refname refsname

#  CERN-TH-2023-158, EFI-23-8  Classifying the ${\mathcal{CP}}$ properties of
the $ggH$ coupling
in $H+2j$ production

Henning Bahl<EMAIL_ADDRESS>University of Chicago, Department of Physics,
5720 South Ellis Avenue, Chicago, IL 60637 USA  Elina Fuchs
<EMAIL_ADDRESS>CERN, Department of Theoretical Physics, 1211 Geneve 23,
Switzerland  Institut für Theoretische Physik, Leibniz Universität Hannover,
Appelstraße 2, 30167 Hannover, Germany  Physikalisch-Technische Bundesanstalt,
Bundesallee 100, 38116 Braunschweig, Germany  Marc Hannig
<EMAIL_ADDRESS>Institut für Theoretische Physik, Leibniz
Universität Hannover, Appelstraße 2, 30167 Hannover, Germany  Marco Menen
<EMAIL_ADDRESS>Institut für Theoretische Physik, Leibniz
Universität Hannover, Appelstraße 2, 30167 Hannover, Germany  Physikalisch-
Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

###### Abstract

The Higgs–gluon interaction is crucial for LHC phenomenology. To improve the
constraints on the ${\mathcal{CP}}$ structure of this coupling, we investigate
Higgs production with two jets using machine learning. In particular, we
exploit the ${\mathcal{CP}}$ sensitivity of the so far neglected phase space
region that differs from the typical vector boson fusion-like kinematics. Our
results suggest that significant improvements in current experimental limits
are possible. We also discuss the most relevant observables and how
${\mathcal{CP}}$ violation in the Higgs–gluon interaction can be disentangled
from ${\mathcal{CP}}$ violation in the interaction between the Higgs boson and
massive vector bosons. Assuming the absence of ${\mathcal{CP}}$-violating
Higgs interactions with coloured beyond-the-Standard-Model states, our
projected limits on a ${\mathcal{CP}}$-violating top-Yukawa coupling are
stronger than more direct probes like top-associated Higgs production and
limits from a global fit.

###### Contents

  1. 1 Introduction
  2. 2 Gluon fusion with two jets in an EFT approach
    1. 2.1 Gluon fusion in association with two jets
    2. 2.2 Effective Higgs-vector boson interactions
  3. 3 Event generation
  4. 4 Analysis strategy
    1. 4.1 Signal-background separation
    2. 4.2 Separating the different ggF2j contributions
  5. 5 Sensitivity to the Higgs–gluon coupling
    1. 5.1 ggF2j signal region
    2. 5.2 VBF signal region
    3. 5.3 Combination
    4. 5.4 Highest-impact observables
  6. 6 Disentangling ${\mathcal{CP}}$ violation in the $ggH$ and $HVV$ couplings
  7. 7 Limits on the ${\mathcal{CP}}$ structure of the top-Yukawa coupling
  8. 8 Conclusions
  9. A Details of the event generation
  10. B Distributions of kinematic variables
  11. C Training uncertainty
  12. D Likelihood evaluation

##  1 Introduction

More than ten years ago, the ATLAS and CMS collaborations announced the
discovery of a new particle at a mass of about $125\,\,\mathrm{GeV}$ during
Run-1 of the Large Hadron Collider (LHC) [1, 2]. Since its discovery in 2012,
the quantum numbers of this Higgs boson have been tested extensively, as well
as its interactions with other SM particles [3, 4]. Up to now, all results are
in agreement with the predictions of the SM within the experimental and
theoretical uncertainties. However, effects from beyond the Standard Model
(BSM) physics could still be hidden in the current uncertainties and may be
unveiled with the large amount of data to be collected during LHC Run-3 and
the high-luminosity phase of the LHC (HL-LHC).

A so far relatively unconstrained property of the discovered Higgs boson is
its behaviour under ${\mathcal{CP}}$ transformations. This is of particular
interest given that the amount of ${\mathcal{CP}}$ violation present in the SM
is — by several orders of magnitude — insufficient to explain the baryon
asymmetry of the Universe [5, 6]. The possibility of the Higgs boson being a
pure ${\mathcal{CP}}$-odd state was already ruled out shortly after its
discovery by analyzing the ${\mathcal{CP}}$ structure of its couplings to
massive vector bosons [7, 8]. It is, nevertheless, still possible that the
discovered Higgs boson is a mixed ${\mathcal{CP}}$-state instead of the
${\mathcal{CP}}$-even state predicted by the SM.

Electric dipole moments (EDMs) are sensitive probes of ${\mathcal{CP}}$
violation beyond the SM [9, 10] and their experimental upper bounds, in
particular of the electron [11, 12], neutron [13] and mercury [14], place
strong constraints on ${\mathcal{CP}}$-violating Higgs interactions [15, 16,
17, 18, 19]. These constraints, however, strongly depend on the assumption
about the first-generation Yukawa couplings [20], which themselves are only
very weakly constrained [21, 22]. It is therefore of great interest to search
for possible ${\mathcal{CP}}$-violating effects also at colliders, which allow
for a distinction between different Higgs couplings.

At the LHC, ${\mathcal{CP}}$-violating Higgs interactions can be constrained
either directly using ${\mathcal{CP}}$-odd observables or indirectly using
${\mathcal{CP}}$-sensitive, but ${\mathcal{CP}}$-even observables. While
measuring a non-zero value for the former is an unambiguous sign of
${\mathcal{CP}}$ violation, deviations in ${\mathcal{CP}}$-even observables
are only indicative for ${\mathcal{CP}}$ violation. On the other hand, it can
be difficult to measure ${\mathcal{CP}}$-odd observables since it often
requires measuring at least four independent momenta associated either with
the Higgs production or with its decay (see e.g. Brehmer:2017lrt). In this
situation, the measurement of ${\mathcal{CP}}$-sensitive observables can
provide valuable complementary information.

The most stringent constraints on ${\mathcal{CP}}$-violating Higgs couplings
so far have been set on the Higgs interactions with massive vector bosons [8,
24, 25, 26, 27, 28, 29, 30, 31, 32]. In most BSM theories, ${\mathcal{CP}}$
violation in these interactions is expected to be loop-suppressed (given that
a pseudoscalar cannot be coupled to massive vector bosons at the tree level).
${\mathcal{CP}}$ violation in the Higgs interactions with fermions can, in
contrast, occur unsuppressed. These are far less constrained with the existing
experimental analyses targeting the Higgs interaction with tau leptons [33,
34] and with top quarks [35, 36, 37, 38, 30, 39, 40].

Since the top-Yukawa coupling is of special interest due to its magnitude,
also many phenomenological studies have been carried out focusing mainly on
top-associated Higgs production as a tree-level probe of the Higgs–top-quark
interaction [41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 49, 50, 51, 50, 52, 53,
54, 53, 55, 56, 54, 57, 58, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 68, 68, 69, 67, 20, 18, 71, 72, 71, 73, 74]. Besides that, the
${\mathcal{CP}}$ character of the top-Yukawa coupling can also be probed by
investigation of the Higgs production via gluon fusion (ggF). While ggF
production alone (without the association of jets) is only sensitive to the
${\mathcal{CP}}$ nature of the top-Yukawa interaction via its total rate, ggF
production in association with one or two jets is more directly sensitive. In
particular, Higgs production in association with two jets (ggF2j) is well
known to be ${\mathcal{CP}}$-sensitive via the difference of the azimuthal
angles of the two jets that enables the construction of a ${\mathcal{CP}}$-odd
observable [75, 76, 77, 47, 78, 74].111For ggF production in association with
one jet, jet substructure information has to be exploited to construct a
${\mathcal{CP}}$-odd observable [79].

Generally, the ggF2j production does not directly probe the ${\mathcal{CP}}$
structure of the top-Yukawa interaction but of the Higgs–gluon interaction.
Conversely, a ${\mathcal{CP}}$-violating Higgs–gluon interaction cannot only
be induced by a ${\mathcal{CP}}$-violating Higgs–top-quark interaction but
also by ${\mathcal{CP}}$-violating Higgs interactions with coloured BSM
states. Yet, given the increasingly strong limits on the mass scale of
coloured BSM states set by experimental searches at the LHC [80, 81, 82, 83,
84, 85], any sign for ${\mathcal{CP}}$ violation in the Higgs–gluon
interaction is plausible to originate at least partially from the Higgs–top-
quark interaction. This makes the ggF2j channel not only interesting in its
own respect, but also in the context of constraining the ${\mathcal{CP}}$
structure of the effective Higgs-gluon interaction and the top-Yukawa
coupling.

Existing experimental measurements so far only put relatively weak constraints
on the ${\mathcal{CP}}$ character of the Higgs–gluon interaction [29, 37, 30].
In this work, we investigate how these limits could be improved. In
particular, we compare the potential of two distinct kinematic regions (a VBF-
like and a ggF2j-like kinematic region) and use well-established machine-
learning techniques (boosted classifiers) to identify whether a given event
originates from a ${\mathcal{CP}}$-even interaction, a ${\mathcal{CP}}$-odd
interaction, or from their interference. This allows us to construct both
${\mathcal{CP}}$-even and ${\mathcal{CP}}$-odd observables. In the ggF2j-like
region, these show a significantly better sensitivity than the difference of
the azimuthal angles of the two jets, which is widely used in the literature.
Moreover, we highlight how the separation into two kinematic regions helps to
distinguish ${\mathcal{CP}}$ violation in the Higgs–gluon interaction from
${\mathcal{CP}}$ violation appearing in the Higgs interaction with massive
vector bosons.

The paper is structured as follows. In Section 2, we review Higgs production
in association with two jets and introduce the effective theory that
parameterizes our BSM couplings. Afterwards, we discuss the event generation
and applied cuts in Section 3. Section 4 deals with the training setup of our
classifiers. In this section, we also define the observables that are used to
probe the ${\mathcal{CP}}$-structure of the Higgs-gluon coupling. We present
expected exclusion limits in Section 5. The interplay of the ${\mathcal{CP}}$
structure of the Higgs–gluon interaction with the ${\mathcal{CP}}$ structure
of the Higgs couplings with massive vector bosons is discussed in Section 6.
The interpretation of our limits in terms of limit on the Higgs–top-quark
interaction and a comparison to current experimental limits is shown in
Section 7. Finally, Section 8 concludes our findings and gives a brief outlook
into possible future studies.

##  2 Gluon fusion with two jets in an EFT approach

Our work focuses on Higgs production with two jets. Here, we give an
introduction to the relevant processes in the SM and then discuss our
parameterization of BSM effects.

###  2.1 Gluon fusion in association with two jets

(a) $gg$-initiated

(b) $gq$-initiated

(c) $qq$-initiated

Figure 1: Example Feynman diagrams for the various ggF2j sub-channels.

(a) VBF

(b) $VH$

Figure 2: Examplary Feynman diagrams for the considered background processes
where $V\in[W^{+},W^{-},Z]$ and $q\in[q,\bar{q}]$.

Higgs production via gluon fusion is the main production channel at the LHC
with its cross section being (at minimum) an order of magnitude higher than
any other Higgs production mode at $\sqrt{s}=13$ TeV. Even if two or more jets
associated with the parton level interaction are required in the final state,
the cross section is still higher than all other Higgs production modes [86]
($\sigma_{\mathrm{ggF}}^{j\geq 2}=7.88~{}\mathrm{pb}$ and
$\sigma_{\mathrm{VBF}}=3.78~{}\mathrm{pb}$ at $\sqrt{s}=13~{}\mathrm{TeV}$).
In the SM, the gluon fusion process is induced mainly by a top-quark loop,
while for our work we consider an effective point-like interaction with
unknown ${\mathcal{CP}}$ state (see Section 2.2).

The ggF2j process can be classified by its initial state, as can be seen in
Fig. 1. The individual initial states with gluons have higher cross sections
due to the larger contribution of the gluons compared to those of the quarks
to the parton distribution functions (PDFs) of the proton. In our setup, we
find that the $gg$ initial state contributes $\sim 72\%$ to the total ggF2j
cross section, while the $gq$ initial state still contributes $\sim 26.5\%$.
The contribution of the $qq$ initial state is very small ($\sim 1.5\%$). We
furthermore note that the interference between the $qq$-induced ggF2j
production and Higgs production via VBF was found to be negligible [87, 88].

Example Feynman diagrams for the considered background (BG) processes — VBF
and $VH$ production — are shown in Fig. 2.

###  2.2 Effective Higgs-vector boson interactions

We perform our study in an effective field theory (EFT) framework. Following
Artoisenet:2013puc, we parameterize the interaction of the Higgs boson with
gluons in the following form,

$\mathcal{L}_{ggH}=-\frac{1}{4v}\bigg{(}-\frac{\alpha_{s}}{3\pi}c_{g}G_{\mu\nu}^{a}G^{\mu\nu,a}+\frac{\alpha_{s}}{2\pi}\tilde{c}_{g}G_{\mu\nu}^{a}\tilde{G}^{\mu\nu,a}\bigg{)}H\,,$
(1)

where $H$ is a Higgs field, $G_{\mu\nu}^{a}$ is the gluon field strength
tensor, $v=246\,\,\mathrm{GeV}$ is the vacuum expectation value and
$\alpha_{s}=g_{s}^{2}/4\pi$ is the strong coupling constant. Here, we assume
that this interaction already includes the Higgs–gluon vertex induced by the
top quark in the infinite top-mass limit.222In other works (see e.g.
CMS:2021nnc,CMS:2022mlq, the Higgs–gluon interaction is written in the form
$\mathcal{L}_{ggH}=-\frac{\alpha_{s}\pi}{v}\bigg{(}c_{gg}G_{\mu\nu}^{a}G^{\mu\nu,a}+\tilde{c}_{gg}G_{\mu\nu}^{a}\tilde{G}^{\mu\nu,a}\bigg{)}H$
without including the effect of the top-quark loop. In the infinite top-quark
mass limit, both parameterizations are related via $c_{g}=1+12\pi^{2}c_{gg}$
and $\tilde{c}_{g}=-8\pi^{2}\tilde{c}_{gg}$. Accordingly, $c_{g}=1$ and
$\tilde{c}_{g}=0$ in the SM. Deviations from the SM can either be induced by a
modified top-Yukawa coupling or by coloured BSM particles.

The prefactors of the operators are chosen such that modifications of the top-
Yukawa interaction directly map to the operators of Eq. 1 if we assume that no
coloured BSM particles affect the Higgs–gluon interaction. In this case, we
have $c_{g}=c_{t}$ and $\tilde{c}_{g}=\tilde{c}_{t}$ in the infinite top-quark
mass limit, if we parameterize the top-Yukawa interaction via

$\mathcal{L}_{\text{top-
Yuk}}=-\frac{y_{t}^{\text{SM}}}{\sqrt{2}}\bar{t}(c_{t}+i\gamma_{5}\tilde{c}_{t})tH,$
(2)

where $y_{t}^{\text{SM}}$ is the SM top-Yukawa interaction and $c_{t}=1$ as
well as $\tilde{c}_{t}=0$ for the SM.

We account for effects due to the finite mass of the top quark by rescaling
the total rate and placing an upper bound on the Higgs transverse momentum
(see Section 3).

The modified Higgs-gluon coupling of Eq. 1 affects ggF2j production. Its
matrix element can be separated into three pieces

$\bigg{|}\mathcal{M}_{\mathrm{ggF2j}}\bigg{|}^{2}=c_{g}^{2}~{}\bigg{|}\mathcal{M}_{\text{even}}\bigg{|}^{2}+2c_{g}\tilde{c}_{g}\mathrm{Re}\bigg{[}\mathcal{M}_{\text{even}}\mathcal{M}_{\text{odd}}^{*}\bigg{]}+\tilde{c}_{g}^{2}~{}\bigg{|}\mathcal{M}_{\text{odd}}\bigg{|}^{2}\,.$
(3)

The first and third terms proportional to the squared values of the coupling
modifiers are ${\mathcal{CP}}$-even, while the second term parameterizing the
interference between the two ${\mathcal{CP}}$-states is ${\mathcal{CP}}$-odd.
${\mathcal{CP}}$ violation in the Higgs-gluon interaction is therefore only
realised for a non-zero value of the interference term. Correspondingly, the
interference term gives a non-zero contribution only for distributions of
${\mathcal{CP}}$-odd observables.

In addition to a modification of the Higgs–gluon interaction, in Section 6 we
will also consider a ${\mathcal{CP}}$-violating interaction of the Higgs boson
with massive vector bosons. We parameterize this interaction as333At dimension
six in the SM effective field theory (SMEFT), three distinct operators can
introduce ${\mathcal{CP}}$ violation in VBF production (see e.g.
Dedes:2017zog). In this work, we concentrate on the
$\mathcal{O}_{\Phi\widetilde{W}}$. We expect similar results for the other
operators.

$\mathcal{L}_{\Phi
VV}=\mathcal{L}_{\text{gauge}}+\frac{c_{\Phi\widetilde{W}}}{\Lambda^{2}}\mathcal{O}_{\Phi\widetilde{W}}\;\;\text{
with
}\;\;\mathcal{O}_{\Phi\widetilde{W}}\equiv\Phi^{\dagger}\Phi\widetilde{W}_{\mu\nu}W^{\mu\nu},$
(4)

where $\mathcal{L}_{\text{gauge}}$ is the SM gauge Lagrangian. $W$ and
$\widetilde{W}$ are the $SU(2)_{L}$ field strength and its dual, respectively.
$\Phi$ is the Higgs doublet and $\Lambda$ denotes the cut-off scale of the
EFT, which we set to 1 TeV. If not stated otherwise, we will for the main part
of this work assume that $c_{\Phi\tilde{W}}=0$ such that the Higgs coupling to
massive vector bosons is SM-like.

##  3 Event generation

As discussed in Section 1, we focus on Higgs production via gluon fusion in
association with two jets, which we refer to as ggF2j in the following. This
production channel offers a unique sensitivity to probe the ${\mathcal{CP}}$
nature of the Higgs–gluon interaction. Moreover, in this work, we concentrate
on the Higgs decay to two photons. While this simplifies our analysis, our
general analysis strategy can be straightforwardly adapted to other Higgs
decay channels.

The main background processes are Higgs production via vector-boson fusion
(VBF) and Higgsstrahlung ($VH$). Additionally, there is a continuous
background from the $qq\to qq$ process with two photons from final state
radiation. However, this background can be subtracted by a fit to the smoothly
falling di-photon mass distribution. In recent measurements of the
$H\to\gamma\gamma$ decay channel, the modelling of this background was
responsible for only an uncertainty of 1.3$\%$ to the total signal rate, which
is minor compared to other experimental uncertainties [91]. Therefore, we
expect the impact of this subtraction on our analysis to be small and neglect
the continuous background in our work.

Events are generated at parton level with MadGraph5_aMC@NLO (version 3.4.0)
[92] and are passed on to Pythia8 (version 8.306) [93] and Delphes3 (version
3.4.2) [94] for parton showering, hadronisation and detector simulation. For
the detector simulation with Delphes3, we employ the ATLAS detector card with
a modified jet cone radius of $\Delta R=0.4$ which is more widely used in
experimental analyses. A jet is then defined as a set of particles in this
cone which in total reach a minimum of $p_{T}^{j}\geq 20~{}\mathrm{GeV}$, with
the particles having passed an electron, muon, photon and neutrino filter
before. Reconstruction of the jet is performed using the anti-$k_{t}$
algorithm [95]. Photons (which are used for reconstructing the Higgs boson)
are identified by observing electrons and muons in a cone with radius $\Delta
R=0.5$ and momentum $p_{T}\geq 0.5~{}\mathrm{GeV}$. More details on the event
generation and the employed UFO model [96, 97] can be found in Appendix A.

| Fraction of accepted events
---|---
Applied cut |  ggF2j $|\mathcal{M}_{\mathrm{even}}|^{2}$ |  ggF2j $\mathrm{Interf.}$ |  ggF2j $|\mathcal{M}_{\mathrm{odd}}|^{2}$ |  VBF |  $VH$
Initial events |  100% |  100% |  100% |  100% |  100%
$N_{j}\geq 2$; $N_{\gamma}\geq 2$ |  48.1% |  50.8 % |  48.1% |  62.6% |  49.8%
$110\,\,\mathrm{GeV}\leq m_{\gamma\gamma}$ |  47.8% |  50.5% |  47.9% |  62.0% |  49.4%
$m_{\gamma\gamma}\leq 140\,\,\mathrm{GeV}$
$p_{T}^{\gamma 1}/m_{\gamma\gamma}\geq 0.35$ |  39.4% |  40.9% |  39.8% |  50.0% |  40.5%
$p_{T}^{\gamma 2}/m_{\gamma\gamma}\geq 0.25$
$p_{T}^{j1}\geq 30~{}\mathrm{GeV}$ |  38.6% |  40.2% |  38.6% |  49.7% |  39.9%
$p_{T}^{j2}\geq 20~{}\mathrm{GeV}$
$|\eta_{j}|\leq 2.5$ |  22.9% |  21.5% |  22.7% |  39.8% |  31.2%
$|\eta_{\gamma}|\leq 2.5$
$p_{T}^{H}\leq 200~{}\mathrm{GeV}$ |  18.6% |  18.4% |  18.3% |  34.4% |  26.8%
Table 1: Cutflow table for the Higgs production mechanisms considered in this
work. Listed are all cuts applied after the event generation, along with the
percentage of events that survives the cut.

We generate data sets for Higgs production from the ggF2j, VBF and $VH$
processes. While the latter two do not receive any BSM contribution in our
model, the event generation of ggF2j production is split into three terms
following the parameterization in Eq. 3. We therefore obtain separate data
sets for the terms proportional to $c_{g}^{2}$ (the ${\mathcal{CP}}$-even
amplitude squared), $\tilde{c}_{g}^{2}$ (the ${\mathcal{CP}}$-odd amplitude
squared), and $c_{g}\tilde{c}_{g}$ (the interference between the
${\mathcal{CP}}$-even and ${\mathcal{CP}}$-odd amplitude). This amounts to
five distinct classes which we further split into independent data sets for
training, validating and testing the classifiers, which will be introduced in
Section 4.

Before training the classifiers, we impose a number of simple baseline cuts
mimicking the event selection typically used by ATLAS and CMS for
$H\to\gamma\gamma$ measurements [35, 36]. A cutflow table detailing the
imposed cuts and the corresponding reduction in the event numbers can be found
in Table 1, where the percentage of acceptance after each cut is displayed for
all signal and background processes examined in this work.

First of all, a cut on $N_{j}$ and $N_{\gamma}$ ensures that at least two jets
and two photons are found in each event, which are needed to reconstruct the
Higgs boson and identify ggF2j production. Only accepting events with
$110~{}\mathrm{GeV}\leq m_{\gamma\gamma}\leq 140~{}\mathrm{GeV}$ ensures that
a Higgs boson is present in the event and suppresses any non-Higgs background.
The cuts on low values of $p_{T}^{\gamma}$ and $p_{T}^{j}$ suppress
misidentified photons and jets that do not originate from the parton level
interaction. Finally, only events with $|\eta_{j}|\leq 2.5$ and
$|\eta_{\gamma}|\leq 2.5$ are accepted to match the pseudorapidity coverage of
the ATLAS inner detector. As an additional requirement, we place an upper
limit of $p_{T}^{H}\leq 200\,\,\mathrm{GeV}$ on the Higgs transverse momentum
(reconstructed out of the two photon momenta). This cut ensures that the top-
quark loop inducing ggF2j production cannot be resolved and ggF2j production
can be reliably interpreted using the effective Lagrangian given in Eq. 1 (see
e.g. discussion in Buschmann:2014twa).

As the cutflow in Table 1 shows, the cut on $p_{T}^{H}$ reduces the number of
surviving events by $14-19\%$. While this reduction is not a critical
limitation for our analysis, in principle one can include the events with
higher $p_{T}^{H}$ beyond the infinite top-quark mass limit by employing the
FT${}_{\textrm{approx}}$ approximation [99]. This approximation combines the
exact top-quark mass and width in the Born, one-loop and real amplitudes with
approximated and re-scaled two-loop virtual contributions, originally applied
to multi-Higgs production. The FT${}_{\textrm{approx}}$ approximation has been
successfully validated against the full NLO calculation in the SM for $H+j$
production [100]. Under the assumption that the FT${}_{\textrm{approx}}$
approximation yields similarly accurate results for Higgs production in
association with a higher multiplicity of jets, Chen:2021azt has also improved
the calculation of differential cross sections for $H+2j$ by complementing the
exact real corrections with the approximated two-loop virtual corrections. The
good agreement between their NLO result in the heavy-top limit (HTL) and in
the FT${}_{\textrm{approx}}$ up to $p_{T}^{H}\leq 300\,\,\mathrm{GeV}$
justifies our treatment in the HTL within the chosen cut on $p_{T}^{H}$. The
relevance of the finite top-quark mass at higher $p_{T}^{H}$ and the success
of FT${}_{\textrm{approx}}$ in $H+j$ motivate a future study of
${\mathcal{CP}}$ properties of the Higgs boson without cutting out the
high-$p_{T}^{H}$ events. For the scope of our work, our robust treatment with
the $p_{T}^{H}$-cut results in conservative limits that might be improved by a
refined event generation.

##  4 Analysis strategy

Figure 3: Outline of the strategy for this analysis. First, a classifier is
trained for signal and background separation for each of the two considered
Higgs production processes X $=$ {ggF2j, VBF}. Afterwards, the obtained data
in the respective kinematic region are used in two classifiers each, where one
of them learns a ${\mathcal{CP}}$-even and the other one a
${\mathcal{CP}}$-odd observable.

Our analysis is structured into two steps. In the first step, two distinct
classifiers are trained to separate the events originating from the different
Higgs production channels. One classifier is trained to separate the signal
events (from ggF2j production) from the relevant background processes, while
the other is trained to recognize VBF events. The scores of these classifiers
are used to define two signal regions (SR), one containing ggF2j-like events
and the other containing VBF-like events. Both SRs offer possible advantages
in the analysis. In the ggF2j-SR, we expect an increased sensitivity as it
contains significantly more events than the VBF-SR. However, the ggF2j events
with a $q\bar{q}$ initial state share the topology of VBF Higgs production and
were shown to carry the most ${\mathcal{CP}}$ information after applying
typical VBF-cuts in an earlier work Hagiwara:2009wt.

Both classifiers are trained on the same input data. Since the two classifiers
are independent, it is possible for events to appear in both or none of the
kinematic regions. For our data set, the percentage of ggF2j data that appear
in both the ggF2j-SR and the VBF-SR is about $8\%$ of the total data set,
while about $11\%$ of the events do not appear in either of them.
Subsequently, the events identified in each category are passed to two
additional classifiers (see description below). These are used to distinguish
the squared ${\mathcal{CP}}$-odd and ${\mathcal{CP}}$-even matrix elements and
the interference term in the ggF2j production (see Eq. 3) and thereby to
construct ${\mathcal{CP}}$-sensitive observables. The analysis steps for each
of the kinematic regions are summarized in Fig. 3.

###  4.1 Signal-background separation

As mentioned above, two classifiers are trained to define a ggF2j- and a VBF-
SR with distinct kinematics. In both cases, we reduce the separation to a
binary classification problem, where the respective signal is trained against
all other types of events. The classifiers are set up with PyTorch and the
same data set is used for training, validation, and testing of both
classifiers.

The following kinematic variables are used as input for the signal-background
separation:

  * •

the energy $E$, the transverse momentum $p_{T}$, the pseudorapidity $\eta$,
and the azimuthal angle $\phi$ of the Higgs boson and the two leading-$p_{T}$
jets,

  * •

the invariant mass of the two leading-$p_{T}$ jets $m_{jj}$, their
pseudorapidity difference $\Delta\eta_{jj}$, and their azimuthal angle
difference $\Delta\phi_{jj}$ (where the sign is chosen such that the $\phi$ of
the jet with the smaller $\eta$ is subtracted from the one with larger
$\eta$),

  * •

the number of jets in the event $N_{j}$, and

  * •

the respective energy $E$ of each jet that is not leading or sub-leading in
$p_{T}$.

Among the most important variables for the signal/background separation are
$N_{j}$, $m_{jj}$, $|\Delta\eta_{jj}|$, and $\Delta\phi_{jj}$ (see e.g.
Demartin:2014fia). It can be seen that the three processes (ggF2j, VBF and
$VH$) show quite distinct topologies which are important for training the NN.
While the usage of higher-level observables (like $|\Delta\eta_{jj}|$ and
$m_{jj}$) does not provide the classifiers with new information, they can
still be useful in order to improve the training process. Such observables
should, however, only be included if they are expected or known to impact the
outcome of the classifier, as they can slow down the training otherwise. In
the present case, $|\Delta\eta_{jj}|$ and $m_{jj}$ are well-known examples for
variables allowing one to distinguish the events stemming from VBF and ggF2j
production.

Figure 4: Scores of the trained classifiers for defining a (a) ggF2j and (b)
VBF signal region for the different production channels.

The classifier creating the ggF2j-SR (VBF-SR) reaches an accuracy of about
$70\%$ ($79\%$). The classifier score, which gives an estimate of the
probability for each event to be a signal event, is calculated for ggF2j, VBF
and $VH$ Higgs production, respectively, and plotted in Fig. 4 (Fig. 4). The
ggF2j production is additionally split up into the three possible initial
states ($qq$, $gq$ and $gg$). We observe that the more quarks are in the
initial state, the more likely it is for an event to be classified as VBF.
Especially for the $qq$ initial state, this can be easily understood by
comparing the example Feynman diagrams in Fig. 1(c) and Fig. 2(a), which only
differ via the exchange of the gluon/heavy vector boson propagators creating
the Higgs. We additionally observe that interference events from ggF2j are
more likely to be identified as VBF-like events than events from the squared
terms are.

For both classifiers, the respective signal process is the dominant process
for a score of $P(\text{signal})\gtrsim 0.5$ and we therefore set
$P(\text{signal})\geq 0.5$ as a cut to define our SRs. All accepted events are
combined into a new data set and subsequently passed on to two additional
classifiers, in order to gain information about their ${\mathcal{CP}}$
structure.

###  4.2 Separating the different ggF2j contributions

Figure 5: The differential cross section as a function of $\Delta\phi_{jj}$ in
the (a) ggF2j- and (b) VBF-SR, plotted for the
$|\mathcal{M}_{\mathrm{even}}|^{2}$ (blue), interference (orange) and
$|\mathcal{M}_{\mathrm{odd}}|^{2}$ (green) contributions.

The three terms in the squared ggF2j amplitude in Eq. 3 can be probed by
exploiting rate information as well as their different kinematics. The
interference term can only be probed by ${\mathcal{CP}}$-odd observables since
its positive and negative contributions cancel out for ${\mathcal{CP}}$-even
observables. One such observable is $\Delta\phi_{jj}$, which has been used as
a ${\mathcal{CP}}$ observable in previous analyses (see e.g. ATLAS:2021pkb).
Fig. 5 shows the contributions of the squared and interference terms to the
differential cross section, plotted against $\Delta\phi_{jj}$. Since
$\Delta\phi_{jj}$ is a ${\mathcal{CP}}$-odd observable, the distribution for
the interference term is antisymmetric, while the distribution of the squared
${\mathcal{CP}}$-even and ${\mathcal{CP}}$-odd matrix elements are symmetric.
We impose a ${\mathcal{CP}}$ flip on the events444This corresponds to flipping
the sign of the particle momenta as well as in case of the interference term
the weight on an event-by-event basis. before the training to ensure the
symmetry of the distributions even for a limited number of Monte-Carlo events.
The two peaks close to $\Delta\phi_{jj}=0$ in the ggF2j-SR originate from
events with low $m_{jj}$ and vanish if a minimal $m_{jj}$ cut is imposed.

For the training of the ${\mathcal{CP}}$ classifiers, we use the full
kinematic information of the two leading jets and the reconstructed Higgs
boson, as well as $m_{jj}$, $\Delta\eta_{jj}$, and $\Delta\phi_{jj}$ as
additional high-level observables. The distributions of all variables used in
the training can be found in Appendix B where they are split up into the three
possible contributions to the ggF2j cross section. Compared to the signal-
background separation, we dropped the information about additional jets, since
these were found to carry little to no information about the ${\mathcal{CP}}$
character of the Higgs–gluon interaction. For the ${\mathcal{CP}}$
classification, we used the Gradient Boosting Classifier from the scikit-learn
[101] package. We train two classifiers independently for each of the signal
regions defined by the signal-background classifiers. Two observables are
constructed from their output in the following way:

  * •

The first classifier only separates between the squared ${\mathcal{CP}}$-even
and the squared ${\mathcal{CP}}$-odd terms. The corresponding output
observable is defined as $P(c_{g}^{2})$, which can be interpreted as a
probability that a given event originates from the
$|\mathcal{M}_{\mathrm{even}}|^{2}$ contribution.

  * •

The second classifier deals with the interference term. It is trained to
separate positive and negative interference events as well as to distinguish
these from events originating from either of the squared terms. Following
Bhardwaj:2021ujv, the observable built from the output is defined as
$P_{+}-P_{-}$ where $P_{+}$ ($P_{-}$) is the probability of an event to
correspond to positive (negative) interference. This observable is
${\mathcal{CP}}$-odd by definition.

Figure 6: Distributions of the two ${\mathcal{CP}}$ discriminants (see text
for the definition) for the classifiers yielding the strongest limits on the
coupling modifiers in the ggF2j-SR.

Fig. 6 shows the output of the two ${\mathcal{CP}}$ classifiers in the
ggF2j-SR. Each classifier has been trained 100 times (see Appendix C) and here
we show the one with the strongest constraints on the Higgs-gluon coupling
modifiers (see Section 5.1). For the $P(c_{g}^{2})$ classifier, this
corresponds to the case in which the difference between the squared terms was
learned best. The classifier learning the interference terms showed only very
slight fluctuations during the training processes. Since its output is a
${\mathcal{CP}}$-odd observable by construction, the squared ggF2j
contributions, as well as BG contributions, show the expected behaviour of
being symmetric around zero. The interference contribution is asymmetric
around zero. Its amplitude compared to the squared contributions is very small
due to its small cross section, an overall lower number of interference events
in the ggF2j-SR from the signal-background separation, and misidentified
events during the ${\mathcal{CP}}$ classification.

Figure 7: Distributions of the two ${\mathcal{CP}}$ discriminants for the
classifiers giving the strongest limits on the coupling modifiers in the VBF-
SR.

The output of the ${\mathcal{CP}}$ classifiers in the VBF-SR is plotted in
Fig. 7. Again, we show the distributions of the classifiers yielding the
strongest limits (see Section 5.2) after 100 training iterations. Here, the
visible statistical fluctuations in the $|\mathcal{M}_{\mathrm{even}}|^{2}$
and $|\mathcal{M}_{\mathrm{odd}}|^{2}$ contributions arise due to the
comparably low number of ggF2j events in this signal region. In contrast, the
asymmetry of the interference term in the $P_{+}-P_{-}$ observable is much
more visible. Even in the VBF-SR, the number of BG events (consisting of VBF
and VH events) is still lower than the number of ggF2j events, which is a
consequence of the smaller cross section.

##  5 Sensitivity to the Higgs–gluon coupling

In this section, we present expected limits on the Higgs–gluon coupling,
individually for the two signal regions as well as in a combined form. Details
on the parameter fit can be found in Appendix D. Furthermore, we evaluate the
impact of individual observables on the expected limits.

###  5.1 ggF2j signal region

Figure 8: Limits from (a) $\Delta\phi_{jj}$, (b) the classifier trained to
distinguish $\big{|}\mathcal{M}_{\text{even}}\big{|}^{2}$ vs.
$\big{|}\mathcal{M}_{\text{odd}}\big{|}^{2}$, (c) the classifier trained to
distinguish positive vs. negative interference, and (d) the combined limits
from both classifiers. All limits are shown for the ggF2j-SR. The 1-, 2- and
3-$\sigma$ contours are marked by white, grey and black dashed lines,
respectively. The SM is marked by an orange cross and the best-fit (BF) point
by a red star.

We first present our results in the ggF2j-SR, where the contributions from
background processes are strongly suppressed by the signal-background
classifier. Expected limits are derived under the assumption that the data is
SM-like for a luminosity of $139\,\,\mathrm{fb}^{-1}$. This fit is based on
the binned distribution of the various observables. The results are then
plotted in the $(c_{g},\tilde{c}_{g})$ parameter plane.

First, we show the limits obtained from the $\Delta\phi_{jj}$ distribution
(see Fig. 5) in Fig. 8. It can be seen that the allowed parameter space is
constrained to the form of an ellipse for which the ggF2j total rate is close
to its SM value. The $\Delta\phi_{jj}$ observable alone is not able to exclude
any region of this ellipse within the $1\sigma$ region.

The limits from the two classifiers are obtained from Fig. 6 and Fig. 6,
respectively. The $P(c_{g}^{2})$ observable provides by far the strongest
constraints among the two observables constructed from the classifiers (see
Fig. 8). Here, the ellipse is split up within the $3\,\sigma$ region, which
constrains the ${\mathcal{CP}}$-odd Higgs–gluon coupling $\tilde{c}_{g}$ to
the interval $[-0.35,0.35]$ at the $1\,\sigma$ level. While the $P_{+}-P_{-}$
observable based on the interference contribution leads to weaker constraints
(see Fig. 8), it still outperforms the $\Delta\phi_{jj}$ variable. Computing
the limits from a two-dimensional histogram with both
${\mathcal{CP}}$-observables shows a small improvement over the $P(c_{g}^{2})$
limit (see Fig. 8), limiting $\tilde{c}_{g}\in[-0.32,0.32]$ at the $1\,\sigma$
level.

###  5.2 VBF signal region

Figure 9: Limits from (a) $\Delta\phi_{jj}$, (b) the classifier trained to
distinguish $\big{|}\mathcal{M}_{\text{even}}\big{|}^{2}$ vs.
$\big{|}\mathcal{M}_{\text{odd}}\big{|}^{2}$, (c) the classifier trained to
distinguish positive vs. negative interference and (d) the combined limits
from both classifiers. All limits are shown for the VBF-SR. The 1-, 2- and
3-$\sigma$ contours are marked by white, grey and black dashed lines,
respectively. The SM is marked by an orange cross and the BF point by a red
star.

Since the VBF-SR contains significantly more VBF events than the ggF2j-SR, the
difference in the rate between the BSM and SM scenarios, which both contain
the same VBF and $VH$ events, is reduced compared to the ggF2j-SR. This is
reflected in Fig. 9 where the ellipse is now wider.

Just like in the ggF2j-SR, the $\Delta\phi_{jj}$ variable in Fig. 9 is not
able to exclude part of the ellipse. In contrast to the previous results, the
$P(c_{g}^{2})$ observable gives very similar limits to $\Delta\phi_{jj}$ (see
Fig. 9). The interference classifier (see Fig. 9) again results in worse
limits than the $P(c_{g}^{2})$ observable, but now even worse limits than
$\Delta\phi_{jj}$. This can be understood by taking into account which
information each of the observables is provided with. While the classifiers
directly use $\Delta\phi_{jj}$ during the training process, the classifier
training $\big{|}\mathcal{M}_{\text{even}}\big{|}^{2}$ vs.
$\big{|}\mathcal{M}_{\text{odd}}\big{|}^{2}$ is missing the information about
the interference contribution. This effect is negligible in the ggF2j-SR due
to the small amplitude of the interference term, but it is non-negligible here
(see Fig. 6 and Fig. 7). On the other hand, the interference classifier does
not differentiate between the $\big{|}\mathcal{M}_{\text{even}}\big{|}^{2}$
and $\big{|}\mathcal{M}_{\text{odd}}\big{|}^{2}$ contributions. Finally, the
combination of both classifiers (see Fig. 9), which uses the full information
about the contributing terms to $\sigma_{\mathrm{ggF2j}}$, puts stronger
constraints on the Higgs–gluon coupling modifiers than $\Delta\phi_{jj}$. In
contrast to the ggF2j-SR, the improvement over $\Delta\phi_{jj}$ is, however,
not so significant. This highlights the high sensitivity of the
$\Delta\phi_{jj}$ variable in the VBF-SR (see also the discussion in Section
5.4). We obtain $\tilde{c}_{g}\in[-0.58,0.58]$ for the best classifier in this
signal region.

###  5.3 Combination

Figure 10: Comparison of the limits on the coupling modifiers in the ggF2j-,
VBF- and combined SRs for the $P(c_{g}^{2})$ (a) and $P_{+}-P_{-}$ (b)
observables, their combination (c) and projections to higher luminosities (d).
The $1\sigma$ regions are shown as dashed lines, while the dotted lines
correspond to the $2\sigma$ confidence levels. The SM is marked by an orange
cross.

We now evaluate how the constraints on the Higgs–gluon coupling modifiers
change when we combine the ggF2j-SR and the VBF-SR. The limits obtained from
the $P(c_{g}^{2})$ and $P_{+}-P_{-}$ classifiers can be found in Fig. 10 and
Fig. 10, respectively. The limits combining the $P(c_{g}^{2})$ and
$P_{+}-P_{-}$ classifiers are depicted in Fig. 10. The $1\,\sigma$
($2\,\sigma$) limit is shown as a dashed (dotted) line for the ggF2j-, the
VBF- and the combined SRs. As the plots display, the combination of both SRs
leads to an advantage over the single ggF2j-SR only for the interference
classifier. In contrast to this, the limits from the $P(c_{g}^{2})$ observable
in the combined region are slightly weaker than in the ggF2j-SR alone. These
opposite effects of the combined SR on the two classifiers originate from a
different fraction of the ggF2j, VBF and $VH$ events when adding the events
from the VBF-SR to the ggF2j-SR: The relative number of both VBF events and
ggF2j interference events is increased, as the latter make up the majority of
misidentified ggF2j events in the VBF-SR. The limits obtained from using both
classifier observables are not affected when combining both signal regions,
see Fig. 10.

The expected limits at different benchmark luminosities corresponding to Run-2
and Run-3 of the LHC and further the HL-LHC are depicted in Fig. 10. We note
that the improvement of the limits with increased luminosity is slightly worse
than naively expected ($\propto 1/\sqrt{\mathcal{L}}$). This is a consequence
of the suppression of the $|\mathcal{M}_{\text{odd}}|^{2}$ contribution close
to the SM point as well as the small impact of the interference term. We
expect the limits at the $2\sigma$ level to improve from
$\tilde{c}_{g}\in[-0.44,0.44]$ at $\mathcal{L}=139\,\,\mathrm{fb}^{-1}$ to
$\tilde{c}_{g}\in[-0.35,0.35]$ at $\mathcal{L}=300\,\,\mathrm{fb}^{-1}$ and
$\tilde{c}_{g}\in[-0.2,0.2]$ at $\mathcal{L}=3000\,\,\mathrm{fb}^{-1}$,
respectively.

In comparison to the expected limits from existing experimental analyses [29,
37, 30], our combined limit is significantly stronger. In particular,
ATLAS:2021pkb,CMS:2021nnc,CMS:2022mlq are not sensitive to the
${\mathcal{CP}}$ structure of the Higgs–gluon interaction at the $2\,\sigma$
level. While these analyses focus on other Higgs decay channels, there are
additional important differences: In CMS:2021nnc,CMS:2022mlq, the analysis is
restricted to a VBF-like signal region. In ATLAS:2021pkb, the limit is based
solely on the $\Delta\phi_{jj}$ distribution. As we have shown, these
restrictions significantly lower the expected sensitivity. On the other hand,
the limits derived here are also expected to degrade once systematic
uncertainties are considered.

###  5.4 Highest-impact observables

Here, we evaluate the impact of the various observables used by the
classifiers on the output of the ${\mathcal{CP}}$ classifier. We evaluate this
impact with the SHAP package [103], which explains the output in terms of
Shapley values from cooperative game theory [104, 105]: Each classifier has a
fixed ”worth” determined from its separation power when all variables are used
in the training. The Shapley value of one variable $x_{i}$ is then determined
by taking the sum over all possible subsets of the parameter space not
containing $x_{i}$ and evaluating the loss in worth compared to the full case.
Taking every possible subspace into account guarantees that possible
correlations between variables do not falsify the result. For more details, we
refer to JMLR:v11:strumbelj10a. In the following, the Shapley values are
referred to as SHAP values to meet the nomenclature of the SHAP package.

The SHAP plots are structured as follows:

  * •

Individual variables are plotted against their SHAP value. A point at a high
absolute SHAP value means that the variable had a large impact on deciding
what label the event has been given by the classifier. The sign of the SHAP
value determines which label was chosen, i.e., for the
$|\mathcal{M}_{\text{even}}|^{2}$ vs. $|\mathcal{M}_{\text{odd}}|^{2}$
classifier, a positive sign signals that $|\mathcal{M}_{\text{even}}|^{2}$ is
more likely to be chosen.

  * •

The color of the points represents the value of the variable itself. Red
points stand for high values of the variable, while blue ones stand for low
ones.

  * •

The variables are ranked by the mean of the absolute SHAP value. Therefore,
variables which have many events (displayed as bulks) deviating from zero are
identified as the most important ones. Single points of variables that reach
high absolute SHAP values are understood as outliers and only have a minor
impact.

Figure 11: SHAP values for the variables used in classifiers separating the
$|\mathcal{M}_{\mathrm{even}}|^{2}$ and $|\mathcal{M}_{\mathrm{odd}}|^{2}$
terms. The classifiers are trained in (a) the ggF2j-SR and (b) the VBF-SR.
High absolute SHAP values indicate a strong influence on the classifier score.
The color of individual points marks the value of the specific variable. The
variables are ordered by taking the mean of the absolute SHAP value. All
variables used in the training which are not shown in the plot are summed up
and marked as “Others”.

We first focus on the classifier separating the squared ggF2j terms in the
respective signal regions in Fig. 11. In the ggF2j-SR (see Fig. 11) the most
important variables are $p_{T}$ and $E$ of the leading-$p_{T}$ jets, as well
as their invariant mass, while the observables associated with the Higgs boson
play a subordinate role. This picture is flipped in the VBF-SR (see Fig. 11)
where now $p_{T}$ and $E$ of the Higgs boson are the most important variables,
followed by the jet kinematics. Note that the overall absolute SHAP values are
relatively low compared to their spread indicating that no single observable
drives the outcome of the classifiers. Instead, the full kinematic information
is needed to separate between the squared ggF2j terms.

Figure 12: Same as Fig. 11, but the SHAP values for the classifiers separating
the positive and negative interference terms are shown.

In the case of the interference classifiers (see Fig. 12), $\Delta\phi_{jj}$
is by far the most important variable for both signal regions. As a
${\mathcal{CP}}$-odd observable by construction, $\Delta\phi_{jj}$ is expected
to give a good separation between the positive and negative parts of the
interference. In the ggF2j-SR (Fig. 12), other important variables are the
$p_{T}$ of the Higgs boson, as well as the pseudorapidity of the jets — in the
form of the combination $\Delta\eta_{jj}$ as well as $\eta^{j_{1}}$ and
$\eta^{j_{2}}$ alone. For the VBF-SR (Fig. 12), it should be noted that the
mean of the $\Delta\phi_{jj}$ distribution peaks at much higher SHAP values
than in the ggF2j-SR. This is in agreement with the limits obtained in Section
5.2, where the limits from the classifier using the full event kinematics
showed only a slight improvement over the limits from $\Delta\phi_{jj}$ alone.
The next two most sensitive variables are $p_{T}^{j_{1}}$ and $p_{T}^{j_{2}}$.
Our findings agree with Butter:2021rvz, in which ${\mathcal{CP}}$-violating
effects in the Higgs coupling to $W$ bosons were studied with a different
machine learning approach, identifying $\Delta\phi_{jj}$, $p_{T}^{j_{1}}$ and
$p_{T}^{j_{2}}$ as the three most sensitive observables for this coupling when
using the product of all three variables.

##  6 Disentangling ${\mathcal{CP}}$ violation in the $ggH$ and $HVV$
couplings

So far, we have concentrated on ${\mathcal{CP}}$ violation in the Higgs–gluon
interaction. However, BSM physics may affect multiple Higgs couplings at once.
As discussed above, VBF production is the main background for investigations
of ggF2j production. While, as mentioned in Section 1, a lot of effort has
been put forward already to investigate the ${\mathcal{CP}}$ structure of the
Higgs couplings to massive vector bosons, we investigate in this section how
the presence of ${\mathcal{CP}}$ violation in the $HVV$ couplings influence
our limits on the Higgs–gluon coupling in the two signal regions.

For this, we generate an additional data set for VBF production in the context
of the SMEFT, where a non-zero value of the $c_{\Phi\tilde{W}}$ Wilson
coefficient introduces ${\mathcal{CP}}$ violation in the Higgs coupling to $W$
bosons (for details see Appendix A). This data set is added to the SM-like VBF
data set which has been used for our analysis so far. To quantify the effects
on our Higgs–gluon coupling limits, we do not re-train our classifiers, but
directly build new observables based on the adjusted events.

Figure 13: Same as Fig. 8, but the limits are shown for the case that
${\mathcal{CP}}$ violation is present in VBF production.

Fig. 13 depicts the limits obtained in the ggF2j-SR when using the VBF data
set with ${\mathcal{CP}}$ violation. Compared to the limits in Fig. 8 in the
same signal region, we obtain slightly weaker limits from the classifier
trained on the $|\mathcal{M}_{\mathrm{even}}|^{2}$ and
$|\mathcal{M}_{\mathrm{odd}}|^{2}$ terms. Now the $1\,\sigma$ contour reaches
$|\tilde{c}_{g}|=0.4$ instead of $|\tilde{c}_{g}|=0.35$, as visible in Fig.
13. In contrast, we obtain slightly tighter constraints from the interference
classifier in comparison to the results without ${\mathcal{CP}}$ violation in
the $HVV$ interaction (Fig. 13). This change is not significant enough to
exclude smaller values of $|\tilde{c}_{g}|$ or affect the combined limits of
the two classifiers (see Fig. 13 and Fig. 8) since these limits are mostly
dominated by $P(c_{g}^{2})$. Accordingly, they get slightly weaker as well
compared to previous results, excluding $|\tilde{c}_{g}|>0.37$ compared to
$|\tilde{c}_{g}|>0.32$ obtained in Section 5.1. In general, both the
comparably low cross section of VBF compared to ggF2j, and the small number of
VBF events surviving the ggF2j signal selection suppress any large changes in
the limits. This makes the ggF2j-SR robust against ${\mathcal{CP}}$ violation
in the $HVV$ coupling.

Figure 14: Same as Fig. 9, but the limits are shown for the case that
${\mathcal{CP}}$ violation is present in VBF production.

The limits based on the VBF-SR with ${\mathcal{CP}}$ violation in the VBF data
are plotted in Fig. 14. The limits obtained individually from the
$P(c_{g}^{2})$ (Fig. 14) or $P_{+}-P_{-}$ (Fig. 14) observables do not differ
significantly compared to the case of SM-like VBF production. We do, however,
observe stronger limits for the combination of both classifiers when compared
to the previous results in Section 5.2 (see Fig. 9). Here, the $1\sigma$
limits from the combined classifier tighten noticeably from
$\tilde{c}_{g}\in[-0.58,0.58]$ to $\tilde{c}_{g}\in[-0.48,0.48]$.

The opposite effects on the limits in the two signal regions most likely stem
from a “washout” of events. Since the classifiers were never trained to
identify ${\mathcal{CP}}$-odd VBF events, a higher (lower) total number of VBF
events are now identified as ggF2j-like (VBF-like) events. Therefore, this
yields more (less) background events in the ggF2j-SR (VBF-SR), leading to the
observed weakened (tightened) limits. We emphasize that consequently the
interpretation of these results should be done carefully, as the “better“
constraints in the VBF-SR on $c_{g}$ and $\tilde{c}_{g}$ are faked from
${\mathcal{CP}}$ violation in the $HVV$ couplings.

Overall we conclude that additional ${\mathcal{CP}}$ violation in the $HVV$
coupling can have different effects depending on the signal region. As
expected, all effects are suppressed in the ggF2j-SR. However, a slight trend
towards weaker limits stemming from the $HVV$ ${\mathcal{CP}}$ violation can
be identified. On the other hand, this additional ${\mathcal{CP}}$ violation
can lead to seemingly stronger limits in the VBF-SR. Together with the overall
better limits in the ggF2j-SR, our findings consolidate the need for a
dedicated signal region with ggF2j-like kinematics.

##  7 Limits on the ${\mathcal{CP}}$ structure of the top-Yukawa coupling

As discussed in Section 2.2, we normalized the coefficients of the effective
Higgs–gluon interaction (see Eq. 1) such that they directly correspond to the
modifiers of the top-Yukawa coupling (see Eq. 2) if no low-mass coloured BSM
particles are present and if we neglect the contributions of the lighter
quarks. Since experimental limits on coloured BSM particles are becoming
increasingly strong [80, 81, 82, 83, 84, 85], these particles can also only
induce a small ${\mathcal{CP}}$-odd Higgs–gluon coupling. Taking as an example
a coloured fermion with a mass of $1\,\,\mathrm{TeV}$ and a Yukawa-type
coupling of ${\mathcal{O}}(1)$ to the Higgs boson, we expect this fermion to
contribute to the Higgs–gluon couplings at the
${\mathcal{O}}(v^{2}/\Lambda^{2})\sim 0.1$.

Assuming that the contribution of coloured BSM particles is negligible, we can
reinterpret our projected limits on the Higgs–gluon coupling as projected
limits on the top-Yukawa coupling (by simply replacing $c_{g}$ by $c_{t}$ and
$\tilde{c}_{g}$ by $\tilde{c}_{t}$). Our findings as well as a comparison to
other approaches are summarized in Table 2 and described in detail in the
following.

Our projected limits on the ${\mathcal{CP}}$ mixing angle
$\alpha^{\mathrm{Htt}}=\mathrm{tan}^{-1}(\tilde{c}_{t}/c_{t})$ for a
luminosity of $139\,\,\mathrm{fb}^{-1}$ result in

$\alpha_{\mathrm{ggF2j}}^{\mathrm{Htt}}\in[-15^{\circ},15^{\circ}],\;\;\alpha_{\mathrm{VBF}}^{\mathrm{Htt}}\in[-25^{\circ},25^{\circ}]\;\;@~{}68\%~{}\mathrm{CL}$
(5)

for the respective signal regions. Our result for the ggF2j-SR is stronger
than existing experimental limits on $\alpha^{\mathrm{Htt}}$ based on ggF2j
production — $\alpha^{\mathrm{Htt}}\in[-26^{\circ},26^{\circ}]$, see
ATLAS:2021pkb,CMS:2022mlq — while our result in the VBF-SR is very similar.
We, however, want to stress several differences between our approach and the
one in CMS:2022mlq that make a direct comparison of the results difficult. In
contrast to our study, the authors worked in the $H\to\tau\tau$ decay channel
and took systematic uncertainties as well as bottom-quark contributions to the
Higgs–gluon loop into account (see also discussion at the end of Section 5.3).

In addition to limits on $\alpha^{\mathrm{Htt}}$ based on ggF2j production,
our projected bound is also tighter than existing experimental limits focusing
on top-associated Higgs production
($\alpha^{\mathrm{Htt}}\in[-35^{\circ},35^{\circ}]$ at 68% $\mathrm{CL}$) [35,
36, 37, 38, 30, 39].

Moreover, our projected limits are stronger than the results of recent global
fits [66, 20]. Besides many other total rate and simplified template cross-
section measurements, these fits take into account total rate information from
Higgs production via gluon fusion and the Higgs decay to two photons, which
are very sensitive, but model-dependent constraints.

process/ SR | our work | literature
---|---|---
68% CL | 95% CL | 68% CL | 95% CL | Ref.
$gg\to Hjj$: ggF2j-SR | $15^{\circ}$ | $25^{\circ}$ | – | – | –
$gg\to Hjj$: VBF-SR | $25^{\circ}$ | – | $26^{\circ}$ | – | [30]
$t\bar{t}H$ | – | – | $35^{\circ}$ | $43^{\circ}$ | [36]
global fit | $21^{\circ}$ | $28^{\circ}$ | see text and Bahl:2020wee,Bahl:2022yrs
Table 2: Upper limits on $|\alpha^{\mathrm{Htt}}|$ at the 68% and 95% CL
obtained from the different signal regions (SR) from our work compared to the
literature. See text regarding caveats of the comparison of these limits.
Figure 15: Comparison of the projected combined ggF2j limits (blue) to the
current global limits from LHC data (green). Shown are the $1\,\sigma$
(dashed) and $2\,\sigma$ regions (dotted) in both cases. The SM is marked by
an orange cross and the BF point of the LHC data by a red star.

To provide an up-to-date comparison, we have updated the fits performed in
Bahl:2020wee,Bahl:2022yrs using the new HiggsSignals which is part of
HiggsTools and contains additional recent experimental results [108, 109,
110].555We are using version 1.1 of the HiggsSignals data set. The result is
shown in the $(c_{t},\tilde{c}_{t})$ parameter plane in Fig. 15. The blue
contours depict the limits from our ggF2j analysis, and the green ones are
based on the global fit to existing experimental results. As discussed in
detail in Bahl:2020wee, the fit constraints are dominated by total rate
measurements of Higgs production via gluon fusion and the Higgs decay to two
photons.666The observed shift of the $1\,\sigma$ and $2\,\sigma$ regions to
lower values of $c_{t}$ is caused by recent experimental measurements of top-
associated Higgs production and Higgs production via gluon fusion with a lower
rate than expected in the SM [33, 111]. By exploiting the kinematic
information, our projected limits are less model-dependent and (slightly)
stronger.

##  8 Conclusions

The ${\mathcal{CP}}$ nature of the Higgs boson is among the few Higgs
properties which are still only loosely constrained. One coupling, which is
especially important for LHC physics, is the Higgs–gluon interaction —
facilitated at the loop level. The ${\mathcal{CP}}$ structure of this
interaction can be probed via Higgs production in association with two jets
(ggF2j).

In this work, we studied how the information about the ${\mathcal{CP}}$
structure of the Higgs–gluon interaction can be best extracted. Focusing on
the Higgs decay to two photons, we first trained a classifier to define a
ggF2j-enriched signal region differentiating it from VBF and $VH$ production
as the relevant backgrounds. In order to fully exploit the quark-initiated
ggF2j channel, which is kinematically very similar to VBF production, we also
defined a VBF-enriched signal region.

For both SRs, we trained two classifiers to separate the different
contributions to the squared ggF2j amplitude: the square of the
${\mathcal{CP}}$-even amplitude, the square of the ${\mathcal{CP}}$-odd
amplitude, and the interference contribution. Out of these classifiers, we
constructed one ${\mathcal{CP}}$-even and one ${\mathcal{CP}}$-odd observable.
Based on the distributions for these observables, we derived expected upper
limits. These constrain the ${\mathcal{CP}}$-odd Higgs–gluon coupling modifier
to $|\tilde{c}_{g}|\leq\\{0.35,0.28,0.15\\}$ at the $2\,\sigma$ level for
integrated luminosities of
$\\{139\,\,\mathrm{fb}^{-1},300\,\,\mathrm{fb}^{-1},3000\,\,\mathrm{fb}^{-1}\\}$,
respectively. We found these to significantly outperform limits based only on
the difference in the azimuthal angle of the two leading jets, which is an
observable commonly employed in the literature. Moreover, the ggF2j-SR has a
significantly higher sensitivity than the VBF-SR, which is expected by the
higher number of ggF2j events. These results suggest that significant
improvements in current experimental limits are possible with well-established
techniques.

We note that our limits could be further improved by including other Higgs
decay channels and removing the cut on the Higgs transverse momentum.
Moreover, advanced analysis techniques like e.g. the matrix-element approach
[112, 113, 114, 115, 116, 117, 118, 119, 54, 120] or machine-learning-based
inference [121, 122, 123, 124, 125] could be used. We leave such improvements
for future work.

In addition, we used interpretable machine learning (in the form of SHAP
values) to investigate which observables have the highest impact on the
classifications. While the azimuthal angle of the two leading jets is by far
the most important observable for separating the interference term from the
squared terms, also other momenta like the transverse momenta of the jets and
the Higgs boson play a sizeable role. For separating the squared
${\mathcal{CP}}$-even and ${\mathcal{CP}}$-odd amplitudes, the transverse
momenta are most decisive.

We also investigated the model dependence of our limits. In particular, we
checked whether ${\mathcal{CP}}$-violating couplings in the VBF channel could
mimic a ${\mathcal{CP}}$-violating Higgs–gluon interaction in our analysis.
Our results show that our analysis — in particular the ggF2j-SR — is very
robust to such a situation and therefore allows to disentangle
${\mathcal{CP}}$-violating Higgs couplings to massive vector bosons and
gluons.

Finally, we reinterpreted our expected limits in terms of limits on the
${\mathcal{CP}}$ character of the top-Yukawa coupling. This reinterpretation
relies on the assumption that no coloured BSM particles contribute sizeably to
Higgs production to gluon fusion, which is motivated by the current limits
from direct searches for such particles. The projected limits on the
${\mathcal{CP}}$ structure of the top-Yukawa coupling from our
reinterpretation are stronger than limits based on existing dedicated analyses
of the top-Yukawa coupling based either on investigations of top-associated
Higgs production or on global fits to Higgs precision measurements using the
Run-2 data of the LHC. Both for the current data and for future runs of the
LHC, this demonstrates the large potential of ggF2j production as a precision
probe of the ${\mathcal{CP}}$ nature of the Higgs boson.

## Acknowledgements

We thank Alberto Carnelli, Frederic Deliot, Christian Grefe, Andrei Gritsan,
Lena Herrmann, Anastasia Kotsokechagia, Andrew Pilkington, Tilman Plehn,
Matthias Saimpert and Laurent Schoeffel for useful discussions. HB
acknowledges support from the Alexander von Humboldt Foundation. EF and MM
acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy –– EXC-2123
“QuantumFrontiers” — 390837967.

##  Appendix A Details of the event generation

The model file used for the event generation is a custom UFO model file [96,
97] which only contains the terms in Eq. 1, as well as an effective Higgs-
photon coupling. The $H\gamma\gamma$ coupling is assumed to be SM-like
throughout this study. It is implemented as an effective coupling to enable
the generation of the ggF2j process at the tree level. With G $\equiv g$, GA
$\equiv\gamma$, H $\equiv$ scalar and A $\equiv$ pseudoscalar, there are four
effective couplings: QGGH, QGGA, QGAGAH and QGAGAA that are used to generate
specific parts of the amplitude. Setting QGGH==1, QGAGAH==1 (QGGA==1,
QGAGAA==1) implies the effective interaction of a scalar (pseudoscalar) Higgs
boson with exactly one pair of gluons and one pair of photons. The parameters
can be set during the event generation, as seen in the following. All events
are generated at leading order and are scaled to NLO by flat K-factors from
Demartin:2014fia.

The syntax for the event generation is “generate p p > a a j j QGGH==1
QGAGAH==1 QED=4 QCD=4” for the $\big{|}\mathcal{M}_{\text{even}}\big{|}^{2}$
term in ggF2j, where the restrictions on the QED and QCD magnitude are set to
exclude di-Higgs production. The interference and
$\big{|}\mathcal{M}_{\text{odd}}\big{|}^{2}$ terms are created by replacing
“QGGH==1” by “QGGH^2==1 QGGA^2==1” and “QGGA==1”, respectively. For the
background processes, the effective Higgs-gluon coupling has been disabled in
the model file. The VBF background is generated via “generate p p > a a j j $$
w+ w- z QGAGAH==1” where heavy vector bosons are forbidden to appear in the
s-channel to exclude contributions from the $VH$ background. The latter is in
turn generated via “generate p p > z > a a j j QGAGAH==1; add process p p > w+
> a a j j QGAGAH==1; add process p p > w- > a a j j QGAGAH==1”.

An additional data set introducing ${\mathcal{CP}}$ violation in the VBF
channel, used for the analysis in Section 6, is generated using the SMEFTsim
3.0 package [126]. Specifically, the U35 model is used in the $m_{W}$-scheme
and we set $c_{\Phi\tilde{W}}=1~{}\mathrm{TeV}^{-1}$ as the only non-zero BSM
parameter of the model. Events are then generated via “generate p p > a a j j
$$ w+ w- z /a QCD=0 NP=1 NP^2==1” and we apply an additional diagram filter to
ensure that only VBF events are generated.

##  Appendix B Distributions of kinematic variables

Fig. 16 shows the distributions of all variables used in the training of the
classifiers (see Section 4.1), separately for the three different
${\mathcal{CP}}$ contributions to the squared ggF2j amplitude. These include
as low-level observables the energy $E$, transverse momentum $p_{T}$ and
pseudorapidity $\eta$ of the reconstructed final state particles, as well as
some high-level observables. An exception to this is the azimuthal angle
$\phi$ of the aforementioned objects as this does not exhibit any separation
between the three contributions due to the rotational invariance of the
production process.

Figure 16: Distributions of the energy (first row), $p_{T}$ (second row) and
$\eta$ (third row) of the Higgs boson (left column), as well as the jets
leading (middle column) and sub-leading (right column) in $p_{T}$. The last
row shows distributions of high-level observables, namely the invariant mass
of the di-jet system $m_{jj}$ (left), as well as the difference in
pseudorapidity $\Delta\eta_{jj}$ (middle) and azimuthal angle
$\Delta\phi_{jj}$ (right) of the two jets. All distributions are split up into
the $\big{|}\mathcal{M}_{\text{even}}\big{|}^{2}$ (blue), interference
(orange) and $\big{|}\mathcal{M}_{\text{odd}}\big{|}^{2}$ (green)
contributions to the total ggF2j cross section.

##  Appendix C Training uncertainty

Figure 17: Distributions of the two ${\mathcal{CP}}$ discriminants for the
mean of 100 trained classifiers in the ggF2j-like kinematic region.

Figure 18: Distributions of the two ${\mathcal{CP}}$ discriminants for the
mean of 100 trained classifiers in the VBF-like kinematic region.

In Fig. 18, the classifier outputs for the two ${\mathcal{CP}}$ discriminants
in the ggF2j-like kinematic region are shown as the mean of 100 individual
training processes, with the standard deviation in each bin plotted as a
shaded region. This allows us to estimate the uncertainty due to the
classifier training. We observe that the largest uncertainties come from bins
close to 0 and 1 in the $P(c_{g}^{2})$ classifier. For the $P_{+}-P_{-}$
classifier, the uncertainty associated with the training process is
negligible. In the VBF-like region (see Fig. 18), the visible statistical
fluctuations in the $|\mathcal{M}_{\mathrm{even}}|^{2}$ and
$|\mathcal{M}_{\mathrm{odd}}|^{2}$ contributions arise due to the comparably
low number of ggF2j events in this kinematic region.

It should be noted that, in the present study, the uncertainty associated with
the training process does not result in an uncertainty on the extraction of
the Higgs–gluon couplings. Each trained classifier represents a slightly
different observable. Consequently, the best-performing classifier can be
chosen without introducing any further uncertainty in the final limits on the
Higgs–gluon coupling.

##  Appendix D Likelihood evaluation

As mentioned above, the output of the ${\mathcal{CP}}$ classifiers is used to
construct different observables. These observables are represented as
histograms filled by all events from the respective test data set. Since each
bin corresponds to a number of events, which follow a Poisson distribution,
the histograms can be used to construct a likelihood

$L_{X}=\frac{e^{-\lambda_{X}}\lambda_{X}^{n}}{n!}\;.$ (6)

Here, $\lambda_{X}$ is the expected number of events for a specific
${\mathcal{CP}}$ hypothesis, while $n$ corresponds to the observed number of
events. We set $n=\lambda_{\mathrm{SM}}$, which corresponds to a perfect
agreement of the data with the SM hypothesis, in order to construct limits on
the Higgs–gluon coupling.

These limits are calculated using a binned likelihood ratio

$\displaystyle t$
$\displaystyle=-2~{}\mathrm{ln}\bigg{(}\frac{L_{\mathrm{tot}}}{L_{\mathrm{SM}}}\bigg{)}=-2~{}\mathrm{ln}\bigg{(}\prod_{i}\frac{e^{-\lambda_{i,\mathrm{tot}}}\lambda_{i,\mathrm{tot}}^{n_{i}}}{e^{-\lambda_{i,\mathrm{SM}}}\lambda_{i,\mathrm{SM}}^{n_{i}}}\bigg{)}$
(7)
$\displaystyle=-2~{}\sum_{i}\bigg{[}\lambda_{\mathrm{SM}}^{i}\cdot\mathrm{ln}\bigg{(}\frac{\lambda_{\mathrm{tot}}^{i}}{\lambda_{\mathrm{SM}}^{i}}\bigg{)}-\lambda_{\mathrm{tot}}^{i}+\lambda_{\mathrm{SM}}^{i}\bigg{]}\,,$

where $t$ is the test statistic and the sum runs over all bins $i$.
$\lambda_{\mathrm{SM}}^{i}=\lambda_{\mathrm{E1}}^{i}+\lambda_{\mathrm{BG}}^{i}$
is the expected SM distribution including ggF2j with $c_{g}=1,\tilde{c}_{g}=0$
and the BG processes, while
$\lambda_{\mathrm{tot}}^{i}=c_{g}^{2}\lambda_{\mathrm{E1}}^{i}+c_{g}\tilde{c}_{g}\lambda_{\mathrm{O}}^{i}+\tilde{c}_{g}^{2}\lambda_{\mathrm{E2}}^{i}+\lambda_{\mathrm{BG}}^{i}$
is the total bin value in the BSM case where $c_{g}$ and $\tilde{c}_{g}$ can
be varied. Following Wilks’ theorem [127], we assume that $t$ follows a
$\chi^{2}$-distribution and can, therefore, be used to construct confidence
intervals.

## References

  * [1] G. Aad  “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC”  In _Phys. Lett._ B716, 2012, pp. 1–29  DOI: 10.1016/j.physletb.2012.08.020
  * [2] S. Chatrchyan  “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC”  In _Phys. Lett._ B716, 2012, pp. 30–61  DOI: 10.1016/j.physletb.2012.08.021
  * [3] “A detailed map of Higgs boson interactions by the ATLAS experiment ten years after the discovery” [Erratum: Nature 612, E24 (2022)]  In _Nature_ 607.7917, 2022, pp. 52–59  DOI: 10.1038/s41586-022-04893-w
  * [4] Armen Tumasyan  “A portrait of the Higgs boson by the CMS experiment ten years after the discovery”  In _Nature_ 607.7917, 2022, pp. 60–68  DOI: 10.1038/s41586-022-04892-x
  * [5] M.. Gavela, P. Hernandez, J. Orloff and O. Pene  “Standard model CP violation and baryon asymmetry”  In _Mod. Phys. Lett. A_ 9, 1994, pp. 795–810  DOI: 10.1142/S0217732394000629
  * [6] Patrick Huet and Eric Sather  “Electroweak baryogenesis and standard model CP violation”  In _Phys. Rev. D_ 51, 1995, pp. 379–394  DOI: 10.1103/PhysRevD.51.379
  * [7] Vardan Khachatryan  “Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton collisions at 7 and 8 TeV”  In _Phys. Rev. D_ 92.1, 2015, pp. 012004  DOI: 10.1103/PhysRevD.92.012004
  * [8] Georges Aad  “Study of the spin and parity of the Higgs boson in diboson decays with the ATLAS detector” [Erratum: Eur.Phys.J.C 76, 152 (2016)]  In _Eur. Phys. J. C_ 75.10, 2015, pp. 476  DOI: 10.1140/epjc/s10052-015-3685-1
  * [9] Maxim Pospelov and Adam Ritz  “Electric dipole moments as probes of new physics”  In _Annals Phys._ 318, 2005, pp. 119–169  DOI: 10.1016/j.aop.2005.04.002
  * [10] Timothy Chupp, Peter Fierlinger, Michael Ramsey-Musolf and Jaideep Singh  “Electric dipole moments of atoms, molecules, nuclei, and particles”  In _Rev. Mod. Phys._ 91.1, 2019, pp. 015001  DOI: 10.1103/RevModPhys.91.015001
  * [11] V. Andreev  “Improved limit on the electric dipole moment of the electron”  In _Nature_ 562.7727, 2018, pp. 355–360  DOI: 10.1038/s41586-018-0599-8
  * [12] Tanya S. Roussy  “An improved bound on the electron’s electric dipole moment”  In _Science_ 381.6653, 2023, pp. adg4084  DOI: 10.1126/science.adg4084
  * [13] Guillaume Pignol and Philipp Schmidt-Wellenburg  “The search for the neutron electric dipole moment at PSI”  In _SciPost Phys. Proc._ 5, 2021, pp. 027  DOI: 10.21468/SciPostPhys\nProc.5.027
  * [14] B. Graner, Y. Chen, E.. Lindahl and B.. Heckel  “Reduced Limit on the Permanent Electric Dipole Moment of ${}^{199}\mathrm{Hg}$”  In _Phys. Rev. Lett._ 116 American Physical Society, 2016, pp. 161601  DOI: 10.1103/PhysRevLett.116.161601
  * [15] Joachim Brod, Ulrich Haisch and Jure Zupan  “Constraints on CP-violating Higgs couplings to the third generation”  In _JHEP_ 11, 2013, pp. 180  DOI: 10.1007/JHEP11(2013)180
  * [16] Joachim Brod and Emmanuel Stamou  “Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order”  In _JHEP_ 07, 2021, pp. 080  DOI: 10.1007/JHEP07(2021)080
  * [17] Elina Fuchs, Marta Losada, Yosef Nir and Yehonatan Viernik  “$CP$ violation from $\tau$, $t$ and $b$ dimension-6 Yukawa couplings - interplay of baryogenesis, EDM and Higgs physics”  In _JHEP_ 05, 2020, pp. 056  DOI: 10.1007/JHEP05(2020)056
  * [18] Joachim Brod, Jonathan M. Cornell, Dimitrios Skodras and Emmanuel Stamou  “Global constraints on Yukawa operators in the standard model effective theory”  In _JHEP_ 08, 2022, pp. 294  DOI: 10.1007/JHEP08(2022)294
  * [19] Joachim Brod, Zachary Polonsky and Emmanuel Stamou  “A Precise Electron EDM Constraint on CP-odd Heavy-Quark Yukawas”, 2023  arXiv:2306.12478 [hep-ph]
  * [20] Henning Bahl et al.  “Constraining the ${\mathcal{C}}{\mathcal{P}}$ structure of Higgs-fermion couplings with a global LHC fit, the electron EDM and baryogenesis”  In _Eur. Phys. J. C_ 82.7, 2022, pp. 604  DOI: 10.1140/epjc/s10052-022-10528-1
  * [21] Georges Aad  “Search for the Higgs boson decays $H\to ee$ and $H\to e\mu$ in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector”  In _Phys. Lett. B_ 801, 2020, pp. 135148  DOI: 10.1016/j.physletb.2019.135148
  * [22] Yotam Soreq, Hua Xing Zhu and Jure Zupan  “Light quark Yukawa couplings from Higgs kinematics”  In _JHEP_ 12, 2016, pp. 045  DOI: 10.1007/JHEP12(2016)045
  * [23] Johann Brehmer, Felix Kling, Tilman Plehn and Tim M.. Tait  “Better Higgs-CP Tests Through Information Geometry”  In _Phys. Rev. D_ 97.9, 2018, pp. 095017  DOI: 10.1103/PhysRevD.97.095017
  * [24] Georges Aad  “Test of CP Invariance in vector-boson fusion production of the Higgs boson using the Optimal Observable method in the ditau decay channel with the ATLAS detector”  In _Eur. Phys. J. C_ 76.12, 2016, pp. 658  DOI: 10.1140/epjc/s10052-016-4499-5
  * [25] Albert M Sirunyan  “Constraints on anomalous Higgs boson couplings using production and decay information in the four-lepton final state”  In _Phys. Lett. B_ 775, 2017, pp. 1–24  DOI: 10.1016/j.physletb.2017.10.021
  * [26] Albert M Sirunyan  “Constraints on anomalous $HVV$ couplings from the production of Higgs bosons decaying to $\tau$ lepton pairs”  In _Phys. Rev. D_ 100.11, 2019, pp. 112002  DOI: 10.1103/PhysRevD.100.112002
  * [27] Albert M Sirunyan  “Measurements of the Higgs boson width and anomalous $HVV$ couplings from on-shell and off-shell production in the four-lepton final state”  In _Phys. Rev. D_ 99.11, 2019, pp. 112003  DOI: 10.1103/PhysRevD.99.112003
  * [28] Georges Aad  “Test of CP invariance in vector-boson fusion production of the Higgs boson in the H→$\tau$$\tau$ channel in proton–proton collisions at s=13TeV with the ATLAS detector”  In _Phys. Lett. B_ 805, 2020, pp. 135426  DOI: 10.1016/j.physletb.2020.135426
  * [29] Georges Aad  “Constraints on Higgs boson properties using $WW^{*}(\rightarrow e\nu\mu\nu)jj$ production in $36.1\,\mathrm{fb}^{-1}$ of $\sqrt{s}=13$ TeV pp collisions with the ATLAS detector”  In _Eur. Phys. J. C_ 82.7, 2022, pp. 622  DOI: 10.1140/epjc/s10052-022-10366-1
  * [30] Armen Tumasyan  “Constraints on anomalous Higgs boson couplings to vector bosons and fermions from the production of Higgs bosons using the $\tau$$\tau$ final state”  In _Phys. Rev. D_ 108.3, 2023, pp. 032013  DOI: 10.1103/PhysRevD.108.032013
  * [31] G. Aad  “Test of CP Invariance in Higgs Boson Vector-Boson-Fusion Production Using the H→$\gamma$$\gamma$ Channel with the ATLAS Detector”  In _Phys. Rev. Lett._ 131.6, 2023, pp. 061802  DOI: 10.1103/PhysRevLett.131.061802
  * [32] Georges Aad  “Test of CP-invariance of the Higgs boson in vector-boson fusion production and its decay into four leptons”, 2023  arXiv:2304.09612 [hep-ex]
  * [33] Armen Tumasyan  “Analysis of the $CP$ structure of the Yukawa coupling between the Higgs boson and $\tau$ leptons in proton-proton collisions at $\sqrt{s}$ = 13 TeV”  In _JHEP_ 06, 2022, pp. 012  DOI: 10.1007/JHEP06(2022)012
  * [34] Georges Aad  “Measurement of the CP properties of Higgs boson interactions with $\tau$-leptons with the ATLAS detector”  In _Eur. Phys. J. C_ 83.7, 2023, pp. 563  DOI: 10.1140/epjc/s10052-023-11583-y
  * [35] Albert M Sirunyan  “Measurements of $\mathrm{t\bar{t}}H$ Production and the CP Structure of the Yukawa Interaction between the Higgs Boson and Top Quark in the Diphoton Decay Channel”  In _Phys. Rev. Lett._ 125.6, 2020, pp. 061801  DOI: 10.1103/PhysRevLett.125.061801
  * [36] Georges Aad  “$CP$ Properties of Higgs Boson Interactions with Top Quarks in the $t\bar{t}H$ and $tH$ Processes Using $H\rightarrow\gamma\gamma$ with the ATLAS Detector”  In _Phys. Rev. Lett._ 125.6, 2020, pp. 061802  DOI: 10.1103/PhysRevLett.125.061802
  * [37] Albert M Sirunyan  “Constraints on anomalous Higgs boson couplings to vector bosons and fermions in its production and decay using the four-lepton final state”  In _Phys. Rev. D_ 104.5, 2021, pp. 052004  DOI: 10.1103/PhysRevD.104.052004
  * [38] Armen Tumasyan  “Search for $CP$ violation in ttH and tH production in multilepton channels in proton-proton collisions at $\sqrt{s}$ = 13 TeV”  In _JHEP_ 07, 2023, pp. 092  DOI: 10.1007/JHEP07(2023)092
  * [39] ATLAS Collaboration  “Probing the $CP$ nature of the top-Higgs Yukawa coupling in $t\bar{t}H$ and $tH$ events with $H\to b\bar{b}$ decays using the ATLAS detector at the LHC”, 2023  arXiv:2303.05974 [hep-ex]
  * [40] Georges Aad  “Observation of four-top-quark production in the multilepton final state with the ATLAS detector”  In _Eur. Phys. J. C_ 83.6, 2023, pp. 496  DOI: 10.1140/epjc/s10052-023-11573-0
  * [41] John F. Gunion and Xiao-Gang He  “Determining the CP nature of a neutral Higgs boson at the LHC”  In _Phys. Rev. Lett._ 76, 1996, pp. 4468–4471  DOI: 10.1103/PhysRevLett.76.4468
  * [42] Ayres Freitas and Pedro Schwaller  “Higgs CP Properties From Early LHC Data”  In _Phys. Rev. D_ 87.5, 2013, pp. 055014  DOI: 10.1103/PhysRevD.87.055014
  * [43] Pankaj Agrawal, Subhadip Mitra and Ambresh Shivaji  “Effect of Anomalous Couplings on the Associated Production of a Single Top Quark and a Higgs Boson at the LHC”  In _JHEP_ 12, 2013, pp. 077  DOI: 10.1007/JHEP12(2013)077
  * [44] Abdelhak Djouadi and Grégory Moreau  “The couplings of the Higgs boson and its CP properties from fits of the signal strengths and their ratios at the 7+8 TeV LHC”  In _Eur. Phys. J. C_ 73.9, 2013, pp. 2512  DOI: 10.1140/epjc/s10052-013-2512-9
  * [45] John Ellis, Dae Sung Hwang, Kazuki Sakurai and Michihisa Takeuchi  “Disentangling Higgs-Top Couplings in Associated Production”  In _JHEP_ 04, 2014, pp. 004  DOI: 10.1007/JHEP04(2014)004
  * [46] Jason Yue  “Enhanced $thj$ signal at the LHC with $h\rightarrow\gamma\gamma$ decay and $\mathcal{CP}$-violating top-Higgs coupling”  In _Phys. Lett. B_ 744, 2015, pp. 131–136  DOI: 10.1016/j.physletb.2015.03.044
  * [47] Federico Demartin et al.  “Higgs characterisation at NLO in QCD: CP properties of the top-quark Yukawa interaction”  In _Eur. Phys. J. C_ 74.9, 2014, pp. 3065  DOI: 10.1140/epjc/s10052-014-3065-2
  * [48] Jung Chang, Kingman Cheung, Jae Sik Lee and Chih-Ting Lu  “Probing the Top-Yukawa Coupling in Associated Higgs production with a Single Top Quark”  In _JHEP_ 05, 2014, pp. 062  DOI: 10.1007/JHEP05(2014)062
  * [49] Xiao-Gang He, Guan-Nan Li and Ya-Juan Zheng  “Probing Higgs boson $CP$ Properties with $t\bar{t}H$ at the LHC and the 100 TeV $pp$ collider”  In _Int. J. Mod. Phys. A_ 30.25, 2015, pp. 1550156  DOI: 10.1142/S0217751X15501560
  * [50] Federico Demartin, Fabio Maltoni, Kentarou Mawatari and Marco Zaro  “Higgs production in association with a single top quark at the LHC”  In _Eur. Phys. J. C_ 75.6, 2015, pp. 267  DOI: 10.1140/epjc/s10052-015-3475-9
  * [51] Fawzi Boudjema, Rohini M. Godbole, Diego Guadagnoli and Kirtimaan A. Mohan  “Lab-frame observables for probing the top-Higgs interaction”  In _Phys. Rev. D_ 92.1, 2015, pp. 015019  DOI: 10.1103/PhysRevD.92.015019
  * [52] Matthew R. Buckley and Dorival Goncalves  “Boosting the Direct CP Measurement of the Higgs-Top Coupling”  In _Phys. Rev. Lett._ 116.9, 2016, pp. 091801  DOI: 10.1103/PhysRevLett.116.091801
  * [53] Federico Demartin et al.  “tWH associated production at the LHC”  In _Eur. Phys. J. C_ 77.1, 2017, pp. 34  DOI: 10.1140/epjc/s10052-017-4601-7
  * [54] Andrei V. Gritsan, Raoul Röntsch, Markus Schulze and Meng Xiao  “Constraining anomalous Higgs boson couplings to the heavy flavor fermions using matrix element techniques”  In _Phys. Rev. D_ 94.5, 2016, pp. 055023  DOI: 10.1103/PhysRevD.94.055023
  * [55] Nicolas Mileo et al.  “Pseudoscalar top-Higgs coupling: exploration of CP-odd observables to resolve the sign ambiguity”  In _JHEP_ 07, 2016, pp. 056  DOI: 10.1007/JHEP07(2016)056
  * [56] Archil Kobakhidze, Ning Liu, Lei Wu and Jason Yue  “Implications of CP-violating Top-Higgs Couplings at LHC and Higgs Factories”  In _Phys. Rev. D_ 95.1, 2017, pp. 015016  DOI: 10.1103/PhysRevD.95.015016
  * [57] D. Azevedo, A. Onofre, F. Filthaut and R. Gonçalo  “CP tests of Higgs couplings in $t\bar{t}h$ semileptonic events at the LHC”  In _Phys. Rev. D_ 98.3, 2018, pp. 033004  DOI: 10.1103/PhysRevD.98.033004
  * [58] Dorival Gonçalves, Kyoungchul Kong and Jeong Han Kim  “Probing the top-Higgs Yukawa CP structure in dileptonic $t\overline{t}h$ with M2-assisted reconstruction”  In _JHEP_ 06, 2018, pp. 079  DOI: 10.1007/JHEP06(2018)079
  * [59] Wei-Shu Hou, Masaya Kohda and Tanmoy Modak  “Probing for extra top Yukawa couplings in light of $t\bar{t}h(125)$ observation”  In _Phys. Rev. D_ 98.7, 2018, pp. 075007  DOI: 10.1103/PhysRevD.98.075007
  * [60] Qing-Hong Cao et al.  “Limiting top quark-Higgs boson interaction and Higgs-boson width from multitop productions”  In _Phys. Rev. D_ 99.11, 2019, pp. 113003  DOI: 10.1103/PhysRevD.99.113003
  * [61] Darius A. Faroughy, Jernej F. Kamenik, Nejc Košnik and Aleks Smolkovič  “Probing the $CP$ nature of the top quark Yukawa at hadron colliders”  In _JHEP_ 02, 2020, pp. 085  DOI: 10.1007/JHEP02(2020)085
  * [62] Jie Ren, Lei Wu and Jin Min Yang  “Unveiling CP property of top-Higgs coupling with graph neural networks at the LHC”  In _Phys. Lett. B_ 802, 2020, pp. 135198  DOI: 10.1016/j.physletb.2020.135198
  * [63] Manfred Kraus, Till Martini, Sascha Peitzsch and Peter Uwer  “Exploring BSM Higgs couplings in single top-quark production”, 2019  arXiv:1908.09100 [hep-ph]
  * [64] Blaž Bortolato, Jernej F. Kamenik, Nejc Košnik and Aleks Smolkovič  “Optimized probes of $CP$ -odd effects in the $t\bar{t}h$ process at hadron colliders”  In _Nucl. Phys. B_ 964, 2021, pp. 115328  DOI: 10.1016/j.nuclphysb.2021.115328
  * [65] Qing-Hong Cao, Ke-Pan Xie, Hao Zhang and Rui Zhang  “A New Observable for Measuring CP Property of Top-Higgs Interaction”  In _Chin. Phys. C_ 45.2, 2021, pp. 023117  DOI: 10.1088/1674-1137/abcfac
  * [66] Henning Bahl et al.  “Indirect $\mathcal{CP}$ probes of the Higgs-top-quark interaction: current LHC constraints and future opportunities”  In _JHEP_ 11, 2020, pp. 127  DOI: 10.1007/JHEP11(2020)127
  * [67] Till Martini, Ren-Qi Pan, Markus Schulze and Meng Xiao  “Probing the CP structure of the top quark Yukawa coupling: Loop sensitivity versus on-shell sensitivity”  In _Phys. Rev. D_ 104.5, 2021, pp. 055045  DOI: 10.1103/PhysRevD.104.055045
  * [68] Rahool Kumar Barman, Dorival Gonçalves and Felix Kling  “Machine learning the Higgs boson-top quark CP phase”  In _Phys. Rev. D_ 105.3, 2022, pp. 035023  DOI: 10.1103/PhysRevD.105.035023
  * [69] Henning Bahl and Simon Brass  “Constraining $\mathcal{CP}$-violation in the Higgs-top-quark interaction using machine-learning-based inference”  In _JHEP_ 03, 2022, pp. 017  DOI: 10.1007/JHEP03(2022)017
  * [70] Dorival Gonçalves, Jeong Han Kim, Kyoungchul Kong and Yongcheng Wu  “Direct Higgs-top CP-phase measurement with $t\overline{t}h$ at the 14 TeV LHC and 100 TeV FCC”  In _JHEP_ 01, 2022, pp. 158  DOI: 10.1007/JHEP01(2022)158
  * [71] Duarte Azevedo, Rodrigo Capucha, António Onofre and Rui Santos  “CP-violation, asymmetries and interferences in $t\overline{t}\phi$”  In _JHEP_ 09, 2022, pp. 246  DOI: 10.1007/JHEP09(2022)246
  * [72] Anja Butter et al.  “Two Invertible Networks for the Matrix Element Method”, 2022  arXiv:2210.00019 [hep-ph]
  * [73] Jona Ackerschott et al.  “Returning CP-Observables to The Frames They Belong”, 2023  arXiv:2308.00027 [hep-ph]
  * [74] Akanksha Bhardwaj, Christoph Englert, Dorival Gonçalves and Alberto Navarro  “Non-linear top-Higgs CP violation”, 2023  arXiv:2308.11722 [hep-ph]
  * [75] G. Klamke and D. Zeppenfeld  “Higgs plus two jet production via gluon fusion as a signal at the CERN LHC”  In _JHEP_ 04, 2007, pp. 052  DOI: 10.1088/1126-6708/2007/04/052
  * [76] Kaoru Hagiwara, Qiang Li and Kentarou Mawatari  “Jet angular correlation in vector-boson fusion processes at hadron colliders”  In _JHEP_ 07, 2009, pp. 101  DOI: 10.1088/1126-6708/2009/07/101
  * [77] Christoph Englert, Dorival Goncalves-Netto, Kentarou Mawatari and Tilman Plehn  “Higgs Quantum Numbers in Weak Boson Fusion”  In _JHEP_ 01, 2013, pp. 148  DOI: 10.1007/JHEP01(2013)148
  * [78] Christoph Englert, Peter Galler, Andrew Pilkington and Michael Spannowsky  “Approaching robust EFT limits for CP-violation in the Higgs sector”  In _Phys. Rev. D_ 99.9, 2019, pp. 095007  DOI: 10.1103/PhysRevD.99.095007
  * [79] Zhite Yu, Kirtimaan A. Mohan and Chien-Peng Yuan  “Determining the $CP$ Property of $ht\bar{t}$ Coupling via a Novel Jet Substructure Observable”, 2022  arXiv:2211.00845 [hep-ph]
  * [80] ATLAS Collaboration, https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2023-025/, 2023
  * [81] CMS Collaboration, https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsSUS, 2023
  * [82] ATLAS Collaboration, https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2023-018/, 2023
  * [83] ATLAS Collaboration, https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2023-006/, 2023
  * [84] ATLAS Collaboration, https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2022-007/, 2023
  * [85] CMS Collaboration, https://twiki.cern.ch/twiki/bin/view/CMSPublic/SummaryPlotsEXO13TeV, 2023
  * [86] D. Florian  “Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector”, 2016  DOI: 10.23731/CYRM-2017-002
  * [87] J.. Andersen, T. Binoth, G. Heinrich and J.. Smillie  “Loop induced interference effects in Higgs Boson plus two jet production at the LHC”  In _JHEP_ 02, 2008, pp. 057  DOI: 10.1088/1126-6708/2008/02/057
  * [88] A. Bredenstein, Kaoru Hagiwara and B. Jager  “Mixed QCD-electroweak contributions to Higgs-plus-dijet production at the LHC”  In _Phys. Rev. D_ 77, 2008, pp. 073004  DOI: 10.1103/PhysRevD.77.073004
  * [89] P. Artoisenet  “A framework for Higgs characterisation”  In _JHEP_ 11, 2013, pp. 043  DOI: 10.1007/JHEP11(2013)043
  * [90] A. Dedes et al.  “Feynman rules for the Standard Model Effective Field Theory in Rξ -gauges”  In _JHEP_ 06, pp. 143  DOI: 10.1007/JHEP06(2017)143
  * [91] G. Aad  “Measurement of the properties of Higgs boson production at $\sqrt{s}=13$ TeV in the $H\to\gamma\gamma$ channel using $139$ fb-1 of $pp$ collision data with the ATLAS experiment”  In _JHEP_ 07, 2023, pp. 088  DOI: 10.1007/JHEP07(2023)088
  * [92] J. Alwall et al.  “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”  In _Journal of High Energy Physics_ 2014.7 Springer ScienceBusiness Media LLC, 2014  DOI: 10.1007/jhep07(2014)079
  * [93] Torbjorn Sjostrand, Stephen Mrenna and Peter Z. Skands  “A Brief Introduction to PYTHIA 8.1”  In _Comput. Phys. Commun._ 178, 2008, pp. 852–867  DOI: 10.1016/j.cpc.2008.01.036
  * [94] J. Favereau et al.  “DELPHES 3: a modular framework for fast simulation of a generic collider experiment”  In _Journal of High Energy Physics_ 2014.2 Springer ScienceBusiness Media LLC, 2014  DOI: 10.1007/jhep02(2014)057
  * [95] Matteo Cacciari, Gavin P. Salam and Gregory Soyez  “The anti-$k_{t}$ jet clustering algorithm”  In _JHEP_ 04, 2008, pp. 063  DOI: 10.1088/1126-6708/2008/04/063
  * [96] Celine Degrande et al.  “UFO - The Universal FeynRules Output”  In _Comput. Phys. Commun._ 183, 2012, pp. 1201–1214  DOI: 10.1016/j.cpc.2012.01.022
  * [97] Luc Darmé  “UFO 2.0: the ‘Universal Feynman Output’ format”  In _Eur. Phys. J. C_ 83.7, 2023, pp. 631  DOI: 10.1140/epjc/s10052-023-11780-9
  * [98] Malte Buschmann et al.  “Resolving the Higgs-Gluon Coupling with Jets”  In _Phys. Rev. D_ 90.1, 2014, pp. 013010  DOI: 10.1103/PhysRevD.90.013010
  * [99] F. Maltoni, E. Vryonidou and M. Zaro  “Top-quark mass effects in double and triple Higgs production in gluon-gluon fusion at NLO”  In _JHEP_ 11, 2014, pp. 079  DOI: 10.1007/JHEP11(2014)079
  * [100] X. Chen et al.  “Top-quark mass effects in H+jet and H+2 jets production”  In _JHEP_ 03, 2022, pp. 096  DOI: 10.1007/JHEP03(2022)096
  * [101] F. Pedregosa et al.  “Scikit-learn: Machine Learning in Python”  In _Journal of Machine Learning Research_ 12, 2011, pp. 2825–2830
  * [102] Akanksha Bhardwaj, Christoph Englert, Robert Hankache and Andrew D. Pilkington  “Machine-enhanced CP-asymmetries in the Higgs sector”  In _Phys. Lett. B_ 832, 2022, pp. 137246  DOI: 10.1016/j.physletb.2022.137246
  * [103] Scott M Lundberg and Su-In Lee  “A Unified Approach to Interpreting Model Predictions”  In _Advances in Neural Information Processing Systems 30_ Curran Associates, Inc., 2017, pp. 4765–4774  URL: http://papers.nips.cc/paper/7062-a-unified-approach-to-interpreting-model-predictions.pdf
  * [104] Guillermo Owen  “Game Theory”  Academic Press, 1982
  * [105] Roger B. Myerson  “Game Theory: Analysis of Conflict”  Harvard University Press, 1997
  * [106] Erik Štrumbelj and Igor Kononenko  “An Efficient Explanation of Individual Classifications using Game Theory”  In _Journal of Machine Learning Research_ 11.1, 2010, pp. 1–18  URL: http://jmlr.org/papers/v11/strumbelj10a.html
  * [107] Anja Butter, Tilman Plehn, Nathalie Soybelman and Johann Brehmer  “Back to the Formula – LHC Edition”, 2021  arXiv:2109.10414 [hep-ph]
  * [108] Philip Bechtle et al.  “$HiggsSignals$: Confronting arbitrary Higgs sectors with measurements at the Tevatron and the LHC”  In _Eur. Phys. J. C_ 74.2, 2014, pp. 2711  DOI: 10.1140/epjc/s10052-013-2711-4
  * [109] Philip Bechtle et al.  “HiggsSignals-2: Probing new physics with precision Higgs measurements in the LHC 13 TeV era”  In _Eur. Phys. J. C_ 81.2, 2021, pp. 145  DOI: 10.1140/epjc/s10052-021-08942-y
  * [110] Henning Bahl et al.  “HiggsTools: BSM scalar phenomenology with new versions of HiggsBounds and HiggsSignals”  In _Comput. Phys. Commun._ 291, 2023, pp. 108803  DOI: 10.1016/j.cpc.2023.108803
  * [111] Georges Aad  “Measurement of Higgs boson decay into $b$-quarks in associated production with a top-quark pair in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector”  In _JHEP_ 06, 2022, pp. 097  DOI: 10.1007/JHEP06(2022)097
  * [112] V.. Abazov  “A precision measurement of the mass of the top quark”  In _Nature_ 429, 2004, pp. 638–642  DOI: 10.1038/nature02589
  * [113] Yanyan Gao et al.  “Spin Determination of Single-Produced Resonances at Hadron Colliders”  In _Phys. Rev. D_ 81, 2010, pp. 075022  DOI: 10.1103/PhysRevD.81.075022
  * [114] J. Alwall, A. Freitas and O. Mattelaer  “The Matrix Element Method and QCD Radiation”  In _Phys. Rev. D_ 83, 2011, pp. 074010  DOI: 10.1103/PhysRevD.83.074010
  * [115] Sara Bolognesi et al.  “On the spin and parity of a single-produced resonance at the LHC”  In _Phys. Rev. D_ 86, 2012, pp. 095031  DOI: 10.1103/PhysRevD.86.095031
  * [116] Paul Avery  “Precision studies of the Higgs boson decay channel H→ZZ→4$\ell$ with MEKD”  In _Phys. Rev. D_ 87.5, 2013, pp. 055006  DOI: 10.1103/PhysRevD.87.055006
  * [117] Jeppe R. Andersen, Christoph Englert and Michael Spannowsky  “Extracting precise Higgs couplings by using the matrix element method”  In _Phys. Rev. D_ 87.1, 2013, pp. 015019  DOI: 10.1103/PhysRevD.87.015019
  * [118] Pierre Artoisenet, Priscila Aquino, Fabio Maltoni and Olivier Mattelaer  “Unravelling $t\overline{t}h$ via the Matrix Element Method”  In _Phys. Rev. Lett._ 111.9, 2013, pp. 091802  DOI: 10.1103/PhysRevLett.111.091802
  * [119] John M. Campbell, R. Ellis, Walter T. Giele and Ciaran Williams  “Finding the Higgs Boson in Decays to $Z\gamma$ using the Matrix Element Method at Next-to-Leading Order”  In _Phys. Rev. D_ 87.7, 2013, pp. 073005  DOI: 10.1103/PhysRevD.87.073005
  * [120] Till Martini and Peter Uwer  “The Matrix Element Method at next-to-leading order QCD for hadronic collisions: Single top-quark production at the LHC as an example application”  In _JHEP_ 05, 2018, pp. 141  DOI: 10.1007/JHEP05(2018)141
  * [121] Johann Brehmer, Gilles Louppe, Juan Pavez and Kyle Cranmer  “Mining gold from implicit models to improve likelihood-free inference”  In _Proc. Nat. Acad. Sci._ 117.10, 2020, pp. 5242–5249  DOI: 10.1073/pnas.1915980117
  * [122] Johann Brehmer, Kyle Cranmer, Gilles Louppe and Juan Pavez  “Constraining Effective Field Theories with Machine Learning”  In _Phys. Rev. Lett._ 121.11, 2018, pp. 111801  DOI: 10.1103/PhysRevLett.121.111801
  * [123] Johann Brehmer, Kyle Cranmer, Gilles Louppe and Juan Pavez  “A Guide to Constraining Effective Field Theories with Machine Learning”  In _Phys. Rev. D_ 98.5, 2018, pp. 052004  DOI: 10.1103/PhysRevD.98.052004
  * [124] Markus Stoye et al.  “Likelihood-free inference with an improved cross-entropy estimator”, 2018  arXiv:1808.00973 [stat.ML]
  * [125] Johann Brehmer, Felix Kling, Irina Espejo and Kyle Cranmer  “MadMiner: Machine learning-based inference for particle physics”  In _Comput. Softw. Big Sci._ 4.1, 2020, pp. 3  DOI: 10.1007/s41781-020-0035-2
  * [126] Ilaria Brivio  “SMEFTsim 3.0 — a practical guide”  In _JHEP_ 04, 2021, pp. 073  DOI: 10.1007/JHEP04(2021)073
  * [127] S.. Wilks  “The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses”  In _The Annals of Mathematical Statistics_ 9.1 Institute of Mathematical Statistics, 1938, pp. 60–62  DOI: 10.1214/aoms/1177732360