling allow an $\sim 0.1\%$ deviation from the the SM
prediction that corresponds to $\Lambda\sim 7~{}\hbox{TeV}$ (for a unit Wilson
coefficient) for group I operators ParticleDataGroup:2022pth . The constraints
on the $ttZ$ coupling (group II) are significantly weaker $\Lambda\sim
1\,\hbox{TeV}$ CMS:2021aly ; 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 $t\bar{t}$ production for a judicious choice of beam
polarization 222As noted above, operators in group IV are generated by
interesting types of new physics; for the current reaction their contributions
do not interfere with the SM, which allows a clean evaluation of the effects
of polarization – they produce $O(1/\Lambda^{4})$ corrections. Operators in
the I, II and III are generated by different types of new physics (e.g.
additional $Z$ gauge bosons) and do interfere with the SM, giving rise to
$O(1/\Lambda^{2})$ contributions. There are also contributions from dimension
8 operators which can generate $O(1/\Lambda^{4})$ corrections as well, but
these occur only in EFT-SM interference terms and do not correspond to the
types of new physics being considered here.. 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

${\cal L}_{\tt
eff}=\frac{c_{1}}{\Lambda^{2}}Q_{lequ}^{\left(1\right)}+\frac{c_{2}}{\Lambda^{2}}Q_{lequ}^{\left(3\right)}+\text{H.c.}\,;$
(6)

where $c_{1,2}$ are dimensionless (Wilson) coefficients, and $\Lambda$ is the
scale of new physics333Within the context of the flipped two-Higgs doublet
model, $\Lambda$ denotes the scale of the heavy scalar and $c_{1}$ the product
of its Yukawa couplings to the leptons and up-type quarks..

The helicity amplitude 444For the helicity amplitude calculation see
Vega:1995cc .
$M(\lambda_{e^{-}},\lambda_{e^{+}};\,\lambda_{t},\lambda_{\bar{t}})$ for this
process is given by

$\displaystyle M(\lambda,\,-\lambda;\lambda^{\prime},\,-\lambda^{\prime})$
$\displaystyle=e_{0}^{2}\left(\frac{\lambda\lambda^{\prime}+\cos\theta}{\beta_{z}^{2}}\right)\left[\frac{2}{3}\beta_{z}^{2}-\left(\frac{4s_{\tt
w}^{2}-1+\lambda}{4s_{\tt 2w}^{2}}\right)\left(1-\frac{8}{3}s_{\tt
w}^{2}-\beta_{t}\lambda^{\prime}\right)\right],$ $\displaystyle
M(\lambda,\,\lambda;\lambda^{\prime},\,-\lambda^{\prime})$
$\displaystyle=\frac{4c_{2}m_{t}\sqrt{s}}{\Lambda^{2}}\,\lambda\sin\theta\,,$
$\displaystyle M(\lambda,\,-\lambda;\lambda^{\prime},\,\lambda^{\prime})$
$\displaystyle=\frac{2e_{0}^{2}m_{t}\lambda^{\prime}\sin\theta}{\sqrt{s}}\left[\frac{2}{3}-\left(\frac{4s_{\tt
w}^{2}-1+\lambda}{4s_{\tt
2w}^{2}\,\beta_{z}^{2}}\right)\left(1-\frac{8}{3}s_{\tt w}^{2}\right)\right],$
$\displaystyle M(\lambda,\,\lambda;\lambda^{\prime},\,\lambda^{\prime})$
$\displaystyle=\frac{s}{2\Lambda^{2}}(\lambda\lambda^{\prime}\beta_{t}-1)\left(c_{1}+4c_{2}\lambda\lambda^{\prime}\,\cos\theta\right);$
(7)

where $\lambda_{i}=\pm 1$ indicates the helicity of particle $i$, $e_{0}$ the
$U(1)_{\tt em}$ coupling constant, $\sqrt{s}$ the CM energy, $s_{\tt
w}=\sin(\theta_{\tt w}),\,s_{\tt 2w}=\sin(2\theta_{\tt w})$ ($\theta_{\tt w}$
denotes the weak mixing angle), $m_{t}$ the top-quark mass, $m_{z}$ the
$Z$-boson mass and

$\beta_{z}=\sqrt{1-\frac{m_{z}^{2}}{s}};\qquad\beta_{t}=\sqrt{1-\frac{4m_{t}^{2}}{s}}.$
(8)

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
$c_{a}\rightarrow{\bf Re}c_{a}+i\lambda_{e}{\bf Im}c_{a}$. We note here that,
for zero electron mass, the SM contributes only to the opposite helicity
amplitudes $\lambda_{e^{-}}=-\lambda_{e^{+}}$, 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. 5.

The corresponding differential cross-section when the $e^{\pm}$ beams have
partial polarizations $P_{e^{\pm}}$ (with $-1\leq P_{e^{\pm}}\leq 1$) is given
by

$\frac{d\sigma(P_{e^{+}},\,P_{e^{-}})}{d\Omega}=\sum_{\lambda_{e}^{+}=\pm
1}\sum_{\lambda_{e}^{-}=\pm
1}\frac{(1+\lambda_{e^{-}}P_{e^{-}})(1+\lambda_{e^{+}}P_{e^{+}})}{4}\left(\frac{d\sigma}{d\Omega}\right)_{\lambda_{e^{-}},\lambda_{e^{+}}}\,;$
(9)

where $\left(d\sigma/d\Omega\right)_{\lambda_{e^{-}},\lambda_{e^{+}}}$ is the
cross section obtained from eq. 7 by summing over
$\lambda_{t},\,\lambda_{\bar{t}}$. For $s\gg m_{t}^{2}$ and $s_{\tt
w}^{2}\simeq 0.25$ it is easy to see that the SM contribution to the total
cross section has the form

$\sigma(P_{e^{+}},\,P_{e^{-}})_{\tt SM}\propto
1-\mathfrak{a}^{2}+\left(\mathfrak{a}-P_{e^{-}}\right)\left(\mathfrak{a}+P_{e^{+}}\right)\,,\quad\mathfrak{a}\sim
0.0652\,;$ (10)

which vanishes when $P_{e^{+}}=P_{e^{-}}=\pm 1$ and has a maximum when
$P_{e^{\pm}}=\pm 1$; the SM polarized cross section lies above the unpolarized
one when $P_{e^{-}}\\!\lesssim\mathfrak{a}$ and
$P_{e^{+}}\\!\gtrsim-\mathfrak{a}$, or $P_{e^{-}}\\!\gtrsim\mathfrak{a}$ and
$P_{e^{+}}\\!\lesssim-\mathfrak{a}$.

Explicitly, the SM contribution is given by

$\displaystyle\frac{d\sigma_{\tt SM}}{d\Omega}$
$\displaystyle=\frac{\alpha^{2}_{0}(1-P_{e^{-}}P_{e^{+}})}{3s}\bigg{\\{}1+{\tt
C}(\xi_{1}-P_{\tt eff}\xi_{2})+\frac{1}{4}(\xi_{1}^{2}-2P_{\tt
eff}\xi_{1}\xi_{2}+\xi_{2}^{2})\bigg{(}{\tt
C}^{2}+\frac{\beta_{t}^{2}}{2-\beta_{t}^{2}}\bigg{)}$
$\displaystyle-\left[\xi_{2}(1+{\tt C}\xi_{1})-\frac{1}{4}P_{\tt
eff}\left(4\xi_{1}-(2\xi_{1}^{2}-\xi_{2}^{2}){\tt
C}\right)\right]\beta_{t}\cos\theta+$ $\displaystyle+\left[1+{\tt
C}\left(\xi_{1}-P_{\tt eff}\xi_{2}\right)+\frac{2{\tt
C}+1}{4}(\xi_{1}^{2}-2P_{\tt eff}\xi_{1}\xi_{2}+\xi_{2}^{2})+\frac{{\tt
C}(2-{\tt C})}{2}P_{\tt
eff}\xi_{1}\xi_{2}\right]\beta_{t}^{2}\cos^{2}\theta\biggr{\\}}\,;$ (11)

where

$P_{\tt
eff}=\frac{P_{e^{-}}-P_{e^{+}}}{1-P_{e^{-}}P_{e^{+}}}\,,\qquad\xi_{1}=\frac{1}{2s^{2}_{\tt
2w}\beta_{z}^{2}},\qquad\xi_{2}=\frac{4s_{\tt w}^{2}-1}{2s^{2}_{\tt
2w}\beta_{z}^{2}},\qquad{\tt C}=3-12s_{\tt w}^{2}\,;$ (12)

and, using the notation of eq. 2, effective operator contribution is
determined by the following choice of $f_{i}$ and $g_{i}$:

$\displaystyle\\{f_{1},\,f_{2},\,f_{3}\\}$
$\displaystyle=\frac{3\beta_{t}s}{256\pi^{2}\Lambda^{4}}\left\\{(1+\beta_{t}^{2}),\,16\beta_{t}\cos\theta,\,32\beta_{t}^{2}\cos^{2}\theta\right\\}\,,$
$\displaystyle\\{g_{1},\,g_{2},\,g_{3}\\}$
$\displaystyle=(1+P_{e^{-}}P_{e^{+}})\left\\{\bigg{(}c_{1}^{2}+16\frac{1-\beta_{t}^{2}}{1+\beta_{t}^{2}}c_{2}^{2}\bigg{)},\,-c_{1}c_{2},\,c_{2}^{2}\right\\}\,.$
(13)

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 $\sqrt{s}$
increases, and the polarized SM cross section for
$P_{e^{\pm}}=_{+80\%}^{-5\%}$ is smaller than the unpolarized one, in
accordance with eq. 10. 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.

Figure 2: Plots of the $e^{+}e^{-}\rightarrow\bar{t}t$ cross-section. Top row:
angular dependence for CM energy $\sqrt{s}=500$ GeV, and different values of
$\Lambda$ for unpolarized (left) and polarized with
$P_{e^{\pm}}=_{+80\%}^{-5\%}$ (right) beams. Second row: total cross section
as a function of $\sqrt{s}$ for $c_{1}=c_{2}=1$ and several values of
$\Lambda$, 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 $\Lambda=4$ 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 $\Lambda$ for unpolarized (left) and
polarized (right) beams; the region labeled “BSM small” corresponds to
$\sigma_{\tt SM+EFT}>2\sigma_{\tt SM}$.

It also evident that as $\Lambda$ increases the total cross-section approaches
the SM value. We identify a value $\Lambda=\Lambda_{\tt boundary}$
corresponding to $\sigma_{\tt tot}=2\sigma_{\tt SM}$, so that the process is
SM-dominated when $\Lambda>\Lambda_{\tt boundary}$; using fig. 2 we find
$\Lambda_{\tt boundary}\sim 2\,\hbox{TeV}$ 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 $t\bar{t}$ production is such an
effective process to probe NP above TeV scale. For our explicit calculations
we use the following collider parameters:

$\Lambda=4\,\hbox{TeV}\\\
\,;\qquad\sqrt{s}=500\,\hbox{GeV}\,;\qquad{\mathfrak{L}}_{\tt
int}=1000\,\text{fb}^{-1}\,.$

###  3.1 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 $\epsilon$ 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. 6. We will consider only the leptonic decays of the
$W$ bosons that follow from the decay of the $\bar{t}\,t$ pair, namely,

$e^{+}e^{-}\longrightarrow\,t\,\bar{t}\rightarrow\left(bW^{+}\right)\,\left(\bar{b}W^{-}\right)\rightarrow\left(bl^{\prime}{}^{+}\nu_{l^{\prime}}\right)\,\left(\bar{b}l^{-}\bar{\nu}_{l}\right)\,;\quad
l,\,l^{\prime}=e,\,\mu\,,$ (14)

(see fig. 3). Thus the signature will be two opposite-sign leptons of
same/different flavors + two b jets+ missing energy ($\not{E}$). The leading
(non-interfering) SM background contributions are generated by $Zh$, $ZZ$ and
$W^{+}W^{-}Z$ production.

Figure 3: Production and decay of top-quark at $e^{+}\,e^{-}$ colliders for $2l~{}+2b$ \+ missing energy signal. BPs | Input model | Cross-section (fb)
---|---|---
$P_{e^{\pm}}=0$ | $P_{e^{\pm}}=_{+80\%}^{-5\%}$
BP1 | $c_{1}=1,\,c_{2}=0$ | 597.8 | 415.4
BP2 | $c_{1}=0,\,c_{2}=1$ | 645.7 | 461.4
BP3 | $c_{1}=1,\,c_{2}=1$ | 651.9 | 467.4
Table 1: Total cross-section at $e^{+}e^{-}$ collider for different benchmark
points for unpolarized and polarized $P_{e^{\pm}}=_{+80\%}^{-5\%}$ beams with
$\sqrt{s}=500$ GeV. All the benchmark points have $\Lambda=4$ TeV.

Figure 4: Invariant di lepton mass ($m_{ll}$) (left), Invariant di b-jet mass
($m_{bb}$) distributions (right) for $2l~{}+2b$ \+ missing energy final state
coming from $t\bar{t}$ signal with EFT ($\Lambda=4$ TeV) as well as dominant
SM backgrounds at the $e^{+}\,e^{-}$ collider with $\sqrt{s}$ = 500 GeV. Top
panel: Unpolarized beams; bottom panel: polarized beams
$P_{e^{\pm}}=^{-5\%}_{+80\%}$.

We follow a standard approach, generating parton-level signal events using
CalcHEP Belyaev:2012qa ; the events are then showered and analyzed using
Pythia Alwall:2014hca . For event reconstruction, and lepton and jet
identification we use the following criteria:

  * •

Events must have two opposite-sign leptons, and two $b$ jets; the $b$-tagging
efficiency is chosen to be $0.6$ in accordance with the ILC TDR Behnke:2013xla
.

  * •

Lepton transverse momentum: $p^{l}_{T}>10\,\hbox{GeV}$.

  * •

Light jet transverse momentum: $p^{j}_{T}>20\,\hbox{GeV}$.

  * •

Leptons and light jets must be isolated: $\Delta R_{ll^{\prime}}>0.2$, $\Delta
R_{jl}>0.4$ and $\Delta R_{jj^{\prime}}>0.4$.

  * •

Exclude events where the dilepton invariant mass is in the range
$75\,\hbox{GeV}<m_{ll}<105\,\hbox{GeV}$ to reduce the
$Z\rightarrow\ell^{+}\ell^{-}$ background.

  * •

Exclude events where the $b-\bar{b}$ invariant mass is in the range
$115\,\hbox{GeV}<m_{bb}<135\,\hbox{GeV}$, to reduce $h\rightarrow b\bar{b}$
contamination.

where we defined $\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$ as the usual
distance in the rapidity ($\eta$) - azimuthal angle ($\phi$) 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 $\epsilon$ as the ratio
of observed cross section $\sigma^{\tt FS}$ to the production cross section
$\sigma^{\tt prod}$:

$\epsilon=\frac{\sigma^{\tt FS}}{\sigma^{\tt prod}}.$ (15)

For the benchmark points in table 1 we find $\epsilon\sim 0.008$, roughly
independent of polarization. In the following we take a conservative approach
and use $\epsilon=0.001$ or $\epsilon=0.005$.

###  3.2 1$\sigma$ surfaces of EFT parameter uncertainties

Using eq. 4, we define the optimal 1$\sigma$ 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: $i)$ $c_{1}^{0}=1,c_{2}^{0}=0$, $ii)$ $c_{1}^{0}=0,c_{2}^{0}=1$, and
$iii)$ $c_{1}^{0}=1,c_{2}^{0}=1$. The 1$\sigma$ regions are plotted in fig. 5
for various choices of input values $c_{i}^{\left(0\right)}$ and beam
polarizations in $\Delta c_{i}-\Delta c_{j}$ plane where $\Delta
c_{i}=c_{i}-c^{0}_{i}$. For this calculation we choose $\sqrt{s}$= 500 GeV and
${\cal L}_{int}$= 1000 $\rm fb^{-1}$. Note that, even though the SM dominates,
the NP effects are significant even for $\Lambda=4\,\hbox{TeV}$.

Figure 5: Optimal 1-$\sigma$ regions in $\Delta c_{1}-\Delta c_{2}$ plane for
various choices of EFT parameters and choices of beam polarization. See figure
inset and heading for details.

The 1$\sigma$ uncertainties for different combination of input values are
listed in table 2, while 1$\sigma$ 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. For
$\left(P_{e^{\pm}}=^{+5\%}_{-80\%}\right)$, 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
$\left(P_{e^{\pm}}=^{-5\%}_{+80\%}\right)$ provides more precise $\Delta
c_{i}-\Delta c_{j}$, which implies that the reduction of SM contribution is
larger than the reduction in BSM contribution. We note that for
$\left(P_{e^{\pm}}=^{+5\%}_{-80\%}\right)$ choice of polarization combination,
NP uncertainties are $\sim 10-20\%$ 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 $t\bar{t}$ production than the
scalar operator. As $\Lambda$ 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 ($\epsilon=0.001$) than the one derived when
ignoring such backgrounds ($\epsilon=0.008$) .

| $P_{e^{\pm}}=0$ | $P_{e^{\pm}}=^{-5\%}_{+80\%}$
---|---|---
Seed parameters | $\epsilon=0.005$ | $\epsilon=0.001$ | $\epsilon=0.005$ | $\epsilon=0.001$
| $\pm\Delta c_{1}$ | $\pm\Delta c_{2}$ | $\pm\Delta c_{1}$ | $\pm\Delta c_{2}$ | $\pm\Delta c_{1}$ | $\pm\Delta c_{2}$ | $\pm\Delta c_{1}$ | $\pm\Delta c_{2}$
$c^{\left(0\right)}_{1}=1$, $c^{\left(0\right)}_{2}=0$ | $+0.52$ | $+0.32$ | $+1.16$ | $+0.71$ | $+0.46$ | $+0.29$ | $+1.03$ | $+0.66$
$-1.41$ | $-0.32$ | $-3.18$ | $-0.71$ | $-1.37$ | $-0.15$ | $-3.06$ | $-0.66$
$c^{\left(0\right)}_{1}=0$, $c^{\left(0\right)}_{2}=1$ | $+0.55$ | $+0.08$ | $+1.24$ | $+0.19$ | $+0.44$ | $+0.07$ | $+1.00$ | $+0.16$
$-0.37$ | $-0.14$ | $-0.84$ | $-0.31$ | $-0.32$ | $-0.11$ | $-0.72$ | $-0.25$
$c^{\left(0\right)}_{1}=1$, $c^{\left(0\right)}_{2}=1$ | $+1.02$ | $+0.10$ | $+2.29$ | $+0.23$ | $+0.55$ | $+0.09$ | $+1.24$ | $+0.19$
$-0.42$ | $-0.33$ | $-0.94$ | $-0.74$ | $-0.37$ | $-0.14$ | $-0.84$ | $-0.31$
Table 2: Optimal 1$\sigma$ statistical uncertainties of the $c_{1},c_{2}$
couplings for unpolarized and polarized $P_{e^{\pm}}=^{-5\%}_{+80\%}$ beams,
and two values of $\epsilon$; we used the parameters in section 3. Figure 6:
1$\sigma$ surfaces in $\Delta c_{1}-\Delta c_{2}$ plane for two choices of
$\Lambda$ and unpolarized (left) and polarized $P_{e^{\pm}}=^{-5\%}_{+80\%}$
(right) beams. See figure inset and heading for details.

###  3.3 Differentiation of hypotheses

$\begin{array}[]{| c | c | c | c | c |}\hline\cr\hbox{\multirowsetup model}&\hbox{\multirowsetup$\epsilon$}&\hbox{\multirowsetup${\cal L}_{\tt int}~{}(\text{fb}^{-1})$}&\lx@intercol\hfil\text{significance($\Delta\sigma$)}\hfil\lx@intercol\vrule\lx@intercol\\\ \cline{4-5}\cr&&&P_{e^{\pm}}=0&P_{e^{\pm}}=^{-5\%}_{+80\%}\\\ \hline\cr\hbox{\multirowsetup\begin{tabular}[]{l}$\bar{c}_{1}=1$\\\ $\bar{c}_{2}=0$\end{tabular}}&0.001&1000&0.29&0.35\\\ &0.001&2000&0.41&0.49\\\ &0.005&1000&0.69&0.78\\\ \hline\cr\hbox{\multirowsetup\begin{tabular}[]{l}$\bar{c}_{1}=0$\\\ $\bar{c}_{2}=1$\end{tabular}}&0.001&1000&2.77&3.36\\\ &0.001&2000&3.92&4.75\\\ &0.005&1000&6.19&7.51\\\ \hline\cr\hbox{\multirowsetup\begin{tabular}[]{l}$\bar{c}_{1}=1$\\\ $\bar{c}_{2}=1$\end{tabular}}&0.001&1000&3.93&4.83\\\ &0.001&2000&5.56&6.82\\\ &0.005&1000&8.79&10.80\\\ \hline\cr\end{array}$ Table 3:  Statistical significance $\Delta\sigma$ (see eq. 16) of several hypotheses $\bar{c}_{1},\,\bar{c}_{2}$ with respect to the SM.

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
$g_{i}^{\left(0\right)}$) can be distinguished from an alternate one
(characterized by $\bar{g}_{i}$). To this end we define the statistical
significance by

$\Delta\sigma^{2}\left(g^{\left(0\right)};\,\bar{g}\right)=\epsilon\sum_{i,j}\left(g^{0}_{i}-\bar{g}_{i}\right)\left(g^{0}_{j}-\bar{g}_{j}\right)\left(V^{-1}_{0}\right)_{ij}\,,\quad
V_{0}=V(g=g^{0})\,;$ (16)

with $V_{0}$ defined in eq. 4, and $g^{0}_{i}=g_{i}(c_{1}^{0},\,c_{2}^{0})$
and $\bar{g}_{i}=g_{i}(\bar{c}_{1},\,\bar{c}_{2})$ as in eq. 13.

As an example, we consider $c^{\left(0\right)}_{1}=c^{\left(0\right)}_{2}=0$
as the base hypothesis (that corresponds to the SM), and various other choices
of the seed values of NP couplings (given in the previous 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 ($\bar{c}_{1}=1,\bar{c}_{2}=0$) is indistinguishable form the SM
even if we choose $\epsilon=0.005$ and
$\mathfrak{L}_{int}=2000~{}\rm{fb^{-1}}$. With $\epsilon=0.001(0.005)$ and
$\mathfrak{L}_{int}=2000~{}\rm{fb^{-1}}$, hypothesis II
($\bar{c}_{1}=0,\bar{c}_{2}=1$) lies beyond 3$\sigma$ (5$\sigma$) the
exclusion (discovery) limit. Due to larger BSM contribution, hypothesis III
($\bar{c}_{1}=1,\bar{c}_{2}=1$) is easier to distinguish than the other two
hypotheses. Appropriate beam polarization, larger $\epsilon$ and high
luminosity help the differentiation become more significant.

Figure 7: Statistical significance (cf. eq. 16) $\Delta\sigma\leq 2,\,3,\,5$
(respectively, red, blue and green areas) of alternate models with respect to
the SM for unpolarized (left) and polarized $P_{e^{\pm}}=^{-5\%}_{+80\%}$
beams (right); also noted the statistical significance of 3 specific models.
See figure inset and heading for details of the parameter choices.

###  3.4 Optimal versus standard $\chi^{2}$

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 $j=1,\,2,\ldots,J$. 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 $N_{j}^{\tt obs}$ the number of events in the $j$-th
bin, and $N_{j}^{\tt theo}(g_{i})$ the theoretical prediction for this number;
using this we define Cowan:1998ji

$\chi^{2}=\sum^{\rm{bins}}_{j}\left(\frac{N_{j}^{\tt obs}-N_{j}^{\tt
theo}(g_{i})}{\sqrt{N^{\tt obs}_{j}}}\right)^{2}.$ (17)

Figure 8: Comparison of 1$\sigma$ surfaces in $\Delta c_{1}-\Delta c_{2}$
plane between OOT (light red) and cut-based analysis as in eq. 17 (in green)
for unpolarized beams.

From this we define the 1$\sigma$ regions in parameter space by the condition
$\chi^{2}\leq 1$. In our case the $N_{j}^{\tt obs}$ 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 $\chi^{2}$ statistic
defined in eq. 17 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 555This choice provides a balance between the
number of events with the statistical fluctuations in each bin. 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. 17 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.

##  4 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
model666For a recent review see Bhattacharyya:2015nca ; Krawczyk:2015xhl ;
Wang:2022yhm . (2HDM). As noted in section 3, the so-called ‘flipped’ 2HDM
Barger:1989fj ; Grossman:1994jb ; Akeroyd:1994ga ; Akeroyd:1996he ;
Akeroyd:1998ui ; Aoki:2009ha ; Wang:2022yhm  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 $H^{\pm}$; we will also assume the
presence of right-handed neutrinos $N_{R}$ and assume that the $N_{R}$ and
$H^{\pm}$ 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.

###  4.1  $t\bar{t}$ 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
$Q_{lequ}^{\left(3\right)}$ in eq. 6, so that $c_{2}=0$. The Yukawa couplings
to the CP-even heavy Higgs ($H$) are given by Wang:2022yhm ,

$-\mathcal{L}_{\tt yuk}=\frac{m_{f}}{v}y_{f}\bar{f}fH\,,$ (18)

where $y_{f}=\cos(\beta-\alpha)-\sin(\beta-\alpha)\kappa_{f}$, with
$\kappa_{f}=\text{cot}\beta=v_{1}/v_{2}$ and $\alpha$ is the angle associated
with the diagonalization of the neutral CP-even scalar mass matrix.

For this model $c_{1}=\frac{m_{e}m_{t}}{v^{2}}k_{f}$. In order to estimate the
the uncertainty of this NP coupling we will consider the resonant production
of $H$ and its subsequent decay $\rightarrow t\bar{t}$. In the numerical
analysis, we consider heavy Higgs mass $m_{H}=500$ GeV,
$\sin(\beta-\alpha)\sim 1$ in the decoupling limit and small
$\tan\beta~{}(\sim 0.5)$. In this case we find $c_{1}=5.7\times 10^{-7}$.
Comparing uncertainty in $c_{1}$, $\Delta c_{1}$ in the flipped-2HDM is
smaller than in the EFT scenario due to a greater NP contribution to
$t\bar{t}$ production.

Figure 9: Left: NP contribution to $e^{+}e^{-}\rightarrow t\bar{t}$ in a
flipped 2HDM; where $H$ denotes a heavy scalar. Right: dependence of
$\chi^{2}$ (eq. 4) on the NP coefficient uncertainty $\Delta c_{1}$, when
$c_{1}^{\left(0\right)}=5.71\times 10^{-7}$, with the 1$\sigma$ bound
indicated.

###  4.2 $H^{+}H^{-}$ production at $e^{+}e^{-}$ colliders

We consider next the pair production of $H^{\pm}$ which provides an additional
channel to probe the NP in this model in which the NP contribution dominates.
We assume that the $H^{\pm}$ have the standard minimal coupling to the photon,

$H^{+}(p_{4})H^{-}(p_{3})\gamma:-~{}ie_{0}(p_{4}^{\mu}-p_{3}^{\mu})\,,$ (19)

and parameterize the leptonic-Yukawa777We assume that the masses of any other
heavy neutrinos are large enough to have negligible effects at the colliders
being considered. and $Z$ couplings as follows:

$\displaystyle H^{+}(p_{4})H^{-}(p_{3})Z:$
$\displaystyle-~{}ia\left(p^{\mu}_{4}-p^{\mu}_{3}\right)~{},$ $\displaystyle
e^{+}N_{R}H^{-}:$ $\displaystyle~{}b;$ (20)

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) Deshpande:1977rw , the
type-II seesaw model (triplet) Schechter:1981cv , and the scotogenic model
Ma:2006km ; these are characterized by888Here, $b$ is a free parameter.,

$\begin{array}[]{l c c}\qquad\text{\bf Model}&a&$\hskip
28.45274pt$b\cr\text{Flipped 2 HDM/Inert-doublet (IDM):}$\hskip
14.22636pt$&e_{0}\cot(2\theta_{\tt w})=0.21&$\hskip
28.45274pt$0\cr\text{Scotogenic:}&e_{0}\cot(2\theta_{\tt w})=0.21&$\hskip
28.45274pt$0.1\cr\text{Type II seesaw:}&e_{0}\tan(\theta_{\tt w})=0.17&$\hskip
28.45274pt$0\end{array}$ (21)

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,
$M^{\prime}(\lambda_{e^{-}},\,\lambda_{e^{+}})$, are easily obtained:
$M^{\prime}(\lambda,\,\lambda)=0$, and

$M^{\prime}(\lambda,\,-\lambda)=i\left[e_{0}^{2}+ae_{0}\left(\frac{4s_{\tt
w}^{2}-1}{2s_{\tt 2w}}-\lambda\frac{1}{2s_{\tt 2w}}\right)\frac{s}{s-m_{\tt
z}^{2}}+\frac{(1+\lambda)b^{2}}{4}\frac{s}{t-m_{\tt
N}^{2}}\right]\beta_{H^{+}}\sin\theta\,;$ (22)

where $\theta$ is the scattering angle of $H^{\pm}$ from the axis of
collision, $\beta_{H^{+}}=\sqrt{1-4m_{H^{+}}^{2}/s}$, and

$t=\frac{s}{2}\left(1+\beta_{H^{+}}\cos\theta\right)+m_{H^{+}}^{2}\,.$ (23)

Figure 10: Leading pair-production mechanism for singly charged scalar pairs
($H^{\pm}$) at an $e^{\pm}$ collider.

In the numerical examples we will choose a charged-Higgs mass of
$m_{H^{\pm}}=200\,\hbox{GeV}$ and the right-handed neutrino mass of
$m_{N}=200\,\hbox{GeV}$. As the $Z$ boson couples strongly to left-handed
(right-handed) electron (positron) compared to right-handed electron, left
polarized electron, and right polarized positron beam enhances the
$H^{+}H^{-}$ cross-section. We study the effect of beam polarization by
comparing the results for unpolarized beams to those with
$P_{e^{\pm}}=^{+30\%}_{-80\%}$ and $P_{e^{\pm}}=^{+10\%}_{-50\%}$. The plots
of the total cross section as a function of the CM energy are presented in
fig. 11.

Figure 11: The total cross-section for charged scalar pair production as a
function of the CM energy ($\sqrt{s}$) 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. 21).

###  4.3 Collider analysis of the inert doublet model

In this section we estimate the efficiency factor $\epsilon$ 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 ${\mathbbm{Z}}_{2}$
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

${\cal L}={\cal L}_{\tt SM}+|D_{\mu}H_{2}|^{2}-V(H_{1},H_{2}),$ (24)

where,

$\displaystyle V(H_{1},H_{2})$ $\displaystyle=$
$\displaystyle-\mu_{H}^{2}(H_{1}^{\dagger}H_{1})+\lambda_{H}(H_{1}^{\dagger}H_{1})^{2}+\mu^{2}(H_{2}^{\dagger}H_{2})+\lambda(H_{2}^{\dagger}H_{2})^{2}+\lambda_{1}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})$
(25) $\displaystyle+$
$\displaystyle\lambda_{2}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\frac{\lambda_{3}}{2}[(H_{1}^{\dagger}H_{2})^{2}+\text{H.c.}].$

We assume $\mu^{2}>0$ that ensures that the vacuum expectation value of the
inert doublet vanishes and guarantees that ${\mathbbm{Z}}_{2}$ 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
$H^{\pm}$, 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

$\displaystyle m_{H^{0}}^{2}$ $\displaystyle=\mu^{2}+\lambda_{L}v^{2},$ (26)
$\displaystyle m_{H^{\pm}}^{2}$
$\displaystyle=\mu^{2}+\frac{1}{2}\lambda_{1}v^{2},$ (27) $\displaystyle
m_{A^{0}}^{2}$
$\displaystyle=\mu^{2}+\frac{1}{2}(\lambda_{1}+\lambda_{2}-2\lambda_{3})v^{2},$
(28)

where $\lambda_{L}=\frac{1}{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})$.

Figure 12: Normalized missing energy distribution of OSL + missing energy
signal for IDM at the $e^{+}\,e^{-}$ collider with $\sqrt{s}=500$ GeV and
unpolarized beams.

The signal we will use to identify the production of an $H^{+}H^{-}$ pair will
be two opposite-sign $e$ or $\mu$ leptons (OSL) plus missing energy
($\not{E}$). The decay chain can be written as

$e^{+}e^{-}\rightarrow H^{+}H^{-},\,\quad H^{\pm}\rightarrow
H^{0}W^{\pm},\,\quad W^{-}\rightarrow l^{-}\bar{\nu}_{l}\,,~{}W^{+}\rightarrow
l^{+}\nu_{l}\,;$ (29)

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., Aoki:2013lhm ; Hashemi:2015swh ; Kalinowski:2018ylg ; Kalinowski:2018kdn
.

BPs | $\Delta m$ | $m_{H^{\pm}}$ | Cross-section (fb)
---|---|---|---
$P_{e^{\pm}}=0$ | $P_{e^{\pm}}=_{-80\%}^{+30\%}$
BP1 | 20 | 90 | 101.2 | 201.2
BP2 | 90 | 160 | 56.5 | 112.4
Table 4: Total cross-section of charged scalar pair-production
($H^{+}\,H^{-}$) for different benchmark points for unpolarized and polarized
($P_{e^{\pm}}=_{-80\%}^{+30\%}$) beams.

We use the criteria for events reconstruction as in section 3.1 in order to
reduce the SM background we impose the following cuts:

  * •

${\cal C}_{1}$: events must contain two opposite sign leptons in the final
state.

  * •

${\cal C}_{2}$: missing energy $\not{E}\leq 370\,(300)$ for BP1 (BP2).

It follows form fig. 12 that ${\cal C}_{1,2}$ strongly reduce the (non-
interfering) SM backgrounds. We also chose charged scalar masses equal to
$m_{H^{\pm}}$ = $90\,\rm{GeV}\,\rm{and}\,160\,\rm{GeV}$ in view of the latest
bounds Pierce:2007ut . Using the the definition of $\epsilon$ in eq. 15, the
efficiency factor for the two benchmark points is tabulated in table 5. These
results justify the conservative choice $\epsilon=0.005$ we used above for all
the different NP scenarios.

| Cross-section | Efficiency factor ($\epsilon$)
---|---|---
| Signal | SM background
Cuts | BP1 | BP2 | $W^{+}\,W^{-}$ | $ZZ$ | $W^{+}\,W^{-}\,Z$ | $\epsilon^{\tt BP1}$ | $\epsilon^{\tt BP2}$
| (fb) | (fb) | (fb) | (fb) | (fb)
$\mathcal{C}_{1}$ | 2.02 | 1.13 | 211.4 | 11.2 | 0.8 | 0.02 | 0.02
$\mathcal{C}_{2}$ | 1.96 | 0.85 | 4.01 | 0.0 | 0.002 | 0.019 | 0.015
Table 5: Event cross-section and efficiency factor ($\epsilon$) for two
different benchmark points of IDM and corresponding SM background estimation
background estimation for unpolarized beams with CM energy ($\sqrt{s}$) = 500
GeV.

###  4.4 $1\sigma$ 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 1$\sigma$ regions are shown in fig. 13 with CM energy
($\sqrt{s}$) = 500 GeV, luminosity ($\mathcal{L}_{int}$) = 1000 $\rm{fb}^{-1}$
and signal efficiency ($\epsilon$) = 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 $b\not=0$, the t-channel diagram
contributes and enhances the cross section, resulting in an increased
sensitivity to $b$. In table 6, 1$\sigma$ uncertainties of NP couplings for
two different charged Higgs masses ($m_{H^{\pm}}=90,160$ GeV) are listed. For
$m_{H^{\pm}}$ =160 GeV, due to smaller production cross-section, the
uncertainties of NP couplings are increased by $\sim$30% compared to the case
of $m_{H^{\pm}}=90$ 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
$P_{e^{\pm}}=^{+30\%}_{-80\%}$ the uncertainty in the parameter $a~{}(b)$ is
reduced by $\sim$ 50-55% ($\sim$25-40%). Finally, as a function of the
luminosity ${\cal L}$ and the efficiency $\epsilon$, the uncertainties scale
as $1/(\sqrt{{\cal L}_{int}\epsilon})$ with the expected result that a larger
luminosity and/or efficiency also lead to reduction of the uncertainties.

Figure 13:  Optimal 1$\sigma$ surfaces for different models with unpolarized and polarized beams $P_{e^{\pm}}=^{+30\%}_{-80\%}$, $\epsilon=0.005$, and $m_{H^{\pm}}=90\,\hbox{GeV}$ (left column) or $m_{H^{\pm}}=160\,\hbox{GeV}$ (right column). | Uncertainties |
---|---|---
$P_{e^{\pm}}=0$ | $P_{e^{\pm}}=^{+30\%}_{-80\%}$ |
Model |  $m_{H^{\pm}}$(GeV) | $\pm\Delta a$ | $\pm\Delta b$ | $\pm\Delta a$ | $\pm\Delta b$ |
Inert doublet | 90 | $+0.025$ | $+0.050$ | $+0.011$ | $+0.038$ |
$-0.077$ | $-0.050$ | $-0.027$ | $-0.038$ |
160 | $+0.033$ | $+0.058$ | $+0.014$ | $+0.046$ |
$-0.117$ | $-0.058$ | $-0.042$ | $-0.046$ |
Scotogenic | 90 | $+0.060$ | $+0.012$ | $+0.028$ | $+0.007$ |
$-0.070$ | $-0.012$ | $-0.028$ | $-0.007$ |
160 | $+0.085$ | $+0.016$ | $+0.041$ | $+0.010$ |
$-0.105$ | $-0.016$ | $-0.042$ | $-0.010$ |
Type-II Seesaw | 90 | $+0.028$ | $+0.050$ | $+0.017$ | $+0.038$ |
$-0.095$ | $-0.050$ | $-0.028$ | $-0.038$ |
160 | $+0.037$ | $+0.058$ | $+0.015$ | $+0.046$ |
$-0.157$ | $-0.058$ | $-0.043$ | $-0.046$ |
Table 6: Optimal 1$\sigma$ statistical uncertainty in the $a,b$ couplings for
both unpolarized and polarized $P_{e^{\pm}}=^{+30\%}_{-80\%}$ beams with
$\epsilon=0.005$.

###  4.5 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 $\\{\bar{a},\,\bar{b}\\}$, respectively, we define the significance as in
eq. 16 (where now
$g^{0}_{i}=g_{i}(a^{0},\,b^{0}),\,\bar{g}_{i}=g_{i}(\bar{a},\,\bar{b})$) and
again assume that these hypotheses can be distinguished at the
$\geq\ell\sigma$ level if $\Delta\sigma\geq\ell$.

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
$\sqrt{s}=500$ GeV, ${\cal L}_{int}=1000\ \text{fb}^{-1}$, $m_{H^{\pm}}=160$
GeV, and $\epsilon=0.005$, 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, $\Delta\sigma<5$), 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 ($P_{e^{\pm}}=^{+30\%}_{-80\%}$) enhances the
production cross-section, which in turn provides a clear distinction (above
5$\sigma$) of the three different models from the base hypothesis. Also note
that with larger luminosity, total number of events increases to provide
higher significance.

Figure 14: Statistical significance of alternate models with respect to the $a^{0}=b^{0}=0$ model (cf. eq. 16), $\Delta\sigma\leq\ell$, for various choices of $\ell$. Cross, square and triangle correspond to the IDM, type-II seesaw and scotogenic models, respectively. Left: unpolarized beams, right: polarized beams $P_{e^{\pm}}=^{+30\%}_{-80\%}$. Models | significance ($\Delta\sigma$)
---|---
$P_{e^{\pm}}=0$ | $P_{e^{\pm}}=_{-80\%}^{+30\%}$
Inert doublet | 4$\sigma$ | 17$\sigma$
Type-II Seesaw | 2.82$\sigma$ | 13$\sigma$
Scotogenic | 17$\sigma$ | 40$\sigma$
Table 7: Statistical significance of three different models with respect to
the $a^{0}=b^{0}=0$ model.

###  4.6 Comparison between optimal and standard $\chi^{2}$

The statistical uncertainties obtained using the OOT can be compared to those
obtained using a basic collider analysis based on eq. 17, as was done in
section 3.4. Here also 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.

Figure 15: Comparison of 1$\sigma$ statistical limit for OOT (pink) and
collider (green) $\chi^{2}$ 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. 21).

The resulting 1$\sigma$ 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 1$\sigma$ 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.

##  5 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 ($t\bar{t}$ 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 $1\sigma$ 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 $5\sigma$ 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 $\epsilon$ 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. 3, 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 $\chi^{2}$ statistic of
eq. 17. 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.

###### Acknowledgements.

SB would like to acknowledge the grant CRG/2019/004078 from DST-SERB, Govt. of
India. SJ thanks Jayita Lahiri for useful discussions.

##  Appendix A OOT for large NP

In this appendix we provide a derivation of the correlation matrix that
determines the optimal statistical uncertainties of the model parameters. We
interpret the differential cross section $d\sigma/d\phi$ (where $\phi$ denote
the relevant phase-space coordinates) as an unnormalized probability density
function; its normalized counterpart is then

$f_{\tt theo}=\frac{1}{\Sigma_{\tt
theo}}\frac{d\sigma}{d\phi}\,;\quad\Sigma_{\tt
theo}=\left\langle\frac{d\sigma}{d\phi}\right\rangle\,,$ (30)

where, for any $X(\phi)$,

$\left\langle X\right\rangle=\int d\phi\,X(\phi)\,.$ (31)

Denoting by ${\mathfrak{L}}_{\tt int}$ the integrated luminosity over a
period, the average event number in that period is

${\mathfrak{N}}={\mathfrak{L}}_{\tt int}\Sigma_{\tt theo}\,.$ (32)

Assuming the events occur at a constant rate and are independent of one
another 999Situations where this may not be true usually occur when the number
of events is very large, so that effects such as Bose enhancement of Pauli
blocking become important., the probability of observing $A$ events is given
by a Poisson distribution

$\frac{{\mathfrak{N}}^{A}}{A!}e^{-{\mathfrak{N}}}\,.$ (33)

Therefore the probability of observing $A$ events in regions $\Delta\phi_{a}$
around phase-space points $\phi_{a},\,a=1,\ldots,A$ is

$\displaystyle{\mathfrak{P}}(A;{\bm{\xi}}_{A})\,\Delta{\bm{\xi}}_{A}=\frac{{\mathfrak{N}}^{A}}{A!}e^{-{\mathfrak{N}}}\prod_{a=1}^{A}f_{\tt
theo}(\phi_{a})\Delta\phi_{a}\,;\quad{\bm{\xi}}_{A}=(\phi_{1},\ldots,\phi_{A})\,,~{}~{}\Delta{\bm{\xi}}_{A}=\prod_{a=1}^{A}\Delta\phi_{a}\,,$
$\displaystyle{\quad\Rightarrow\quad}{\mathfrak{P}}(A;{\bm{\xi}}_{A})=\frac{e^{-{\mathfrak{N}}}}{A!}\prod_{a=1}^{A}{\mathfrak{f}}_{\tt
theo}(\phi_{a})\,;\quad{\mathfrak{f}}_{\tt theo}={\mathfrak{N}}_{\tt
theo}f_{\tt theo}={\mathfrak{L}}_{\tt int}\frac{d\sigma_{\tt theo}}{d\phi}\,.$
(34)

Now, for all cases of interest the cross-section can be written in form

$\frac{d\sigma_{\tt
theo}}{d\phi}=\\!{\textbf{{g}}}\cdot\\!{\textbf{{f}}}(\phi)\,,$ (35)

where $\\!{\textbf{{g}}}=(g_{1},\,\ldots,\,g_{n})$ are a set of parameters to
be determined through observations, and f a set of convenient linearly-
independent phase-space functions 101010The choice of $f_{i}$ is not unique,
but all observable results are unambiguous Diehl:1993br . The coefficients
$g_{i}$ will be (not necessarily linear) functions of the Lagrangian
parameters and, for processes that receive SM contributions, some or all of
them will depend on the SM parameters. We make no additional assumptions on
the magnitude of the $g_{i}$.

The goal is to obtain a set observables that allow the determination of the g
with the smallest statistical uncertainty possible. To do this we first choose
a generic set of observables $\mathpzc{o}_{i}(\phi),\,(i=1,\,\ldots,\,n)$ and
let

$\mathpzc{O}_{i}(A;{\bm{\xi}}_{A})=\sum_{a=1}^{A}\mathpzc{o}_{i}(\phi_{a})\,.$
(36)

Then

$\left\lgroup\\!{\mathfrak{P}}\mathpzc{O}_{i}\\!\right\rgroup=\sum_{A=1}^{\infty}\frac{e^{-{\mathfrak{N}}}}{A!}{\mathfrak{N}}^{A-1}A\left\langle{\mathfrak{f}}_{\tt
theo}\mathpzc{o}_{i}\right\rangle=\left\langle{\mathfrak{f}}_{\tt
theo}\mathpzc{o}_{i}\right\rangle={\mathfrak{L}}_{\tt
int}\left\langle\mathpzc{o}_{i}f_{j}\right\rangle g_{j}\,,$ (37)

where, given a function ${\cal U}({\bm{\xi}}_{A})$,

$\left\lgroup\\!{\cal
U}\\!\right\rgroup=\sum_{A=1}^{\infty}\frac{e^{-{\mathfrak{N}}}}{A!}\int
d{\bm{\xi}}_{A}\,{\cal U}({\bm{\xi}}_{A})\,.$ (38)

Let then

$\Gamma_{i}=C^{-1}_{ij}\mathpzc{O}_{j}\,,\qquad C_{ij}={\mathfrak{L}}_{\tt
int}\left\langle\mathpzc{o}_{i}f_{j}\right\rangle{\quad\Rightarrow\quad}g_{i}=\left\lgroup\\!{\mathfrak{P}}\,\Gamma_{i}\\!\right\rgroup\,,$
(39)

so the $\Gamma_{i}$ can be used to determine the $g_{i}$. The correlation
matrix of these observables is given by

$V_{ij}=\left\lgroup\\!{\mathfrak{P}}\,\Gamma_{i}\Gamma_{j}\\!\right\rgroup-
g_{i}g_{j}\,,$ (40)

which we extremize as a function of the $\mathpzc{o}_{i}$:

$\frac{\delta V_{ij}}{\delta\mathpzc{o}_{k}}=0\,.$ (41)

A straightforward calculation shows that the solution is

$\mathpzc{o}_{k}=\frac{f_{k}}{{\mathfrak{f}}_{\tt theo}}\,,$ (42)

for which the condition $\left\langle f_{k}\right\rangle=C_{kl}g_{l}$ is
satisfied and

$V=\frac{1}{{\mathfrak{L}}_{\tt int}}C^{-1}\,;\qquad
C_{ij}={\mathfrak{L}}_{\tt
int}\left\langle\frac{f_{i}f_{j}}{{\mathfrak{f}}_{\tt
theo}}\right\rangle=\left\langle\frac{f_{i}f_{j}}{(d\sigma/d\phi)}\right\rangle\,.$
(43)

If the cross section receives a contribution from both the SM and the new
physics, and when the relevant SM parameters are well measured we write

$\frac{d\sigma}{d\phi}=\frac{d\sigma_{\tt
MS}}{d\phi}+\\!{\textbf{{g}}}\cdot\\!{\textbf{{f}}}(\phi)$ (44)

where the second term contains pure NP contributions, and, in general, SM-NP
interference terms as well. In this case a similar derivation to the one above
gives again eq. 43.

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