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In this chapter the \wwy cross section measurement is interpreted in the context of an EFT with dimension-8 operators, discussed in \Cref{sec:theory:eft}. The EFT model provides a useful framework for parameterizing potential deviations from the SM in LHC data.

\section{EFT MC sample decomposition}
\label{sec:aqgc:sample_decomposition}

The scattering matrix element $\mathcal{M} = \mel{\wwy}{S}{pp}$ for the $S$-matrix derived from the EFT Lagrangian (\Cref{eqn:theory:eft_lagrangian}) can be decomposed into four terms
\begin{equation} \label{eqn:aqgc:decomposition}
  |\mathcal{M}|^{2} = \underbrace{\vphantom{\sum_{i}}|\mathcal{M}_\mathrm{SM}|^{2}}_\mathrm{SM} + \underbrace{\sum_{i} 2c_{i} \mathrm{Re}(\mathcal{M}_\mathrm{SM}^{*}\mathcal{M}_{i})}_\mathrm{SM-EFT\ interference} + \underbrace{\sum_{i} c_{i}^{2} |\mathcal{M}_{i}|^{2}}_\mathrm{Pure\ EFT} + \underbrace{\sum_{i}\sum_{j \neq i} 2c_{i}c_{j} \mathrm{Re}(\mathcal{M}_{i}^{*}\mathcal{M}_{j}^{*})}_\mathrm{EFT-EFT\ interference}
\end{equation}
The first term represents the SM contribution, the second term (linear in $c_{i}$) represents interference between the SM and EFT, the third term (quadratic in $c_{i}$) represents contributions from pure EFT, and the fourth term represents interference between pairs of EFT operators. We neglect the EFT-EFT interference term in this analysis as it is expected to be very small compared to the other three.

Generating MC samples under the assumption of the EFT requires a specific value of the Wilson coefficient $c_{i}$ for the generator to compute the matrix element. In light of \Cref{eqn:aqgc:decomposition}, however, the MC samples generated at a specific Wilson coefficient value $c_{i}=f$ can be re-scaled to arbitrary Wilson coefficient value $f'$ by scaling the pure EFT part by the factor $(f'/f)^{2}$ and the SM-EFT interference part by the factor $f'/f$. Therefore, for each EFT operator under study we generated two MC samples, representing pure EFT (called QUAD) and SM-EFT interference (called SMINT), and applied these factors to the weights of the MC events to obtain distributions at arbitrary Wilson coefficient values. \Cref{fig:aqgc:M0_gradient} shows the effect of scaling the Wilson coefficient of the M0 operator on the \pTgamma distribution in the \emuy region.

\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{figures/aqgc/emuy_M0_gradient}
  \caption{Distribution of \pTgamma in the \emuy region with M0 Wilson coefficient set to various values. Uncertanties are statistical only. Last bin includes overflow.}
  \label{fig:aqgc:M0_gradient}
\end{figure}

To correct for differences between the nominal \wwy sample produced with \textsc{Sherpa} and the EFT samples produced with \textsc{MadGraph}, an alternate \wwy SM sample was produced with \textsc{MadGraph}. The differences between the two \wwy SM samples are plotted in \Cref{fig:aqgc:Sherpa_MG}. The ratio of \textsc{Sherpa}/\textsc{MadGraph}, binned in \pTgamma, is taken as a scale factor to reweight EFT events under the assumption that differences listed in \Cref{sec:samples:samples} would affect the EFT samples similarly to how they affect the SM samples.

\begin{figure}
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/emuy_SherpaMG_ph1Pt}
  \includegraphics[width=.45\textwidth]{figures/aqgc/emuy_SherpaMG_lep1Pt}
  \includegraphics[width=.45\textwidth]{figures/aqgc/emuy_SherpaMG_mLep1Lep2Ph1}
  \caption{Distributions of \pTgamma (upper left), $\pT(\ell_{1})$ (upper right), and $m(e,\mu,\gamma)$ (bottom) in the inclusive \emuy region comparing the \wwy SM \textsc{Sherpa} and \textsc{MadGraph} samples. Uncertanties are statistical only. Last bin includes overflow.}
  \label{fig:aqgc:Sherpa_MG}
\end{figure}

\section{EFT SR definition}

The distributions of \pTgamma for the 13 operators under study, decomposed into QUAD and SMINT components, are plotted in \Cref{fig:aqgc:emuy_ph1Pt}. \pTgamma is a good discriminant between EFT signal and background, so an alternate SR for the EFT analysis was defined with the same criteria as the nominal \emuy SR but requiring $\pTgamma >$ 500 GeV. Event yields attributed to the EFT signal and SM backgrounds in the SR are listed in \Cref{tab:aqgc:SR_yields}. The remaining events with $\pTgamma <$ 500 GeV define the \emuy CR used to constrain the normalization of SM \wwy in the EFT fit. BDT distributions in the CRs for each EFT operator are plotted in \Cref{fig:aqgc:cr_emuy,fig:aqgc:cr_tty,fig:aqgc:cr_vv}. The contribution of EFT signal in the \Zy CR is neglected because of the generator filter (see \Cref{sec:samples:samples}) requiring at least one electron at least one muon in the EFT samples at generator level, however the contribution to this CR is expected to be very small because \wwy SM contribution is already very small.

\begin{table}
  \centering
  \singlespacing
  \begin{tabular}{c|c}
    \hline
    & EFT SR \\
    \hline
    M0 & 1.90 \\
    M1 & 1.86 \\
    M2 & 1.91 \\
    M3 & 1.89 \\
    M4 & 1.91 \\
    M5 & 1.91 \\
    M7 & 1.91 \\
    T0 & 1.89 \\
    T1 & 1.88 \\
    T2 & 1.86 \\
    T5 & 1.90 \\
    T6 & 1.90 \\
    T7 & 1.91 \\
    \hline
    \wwy SM & 1.23 \\
    \tty & 0.37 \\
    \jtoy & 0.29 \\
    \Zy & 0.24 \\
    \VZy & 0.09 \\
    \etoy & 0.01 \\
    \hline
    Total bkg. & 2.23 \\
    \hline
  \end{tabular}
  \caption{EFT signal and SM background contributions to the EFT SR, when the EFT signal samples' Wilson Coefficients are set to the expected upper limit as determined in \Cref{sec:aqgc:limits}.}
  \label{tab:aqgc:SR_yields}
\end{table}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M3}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M4}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_M7}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_T0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_T1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_T2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_T5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_T6}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/emuy_T7}
  \caption{\footnotesize \pTgamma distributions of EFT samples in the inclusive \emuy region when Wilson coefficient is set to its expected upper limit as determined in \Cref{sec:aqgc:limits}. Each sample's QUAD and SMINT components is also shown. Uncertanties are statistical only. Last bin includes overflow.}
  \label{fig:aqgc:emuy_ph1Pt}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M3}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M4}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_M7}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_T0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_T1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_T2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_T5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_T6}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_emuy_T7}
  \caption{\footnotesize BDT distributions of EFT samples in the \emuy CR when Wilson coefficient is set to its expected upper limit as determined in \Cref{sec:aqgc:limits}. Each sample's QUAD and SMINT components is also shown. Uncertanties are statistical only. Last bin includes overflow.}
  \label{fig:aqgc:cr_emuy}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M3}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M4}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_M7}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_T0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_T1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_T2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_T5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_T6}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_tty_T7}
  \caption{\footnotesize BDT distributions of EFT samples in the \tty CR when Wilson coefficient is set to its expected upper limit as determined in \Cref{sec:aqgc:limits}. Each sample's QUAD and SMINT components is also shown. Uncertanties are statistical only. Last bin includes overflow.}
  \label{fig:aqgc:cr_tty}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M3}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M4}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_M7}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_T0}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_T1}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_T2}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_T5}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_T6}
  \includegraphics[width=.27\textwidth]{figures/aqgc/distributions/cr_vv_T7}
  \caption{\footnotesize BDT distributions of EFT samples in the $VV$ CR when Wilson coefficient is set to its expected upper limit as determined in \Cref{sec:aqgc:limits}. Each sample's QUAD and SMINT components is also shown. Uncertanties are statistical only. Last bin includes overflow.}
  \label{fig:aqgc:cr_vv}
\end{figure}

\section{Limits on Wilson coefficients}
\label{sec:aqgc:limits}

Upper and lower limits on the Wilson coefficients are determined by a maximum likelihood fit to the one-bin EFT SR and binned BDT distributions of the CRs. An independent fit was performed for each of the 13 Wilson coefficients. \TRExFitter was used to perform the fit with the same likelihood and fit settings as described in \Cref{sec:wwy:xsec} but with some key differences:
\begin{enumerate}
  \item The EFT processes are considered the signal in the EFT fits.
  \item The SM \wwy process, considered signal in the cross section measurement, is considered background in the EFT fit.
  \item The SM \wwy normalization factor is considered a nuisance parameter.
  \item A new unconstrained parameter representing the Wilson coefficient is defined as the POI.
  \item The Asimov dataset is defined as the SM background expectation with NP values set to the results of the CR only fit of \Cref{sec:wwy:xsec}.
\end{enumerate}
The normalization factor of the SMINT component of the signal is defined to depend linearly on the POI, and the normalization factor of the QUAD component is defined to depend quadratically on the POI. The EFT MC samples were generated with Wilson coefficient = 1 \WilsonCoeffUnit, so with this prescription the POI is essentially the Wilson coefficent itself in units of \WilsonCoeffUnit. The fit results are presented for the M0 operator in \Cref{fig:aqgc:norm_factors_M0,fig:aqgc:corr_matrix_M0,fig:aqgc:pulls_0_M0,fig:aqgc:pulls_1_M0,fig:aqgc:pulls_2_M0,fig:aqgc:pulls_3_M0,fig:aqgc:pulls_4_M0,fig:aqgc:pulls_5_M0,fig:aqgc:pulls_6_M0}.

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.8\textwidth]{figures/aqgc/NormFactors_M0}
  \caption{Best-fit normalization factors for the M0 Wilson coefficeint, \wwy, \tty, \Zy, \etoy fakes with $\pm 1\sigma$ uncertainties.}
  \label{fig:aqgc:norm_factors_M0}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=\textwidth]{figures/aqgc/CorrMatrix_M0}
  \caption{Correlations of fit parameters in the M0 EFT fit.}
  \label{fig:aqgc:corr_matrix_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Luminosity_M0}
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Pileup_M0}
  \caption{Luminosity and pile-up nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_0_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Signal_modelling_M0}
  \caption{Signal modeling nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_1_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Flavor_tagging_M0}
  \caption{Flavor tagging nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_2_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Leptons_M0}
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Photons_M0}
  \caption{Lepton and photon nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_3_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_DataDriven_M0}
  \caption{Data driven nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_4_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.60\textwidth]{figures/aqgc/NuisPar_Background_modelling_M0}
  \caption{Background modeling nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_5_M0}
\end{figure}

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_Jets_M0}
  \includegraphics[width=0.80\textwidth]{figures/aqgc/NuisPar_MET_M0}
  \caption{Jets and MET nuisance parameter pulls and constraints in the M0 EFT fit.}
  \label{fig:aqgc:pulls_6_M0}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=1.0\textwidth]{figures/aqgc/Ranking_M0}
  \caption{Pre-fit and post-fit impacts of nuisance parameters on the M0 Wilson coefficient (POI).}
  \label{fig:aqgc:ranking_M0}
\end{figure}

The best-fit value of the Wilson coefficient, from \Cref{fig:aqgc:norm_factors_M0}, is of course 0 because the Asimov dataset is defined as the SM expectation. The uncertainties tell us that Wilson coefficient values in the range [-5.16, 5.16] are consistent with the SM at $\pm 1\sigma$ level. The expected upper and lower limits at the 95\% confidence level (CL) are computed using Wilks' theorem\cite{10.1214/aoms/1177732360}, which states that $-2\Delta\ln\mathcal{L}$ asymptotically (in the limit of many events) converges to a $\chi^{2}$ distribution. A $p$-value of 0.05 for a $\chi^{2}$ distribution with one degree of freedom corresponds to $\chi^{2}=3.84$, hence the upper and lower expected limits are extracted from the likelihood curve where it intersects $-\Delta\ln\mathcal{L}=3.84/2$. The likelihood curves for the 13 Wilson coefficients are plotted in \Cref{fig:aqgc:LHscans_0,fig:aqgc:LHscans_0,fig:aqgc:LHscans_2} and the expected limits are listed in \Cref{tab:aqgc:limits}.

\begin{table}[!ht]
  \centering
  \singlespacing
  \begin{tabular}{|c|c|c|c|}
    \hline
    & Limits @ 95\% CL & Limits from ATLAS run-1\cite{STDM-2016-05} & Best limits to date \\
    \hline
    M0 & [--7.85, 7.85] & [-300, 300] & [-0.69, 0.70] \nocite*{CMS-SMP-18-006} \\
    M1 & [--12.83, 12.47] & [-500, 500] & [-2.0, 2.1] \nocite*{CMS-SMP-18-006} \\
    M2 & [--3.07, 3.07] & [-1800, 1800] & [-1.9, 1.9] \nocite*{CMS-SMP-21-011} \\
    M3 & [--4.97, 4.91] & [-3100, 3100] & [-2.7, 2.7] \nocite*{CMS-SMP-21-011} \\
    M4 & [--7.72, 7.72] & [-1100, 1100] & [-3.7, 3.6] \nocite*{CMS-SMP-21-011} \\
    M5 & [--6.22, 6.22] & [-1700, 1700] & [-3.9, 3.9] \nocite*{CMS-SMP-21-011} \\
    M7 & [--25.37, 25.37] & [-1100, 1100] & [-3.4, 3.4] \nocite*{CMS-SMP-18-006} \\
    T0 & [--1.40, 1.38] & [-100, 100] & [-0.12, 0.11] \nocite*{CMS-SMP-18-006} \\
    T1 & [--1.68, 1.65] & [-200, 200] & [-0.12, 0.13] \nocite*{CMS-SMP-18-006} \\
    T2 & [--3.82, 3.72] & [-400, 400] & [-0.85, 1.0] \nocite*{CMS-SMP-21-011} \\
    T5 & [--1.09, 1.08] & [-1500, 1600] & [-0.31, 0.35] \nocite*{CMS-SMP-21-011} \\
    T6 & [--1.31, 1.31] & [-1900, 1900] & [-0.25, 0.27] \nocite*{CMS-SMP-21-011} \\
    T7 & [--2.92, 2.92] & [-4300, 4300] & [-0.67, 0.73] \nocite*{CMS-SMP-21-011} \\
    \hline
  \end{tabular}
  \caption{Expected limits, in units of \WilsonCoeffUnit, at the 95\% CL of the 13 Wilson coefficients.}
  \label{tab:aqgc:limits}
\end{table}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M0}
  \caption{Negative log likelihood vs.\ Wilson coefficient (POI) with and without systematic uncertainties.}
  \label{fig:aqgc:LHscans_0}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M1}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M2}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M3}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M4}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M5}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_M7}
  \caption{Negative log likelihood vs.\ Wilson coefficient (POI) with and without systematic uncertainties.}
  \label{fig:aqgc:LHscans_1}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_T0}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_T1}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_T2}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_T5}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_T6}
  \includegraphics[width=.45\textwidth]{figures/aqgc/LH_scan_T7}
  \caption{Negative log likelihood vs.\ Wilson coefficient (POI) with and without systematic uncertainties.}
  \label{fig:aqgc:LHscans_2}
\end{figure}

\clearpage
\section{Unitarity restoration}
\label{sec:aqgc:unitarization}

As discussed in \Cref{sec:theory:eft}, the EFT operators are known to be non-unitary at high \sqrtS. The unitarity bounds of \Cref{tab:theory:unitarity_bounds} at \sqrtS = 3 TeV where the distribution of $m(\wwy)$ at parton level (\Cref{fig:theory:mWWy_truth}) peaks are smaller than the expected limits of \Cref{tab:aqgc:limits}, indicating likely unitary violation. We implemented two model-dependent procedures to restore unitarity to the EFT model, and computed the dependence of the 95\% CL limits on the model parameters.

\subsection*{Clipping method}

The first method to restore unitarity to the MC samples is simply to veto events with parton-level $m(\wwy)$ greater than some energy threshold. The energy threshold is an arbitary model parameter, as we do not know \textit{a priori} what values of \sqrtS violate unitarity in nature. The upper and lower expected limits at the 95\% CL vs.\ the energy threshold are presented in \Cref{fig:aqgc:unitarity_clipping_0,fig:aqgc:unitarity_clipping_1,fig:aqgc:unitarity_clipping_2}.

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M0}
  \caption{95\% CL upper and lower expected limits on Wilson coefficients vs.\ clipping energy.}
  \label{fig:aqgc:unitarity_clipping_0}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M1}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M2}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M3}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M4}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M5}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_M7}
  \caption{95\% CL upper and lower expected limits on Wilson coefficients vs.\ clipping energy.}
  \label{fig:aqgc:unitarity_clipping_1}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_T0}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_T1}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_T2}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_T5}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_T6}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_clipping_T7}
  \caption{95\% CL upper and lower expected limits on Wilson coefficients vs.\ clipping energy.}
  \label{fig:aqgc:unitarity_clipping_2}
\end{figure}

\clearpage
\subsection*{Dipole form factor method}

An alternative method of restoring unitarity is to introduce a form factor term\cite{Eboli:2003nq} to the Wilson coefficient:
\begin{equation}
c_{i} \rightarrow \frac{c_{i}}{\left( 1 + \frac{\sqrt{s}^{2}}{\Lambda_\mathrm{FF}^{2}} \right)^{n}}
\end{equation}
where $\Lambda_\mathrm{FF}$ and $n$ are arbitrary model parameters. It is conventional to choose $n=2$ for a dipole form factor, but $\Lambda_\mathrm{FF}$ remains an arbitary parameter. The form factor is implemented by re-weighting EFT events with the dipole form factor as a coefficeint, with parton-level $m(\wwy)$ representing \sqrtS. \Cref{fig:aqgc:unitarity_FF_0,fig:aqgc:unitarity_FF_1,fig:aqgc:unitarity_FF_2} presents the $\Lambda_\mathrm{FF}$ dependence of the expected 95\% CL upper and lower limits.

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M0}
  \caption{95\% CL upper and lower expected limits on Wilson coefficients vs.\ $\Lambda_\mathrm{FF}$.}
  \label{fig:aqgc:unitarity_FF_0}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M1}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M2}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M3}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M4}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M5}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_M7}
  \caption{95\% CL upper and lower expected limits on Wilson coefficients vs.\ $\Lambda_\mathrm{FF}$.}
  \label{fig:aqgc:unitarity_FF_1}
\end{figure}

\begin{figure}[!ht]
  \centering
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_T0}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_T1}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_T2}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_T5}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_T6}
  \includegraphics[width=.45\textwidth]{figures/aqgc/unitarity_FF_T7}
  \caption{95\% CL upper and lower expected limits on Wilson coefficients vs.\ $\Lambda_\mathrm{FF}$.}
  \label{fig:aqgc:unitarity_FF_2}
\end{figure}
