text:
\section{Introduction}
\label{sec.intro}

Hadron colliders are typically perceived as wonderful machines for direct searches of new particles, but with a limited impact on precision measurements, especially regarding electroweak (EW) observables. There are, however, noteworthy exceptions to this statement. Thanks to the interplay between the large accessible energy
range and a clever selection of channels with relatively low uncertainties, one can perform precision measurements at hadron colliders that can compete with the ones possible at lepton machines such as LEP~\cite{Farina:2016rws,deBlas:2013qqa}.

The key ingredient that allows for this enhanced precision is that
new physics effects tend to grow with the center of mass energy.
If we parametrize the deviations within an effective field theory (EFT) formalism, the leading SM deformations, which generically correspond to operators of dimension six, give rise to amplitudes that can grow up to quadratically with the energy of the process.
In such a situation, having access to the high-energy tails of the kinematic distributions can significantly enhance the achievable precision.

It has been shown that, at the LHC, several simple two-body production channels can be exploited to obtain precision measurements~\cite{Farina:2016rws,deBlas:2013qqa,Domenech:2012ai,Farina:2018lqo,Biekotter:2018rhp,Baglio:2020oqu,Alioli:2020kez,Boughezal:2020uwq}. Among them, diboson production processes, featuring EW gauge bosons or the Higgs boson,
play a privileged role since they can be used to indirectly test the high-energy Higgs dynamics~\cite{Biekotter:2018rhp,Baglio:2020oqu,Falkowski:2015jaa,Butter:2016cvz,Azatov:2017kzw,Franceschini:2017xkh,Liu:2019vid,Bellazzini:2018paj,Banerjee:2018bio,Grojean:2018dqj,Baglio:2018bkm,Almeida:2018cld,Azatov:2019xxn,Banerjee:2019pks,deBlas:2019wgy,Brehmer:2019gmn,Henning:2019vjr,Chiu:2019ksm,Baglio:2019uty,Banerjee:2019twi,Freitas:2019hbk}.

In this paper, we will focus on a specific diboson channel, $Wh$, where the $W$ decays leptonically. Figure~\ref{fig:feynman} shows the leading order SM Feynman diagram (leftmost diagram). At the LHC, this channel can be exploited~\cite{Butterworth:2015bya,Franceschini:2017xkh,Liu:2019vid,Banerjee:2019twi} for precision measurements by only considering decays of the Higgs into a pair of bottom quarks, especially thanks to jet substructure analysis~\cite{Butterworth:2008iy}.
\begin{figure}[t]
    \centering
    \includegraphics{figs/feynman.pdf}
    \caption{Representative Feynman diagrams for $q\,q'\to Wh$ at leading order. The leftmost diagram shows the SM process while the gray circles in the other diagrams represent one insertion of a dimension-6 operator.}
    \label{fig:feynman}
\end{figure}
To give an idea of how this situation will change when the next generation colliders are ready, we show in Table~\ref{tab:events_comparison} the approximate number of $Wh$ events expected for different Higgs decay channels at the LHC and future hadron colliders. These results correspond to the leading order SM prediction for the number of events with high Higgs transverse momentum ($p_T^{h} > 550\, {\rm GeV}$). The $W$ is assumed to decay to first and second generation leptons and only detector acceptance cuts were applied (see upper part of Table~\ref{tab:gen_cuts}). We considered three benchmark colliders: the high-luminosity LHC (HL-LHC), at $14\,$TeV and $3\,{\rm ab}^{-1}$,
the high-energy LHC (HE-LHC), at $27\,$TeV and $15\,{\rm ab}^{-1}$, and the FCC-hh at $100\,$TeV and $30\,{\rm ab}^{-1}$.

One can see that rare channels, such as the final states with the Higgs decaying into two photons or two muons, have branching ratios that are too small to populate the high-energy tail at HL-LHC.
At future high-energy colliders, the situation will improve drastically thanks to a big increase in the production cross section ($\sim 30\times$) and the possibility to collect significantly more integrated luminosity ($\sim 10\times$).
For instance, at FCC-hh, the $\gamma \gamma$ channel is expected to provide $\sim 700$ events, which can allow one to probe new physics effects at the $5$--$10\%$ level.

A clear advantage of these rare decay channels is the fact that the final-state configuration can be easily reconstructed and  background processes are small. In such cases, very simple analysis strategies can give competitive results.
In this paper, we study the Higgs to two photon channel at the FCC-hh.
The complementary $Z(h\to\gamma\gamma)$ channel with $Z\to\ell\ell$ and $Z\to\nu\nu$ also becomes accessible at the FCC-hh but we leave its investigation for future work.

\begin{table}[t]
\centering
$\begin{array}{c c c c c}
\hline
\text{Higgs decay} \quad \quad &  \text{Higgs BR} \quad \quad & n_\textsc{hl-lhc}  \quad \quad & n_\textsc{he-lhc}  \quad \quad & n_{\textsc{fcc}\textrm{\tiny -hh}}  \quad \quad  \\
\hline
\vrule height 13pt depth 5pt width 0pt
\bar{b} b		& 0.6			& 600 		& 1\cdot10^4 	& 2 \cdot 10^5 \\
\tau \tau 		&  6\cdot10^{-2}			&  60 		&  1 \cdot 10^3 	& 2 \cdot 10^4 \\
\gamma \gamma	& 2 \cdot 10^{-3} 	& 2		& 40	& 700\\
\mu \mu 		& 2\cdot 10^{-4} 	& 0.2 	& 4		& 70 \\
4 \ell			& 1\cdot 10^{-4} 	& 0.1	& 2		& 40 \\
\hline
\end{array}$
\caption{Number of $Wh \to \ell \nu \, XX$ events predicted by the SM at LO for different Higgs decay channels and with
a cut $p_T^{h} > 550\,{\rm GeV}$. The results correspond to $3 \, \text{ab}^{-1}$, $|\eta|<2.5$ for the HL-LHC, $15 \, \text{ab}^{-1}$, $|\eta|<6$ for the HE-LHC and $30 \, \text{ab}^{-1}$, $|\eta|<6$ for the FCC-hh.}
\label{tab:events_comparison}
\end{table}

\medskip

The paper is structured as follows. In Section~\ref{sec:EFT}, we discuss the general features
of the $Wh$ production channel and the main new physics effects that can be tested through
its study. Also in Section~\ref{sec:EFT}, we estimate the expected size of the dimension-six Wilson coefficients in generic BSM scenarios.
In Section~\ref{sec:signal_background}, the details of our analysis are presented. In particular, we discuss the features of the signal and background processes and the cut-flow we devised to enhance the sensitivity to new physics effects.
The results of the analysis are collected in Section~\ref{sec.results}, while the summary of our work and some future directions are discussed in Section~\ref{sec:conclusions}.
Finally, we collect in Appendices~\ref{app:HelAmps}, \ref{app:mc_evt_gen}, and~\ref{app:other} some additional details that were not included in the main text.

\section{Theoretical background}
\label{sec:EFT}

\subsection{High energy sensitivity and interference patterns}\label{sec:interference}

In order to parametrize new physics effects we adopt the EFT formalism, focusing on the leading SM deformations
corresponding to dimension-6 operators. We restrict our attention to operators that induce a growth in the $Wh$ amplitude in the high energy limit. We further assume that new physics obeys the minimal flavor violation hypothesis~\cite{DAmbrosio:2002vsn,Chivukula:1987py,Hall:1990ac,Buras:2003jf,Buras:2000dm}.
Hence, we neglected dipole operators or those generated by right-handed charged currents, since they are suppressed by light Yukawa couplings (see Refs~\cite{Alioli:2017ces,Alioli:2018ljm} for a study of scenarios where right-handed charged currents are not subject to MFV suppression).
In the Warsaw basis~\cite{Grzadkowski:2010es}, we can therefore restrict our attention to the three operators:
\begin{eqnarray}
{\cal O}_{\varphi q}^{(3)} &=&\left(\overline{Q}_{L} \sigma^{a} \gamma^{\mu} Q_{L}\right)\left(i H^{\dagger} \sigma^{a} \stackrel{\leftrightarrow}{D}_{\mu} H\right)\,, \\
\mcO_{\varphi {\textsc w}} &=& H^\dagger H\, W^{a, {\mu\nu}} W^a_{\mu\nu}\,, \\
\mcO_{\varphi \widetilde {\textsc w}} &=& H^\dagger H\, W^{a, {\mu\nu}} \widetilde {\textsc W}^a_{\mu\nu}\,.
\end{eqnarray}
where $\sigma^a$ are the Pauli matrices and $\widetilde W^{a, {\mu\nu}} \equiv 1/2\, \epsilon^{\mu\nu\rho\sigma} W^a_{\rho \sigma}$. We define dimensionless Wilson coefficients for the effective operators by introducing explicit powers of the cutoff $\Lambda$. For instance, the coefficient of $\mcO_{\varphi q}^{(3)}$ is $c_{\varphi q}^{(3)}/\Lambda^2$; analogous conventions are used for the other operators.

In our analysis we neglect any modification of the Higgs branching ratio to
$\gamma \gamma$.\footnote{Notice that since $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ modify the $H \to \gamma\gamma$ branching ratio, for large values of the corresponding Wilson coefficients, some cancellation must take place. For instance additional contributions coming from ${\cal O}_{\varphi B}$ could provide such a cancellation, as happens in minimally coupled models.
}

This is justified because by the end of the HL-LHC the bound on the effective $h \gamma \gamma$ coupling is expected to be below $2 \%$, and below $0.3 \%$ after FCC-ee+FCC-hh, from a global analysis of Higgs data~\cite{deBlas:2019rxi}.
Furthermore, the CP-odd operator, $\mcO_{\varphi \widetilde {\textsc w}}$, is strongly constrained by EDM measurements~\cite{Dekens:2013zca,Panico:2018hal,Cirigliano:2019vfc} as we discuss in Section~\ref{sec.results}.

In order to maximize the sensitivity of our analysis to BSM effects,
it is useful to analyze the interference between the SM and the new physics contributions. We recall here that the presence of interference between the SM and the BSM contributions is a key ingredient to enhance the
sensitivity, since in our case of study the SM term always dominates over the BSM contribution. Indeed, in the absence of interference, the BSM contributions come
uniquely from the square of the new physics amplitude and become visible only for very large values of the Wilson coefficients.\footnote{Such a situation is also problematic because additional contributions from dimension-8 operators, which we do not include in our analysis, could play an important role, making the bounds more model dependent~\cite{Contino:2016jqw}.}
\begin{table}[t]
\begin{center}
\begin{tabular}{c | cccc}
  \toprule
  $W$ polarization & SM & ${\cal O}_{\varphi q}^{(3)} $ & ${\cal O}_{\varphi {\textsc w}}$ & ${\cal O}_{\varphi \widetilde{\textsc w}}$ \\
  \midrule
  $\lambda=0$ & $1$  &	 $\dfrac{\hat{s}}{\Lambda^2}$  & $\dfrac{M_W^2}{\Lambda^2}$ & 0 \\[4mm]
  $\lambda=\pm$ &$ \dfrac{M_W}{\sqrt{\hat{s}}} $ & $\dfrac{\sqrt{\hat{s}}\, M_W}{\Lambda^2}$ & $\dfrac{\sqrt{\hat{s}}\, M_W}{\Lambda^2}$ &
  $\dfrac{\sqrt{\hat{s}}\,M_W}{\Lambda^2}$\\
  \bottomrule
\end{tabular}
\end{center}
\caption{High energy behavior of the SM and BSM helicity amplitudes for $pp\rightarrow Wh$.
}
\label{tab:Operator_growth}
\end{table}

The leading high-energy behavior of each helicity amplitude is shown in Table~\ref{tab:Operator_growth}. The leading SM amplitude is the one with a longitudinally-polarized $W$ boson, which behaves as a constant at high center-of-mass energy of the process, $\sqrt{\hat s}$, whereas the transverse polarization channels are suppressed at high energy. This contrasts with the case of $WZ$ production, where the $(+,-)$ and $(-,+)$ polarization channels, which are obviously absent for $Wh$, have the same energy behaviour as the longitudinal one and constitute the main source of background for the longitudinal BSM signal at high energy (at least for leptonic $WZ$ decays).
The only new physics operator that induces a growth of order $\hat s/\Lambda^2$ in the $Wh$ amplitude is ${\cal O}_{\varphi q}^{(3)}$, while $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ generate amplitudes that grow at most with $\sqrt{\hat s}/\Lambda$. It is interesting to notice that the $\mcO_{\varphi q}^{(3)}$ operator mainly contributes to the longitudinal $W$ channel
and therefore can lead to a strong interference with the SM amplitude.
On the contrary, the leading contributions from the $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ operators
are in the transverse $W$ channels, which are subleading for the SM.

\subsubsection*{Differential analysis in $p_T^h$}
If one performs an analysis by integrating over the $W$ decay products,
only amplitudes with the same $W$ polarizations can interfere with each other in the high energy limit. In this case the SM squared amplitude and the leading interference terms have the following $\sqrt{\hat s}$ and $\theta$ dependence
\begin{eqnarray}
\left| \mathcal{M}_{SM} \right|^2 &\sim& \sin^2 \theta \,, \hspace{1.6cm} {\rm Re\,}\mathcal{M}_{SM} \, \mathcal{M}_{\varphi {\textsc w}}^* \sim \frac{M_W^2}{\Lambda^2}\,, \nonumber \\
{\rm Re\,}\mathcal{M}_{SM} \, \mathcal{M}_{\varphi q}^{(3) \, *} &\sim& \frac{\hat s}{\Lambda^2} \sin^2 \theta \,,
 \hspace{1.1cm} {\rm Re\,}\mathcal{M}_{SM} \, \mathcal{M}_{\varphi \widetilde {\textsc w}}^* =0 \,,
 \label{eq:inclusiveInterf}
\end{eqnarray}
where $\theta$ is the scattering angle of the $W$ boson (see Fig.~\ref{fig:decay_angles}). We provide the full expressions for the helicity amplitudes in Appendix~\ref{app:amps_wh}.

The interference term between $\mcO_{\varphi {\textsc w}}$ and the SM is constant and no enhancement with respect to the SM amplitude is present, while the interference between ${\cal O}_{\varphi q}^{(3)}$ and the SM goes like $\hat s/\Lambda^2$. Finally, the $\mcO_{\varphi \widetilde {\textsc w}}$ amplitude does not interfere with the SM amplitude because it is CP odd and can only enter quadratically in the cross section.

An inclusive analysis in the $W$ decay products and differential in $p_T^h$ (which is correlated with $\sqrt{\hat s}$) is therefore expected to provide a good sensitivity to the $\mathcal{O}_{\varphi q}^{(3)}$ operator, but to be rather inefficient in
probing $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$.

\subsubsection*{Double differential analysis in $p_T^h$ and $\phi_W$}

If one considers differential distributions in the decay
angles of the $W$ boson products, the interference between different helicity channels (and CP parities) can be restored~\cite{Panico:2017frx,Azatov:2017kzw,Azatov:2019xxn,Banerjee:2019pks}.
This can be checked explicitly by computing the fully differential amplitudes for $p p  \to W h \to \ell \nu h$ and looking at the interference terms.
For reference, we write down, schematically, the behavior of the squared SM amplitude and the interference terms, see Appendix \ref{app:amps_squared} for the full expressions.
The leading terms in the $M_W/\sqrt{\hat s}$ expansion that depend on the $W \to \ell \nu$ decay angles are
\begin{equation}
\begin{split}\label{eq:sqrd_amps}
\left| \mathcal{M}_{SM} \right|^2 & \sim \frac{1}{4}\sin ^2\theta \sin ^2\theta_W  \
+\frac{M_W}{\sqrt{\hat s}}  {\cal F}(\theta, \theta_W) \cos\phi_W \,, \\
{\rm Re\,}\mathcal{M}_{SM} \, \mathcal{M}_{\varphi q}^{(3)*}& \sim\frac{\hat s}{\Lambda^2} \left[
    \frac{1}{4}\sin ^2\theta \sin ^2\theta_W
    +\frac{M_W}{\sqrt{\hat s}}  {\cal F}(\theta, \theta_W) \cos\phi_W \right]\,,\\
{\rm Re\,}\mathcal{M}_{SM} \, \mathcal{M}_{\varphi {\textsc w}}^* &\sim \frac{\sqrt{\hat s}\,M_W}{\Lambda^2} {\cal F}(\theta, \theta_W) \cos \phi_W\,,\\
{\rm Re\,} \mathcal{M}_{SM} \, \mathcal{M}_{\varphi \widetilde {\textsc w}}^* &\sim \frac{\sqrt{\hat s} \,M_W}{\Lambda^2} {\cal F}(\theta, \theta_W) \sin \phi_W\,,
\end{split}
\end{equation}
where ${\cal F}(\theta, \theta_W) = \left(1-\cos\theta\cos\theta_W\right) \sin\theta  \sin\theta_W$. Since integration over the polar decay angle $\theta_W$ does not destroy the interference terms for $\mcO_{\varphi {\textsc w}}$, $\mcO_{\varphi \widetilde {\textsc w}}$, we choose for the double differential analysis a binning in the azimuthal angle $\phi_W$ and $p_T^h$, see Section~\ref{sec:binning}.

The definition of each of the scattering and decay angles is shown in Fig.~\ref{fig:decay_angles}, where the reference vector $\hat r$ is defined as the direction of the boost of the $Wh$ system in the lab frame.
The scattering angle $\theta$ is defined as the angle between the $W$ boson momentum and $\hat r$ (which for a $2\to2$ process is parallel to the beam axis).
The positive helicity lepton decay angles, $\theta_W$ and $\phi_W$, are defined in the $W$ rest frame. The positive helicity lepton corresponds to $\ell$ or $\nu$ depending on the sign of the charged lepton. See Ref.~\cite{Panico:2017frx} for more details.
\begin{figure}[t]
	\centering
	\includegraphics[width=0.5\linewidth]{figs/decay_angles.pdf}
	\caption{Scattering and decay angles. $f_{\pm}$ denotes the $\pm$ helicity lepton from the $W$ decay.}
	\label{fig:decay_angles}
\end{figure}

From Eq.~\eqref{eq:sqrd_amps}, we find that the leading interference terms involving $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ grow with $\sqrt{\hat s}/\Lambda$. This is true provided that we do not integrate over the azimuthal angle, $\phi_W$.
For the $\mcO_{\varphi {\textsc w}}$ operator, the leading interference terms after integrating over the azimuthal angle are constant, see Eq.~\eqref{eq:cp-even_interf_nu}.
On the other hand the interference involving $\mcO_{\varphi \widetilde {\textsc w}}$ vanishes exactly since the amplitudes have opposite parity.
This is not the case for $\mcO_{\varphi q}^{(3)}$, since this operator mainly contributes to the
longitudinal $W$ amplitude, which is also the leading SM channel.

There is, however, a subtlety connected to the reconstruction of the neutrino. The missing transverse momentum and the kinematics of the charged lepton can be exploited to reconstruct the neutrino momentum only up to a two-fold ambiguity. This ambiguity, in the limit of high neutrino $p_T$, corresponds to a phase shift,
$\phi_W \rightarrow \pi - \phi_W$~\cite{Panico:2017frx}.
Because of this, at high energies, all the terms proportional to $\cos \phi_W$ in Eq.~\eqref{eq:sqrd_amps} vanish, while the ones proportional to $\sin \phi_W$ do not. In particular, the leading interference between the $\mcO_{\varphi {\textsc w}}$ amplitude and the SM, and the subleading terms in the squared and interference amplitudes for the SM and $\mcO_{\varphi q}^{(3)}$ average to zero.\footnote{The interference for the $\mcO_{\varphi {\textsc w}}$ operator could be restored by considering hadronic $W$ decay channels, in which case the decay angles can be reconstructed up to an ambiguity
$(\theta_W, \phi_W) \rightarrow (\pi - \theta_W, \pi - \phi_W)$~\cite{Panico:2017frx}. This channel, however, has much larger backgrounds, so we do not consider it in our analysis.} On the other hand, the leading interference term for $\mcO_{\varphi \widetilde {\textsc w}}$ is unaffected by the ambiguity. It is important to notice that the neutrino ambiguity will not have a significant impact on the bounds of $\mcO_{\varphi q}^{(3)}$, since as shown in Eq.~\eqref{eq:sqrd_amps} its leading interference with the SM is insensitive to $\phi_W$.
\begin{figure}[t]
	\centering
	\includegraphics[width=0.49\linewidth]{figs/dsdfw_ambiguo.pdf}
	\hfill
	\includegraphics[width=0.49\linewidth]{figs/dsdfw_perfect.pdf}
	\caption{Distributions in the azimuthal angle of the $W$, $\sqrt{\hat s} = 1 \, {\rm TeV}$ and integrated over $\theta$ and $\theta_W$.  Gray lines correspond to the SM distribution, blue to $c_{\varphi q}^{(3)} = - 0.03$, red to $c_{\varphi {\textsc w}} = -0.05$,
	and green to $c_{\varphi \widetilde {\textsc w}} = 0.1$. Dotted lines do not include the squared BSM terms, whereas solid lines correspond to the full amplitude. {\bf Left:} distribution taking into account the ambiguity in the neutrino reconstruction
	($\phi_W \rightarrow \pi - \phi_W$). {\bf Right:} distribution with perfect neutrino reconstruction.
    Note that the values of the Wilson coefficients here were chosen to make the figure legible and are not necessarily within the bounds reported later.
	}
	\label{fig:distributions}
\end{figure}

After taking into account the neutrino ambiguity, the first non-vanishing interference term for $\mcO_{\varphi {\textsc w}}$ does not grow with $\sqrt{\hat s}$ anymore; rather, it is constant. Its explicit analytic expression is:
\begin{equation}\label{eq:cp-even_interf_nu}
{\rm Re\,}\mathcal{M}_{SM} \, \mathcal{M}_{\varphi {\textsc w}}^* \sim \frac{M_W^2}{\Lambda^2}
\left[(1 - \cos \theta \cos \theta_W)^2 + \frac{1}{2} \sin^2 \theta \sin^2 \theta_W (1 + \cos 2 \phi_W)\right]\,.
\end{equation}
This expression contains two contributions.
One of them comes from
the interference between the SM and the BSM amplitude with the same $W$ polarization. This term has no $\phi_W$ dependence.
The second contribution comes from an interference between opposite transverse $W$ polarizations. This gives rise to a modulation in the azimuthal angle proportional to $\cos 2 \phi_W$, and vanishes if we integrate over $\phi_W$. A similar modulation in $\cos 2 \phi_W$ can be derived for the SM and $\mcO_{\varphi q}^{(3)}$ distributions looking at the equations in Appendix \ref{app:amps_squared} and averaging them over the neutrino ambiguity.

We show the differential distributions with respect to $\phi_W$ for each BSM operator, and the SM, in Fig. \ref{fig:distributions}. We set $\sqrt{\hat s} = 1$ TeV and integrate over all other kinematical variables ($\theta$, $\theta_W$). On the right panel we show the distributions without taking into account the neutrino ambiguity, while on the left panel we plot the same distributions averaging over $\phi_W$ and $\pi - \phi_W$. As discussed above, the SM, $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi q}^{(3)}$ distributions are almost flat when considering the neutrino ambiguity and only show a mild modulation proportional to $\cos 2 \phi_W$, while the $\mcO_{\varphi \widetilde {\textsc w}}$ distribution is unchanged and goes like $\sin \phi_W$.

\subsection{Power-counting considerations}\label{sec:power-counting}

Before entering into the actual analysis, we briefly provide some estimates of the size of the Wilson coefficients in common BSM scenarios.

In many models, the largest contributions are expected to be associated to the $\mcO_{\varphi q}^{(3)}$ operator which can be easily generated at tree-level through the exchange of fermionic partners of the top or vector resonances charged under the EW group. According to the SILH~\cite{Giudice:2007fh} power counting we expect\footnote{Note that in the SILH basis, $c_{\varphi q}^{(3)} = \frac{\Lambda^2}{v^2}(c'_{Hq} + c_W + c_{HW} + 2 c_{2W})$, see Ref.~\cite{Falkowski:2001958}.}
\begin{equation}
c_{\varphi q}^{(3)} \sim g^2\,,
\end{equation}
where $g$ is the EW gauge coupling. This result is valid in theories in which new physics is either weakly coupled or not directly coupled to the SM fields. This happens, for instance, in composite Higgs scenarios, where the new vector resonances interact with the SM fermions through a mixing with the SM gauge fields.

The power counting estimate changes if new physics is strongly coupled to the SM (both to the SM quarks and the Higgs). In this case the estimate becomes
\begin{equation}
c_{\varphi q}^{(3)} \sim g_*^2\,,
\end{equation}
where $g_*$ is the typical size of the new physics coupling. In the fully strongly-coupled case $g_* \sim 4 \pi$.

The $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ operators are instead typically more suppressed, since they are often generated at loop level~\cite{Giudice:2007fh}.
In the case of weakly coupled new physics one finds
\begin{equation}
c_{\varphi {\textsc w}} \sim c_{\varphi \widetilde {\textsc w}} \sim \frac{g^4}{16 \pi^2}\,.
\label{eq:weakChw}
\end{equation}
If the new physics in the loop are strongly-coupled, then the estimate becomes instead $c_{\varphi {\textsc w}} \sim c_{\varphi \widetilde {\textsc w}} \sim g^2 g_*^2/16\pi^2$. Larger effects in these operators could be present in theories with ``remedios''-like power counting, in which the transverse components of the gauge fields are strongly coupled to new physics~\cite{Franceschini:2017xkh,Liu:2016idz}. In this case the values of the Wilson coefficients could become as large as $c_{\varphi {\textsc w}} \sim c_{\varphi \widetilde {\textsc w}} \sim g_*^4/({16 \pi^2})$.

For completeness, let us mention that $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ could be generated at tree-level by massive vector fields ${\cal L}_1$ in the ${\bf 2}_{-1/2}$ representation of ${\rm SU}(2)_L \times {\rm U}(1)_Y$, via mixing with the Higgs ${\cal L}_{1\mu}^\dagger D^\mu H + {\rm h.c.}$~\cite{deBlas:2017xtg}.
Moreover, this multiplet does not give rise to a tree-level contribution to $\mcO_{\varphi q}^{(3)}$. If the ${\cal L}_1$ multiplet is the lightest new physics state, the $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ operators could therefore receive larger contributions than $\mcO_{\varphi q}^{(3)}$.
This situation, however, does not arise in the most common BSM scenarios.

In any case where $c_{\varphi {\textsc w}}$ or $c_{\varphi \widetilde {\textsc w}}$ is parametrically enhanced with respect to Eq.~\eqref{eq:weakChw}, minimal coupling imposes structural cancellations in the contributions to $h \to \gamma \gamma$~\cite{Giudice:2007fh}. Hence one should be careful when setting bounds on these operators from Higgs data.

\section{Event generation and analysis}\label{sec:signal_background}

Our signal process is $pp\to W(\to\ell\nu)\,h(\to\gamma\gamma$).
The main backgrounds for this process are $W\gamma\gamma$, $W\gamma j$, and $W j j$, with the jets faking a photon.\footnote{We assume that, at the FCC-hh, the rapidity coverage of the detector is large enough such that the background due to $pp\to Z(\to\ell\ell)\,h(\to\gamma\gamma)$ with one missing lepton is negligible.}
We take the rate for a jet to fake a photon to be $P_{j\to\gamma}=10^{-3}$. Moreover, while this rate is conservative with respect to the fake rates reported in~\cite{Contino:2016spe,Abada:2019lih}, see Appendix~\ref{app:mc_evt_gen}, reducing it further does not have a significant impact on the bound we obtain. We simulated the events with \textsc{MadGraph5\_aMC@NLO} v.2.6.5~\cite{Alwall:2014hca} using the \texttt{NNPDF23\_LO} parton distribution functions~\cite{Ball:2013hta}. We used \textsc{Pythia8.2}~\cite{Sjostrand:2014zea} to model the parton shower and to decay the Higgs into two photons. Detector effects were modeled with \textsc{Delphes} v.3.4.1~\cite{deFavereau:2013fsa,Selvaggi:2014mya,Mertens:2015kba,Cacciari:2011ma,Cacciari:2005hq,Cacciari:2008gp} using its FCC-hh card.
For a detailed discussion of the event generation, the applied generation cuts, and QCD and EW radiative corrections, see Appendix~\ref{app:mc_evt_gen}.
\subsection{Selection cuts}
\label{sec:cuts}
To reconstruct the signal, we require at least one electron or muon with $p_T > 30$ GeV, missing
transverse momentum, $\slashed{E}_T>100$ GeV, and at least two photons, each with
$p_T>50$ GeV and with an invariant mass $m_{\gamma \gamma} \in [120,130]$ GeV. If more
than two pairs of photons satisfy this condition, we select the pair with the
smallest distance between the two photons defined as $\Delta R^{\gamma\gamma} =
\sqrt{(\Delta \eta^{\gamma \gamma})^2 + (\Delta \phi^{\gamma \gamma})^2}$.
These selection cuts are summarized in Table~\ref{tab:sel_cuts}.

To further reduce the backgrounds, we apply two additional cuts: first, we impose a maximum separation cut on the diphoton pair, $\Delta R^{\gamma\gamma}_{\max}$;
since the background pair is non-resonant, it tends to have a larger $\Delta R^{\gamma\gamma}$ than the one that originates from the boosted-Higgs.
Second, we impose a maximum cut on the transverse momentum of the reconstructed $Wh$ system,
\begin{equation}
p_T^{Wh} <  p_{T,\max}^{Wh}\,.
\end{equation}
This cut is motivated by the fact that we only expect large $Wh$ transverse momenta from processes recoiling against hard QCD jets and not from the growth with energy we expect from BSM physics.
Moreover, to increase the efficacy of the $\Delta R^{\gamma\gamma}$ and $p_T^{Wh}$  cuts, their values depend on the choice of binning for $p_{T}^{h}$ which is described in the next section.

\begin{table}[t]
	\centering{
		\renewcommand{1.3}{1.25}
		\begin{tabular}{ c @{\hspace{.5em}} | @{\hspace{.5em}} c   }
			\toprule
			& Selection cuts  \\\midrule
			$p_{T,\min}^{\ell}$ [GeV] &  30 \\
			$p_{T, \min}^{\gamma}$  [GeV] & 50 \\
			$\slashed{E}_{T, \, \min}$ [GeV] & 100  \\
			$m_{\gamma \gamma}$ [GeV] & $[120,130]$ \\
			\hline
			$\Delta R^{\gamma\gamma}_{\max}$ & $\{1.3, 0.9, 0.75, 0.6, 0.6\}$\\
			$p_{T,\max}^{Wh}$ [GeV] &  $\{300,500,700, 900, 900\}$  \\
			\bottomrule
		\end{tabular}}
		\caption{
		A summary of the selection cuts applied to the Monte Carlo events. The entries for $\Delta R^{\gamma\gamma}_{\max}$ and $p_{T,\max}^{Wh}$ correspond to the cuts performed in each $p_{T}^{h}$ bin, see Section~\ref{sec:binning} for the definition of the bins.
		}
		\label{tab:sel_cuts}
	\end{table}

\subsection{Binning of the double differential distribution}
\label{sec:binning}

As discussed in Section~\ref{sec:interference}, to maximize the sensitivity to the three operators of interest we need to select events with large $Wh$ invariant mass and be differential in the azimuthal angle of the leptons, $\phi_W$, in order to have linear sensitivity to the
CP-odd operator. Since new physics effects are largely in the central scattering region, it is convenient to perform a binning in the $p_T^{h}$ variable, which is also correlated with $\sqrt{\hat s}$.
In our analysis we use the following binning,
\begin{equation}
\begin{split}
p_T^{h} &\in \{200, 400, 600, 800, 1000, \infty\}\,{\rm GeV}\,,\\
\phi_W &\in [-\pi,0]\,,\; [0,\pi]\,.
\end{split}
\label{eq:binning}
\end{equation}
With this choice, the overflow $p_T^h$ bin contains ${\cal O}(10)$ SM events for $30 \, \text{ab}^{-1}$ of integrated luminosity.
The number of events per $p_T^h$ bin after all the cuts is shown in Fig.~\ref{fig.pTh} for both the signal and the backgrounds.

As explained in Section~\ref{sec:interference}, the binning in $\phi_W$ is chosen to enhance the sensitivity to $\mcO_{\varphi \widetilde {\textsc w}}$ by recovering the interference with the leading SM amplitude.
On the other hand, no improvement is possible for $\mcO_{\varphi {\textsc w}}$ since the neutrino ambiguity
cancels the leading modulation in $\phi_W$, as can be seen in the left panel of Fig.~\ref{fig:distributions}.

To efficiently populate the five $p_{T}^{h}$ bins in Eq.~\eqref{eq:binning}, we generated four runs with a cut on $p_{T}^{h}$ or a proxy for it in the case of backgrounds, see Appendix~\ref{app:mc_evt_gen}.

\subsection{Cut efficiencies}
To evaluate the effectiveness of the selection cuts described in Section~\ref{sec:cuts} in suppressing the backgrounds, we show in Table~\ref{tab.cutflow} the cutflow analysis for the third $p_{T}^{h}$ bin.
Due to small differences in the generation-level cuts (see Table~\ref{tab:gen_cuts}), the starting phase space volumes differ slightly. Nevertheless, the analsyis gives a good idea of which cuts are the most effective in suppressing the backgrounds.
In particular, apart from the efficiency due to the $j\to\gamma$ fake rate, $P_{j\to\gamma}=10^{-3}$, Table~\ref{tab.cutflow} shows the effectiveness of the $m_{\gamma\gamma}$ window cut which reduces the backgrounds by more than one order of magnitude. Furthermore, the $p_T^{Wh}$ cut reduces the $W\gamma\gamma\,(Wj\gamma)$ by a factor of $5\,(3)$ respectively.

\begin{table}
\centering\renewcommand*{1.3}{1.5}
\begin{tabular}{c|>{\centering\arraybackslash}p{0.1\textwidth}|>{\centering\arraybackslash}p{0.1\textwidth}|>{\centering\arraybackslash}p{0.1\textwidth}|>{\centering\arraybackslash}p{0.1\textwidth}}
     \toprule
     \rule[-5pt]{0pt}{.5em}Selection cuts / efficiency  & $\xi_{h \to \gamma \gamma}^{(3)}$ & $\xi_{\gamma \gamma}^{(3)}$ & $\xi_{j \gamma}^{(3)}$ & $\xi_{j j}^{(3)}$ \\
     \hline
     $\geq 1 \ell^\pm$ with $p_{T} > 30$ GeV & $0.86$ & $0.46$ &$0.94$ & $0.94$ \\
     $\geq 2 \gamma$ each with $p_{T}> 50$ GeV& $0.50$ & $0.18$ &$5.7\cdot10^{-3}$ & $8.7\cdot10^{-7}$ \\
     $\slashed{E}_T > 100$\,GeV & $0.49$ & $0.16$ & $5.1\cdot10^{-3}$&  $8.5\cdot10^{-7}$\\
     $120\,\text{GeV} < m_{\gamma\gamma} < 130\,\text{GeV} $ & $0.46$ & $6\cdot10^{-3}$ & $2\cdot10^{-4}$ & $8.2\cdot10^{-8}$ \\
     $\Delta R^{\gamma\gamma}< \Delta R_{max}$ & $0.45$ & $4\cdot10^{-3}$ & $3.1\cdot10^{-5}$ & $6.4\cdot10^{-8}$ \\
     $p_T^{Wh} < p_{T,max}^{Wh}$ & $0.41$ & $7\cdot10^{-4}$ & $1.1\cdot10^{-5}$ & $4.7\cdot10^{-8}$ \\
     \bottomrule
     \end{tabular}%
     \caption{Cut-flow efficiency for the selection cuts. The superscript, $(3)$, refers to the third $p_{T}^{h}$ bin at generation level.
     }
   \label{tab.cutflow}%
\end{table}

The number of SM events in each bin for the signal and background channels after applying the selection cuts are shown in Fig.~\ref{fig.pTh}.
This figure shows that the dominant backgrounds are $W\gamma\gamma$ and $Wj\gamma$. Their size is roughly one third of the signal in the first bin and gradually reduces to roughly $10\%$ of the signal in the last bin.
The $Wjj$ background, however, is at least one to two orders of magnitude smaller than the signal and it can, therefore, be safely neglected.

Let us stress that the exact background projections for $j \gamma$ and $jj$ crucially depend on the jet fake rate into photons and therefore is highly sensitive to the detector performance. Nonetheless, given that even after taking a very conservative estimate for the fake rate the backgrounds are much smaller than the signal; we do not expect our bounds to change much even if the fake rate is further reduced.

In summary, the selection cuts used in our analysis are very efficient in reducing the backgrounds while preserving most of the signal, rendering the $p p \to Wh \to \ell \nu \gamma \gamma$ channel essentially background-free.

\begin{figure}
	\centering
	\includegraphics[width=0.74\linewidth]{figs/ANR_Plot_Paper_Wh_aa_3_Bars_FO-LO_FCC_30ab_SM_mHW5_FadyCode_pTWhCut_5bins_NoFrame_8_5.pdf}
	\caption{Number of SM events per $p_T^{h}$ bin after selection cuts for the signal and backgrounds at the FCC-hh assuming 30 ab$^{-1}$.}
	\label{fig.pTh}
\end{figure}

\section{Results}
\label{sec.results}
In this section, we present the bounds on the Wilson coefficients of the three operators in question. As a first step,  in Section~\ref{sec:exclusive_fit}, we focus exclusively
on the $\mcO_{\varphi q}^{(3)}$ operator. This simplification is justified because, in many BSM scenarios, the $\mcO_{\varphi q}^{(3)}$ operator is generated at tree level while $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ are generated at loop level, see Section~\ref{sec:power-counting}.
Such an exclusive analysis is also sensible in BSM models where the three Wilson coefficients are of the same size.
In fact, the $\mcO_{\varphi q}^{(3)}$ operator induces larger deviations in the $Wh$ distributions and can be tested with much higher accuracy than $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$.
We verify this statement quantitatively in Section~\ref{sec:global_fit} where we perform a combined three-operator analysis.

\subsection{Single operator analysis: $\mcO_{\varphi q}^{(3)}$}\label{sec:exclusive_fit}
As we discussed in Section~\ref{sec:EFT}, the $\mcO_{\varphi q}^{(3)}$ operator contributes
to the amplitude with a longitudinally-polarized $W$ boson. It directly interferes with the leading SM amplitude giving a quadratic growth with energy. In this case, an analysis strategy with a simple binning in the transverse momentum of the Higgs is adequate to extract the bounds on $c_{\varphi q}^{(3)}$. Moreover, further binning in the leptonic azimuthal angle, $\phi_W$, does not appreciably improve the bound because the leading modulation is destroyed by the neutrino ambiguity.
Nevertheless, to be consistent with the double-differential three-operator analysis that we present in the next subsection, we also use the double binning defined in Eq.~\eqref{eq:binning} for the one-operator analysis.

For the statistical analysis, we assume that the likelihood function is Gaussian for simplicity and do a $\chi^2$ analysis. The Gaussian assumption is justified since we do not have any bins with less than $\mathcal{O}(10)$ events for which the Gaussian approximation is already very good. To construct the $\chi^2$ function, then, we need the number of expected signal events as a function of the Wilson coefficients in each bin. The fits as a function of $c_{\varphi q}^{(3)}$ are given in Table~\ref{tab.fitperbin} for each $p_{T}^{h}$ bin while the fits in both $p_{T}^{h}$ and $\phi_W$ bins are given in Table~\ref{tab:sigma_full} in Appendix \ref{app:other}.

\begin{table}
\begin{centering}
\begin{tabular}{c|c|c}
\toprule
\rule[-5pt]{0pt}{.5em}
\multirow{2}{*}{$p_T^{h}$ bin} & \multicolumn{2}{c}{Number of expected events} \tabularnewline
\cline{2-3} \cline{3-3} & Signal & Background \tabularnewline
\hline
\rule[-10pt]{0pt}{2.5em}$[200 - 400]$ GeV & $2620 + 20900\,c_{\varphi q}^{(3)} + 52700\,\big(c_{\varphi q}^{(3)}\big)^{2}$ & $1790$ \tabularnewline
\rule[-10pt]{0pt}{2.5em}$[400 - 600]$ GeV & $566 + 11700\,c_{\varphi q}^{(3)} + 71800\,\big(c_{\varphi q}^{(3)}\big)^{2}$ & $248$ \tabularnewline
\rule[-10pt]{0pt}{2.5em}$[600 - 800]$ GeV & $140 + 5600\,c_{\varphi q}^{(3)} + 67200\,\big(c_{\varphi q}^{(3)}\big)^{2}$ & $43$ \tabularnewline
\rule[-10pt]{0pt}{2.5em}$[800 - 1000]$ GeV & $29 + 1890\,c_{\varphi q}^{(3)} + 36100\,\big(c_{\varphi q}^{(3)}\big)^{2}$ & $8$ \tabularnewline
\rule[-10pt]{0pt}{2.5em}$[1000 - \infty] $ GeV & $7 + 854\,c_{\varphi q}^{(3)} + 33000\,\big(c_{\varphi q}^{(3)}\big)^{2}$ & $3$ \tabularnewline
\bottomrule
\end{tabular}
\par\end{centering}
\caption{Number of expected signal and backgrounds events at FCC-hh with 30 ab$^{-1}$. The signal events number is given as a function of $c_{\varphi q}^{(3)}$ (fixing $\Lambda = 1 \, \text{TeV}$) and the contribution of $Wjj$ to the background is disregarded.
}
\label{tab.fitperbin}
\end{table}

Since we do not know exactly the size of possible systematic errors, we consider three benchmark scenarios with $1\%$, $5\%$, and $10\%$ systematic uncertainty. The $1\%$ benchmark is meant to give an estimate of the maximal sensitivity achievable by removing all possible sources of systematic uncertainty while the $5\%$ one should provide a more realistic assumption, being roughly comparable with the present LHC systematics for production processes with leptonic final states~\cite{Franceschini:2017xkh}. Instead, the $10\%$ benchmark should be considered as a worst-case scenario.

The $\Delta \chi^2 = 3.84$ ($\simeq 95\%$ C.L.) bounds for $c_{\varphi q}^{(3)}$ are shown in Fig.~\ref{fig:fit_cphiq} for $\Lambda = 1$ TeV, as a function of the cutoff of the EFT, $M$, with 1\%, 5\%, and 10\% systematic uncertainty.
The cutoff is enforced by selecting events with $m_{Wh} \leq M$, where $m_{Wh}$ is the $Wh$ invariant mass, for each value of $M$.\footnote{To reconstruct the invariant mass we randomly chose one of the two neutrino solutions. We checked that this procedure gives the same bound as always choosing for a given event, the solution that gives the highest $\sqrt{\hat s}$, i.e., the conservative case where we reject more events.} This result can give an idea of the
dependence of the bound on the cutoff of the EFT, since we can roughly identify $\Lambda$ with $M$.
The bound saturates for $M \sim 4\,{\rm TeV}$ and there is only a mild degradation for $M \simeq 2\,{\rm TeV}$, below which it rapidly becomes much worse. We find then that for $M \gtrsim 4$ TeV, the $95\%$ C.L.~bounds for $c_{\varphi q}^{(3)}$ (with $\Lambda = 1 \, \text{TeV}$) are:
\begin{equation}\label{eq:oq3_bounds}
\begin{split}
c_{\varphi q}^{(3)} &\in [-2.7, 2.5] \times 10^{-3}\quad 1\%\;{\rm syst.},\\
\rule{0pt}{1.75em}c_{\varphi q}^{(3)} &\in [-3.3, 2.9] \times 10^{-3}\quad  5\%\;{\rm syst.},\\
\rule{0pt}{1.75em}c_{\varphi q}^{(3)} &\in [-4.0, 3.4] \times 10^{-3}\quad  10\%\;{\rm syst.}
\end{split}
\end{equation}
The symmetry between the positive and negative bounds on the Wilson coefficients indicates that either the linear interference term between the SM and BSM dominates the bound or that the squared BSM one does.
One can check that in fact it is the linear term that dominates by comparing the quadratic and linear terms in Table~\ref{tab.fitperbin} setting $c_{\varphi q}^{(3)} \sim {\textit few}\times 10^{-3}\,$.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.65\linewidth]{figs/ANR_Chi2vsCutoff_1D_CpQ3_PhiBin_DiffSyst_95CL_8_5.pdf}
	\caption{Expected bounds on $c_{\varphi q}^{(3)}$, setting $\Lambda = 1$\,TeV, at the FCC-hh with 30\,ab$^{-1}$ as a function of the maximal invariant mass cut $M$. The bounds correspond to $\Delta \chi^2 = 3.84$ ($\simeq 95\%$ C.L.) and are obtained from a single parameter fit to the $\mcO_{\varphi q}^{(3)}$ operator. The dashed, solid and dotted blue lines show the bounds for $1\%$, $5\%$ and $10\%$ systematics. The orange shaded area shows the expected bound from a global fit at FCC-ee~\cite{deBlas:2019wgy}, while the shaded red area delimited by a solid (dashed) line corresponds to the HL-LHC with $3 \, \text{ab}^{-1}$  (LHC run 1) bounds from leptonically decaying $WZ$~\cite{Franceschini:2017xkh}. The light blue shaded are corresponds to the bound obtained by LEP~\cite{LEP:2003aa}. The diagonal dashed and solid gray lines show the values of the Wilson coefficient expected in weakly-coupled ($c_{\varphi q}^{(3)}\sim g^2/(4M^2)$) and strongly-coupled ($c_{\varphi q}^{(3)}\sim (4\pi)^2/(4M^2)$) new physics models~\cite{Franceschini:2017xkh}.
	}
	\label{fig:fit_cphiq}
\end{figure}

The diagonal dashed and solid gray lines show the values of the Wilson coefficient that are expected in weakly-coupled new physics models (labeled `Weak' in the plot), with $c_{\varphi q}^{(3)}\sim g^2/(4M^2)$, and strongly-coupled ones (labeled `Strong'), with $c_{\varphi q}^{(3)}\sim (4 \pi)^2/(4M^2)$; see discussion in Section~\ref{sec:power-counting}.
The extra factor of $1/4$ is included to match the conventions of Ref.~\cite{Franceschini:2017xkh}; it also arises in the matching of vector-like-quark extensions of the SM, see Ref.~\cite{deBlas:2017xtg}.

For comparison, the projections obtained for the leptonic $WZ$ at the FCC-hh, assuming $5\%$ systematics give a bound (for $M \gtrsim 5$ TeV)~\cite{Franceschini:2017xkh}
\begin{equation}
\label{eq:FCChh_bounds}
\begin{array}{l@{\hspace{.5em}}l@{\hspace{2.em}}l@{\hspace{2.em}}l}
    \textrm{FCC-hh} & {\small (20\;{\rm ab}^{-1})} & c_{\varphi q}^{(3)} \in [-1.8, 1.4] \times 10^{-3}\; & 5\%\;{\rm syst.},
\end{array}
\end{equation}
which is about $2$ times stronger than the one we find from $W(h \to \gamma \gamma)$.\footnote{The $WZ$ channel fit from Ref.~\cite{Franceschini:2017xkh} exploited a simple binned analysis similar to the one used by us. It neglected possible backgrounds which, however, are not expected to be large.}

To give an idea of the FCC-hh constraining power, in Fig.~\ref{fig:fit_cphiq} we also show the current bounds from LEP and LHC run 1 and the expected bounds from leptonic $WZ$ production at HL-LHC~ and from a global fit at future lepton colliders. We summarize these bounds in the following:
\begin{equation}\label{eq:bounds_1}
\def1.3{1.3}
\begin{array}{l@{\hspace{.8em}}l@{\hspace{2.em}}l@{\hspace{2.em}}l}
\textrm{LEP \cite{LEP:2003aa}} &  & c_{\varphi q}^{(3)} \in [-5.7, 5.7] \times 10^{-1}\;, \\
\textrm{HL-LHC \cite{Franceschini:2017xkh}} & {\small (3\;{\rm ab}^{-1})} & c_{\varphi q}^{(3)} \in [-1.2, 1.0] \times 10^{-2}\; & 5\%\;{\rm syst.}, \\
\textrm{HE-LHC \cite{Franceschini:2017xkh}} & {\small (27\;{\rm TeV},\, 10\;{\rm ab}^{-1})} & c_{\varphi q}^{(3)} \in [-4.0, 3.3] \times 10^{-3}\; & 5\%\;{\rm syst.},\\
\textrm{ILC \cite{deBlas:2019wgy}} & {\small} & c_{\varphi q}^{(3)} \in [-6.0, 6.0] \times 10^{-3}\;, \\
\textrm{CEPC \cite{deBlas:2019wgy}} & {\small} & c_{\varphi q}^{(3)} \in [-1.0, 1.0] \times 10^{-2}\;, \\
\textrm{CLIC \cite{deBlas:2019wgy}} & {\small} & c_{\varphi q}^{(3)} \in [-8.0, 8.0] \times 10^{-3}\;, \\
\textrm{FCC-ee \cite{deBlas:2019wgy}} & {\small} & c_{\varphi q}^{(3)} \in [-6.4, 6.4] \times 10^{-3}\;, \\
\end{array}
\end{equation}
where the lepton collider bounds cited above are valid for $M \gtrsim M_Z$ while the hadron collider ones only for $M \gtrsim 5$ TeV \cite{Franceschini:2017xkh}.

Comparing these results with the ones in Eqs.~(\ref{eq:oq3_bounds}) and~\eqref{eq:FCChh_bounds}, we find that the HE-LHC projections are slightly weaker than ours, and the FCC-ee ones are worse by a factor $\sim 3$. It must, however, be noted that the FCC-ee center of mass energy is much lower than the one at FCC-hh and other hadron colliders. Therefore, the corresponding bound on $c_{\varphi q}^{(3)}$ is also valid for low cutoffs, which cannot be tested at hadron machines.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.47\linewidth]{figs/ANR_Chi2_2D_PhiBin_cpQ_cpW_Profil_8_5.pdf}
	\includegraphics[width=0.47\linewidth]{figs/ANR_Chi2_2D_PhiBin_cpQ_cpWtil_Profil_8_5.pdf} \\
	\vspace{0.3cm}
    \includegraphics[width=0.47\linewidth]{figs/ANR_Chi2_2D_PhiBin_cpW_cpWtil_Profil_8_5.pdf}
	\includegraphics[width=0.47\linewidth]{figs/ANR_Chi2_2D_PhiNoPhi_cpW_cpWtilde_Profil_Syst5_8_5.pdf}
	\caption{Expected $95\%$ C.L. bounds on $c_{\varphi {\textsc w}}$, $c_{\varphi \widetilde {\textsc w}}$, $c_{\varphi q}^{(3)}$ at the FCC-hh for $30 \, \text{ab}^{-1}$. Bounds in green, red, blue, assume 1\%, 5\% and 10\% systematic error. Solid (dashed) lines in top and bottom left panels correspond to the bounds when profiling over (setting to zero) the Wilson coefficient not appearing in the plot {\bf Top Left:} Bounds on the $c_{\varphi {\textsc w}}$, $c_{\varphi q}^{(3)}$ plane. {\bf Top Right:} Bounds on the $c_{\varphi \widetilde {\textsc w}}$, $c_{\varphi q}^{(3)}$ plane.  {\bf Bottom Left:} Bounds on the $c_{\varphi {\textsc w}}$, $c_{\varphi \widetilde {\textsc w}}$ plane. {\bf Bottom Right:} In solid red (dashed red) the bounds when using (not using) the double differential binning in $\phi_W$ assuming $5\%$ systematics in the plane $c_{\varphi {\textsc w}}, \, c_{\varphi \widetilde {\textsc w}}$.
	}
	\label{fig:bounds2d_cphi}
\end{figure}

\subsection{Three operator analysis: {\normalfont $\mcO_{\varphi {\textsc w}}$}, {\normalfont $\mcO_{\varphi \widetilde {\textsc w}}$}, $\mcO_{\varphi q}^{(3)}$}\label{sec:global_fit}
We now extend our analysis to all the three operators, $\mcO_{\varphi q}^{(3)}$, $\mcO_{\varphi {\textsc w}}$, and $\mcO_{\varphi \widetilde {\textsc w}}$, simultaneously.
The number of events in each bin as a function of the Wilson coefficients is reported in Table~\ref{tab:sigma_full} in Appendix~\ref{app:other}.

The $95\%$ C.L. constraints
in the $(c_{\varphi q}^{(3)} - c_{\varphi {\textsc w}})$,
$(c_{\varphi q}^{(3)} - c_{\varphi \widetilde {\textsc w}})$, and $(c_{\varphi {\textsc w}} - c_{\varphi \widetilde {\textsc w}})$ planes are shown in Fig.~\ref{fig:bounds2d_cphi} in the top-left, top-right, and bottom-left panels respectively. The bottom-right panel of the figure shows the effect of binning in $\phi_W$ on the bound on $c_{\varphi \widetilde {\textsc w}}$.
We present two sets of results here. The first one is obtained by profiling over the additional Wilson coefficient and is delineated by solid contours in the plots. The second set of results, delineated by dashed contours, is obtained by setting the remaining Wilson coefficient to zero. All the results are given for the three benchmark scenarios with $1\%$, $5\%$ and $10\%$ systematic uncertainty (green, red, and blue contours, respectively).

As expected, the constraints on $c_{\varphi q}^{(3)}$ are stronger, by roughly one order of magnitude, than the ones on $c_{\varphi {\textsc w}}$ and $c_{\varphi \widetilde {\textsc w}}$. This confirms the
expectation that a one-operator analysis for $c_{\varphi q}^{(3)}$ is fully justified even in BSM scenarios in which
the contributions to all three effective operators are of the same order.

The top left panel in Fig.~\ref{fig:bounds2d_cphi} shows that the $c_{\varphi q}^{(3)}$ and $c_{\varphi {\textsc w}}$ operators are correlated, mainly due to the fact that both operators can only be distinguished by a different growth in the $p_T^{h}$ distribution.
On the other hand, $c_{\varphi \widetilde {\textsc w}}$ is basically uncorrelated with the other two Wilson coefficients.
This is because the linear sensitivity to $c_{\varphi \widetilde {\textsc w}}$ derives from the $\phi_W$ binning which allows for interference with the SM.
The effect of the $\phi_W$ binning can be clearly seen in the bottom right panel of Fig.~\ref{fig:bounds2d_cphi}.
For the 5\% systematic uncertainty benchmark, the bound is $\sim 3$ times stronger with only two $\phi_W$ bins in comparison with no binning. The improvement is even larger for the 1\% systematic uncertainty case.
Meanwhile, the binning in $\phi_W$ has a very mild effect on the bound on $c_{\varphi {\textsc w}}$.

The impact of the systematic error on the fits is also quite strong. For $1\%$ systematics, the bounds are mainly driven by the linear SM-BSM interference terms in the cross section for all three operators.
However, for the $5\%$ and $10\%$ benchmarks, quadratic terms clearly play an important role, significantly worsening and distorting the constraints.

It is also interesting to compare the results obtained by our two analysis procedures, i.e. profiling over versus setting to zero the remaining Wilson coefficient. From the upper left plot in Fig.~\ref{fig:bounds2d_cphi},
one can see that the constraints in the $(c_{\varphi q}^{(3)} - c_{\varphi {\textsc w}})$
plane remain almost unchanged. This is expected, since the correlation between these Wilson coefficients and $c_{\varphi \widetilde {\textsc w}}$
is very small. On the contrary, the correlation between $c_{\varphi q}^{(3)}$ and $c_{\varphi {\textsc w}}$ leads to a significant weakening of the bounds on each of these coefficients when we profile over the other one.
Profiling over $c_{\varphi q}^{(3)}$ or $c_{\varphi {\textsc w}}$ has instead only a minor impact on the determination of $c_{\varphi \widetilde {\textsc w}}$.

\begin{table}
\begin{centering}
\begin{tabular}{c|c|c}
\toprule
Coefficient & Profiled Fit & One Operator Fit \tabularnewline
\hline
$c_{\varphi q}^{(3)}$ &
\begin{tabular}{ll}
\rule{0pt}{1.25em}$[-5.1,\,3.4]\times10^{-3}$ & $1\%$ syst.\\
\rule{0pt}{1.25em}$[-11.6,\,3.8]\times10^{-3}$ & $5\%$ syst.\\
\rule[-.65em]{0pt}{1.9em}$[-20.6,\,4.1]\times10^{-3}$ & $10\%$ syst.
\end{tabular}
&
\begin{tabular}{ll}
\rule{0pt}{1.25em}$[-2.7,\,2.5]\times10^{-3}$ & $1\%$ syst.\\
\rule{0pt}{1.25em}$[-3.3,\,2.9]\times10^{-3}$ & $5\%$ syst.\\
\rule[-.65em]{0pt}{1.9em}$[-4.0,\,3.5]\times10^{-3}$ & $10\%$ syst.
\end{tabular}
\tabularnewline

\hline
$c_{\varphi {\textsc w}}$ &
\begin{tabular}{ll}
\rule{0pt}{1.25em}$[-7.1,\,7.9]\times10^{-2}$ & $1\%$ syst.\\
\rule{0pt}{1.25em}$[-13.0,\,17.5]\times10^{-2}$ & $5\%$ syst.\\
\rule[-.65em]{0pt}{1.9em}$[-20.0,\,25.2]\times10^{-2}$ & $10\%$ syst.
\end{tabular}
&
\begin{tabular}{ll}
\rule{0pt}{1.25em}$[-5.3,\,4.3]\times10^{-2}$ & $1\%$ syst.\\
\rule{0pt}{1.25em}$[-12.1,\,6.8]\times10^{-2}$ & $5\%$ syst.\\
\rule[-.65em]{0pt}{1.9em}$[-18.8,\,9.0]\times10^{-2}$ & $10\%$ syst.
\end{tabular}
\tabularnewline

\hline
$c_{\varphi \widetilde {\textsc w}}$ &
\begin{tabular}{ll}
\rule{0pt}{1.25em}$[-6.4,\,6.4]\times10^{-2}$ & $1\%$ syst.\\
\rule{0pt}{1.25em}$[-9.0,\,8.8]\times10^{-2}$ & $5\%$ syst.\\
\rule[-.65em]{0pt}{1.9em}$[-13.5,\,14.2]\times10^{-2}$ & $10\%$ syst.
\end{tabular}
&
\begin{tabular}{ll}
\rule{0pt}{1.25em}$[-6.1,\,6.1]\times10^{-2}$ & $1\%$ syst.\\
\rule{0pt}{1.25em}$[-8.1,\,8.1]\times10^{-2}$ & $5\%$ syst.\\
\rule[-.65em]{0pt}{1.9em}$[-10.1,\,10.1]\times10^{-2}$ & $10\%$ syst.\\
\end{tabular} \\
\bottomrule

\end{tabular}
\par\end{centering}
\caption[caption]{Bounds at $95\%$ C.L.~on the coefficients of the $\mcO_{\varphi q}^{(3)}$, $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ operators setting $\Lambda = 1 \, \text{TeV}$. {\bf Left column:} bounds profiling over the other two coefficients. {\bf Right column:} bounds with a one operator fit, i.e. setting the other two coefficients to zero.}
\label{tab:bounds_summary}
\end{table}

To get a quantitative idea of the sensitivity to each Wilson coefficient, we report in
Table~\ref{tab:bounds_summary} the bounds on $c_{\varphi q}^{(3)}$, $c_{\varphi {\textsc w}}$ and
$c_{\varphi \widetilde {\textsc w}}$ at the FCC-hh with 30 ab$^{-1}$. We list both the fits obtained through profiling and the ones that take into account each operator separately.

We already compared the one-operator fit bounds on $c_{\varphi q}^{(3)}$ with the ones from other colliders in the previous subsection. Here we notice that the bounds from the profiled fit become significantly worse especially for negative values of the Wilson coefficient, whereas they are relatively stable for positive values. For the $5\%$ systematics benchmark, the profiled bounds are worse than the
ones expected from the $WZ$ channel at HE-LHC and roughly comparable with the FCC-ee ones, see Eq.~(\ref{eq:bounds_1}).

For completeness, we compare our constraints on $c_{\varphi {\textsc w}}$ and $c_{\varphi \widetilde {\textsc w}}$ with projected and current bounds. We compare our constraints on $c_{\varphi {\textsc w}}$ with the projections at the HL-LHC, future lepton colliders and FCC-hh. At 95\% C.L., the bounds are expected to be~\cite{deBlas:2019wgy,deBlas:2019rxi}
\begin{equation}
\begin{array}{l@{\hspace{.8em}}l@{\hspace{3.em}}l@{\hspace{2.em}}l}
\textrm{HL-LHC } \ &{\small (3\;{\rm ab}^{-1})}&c_{\varphi \textsc{w}} \in [-0.4, 0.4]\,,  \\
\textrm{FCC-ee / CEPC / ILC}& &c_{\varphi \textsc{w}}  \in [-0.02, 0.02]\,,\\
\textrm{CLIC}& &c_{\varphi \textsc{w}}  \in [-0.01, 0.01]\,,\\
\textrm{FCC-hh}&{\small (30\;{\rm ab}^{-1})} &c_{\varphi \textsc{w}}  \in [-0.01, 0.01]\,,
\end{array}
\end{equation}
for $\Lambda = 1$ TeV, which for future lepton colliders  and FCC-hh are significantly stronger than our results.\footnote{Recall that in minimally coupled models, large single-operator contributions to $h \to \gamma \gamma$ are structurally correlated, cancelling their contribution.}

The situation is very different for $\mcO_{\varphi \widetilde {\textsc w}}$, which can be indirectly tested
through the contributions it induces to the electric dipole moment of the electron. Barring accidental cancellations with the contributions from other CP-violating operators, the current experimental results give a constraint $c_{\varphi \widetilde{\textsc w}} \lesssim 2 \cdot 10^{-5}$(with $\Lambda = 1$ TeV)~\cite{Panico:2018hal}, which is three orders of magnitudes stronger than our bound.
The current direct bounds on $\mcO_{\varphi \widetilde {\textsc w}}$ at the LHC (with $36$ fb$^{-1}$) are $|c_{\varphi \widetilde{\textsc w}}| \lesssim 11$
and are expected to reach the level $|c_{\varphi \widetilde{\textsc w}}| \lesssim 1$ (with $\Lambda = 1$ TeV) at the HL-LHC~\cite{Bernlochner:2018opw,biektter2020constraining}, which is worse than the bound we obtain. Nonetheless, we expect that the extrapolation of this differential analysis to FCC-hh will overpass the bound derived from our $WH$ analysis.

\subsection{Connection to aTGCs}
\label{sec:Higgs_EDM_aTGC}

We mention that $c_{\varphi q}^{(3)}$ can be written as a combination of vertex corrections and the anomalous triple gauge coupling, $\delta g_{1z}$
\begin{equation}
    c_{\varphi q}^{(3)} = \frac{\Lambda^2}{m_W^2} g^2(\delta g_L^{Zu} - \delta g_L^{Zd} - c_\theta^2 \, \delta g_{1z}) \,,
\end{equation}
where $c_{\theta}$ is the cosine of the Weinberg angle. Therefore, for theories where the vertex corrections are small, the bound on $c_{\varphi q}^{(3)}$ can be recast as a bound on  $\delta g_{1z}$. For universal theories, where $\delta g_L^{Zu}$, $\delta g_L^{Zd}$ depend only on a combination of the oblique parameters $S$, $T$, $W$, $Y$~\cite{Franceschini:2017xkh, Grojean:2018dqj}, this is especially justified, since the oblique parameters are expected to be constrained with excellent accuracy through a variety of measurements at the FCC-ee and FCC-hh, making $\delta g_L^{Zu}$, $\delta g_L^{Zd}$ negligible.

In this way, from the one-operator fit assuming 5\% systematics and setting $\Lambda = 1$~TeV, we obtain
\begin{equation}\label{eq:TGCbound_exclusive}
\delta g_{1z} \in [-5.7, 6.5] \times 10^{-5}\,,
\end{equation}
whereas from the profiled bound we get
\begin{equation}\label{eq:TGCbound_profiled}
\delta g_{1z} \in [-7.5, 22.9]\times 10^{-5}\,.
\end{equation}
For comparison, we collect the current and future estimated bounds on $\delta g_{1z}$:
\begin{equation}
\begin{array}{l@{\hspace{.8em}}l@{\hspace{3.em}}l@{\hspace{2.em}}l}
\textrm{LEP \cite{ALEPH:2003tss}} \ & & \delta g_{1z} \in [-5.1, 3.4]\times 10^{-2}\,,  \\
\textrm{LHC \cite{Grojean:2018dqj}}& &\delta g_{1z}  \in [-15, 1]\times 10^{-3}\,,\\
\textrm{HL-LHC \cite{Franceschini:2017xkh}}& &\delta g_{1z}  \in [-1, 1]\times 10^{-3}\,,\\
\textrm{FCC-ee \cite{deBlas:2019wgy}}& &\delta g_{1z}  \in [-5, 5]\times 10^{-4}\,,
\end{array}
\end{equation}
for $\Lambda = 1$ TeV. The bounds from LHC and HL-LHC were obtained using the diboson production process $pp\rightarrow WV$, with $V=W,\,Z$. It is clear that our results improve the existent and HL-LHC bounds and are even better than the expected bound from FCC-ee.
\section{Summary and conclusions} \label{sec:conclusions}
\begin{figure}[t]
	\centering
	\includegraphics[width=0.8\linewidth]{figs/ANR_AllBounds_1D_Profiled_DiffSyst_95CL_8_5.pdf}
	\caption{$95\%$ C.L bounds on $c_{\varphi q}^{(3)}$, $c_{\varphi {\textsc w}}$ and $c_{\varphi \widetilde {\textsc w}}$. In blue our bounds from $Wh \to \ell \nu \gamma \gamma$ for FCC-hh with 30 ab$^{-1}$ for different systematics for a three operator fit. The black lines with a triangle on top represent the the bound for a one operator fit instead. In light orange the current LEP~\cite{Franceschini:2017xkh} bound for $c_{\varphi q}^{(3)}$. In lighter and darker green for $c_{\varphi q}^{(3)}$, the LHC and HL-LHC bounds using leptonic $WZ$ (5\% syst)~\cite{Franceschini:2017xkh}. In medium green the current bounds on $c_{\varphi q}^{(3)}$ from a global fit~\cite{Ellis:2018gqa}. In darker green also, the HL-LHC projections for $c_{\varphi {\textsc w}}$~\cite{deBlas:2019wgy,deBlas:2019rxi} (obtained from a global fit) and $c_{\varphi \widetilde {\textsc w}}$~\cite{Bernlochner:2018opw,biektter2020constraining} (obtained from CP-odd observables in single Higgs production). Assuming no structural cancellations, we show the bound on $c_{\varphi {\textsc w}}$ from a one parameter fit using $\delta \kappa_\gamma$ in~\cite{deBlas:2019rxi,Cepeda:2019klc} as a triangle on the green bar. In darker orange the FCC-ee bounds from a global fit in~\cite{deBlas:2019wgy} with the configuration FCC-ee Z/WW/$240$~GeV/$365$~GeV. In yellow the EDM bound taken from~\cite{Panico:2018hal}.
	%
 }
	\label{fig:comparison_bounds}
\end{figure}

In this work, we analyzed the $p p \to Wh \to \ell \nu \gamma \gamma$ channel at FCC-hh as a way
to perform precision EW measurements. We focused on new physics effects that grow with energy,
parametrized by dimension-6 effective operators within the SMEFT framework. In particular, we
identified three operators that induce a growth with energy in the amplitude. Adopting the
Warsaw basis convention, they are: $\mcO_{\varphi q}^{(3)}$, which induces a $\hat s$ growth in the longitudinally polarized amplitude, and
$\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$, which induce a $\sqrt{\hat s}$ growth in the transverse ones.

We found that a simple analysis strategy exploiting a binning in the $p_T$ of the Higgs boson
is already sufficient to obtain a good sensitivity to $\mcO_{\varphi q}^{(3)}$.
Testing the $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ operators is more challenging, since they do not interfere with the
leading SM amplitude if we simply use a $p_T^h$ binning.
We found, nonetheless, that a double
differential distribution which also takes into account the azimuthal decay angle of the lepton, $\phi_W$, can be used to recover the interference for $\mcO_{\varphi \widetilde {\textsc w}}$ and significantly strengthen the
bounds. The double binning, however, only has a minor impact on the determination of $\mcO_{\varphi {\textsc w}}$.

We show a summary of the projected 95\% C.L. bounds on the three operators in Fig.~\ref{fig:comparison_bounds}. We give two sets of results. The first, given by the blue
bars, corresponds to the constraints derived from the profiling of a fit including all three effective operators. The second set, shown by the black lines with a triangle on top, corresponds
to the fits including one operator at a time. For each of our fits we consider three benchmark scenarios characterized by different systematic uncertainties: $1\%$ (lighter shading), $5\%$ (medium shading) and $10\%$ (darker shading).
One can see that systematics play a significant role in the bounds obtained. We believe that the $5\%$
benchmark could be a realistic estimate, since it is comparable to the present LHC systematics for similar processes with leptonic final states.

Another interesting feature of our results is the impact of a three-operator fit instead of a single-operator one. We find that the bounds on $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ are almost unchanged, whereas the bound on $\mcO_{\varphi q}^{(3)}$ is strongly affected. This feature comes from a correlation between $c_{\varphi q}^{(3)}$ and $c_{\varphi \textsc{w}}$. However, it is important to stress that, in a majority of BSM scenarios, the deformations parametrised by $\mcO_{\varphi {\textsc w}}$ and $\mcO_{\varphi \widetilde {\textsc w}}$ are expected to be subleading with respect to the
ones to $\mcO_{\varphi q}^{(3)}$. In this class of theories, the correct bound on $\mcO_{\varphi q}^{(3)}$ is the one obtained from the one operator fit.

Regarding the comparison with present and future bounds from other collider experiments, we find that
the one-operator $\mcO_{\varphi q}^{(3)}$ constraints are significantly stronger than the ones achievable at the HL-LHC through leptonic $WZ$ production. They are also marginally better than the expected ones at HE-LHC and FCC-ee. The $WZ$ production channel at FCC-hh could instead provide stronger bounds,
although only by a factor~$\sim 2$, see~Eq.~\eqref{eq:FCChh_bounds}.

We find that our bounds on $\mcO_{\varphi {\textsc w}}$ are competitive with a projected global fit to Higgs data anticipated by the end of the HL-LHC but not at FCC-ee nor FCC-hh.

Our results for the $\mcO_{\varphi \widetilde {\textsc w}}$ determination are one order of magnitude stronger than the ones
achievable at the HL-LHC obtained using CP-sensitive observables from VBF and gluon fusion \cite{Bernlochner:2018opw, biektter2020constraining}. One feature of these observables is that they are linearly sensitive to CP-odd operators, as opposed to inclusive Higgs data. This is also true for the observable we constructed in our analysis. Nevertheless, indirect constraints coming from the current electron EDM bounds are already three orders of magnitude stronger than the expected bound we find.

We conclude by mentioning a few connected research directions that are worth exploring as a continuation of this work. At present we focused on the rare final state $h \rightarrow \gamma \gamma$ since it
provides a cleaner and simpler to analyze channel. Other final states with larger cross section, chiefly among them $h \rightarrow b\bar{b}$, are, however, worth investigating. These channels are more challenging due to the much larger backgrounds, however they could provide access to a significantly larger energy range, allowing us to exploit more efficiently the energy-growing new physics effects.

It is also worth mentioning that $Wh$ is not the only channel that can be exploited for precision measurements. A closely related one is $Zh$ production, that has similar features at FCC-hh. Although this channel has a smaller cross section and is expected to provide weaker bounds on $\mcO_{\varphi q}^{(3)}$, see Ref.~\cite{Banerjee:2018bio}, it allows us to access an additional operator that induces an $\hat s$ growth,
namely ${\cal O}_{\varphi q}^{(1)}$~\cite{Franceschini:2017xkh}.

\section*{Acknowledgments}

We thank Jorge de Blas, Jiayin Gu, Rick S. Gupta, Ayan Paul, J\"urgen Reuter, and Raoul R{\"o}ntsch for useful discussions. M.M.~was supported by the Swiss National Science Foundation, under Project Nos.~PP00P2176884. G.P.~was supported in part by the MIUR under contract 2017FMJFMW (PRIN2017). F.B.,~P.E.,~C.G.~and A.R.~acknowledge support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence
Strategy – EXC 2121 "Quantum Universe" – 390833306. F.B. was also supported by the ERC Starting Grant NewAve (638528). This work was partially performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

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text:
Showing results for tags 'characters'.
Human Female - Imperial - SWTOR era
GreaterRealms posted a gallery image in Icons
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Twilek
Chiss Male - Untextured Clothing
Female Elf
Membership Have you ever had a dream about yours or someone else's character?
Lusitano posted a topic in Member Confessions
Ok so I'm just curious as to whether I'm the only freak here that's had a dream about a character or not. In my case last night, I had a dream about someone else's character. I used to role play with this girl a while ago and she had this character who was severely anti-social. I had a dream about him, his name was Punk. Wasn't sure whether I should run from him or not. Anyone know the character from Lee Child novels (or the movie) Jack Reacher? Kind of like him but with more of an inkling to murder people except like, just a very select couple people.
The Fair Way to Kill a Character?
Kajes posted a topic in Roleplay Chat
Here's the hypothetical... You're playing on an RP forum that is sandbox-y but very character driven. All the players, new and old, have only one or two characters each, which they have invested a lot of time and energy into. Some of these characters were earned through IC means, meaning a player cannot easily create a character of a similar nature without investing significant time into the RP. There are guilds, families, kingdoms, and all sorts of drama-heavy and impactful plots abound between the whole character-base. The players are so interconnected on an IC level, that removing any one character - while not ruining the RP - would cause IC ripples that may become waves. Felt deeply and for a prolong period of time. The setting has some combat-focused elements, but the RP is not at all combat-centric. Yet, characters may kill other players' characters without OoC permission. Meaning, the character you have invested many hours and threads into, that you and others love very dearly, can be killed by another character before you're ready for their story to end. There is no means of resurrecting a dead character. In this hypothetical, what sort of rules would you want to stand between a character and death? Throwing out the "OoC permission" rule altogether. I've contemplated this question for a long while, and am curious to see what other people think! Leave it as a free for all, perhaps stressing "IC actions = IC consequences?" Perhaps a "no-murder" time period for new players or characters, so people have to be invested in the RP before they can kill off well-established characters? A form of voting system, if the character should be allowed to kill another in a given scenario? (This thought sounds far too much like a popularity contest type thing, and makes me cringe and twitch.) A Game Master who (impartially, I hope) determines the "validity" of a given scenario, determining if the character should live or die? A set of fair combat or "killing" rules, mechanics, etc.? Would the possibility of IC or OoC ostracism, or even revenge, keep you - and others - from ever killing another character at all? The social pressure making it a non-issue? I'd love to hear y'all's thoughts!
Female Dwarven Cleric / No Background
Female Dwarven Cleric
Characters you love so much but would/could never play...
Thyme posted a topic in Member Confessions
So I was just scrolling through tumblr and I saw this collage of images of Frodo Baggins and I just stopped for a second and realized that Frodo had been one of my favorite characters since I was in elementary school, but I never have (nor would I) play him on a site -- even though I do play fandom characters... I thought it might be interesting to know if other people have characters they adore beyond pieces but for whatever reason (feel free to divulge), they never have (maybe never would) play them. -- or maybe even characters you tried, but just couldn't do. I'll start us off: Frodo Baggins (LotR) Wolverine (Marvel) Magnus Bane (The Mortal Instruments) Dr. Strange (Marvel) River Tam (Firefly) Delenn (Babylon 5) Natasha Romanoff (Marvel)
Female - Chiss - Star Wars the Old Republic Era
Female - Smiling - Wink
Svirfneblin/Deep Gnome Bard, Male
Female Sith Blue Twilek
Female Zabrak
Springhole RPG Generators
Morrigan posted a resource in Resources
http://www.springhole.net/writing_roleplaying_randomators/index.html
Membership Character Creation Patterns
Dragon posted a topic in Roleplay Chat
Are there any patterns you catch yourself falling into when creating new characters? Things like gender, physical/personality traits or things like birth dates/zodiac signs? Whether or not they keep pets, or the kinds of pets they keep. Any sort of thing, really. Are there any kinds of traits that you intentionally incorporate into all characters? As an example (arguably a particularly odd one), I was once on a site with a player who would literally only play characters with a name that began with C. So what are your patterns and quirks with character building? And, more importantly, is there any specific reason why?
character making
Membership Characters We Love and Love to Hate
StormWolfe posted a topic in Roleplay Chat
This topic was inspired by @MissQ's comment in I'm so tired of.... I am sure it has been touched on in other topics or might have been a topic all on its own at some point (a search did not turn up any results though). Although some ranting and venting is okay, that's not the purpose of this topic. The questions are just guidelines. Please don't think you need to answer all of them or answer them in order. They can be combined, ignored, etc. This is a free-for-all discussion for fun and elucidation. I'll weigh in with my thoughts once we get going. What character concepts do you love? What traits make you avoid a specific type of character? What are your character concept preferences when you create a new character? Please clarify if genre-specific. What traits and concepts do you prefer RPing with (perhaps opposite the type of character you are writing)? When looking over a new RPG to join, what do you need to see to make you drool over creating a character? [Clarification: based on active character profiles, wanteds, etc.]
Membership What do you like in your development?
omgwes posted a topic in Roleplay Chat
Hey y'all. What kind of extras do you like to stuff in those 'free for all' places to enhance your character development? I'm coding tabs for our development threads with different sections for people to put things in. So I'm trying to figure out what kind of things people may want to put in there... I've seen development used for playlists, extra photos / mood boards, thread trackers.... EDIT: For context, I'm adding a screen cap of the code so y'all can see what I mean by tabs / sections. I'm making my code into a tutorial so people can customize it and add whatever they want. Still want ideas on what I could add! Thanks guys!
Dancing Female Twilek - Pink / No background
Female Pink Twi'lek
Membership Sailing both sides of the 'ship
When I was debating what characters I wanted to bring back with Tally, I almost decided not to pick up one of my old favourites. Why? Because I also play his wife. Is that a weird thing? Do you get put off by players who play characters on both sides of a relationship? It was weird when it happened. The characters had some shared backstory, but they were so different. Both were also in relationships with other characters, and one even had a character made for them. Problem was: none of it would stick. The more scenes I wrote that included these two characters, the more they sort of... moved together. It wasn't a conscious thing on my part it just happened. I was so confused by it. I'm writing both halves here, so shouldn't I have full control? Eventually I gave in, and let them be. But I do wonder what people think when they realise that I write both of them, and whether it raises eyebrows. I don't really do scenes with just them, I tend to write short stories (it flows better) to fill in gaps/provide background on what they're doing. They still have active lives outside each other. But... I don't know. It could be weird. It could be seen as playing with myself. I'm not sure. What are your thoughts?
playing with yourself
Membership Character Prompts
We all get into those slumps. The one where writers block just slams us and we cannot figure out how to reply, how to start a new thread, how to get our character to interact with another. It can be difficult to get out of them. One of my favorite ways to do this is to come up with a random list of character prompts. Just small one sentence thread descriptions that could be fun, random, silly, hectic, chaotic, or insane for anyone to be a part of. What are some of your favorite thread prompts?

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text:
Q: Logging in an asynchronous Tornado (python) server I am working on an application in which I may potentially need to log the entire traffic reaching the server. This feature may be turned on or off, or may be used when exceptions are caught.
In any case, I am concerned about the blocking nature of disk I/O operations and their impact on the performance of the server. The business logic that is applied when a request is handled (mostly POST http requests), is asynchronous in such that every network or db calls are asynchronously executed.
On the other hand, I am concerned about the delay to the thread while it is waiting for the disk IO operation to complete. The logged messages can be a few bytes to a few KBs but in some cases a few MBs. There is no real need for the thread to pause while data is written to disk, the http request can definitely complete at that point and there is no reason that the ioloop thread not to work on another task while data is written to disk.
So my questions are:

*

*am I over-worried about this issue? is logging to standard output
and    later redirecting it to a file "good enough"?

*what is the    common    approach, or the one you found most practical for logging    in    tornado-based applications? even for simple logging and not    the    (extreme) case I outlined above?

*is this basically an ideal case for    queuing the logging messages and consume them from a  dedicated    thread?

*Say I do offload the logging to a different thread (like    Homer Simpson's "Can't Someone Else Do It?"), if the thread that    performs the disk logging is waiting for the disk io operation to    complete, does the linux kernel takes that point as an opportunity a    context switch?

Any comments or suggestion are much appreciated,
Erez

A: For "normal" logging (a few lines per request), I've always found logging directly to a file to be good enough. That may not be true if you're logging all the traffic to the server. The one time I've needed to do something like that I just captured the traffic externally with tcpdump instead of modifying my server.
If you want to capture it in the process, start by just writing to a file from the main thread. As always, measure things in your own environment before taking drastic action (IOLoop.set_blocking_log_threshold is useful for determining if your logging is a problem).
If writing from the main thread blocks for too long, you can either write to a queue that is processed by another thread, or write asynchronously to a pipe or socket to another process (syslog?).

A:
" write asynchronously to a pipe or socket to another process
  (syslog?"

How can it be? log_requestis a normal function - not a coroutine and all default python handlers are not driven by asyncio event loop so they are not truly asynchronous. This is imho one of the factors that make Tornado less performant than ie. aiohttp. Writing to the memory or using udp is fast but it is not async anyway.

A: Historic reference
I have ops experience from the time of the question circa 2016 with a Python 3.4 Tornado 4 application running on a decent bare-metal machine. The application interacted with few 3rd-party HTTP APIs, and logged some of the interactions for potential troubleshooting in the future (which is similar to OP's requirements). The machine had a RAID of HDDs. As far as I can recall the application wasn't high-traffic.
Tornado 4 had its own IO loop implementation (Tornado 5+ uses asyncio's now), and there was an interesting code instrumentation, controlled by IOLoop.set_blocking_log_threshold. Basically it logged a WARNING record with the stack trace whenever the loop was blocked longer than the threshold seconds. I can find a couple of screenshots from that time from the Sentry timeline for the very warning where the threshold was set to 1 second.

Most of the warnings had stack traces ending on the logging file handler's flush.  It was a rotating and gzipping file handler. The later may explain what might take longer that a second, but anyway for the application it was desired to keep full responsibility over logging. The solution was the stdlib pair of logging.handlers.QueueHandler and logging.handlers.QueueListener.

Logging queue
Python logging cookbook has a dedicated section on Dealing with handlers that block. Here's the example from it (where listener.start starts a thread that read off the queue and delegates the records to the handler):
que = queue.Queue(-1)  # no limit on size
queue_handler = QueueHandler(que)
handler = logging.StreamHandler()
listener = QueueListener(que, handler)
root = logging.getLogger()
root.addHandler(queue_handler)
formatter = logging.Formatter('%(threadName)s: %(message)s')
handler.setFormatter(formatter)
listener.start()
# The log output will display the thread which generated
# the event (the main thread) rather than the internal
# thread which monitors the internal queue. This is what
# you want to happen.
root.warning('Look out!')
listener.stop()

For a real-world reference of a QueueHanlder implementation that covers the edge cases, chronologer.client.QueueProxyHandler can be used.
asyncio instrumentation
asyncio has a debug mode.

By default asyncio runs in production mode. In order to ease the development asyncio has a debug mode. [...] When the debug mode is enabled:

*

*asyncio checks for coroutines that were not awaited and logs them; this
mitigates the "forgotten await" pitfall.

*Many non-threadsafe asyncio APIs (such as loop.call_soon() and
loop.call_at() methods) raise an exception if they are called from a
wrong thread.

*The execution time of the I/O selector is logged if it takes too long to
perform an I/O operation.

*Callbacks taking longer than 100ms are logged. The
loop.slow_callback_duration attribute can be used to set the minimum
execution duration in seconds that is considered "slow".

It may look richer than what Tornado 4 had, but in fact it's not. First, it's not intended for production (and a pretty important metric is missing). Moreover, it's an after the fact warning without a stack trace, whereas Tornado's implementation was based on signal.SIGALRM and provided the stack trace at the threshold hit.
For the curious
Have you noticed that the warnings didn't go away completely? But I can assure you that the logging issue was fixed. What caused these rare issues, to my surprise, was uuid.uuid4, which can be blocking on a machine with empty entropy pool, but that's a another story.
Further reading

*

*Python-tulip group discussion between asyncio maintainers about
asynchronous file IO, logging and aiofiles library

*A section in Trio documentation that explains the theory and
trafeoffs behind async file I/O

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