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\title{Composite Higgs Model parameter determination at the FCC-ee}
%% %simple case: 2 authors, same institution
%% \author{A. Uthor}
%% \author{and A. Nother Author}
%% \affiliation{Institution,\\Address, Country}
\author{Stefania De Curtis$^{(a)}$, Patrick Janot$^{(b)}$, Stefano Moretti$^{(c)}$ \\
{\it \footnotesize{
(a) INFN, Sezione di Firenze \& Dept. of Physics and Astronomy, University of Florence, Via G.~Sansone 1, 50019 Sesto Fiorentino, Italy \\
(b) CERN, EP Department, Geneva, Switzerland \\
(c) School of Physics \& Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK}}}
% more complex case: 4 authors, 3 institutions, 2 footnotes
%\author[a,b,1]{F. Irst,\note{Corresponding author.}}
%\author[c]{S. Econd,}
%\author[a,2]{T. Hird\note{Also at Some University.}}
%\author[a,2]{and Fourth}
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%\affiliation[a]{One University,\\some-street, Country}
%\affiliation[b]{Another University,\\different-address, Country}
%\affiliation[c]{A School for Advanced Studies,\\some-location, Country}
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%\emailAdd{first@one.univ}
%\emailAdd{second@asas.edu}
%\emailAdd{third@one.univ}
%\emailAdd{fourth@one.univ}
\abstract{\begin{abstract}
We assess the scope of a Future Circular Collider operating in $e^+e^-$ mode (FCC-ee) in accessing the parameter space of the 4-Dimensional Composite Higgs Model (4DCHM), which represents a realistic implementation of
Electro-Weak Symmetry Breaking (EWSB) dynamics triggered by a pseudo-Nambu-Goldstone Boson (pNGB) emerging
from the breaking of a global symmetry group $SO(5)\to SO(4)$ describing new strong interactions eventually resposible for
Higgs compositeness. In fact, in such a framework, other composite states exist, like heavy spin-1 bosons ($W'$s and $Z'$s) and spin-1/2 fermions ($b'$s and $t'$s). Herein, we concentrate initially on the neutral gauge boson sector of this scenario,
by attempting to extract the (modified) couplings of the Standard Model (SM) $Z$ state as well as the masses and quantum numbers of the additional $Z'$ objects present in the 4DCHM. We establish the sensitivity of an FCC-ee to
such parameters via the processes $e^+e^-\to \mu^+\mu^-$ and $e^+e^-\to t\bar t$ for a varieties of foreseen
energies and luminosities, by exploiting both cross section and asymmetry observables. We finally combine the results
obtained in the $Z'$ sector of the 4DCHM with those emerging from foreseen Higgs measurements, as previously assessed, so
as to delineate an analysis programme aimed at confirming or disproving once and for all the validity of the compositeness paradigm.
\end{abstract}
\newpage}
\begin{document}
\maketitle
\flushbottom
\pagestyle{empty}
\noindent
A future $e^+e^-$ collider will be capable to show the imprint of composite Higgs scenarios encompassing partial compositeness. Amongst the possible designs of such a machine, a Future Circular Collider of $e^+e^-$ beams (FCC-ee)
has become a frontrunner project in terms of cost effectiveness, precision and search reach~\hyperref[csl:1]{(Bicer et al. 2014)}.
Besides the detailed study of the Higgs boson properties, based upon the analysis of the Bjorken production
channel $e^+e^-\to ZH$ at an energy of about 240 GeV,
such a machine will have a rich programme also covering top-quark physics (at the energies of 350 to 370 GeV)
and revisiting the typical LEP1/SLC and LEP2 energy ranges (from $M_Z$ to $2M_W$) with significantly increased
luminosity. Of particular relevance for our purposes is the FCC-ee ability to afford one with a very accurate determination
of the top-quark properties, chiefly, its mass, width and couplings to SM objects. This is because the top quark is the natural carrier of New Physics (NP) phenomena associated to the partial compositeness mechanism.
Herein we discuss such a possibility by using a particular realisation of the latter, namely, the 4-Dimensional Composite Higgs Model (4DCHM) of Ref. \hyperref[csl:2]{(Curtis et al. 2012)}. This describes the intriguing possibility that the Higgs particle may be a composite state arising from some strongly interacting dynamics at a high scale. This will solve the hierarchy problem owing to compositeness form factors taming the divergent growth of the Higgs mass upon quantum effects. Furthermore, the now measured light mass of the Standard Model (SM)-like Higgs state discovered at the Large Hadron Collider (LHC) in 2012
could well be consistent with the fact that such a (now composite) object arises as a pseudo Nambu-Goldstone Boson (pNGB) from a particular coset of a global symmetry breaking.
Models with a Higgs state as a pNGB generally also predict modifications of its couplings to both bosons and fermions of the SM, hence the measurement of these quantities represents a powerful way to test its possible non-fundamental nature. Furthermore, the presence of additional particles in the spectrum of generic Composite Higgs Models (CHMs) leads to both mixing effects with the SM states and new Feynman diagram topologies, both of which would represent a further source of deviations from the SM expectations.
In the near future, the LHC will be able to test Beyond the SM (BSM) scenarios more extensively, probing the existence of new particles predicted herein to an unprecedented level, potentially also at a High Luminosity (HL-LHC) option
of the CERN machine \hyperref[csl:3]{(Gianotti et al. 2005)}, whose approval is presently being discussed. Nevertheless, the expected bounds, though severe, might not be conclusive to completely exclude natural scenarios for the Fermi scale, such as CHMs. As an example, new gauge bosons predicted by CHMs with mass larger than $\sim$ 2~TeV could escape detection at the LHC in the Drell-Yan channel owing to the small $W'/Z'$ couplings to both light quarks and leptons \hyperref[csl:4]{(Accomando et al. 2016)}. Likewise, while new $b'/t'$ states, presently bounded
to have a mass above 800 GeV or so, may be within the reach of the current and future runs of the LHC, their peculiar
decay patterns may prevent their discovery \hyperref[csl:5]{(Barducci et al. 2014)}. Furthermore, concerning Higgs boson properties, the LHC will not be able to measure the Higgs couplings to better than few \% thus leaving room for scenarios, like CHMs, which predict deviations within the foreseen experimental accuracy for natural choices of the compositeness scale $f$ (namely in the TeV range) and of the (new) strong coupling constant $g_\rho$ (namely, of order 1). For these reasons, we will tackle here the case in which LHC will discover neither a $W'/Z'$ (or it will not be able to clearly assess its properties if it does) nor a $b/,t'$ (or it will discover it but without any other hints about the theory to which it belongs). In this situation, a
high precision lepton collider like the FC-ee would have a great power in enlightening indirect effects of such a BSM physics
scenario.
The main Higgs production channels within the 4DCHM were considered in \hyperref[csl:5]{(Barducci et al. 2014)} for three possible energy stages and different luminosity options of various proposed $e^+e^-$ machines, including the linear prototypes (the International Linear Collider (ILC)
\hyperref[csl:6]{(Abramowicz and others 2013}; \hyperref[csl:7]{Adolphsen et al. 2013}; \hyperref[csl:8]{Adolphsen et al. 2013}; \hyperref[csl:9]{Behnke et al. 2013}; \hyperref[csl:10]{Baer et al. 2013)} and
the Compact Linear Collider (CLiC)~\hyperref[csl:11]{(Aicheler et al. 2012)}) alongside FCC-ee,
and the results were confronted to the expected experimental accuracies in the various Higgs decay channels. Moreover the potential of such colliders in revealing partial compositeness in the top-quark sector through an accurate determination of the top properties at the $t \bar t$ production threshold were analysed in \hyperref[csl:12]{(Barducci et al. 2015)}. Borrowing results from these previous
studies and others, we now set the stage for the present one. This is done in
Figs.~\ref{fig:deviations} and~\ref{fig:tLR}, where we compare the deviations for the $HZZ$ and $Hbb$ couplings and for the $Z t_L\bar t_L$ and $Z t_R\bar t_R$ couplings in the 4DCHM with respect to the relative precision expected at the HL-LHC, ILC and FCC-ee \hyperref[csl:13]{(Janot 2015}; \hyperref[csl:12]{Barducci et al. 2015}; \hyperref[csl:14]{Janot 2015}; \hyperref[csl:15]{Andreazza and others 2015)}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/FIG1/FIG1}
\caption{{Deviations for the $HZZ$ and $Hb\bar b$ couplings in the 4DCHM (black points) compared with the relative precision expected at HL-LHC, ILC, FCC-ee \protect\hyperref[csl:13]{(Janot 2015)}. \label{fig:deviations}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/FIG1-Updated1/FIG1-Updated1}
\caption{{Typical deviations for the $Zt_L \bar t_L$ and $Z t_R \bar t_R$ couplings in various NP models represented by purple points (see~\protect\hyperref[csl:16]{(Richard 2014)}) and in the 4DCHM (black points). Also shown are the sensitivities expected after LHC-13 with 300 fb$^{-1}$, (region inside the red-dashed lines), after HL-LHC with 3000 fb$^{-1}$ (region inside the inner red-dashed lines), from ILC-500 with polarised beams (region inside the blue-dashed lines) and from FCC-ee (region inside the green lines: the continuous(dashed) line indicates the bounds extracted from the angular and energy distribution of leptons($b$-quarks))~\protect\hyperref[csl:12]{(Barducci et al. 2015}; \hyperref[csl:14]{Janot 2015}; \hyperref[csl:15]{Andreazza and others 2015)}.
The 4DCHM black points correspond to a scan with $0.752$ TeV and $M_{t'}>800$ GeV with $t'$ the lightest extra fermion (indeed a heavy top).
\label{fig:tLR}%
}}
\end{center}
\end{figure}
From the Higgs coupling measurements, it is clear that the FCC-ee will be able to discover CHMs with a very large significance, also for values of $f$ larger than 1 TeV for which the LHC measurements will not be sufficient to detect deviations from the SM{\textcolor{red}{\footnote{\textcolor{red}{We can insert here the determination of the Higgs total width in the 4DCHM, and also the dependence of the Higgs coupling extraction from the $Z'$ exchanges}}}} .This is true {\it a fortiori} for the top EW coupling measurements. In fact, an $e^+e^-$ collider can separately extract the left- and right-handed components of such couplings. This is particularly relevant for models with a partially composite top quark, like the 4DCHM, where the $Zt\bar t$ coupling modifications come not only from the mixing of the $Z$ with the $Z'$s but also from the mixing of the top (antitop) with the extra fermions (antifermions). For the rather ``natural" scan described in the caption of Fig.~\ref{tLR}, the typical deviations lie within the region uncovered by the HL-LHC but are well inside the reach of a polarized ILC running at 500 GeV (ILC-500) and the FCC-ee where the lack of initial polarisation is compensated by the presence of a substantial final state polarisation combined with a larger integrated luminosity \hyperref[csl:12]{(Barducci et al. 2015}; \hyperref[csl:14]{Janot 2015}; \hyperref[csl:15]{Andreazza and others 2015)}.
But this is not the end of the story, in fact, in CHMs the modifications to the $e^+e^-\to H Z $ and $e^+e^-\to t \bar t$ processes arise not only via the alterations to the $HZZ$ coupling for the former and to the $Z t \bar t$ vertex for the latter, but also from the presence of new particles, namely the $s$-channel exchange of $Z'$s, which can be sizeable also for large $Z'$ masses (compared to the $e^+e^-$ energy)
due to their interference with the SM states ($\gamma$ and $Z$). This effect can thus be crucial at high
Center-of-Mass (CM) energies
of the collider but also important at moderate $\sqrt{s}$ values \hyperref[csl:12]{(Barducci et al. 2015)}. In particular, it is impressive how the FCC-ee with $\sqrt{s}=365$ GeV and 2.6 ab$^{-1}$ (corresponding to 3 years of operation) could discover the presence of extra $Z'$ particles through their (effective) contribution to the EW top coupling deviations. This result emerges from the
optimal-observable analysis of the lepton angular and energy distributions from top-quark pair production with semi-leptonic decays \hyperref[csl:17]{(Grzadkowski and Hioki 2000}; \hyperref[csl:18]{Grzadkowski and Hioki 2015}; \hyperref[csl:14]{Janot 2015}; \hyperref[csl:13]{Janot 2015)}. Things go as follow.
The $Vt\bar t$ ($V=\gamma,Z$) vertices can be expressed in terms of 8 form factors:
\begin{eqnarray}
\Gamma_{Vt \bar t}^\mu=\frac g 2 \bar u(p_t)[\gamma^\mu\{A_V+\delta A_V-(B_V+\delta B_V)\gamma_5\}+\frac{(p_t-p_{\bar t})^\mu}{2 m_t}(\delta C_V- \delta D_V \gamma_5)]v(p_{\bar t})
\label{asym}
\end{eqnarray}
and the differential cross section for the process $e^+e^- \to t \bar t\to (bW^+)(\bar b W^-)\to(bq q')(\bar b l \nu)$ can be ``expanded'' around the SM prediction
\begin{eqnarray}
\frac{d^2 \sigma}{dx d\cos\theta}\sim S^0(x,\theta)+\sum_{i=1}^8 \delta_i f^i(x,\cos\theta), ~ \delta_i=\delta(A,B,C,D)_V,~f^i=f^{A,B,C,D}_V(x,\cos\theta).
\label{xsec}
\end{eqnarray}
Here, $S^0$ gives the SM contribution, $x$ and $\theta$ are the lepton reduced energy and the polar angle.
By considering only the 6 CP-conserving form factors ($A_V,B_V,C_V$), the elements of the covariance matrix (the statistical uncertainties) are derived from a likelihood fit to the lepton angular/energy distributions and the total event rate \hyperref[csl:14]{(Janot 2015)}. The result for the top-quark left- and right-handed couplings to the $Z$ is represented by the continous green ellipses in Fig. \ref{fig:tLR}.
In order to compare these uncertainties with the deviations expected in CHMs (deviations in the form factors due not only to coupling modifications but also to $Z'$s exchanges) we have considered one representative benchmark point of the 4DCHM (hereafter, point-A) corresponding to $f=1.3$ TeV, $g_\rho$=1.5, $M$=1.4 TeV ($M$ is the scale of the extra fermion masses). This scenario describes two nearly degenerate $Z'$s, with mass $\sim$ 2.1 and 2.2 TeV, respectively, which are active in top-quark pair production. The deviations in the $Zt_L \bar t_L$ and $Z t_R \bar t_R$ couplings are $\delta g^Z_L/g^Z_L=-2.8\%$ and
$\delta g^Z_R/g^Z_R=6.2\%$ as shown by the yellow point in the top-left panel of Fig.~\ref{fig:tLR4DHCM} , while $\delta g^\gamma_L/g^\gamma_L=\delta g^\gamma_R/g^\gamma_R=0$.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/bench1/bench1}
\caption{{Deviations in the $Zt_L \bar t_L$ and $Z t_R \bar t_R$ couplings for the 4DCHM benchmark point-A (yellow point). \label{fig:tLR4DHCM}%
}}
\end{center}
\end{figure}
As an exercise, we have evaluated the double differential cross section of Eq.~\ref{xsec} within the 4DCHM without including the $Z'$ exchanges and extracted the deviations in the $Z$ and photon left- and right-handed couplings to the top quark by performing a 4 parameter fit (i.e., by fixing the other two CP-conserving form factors, $C_V$, to their SM values). The result is shown in Figs.~\ref{fig:ZttNoZZ} and~\ref{fig:gttNoZZ}, wherein the ellipses correspond to 1,2,3 $\sigma$. As expected, the deviations in the photon couplings are fully compatible with zero. Also, the central values of the $Z$ coupling deviations reproduce very well the deviations corresponding to point-A: $\delta g^Z_{L(R)} {}^{\rm {point-A}}=-0.00713(-0.00708)$ to be compared with $\delta g^Z_{L(R)}{}^{\rm {Fit}}=-0.00717\pm 0.00475(-0.00701\pm0.00358)$ and $\delta g^\gamma_{L(R)} {}^{\rm {point-A}}=0$ to be compared with $\delta g^\gamma_{L(R)}{}^{\rm {Fit}}=-0.00056\pm 0.00224(0.00035\pm 0.00201)$ where we have included the marginalised uncertainties. The fit is very good in reproducing the theoretical deviations for the couplings, but, as said, this is an exercise. In fact, the $Z'$s were artificially removed.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/gz-noz-1/gz-noz-1}
\caption{{Determination of the $Zt_L \bar t_L$ and $Z t_R \bar t_R$ couplings from a 4 parameter fit of the optimal observable analysis for the point-A without the $Z'$ exchanges. \label{fig:ZttNoZZ}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/ggamma-noz-1/ggamma-noz-1}
\caption{{Same as above for $\gamma t_L \bar t_L$ and $\gamma t_R \bar t_R$ couplings. \label{fig:gttNoZZ}%
}}
\end{center}
\end{figure}
The double differential cross section within the full 4DCHM normalised to the SM one is shown in the bottom-left panel of Fig.~\ref{fig:doubleDiff}, while the deviations in the $Z$ and photon left- and right-handed (effective) couplings to the top quark extracted by performing the aforementioned 4 parameter fit are shown in Figs.~\ref{fig:ZttWithZZ} and~\ref{fig:gttWithZZ}. We called them "effective" couplings because they include the effects of the interference of the $Z'$s with the SM gauge bosons (and, marginally, their individual contributions). This is clearly evident by looking at the deviations of the photon couplings, which are entirely due to these interference effects.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/diffxsect1/diffxsect1}
\caption{{Double differential cross section with respect to the reduced lepton energy and the lepton polar angle within the 4DCHM normalized to the SM one. \label{fig:doubleDiff}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/gz1/gz1}
\caption{{Determination of the $Zt_L \bar t_L$ and $Z t_R \bar t_R$ couplings from a 4 parameter fit of the optimal observable analysis for the point-A with the $Z'$ exchanges. \label{fig:ZttWithZZ}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/ggamma1/ggamma1}
\caption{{Same as above for $\gamma t_L \bar t_L$ and $\gamma t_R \bar t_R$ couplings. \label{fig:gttWithZZ}%
}}
\end{center}
\end{figure}
As a check, we have considered the general expression of the polarised squared matrix element for the process $f \bar f \to Z'_i/Z'_j \to F \bar F$ where $f$ and $F$ are massless and massive fermions respectively and $Z'_i$ denotes a vector boson with mass $m_i$ and width $\Gamma_i$.
Assuming chiral interaction terms $Z'_i \bar f f$ and $Z'_i \bar F F$ of the form
\begin{eqnarray}
\frac 1 2 Z'_{i\mu} \{\bar f \gamma^\mu[(1-\gamma^5)f^i_L+(1+\gamma^5)f^i_R] f + \bar F \gamma^\mu[(1-\gamma^5)F^i_L+(1+\gamma^5)F^i_R] F \}
\label{coup}
\end{eqnarray}
and using the helicity projection operators, we get the squared matrix element ${\cal{M}}(\lambda_F,\lambda_{\bar F})$ in terms of the helicities of the final state fermions,
%
\begin{eqnarray}
{\cal{M}}(\lambda_F,\lambda_{\bar F})=C_F \frac{s^2}{8} \sum_{i,j}\Delta_{ij}{\cal{\overline M}}(\lambda_F,\lambda_{\bar F}),
\label{ME}
\end{eqnarray}
where
\begin{eqnarray}
{\cal{\overline M}}(\pm,\pm)&=&(1-\beta^2)s^2_{\theta_{p}}(f^i_L f^j_L+f^i_R f^j_R)(F^i_L+F^j_R)(F^i_L+F^j_R), \\
{\cal{\overline M}}(\pm,\mp)&=&[(1+c_{\theta_p})^2 f^i_L f^j_L+(1-c_{\theta_p})^2 f^i_R f^j_R] [(1\mp \beta)F^i_L+(1\pm \beta) F^i_R]\\ \nonumber
&\times&[(1\mp \beta)F^j_L+(1\pm \beta) F^j_R)],\nonumber
\label{ME2}
\end{eqnarray}
with $C_F$ a color factor for the final state fermions (equal to 1 for leptons and 3 for quarks), $\theta_p$ is the polar angle of $F$ with respect to $f$, $s$ is the CM energy squared, $\beta^2=1-4 m_F^2/s$ and $\Delta_{ij}$ the propagator squared
\begin{eqnarray}
\Delta_{ij}=\frac{(s-m_i^2)(s-m_j^2)+m_i m_j \Gamma_i \Gamma_j}{((s-m_i^2)^2+m_i^2 \Gamma_i^2)((s-m_j^2)^2+m_j^2\Gamma_j^2)}.
\label{Deltaij}
\end{eqnarray}
In our case we must include the $s$-channel exchange of four spin-1 particles: $\gamma,~Z,~Z'_2,~Z'_3$
(let us label these as $i=0,1,2,3$, respectively). We can then expand $\Delta_{ij}$ in the limit $s>> M_{Z'}^2$. At the first order in $s/M_{Z'}^2$ we get:
\begin{eqnarray}
&&\Delta_{00}=\frac 1{s^2}, ~~~~ \Delta_{11}=\frac 1{(s-m_1^2)^2+\Gamma^2_1 m^2_1},~~~~ \Delta_{01}=\frac {s-m_1^2}{s[(s-m_1^2)^2+\Gamma^2_1 m^2_1]}\label{DD}\\
&&\Delta_{02}\sim - \frac 1{s m_2^2} \sim -\frac{s}{m_2^2} \Delta_{00}, ~~~~ \Delta_{03}\sim - \frac 1{s m_3^2} \sim -\frac{s}{m_3^2} \Delta_{00},~~~ \Delta_{22}\sim \Delta_{33}\sim \Delta_{23}\sim 0,\nonumber\\
&&\Delta_{12}\sim -\frac {s-m_1^2}{m_2^2 [(s-m_1^2)^2+\Gamma^2_1 m^2_1]}\sim -\frac{s}{m_2^2} \Delta_{01},\nonumber \\
&& \Delta_{13}\sim -\frac {s-m_1^2}{m_3^2 [(s-m_1^2)^2+\Gamma^2_1 m^2_1]}\sim -\frac{s}{m_3^2} \Delta_{01},\nonumber
\end{eqnarray}
with $\Gamma_1=\Gamma_Z$, $m_1=M_Z$, $m_2=M_{Z'_2}$ and $m_3=M_{Z'_3}$.
As intimated, the left- and right-handed EW couplings of the $Z$ boson to the top quark get modified in tne 4DCHM and this is mainly due to the partial compositeness of the top itself. On the contrary, the modifications of the couplings to leptons are quite small and we can neglect them here\footnote{This will no longer be possible though in the forthcoming LEP and SLC
analysis.}. The coupling of the photon to all the fermions is given by the Electro-Magnetic (EM) interaction and determined by the electric charge. Referring to eq.~\ref{coup} we will parameterise
\begin{eqnarray}
&& f^0_{L,R}=g^\gamma_{L,R}, ~~~f^1_{L,R}=g^Z_{L,R}\label{DG},\\
&&F^0_{L,R}=G^\gamma_{L,R}~~~F^1_{L,R}=G^Z_{L,R}+ \Delta G_{L,R},\nonumber
\end{eqnarray}
where $g^{\gamma,Z}$ are the SM couplings to the electron, $G^{\gamma,Z}$ are the SM couplings to the top-quark and $\Delta G$ is the modification of the $Z$ coupling. We will show that the $Z'_{2,3}$ $s$-channel exchange can be expressed as an effective modification of the couplings.
Let us insert eqs.~(\ref{DD})-(\ref{DG}) in eq.~\ref{ME} and expand to first order in $\Delta G$.
For simplicity, let us consider the influence of the $Z'_{2,3}$ exchanges in the ${\cal{M}}(+,+)$ squared amplitude:
\begin{eqnarray}
{\cal{M}}(+,+)&\simeq& {\cal{M}}(+,+)^{\rm SM}+\frac{s^2}{4}(1-\beta^2)s^2_{\theta_{p}} \Big[c (\Delta G_L+\Delta G_R) \Delta_{01}\\
&&+ d (\Delta G_L+\Delta G_R) \Delta_{11}
+\sum_{i=2,3} \alpha_i (F^i_L+F^i_R) \Delta_ {0i} + \beta_i (F^i_L+F^i_R) \Delta_{1i} \Big]\nonumber\\
&\simeq& {\cal{M}}(+,+)^{\rm SM}+\frac{s^2}{4}(1-\beta^2)s^2_{\theta_{p}}\Big [\sum_{i=2,3} \alpha_i(-\frac{s}{m_i^2}) (F^i_L+F^i_R) \Delta_ {00} \nonumber
\\&& + [c (\Delta G_L+\Delta G_R)+\sum_{i=2,3} \beta_i(-\frac{s}{m_i^2}) (F^i_L+F^i_R) \Delta_ {01}
+ d (\Delta G_L+\Delta G_R) \Delta_{11}\Big]\nonumber
\label{m++}
\end{eqnarray}
with
\begin{eqnarray}
&& c=(g^\gamma_L g^ Z_L+ g^\gamma_R g^ Z_R)(G^ \gamma_L+G^ \gamma_R), \\
&&d=((g^Z_L)^2+(g^Z_R)^2)(G^Z_L+G^Z_R), \\
&&\alpha_i=(g^\gamma_L f^i_L+g^\gamma_R f^i_R)(G^\gamma_L+G^\gamma_R),\\
&&\beta_i=(g^Z_L f^i_L+g^Z_R f^i_R)(G^Z_L+G^Z_R).
\end{eqnarray}
The deviation with respect the SM value can be recast in a form as due to an effective coupling correction, namely,
\begin{eqnarray}
{\cal{M}}(+,+)&\simeq& {\cal{M}}(+,+)^{\rm SM}+\frac{s^2}{4}(1-\beta^2)s^2_{\theta_{p}} \Big[a (\Delta G^{\gamma {\rm eff}}_L+\Delta G^{\gamma {\rm eff}}_R) \Delta_{00} \\&&
\hspace{-1cm}+ [b(\Delta G^{\gamma {\rm eff}}_L+\Delta G^{\gamma {\rm eff}}_R) + c (\Delta G^{Z {\rm eff}}_L+\Delta G^{Z {\rm eff}}_R)] \Delta_{01}+d (\Delta G^{Z {\rm eff}}_L+\Delta G^{Z {\rm eff}}_R) \Delta_{11}\Big]
\label{eff}
\end{eqnarray}
with
\begin{eqnarray}
&& a=((g^\gamma_L)^2+( g^\gamma_R)^2 )(G^ \gamma_L+G^ \gamma_R), \\
&& b=(g^\gamma_L g^ Z_L+ g^\gamma_R g^ Z_R)(G^ Z_L+G^ Z_R).
\end{eqnarray}
By comparing eq.~(\ref{m++}) with eq.~(\ref{eff}) we get the expression for the effective coupling corrections:
\begin{eqnarray}
&& \Delta G^{\gamma {\rm eff}}_{L,R}=\sum_{i=2,3}\frac{\alpha_i G^i_{L,R} (-s/m^2_i)}{a+b \Delta_{01}/\Delta_{00} },\nonumber\\
&&\Delta G^{Z {\rm eff}}_{L,R}=\Delta G^Z_{L,R}+\sum_{i=2,3}\frac{\beta_i G^i_{L,R} (-s/m^2_i)}{c+d \Delta_{11}/\Delta_{01}}.
\label{eff1}
\end{eqnarray}
Notice that the expressions for the effective modifications depend on the CM energy of the process and are obtained for $s<