text:
\section{Introduction}
\label{sect:cp_intro}

The search for $CP$ violation is an important research direction of future experiments in particle physics.
$CP$ violation is one of the requirements for baryogengesis~\cite{Sakharov:1967dj}. So far the only experimental
evidence for $CP$ violation comes from quark flavor physics, which is consistent with the CKM mechanism appearing
in the Standard Model (SM) of particle physics~\cite{Kobayashi:1973fv}.
This SM mechanism is believed to be insufficient for generating the observed predominance
of baryon matter over antimatter on a cosmological scale~\cite{Shaposhnikov:1987tw}.
Therefore, the search for $CP$ violation in interactions of the Higgs boson ($H$) with
either fermions or bosons is an interesting path to search for a new mechanism.

Through the study of the $HVV$ and $H\!f\!\bar{f}$ tensor structure of interactions of the $H$~boson\xspace with vector bosons
($Z, W, \gamma, g$) and fermions ($t$, $\tau$), the CMS and ATLAS experiments on LHC have established that
the $J^{PC}$ quantum numbers of the $H$~boson\xspace should be $0^{++}$, if this boson has definite $P$ and $C$~\cite{Chatrchyan:2012jja,
Aad:2013xqa,Chatrchyan:2013mxa,Khachatryan:2014kca,Aad:2015mxa,Khachatryan:2016tnr,
Aad:2016nal,Sirunyan:2017tqd,Aaboud:2017oem,Aaboud:2017vzb,Aaboud:2018xdt,Sirunyan:2019twz,
Sirunyan:2019nbs,Sirunyan:2020sum,ATLAS:2020ior,Aad:2020mnm,Sirunyan:2021fpv,CMS:2021sdq,
ATLAS:2021pkb,CMS:2022ley,Collaboration:2022mlq,CMS-PAS-HIG-21-006,ATLAS-CONF-2022-016}.
This observation is consistent with the  Standard Model (SM) expectation for these quantum numbers to be that of
the vacuum. However, small violation of $CP$ symmetry in those interactions cannot be excluded within the experimental
precision of current measurements. Squeezing the allowed range of $CP$-violating parameters, or, alternatively,
discovering non-zero $CP$ violation in the $H$~boson\xspace interactions, becomes an important target of experimental
measurements~\cite{Nelson:1986ki,Soni:1993jc,Chang:1993jy,Barger:1993wt,
Arens:1994wd,BarShalom:1995jb,Gunion:1996xu,Han:2000mi,Plehn:2001nj,Choi:2002jk,Buszello:2002uu,Hankele:2006ma,
Accomando:2006ga,Godbole:2007cn,Hagiwara:2009wt,Gao:2010qx,DeRujula:2010ys,Christensen:2010pf,
Gainer:2011xz,Bolognesi:2012mm,Ellis:2012xd,Chen:2012jy,Gainer:2013rxa,Artoisenet:2013puc,Anderson:2013afp,Chen:2013waa,
Chen:2013ejz,Gainer:2014hha,Gonzalez-Alonso:2014eva,Dolan:2014upa,Demartin:2014fia,Chen:2014gka,Buckley:2015vsa,
Greljo:2015sla,Gritsan:2016hjl,deFlorian:2016spz,Hartmann:2015aia,Dawson:2018pyl,Dedes:2018seb,Dawson:2018liq,
Brivio:2019myy,Gritsan:2020pib,Martini:2021uey,Davis:2021tiv,Barman:2021yfh,Barman:2022pip}.

Future high-energy physics experiments, either planned or proposed, have unique features for testing $CP$ violation
in the $H$~boson\xspace interactions. For example, photon and muon colliders with a beam polarization scan
could provide a unique opportunity to search for $CP$ violation in couplings to either photons or muons.
The electron-positron collider is positioned uniquely to search for $CP$ violation in $HVV$ interactions, with vector bosons
appearing in electron-positron annihilation, and allow for $CP$ studies in decay. Proton colliders provide an array
of opportunities for $HVV$ and $H\!f\!\bar{f}$ studies in both production and decay, as already demonstrated by the LHC experiments.

This makes $CP$ violation studies with the $H$~boson\xspace an attractive reference measurement in the Particle Physics
Community Planning Exercise (a.k.a. ``Snowmass"), organized by the US High Energy Physics community
to set directions in the field of particles physics for the next decade and beyond. Benchmark measurements of
$CP$ violation in interactions of the $H$~boson\xspace with SM particles provide a limited and well-defined set of parameters
that could be tested at future high-energy physics colliders. Moreover, these measurements can also be achieved
through study of virtual effects in quark flavor physics and electric dipole moment (EDM) measurements.
These $CP$ violation effects are tiny in the SM, and they therefore become excellent null tests for comparing
performance of future facilities. Beyond-the-SM (BSM) theories predict sizable $CP$ violation effects, which could
have profound implications for the future of particle physics, if discovered.

\section{Framework of the Higgs $CP$ study for Snowmass}
\label{sect:cp_framework}

Since the discovery of the $H$~boson\xspace by the ATLAS and CMS experiments on the Large Hadron Collider
(LHC)~\cite{Aad:2012tfa,Chatrchyan:2012xdj}, the search for $CP$ violation in its interactions started
immediately~\cite{Chatrchyan:2012jja,Aad:2013xqa,Chatrchyan:2013mxa}. The $CP$ violation parameters
were identified as benchmark measurements in the  Snowmass-2013 Particle Physics Community Planning
Exercise~\cite{Dawson:2013bba}. In that study, $CP$-violating parameters were defined in the coupling
of the $H$~boson\xspace to massive vector bosons ($HVV$), to massless vector bosons
($H\gamma\gamma$, $HZ\gamma$ , $H$gg), and to fermions ($Ht\bar{t}$, $H\tau\tau$, $H\mu\mu$).
In this work, we build on that study, take advantage of both experimental and theoretical progress
in the study of the $H$~boson\xspace interactions over the past decade, and make assessment of prospects
of the $H$~boson\xspace $CP$ study at the future facilities, both proposed and planned.

On the experimental front, measurements of $CP$ violation parameters have been achieved on LHC
experiments~\cite{Chatrchyan:2012jja,
Aad:2013xqa,Chatrchyan:2013mxa,Khachatryan:2014kca,Aad:2015mxa,Khachatryan:2016tnr,
Aad:2016nal,Sirunyan:2017tqd,Aaboud:2017oem,Aaboud:2017vzb,Aaboud:2018xdt,Sirunyan:2019twz,
Sirunyan:2019nbs,Sirunyan:2020sum,ATLAS:2020ior,Aad:2020mnm,Sirunyan:2021fpv,CMS:2021sdq,
ATLAS:2021pkb,CMS:2022ley,Collaboration:2022mlq}.
This allows us to make realistic quantitative projections to the HL-LHC.
The main change on the theoretical front has been development of the Effective Field Theory (EFT)
framework, and in particular SMEFT, where $CP$ violation naturally appears from a sub-set of
higher-dimension operators. Both developments have been discussed within the LHC Higgs and
LHC EFT Working Groups~\cite{lhcxwg,lhceft}, where we rely on some of their efforts.

One could consider the study of $CP$ violation to be redundant with respect to the larger
project of EFT global fits. However, there are two reasons which make this consideration less
reliable for the Snowmass studies. First of all, the global EFT fits are very complex with many parameters
and assumptions invoked to reduce the number of those parameters. One of the common constraints
applied in the current global EFT fits is the lack of $CP$-odd operators, effectively setting all
$CP$ violation effects to zero. Second, even when such $CP$ constraints are not invoked, the
actual measurements are often based on experimental information which is not necessarily
truly $CP$-sensitive. This means that the measurement may be sensitive to the presence of
the higher-dimension operators, but may not distinguish well between $CP$-odd and $CP$-even terms.
Therefore, the idea of the dedicated $CP$-sensitive measurements of the $H$~boson\xspace for the Snowmass
studies is to provide simple but at the same time reliable benchmarks which could serve as a guide
to compare future facilities.

As an example of the future measurement projections based on the global EFT fits, let us refer to the
$H$~boson\xspace studies performed for the 2020 European Strategy for Particle Physics Update~\cite{deBlas:2019rxi}.
In the EFT description of the $H$~boson\xspace couplings, either 18 (with flavor universality) or
30 (with neutral diagonality) $CP$-even operators in the so-called Higgs basis were considered
within the SMEFT framework, which invokes the SU(2)$\times$U(1) EW symmetry.
To assess the sensitivity to deviations from the SM in a basis-independent way
the results of the fit were projected onto the following $H$~boson\xspace effective couplings:
\begin{equation}
{\rm g}^{\rm eff~2}_{HX}\equiv
\frac{\Gamma_{H\to X}}{\Gamma^{\rm SM}_{H\to X}}
\,.
\label{eq:gHX}
\end{equation}
These parameters are convenient to compare different studies in a straightforward manner.
However, these parameters do not allow for the $CP$ structure in the $HX$ interaction.
Therefore, we expand this set of $CP$-conserving parameters with the following set,
allowing for $CP$ violation in each $HX$ interaction:
\begin{equation}
f_{CP}^{HX}\equiv
\frac{\Gamma^{CP\rm\,odd}_{H\to X}
}{\Gamma^{CP\rm\,odd}_{H\to X}
+\Gamma^{CP\rm\,even}_{H\to X}}
\,,
\label{eq:fCP}
\end{equation}
where the partial decay $H\to X$ width is calculated with either the $CP$-odd or $CP$-even part of the amplitude.
This definition is consistent with the $CP$-sensitive parameters $f_{CP}$ defined for the Snowmass-2013
study~\cite{Dawson:2013bba}. These $f_{CP}$ parameters have been adopted in the LHC measurements
as well, for a recent summary refer to Ref.~\cite{Sirunyan:2021fpv}.
Therefore, we adopt Eq.~(\ref{eq:fCP}) for the benchmark parameter measurements.

We note that Eq.~(\ref{eq:fCP}) is defined in decay of the $H$~boson\xspace. For example,
the general scattering amplitude that describes the interaction of the $H$~boson\xspace
with the fermions, such  as $\tau^+\tau^-$, $\mu^+\mu^-$,  $b\bar{b}$, and $t\bar{t}$,
can be written as
\begin{eqnarray}
&& A( H \to f\bar f) = \frac{m_f}{v}
\bar u_2 \left ( b_{1}^{H\!f\!\bar{f}} + i b_{2}^{H\!f\!\bar{f}} \gamma_5 \right ) u_1
\,.
\label{eq:ampl-spin0-qq}
\end{eqnarray}
Therefore, the $CP$-sensitive parameter takes the form
\begin{equation}
f_{CP}^{H\!f\!\bar{f}}\equiv
\frac{|b_2^{H\!f\!\bar{f}}|^2}{|b_1^{H\!f\!\bar{f}}|^2+|b_2^{H\!f\!\bar{f}}|^2} = \sin^2\left( \alpha^{H\!f\!\bar{f}} \right)
\,.
\label{eq:fHff}
\end{equation}
Technically, Eq.~(\ref{eq:fCP}) does not cover $Ht\bar{t}$ interactions, because the decay $H\to t\bar{t}$ is not possible.
However, we expand the definition in Eq.~(\ref{eq:fHff}) to all fermion couplings. The effective mixing angle $\alpha^{H\!f\!\bar{f}}$,
introduced in Eq.~(\ref{eq:fHff}), is often used in describing the $CP$-odd amplitude contribution. However, we adopt a
more general parameterization with effective cross-section fractions because they allow more than two amplitude
contributions, as this becomes important in description of the $HVV$ interactions, discussed below.

For the coupling to the gauge bosons, such as $WW$, $ZZ$, $Z\gamma$, $\gamma \gamma$, or gg,
the scattering amplitude can be written as
\begin{equation}
A(H \to V_1V_2) = v^{-1} \left (
  a^{HVV}_{ 1} m_{V}^2 \epsilon_1^* \epsilon_2^*
+ a^{HVV}_{ 2} f_{\mu \nu}^{*(1)}f^{*(2),\mu \nu}
+ a^{HVV}_{ 3}  f^{*(1)}_{\mu \nu} {\tilde f}^{*(2),\mu  \nu}
\right )\,,
\label{eq:fullampl-spin0}
\end{equation}
where $a_i^{HVV}$ are generally $q^2$-dependent coefficients scaling the three unique Lorentz structures,
described with the help of the (conjugate) field strength tensor $f^{(i),\mu \nu}$ (${\tilde f}^{(i),\mu  \nu}$)
of a gauge boson with momentum $q_i$ and polarization vector $\epsilon_i$.
In the following, we will keep only the first-order $q^2$-expansion of Eq.~(\ref{eq:fullampl-spin0})
with constant coefficients $a_{i}$,
which correspond to dimension-six operators in the effective Lagrangian formulation.
The presence of the $CP$-odd contribution $a^{HVV}_{3}$, which can be treated as constant
in this expansion, indicates $CP$ violation, and the $CP$-sensitive parameter takes the form
\begin{eqnarray}
&& f_{CP}^{HVV} =  \frac{|a_{3}^{HVV} |^2}{\sum |a_{i}^{HVV} |^2 (\sigma^{HVV} _i/\sigma^{HVV} _3)} \,,
\label{eq:fractions}
\end{eqnarray}
where $\sigma_i$ is the effective cross-section of the $H\to VV$
decay process corresponding to $a_{i}=1, a_{j \ne i}=0$.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\captionsetup{justification=centerlast}
\caption{
List of expected precision (at 68\% C.L.) of ${CP}$-sensitive measurements of the parameters $f_{CP}^{HX}$ defined in Eq.~(\ref{eq:fCP}).
Numerical values are given where reliable estimates are provided,
$\checked$ mark indicates that feasibility of such a measurement could be considered.
}
\begin{center}
\begin{tabular}{|l|ccccccccc|c|}
\hline\hline
Collider                     &   $pp$       &   $pp$    &   $e^+e^-$    &   $e^+e^-$    &  $e^+e^-$    &  $e^+e^-$    & $\gamma\gamma$ &  $\mu^+\mu^-$  & $\mu^+\mu^-$  & target \\
E (GeV)                     &   14,000   &   14,000     &  250          &  350                 & 500             &   1,000       &            125               &    125          & $\ge500$                        &   (theory) \\
${\cal L}$ (fb$^{-1}$) & 300  & 3,000 & 250          &  350                  & 500              &    1,000       &          250                  &                                  &   &     \\
\hline
\hline
 $HZZ/HWW$ &  $2\!\cdot\!10^{-4}$ &  $0.5\!\cdot\!10^{-4}$& $7\!\cdot\!10^{-4}$ & $1.1\! \cdot\!10^{-4}$ & $4\!\cdot \!10^{-5}$ & $8\!\cdot\!10^{-6}$  & \checked  &  \checked  &  \checked  & $<10^{-5}$  \\
\hline
\hline
  $H\gamma\gamma$ & --  & 0.50 & -- & -- & -- & -- & 0.06 & -- & --  & $<10^{-2}$  \\
\hline
  $HZ\gamma$ & --  & $\sim\!1$ &-- & -- & -- & --  & -- & --  & --  & $<10^{-2}$   \\
\hline
  $H$gg & 0.20 & 0.06 & -- & -- & -- & -- & -- & -- &  -- & $<10^{-2}$   \\
\hline
\hline
 $Ht\bar{t}$ & 0.24  & 0.05 & -- & -- & 0.29 & 0.08  & -- & --  & \checked & $<10^{-2}$  \\
\hline
 $H\tau\tau$ &  0.07 & 0.008 & $0.01$ & $0.01$  &  ${0.02}$  & ${0.06}$  & \checked & \checked & \checked & $<10^{-2}$  \\
\hline
 $H\mu\mu$ & --  & -- & -- & -- & -- & --  & -- & \checked & -- & $<10^{-2}$   \\
\hline
\hline
\end{tabular}
\end{center}
\label{table-cpscenarios}
\end{table}

This brings us to the summary of possible $CP$-sensitive measurements in the $H$~boson\xspace interactions in Table~\ref{table-cpscenarios}.
In the following, we will review unique features of the photon, muon, hadron, and electron-positron colliders.
For example, beam polarization in the photon and muon colliders would be essential for $CP$ measurements in the
$H\gamma\gamma$ and $H\mu\mu$ couplings, which we discuss next.

\section{Prospects of Higgs $CP$ measurements at a photon collider}
\label{sect:photon}

The photon collider has a unique feature in that it can be used to study the $H$~boson\xspace couplings to photons in direct production
$\gamma\gamma\to H$. It is also possible to study the $H$~boson\xspace couplings in decay, such as $CP$ structure
in $H\to \tau^+\tau^-$ or $H\to 4f$. However, the decay measurements critically depend on the number
of produced $H$ bosons, and a Higgs factory in either lepton or proton collisions is better positioned
to make those measurements. In this Section, therefore, we focus on the $H\gamma\gamma$ measurements,
which are unique to the photon collider.

The coupling of the $H$~boson\xspace to two photons cannot happen at tree level, but can be generated
by loops of any charged particles. In the SM, those are the charged fermions and W boson.
In the SM, $CP$ violation is tiny, as it can be generated only at three-loop level.
In BSM theories, new heavy states can contribute to the loop, and could generate sizable $CP$ violation.
Alternatively, $CP$ violation in the $H$~boson\xspace couplings to SM particles could also generate $CP$-odd
contributions to the $H\gamma\gamma$ loop.
Both $H\to\gamma\gamma$ decay and $\gamma\gamma\to H$ production can be
parameterized with the $CP$-even $a_2^{H\gamma\gamma}$ and $CP$-odd $a_3^{H\gamma\gamma}$
contributions in Eq.~(\ref{eq:fullampl-spin0})
with the ratio $\sigma^{H\gamma\gamma} _2/\sigma^{H\gamma\gamma} _3=1$ in Eq.~(\ref{eq:fractions}).
However, without access to the photon polarization, it is not possible to distinguish between the two
contributions in the $H\to\gamma\gamma$ decay.\footnote{An attempt to measure photon polarization
in its conversion is possible~\cite{Bishara:2013vya}, but it suffers from a significant loss of statistical precision. We will discuss
the photon polarization measurements in the $H\to\gamma^*\gamma^*\to 4f$ process in Section~\ref{sect:hadron}.}
Therefore, variation of the photon polarization in the photon collider becomes a unique approach to
study the $CP$ structure of the $H\gamma\gamma$ vertex.

Three parameters ${\cal A}_1, {\cal A}_2, {\cal A}_3$ sensitive to $C\!P$ violation
have been defined in the context of the photon collider~\cite{Grzadkowski:1992sa, Kramer:1993jn, Gunion:1994wy}.
The ${\cal A}_1$ parameter can be measured as an asymmetry in the $H$~boson\xspace production cross-section
between the $A_{++}$ and $A_{--}$ circular polarizations of the beams. This asymmetry
is the easiest to measure, but it is proportional to $\Im{m}(a_2^{H\gamma\gamma}a_3^{H\gamma\gamma~*})$ and is zero
when $a_2^{H\gamma\gamma}$ and $a_3^{H\gamma\gamma}$ are real,
as expected for the two loop-induced couplings with heavier particles in the loops.
A more interesting parameter,
\begin{eqnarray}
{\cal A}_3
= \frac{|A_{\parallel}|^2-|A_{\perp}|^2}{|A_{\parallel}|^2+|A_{\perp}|^2}
= \frac{2\Re{e}(A_{--}^*A_{++})}{|A_{++}|^2+|A_{--}|^2}
= \frac{|a_2^{H\gamma\gamma}|^2-|a_3^{H\gamma\gamma}|^2}{|a_2^{H\gamma\gamma}|^2+|a_3^{H\gamma\gamma}|^2}
= (1-2f_{CP}^{H\gamma\gamma}),
\label{eq:photoncp}
\end{eqnarray}
can be measured as an asymmetry between two configurations with the linear
polarization of the photon beams, one with parallel and the other with orthogonal
polarizations.

In Ref.~\cite{Asner:2001ia}, a careful simulation of the process has been performed.
The degree of linear polarization at the maximum energies is 60\% for an electron
beam of energy $E_0 \approx 110$\,GeV and a laser wavelength $\lambda \approx 1\, \mu{m}$.
The expected uncertainty on ${\cal A}_3$ is 0.11 for $2.5\cdot 10^{34} \times
10^7$ = 250\,fb$^{-1}$ integrated luminosity and $m_H=120$\,GeV.
This translates to a $f_{CP}^{H\gamma\gamma}$ uncertainty of 0.06,
which we enter as an estimate in Table~\ref{table-cpscenarios}.

The $CP$ mixture study at a photon collider was also shown based on a sample of 50,000 raw
$\gamma\gamma\to H$ events assuming 80\% circular polarization of both electron beams~\cite{Chou:2013xaa}.
This study corresponds to a ${\cal A}_1$ asymmetry measurement,
with expected precision on ${\cal A}_1$ of about 1\%.
However, this asymmetry is expected to be zero with real coupling constants $a_2^{H\gamma\gamma}$ and
$a_3^{H\gamma\gamma}$ and is therefore of limited interest compared to $f_{CP}^{H\gamma\gamma}$.

\section{Prospects of Higgs $CP$ measurements at a muon collider}
\label{sect:muon}

Similarly to the photon collider, we focus on a unique feature of the muon collider
operating at the $H$~boson\xspace pole. This allows one to measure the $CP$ structure of the $H\mu\mu$ vertex
with the beam polarization in the $\mu^+\mu^-\to H$ process.
It is not possible to study the $CP$ structure in the $H\to \mu^+\mu^-$ decay because the muon polarization is not accessible.
The muon collider may become the only facility allowing a measurement of $CP$ structure
in the $H$~boson\xspace's connection to the second-family fermions.
At a muon collider operating both at the $H$~boson\xspace pole and at higher energy,
analysis of the $H$~boson\xspace decays is also possible. However, this analysis is similar to
the studies performed at other facilities and depends critically on the number of the $H$ bosons
produced and their purity.

\subsection{Muon collider at the $H$~boson\xspace pole}
\label{sect:mhmuon}

At a muon collider operating at the resonance pole, the $CP$ quantum numbers of the states can be determined
if the muon beams can be transversely polarized.
The cross section for production of a resonance takes the form~\cite{Grzadkowski:2000hm}
\begin{equation}
  \sigma_{\rm pol}(\zeta) = \sigma_{\rm unpol} \left( 1 + P_L^+ P_L^-
  + P_T^+ P_T^- \left[ \frac{(b_1^{H\mu\mu})^2 - (b_2^{H\mu\mu})^2}{(b_1^{H\mu\mu})^2 + (b_2^{H\mu\mu})^2} \cos\zeta
    - \frac{2b_1^{H\mu\mu}b_2^{H\mu\mu}}{(b_1^{H\mu\mu})^2 + (b_2^{H\mu\mu})^2} \sin\zeta \right] \right),
\end{equation}
which depends on $P_T$ ($P_L$), the degree of transverse (longitudinal)
polarization of each of the beams and $\zeta$ is the angle of the
$\mu^+$ transverse polarization relative to that of the $\mu^-$
measured using the direction of the $\mu^-$ momentum as the $z$ axis.
In particular, muon beams polarized in the same transverse direction
selects out the $C\!P$-even state, while muon beams polarized in opposite
transverse directions (i.e., with spins $+1/2$ and $-1/2$ along one
transverse direction) selects out the $C\!P$-odd state.
A quantitative estimate of the muon collider precision in the measurement of
$CP$ structure of the $H\mu\mu$ vertex is left for future studies, which we indicate
with a checkmark in Table~\ref{table-cpscenarios}.

\subsection{High-energy muon collider}
\label{sect:hemuon}

Operation of the muon collider at higher energies will allow access to associated production
of the $H$~boson\xspace and study $CP$ properties in those processes. Such studies would
be similar to those discussed in Sections~\ref{sect:hadron} and~\ref{sect:lepton}
and would depend on achieved performance of the muon collider.
At energies around 1\,TeV, VBF production of the $H$~boson\xspace dominates, similarly to the $e^+e^-$ collider.
The dominant channel $\mu^+\mu^-\to \nu_\mu\bar\nu_\mu(W^+W^-)\to \nu_\mu\bar\nu_\mu H$
does not provide kinematic information to analyze the final state with missing neutrinos,
but provides $H$ bosons for analysis of their decay.
The momentum of the $H$~boson\xspace in this VBF production provides sensitivity to the higher-dimension
operators, but does not allow one to separate $CP$-odd and $CP$-even contributions.
The other channel $\mu^+\mu^-\to \mu^+\mu^-(ZZ/Z\gamma^*/\gamma^*\gamma^*)\to \mu^+\mu^- H$
provides sufficient information to analyze potential $CP$ structure in the $HZZ/HZ\gamma/H\gamma\gamma$ couplings.
The $t\bar{t}H$ production allows access to $CP$ in the $Ht\bar{t}$ coupling, which is accessible at energies above 0.5\,TeV.
It is pointed out in Ref.~\cite{Barman:2022pip} that at energies around 10\,TeV, VBF production of $t\bar{t}H$ and $t\bar{q}H$
becomes important.
According to Ref.~\cite{Barman:2022pip}, it is expected to achieve constraints on $f_{CP}^{Ht\bar{t}}<0.67, 0.024$, and~$0.003$ in three scenarios
of the muon collider at 1\,TeV with 0.1\,ab$^{-1}$ of data, 10\,TeV with 10\,ab$^{-1}$, and 30\,TeV with 10\,ab$^{-1}$.
We do not enter numerical values in Table~\ref{table-cpscenarios} due to uncertain muon collider scenarios,
but point to the possible measurements with the checkmarks.

\section{Prospects of Higgs $CP$ measurements at a hadron collider}
\label{sect:hadron}

Hadron colliders provide essentially the full spectrum of possible measurements sensitive to $CP$ violation
in the $H$~boson\xspace interactions, as outlined in Table~\ref{table-cpscenarios}, with the exception of the $H\mu\mu$ vertex.
Here we discuss applications to the LHC experiment, and its high-luminosity upgrade, with a proton-proton
collision energy around 14 TeV. In the following, we review various processes accessible in hadron collisions.

\subsection{Gluon fusion process at a hadron collider}
\label{sect:ggH}

The LHC could be considered a gluon collider, as the dominant
$H$~boson\xspace production mechanism is the gluon fusion gg~$\to H$ process.
Many aspects discussed in Section~\ref{sect:photon} in application to the photon couplings apply here as well.
The coupling of the $H$~boson\xspace to two massless gluons is generated by the loops of any massive particles with
color charge, which are quarks in the SM. In BSM, new heavy states, either fermions or bosons, could
contribute to the loop and generate $CP$ violation. While there is a sizable decay rate $H\to$~gg, study of
the gluon polarization is difficult, and within the hadron collider environment this decay mode is hard to distinguish
from the dominant QCD background. However, study of the gluon fusion process in the scattering topology
of the $H$~boson\xspace production in association with two hadronic jets allows access to the $CP$ property of the
$H$gg vertex~\cite{Hankele:2006ma}. This VBF topology is illustrated in the first diagram of Fig.~\ref{fig:decay},
where $V^*=$\,g.

Similarly to the photon couplings, the gluon fusion process can be characterized by the two
couplings $CP$-even $a_2^{H\rm gg}$ and $CP$-odd $a_3^{H\rm gg}$, which absorb both
SM and heavy BSM particles in the loop.
While in the EFT approach, these contributions could be disentangled in a global fit of multiple
processes, this is not possible with the gluon fusion process alone. Therefore, we parameterize the
$CP$ violation effects with a single parameter $f_{CP}^{H\rm gg}$. In order to isolate $CP$-sensitive
effects, no constraint on the process rate is applied, which is proportional to
$|a_2^{H\rm gg}|^2+|a_3^{H\rm gg}|^2$.

The Snowmass-2013 projection~\cite{Dawson:2013bba} was based on the study of the gg~$\to H$
process with $H\to4\ell$~\cite{Anderson:2013afp}. Since then, this approach was successfully applied
on LHC~\cite{Sirunyan:2021fpv,Collaboration:2022mlq,ATLAS:2021pkb} using 140\,fb$^{-1}$ of data
with results in good agreement with the above expectation. For example, the constraint
$f_{CP}^{H\rm gg}<0.3$ is expected with $H\to\tau\tau$ and $4\ell$ combined~\cite{Collaboration:2022mlq},
which would likely scale to $f_{CP}^{H\rm gg}<0.2$ with 300\,fb$^{-1}$, where the improvement
with respect to the Snowmass-2013 projection is due to an additional $H$~boson\xspace decay channel analyzed.
Somewhat more conservative results are projected in Ref.~\cite{Gritsan:2020pib}.
Therefore, in Table~\ref{table-cpscenarios},
we estimate that $f_{CP}^{H\rm gg}<0.06$ could be achieved with 3,000\,fb$^{-1}$,
but note that further improvement is very likely, both from inclusion of multiple decay channels and
from improvements in experimental analyses.

\begin{figure}[b]
\centerline{
\setlength{\epsfxsize}{0.31\linewidth}\leavevmode\epsfbox{paper_cp/angles-HZZVBF.pdf}
\setlength{\epsfxsize}{0.31\linewidth}\leavevmode\epsfbox{paper_cp/angles-ZZHbb.pdf}
\setlength{\epsfxsize}{0.31\linewidth}\leavevmode\epsfbox{paper_cp/angles-HZZ4l.pdf}
}
\captionsetup{justification=centerlast}
\caption{
Illustrations of the $H$~boson\xspace kinematic observables in $pp$, $e^+e^-$, or $\mu^+\mu^-$ collision in
(1) the VBF process: $e^+e^-(q{q^\prime})\to e^+e^-(q{q^\prime})H \to e^+e^-(q{q^\prime}) b\bar{b}$;
(2) the $VH$ process: $e^+e^-(q\bar{q})\to V^*\to V^*H\to f\bar{f}b\bar{b}$;
and (3) the decay: gg\,$\to H\to V^*V^*\to 4f$.
Five angles fully characterize the orientation of the production and decay chain and are defined
in the suitable rest frames.
The diagrams are adopted from Refs.~\cite{Gao:2010qx,Anderson:2013afp}.
}
\label{fig:decay}
\end{figure}

\subsection{The $H\gamma\gamma$ and $HZ\gamma$ couplings at a hadron collider}
\label{sect:HAA}

While the $H\to\gamma\gamma$ decay was one of the two primary $H$~boson\xspace discovery channels, this decay
process does not allow access to $CP$ structure of the photon couplings, as discussed in Section~\ref{sect:photon}.
Similarly, the $H\to Z\gamma$ decay does not provide access to $CP$ structure of the $HZ\gamma$ vertex
when the couplings in Eq.~(\ref{eq:fullampl-spin0}) are real. Complex couplings could be generated with
light particles in the loop, for example, and could generate forward-backward asymmetry in the polar angle,
as shown in Ref.~\cite{Anderson:2013afp}, but we do not consider such a possibility here.
Another approach, suggested in Ref.~\cite{Chen:2014ona}, relies on complex phases generated through
different Breit-Wigner propagators in the interference of the $\gamma^*$ and $Z$ contributions in the decay
$H\to \ell^+\ell^-\gamma$ process. However, this approach is not necessarily better than $H\to4\ell$,
which includes $H\to \ell^+\ell^-(\gamma^*/Z)$, and would require further feasibility studies.

However, several other topologies may allow access to the $CP$ structure of the $H\gamma\gamma$
and $HZ\gamma$ couplings. These include
\begin{enumerate}
\item the VBF process: $qq^\prime\to qq^\prime(\gamma^*\gamma^*/Z\gamma^*)\to qq^\prime H$,
\item the $VH$ process: (a) $q\bar{q}\to\gamma^*/Z\to\gamma^*H/ZH\to (2f)H$, ~~~(b) $q\bar{q}\to\gamma^*/Z\to\gamma H$,
\item the 4/3/2-body decays: (a) $H\to \gamma^*\gamma^*/Z\gamma^*\to 4f$, ~~~(b) $H\to \gamma\gamma^*/\gamma Z\to\gamma(2f)$, ~~~(c) $H\to\gamma\gamma$.
\end{enumerate}
It is important to note that the above processes do not appear in isolation and whenever $\gamma^*$ appears in
the intermediate state, $Z^*$ appears as well, leading to interference. For example, the full analysis of the process
$H\to \gamma^*\gamma^*/\gamma^*Z/ZZ\to 4f$ may allow access to $CP$ violation through interference
of the $CP$-odd $a_3^{HZ\gamma}$ term with the dominant $CP$-even $a_1^{HZZ}$ term
appearing at tree level, and this needs to be disentangled from the possible $a_3^{H\gamma\gamma}$
and $a_3^{HZZ}$ terms. Therefore, the full analysis of each process requires accounting for all contributions,
including the $HZZ$ couplings discussed in Section~\ref{sect:HVV}.

The three LHC topologies which involve the $H\gamma\gamma$ and $HZ\gamma$ couplings are shown in Fig.~\ref{fig:decay},
where $V^*=\gamma^*$ or $Z$. The processes with onshell photons can also be represented by the diagrams (2) and (3)
in Fig.~\ref{fig:decay} with $V^*=\gamma$, but with no subsequent decay of the photon. However, these
processes with onshell photons do not allow access to $CP$ effects without the measurement of
the onshell photon polarization, unless complex anomalous couplings are considered.
Illustration of $CP$-sensitive effects with complex couplings appearing in the forward-backward asymmetry
of the angular distributions can be found for $q\bar{q}\to Z^*\to\gamma H$ in Ref.~\cite{Davis:2021tiv}
and $H\to \gamma Z\to\gamma(2f)$ in Ref.~\cite{Anderson:2013afp}.
An angle $\Phi$ identified in all three diagrams in Fig.~\ref{fig:decay} is an angle between the decay or
production planes defined by the four four-momenta and is the primary $CP$-odd observable in
each process. However, a multivariate analysis of the full kinematic information leads to the most
optimal amplitude analysis, which is sensitive to both squared and interference of the $CP$-odd
and $CP$-even terms.

An attempt to study the $CP$ structure of the $H\gamma\gamma$ and $HZ\gamma$ couplings in the golden
channel $H\to4\ell$ was performed at the LHC in Ref.~\cite{Khachatryan:2014kca}, where it became clear that
reaching an interesting level of sensitivity will require very high luminosity. The $H\gamma\gamma$
couplings in the $H\to4\ell$ decay were considered phenomenologically in Ref.~\cite{Chen:2014gka}.
More recently, a joint analysis of the three processes VBF, $VH$, and decay $H\to4\ell$
was investigated in Ref.~\cite{Davis:2021tiv}, where it was shown that while
the decays $H\to\gamma\gamma$ and $H\to Z\gamma$ are most sensitive to the overall strength
$|a_2^{H\gamma\gamma}|^2+|a_3^{H\gamma\gamma}|^2$ and $|a_2^{HZ\gamma}|^2+|a_3^{HZ\gamma}|^2$,
the decay $H\to4\ell$ process is most sensitive to study the tensor structure of the $H\gamma\gamma$ and $HZ\gamma$ couplings,
relevant for $CP$ violation measurements. We should note that in all the above studies, the effective values of
$a_2^{H\gamma\gamma}$ and $a_2^{HZ\gamma}$ which reproduce the SM rate of $H\to\gamma\gamma$ and $H\to Z\gamma$
decays were used to approximate the SM processes $H\to \gamma^*\gamma^*/Z\gamma^*\to 4\ell$, VBF, and $VH$.
While this prescription is not technically correct to represent the SM rate due to $q^2$ dependence of the couplings,
this simulated value of $a_2$ is a good benchmark for the Snowmass exercise as it is used only as
a reference to estimate sensitivity to $a_3$.
The results of the study in Ref.~\cite{Davis:2021tiv} are reinterpreted in terms of
$f_{CP}^{H\gamma\gamma}$ and $f_{CP}^{HZ\gamma}$ in Appendix~\ref{app:A}
and are entered in Table~\ref{table-cpscenarios}. The full dataset of the HL-LHC will be at the
boundary to start setting meaningful constraints on these parameters.

Given the difficulty to set $CP$ constraints on the $H\gamma\gamma$ and $HZ\gamma$ couplings at HL-LHC,
exploring other options might be useful. A possible study of $H\to\gamma\gamma\to4e$ with photon polarization
in its conversion has been suggested in Ref.~\cite{Bishara:2013vya}, but this study suffers from a significant loss
of statistical precision, and a more detailed study of experimental aspects, such as reconstruction of displaced
and boosted $e^+e^-$ pairs with a small opening angle, may be required.

\subsection{The $HZZ$ and  $HWW$ couplings at a hadron collider}
\label{sect:HVV}

The $HZZ$ and $HWW$ couplings appear at tree level in the SM and the decays $H\to ZZ\to 4f$ and $H\to W^+W^-\to 4f$
provided rich kinematic information for studies of spin and $CP$ properties of the $H$~boson\xspace in the early days after the $H$~boson\xspace
discovery and have historically been studied extensively on LHC experiments~\cite{Chatrchyan:2012jja,
Aad:2013xqa,Chatrchyan:2013mxa,Khachatryan:2014kca,Aad:2015mxa,Khachatryan:2016tnr,
Aad:2016nal,Sirunyan:2017tqd,Aaboud:2017oem,Aaboud:2017vzb,Sirunyan:2019twz,
Sirunyan:2019nbs,Aad:2020mnm,Sirunyan:2021fpv,ATLAS:2021pkb,CMS:2022ley,Collaboration:2022mlq}.
However, with the growing significance of the $H$~boson\xspace electro-weak production (VBF and $VH$),
the larger $q^2$ values tested lead to stronger constraints of $CP$ effects in these production modes.
The three main topologies involving the $HZZ$ and $HWW$ couplings follow closely those in Section~\ref{sect:HAA}
and appear in Fig.~\ref{fig:decay}, with $V^*=Z$ or $W$, as
\begin{enumerate}
\item the VBF process: $qq^\prime\to qq^\prime(W^+W^-/ZZ)\to qq^\prime H$,
\item the $VH$ process: (a) $q\bar{q}\to Z\to ZH\to (2f)H$, ~~~(b) $q\bar{q}^\prime\to W^\pm\to W^\pm H\to (2f)H$,
\item the 4-body decay: (a)  $H\to ZZ\to 4f$, ~~~(b) $H\to W^+W^-\to 4f$.
\end{enumerate}
All processes with the $Z$ boson interfere with the same processes involving $\gamma^*$ in its place,
as listed in Section~\ref{sect:HAA}. We note that the process gg~$\to VH$ also receives attention due to the large
gluon luminosity in proton collisions. This channel provides an interesting interplay of $H\!f\!\bar{f}$ and $HVV$ couplings,
but does not have contribution from the $CP$-odd terms with $a_3^{HZZ}, a_3^{HZ\gamma},$ or
$a_3^{H\gamma\gamma}$~\cite{Gritsan:2020pib}, and therefore is not suitable for studies of $CP$ violation in
$HZZ, {HZ\gamma}$, or ${H\gamma\gamma}$ interactions.

Even though the $HZZ$ and $HWW$ couplings can be easily analyzed separately in the $ZH$ vs. $WH$ production
with leptonic $Z$ or $W$ decay, or in $H\to ZZ$ vs. $H\to WW$ decays, it is essentially impossible to disentangle those
in the VBF production, where all kinematic features are nearly identical.
The tree-level couplings $HZZ$ and $HWW$ can be related through
custodial symmetry, leading to $a_1^{HZZ}=a_1^{HWW}$. Within the precision of the $H$~boson\xspace measurements,
this relationship is not significantly affected by the recent tension in the $W$ mass measurements.
The anomalous $HZZ$ and $HWW$ couplings,
such as $a_3^{HZZ}$ and $a_3^{HWW}$, could also be related through symmetry, such as $SU(2)\times U(1)$.
For example, $a_3^{HWW}=a_3^{HZZ}\!\cdot\!\cos^2\theta_W$ if contributions
of the $H\gamma\gamma$ and $HZ\gamma$ couplings are neglected.
Most of the experimental studies on LHC
and projections of the feasibility studies have been performed under such or a similar relationship
of the $HZZ$ and $HWW$ couplings. Therefore, in Table~\ref{table-cpscenarios} we estimate
precision on $f_{CP}^{HVV}$ which represents $V=Z$ and $W$ combined.\footnote{Technically, $f_{CP}^{HVV}$
is defined for $H\to ZZ\to 2e2\mu$ with $\sigma_1^{HZZ}/\sigma_3^{HZZ}=6.54$ in Eq.~(\ref{eq:fractions}),
but the measurement relies on both $HZZ$ and $HWW$ couplings.}
Precision of the separate measurements would be less, but similar.
We should also note that since $a_1^{HZZ}$  and $a_1^{HWW}$ are generated at tree level in the SM,
they are expected to be much larger than $a_3^{HZZ}$ and $a_3^{HWW}$, which appear at loop level,
similar to the photon couplings. Therefore, the interesting values of $f_{CP}^{HVV}$ are much smaller
than those for $f_{CP}^{H\rm gg}$, $f_{CP}^{H\gamma\gamma}$, and $f_{CP}^{HZ\gamma}$,
as reflected in the last column of Table~\ref{table-cpscenarios}.

The Snowmass-2013 projections~\cite{Dawson:2013bba} were split into the study of the $HVV$
couplings in three processes: $H\to4\ell$ decay, VBF production with $H\to\gamma\gamma$, and
$VH$ production with $H\to b\bar{b}$~\cite{Anderson:2013afp}, where the most powerful channels
were picked in each case. In the present study, we do not separate the channels and
consider the combined or best performance, assuming that the effective field-theoretic description
does not breakdown with the $q^2$ growth. Several experimental updates with 140\,fb$^{-1}$
of LHC data have appeared since then, some of the recent ones include
Refs.~\cite{Sirunyan:2021fpv,Collaboration:2022mlq}, where the constraint $f_{CP}^{HVV}<6\times 10^{-4}$
is expected at 68\% C.L. from analysis of electroweak production information in the $H\to\tau\tau$ and $4\ell$ channels.
The $H$~boson\xspace physics projections at the HL-LHC and HE-LHC were revised in Ref.~\cite{Cepeda:2019klc},
where Fig.~38 indicates $f_{CP}^{HVV}<0.037$ from $H\to4\ell$ and Fig.~39 indicates
$f_{CP}^{HVV}<1.8\times 10^{-4}$ at 95\% C.L. from production with $H\to4\ell$ at 3,000\,fb$^{-1}$.
We use these studies to indicate that $f_{CP}^{HVV}<0.5\times 10^{-4}$ at 3,000\,fb$^{-1}$
and $f_{CP}^{HVV}<2\times 10^{-4}$ at 300\,fb$^{-1}$ are achievable at 68\% C.L.
Further improvement are likely from inclusion of multiple decay channels and
from improvements in experimental analyses.

\subsection{The $Ht\bar{t}$ coupling at a hadron collider}
\label{sect:Htt}

The $CP$ structure of the $H$~boson\xspace couplings to fermions is particularly interesting because
both $CP$-even and $CP$-odd components can appear at tree level, and therefore the
$f_{CP}^{H\!f\!\bar{f}}$ values do not necessarily need to be very small.
(This is in contrast to $f_{CP}^{HVV}$, for example.)
One could get access to the ${H\!f\!\bar{f}}$ interactions through loops appearing in the
${H\rm gg}$, $H\gamma\gamma$, and $HZ\gamma$, but we treat those separately
in Sections~\ref{sect:ggH} and \ref{sect:HAA} because
one cannot disentangle loop contributions without a global analysis.
A measurement of the $CP$ structure of the $H$~boson\xspace couplings to the first- and
second-family fermions is essentially impossible at a hadron collider, as there are no channels
where polarization measurements could be performed. A $CP$ measurement of the $H b\bar{b}$
vertex is also impossible, as neither $H\to b\bar{b}$ decay nor $b\bar{b}H$ production
allows access to $CP$~\cite{Gritsan:2016hjl}. This leaves only the $Ht\bar{t}$ and $H\tau\tau$
couplings with $CP$ structure that can be measured at a hadron collider.

The $t\bar{t}H$ production process has received the primary attention on LHC as the channel
to study $CP$ the structure of the $Ht\bar{t}$
coupling~\cite{Gunion:1996xu,Demartin:2014fia,Buckley:2015vsa,Gritsan:2016hjl,Martini:2021uey,Barman:2021yfh,Barman:2022pip},
while the $tqH$ and $tWH$ processes also allow access to $CP$ in this coupling.
The cross sections of the latter channels are smaller, but they feature interference of the
$Ht\bar{t}$ and $HVV$ couplings, which help in resolving the sign ambiguity in the relative phase,
and a joint analysis of all these channels is often required due to cross-feed of events in analysis of the data.
There is also a proposal to access $CP$-violating effects in $Ht\bar{t}$ couplings through loop
effects in $t\bar{t}$ production~\cite{Martini:2021uey}, but the precision of such constraints does not
alter our conclusion drawn from channels with associated $H$~boson\xspace production.
There is rich kinematic information in the sequential decay of the particles produced in
association with the $H$~boson\xspace in the $t\bar{t}H$, $tqH$, and $tWH$ processes,
as indicated in the two diagrams in Fig.~\ref{fig:Hff}. However, most information
is sensitive to the square of the $CP$-odd and $CP$-even amplitudes.

It is also possible to construct $CP$-odd observables that are sensitive to the interference term in the amplitude
by exploring the $t\bar{t}$ spin correlations. These spin correlations can be traced back from the decay products
of $t$ and $\bar{t}$, since the top-quark lifetime~($\sim 10^{-25}~\mathrm{s}$~\cite{ParticleDataGroup:2020ssz})
is much shorter than the time required for spin decorrelation effects
to actualize~($\sim 10^{-21}~\mathrm{s}$)~\cite{Mahlon:2010gw}.
Thus, $CP$-odd observables can be constructed from antisymmetric tensor products of the four-momenta
of the top, anti-top, and their respective decay products $i$ and $k$,
$\epsilon(p_{t}, p_{\bar{t}}, p_{i}, p_{k}) \equiv \epsilon_{\mu \nu \rho\sigma}p_{t}^{\mu}p_{\bar{t}}^{\nu}p_{i}^{\rho}p_{k}^{\sigma}$.
In the $t\bar{t}$ rest frame, this antisymmetric tensor product can be simplified to
$\vec{p}_{t}\cdot\left(\vec{p}_{i} \times \vec{p}_{k} \right)$, which can be used to define genuine
$CP$-sensitive azimuthal angle differences~\cite{Goncalves:2018agy,Goncalves:2021dcu}.
Correlations between two decay products, one from $t$ and the other from $\bar{t}$, scale with the spin
analyzing power ($\beta_{i}$) associated with the decay product~\cite{Bernreuther:2010ny}.
Charged leptons and down-type quarks exhibit the highest spin correlations $|\beta_{i}| = 1$,
followed by bottom quarks and W bosons with $|\beta_{i}| = 0.4$, and neutrinos and up-type quarks
with $|\beta_{i}| = 0.3$. Therefore, one would require access to the flavors of the fermions and
anti-fermions in the subsequent $t/W^+$ and $\bar{t}/W^-$decays to probe the interference.
This is possible in the leptonic decays of both $W$'s in the $t\bar{t}$ decay, but statistical precision
in this fully leptonic channel is significantly weaker than in the semi-leptonic and fully-hadronic channels.

Both the CMS and ATLAS experiments~\cite{Sirunyan:2020sum,ATLAS:2020ior,Sirunyan:2021fpv,
CMS-PAS-HIG-21-006,ATLAS-CONF-2022-016}
have performed an amplitude analysis of the $CP$-even and $CP$-odd components of
the $Ht\bar{t}$ coupling analyzing both $t\bar{t}H$ and $tH$ processes. One of the dominant
$H$~boson\xspace decay channels is $H\to\gamma\gamma$, but other decays can also make significant
contributions. Only semi-leptonic and fully-hadronic top decays in the $t\bar{t}H$ channels
have been used, therefore limiting $CP$ analysis to the square of the amplitudes.
With about 140\,fb$^{-1}$ of LHC data, a single experiment obtained an expected sensitivity
of $f_{CP}^{Ht\bar{t}}<0.5$ at 68\% C.L. with $H\to\gamma\gamma$~\cite{Sirunyan:2020sum} and
of $f_{CP}^{Ht\bar{t}}<0.35$ in combination with the multi-lepton $H$~boson\xspace decays~\cite{CMS-PAS-HIG-21-006}.
Phenomenological studies indicate
$f_{CP}^{Ht\bar{t}}<0.5$ at 300\,fb$^{-1}$ in Ref.~\cite{Gritsan:2016hjl} using the $H\to\gamma\gamma$ channel.
Exploring the same $H$~boson\xspace final state together with combined searches in the semi-leptonic, di-leptonic, and
fully-hadronic top quark pair decays, the study in Ref.~\cite{Barman:2021yfh} indicates a sensitivity of
$f_{CP}^{Ht\bar{t}}<0.05$ at the HL-LHC with 3,000\,fb$^{-1}$. Despite larger rates, searches in the
$pp \to t\bar{t}(H \to b\bar{b})$ channel leads to typically weaker projections due to an imposing QCD background,
which is also marred by substantial systematic uncertainties~\cite{Goncalves:2018agy,Goncalves:2021dcu}.
The diphoton channel stands at a vantage point due to controlled backgrounds facilitated by data-driven side-bands.
We therefore enter $f_{CP}^{Ht\bar{t}}<0.24$ at 300\,fb$^{-1}$ and $<0.05$ at 3,000\,fb$^{-1}$
in Table~\ref{table-cpscenarios} and note that further improvements are expected
from analysis of other $H$~boson\xspace decay channels.

\begin{figure}[t]
\centerline{
\setlength{\epsfxsize}{0.31\linewidth}\leavevmode\epsfbox{paper_cp/angles-ttH1.pdf}
\setlength{\epsfxsize}{0.31\linewidth}\leavevmode\epsfbox{paper_cp/angles-ttH2.pdf}
\setlength{\epsfxsize}{0.31\linewidth}\leavevmode\epsfbox{paper_cp/angles-HTT.pdf}
}
\captionsetup{justification=centerlast}
\caption{
Illustrations of the $H$~boson\xspace kinematic observables in $pp$, $e^+e^-$, or $\mu^+\mu^-$ collision
in (left and middle) the $t\bar{t}H$ process with sequential decay;
(right) the $H\to \tau^+\tau^-$ decay process. Subsequent $W$ decay is not shown.
The $b$ and $W$ pairing in the $t\bar{t}$ decays is switched to enhance visibility
of $CP$ effects in individual angular observables.
The diagrams are adopted from Ref.~\cite{Gritsan:2016hjl}.
}
\label{fig:Hff}
\end{figure}

\subsection{The $H\to \tau^+\tau^-$ process at a hadron collider}
\label{sect:HTT}

The $H\to \tau^+\tau^-$ decay is an excellent probe of spin correlation in the sequential decay of the two taus.
For example, an angle between the two decay planes indicated in Fig.~\ref{fig:Hff} is sensitive to $CP$ in the
$H\tau\tau$ interaction. However, this angle cannot be measured directly due to missing neutrinos, and the
experimental challenge is to approximate it with available information. For example, the pion is preferably
emitted in the direction of the $\tau$ spin in the $\tau$ rest frame, and additional information, such as the tau
decay impact parameter, help to reconstruct $CP$-sensitive observables.

At the time of the Snowmass-2013 studies~\cite{Dawson:2013bba}, it was believed that this reconstruction
would be challenging, though possible, in the hadron collider environment~\cite{Berge:2011ij}.
Most studies were focussed on the cleaner $e^+e^-$ collider environment, discussed in Section~\ref{sect:eeHTT}.
A study in Ref.~\cite{Harnik:2013aja} using an optimal observable based on the internal substructure of
$\tau^\pm\to\pi^\pm\pi^0\nu$ indicated sensitivity to $f_{CP}^{H\tau\tau}<0.04$ at 3,000\,fb$^{-1}$ integrated luminosity of HL-LHC.
However, it was found in Ref.~\cite{Askew:2015mda} that detector effects would be more important than originally suggested.
A realistic study by the ATLAS collaboration~\cite{ATL-PHYS-PUB-2019-008} was based on analysis of
$\tau^\pm\to\pi^\pm\pi^0\nu$ and indicated that at HL-LHC the statistical precision
on $f_{CP}^{H\tau\tau}$ would range between 0.10 and 0.30,
depending on the precision of the $\pi^0$ reconstruction.
Finally, a very detailed study of multiple $\tau$ decay channels by the CMS experiment~\cite{CMS:2021sdq}
achieved an expected precision of $f_{CP}^{H\tau\tau}<0.13$ at 68\% C.L. with about 140\,fb$^{-1}$.
The CMS experiment provided projection to 3,000\,fb$^{-1}$ as supplemental materials~\cite{CMS-HIG-20-006}
to Ref.~\cite{CMS:2021sdq}, from which we expect $f_{CP}^{H\tau\tau}<0.07$ at 300\,fb$^{-1}$
and $<0.008$ at 3,000\,fb$^{-1}$, which are entered in Table~\ref{table-cpscenarios}.

\section{Prospects of Higgs $CP$ measurements at an electron-positron collider}
\label{sect:lepton}

Many of the approaches to the $H$~boson\xspace $CP$ measurements at an electron-positron collider are similar to those
at a hadron collider, but with several notable features. First, the $e^+e^-$ collider environment is much cleaner,
and therefore even with a smaller number of $H$ bosons produced, essentially every final state of its decay
may be used for tagging. Second, certain final states, most notably $\tau^+\tau^-$, could be reconstructed and
analyzed for $CP$ structure with better efficiency. Third, the fixed initial-state energy in the $e^+e^-\to V^*\to VH$
production allows control over the $q^2$ of the initial $V^*$. Similarly, possible polarization of the colliding beams
may give additional control in polarization measurements.

The $CP$ structure of the $H$~boson\xspace couplings to gluons cannot be easily measured
at a lepton collider, because the decay to two gluons does not allow easy access to gluon polarization.
On the other hand, most other processes could be studied at an $e^+e^-$ collider, especially with the beam
energy above the $t\bar{t}H$ threshold.

\subsection{The $VH$ process at an electron-positron collider}
\label{sect:eeVH}

The $e^+e^-\to ZH/\gamma^*H\to (2f)H$ process is the dominant SM process at lower energies
with cross section of about $240/129/57/13$\,fb at $\sqrt{s}=250/350/500/1,000$\,GeV.
Full angular analysis of the final state allows access to $CP$ information.
Similarly to the hadron collider, the process $e^+e^-\to \gamma H$ is possible to study,
but does not allow access to $CP$ properties from the angular analysis. This channel
has been used at LEP to set constraints on the $H$~boson\xspace production with possible
anomalous $HZ\gamma$ and $H\gamma\gamma$ couplings.

An early feasibility study of spin-parity determination and analysis of the $HZZ$ and $HZ\gamma$ coupling
tensor structure in the $VH$ process at an $e^+e^-$ collider was performed as part of the TESLA
design~\cite{ECFADESYLCPhysicsWorkingGroup:2001igx}
based on 300 fb$^{-1}$ at a centre-of-mass energy of 500 GeV and $m_H=120$ GeV.
The Snowmass-2013 studies~\cite{Dawson:2013bba} relied on Ref.~\cite{Anderson:2013afp}, which compared the expected performance
of an $e^+e^-$ collider and the LHC, where the $H\to b\bar{b}$ and $Z\to\ell\ell$ decays were used in the former case.
Precision on the fraction of the $CP$-odd cross-section contribution of about 0.03 was obtained
across the four energy and luminosity scenarios, which correspond to the values of $f_{CP}^{HVV}$
entered in Table~\ref{table-cpscenarios}. The significant reduction in the $f_{CP}^{HVV}$ uncertainties
with energy is due to the increase of the $q^2$ of intermediate $Z$, and therefore higher relative contribution
of the higher-dimension operators to the production cross section, where it is assumed that no strong momentum
dependence of couplings occurs at these energies.
There wer no separate studies of precision on $f_{CP}^{HZ\gamma}$ or $f_{CP}^{H\gamma\gamma}$
at that time, but a recent update of these studies in Appendix~\ref{app:B} indicate that it is not feasible
to constrain $f_{CP}^{HZ\gamma}$ or $f_{CP}^{H\gamma\gamma}$ at an $e^+e^-$ collider with
parameters listed in Table~\ref{table-cpscenarios}.

The $CP$-odd $HZZ$ and $HZ\gamma$ couplings have been investigated with 5.6\,ab$^{-1}$ at 240\,GeV
with $Z\to\mu^+\mu^-$ and $H\to b\bar{b}, c\bar{c}$, gg in Ref.~\cite{Sha:2022bkt}, which follows closely
similar earlier studies in Refs.~\cite{Craig:2015wwr,Beneke:2014sba}.
The expected constraint on the $CP$-odd $HZ\gamma$ coupling $a_3^{Z\gamma}$
is about a factor of six larger than the SM $a_2^{Z\gamma}$ expectation~\cite{Davis:2021tiv},
which indicates that it is hard to constrain photon couplings in this process.
The expected constraint on the $CP$-odd $HZZ$ coupling $a_3^{ZZ}$
requires careful investigation to translate it to a limit on $f_{CP}^{HVV}$ and
for a luminosity scenario about a factor of 20 smaller, as expected in Table~\ref{table-cpscenarios}.

\subsection{The VBF process at an electron-positron collider}
\label{sect:eeVBF}

VBF production with charged boson fusion $e^+e^-\to \nu_e\bar\nu_e(W^+W^-)\to \nu_e\bar\nu_e H$
is the dominant SM process at higher energies with cross section of $21/34/72/210$\,fb at $\sqrt{s}=250/350/500/1,000$\,GeV.
However, there is essentially no kinematic information to analyze in the final state with missing neutrinos,
with the exception of the momentum of the $H$~boson\xspace, which provides sensitivity to the higher-dimension
operators, but does not allow one to separate $CP$-odd and $CP$-even contributions.
Therefore, this channel is useful to study $CP$ in the subsequent $H$~boson\xspace decays, though the lack of
a vertex from associated particles makes certain techniques less reliable, as discussed in application
to $H\to \tau^+\tau^-$ for example.

VBF production with neutral boson fusion $e^+e^-\to e^+e^-(ZZ/Z\gamma^*/\gamma^*\gamma^*)\to e^+e^- H$
cross section is smaller than of the main VBF channel with associated neutrions, but is still sizable at higher energies
with the SM cross section of $0.7/3/7/21$\,fb at $\sqrt{s}=250/350/500/1,000$\,GeV.
Full angular analysis allows access to $CP$ information.
For example, an ongoing study of the $ZZ$-fusion process at 1.4 TeV CLIC and 1 TeV ILC
are mentioned in Ref.~\cite{Bozovic-Jelisavucic:2022ivd}, and concrete results are expected to appear later.
While there is no dedicated study of the $CP$-odd $H\gamma\gamma$ and $HZ\gamma$ interactions
in VBF production $e^+e^-\to e^+e^-(ZZ/Z\gamma^*/\gamma^*\gamma^*)\to e^+e^- H$,
an analogy has been drawn to the VBF process at LHC in Appendix~\ref{app:B}, which indicates that
it is unlikely that $f_{CP}^{HZ\gamma}$ or $f_{CP}^{H\gamma\gamma}$ could be constrained
at an $e^+e^-$ collider.

\subsection{The $t\bar{t}H$ process at an electron-positron collider}
\label{sect:eettH}

$H$~boson\xspace production in association with top quarks $e^+e^-\to t\bar{t}H$ is the fourth production channel for energies
above the threshold around 500\,GeV, with cross section of $0.27/2.0$\,fb at $\sqrt{s}=500/1,000$\,GeV.
Many of the techniques used at the LHC in Section~\ref{sect:Htt} and at a muon collider in Section~\ref{sect:hemuon}
can be employed at an $e^+e^-$ collider, with the diagram in Fig.~\ref{fig:Hff} representating
kinematic information in the process.

A study of ${C\!P}$-odd contribution in the $Ht\bar{t}$ coupling has been studied in the context
of ILC~\cite{Dawson:2013bba}. Cross-section dependence on the coupling has been employed
and an uncertainty on $f_{CP}^{Htt}$ of 0.08 (0.29) at 1,000 (500) GeV center-of-mass energy has been estimated.
A beam polarization of $(+0.2,-0.8)$~\cite{Price:2014oca} and $(+0.3,-0.8)$ is assumed at 1,000 and 500 GeV, respectively.
A more recent study indicates sensitivity to $f_{CP}^{Htt}$ of about 0.07 expected with 2,000\,fb$^{-1}$
and 1,400\,GeV~\cite{CLICdp:2018esa}, which employs a similar cross section dependence.
Interpretation of a cross-section deviation as an indication of ${C\!P}$-odd coupling contribution
is strongly model-dependent, but allows access to anomalous $Ht\bar{t}$ couplings.
An analysis of the full kinematic information could proceed in a manner similar to that
employed at LHC and would benefit from the clean $e^+e^-$ collider environment with
the beam energy constraints available.
An improvement from using the differential information has been observed in Ref.~\cite{thesis-Zhang}.

\subsection{The $H\to \tau^+\tau^-$ and other decay processes at an electron-positron collider}
\label{sect:eeHTT}

At the time of the Snowmass-2013 exercise~\cite{Dawson:2013bba}, most $CP$ studies with $H\to \tau^+\tau^-$
were performed in a clean $e^+e^-$ environment, either in the decays $\tau\to\pi\pi\nu$~\cite{Desch:2003rw,Harnik:2013aja},
or in all final states~\cite{Berge:2012wm,Berge:2013jra}. All studies agree on a similar $f_{CP}^{H\tau\tau}$
precision of about $0.01$ for the typical scenarios in Table~\ref{table-cpscenarios}.
The precision becomes somewhat worse with an increased collider energy due to the reduced $ZH$ production
cross-section, and this technique relies on the knowledge of the $Z$ vertex.
A recent full simulation study of the ILC physics reach with 1,000\,fb$^{-1}$ at 250\,GeV indicates a very similar $f_{CP}^{H\tau\tau}$
precision of about $0.01$ with $\tau^\pm\to\pi^\pm\nu$ and $\tau^\pm\to\pi^\pm\pi^0\nu$\cite{Jeans:2018anq, Bozovic-Jelisavucic:2022ivd},
but additional $\tau$ lepton decays may bring an increase in sensitivity.
We therefore leave the estimates in Table~\ref{table-cpscenarios} the
same as in the Snowmass-2013 projection. Further improvements could be achieved using the
lessons learned from the realistic analysis of the $H\to \tau^+\tau^-$ channel at LHC, as
discussed in Section~\ref{sect:HTT}.

Analysis of the other decay channels, most notably $H\to 4f$, could be performed at an $e^+e^-$ collider.
The clean collider environment would allow exploration of multiple final states, beyond just the golden
channels with charged leptons used at LHC. However, as noted in Appendix~\ref{app:B}, the number
of produced $H\to ZZ\to 4f$ events at an $e^+e^-$ collider would be significantly smaller than the number
of $H$ bosons produced in the golden clean channel $H\to 4\ell$ at a proton collider.

\section{Comparison to EDM measurements}
\label{sect:edm}

A dedicated Snowmass-2022 study of EDM measurements can be found in Ref.~\cite{Alarcon:2022ero}.
Asymmetry in the charge distribution along the particle's spin requires $T$ violation,
which is equivalent to $CP$ violation when invoking the $CPT$ theorem.
The EDMs of atoms and molecules are sensitive to $CP$ violation in interactions of the $H$~boson\xspace through loop effects.
The SM values of these EDMs are beyond the current or planned experimental reach, which allows excellent null tests in the SM.
The EDM constraints on $CP$-odd $H$~boson\xspace couplings are typically stronger than those from direct $H$~boson\xspace
measurements~\cite{Pospelov:2005pr,Li:2010ax,Cirigliano:2016nyn,Cirigliano:2019vfc}.
However, these constraints are set under an assumption that only one modification
of the $H$~boson\xspace coupling is present in the loop, and therefore no cancellation effect is allowed.
With multiple $CP$-odd EFT operators, the EDM measurements set constraints on certain linear
combinations of these operators, and direct constraints on the $CP$-odd operators in the $H$~boson\xspace
measurements provide complementary information.

For example, the $H\gamma\gamma$, $HZ\gamma$, $HZZ$, as $H$gg induce EDMs through one-loop diagrams.
Replacing the $HVV$ vertex with a fermion loop in these diagrams leads to two-loop graphs, through which
the $Hff$ couplings can contribute. One can also analyze these interactions with simultaneous contributions of
loops of SM particles together with BSM interactions and point-like $HVV$ interactions generated by heavy BSM states.
At the same time, the second vertex of the $H$~boson\xspace involves $Hu\bar{u}$, $Hd\bar{d}$, or $Hee$ interactions,
where $CP$ violation could be introduced as well. Therefore, in general, EDMs receive contributions from
a large number of $CP$-odd interactions, allowing for the possibility of cancellations.
While this brings complications, the EDM measurements may also allow the only access to $CP$ violation
in the $Hee$, $Hu\bar{u}$, and $Hd\bar{d}$ interactions.
Resolving all constraints simultaneously will require direct measurements of the $H$~boson\xspace couplings
in combination with EDM measurements.
Moreover, it has not been experimentally established if the $H$~boson\xspace couples to the first-family
fermions. In case these couplings are absent or significantly suppressed, EDM measurements
provide no constraints of $CP$ violation in $H$~boson\xspace interactions.

As part of this Snowmass study, we examine EDM constraints on parameters in Table~\ref{table-cpscenarios}
and add the $Hee$, $Hu\bar{u}$, and $Hd\bar{d}$ couplings in Table~\ref{tab:edm}.
These constraints from the present EDM measurements are obtained in Appendix~\ref{app:C}.
In Table~\ref{tab:edm}, only one $CP$-odd $HX$ coupling is allowed to be present at a time.
As it can also be seen in Fig.~\ref{fig:edm} of Appendix~\ref{app:C}, constraints on individual couplings
$H\gamma\gamma$, $HZ\gamma$, $HZZ$ are essentially lost if two other couplings are allowed to be present,
but a big part of parameter space is still excluded from a correlated measurement.
Most constraints on the parameters in Table~\ref{tab:edm} are dominated by the
the current limit on electron EDM $d_e<1.1\times 10^{-29}$\,$e$\,cm~\cite{ACME:2018yjb}
from the ThO measurement,
while the $CP$-odd $H$gg, $Hu\bar{u}$, and $Hd\bar{d}$ couplings are constrained by
the neutron~\cite{Abel:2020pzs} and mercury~\cite{Graner:2016ses} EDMs.
The limit on the neutron EDM is $d_n<1.8\times 10^{-26}$\,$e$\,cm~\cite{Abel:2020pzs},
and the mercury EDM constraint is equivalent to a similar limit on $d_n$~\cite{Graner:2016ses},
for the couplings under consideration here.

Over the next two decades, one could expect an order of magnitude increase in the precision
of the electron EDM every 5-6 years, e.g. Fig.~5 in Ref.~\cite{Alarcon:2022ero}.
There is also a dramatic increase possible in the nucleon EDM measurements,
e.g. Fig.~8 in Ref.~\cite{Alarcon:2022ero}.
There is a proposal to reach a precision on the proton EDM $d_p< 10^{-29}$\,$e$\,cm
using the proton storage ring within the next decade~\cite{Alexander:2022rmq}, which would
be a big improvement over the current neutron EDM constraint.
This may lead to an improvement by $10^3$ in constraints on $CP$-odd $H$gg, $Hu\bar{u}$, and $Hd\bar{d}$
couplings, and potentially to an improvement by $10^6$ in constraints on corresponding~$f_{CP}^{HX}$.
We note that even under the assumption of one $CP$-odd contribution to EDM, the
expected constraint on $f_{CP}^{H\rm gg}$ at the HL-LHC in Table~\ref{table-cpscenarios}
is stronger than the present EDM constraint in Table~\ref{tab:edm}.
With the above potential improvement on the proton EDM using the proton storage ring,
this will change. However, the HL-LHC constraints will be essential in order to analyze
all $CP$-violating couplings in Table~\ref{tab:edm} simultaneously.

\begin{table}[t!]
\center\small
\renewcommand{\arraystretch}{1.3}
\captionsetup{justification=centerlast}
\caption{Constraints on the parameter $\Big|\frac{f_{CP}^{HX}}{1-f_{CP}^{HX}}\Big|$ at $68\%$ C.L.\
from EDM measurements, assuming only one $CP$-odd $HX$ coupling is nonzero at a time.
Refer to Appendix~\ref{app:C} for more details.
}
\begin{tabular}{|ccccccccccc|}
	\hline\hline
$HX$ coupling & ~~${H\rm gg}$~~ & ~~${H\gamma\gamma}$~~ & ~~${HZ\gamma}$~~ & ~~${HZZ}$~~ & ~~${Ht\bar{t}}$~~ & ~~${Hu\bar{u}}$~~ & ~~${Hd\bar{d}}$~~ & ~~${H\tau\tau}$~~ & ~~${H\mu\mu}$~~ & ~~${Hee}$~~
	\\
	\hline
${f_{CP}^{HX}}/{(1-f_{CP}^{HX})}<$
& $0.12 $&	$2.4\cdot 10^{-8}$&$	4.4\cdot 10^{-8}$&$	1.2\cdot 10^{-13}$&	$4.3\cdot 10^{-7}$& $0.72$ & $0.039$ &  $	2.2\cdot 10^{-2} $& $36$ & $1.1\cdot 10^{-6}$
\\
	\hline\hline
\end{tabular}
\label{tab:edm}
\end{table}

\section{Summary}
\label{sect:summary}

We have reviewed the current status and prospects of the search for $CP$ violation in interactions
of the Higgs boson with either fermions or bosons. These studies provide several well-defined
and important benchmark measurements considered in the Particle Physics Community Planning
Exercise (a.k.a. ``Snowmass"). These benchmarks are compared between the proton, electron-positron,
photon, and muon colliders in Table~\ref{table-cpscenarios}.
Connection is made to the study of virtual effects in the electric dipole moment measurements.

\bigskip

\noindent
{\bf Acknowledgments}:
We would like to thank all contributors of individual studies and participants of the ``Snowmass" community exercise.
A.V.G., J.D., L.S.M.G., and S.K. thank the United States National Science Foundation for the financial support, under grant number PHY-2012584.
R.K.B and D.G. thank the United States Department of Energy for the financial support, under grant number DE-SC0016013.

\clearpage

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Model-Based Reinforcement Learning (MBRL) is proposed to reduce sample complexity introduced by model-free Deep Reinforcement Learning (DRL) algorithms \cite{nagabandi2018neural}.
Specifically, MBRL approximates the system dynamics with a parameterized model, which can be utilized for policy optimizing when the training data is very limited or costly to obtain in the real world.

Implementing a RL experiments from scratch can be tedious and bug-introducing.
Luckily, many open-source frameworks have been developed to facilitate DRL research, including baselines \cite{baselines}, rllab \cite{duan2016benchmarking}, Coach \cite{coach}, and Horizon \cite{gauci2018horizon}. However, these frameworks are mainly implemented for model-free DRL methods, and lack enough support for MBRL.

Existing model-based frameworks are few and have some shortcomings. The work in \cite{DBLP:journals/corr/abs-1907-02057} gives a comprehensive benchmark over state-of-the-art MBRL algorithms, but the implementations are scattered across different codebases without a unified implementation, posing obstacles to conduct experiments with it. The work in \cite{finn2016guided} provides the implementations for Guided Policy Search(GPS) \cite{levine2016end}, which supports robotics controlling tasks. But it lacks support for other MBRL algorithms. Thus, a unified MBRL open-source framework is in need.
To fill this gap, we design and implement a unified MBRL framework, Baconian, by trading off the diversity of included MBRL algorithms against the complexity of the framework.
Users can reproduce benchmark results or prototype their idea easily with it by a minimal amount of codes without understanding the detailed implementations.
Moreover, the design of Baconian not only benefits the research of MBRL,  but is also applicable to other types of RL algorithms including model-free algorithms.
The codebase is available at \url{https://github.com/cap-ntu/baconian-project}. The demo video is available at \url{https://youtu.be/J6xI6qI3CvE}.
\begin{figure}
	\centering
	\includegraphics[width=0.85\columnwidth]{features_tab.pdf}
	\caption{Feature list of Baconian.}
	\label{fig:feature}
\end{figure}
\section{Main Features}
Baconian supports many RL algorithms, test environments, and experiment utilities. We summarize the main features in Fig. \ref{fig:feature}.

\textbf{State-of-the-Art RL Algorithms}. We implement many widely used RL algorithms. For model-based algorithms, we implement Dyna\cite{sutton1991dyna}, ME-TRPO (Model-ensemble Trust Region Policy Optimization) \cite{kurutach2018model}, MPC (Model Predictive Control)\cite{richards2005robust}, iLQR (Iterative Linear Quadratic Regulator)\cite{tassa2012synthesis}, etc. Since many model-based algorithms are built upon model-free algorithms, we also implement some popular model-free algorithms including DQN\cite{DBLP:journals/corr/HosuR16}, DDPG\cite{DBLP:journals/corr/LillicrapHPHETS15}, and PPO\cite{DBLP:journals/corr/SchulmanWDRK17} in Baconian.

\textbf{Supported Test Environments}. To evaluate the performance of RL algorithms, it is a must to support a wide range of test environments. Baconian support OpenAI Gym\cite{gym}, RoboSchool\cite{klimov2017roboschool}, DeepMind Control Suite\cite{deepmindcontrolsuite2018}. These test environments cover most essential tasks in RL community.

\textbf{Experiment Utilities}. Baconian provides many utilities to reduce users' efforts on experiment set-up, hyper-parameter tuning, logging, result visualization, and algorithms diagnosis. We provide integration of TensorFlow to support neural network building, training, and managing. As the hyper-parameters play a critical role in RL experiments, we provide user-friendly parameter management utility to remove the tedious work of setting, loading, and saving these hyper-parameters.

\textbf{Open-source Support}. Baconian provide detailed user guides and API references\footnote{The documentation can be found at \url{https://baconian-public.readthedocs.io/en/latest/API.html}.}, so users can hand on Baconian easily and conduct novel MBRL research upon it. We also release some preliminary benchmark results in the codebase.

\section{Design and Implementation}
Baconian consists of three major components, namely, Experiment Manager, Training Engine, and Monitor.
The system overview of Baconian is shown in Fig. \ref{fig:system-diagram}. Various design patterns are applied to decouple the complicated dependencies across different modules to enable the easily extension and programming over the framework.

\begin{figure}
	\centering
	\includegraphics[width=1.0\columnwidth]{baconian-system-design.pdf}
	\caption{The system design of Baconian. The system is divided into three main modularized components to minimize the coupling for flexibility and maintainability.}
	\label{fig:system-diagram}
\end{figure}

\subsection{Experiment Manager}
The Experiment Manager consists of Experiments Settings, Status Collector, and Experiment Recorder. Experiments Settings manages the creating and initialization of each module. Status Collector controls the status information collected across different modules to compose a globally shared status that can be used including learning rate decay, exploration strategy scheduling, etc.
Experiment Recorder will record the information generated from the experiment, such as loss, rewards. Such information will be handed to the Monitoring layer for rendering or saving.

\subsection{Training Engine}
 Training Engine handles the training process of the MBRL algorithms. The novelty of the design lies in abstracting and encapsulating the training process as a Control Flow module, which controls the execution processes of the experiment based on the user's specifications, including the agent's sampling from environment, policy model and dynamics model optimization, and testing.
 %
 MBRL algorithms can be complicated\cite{sutton1991dyna, nagabandi2018neural}. Such abstractions can decouple the tangled and complicated MBRL training processes into some independent tasks, which are further encapsulated as the sub-modules of Control Flow module for providing flexibility.
\subsection{Monitor}
Monitor is responsible for monitoring and recording of the experiment as it proceeds. This includes recording necessary loggings, printing information/warning/error, and rendering the results.
\begin{figure}
	\centering
	\includegraphics[width=0.8\columnwidth]{process.pdf}
	\caption{The procedure to create an MBRL experiment in Baconian. Each module is replaceable and configurable to reduce the effort of building from scratch.}
	\label{fig:process}
\end{figure}

\section{Usage}
This section presents the procedures to create a MBRL experiment with the essential modules in Baconian. The procedures are shown in Fig. \ref{fig:process}\footnote{For more details of how to configure these modules, please see the documentation page \url{https://baconian-public.readthedocs.io/en/latest/step_by_step.html} due to the page limit.}. For high flexibility, most of modules are customizable.
Meanwhile, user can directly adopt built-in benchmark module or codes if customization is unnecessary.

First, the user should create a environment and a RL algorithm module with necessary hyper-parameters configured, e.g., neural network size, learning rate. Algorithm module is usually composed of a policy module and a dynamics model module depending on different algorithms.
Then, the user needs to create an agent module by passing the algorithm module, and the exploration strategy module if needed, into it.

Second, the user should create a control flow module that defines how the experiments should be proceeded and the stopping conditions. This includes defining how much samples should be collected for training at each step,
and what condition to indicate the completion of an experiment, etc.
Some typical control flows have already been implemented in Baconian to meet the users' needs.

Finally, an experiment module should be created by passing the agent, environment, control flow modules into it, and then launched. After that, the Baconian will handle the experiment running, monitoring, and results saving/logging, etc.

\section{Conclusion}
This paper presented a unified, reusable, and flexible framework, Baconian, for MBRL research.
It can reduce users' effort to conduct MBRL experiments and prototype new MBRL algorithms.
In the future, we will implement more state-of-the-art MBRL algorithms and benchmark them on different tasks.