refname refsname

#  ${\cal CP}$-sensitive simplified template cross-sections for $t\bar{t}H$

Henning Bahl<EMAIL_ADDRESS>Institut für Theoretische Physik,
Universität Heidelberg, Philosophenweg 16, 61920 Heidelberg, Germany  Alberto
Carnelli<EMAIL_ADDRESS>Université Paris-Saclay, CEA, Département de
Physique des Particules, 91191, Gif-sur-Yvette, France  Frédéric Déliot
<EMAIL_ADDRESS>Université Paris-Saclay, CEA, Département de Physique
des Particules, 91191, Gif-sur-Yvette, France  Elina Fuchs<EMAIL_ADDRESS>Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2,
30167 Hannover, Germany  Physikalisch-Technische Bundesanstalt, Bundesallee
100, 38116 Braunschweig, Germany  Anastasia Kotsokechagia
<EMAIL_ADDRESS>Université Paris-Saclay, CEA, Département de
Physique des Particules, 91191, Gif-sur-Yvette, France  Tanguy Lucas Marsault
<EMAIL_ADDRESS>Université Paris-Saclay, CEA, Département de Physique
des Particules, 91191, Gif-sur-Yvette, France  Marco Menen
<EMAIL_ADDRESS>Institut für Theoretische Physik, Leibniz
Universität Hannover, Appelstraße 2, 30167 Hannover, Germany  Physikalisch-
Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
Laurent Schoeffel<EMAIL_ADDRESS>Université Paris-Saclay, CEA,
Département de Physique des Particules, 91191, Gif-sur-Yvette, France
Matthias Saimpert<EMAIL_ADDRESS>Université Paris-Saclay, CEA,
Département de Physique des Particules, 91191, Gif-sur-Yvette, France

###### Abstract

The ${\cal CP}$ structure of the Higgs boson is a fundamental property which
has not yet been constrained with high precision. ${\cal CP}$ violation in the
Yukawa coupling between the Higgs boson and top quark pair can be probed
directly at the Large Hadron Collider by measuring top-quark-associated Higgs
production. Multivariate analysis techniques commonly developed so far by the
experiments are designed for a specific signal model and, therefore,
complicate reinterpretations and statistical combinations. With this
motivation in mind, we propose a ${\cal CP}$-sensitive extension of the
simplified template cross-section (STXS) framework. Considering multiple Higgs
decay channels, we perform an in-depth comparison of ${\cal CP}$-sensitive
observables and combinations thereof. Our resulting proposal is to extend the
existing binning in the transverse momentum of the Higgs boson $p_{T,{H}}$ by
either the pseudorapidity difference of the two top-quarks
$\Delta\eta_{t\bar{t}}$, or a variable that is based on the top quark momenta,
namely $b_{2}$ or the Collins-Soper angle $|\cos\theta^{*}|$. We demonstrate
that this variable selection provides close to optimal sensitivity to the
${\cal CP}$ mixture in the top Yukawa coupling for an integrated luminosity of
$300\,\,\mathrm{fb}^{-1}$, by comparing it to the results of a multivariate
analysis. Our results also suggest a benefit of the two-dimensional STXS
extension at 3000$\,\,\mathrm{fb}^{-1}$.

###### Contents

  1. 1 Introduction
  2. 2 ${\cal CP}$-sensitive observables in $t\bar{t}H$ events
    1. 2.1 Event generation
    2. 2.2 Observable definition
    3. 2.3 Distributions at the parton-level
  3. 3 Evaluation of the observable performance
    1. 3.1 Detector effects and selection efficiency
    2. 3.2 Significance computation
  4. 4 Results
    1. 4.1 Results with default binning
    2. 4.2 Final results with optimised binning
    3. 4.3 Comparison to a multivariate analysis
    4. 4.4 Expected exclusion limits
  5. 5 Extending the STXS framework
    1. 5.1 Further background considerations
    2. 5.2 Proposal for $t\bar{t}H$ STXS extension
  6. 6 Conclusion
  7. A Supplementary parton-level distributions for $t\bar{t}H$
    1. A.1 Higgs boson rest frame
    2. A.2 $t\bar{t}$ rest frame
    3. A.3 $t\bar{t}H$ rest frame
  8. B Full two-dimensional significance tables

##  1 Introduction

The Standard Model (SM) of particle physics does not contain sufficient ${\cal
CP}$ violation to explain the observed baryon asymmetry of the Universe (BAU)
[1, 2]. Therefore, new sources of ${\cal CP}$ violation are needed and the
search for corresponding ${\cal CP}$-violating interactions is an important
target for searches beyond the SM (BSM) at the LHC.

The investigation of the Higgs sector is especially important in this context.
First, the ${\cal CP}$ structure of the Higgs couplings is still comparably
unexplored. Second, the BAU can be linked to the Higgs sector via the
mechanism of electroweak baryogenesis [3] (for reviews, see e.g.
Trodden:1998ym, Cline:2006ts,Morrissey:2012db,White:2016nbo,Bodeker:2020ghk).
While the ${\cal CP}$ structure of the Higgs couplings to massive gauge bosons
is already relatively tightly constrained [9, 11, 12, 13, 14, 15, 16, 17, 18,
10], the test of the ${\cal CP}$ structure of the Higgs–fermion interactions
at the LHC only started recently [19, 20, 21, 22, 23, 17, 24, 25]. Moreover,
most BSM theories predict a larger amount of ${\cal CP}$ violation in the
Yukawa couplings than in the interactions with massive vector bosons, since
the latter are loop-suppressed.

The top-Yukawa coupling has a special relevance because of its large size. It
can be parameterized in the following way in the Higgs Characterization Model
[26],

$\displaystyle\mathcal{L}_{\text{top-
Yuk}}=\frac{y_{t}^{\text{SM}}g_{t}}{\sqrt{2}}\bar{t}\left(\cos\alpha_{t}+i\gamma_{5}\sin\alpha_{t}\right)tH\,,$
(1)

where $y_{t}^{\text{SM}}$ is the SM top-Yukawa coupling, $g_{t}$ is the (real)
modifier of the strength of the top-Yukawa coupling, and $\alpha_{t}$ is the
${\cal CP}$-mixing angle. The SM corresponds to $g_{t}=1$, $\alpha_{t}=0$.
Equivalently, this interaction can be parameterized in terms of a ${\cal
CP}$-even and a ${\cal CP}$-odd coupling, $c_{t}\equiv g_{t}\cos\alpha_{t}$
and $\tilde{c}_{t}\equiv g_{t}\sin\alpha_{t}$, respectively.

In an EFT framework (e.g. the SM effective field theory), this gauge-invariant
modification is generated by the operator
$(\Phi^{\dagger}\Phi)Q_{L}u_{R}\tilde{\Phi}$ (where $\Phi$ is the Higgs
doublet, $Q_{L}$ is the left-handed quark doublet, and $u_{R}$ is the right-
handed up-type quark singlet) and the coefficients
$y_{t}^{\text{SM}}g_{t}\cos\alpha_{t}$ and
$y_{t}^{\text{SM}}g_{t}\sin\alpha_{t}$ in Eq. 1 can be related to the real and
imaginary parts of the corresponding Wilson coefficient (see e.g.
Fuchs:2020uoc,AharonyShapira:2021ize).111In the SM effective field theory, the
operator $(\Phi^{\dagger}\Phi)Q_{L}u_{R}\tilde{\Phi}$ also generates
additional multi-Higgs and Goldstone boson interactions with top quarks, which
are in general needed to guarantee gauge invariance. These are, however, not
relevant to top-associated Higgs production studied in this work.

This general coupling structure can be probed at the LHC but also via
measurements of electric dipole moments (EDMs) of particles like the electron
or the neutron, for which so far only upper bounds exist [29, 30]. If one
assumes that the first-generation Yukawa couplings do not deviate from their
SM predictions, these EDM bounds put strong constraints on the ${\cal CP}$-odd
part of the top-Yukawa coupling, constraining $\tilde{c}_{t}\lesssim 10^{-3}$
at the 90% CL [31, 27, 32]; this translates into $\alpha_{t}\lesssim
0.06^{\circ}$ for $g_{t}=1$. Since the first-generation Yukawa couplings are
only very weakly constrained [33, 36, 37, 38, 39, 40, 41, 42, 43, 42, 34, 35],
this assumption, however, does not necessarily hold, resulting in relaxed EDM
bounds. Therefore, it is crucial to measure the ${\cal CP}$ properties of the
top-Yukawa coupling directly, i.e. without a strong dependence on the other
Yukawa couplings. This can be achieved at the LHC where the measurement of
distinct production and decay channels provides access to specific couplings,
whereas the measured value of an EDM results from the sum over all ${\cal
CP}$-violating contributions.

Bounds on the ${\cal CP}$ structure of the top-Yukawa interaction at the LHC
can be derived from rate measurements [44, 51, 52, 53, 54, 55, 56, 57, 58, 45,
46, 47, 27, 48, 49, 32, 50]. In this case, the most stringent constraints
arise from Higgs production via gluon fusion when the Higgs boson decays to
two photons (constraining $\alpha_{t}\lesssim 28^{\circ}$ at the 95% CL
assuming that $g_{t}=1$ [50]). These constraints are, however, highly model-
dependent since other particles also contribute to the respective loop
diagrams. A more direct probe is top-associated Higgs production (consisting
of the three sub-channels $t\bar{t}H$, $tHq$, and $tWH$). Top-associated Higgs
production can be used to constrain the ${\cal CP}$ properties of the top-
Yukawa coupling via rate measurements (constraining $\alpha_{t}\lesssim
72^{\circ}$ at the 95% CL [48] assuming that $g_{t}=1$), kinematic
distributions, and ${\cal CP}$-odd observables. While rate measurements are
again model-dependent (e.g., the Higgs-gluon coupling contributes to top-
associated Higgs production at next-to-leading order), ${\cal CP}$-odd
observables in principle offer a way to unambiguously probe ${\cal CP}$. For
top-associated Higgs production, they are, however, very difficult to measure
due to the necessity of reconstructing the polarization of the top quark [59,
60, 61, 62, 63, 64, 65]. Specifically, in hadronic decays, it requires to
differentiate between up- and down-type quarks. While this might change during
the high-luminosity phase of the LHC, at the moment they certainly only
provide little sensitivity for testing the ${\cal CP}$ character of the top-
Yukawa interaction. On the other hand, kinematic distributions of ${\cal
CP}$-even observables offer a good compromise between the ${\cal CP}$-odd
observables and rate measurements. After the observation of top-associated
Higgs production in 2018 [66, 67], the additional data collected and to be
collected at the LHC allows now for differential measurements even for this
comparably rare process, making it a promising way to probe ${\cal CP}$ during
over the next few years.

Kinematic information can be best exploited using multivariate analysis
techniques (see e.g. Ren:2019xhp,Barman:2021yfh,Bahl:2021dnc,Esmail:2024gdc)
or the matrix element method (see e.g.
Martini:2021uey,Kraus:2019myc,Gritsan:2016hjl,Goncalves:2018agy,Butter:2022vkj).
These methods have been used by the experimental collaborations to put
constraints on $\alpha_{t}$, constraining at the 95% CL $\alpha_{t}\lesssim
43^{\circ}$ (ATLAS using the $H\to\gamma\gamma$ channel [20]; in addition, see
also the currently less sensitive $H\to b\bar{b}$ channel in ATLAS:2023cbt)
and $\alpha_{t}\lesssim 45^{\circ}$ (CMS using
$H\to\gamma\gamma,4\ell,\tau\tau$ [19, 22, 17, 23])222In the CMS papers, the
limits are set on $|f_{\cal
CP}^{Htt}|\equiv\nicefrac{{\tilde{c}_{t}^{2}}}{{(c_{t}^{2}+\tilde{c}_{t}^{2})}}=\sin^{2}\alpha_{t}$.
assuming that $g_{t}=1$. For an overview of future prospects, see
Gritsan:2022php. These techniques are typically tailored to specific Higgs
decay channels, detector environments, and background assumptions. Moreover,
they are designed for specific signal models and intrinsically depend on the
assumptions about the other couplings involved. As a consequence, the
combination of different decay channels (see e.g. CMS:2021nnc,CMS:2022dbt) and
across experiments is non-trivial. The usage of such approaches also makes the
reinterpretation of the analyses difficult, as it usually requires the signal
efficiency as a function of the final discriminating algorithm output score.
To mitigate such issues — before providing likelihoods encoding all relevant
dependencies becomes feasible —, the simplified template cross-section (STXS)
framework [77, 78, 79, 80] has been established. This study, focusing on
${\cal CP}$-sensitive but ${\cal CP}$-even observables, aims to enhance the
${\cal CP}$ sensitivity of this framework for $t\bar{t}H$ production by taking
the interplay of two variables into account. As a complementary example, also
for Drell-Yan production, double- and triple-differential distributions have
recently been found to yield a higher sensitivity to SMEFT operators compared
to a finer binning in a single variable [81].

We start in Section 2 with a comparison of the kinematic shapes of various
${\cal CP}$-sensitive observables evaluated in different reference frames
building upon existing proposals in the literature [82, 53, 85, 86, 55, 57,
87, 58, 73, 83, 74, 84, 71, 63, 69, 64, 65]. In Section 3, we evaluate how
these observables could be impacted by detector effects using a simplified
model and describe how their performance is evaluated. In Section 4, we
perform a sensitivity study based on one- and two-dimensional distributions,
optimize the binning of the most sensitive observables assuming
$300\,\,\mathrm{fb}^{-1}$ of data (which should correspond to the available
LHC dataset by the end of 2025), and compare the outcome to a multivariate
analysis based on a boosted decision tree (BDT). Based on these findings, we
make a proposal for a ${\cal CP}$-sensitive extension of the latest version of
the STXS framework (v1.2) in Section 5. It also includes a qualitative
discussion about the Higgs and non-Higgs backgrounds and a comparison between
the expected ${\cal CP}$ sensitivity of the STXS framework v1.2 and our
proposed extension. We present conclusions in Section 6.

##  2 ${\cal CP}$-sensitive observables in $t\bar{t}H$ events

The $t{\bar{t}}H$ production process possesses many final state particles and
therefore also numerous kinematic observables. In experimental analyses
targeting ${\cal CP}$ violation in the top-Yukawa coupling, a number of such
observables have been used already (see e.g. ATLAS:2023cbt). These are,
however, only optimized for a specific channel and selection. In this work, we
aim to find observables that are optimal to use in the context of an STXS
extension and should therefore be independent of the analysis details.

We describe the generation of the $t{\bar{t}}H$ events used in this study in
Section 2.1. We then introduce an extended number of kinematic observables in
various rest frames in Section 2.2. The parton level distributions of these
observables are shown in Section 2.3.

###  2.1 Event generation

We generate parton-level events for the $pp\rightarrow t{\bar{t}}H$ process at
a center-of-mass energy of $\sqrt{s}=13\,\,\mathrm{TeV}$ using the event
generator MadGraph5_aMC@NLO [88] (version 3.3.2) and the NNPDF23_lo_
as_0130_qed PDF set [89]. The renormalization and factorization scales are set
to $1/2\cdot(2m_{t}+m_{H})$. Events are generated using the “Higgs
Characterization” (HC) UFO model [26] for $g_{t}=1$ and two different values
for the ${\cal CP}$ phase $\alpha_{t}$: $0^{\circ}$ (pure scalar,
corresponding to the SM) and $90^{\circ}$ (pure ${\cal CP}$-odd case). All
simulated samples were generated at leading order (LO) and include one million
events each.

While the HC model allows to generate events at next-to-leading order (NLO) in
QCD, such calculation makes the implicit assumption of no additional BSM
Higgs–gluon coupling. As discussed in Section 1, in this case Higgs production
via gluon fusion allows to set much stronger (indirect) constraints on the
${\cal CP}$ character of the top-Yukawa coupling with respect to top-
associated Higgs production. Therefore, for self-consistency, all events are
generated at LO (see also the discussion in Bahl:2020wee). The NLO correction
on the $t\bar{t}H$ total cross-section depends only very weakly on the ${\cal
CP}$ character of the top-Yukawa coupling [86, 48]. Similarly, we showed
recently that this observation still holds for kinematic distributions.333In
Bahl:2020wee, the NLO $N_{\text{jet}}$ distribution for $tHq$ production was
found to significantly differ from the LO distribution. This is not relevant
to the discussion in this paper which focuses on $t\bar{t}H$ production.
Hence, a simple scaling factor of 1.14 [86] is used as NLO correction in the
following.

Since only ${\cal CP}$-even observables are considered in this study, signal
yields for mixed-${\cal CP}$ scenarios are computed by combining the yields
from the SM and the pure ${\cal CP}$-odd samples. The following formula is
used :

$\displaystyle
N(g_{t},\alpha_{t})=g_{t}^{2}\left[N_{\mathrm{SM}}\cos^{2}\alpha_{t}+N_{\mathrm{odd}}\sin^{2}\alpha_{t}\right],$
(2)

where $N(g_{t},\alpha_{t})$ is the expected yield for the signal corresponding
to the parameters $(g_{t},\alpha_{t})$, with $N_{\mathrm{SM}}=N(1,0)$ and
$N_{\mathrm{odd}}=N(1,90^{\circ})$. The yields for $\alpha_{t}=35^{\circ}$ and
$\alpha_{t}=45^{\circ}$ obtained from Eq. 2 were validated against the yields
obtained from dedicated samples generated with these $\alpha_{t}$ values and
an agreement within 1% was obtained.

The SM background processes discussed in Section 5.1, i.e.
$t\bar{t}\gamma\gamma$, $t\bar{t}W$ and $t\bar{t}b\bar{b}$, were generated
using a similar setup. Half a million events are generated at LO for each
process using the NNPDF23_lo_as_0130_qed PDF set [89] and the renormalization
and factorization scales were set to the sum of the transverse mass of all the
final state particles divided by a factor of two (corresponding to
dynamical_scale_choice $=3$ in MadGraph5_aMC@NLO).

###  2.2 Observable definition

We consider a right-handed coordinate system with its origin at the nominal
interaction point (IP). In the laboratory frame, see Fig. 1, top-left, the
$z$-axis is defined along the beam axis, and $\boldsymbol{n}$ denotes the
associated unit vector. The $x$-axis points from the IP to the centre of the
LHC ring, and the $y$-axis points upwards. Cylindrical coordinates $(r,\phi)$
are used in the transverse ($x,y$) plane, $\phi$ being the azimuthal angle
around the $z$-axis. The pseudorapidity is defined in terms of the polar angle
$\theta$ as $\eta=-\ln\tan(\theta/2)$. Four-momenta are denoted by $p$ and
their corresponding three-momenta are denoted by $\boldsymbol{p}$. The
magnitudes of the components of the three-momenta in the transverse plane and
along the $x$ and $z$-axes are labeled as $p_{\text{T}}$, $p^{x}$ and $p^{z}$,
respectively. The subscripts $t$, $\bar{t}$, $H$, $p_{1}$ and $p_{2}$ in the
labels indicate to which particle they refer to, i.e. top quark ($t$), anti-
top quark ($\bar{t}$), Higgs boson ($H$) or the colliding protons ($p_{1}$,
$p_{2}$). We consider four possible reference frames for the computation of
the observables:

  * •

the laboratory frame (lab frame),

  * •

the $t\bar{t}$ rest frame, where
$\boldsymbol{p}_{t}+\boldsymbol{p}_{\bar{t}}=\boldsymbol{0}$ ($t\bar{t}$
frame),

  * •

the $H$ rest frame, where $\boldsymbol{p}_{H}=\boldsymbol{0}$ ($H$ frame),

  * •

the $t\bar{t}H$ rest frame, where
$\boldsymbol{p}_{t}+\boldsymbol{p}_{\bar{t}}+\boldsymbol{p}_{H}=\boldsymbol{0}$
($t\bar{t}H$ frame).

To unambiguously define these frames, we start in the lab frame. To move to
the frame $X$ (defined by $\boldsymbol{p}_{X}=\boldsymbol{0}$, where $X=H$,
$t\bar{t}$, $t\bar{t}H$), we first apply a rotation around the $z$-axis such
that $\boldsymbol{p}_{T,X}$ is parallel to the $x$-axis, see Fig. 1, left.
Then, a boost along the transverse direction is performed such that
$\boldsymbol{p}_{T,X}=0$. Finally, a boost along the longitudinal direction
ensures that $\boldsymbol{p}_{X}=0$. These last two steps are illustrated in
Fig. 1, right. In the resulting coordinate system, the proton momenta lie in
the $x-z$ plane and both form the same angle with the $z$-axis.

Figure 1: Sketch illustrating the rest frame definition adopted in this work.
The rest frame $X$ is shown here, defined by
$\boldsymbol{p}_{X}=\boldsymbol{0}$, where $X=H$, $t\bar{t}$, $t\bar{t}H$.

The ${\cal CP}$-sensitive observables considered in this work are summarised
in Table 1. All of these are ${\cal CP}$-even observables. We analyse them in
all applicable rest frames listed above. Any new observable candidate for the
extended STXS framework should be defined for all $t\bar{t}H$ events,
regardless of the Higgs boson and top quark pair decay mode. Therefore, in the
following, we consider that the Higgs boson, the top quark and anti-top quark
are reconstructed experimentally such that their momenta are accessible. In
practice, we compute the observables listed in Table 1 at the parton level and
apply specific acceptance and smearing factors to mimic detector effects, as
described in Section 3.1. Note that none of the observables listed in Table 1
requires a distinction between the top and the anti-top quark, which would be
very challenging experimentally.

observable |  definition |  frame |  reference
---|---|---|---
$p_{T,{H}}$ |  - |  lab, $t\bar{t}$, $t\bar{t}H$ |  -
$\Delta\eta_{t\bar{t}}$ |  $|\eta_{t}-\eta_{\bar{t}}|$ |  lab, $H$, $t\bar{t}H$ |  -
$\Delta\phi_{t\bar{t}}$ |  $|\phi_{t}-\phi_{\bar{t}}|$ |  lab, $H$, $t\bar{t}H$ |  -
$m_{t\bar{t}}$ |  $(p_{t}+p_{\bar{t}})^{2}$ |  frame-invariant |  -
$m_{t\bar{t}H}$ |  $(p_{t}+p_{\bar{t}}+p_{H})^{2}$ |  frame-invariant |  -
$|\cos\theta^{*}|$ |  $\frac{|\boldsymbol{p}_{t}\cdot\boldsymbol{n}|}{|\boldsymbol{p}_{t}|\cdot|\boldsymbol{n}|}$ |  $t\bar{t}$ |  [90, 74]
$b_{1}$ |  $\frac{(\boldsymbol{p}_{t}\times\boldsymbol{n})\cdot(\boldsymbol{p}_{\bar{t}}\times\boldsymbol{n})}{p_{T,{t}}p_{T,{\bar{t}}}}$ |  all |  [82]
$b_{2}$ |  $\frac{(\boldsymbol{p}_{t}\times\boldsymbol{n})\cdot(\boldsymbol{p}_{\bar{t}}\times\boldsymbol{n})}{\left|\boldsymbol{p}_{t}\right|\ \left|\boldsymbol{p}_{\bar{t}}\right|}$ |  all |  [82]
$b_{3}$ |  $\frac{p_{t}^{x}\ p_{\bar{t}}^{x}}{p_{T,{t}}p_{T,{\bar{t}}}}$ |  all |  [82]
$b_{4}$ |  $\frac{p_{t}^{z}\ p_{\bar{t}}^{z}}{\left|\boldsymbol{p}_{t}\right|\ \left|\boldsymbol{p}_{\bar{t}}\right|}$ |  all |  [82]
$\phi_{C}$ |  $\arccos\bigg{(}\frac{|(\boldsymbol{p}_{p_{1}}\times\boldsymbol{p}_{p_{2}})\cdot(\boldsymbol{p}_{t}\times\boldsymbol{p}_{\bar{t}})|}{\left|\boldsymbol{p}_{p_{1}}\times\boldsymbol{p}_{p_{2}}\right|\ \left|\boldsymbol{p}_{t}\times\boldsymbol{p}_{\bar{t}}\right|}\bigg{)}$ |  $H$ |  [84]
Table 1: Overview of the ${\cal CP}$-sensitive observables considered in this
work, including their definition, the rest frames in which they are analysed,
and references where they are discussed in more detail.

###  2.3 Distributions at the parton-level

The distributions of the observables defined in Section 2.2 which will be
discussed further in this work are shown in this section, while the others,
which were found to have a lower sensitivity, are shown in Appendix A. As a
baseline cut to mimic the ATLAS and CMS tracker acceptance [91, 92], all
particles are required to satisfy $|\eta|<2.0$.444the ATLAS and CMS tracker
acceptance is $|\eta|<2.5$, however a tighter cut of $|\eta|<2.0$ is applied
to the Higgs boson, the top and anti-top quarks to account for the angular
distribution of their decay products. The Higgs boson, top and anti-top quarks
being massive particles, no minimum $p_{\text{T}}$ cut is applied.

The distributions of the Higgs $p_{\text{T}}$, azimuthal angle difference and
pseudorapidity difference between the $t\bar{t}$ pair in the laboratory frame
are shown in Fig. 2, together with the invariant mass of the $t\bar{t}$ pair,
which is frame-invariant. The distributions of the $b_{1}$, $b_{2}$, $b_{3}$
and $b_{4}$ observables in the laboratory frame are shown in Fig. 3, while the
distributions of the $\phi_{C}$ angle in the Higgs boson frame and the cosine
of the scattering angle with respect to the direction of the proton beams in
the $t\bar{t}$ frame ($|\cos\theta^{*}|$, the so-called Collins-Soper angle)
are shown in Fig. 4.

Large shape differences are observed between the SM and the pure ${\cal
CP}$-odd scenario, whereas the shape differences between the SM and ${\cal
CP}$ models closer to the current experimental limit ($\alpha_{t}=45^{\circ}$
or $35^{\circ}$) are much reduced, yet visible. Larger tails at high (low)
value are observed in case of the presence of a ${\cal CP}$-odd component for
$p_{T,{H}}$, $\Delta\eta_{t\bar{t}}$, $m_{t\bar{t}}$, $b_{1}$, $b_{2}$,
$b_{3}$ and $|\cos\theta^{*}|$ ($b_{4}$, $\phi_{C}$ and
$\Delta\phi_{t\bar{t}}$).

The effect from the $|\eta|<2.0$ cut on the top and anti-top quarks is
particularly visible in the $\Delta\eta_{t\bar{t}}$ distribution, which
consequently has a sharp cut-off at $\Delta\eta_{t\bar{t}}=4$. The impact of
this cut is also clearly visible on the $\phi_{C}$ angle and $\cos\theta^{*}$
distributions, as it breaks the approximate, accidental flat distribution
observed for these two observables in the SM by depleting the $\phi_{C}<\pi/4$
and $|\cos\theta^{*}|>\cos(\pi/4)$ regions. Let us also note here the large
peak in the $b_{1}$ observable distribution in the laboratory frame between
the lower bound value of $-1$ and $-0.9$ that occurs irrespective of the
${\cal CP}$ hypothesis, but is more pronounced for lower $\alpha_{t}$.

Figure 2: Distributions of the (top, left) Higgs $p_{\text{T}}$, (top, right)
pseudorapidity difference and (bottom, left) azimuthal angle difference
between the $t\bar{t}$ pair in the laboratory frame, as well as (bottom,
right) the invariant mass of the $t\bar{t}$ pair, which is frame-invariant.
$t\bar{t}H$ events generated at parton-level with $g_{t}=1$ and various values
of ${\cal CP}$ phase $\alpha_{t}$ are considered following the event
generation described in Section 2.1. All distributions are normalised to
unity.

Figure 3: Distributions of the $b_{1}$, $b_{2}$, $b_{3}$ and $b_{4}$
observables in the laboratory frame. $t\bar{t}H$ events generated at parton-
level with $g_{t}=1$ and various values of ${\cal CP}$ phase $\alpha_{t}$ are
considered following the event generation described in Section 2.1. All
distributions are normalised to unity.

Figure 4: Distributions of (left) the $\phi_{C}$ and (right) the
$|\cos\theta^{*}|$ observables in the Higgs and $t\bar{t}$ frame. $t\bar{t}H$
events generated at parton-level with $g_{t}=1$ and various values of ${\cal
CP}$ phase $\alpha_{t}$ are considered following the event generation
described in Section 2.1. All distributions are normalised to unity.

##  3 Evaluation of the observable performance

This section focuses on quantifying the discriminating power of each
observable to distinguish between a mixed ${\cal CP}$ state of the top-Yukawa
coupling and the SM case. With the goal of extending the STXS framework in
mind, we examine specifically the three channels explored so far in the
experimental $t\bar{t}H$ analyses at the LHC. We discuss in Section 3.1 the
limitations in resolution and reconstruction of the top quarks and Higgs boson
in each channel, while the formula used for performance evaluation is
explained in Section 3.2.

###  3.1 Detector effects and selection efficiency

The $t\bar{t}H$ process at the LHC has been studied so far through three main
experimental channels. The first two channels target $H\to\gamma\gamma$ and
$H\to b\bar{b}$ decays and will be labeled $t\bar{t}H(\to\gamma\gamma)$ and
$t\bar{t}H(\to bb)$ in the following, while the third channel selects events
with multiple leptons in the final state and will be labeled
$t\bar{t}H$(multilep.). The latest experimental results from the ATLAS and CMS
collaborations for each of these channels are available in
ATLAS:2020ior,ATLAS:2023cbt,CMS:2020cga,CMS:2021nnc,CMS:2022mlq,CMS:2022dbt.
In order to compare the performance of the various observables defined in
Section 2 in discriminating the top-Yukawa ${\cal CP}$ properties in a
meaningful way, it is necessary to account for the limited efficiency of the
event selection performed in experimental data analyses as well as the finite
resolution and acceptance of the detectors. The $t\bar{t}H(\to\gamma\gamma)$,
$t\bar{t}H(\to bb)$ and $t\bar{t}H$(multilep.) channels differ widely in terms
of final state objects and cross-section times branching ratio; taking into
account these detector and acceptance effects is highly non-trivial, such that
a simplified approach is adopted in this work. All the input parameters to our
model are shown in Table 2, while more details are given below.

The $t\bar{t}H(\to\gamma\gamma)$ channel provides the smallest cross-section
times branching ratio, but the best signal-over-background ratio due to the
excellent detector resolution on the reconstructed Higgs mass. If any, all
high-$p_{\text{T}}$ neutrinos can be assumed to originate from the top or
anti-top decays, which eases the $t\bar{t}$ pair reconstruction. Following the
$t\bar{t}H$ event yields in signal-enriched regions provided in ATLAS:2020ior,
we scale the $t\bar{t}H$ yields at the parton-level by the branching ratio of
$H\to\gamma\gamma$ in the SM times a factor ranging from 0.25 to 0.32
depending on the ${\cal CP}$ hypothesis to account for the limited efficiency
of the event selection. The Higgs boson and top/anti-top quark transverse
momenta are smeared using a Gaussian distribution with a width of 4 and
$40\,\,\mathrm{GeV}$, respectively, while the pseudo-rapidity of the top/anti-
top quark is smeared by $0.5$. These last numbers were tuned to match the
ATLAS photon energy resolution [93] and the top quark mass distribution
published in ATLAS:2020ior.

The $t\bar{t}H(\to bb)$ channel provides the highest cross-section times
branching ratio, but the lowest signal-over-background ratio due to the large
background originating from $t\bar{t}$ pairs produced in association with
$b$-quarks and the comparatively poor detector energy resolution on jet
measurements with respect to photons. Similarly to
$t\bar{t}H(\to\gamma\gamma)$, if any, all high-$p_{\text{T}}$ neutrinos shall
originate from the top or anti-top decays. However, jets originating from
$b$-hadrons can come from either the Higgs boson decay or the top/anti-top
decays, making the top/anti-top reconstruction more challenging with respect
to $t\bar{t}H(\to\gamma\gamma)$. Following the $t\bar{t}H$ event yields in
signal-enriched regions provided in ATLAS:2023cbt, we apply on top of the
$H(\to bb)$ SM branching ratio a scaling factor ranging from 0.005 to 0.0065
depending on the ${\cal CP}$ hypothesis to the $t\bar{t}H$ yields at the
parton-level. The Higgs boson and top/anti-top quark transverse momenta are
smeared using a Gaussian distribution with a width of 80 and
$70\,\,\mathrm{GeV}$, respectively, while the pseudo-rapidity of the top/anti-
top quark is smeared by $0.8$ and the azimuth angle by $20^{\circ}$. The
smearing parameters were tuned to match the Higgs boson candidate mass
distribution shown in HIGG-2020-23 and the top reconstruction performance
described in [95, 96].

Finally, the $t\bar{t}H$(multilep.) channel lies in between the
$t\bar{t}H(\to\gamma\gamma)$ and $t\bar{t}H(\to bb)$ channels in terms of
cross-section times branching ratio and signal-over-background ratio. A large
number of combinations of top, anti-top quarks and Higgs boson decays lead to
multi-lepton final states. In this work, we adopt a simplified approach by
considering only the two most sensitive experimental sub-channels, which
include the final states with two electrons or muons with the same sign or at
least three electrons or muons. The resulting branching ratio is $6.79\cdot
10^{-2}$, to which we apply an additional acceptance and efficiency factor
ranging from 0.036 to 0.042 depending on the ${\cal CP}$ hypothesis (based on
CMS:2022dbt). The smearing parameters are the same as the ones considered for
the $t\bar{t}H(\to bb)$ channel, except for the Higgs $p_{\text{T}}$ which is
smeared by a larger value of $120\,\,\mathrm{GeV}$ instead of
$80\,\,\mathrm{GeV}$. Indeed, despite the lack of published experimental data,
we anticipate lower performance in the Higgs boson reconstruction due to the
multiple high-$p_{\text{T}}$ neutrinos in these events, which can typically
originate from either the Higgs or the top/anti-top decays.

| $t\bar{t}H(\to\gamma\gamma)$ | $t\bar{t}H(\mathrm{multilep.})$ | $t\bar{t}H(\to b\bar{b})$
---|---|---|---
BR | $2.27\cdot 10^{-3}$ | $6.79\cdot 10^{-2}$ | $5.81\cdot 10^{-1}$
Acceptance/efficiency scaling factors
$\alpha_{t}=0^{\circ}$ | $2.5\cdot 10^{-1}$ | $3.6\cdot 10^{-2}$ | $5.0\cdot 10^{-3}$
$\alpha_{t}=35^{\circ}$ | $2.5\cdot 10^{-1}$ | $3.6\cdot 10^{-2}$ | $5.2\cdot 10^{-3}$
$\alpha_{t}=45^{\circ}$ | $2.7\cdot 10^{-1}$ | $3.8\cdot 10^{-2}$ | $5.4\cdot 10^{-3}$
$\alpha_{t}=90^{\circ}$ | $3.2\cdot 10^{-1}$ | $4.2\cdot 10^{-2}$ | $6.5\cdot 10^{-3}$
Smearing factors
$\Delta p_{T,{H}}$ [GeV] | 4 | 120 | 80
$\Delta p_{T,t}$ [GeV] | 40 | 70 | 70
$\Delta\eta_{t}$ | $0.5$ | $0.8$ | $0.8$
$\Delta\phi_{t}$ [∘] | – | 20 | 20
Final $t\bar{t}H$ event yields at 300 fb-1
$\alpha_{t}=0^{\circ}$ | $86$ | $372$ | $442$
$\alpha_{t}=35^{\circ}$ | $70$ | $302$ | $373$
$\alpha_{t}=45^{\circ}$ | $67$ | $281$ | $341$
$\alpha_{t}=90^{\circ}$ | $47$ | $185$ | $245$
Table 2: Summary of the input parameters to the simplified model used to
emulate the detector effects and the analysis selection efficiency in each
experimental channel. The first block (from top to bottom) shows the scaling
factors accounting for the branching ratio, the second block shows the scaling
factors accounting for the limited acceptance of the detector and the limited
efficiency of the analysis selection, while the third and fourth blocks show
the Gaussian smearing factors applied on the Higgs boson and top/anti-top
momenta and the final $t\bar{t}H$ event yields obtained for an integrated
luminosity of $300\,\,\mathrm{fb}^{-1}$, respectively. As a reminder, all
cross-sections are obtained directly from MadGraph5_aMC@NLO to which a
k-factor of $1.14$ is applied to correct for NLO effects.

###  3.2 Significance computation

In order to quantify the sensitivity of the various observables discussed in
Section 2 to ${\cal CP}$ violation in the top-Yukawa coupling, we assume to
have at our disposal the measurement of the corresponding $t\bar{t}H$
distributions in each experimental channel. The scaling and smearing factors
described in the previous section are applied to $t\bar{t}H$ events generated
at the parton level to emulate detector effects and obtain realistic yields.

The sensitivity of an observable to a given BSM model is evaluated by
computing a significance $S$. This significance reflects the discriminating
power to reject the BSM hypothesis (parameterized by $g_{t},\alpha_{t}$)
against the null hypothesis of $g_{t}=1$, $\alpha_{t}=0$ (corresponding to the
SM). Formally, it is defined as the sum in quadrature of the significance
computed in each bin of the observable, $(S_{i})_{i=1..N_{\text{bins}}}$. The
observed $t\bar{t}H$ event yields are assumed to match the central SM
predictions. If we make the additional assumption that they are Poisson
distributed and not impacted by additional sources of uncertainty, the $S_{i}$
can then be evaluated using the Wilk’s theorem:

$\displaystyle
S_{i}=\sqrt{-2\left(n_{i}\ln\frac{m_{i}}{n_{i}}+n_{i}-m_{i}\right)}\,,$ (3)

where $n_{i}$ and $m_{i}$ are the numbers of events in the $i$th bin of the
observable as predicted by the SM and the BSM model under consideration,
respectively. Accordingly, the significance $S$ results as

$\displaystyle
S=\sqrt{\sum_{i=1}^{N_{\text{bins}}}S_{i}^{2}}=\sqrt{-2\sum_{i=1}^{N_{\text{bins}}}\left(n_{i}\ln\frac{m_{i}}{n_{i}}+n_{i}-m_{i}\right)}\,.$
(4)

In the following, all bins with $n_{i}<2$ are merged with their neighbouring
bins until the merged bin contains more than two events, starting with left
neighbours, to ensure the validity of Wilk’s theorem for significance
evaluations. We checked that the signifiance values are stable within a few
percent if $n_{i}<5$ is used instead.

The previous equations rely on two important assumptions, first that the
measured $t\bar{t}H$ event yields are Poisson distributed and second that they
are not impacted by additional sources of uncertainty. In reality,
non-$t\bar{t}H$ events are expected to enter the measurement selection such
that

$\displaystyle n_{i}=n^{\mathrm{data}}_{i}-n^{\mathrm{bkg}}_{i},$ (5)

where $n^{\mathrm{data}}_{i}$ and $n^{\mathrm{bkg}}_{i}$ are the number of
observed data events and the number of predicted non-$t\bar{t}H$ events in the
$i$th bin, respectively. The evaluation of $n^{\mathrm{bkg}}_{i}$ is part of
the experimental analysis work. It is impacted by systematic uncertainties
related to the detector calibration and physics modelling. Therefore the
assumptions used to derive Eqs. 3 and 4 are valid only in the limit of high
signal over background ratio ($S/B$), i.e. $n_{i}/n^{\mathrm{bkg}}_{i}\gg 1$.
To take account of these deviations from our initial assumptions, the Eqs. 3
and 4 are corrected in the following way [97]:

$\displaystyle S_{i}$
$\displaystyle=\sqrt{-2\bigg{(}n_{i}\ln\frac{m_{i}(n_{i}+\sigma_{i}^{2})}{n_{i}^{2}+m_{i}\sigma_{i}^{2}}-\frac{n_{i}^{2}}{\sigma_{i}^{2}}\ln\bigg{[}1+\frac{\sigma_{i}^{2}(m_{i}-n_{i})}{n_{i}(n_{i}+\sigma_{i}^{2})}\bigg{]}\bigg{)}},$
(6) $\displaystyle S$
$\displaystyle=\sqrt{-2\sum_{i=1}^{N_{\text{bins}}}\bigg{(}n_{i}\ln\bigg{[}\frac{m_{i}(n_{i}+\sigma_{i}^{2})}{n_{i}^{2}+m_{i}\sigma_{i}^{2}}\bigg{]}-\frac{n_{i}^{2}}{\sigma_{i}^{2}}\ln\bigg{[}1+\frac{\sigma_{i}^{2}(m_{i}-n_{i})}{n_{i}(n_{i}+\sigma_{i}^{2})}\bigg{]}\bigg{)}},$
(7)

where $\sigma_{i}$ is a newly introduced term corresponding to an extra
uncertainty in $n_{i}$. It is computed as the sum in quadrature of two terms,

$\displaystyle\sigma_{i}^{2}=(\sigma_{i}^{\mathrm{stat}})^{2}+(\sigma_{i}^{\mathrm{syst}})^{2}.$
(8)

The first term, $\sigma_{i}^{\mathrm{stat}}$, accounts for the additional
Poisson-like uncertainty originating from the fact that
$n^{\mathrm{bkg}}_{i}\neq 0$ and can be approximated by
$\sqrt{n^{\mathrm{bkg}}_{i}}$. The
$(n^{\mathrm{bkg}}_{i})_{i=1..N_{\text{bins}}}$ largely depend on the analysis
design and the observable considered and therefore are in general not trivial
to evaluate in the context of this work. Though we note that they are
connected to the $S/B$ in each bin such that
$\sigma_{i}^{\mathrm{stat}}~{}=~{}\sqrt{\frac{n_{i}}{S/B}}$. The $S/B$
typically varies bin-by-bin in the experimental data analyses, however, a
single value per channel is considered here for simplification, taken from the
typical values observed in the $t\bar{t}H$-enriched regions defined in the
corresponding ATLAS and CMS analyses [23, 25, 20]. $S/B=1$, 0.4, and 0.1 is
assumed in the $t\bar{t}H(\to\gamma\gamma)$, $t\bar{t}H$(multilep.), and
$t\bar{t}H(\to bb)$ channels, respectively. The validity of this simplifying
assumption depends on the shape differences between signal and background for
the observables under consideration, which will be discussed further in
Section 5.1.

The second term, $\sigma_{i}^{\mathrm{syst}}$, accounts for the systematic
uncertainty in $n^{\mathrm{bkg}}_{i}$. We define it so that it contains the
uncertainty in the shape of the non-$t\bar{t}H$ events. We take
$\sigma_{i}^{\mathrm{syst}}=n_{i}~{}\cdot 0.2$ $[0.5]$ for the
$t\bar{t}H$(multilep.) [$t\bar{t}H(\to bb)$] channel, following
CMS:2022dbt,ATLAS:2023cbt, while $\sigma_{i}^{\mathrm{syst}}=0$ is assumed for
the $t\bar{t}H(\to\gamma\gamma)$ channel as this channel is typically
dominated by statistical uncertainties [20]. The last source of uncertainty to
take into account is the uncertainty in the overall background normalisation.
It is significant in the $t\bar{t}H(\to bb)$ and $t\bar{t}H$(multilep.)
channels only. For these two channels, we decided to account for it by
rescaling the total BSM $t\bar{t}H$ event yield to match the total SM
$t\bar{t}H$ event yields prior to the computation of the significance. This
way no discrimination power is obtained from differences in the total rate
between SM and BSM in these two channels. In practice, the $m_{i}$ are
replaced in Eqs. 6 and 7 by $m^{\prime}_{i}$, which corresponds to the number
of events in the $i$th bin of the observable as predicted by the BSM model
under consideration rescaled so that
$\sum_{i=1}^{N_{\text{bins}}}m^{\prime}_{i}=\sum_{i=1}^{N_{\text{bins}}}n_{i}$.

##  4 Results

We present results for a luminosity of $300\,\,\mathrm{fb}^{-1}$, targeting
the end of LHC Run-3. Since the current experimental limits exclude values of
$g_{t}=1,~{}\alpha_{t}\gtrsim 43^{\circ}$ at the $95\%~{}\mathrm{CL}$ using
$139\,\,\mathrm{fb}^{-1}$ [20], we choose $g_{t}=1,~{}\alpha_{t}=35^{\circ}$
as a benchmark. In Section 4.1, we calculate the significance for all the
observables mentioned in Section 2.2 as well as for the associated two-
dimensional observable combinations. This is done with a fixed number of
evenly spaced bins. In Section 4.2, we optimize the binning individually for
the most promising variables based on the combined significance of the three
experimental channels. We compare our findings to a multivariate approach in
Section 4.3. Finally, we show in Section 4.4 the expected sensitivity in the
full two-dimensional $(g_{t},\alpha_{t})$ parameter plane.

###  4.1 Results with default binning

We first evaluate the sensitivity of each observable separately starting from
a default binning of $14$ evenly distributed bins. In addition, we evaluate
the sensitivity of the combination of two observables starting from a two-
dimensional binning of $6\times 6$ evenly distributed bins. These (not yet
optimised) binnings were chosen to maximise the number of bins to avoid losing
valuable kinematic shape information while populating most of the bins with at
least a few events. The bin merging method described in the previous section
is then applied to guarantee at least two SM $t\bar{t}H$ events per bin. In
total, $31$ different observables (and their combinations) are considered
across the four different rest frames defined in Section 2.2. While the
results for all 465 different combinations can be found in Appendix B, we
focus in the following on the most promising observables and related
combinations.

The sensitivity of $p_{\mathrm{T},H}$ and $\Delta\phi_{t\bar{t}}$ in the
laboratory frame to a BSM signal corresponding to
$g_{t}=1,~{}\alpha_{t}=35^{\circ}$ is shown in Fig. 5 for an integrated
luminosity of 300 fb-1, separately (emphasized column) and in combination with
a second observable (other columns). The significance in each channel as well
as the combined significance are shown. The $p_{\mathrm{T},H}$ observable was
chosen as it shows high sensitivity and corresponds to the observable used in
the STXS binning v1.2 [77, 78, 79, 80], whereas $\Delta\phi_{t\bar{t}}$
provides the best significance when combined with other observables.

The sensitivity of the $t\bar{t}H(\to\gamma\gamma)$ channel largely comes from
the difference in terms of total event rate between the SM and BSM hypotheses,
and therefore does not depend strongly on the choice of observable. On the
other hand, the other channels suffer from a larger background and a less
precise event reconstruction, such that no discrimination from the total event
rate was considered (see Section 3.2). Hence, the choice of the best
observable for the combined three channels is mostly driven by the expected
performance in the $t\bar{t}H(\to bb)$ and $t\bar{t}H$(multilep.) channels.
The highest significance is obtained when combining $\Delta\phi_{t\bar{t}}$
with $b_{4}$ in the lab frame and reaches $S=1.91$. The best combinations
involving $p_{T,{H}}$ reach significances ranging from $1.82$ to $1.87$, and
are obtained with $\Delta\phi_{t\bar{t}}$, $b_{1}$, and $b_{2}$ in the lab
frame, as well as $\Delta\eta_{t\bar{t}}$, $|\cos\theta^{*}|$, and $b_{2}$ in
the $t\bar{t}$ frame.

While $p_{T,{H}}$ alone yields a significance considerably weaker than the
best combinations, the difference between the 1D $\Delta\phi_{t\bar{t}}$
result, the best 2D combinations involving $p_{T,{H}}$ and any 2D combination
involving $\Delta\phi_{t\bar{t}}$ is smaller. This result brings some
flexibility in the final choice of observables. Since the $t\bar{t}H$ process
is split into $p_{T,{H}}$ bins in the current STXS binning, we favour for
practical reasons a reduced set of observables which reach the highest
significances when paired with $p_{T,{H}}$. For comparison, we also show in
Fig. 5 the best combinations involving $\Delta\phi_{t\bar{t}}$. As we will
see, given the small difference to the combinations with
$\Delta\phi_{t\bar{t}}$, we understand our chosen combinations with
$p_{T,{H}}$ as close to optimal.

Figure 5: Significances evaluated from the (top) $p_{\mathrm{T},H}$ and
(bottom) $\Delta\phi_{t\bar{t}}$ distributions in the laboratory frame for a
BSM signal corresponding to $g_{t}=1,~{}\alpha_{t}=35^{\circ}$. The emphasised
column shows the significance obtained from the one-dimensional distribution
while the others show the significances obtained when combining with a second
observable. The significances are shown per experimental channel and after
their statistical combinations.

###  4.2 Final results with optimised binning

We now optimize the binning of the observables preselected in the previous
section for an integrated luminosity of $300\,\,\mathrm{fb}^{-1}$. In
combination with $p_{T,{H}}$, the $b_{2}$ variable yields the same
significance in the lab and $t\bar{t}$ frame. Taking into account that
$p_{T,{H}}$ is given in the lab frame, we will consider the $b_{2}$ observable
only in the lab frame in the following. For practical reasons, we follow the
STXS v1.2 framework for the Higgs transverse momentum, which specifies the
following six bins:

  * •

$p_{T,{H}}$: $[0,60,120,200,300,450,+\infty]$ (GeV)

For the remaining five observables, i.e. $\Delta\phi_{t\bar{t}}$, $b_{1}$, and
$b_{2}$ in the lab frame, as well as $\Delta\eta_{t\bar{t}}$ and
$|\cos\theta^{*}|$ in the $t\bar{t}$ frame, we define six bins as well such
that the less populated regions in the tails of the distributions get wider
bins555This adjustment of the bins is what we refer to as optimised binning
without strictly employing an optimization algorithm.:

  * •

$\Delta\phi_{t\bar{t}}^{\mathrm{lab}}$: [0, $\pi/4$, $\pi/2$, $2\pi/3$,
$5\pi/6$, $11\pi/12$, $\pi$] (rad.),

  * •

$b_{1}^{\mathrm{lab}}$: [-1, -0.95, -0.8, -0.2, 0.3, 0.8, 1.0],

  * •

$b_{2}^{\mathrm{lab}}$: [-1, -0.6, -0.4, -0.2, 0., 0.3, 1.0],

  * •

$\Delta\eta_{t\bar{t}}^{t\bar{t}}$: [0, 0.5, 1, 1.5, 2, 3, 5],

  * •

$|\cos\theta^{*}|$: [0, 0.2, 0.4, 0.55, 0.7, 0.85, 1].

The normalized distributions of these observables for $\alpha_{t}=0^{\circ}$
and $35^{\circ}$ are shown in Figs. 6 and 7. The distributions are shown for
each observable at the parton level and for the various experimental channels
including the smearing factors discussed in Section 3.1. The significances
obtained for each preselected observable in a two-dimensional combination with
$p_{T,{H}}$ using the optimized binning are shown in Fig. 8. The same table
considering combinations with $\Delta\phi_{t\bar{t}}$ is also displayed.

We observe that the optimised binning using just 6 bins almost reproduces the
significance based on the default binning after bin merging in the case of the
1D distributions of either $p_{T,{H}}$ or $\Delta\phi_{t\bar{t}}$. For the
case of the 2D distributions, the optimised binning yields a very similar
performance as the default binning, with an improvement typically at the few-
percent level. The similarity of the significances obtained in the default and
the optimised binning also indicates a robustness of the method of 2D
distributions against the choice of the binning as long as one makes sure that
each bin is populated by at least a few events. Therefore we consider the
optimised binning as validated and will from now on refer to significances
computed from these exclusively.

Figure 6: Distributions (from top to bottom) of $p_{T,{H}}$,
$\Delta\phi_{t\bar{t}}$, $b_{1}$ and $b_{2}$ in the lab frame with optimised
binning. Each canvas is split into four panels, from left to right the
following distributions are shown: parton-level and smeared distributions for
the $t\bar{t}H(\to\gamma\gamma)$, $t\bar{t}H(\mathrm{multilep.})$, and
$t\bar{t}H(\to b\bar{b})$ channels. The SM distributions are represented as
solid black lines, while the distributions for ${\alpha_{t}=35^{\circ}}$ are
shown as dashed orange lines.

Figure 7: Distributions of (top) $\Delta\eta_{t\bar{t}}$ in the $t\bar{t}$
frame and (bottom) $|\cos\theta^{*}|$ with optimized binning. Each canvas is
split into four panels, from left to right the following distributions are
shown: parton level and smeared distributions for the
$t\bar{t}H(\to\gamma\gamma)$, $t\bar{t}H(\mathrm{multilep.})$, and
$t\bar{t}H(\to b\bar{b})$ channels, respectively. The SM distributions are
represented as solid black lines, while the distributions for
${\alpha_{t}=35^{\circ}}$ are shown as dashed orange lines.

Figure 8: Significances evaluated from the (left) $p_{\mathrm{T},H}$ and
(right) $\Delta\phi_{t\bar{t}}$ distributions in the lab frame for a BSM
signal corresponding to $g_{t}=1,~{}\alpha_{t}=35^{\circ}$. The emphasised
column shows the significance obtained from the one-dimensional distribution
while the others show the significance obtained when combining with a second
observable. All observables are considered in their optimised binning. The
significances are shown per experimental channel and after their statistical
combinations.

###  4.3 Comparison to a multivariate analysis

In this section, we compare the significance obtained from the two-dimensional
distributions reported in the previous section to the one obtained from the
output of a multivariate analysis algorithm trained on the full set of
observables666We checked that providing additional low-level input variables
leads to a negligible performance improvement (below the permille level).
introduced in Section 2. We assume that such a multivariate analysis would be
able to exploit the kinematic information of the observables (almost) fully
and so we use this comparison to evaluate the potential sensitivity drop due
to the practical usage of only two observables against the reference of the
more complete multivariate analysis.

We implement a Boosted Decision Tree (BDT) based on the XGBoost algorithm
[98]. The goal of the BDT is to distinguish a signal characterized by
$\alpha_{t}\neq 0$ from the $\alpha_{t}=0$ hypothesis. Separate trainings are
performed for the three $t\bar{t}H$ channels. The scaling factors and
smearings described in Table 2 are applied to $t\bar{t}H$ events generated at
the parton level to emulate detector effects and obtain realistic yields in
each channel. The SM $t\bar{t}H$ sample is used as ‘background’ and the
$g_{t}=1,~{}\alpha_{t}=90^{\circ}$ sample is used as ‘signal’ in the training
as this choice leads to the best performance in the full parameter space
[99].777All input variables are ${\cal CP}$-even such that the BDT is not
sensitive to the interference term. To leverage the statistics of the entire
dataset, we employ a $k$-fold strategy [100] with $k=5$. 80% of the dataset is
dedicated to training, while the remaining 20% serves as testing subset. This
operation is repeated five times to test the full dataset. The performance of
the BDT is evaluated by computing the significance from the BDT output score
based on Eqs. 6 and 7.

The output BDT score distributions for the SM scenario and the
$g_{t}=1,~{}\alpha_{t}=35^{\circ}$ signal sample in each $t\bar{t}H$ channel
are shown in Fig. 9. The bin merging method described in Section 3.2 is
applied to guarantee at least two SM $t\bar{t}H$ events per bin. The
significance values computed from these distributions are summarized and
compared to the best significance values obtained from two-dimensional
distributions in Table 3. As expected, the significances obtained from the BDT
are higher, but only to a limited extent ($\sim 10\%$ for $H\to\gamma\gamma$,
$\sim 17\%$ for $H\to\text{multilep.}$ and $\sim 25\%$ for $H\to b\bar{b}$),
with a combined significance reaching $2.21$ compared to $1.91$ for the two-
dimensional method. The improvement in sensitivity of the BDT is larger for
the $t\bar{t}H(\mathrm{multilep.})$ and $t\bar{t}H(\to b\bar{b})$ channels, as
these channels select more $t\bar{t}H$ events (see Table 2) and therefore
allow better use of kinematic shape information. We conclude that the approach
taken so far consisting of using two-dimensional distributions to probe ${\cal
CP}$ violation in the top-Yukawa coupling is sufficient to reach a close-to-
optimal sensitivity with 300 fb-1 of data, especially since the combined
sensitivity is driven by statistically limited $H\to\gamma\gamma$ channel.

Figure 9: Distributions of the output BDT score for the (left) $t\bar{t}H(\to\gamma\gamma)$, (center) $t\bar{t}H$(multilep.), and (right) $t\bar{t}H(\to bb)$ channels, evaluated for the (black) SM and (orange) $g_{t}=1,~{}\alpha_{t}=35^{\circ}$ sample. Channel | Significance (BDT) | Significance (2D)
---|---|---
$t\bar{t}H(\to\gamma\gamma)$ | 1.75 | 1.57
$t\bar{t}H$(multilep.) | 1.17 | 0.94
$t\bar{t}H(\to bb)$ | 0.69 | 0.55
Combined | 2.21 | 1.91
Table 3: Significance obtained from the output BDT score distributions and
best significance obtained from two-dimensional distributions in Section 4.2.

###  4.4 Expected exclusion limits

Only results for $\alpha_{t}=35^{\circ}$ and $g_{t}=1$ have been presented so
far. In this section, we show and discuss the expected exclusion limits at the
95% confidence-level (using the Gaussian limit, i.e. $S=1.96$) in the full
$(\alpha_{t},g_{t})$ plane for an integrated luminosity of
$300\,\,\mathrm{fb}^{-1}$. The significance at any given point of the
parameter space is computed based on the distribution of the observables under
consideration for both the SM and BSM cases, where the BSM $t\bar{t}H$ yields
are obtained from Eq. 2. The resulting two-dimensional contours are shown for
each $t\bar{t}H$ channel as well as their statistical combination in Fig. 10.
The contours for the $t\bar{t}H$(multilep.) and $t\bar{t}H(\to bb)$ channels
are not closed at high values of $g_{t}$, which is due to the fact that any
discrimination power arising from differences in the total rate between SM and
BSM is neglected in these two channels (see Section 3.2).

In the current STXS $p_{T,{H}}$ setup (i.e., $p_{T,{H}}$ as the single
discriminating variable with 6 bins), the discriminating power between SM and
BSM hypotheses in the $t\bar{t}H(\to\gamma\gamma)$ channel originates mainly
from differences in the total event rate (see Section 4.1). This channel
almost entirely drives the sensitivity and allows to constrain
$|\alpha_{t}|\lesssim 43^{\circ}$ for $g_{t}=1$, while the combination of the
three channels improves the constraint marginally to $|\alpha_{t}|\lesssim
41^{\circ}$ for $g_{t}=1$. In this scenario, the $t\bar{t}H(\to bb)$ channel
contribution to the combined sensitivity in the entire parameter space is
rather small. The $t\bar{t}H$(multilep.) channel contributes to the combined
sensitivity more strongly for $g_{t}>1$, where higher $t\bar{t}H$ yields are
expected such that a higher discrimination power from the $p_{T,{H}}$
kinematic shape is obtained. In this region, an improvement up to $36$% is
observed in the combined limit on $|\alpha_{t}|$ due to
$t\bar{t}H$(multilep.), obtained for $g_{t}=1.27$.

If the current STXS $p_{T,{H}}$ setup is extended by splitting further
$t\bar{t}H$ events according to their $|\cos\theta^{*}|$ value, the combined
limit at $g_{t}=1$ is improved by 12% to reach $|\alpha_{t}|\lesssim
36^{\circ}$. The best improvement of the combined limit on $|\alpha_{t}|$ over
the $t\bar{t}H(\to\gamma\gamma)$ standalone limit is $40$% and is reached at
$g_{t}=1.24$. The improvement originates mainly from the improved sensitivity
of the $t\bar{t}H$(multilep.) channel, although the sensitivity of the
$t\bar{t}H(\to bb)$ channel is also significantly improved and contributes to
the combination. In this scenario, a higher sensitivity from the
$t\bar{t}H$(multilep.) is expected with respect to the
$t\bar{t}H(\to\gamma\gamma)$ channel for $1.10<g_{t}<1.31$. We obtain
comparable results when combining $p_{T}^{H}$ with one of the other promising
candidates identified in the previous sections, i.e. $\Delta\phi_{t\bar{t}}$,
$b_{1}$ or $b_{2}$ in the lab frame.

Fig. 10 also shows the exclusion contours for $\Delta\phi_{t\bar{t}}$ instead
of $p_{T,{H}}$ as single variable, as well as for the two-dimensional binning
of $\Delta\phi_{t\bar{t}}$ and $|\cos\theta^{*}|$, which was among the best
combinations not involving $p_{T,{H}}$ identified in Section 4.1. As expected
from Fig. 8, the $\Delta\phi_{t\bar{t}}$ observable alone performs better than
$p_{T,{H}}$ alone mainly due to its higher level of kinematic shape
discrimination, which leads to an increased sensitivity in the
$t\bar{t}H$(multilep.) and $t\bar{t}H(\to bb)$ channels. Regarding the two-
dimensional observable combination $(\Delta\phi_{t\bar{t}},|\cos\theta^{*}|$),
the picture is qualitatively very similar to the
($p_{T,{H}}$,$|\cos\theta^{*}|$) case and compatible limits are found after
statistical combinations of the three channels ($|\alpha_{t}|\lesssim
36^{\circ}$ for $g_{t}=1$).

Finally we show in Fig. 10 the exclusion contours resulting from the
significance computed from the output BDT score described in Section 4.3. We
assume the BDT to close to fully exploit the kinematic information contained
in the set of observables of Table 1. Still, only a slight improvement over
the limits obtained from simple two-dimensional variable combinations is
observed overall. In particular, the expected limit of $|\alpha_{t}|\lesssim
33^{\circ}$ for $g_{t}=1$ from the BDT is compatible with the corresponding
result obtained from the ($p_{T,{H}}$,$|\cos\theta^{*}|$) distribution.

To summarize, we found comparable performance between the best
$p_{T,{H}}$-based and $\Delta\phi_{t\bar{t}}$-based two-dimensional variable
combinations and the output from a multivariate analysis based on the full set
of observables shown in Table 1.

Figure 10: Expected limits at the 95% confidence level in the
$(\alpha_{t},g_{t})$ plane for ${\mathcal{L}=300\,\,\mathrm{fb}^{-1}}$ when
considering (upper left) the current $p_{T,{H}}$ STXS setup and (upper right)
the $\Delta\phi_{t\bar{t}}^{\text{lab}}$ observable alone. On the second row,
the results from the two-dimensional observables
$(p_{T,{H}},|\cos\theta^{*}|)$ and
$(\Delta\phi_{t\bar{t}}^{\text{lab}},|\cos\theta^{*}|)$ are shown. The
expected limits obtained from the BDT output score described in Section 4.3
are also shown for comparison.

##  5 Extending the STXS framework

The additional data from the ongoing Run 3 of the LHC is expected to increase
the size of the proton-proton dataset of the ATLAS and CMS collaborations from
$140\,\,\mathrm{fb}^{-1}$ to about $300\,\,\mathrm{fb}^{-1}$. The increased
statistics will open the possibility to extract more precise information from
the $t\bar{t}H$ kinematic shapes. As of now, the sensitivity to the ${\cal
CP}$ nature of the top-Yukawa coupling is largely driven by the
$t\bar{t}H(\to\gamma\gamma)$ channel. However, based on our results, we expect
a significant increase in sensitivity of the $t\bar{t}H$(multilep.) and
$t\bar{t}H(\to bb)$ channels by the end of Run 3. If confirmed, the
statistical combination of the various $t\bar{t}H$ channels will become key to
constrain further the ${\cal CP}$ nature of the top-Yukawa coupling over the
next few years. We propose in Section 5.1 a qualitative discussion regarding
the validity of our background assumptions used to evaluate the significance
for the best candidate observables. This allows us to reduce further our list
of candidates and to formulate our final proposal for a ${\cal CP}$-sensitive
STXS extension in Section 5.2.

###  5.1 Further background considerations

The significance computation used to evaluate the sensitivity of each
observable combination relies on some assumptions which were discussed in
Section 3.2. In particular, a constant signal ($t\bar{t}H$) over background
(non-$t\bar{t}H$) ratio is assumed throughout all the bins considered in our
study and $S/B=1$, $0.4$, and $0.1$ are assumed in the
$t\bar{t}H(\to\gamma\gamma)$, $t\bar{t}H$(multilep.), and $t\bar{t}H(\to bb)$
channels, respectively. These values are based on typical values observed in
the corresponding ATLAS and CMS analyses. Qualitatively, this assumption
should be conservative under the condition that the background shape is not
peaking around the highest significance bins, which we propose to verify here.

#### Higgs background processes

We start with a short discussion of Higgs background processes, i.e. mainly
$tHq$ and $tWH$ events in which a second top quark could be wrongly
reconstructed. In the SM, the cross-sections of the $tHq$ and $tWH$ processes
are comparably low with respect to $t\bar{t}H$, with
$\sigma_{{\text{SM}}}^{tHq}/\sigma_{{\text{SM}}}^{t\bar{t}H}\approx 0.15$ and
$\sigma_{{\text{SM}}}^{tWH}/\sigma_{{\text{SM}}}^{t\bar{t}H}\approx 0.03$.
Both Higgs background processes become more important for higher fractions of
${\cal CP}$-odd top-Yukawa coupling, however, at
$(g_{t},\alpha_{t})=(1,35^{\circ})$, we still find
$\sigma^{tHq}/\sigma^{t\bar{t}H}\approx 0.24$ and
$\sigma^{tWH}/\sigma^{t\bar{t}H}\approx 0.05$. Given the low cross-sections in
the parameter space of interest and the additional yield reduction expected
from the analysis selection, we conclude that the $tHq$ and $tWH$ processes
can only play a minor role in the measurement of the $t\bar{t}H$ kinematic
distributions and therefore satisfy the assumptions made in Section 3.2.

#### Non-Higgs background processes

We discuss here the most relevant non-Higgs background process in each
channel: $t\bar{t}\gamma\gamma$ for the $H\to\gamma\gamma$ channel,
$t\bar{t}W$ for the multi-lepton channel, and $t\bar{t}b\bar{b}$ for the $H\to
b\bar{b}$ channel, for which samples were generated at the parton-level based
on the setup described in Section 2.1. For each background process, we assume
that the non-top-quark particles are misidentified as a Higgs boson to compute
the observables introduced in Section 2. In practice, we implement this by
setting $p_{H}=p_{\gamma\gamma}$ for $t\bar{t}\gamma\gamma$, $p_{H}=p_{W}$ for
$t\bar{t}W$, and $p_{H}=p_{b\bar{b}}$ for $t\bar{t}b\bar{b}$. Following these
assignments, we show in Fig. 11 the $t\bar{t}\gamma\gamma$, $t\bar{t}W$ and
$t\bar{t}b\bar{b}$ distributions for the various observables selected in
Section 4. We also show in the same Figure the combined significance obtained
in each bin for the $g_{t}=1$, $\alpha_{t}=35^{\circ}$ sample. This gives us a
qualitative estimate of whether our $S/B$ assumptions about the background
uncertainties from Section 3.2 are conservative or not. If the non-Higgs
backgrounds for a specific variable peaks in the bin which mainly drives the
sensitivity, we consider this variable’s sensitivity to be possibly inflated
in Section 4 and discard it in the final proposal.

The $\Delta\phi_{t\bar{t}}$ and $b_{1}$ distributions in the lab frame show
the same behaviour, with a clear peak at one end of the distribution in both
the background distributions and the significance. The peak is particularly
pronounced for the $t\bar{t}\gamma\gamma$ background and has the same location
as the peak in the significance. Consequently, the sensitivity of these
observables may be overestimated in our results shown in Section 4 and they
are excluded from the final shortlist. The distribution of $b_{2}$ in the lab
frame has a peak of the background in the third bin, which is, however, not
the bin dominating the total significance. For the $\Delta\eta_{t\bar{t}}$ and
$|\cos\theta^{*}|$ distributions in the $t\bar{t}$ frame, the sensitivity as
well as the background distributions are relatively flat across the full
range. Therefore we expect our sensitivity estimates to be more robust for
these three last observables, which are the final ones selected.

Figure 11: Distributions of the (top, left) azimuthal angle difference between
the $t\bar{t}$ pair, (top, right) $b_{1}$ observable, and (center, left)
$b_{2}$ observable in the lab frame, and distribution of the (center, right)
pseudorapidity difference between the $t\bar{t}$ pair and (bottom)
$|\cos\theta^{*}|$ observable in the $t\bar{t}$ frame. The distributions for
SM $t\bar{t}\gamma\gamma$, $t\bar{t}W$ and $t\bar{t}b\bar{b}$ events are shown
in each upper panel plot. The observables are reconstructed by setting
$p_{H}=p_{\gamma\gamma}$ for $t\bar{t}\gamma\gamma$, $p_{H}=p_{W}$ for
$t\bar{t}W$, and $p_{H}=p_{b\bar{b}}$ for $t\bar{t}b\bar{b}$. The events are
generated at the parton-level following the setup described in Section 2.1.
All distributions are normalised to unity. We also show in each case the per-
bin combined significance obtained for the $t\bar{t}H$ sample at $g_{t}=1$,
$\alpha_{t}=35^{\circ}$ in a lower panel plot.

###  5.2 Proposal for $t\bar{t}H$ STXS extension

The findings of our analysis are the following, considering as a benchmark the
$g_{t}=1$, $\alpha_{t}=35^{\circ}$ sample with 300 fb-1 of data:

  * •

The combination of $p_{T,{H}}$ with $\Delta\phi_{t\bar{t}}^{\text{lab}}$,
$b_{1}^{\text{lab}}$, $b_{2}^{\text{lab}}$,
$\Delta\eta_{t\bar{t}}^{t\bar{t}}$, or $|\cos\theta^{*}|$ feature compatible
sensitivity within $5$% of the best combination. We favour combinations with
$p_{T,{H}}$ due to the existing STXS binning.

  * •

When using the existing STXS binning in $p_{T,{H}}$ (6 bins) adding any second
observable composed of 6 bins with optimized boundaries, the two-dimensional
combinations including $p_{T,{H}}$ listed above yield compatible sensitivity.

  * •

If we compare these results with the sensitivity that we obtain from a
multivariate analysis trained on all the observables introduced in Section 2,
a limited gain of $16$% is found when using the multivariate analysis (after
combining all channels).

  * •

A qualitative comparison of the main non-Higgs background shapes with respect
to the significance in each of the tested bins shows that
$\Delta\phi_{t\bar{t}}^{\text{lab}}$ and $b_{1}^{\text{lab}}$ are disfavored.
This leads to the exclusion of the respective combinations with $p_{T,{H}}$.

  * •

The final shortlist of candidate observables to be combined with $p_{T,{H}}$
include $b_{2}^{\text{lab}}$, $\Delta\eta_{t\bar{t}}^{t\bar{t}}$, and
$|\cos\theta^{*}|$. Our analysis provides no support to favour one over
another.

Based on these findings, our final proposal is to extend the current STXS
binning by a second dimension that could either be
$b_{2}^{\mathrm{lab}}$,$\Delta\eta_{t\bar{t}}^{t\bar{t}}$, or
$|\cos\theta^{*}|$, as illustrated in Fig. 12.

Figure 12: Proposal for an extension of the current STXS binning for the
$t\bar{t}H$ production mode. Each bin in $p_{T,{H}}$ is further split in bins
of either $b_{2}^{\mathrm{lab}}$$,\Delta\eta_{t\bar{t}}^{t\bar{t}}$, or
$|\cos\theta^{*}|$.

To further support our proposal, we show in Fig. 13 a comparison of the
expected two-dimensional limits in the ($g_{t}$, $\alpha_{t}$) parameter space
that we obtain from the current STXS binning for $t\bar{t}H$ production, a
finer one-dimensional binning including 36 bins, and our proposal taking the
example of $|\cos\theta^{*}|$ as the second observable. This comparison is
shown for $300\,\,\mathrm{fb}^{-1}$ and $3000\,\,\mathrm{fb}^{-1}$ (without
any other modification in the analysis). Although the
$3000\,\,\mathrm{fb}^{-1}$ scenario may be conservative as the moderate
increase of center-of-mass energy from $\sqrt{s}=13\,$TeV to $14\,$TeV,
improved acceptance and potential systematic uncertainty reduction are not
taken into account, it still shows that our proposal allows to improve the
sensitivity of the STXS measurement in the longer term. The sensitivity
improvement from the introduction of a second observable due to the improved
kinematic shape discrimination with higher number of $t\bar{t}H$ events goes
up to +$33$% (+$28$%), reached at $g_{t}=1.24~{}(1.08)$ for
$300\,\,\mathrm{fb}^{-1}$ ($3000\,\,\mathrm{fb}^{-1}$) of data. On the other
hand, the introduction of more $p_{T,{H}}$ bins lead to lower sensitivity
improvements, going up to +$24$% (+$15$%), reached at $g_{t}=1.26~{}(1.10)$
for $300\,\,\mathrm{fb}^{-1}$ ($3000\,\,\mathrm{fb}^{-1}$) of data.888The
sensitivity improvements due to the introduction of more bins are slightly
reduced at $3000\,\,\mathrm{fb}^{-1}$ with respect to
$300\,\,\mathrm{fb}^{-1}$. This effect originates from the fact that the
discrimination power from the differences in the total rate between SM and BSM
is enhanced due to the larger number of $t\bar{t}H(\to\gamma\gamma)$ events,
and therefore it reduces the sensitivity boost due to additional kinematic
shape information.

Figure 13: Constraints in the $(g_{t},\alpha_{t})$ plane for (blue)
$\mathcal{L}=300\,\,\mathrm{fb}^{-1}$ and (red)
${\mathcal{L}=3000\,\,\mathrm{fb}^{-1}}$ at the $95~{}\%~{}\mathrm{CL}$ using
the one-dimensional $p_{T,{H}}$ distribution — evaluated using 6 (dotted line)
and 36 (dashed line) bins — as well as the two-dimensional
$(p_{T,{H}},|\cos\theta^{*}|)$ distributions (solid line, $6\times 6$ bins).
The contour for $\mathcal{L}=3000\,\,\mathrm{fb}^{-1}$ is shown alone on the
right-hand side and together with the $\mathcal{L}=300\,\,\mathrm{fb}^{-1}$
contour on the left-hand side for better comparison.

Our proposal for the extension of the STXS framework assumes a common
definition of the top quark. This definition should guarantee the possibility
to calculate higher-order loop corrections in a well-defined way. A pseudo-
top-quark definition has already been worked out in the context of the LHC Top
Working Group [101]. We leave a study of the applicability of this definition
to the $t\bar{t}H$ process for future work.

##  6 Conclusion

${\cal CP}$ violation in the Higgs sector is a possibility with far-reaching
phenomenological and cosmological consequences such as for electroweak
baryogenesis, which should be thoroughly explored. In this work, we focused on
constraining the ${\cal CP}$ structure of the top-Yukawa coupling via top-
associated Higgs production. While multivariate analysis techniques allow to
best exploit the experimental data, measurements of one- or two-dimensional
contributions allow for a clearer reinterpretation of the data and an easier
combination of different Higgs decay channels and across experiments. With
this motivation, we have studied in this paper which combinations of
observables are most sensitive to the ${\cal CP}$ nature of the top-Yukawa
coupling.

Our study considers eleven observables (in various reference frames) already
identified as being ${\cal CP}$-sensitive in the literature. For the
comparison of their performance, we investigated the three most important
experimental channels measured at the LHC: the channel when the Higgs boson
decays into two photons, the channel where the Higgs boson decays into a
bottom quark and antiquark, and the $t\bar{t}H$ events leading to multi-lepton
final states. For each of these channels, we included detection effects and
selection efficiencies to mimic the data efficiency and resolution from the
experimental analysis results. Based on the resulting distributions, we
evaluated the sensitivity of each (combination of) observable(s) to
distinguish between a mixed ${\cal CP}$ state of the top-Yukawa coupling and
the SM case via the computation of significances based on a benchmark signal
corresponding to $g_{t}=1$ and $\alpha_{t}=35^{\circ}$ and
$300\,\,\mathrm{fb}^{-1}$ of data. The best sensitivity is found for
combinations of observables with the azimuthal angle difference between the
top quarks, $\Delta\phi_{t\bar{t}}$, while combinations with the Higgs
transverse momentum, $p_{T,{H}}$, show comparable performance.

With an extension of the current STXS v1.2 binning in $p_{T,{H}}$ in mind, we
selected five observables which have the highest sensitivity in combination
with $p_{T,{H}}$: $\Delta\phi_{t\bar{t}}$, $b_{1}$ and $b_{2}$ in the
laboratory frame, and $\Delta\eta_{t\bar{t}}$ and $|\cos\theta^{*}|$ in the
$t\bar{t}$ frame (see Section 2 for exact definitions). For these observables,
we optimized the binning for the final sensitivity comparison with six bins
per observable to match the number of bins used in the current STXS
$p_{T,{H}}$ binning. We found that all five observables show very similar
performance when used in combination with $p_{T,{H}}$. As a comparison, we
also trained a boosted decision tree with all variables considered in our
study. Overall, we found that for $300\,\,\mathrm{fb}^{-1}$ of data, the
exclusion limits are only slightly improved with the multivariate approach
over our best two-dimensional combinations of observables.

For the final selection of an observable combination, we also qualitatively
investigated the behaviour of the various non-Higgs backgrounds. For
$\Delta\phi_{t\bar{t}}$ and $b_{1}$, we found enhanced backgrounds in the bins
which are most sensitive to the ${\cal CP}$ nature of the top-Yukawa
interactions, which make these observables less favorable. This is not the
case for $\Delta\eta_{t\bar{t}}$, $|\cos\theta^{*}|$, and $b_{2}$. Their two-
dimensional combinations with $p_{T,{H}}$ constitute our final proposal for a
${\cal CP}$-sensitive STXS extension. Their sensitivities are very similar and
the best choice likely depends on more specific analysis details not
considered in this work. Finally, we compared the expected limits in the
$(g_{t},\alpha_{t})$ plane obtained with the current STXS binning, an extended
one-dimensional $p_{T,{H}}$ binning, and our proposal. Both with
$300\,\,\mathrm{fb}^{-1}$ and $3000\,\,\mathrm{fb}^{-1}$ of data, our proposal
of two-dimensional binning exceeds the sensitivity of the other alternative
STXS binning scenarios.

It is important to note that all the proposed STXS extensions in this work
require the reconstruction of both top quarks per event. Therefore, a thorough
discussion within the community is needed to agree beforehand on a common top-
quark definition.

## Acknowledgements

We thank Sarah Heim and Frank Tackmann for helpful discussions. This work was
supported by ANR PIA funding: ANR-20-IDEES-0002. H.B. acknowledges support
from the Alexander von Humboldt foundation. E.F. and M.M. acknowledge funding
via Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers – 390837967.
E.F. also thanks the CERN TH Department for support during the initial phase
of this project.

##  Appendix A Supplementary parton-level distributions for $t\bar{t}H$

###  A.1 Higgs boson rest frame

The supplementary kinematic distributions for observables defined in the Higgs
boson rest frame are shown in Fig. 14.

Figure 14: Distributions of the observables defined in Section 2 in the Higgs
boson rest frame. $t\bar{t}H$ events generated at parton-level with $g_{t}=1$
and various values of ${\cal CP}$ phase $\alpha_{t}$ are considered following
the event generation described in Section 2.1. All distributions are
normalised to unity.

###  A.2 $t\bar{t}$ rest frame

The supplementary kinematic distributions for observables defined in the
$t\bar{t}$ rest frame are shown in Fig. 15.

Figure 15: Distributions of the observables defined in Section 2 in the
$t\bar{t}$ rest frame. $t\bar{t}H$ events generated at parton-level with
$g_{t}=1$ and various values of ${\cal CP}$ phase $\alpha_{t}$ are considered
following the event generation described in Section 2.1. All distributions are
normalised to unity.

###  A.3 $t\bar{t}H$ rest frame

The supplementary kinematic distributions for observables defined in the
$t\bar{t}H$ rest frame are shown in Fig. 16.

Figure 16: Distributions of the observables defined in Section 2 in the
$t\bar{t}H$ rest frame. $t\bar{t}H$ events generated at parton-level with
$g_{t}=1$ and various values of ${\cal CP}$ phase $\alpha_{t}$ are considered
following the event generation described in Section 2.1. All distributions are
normalised to unity.

##  Appendix B Full two-dimensional significance tables

The full two-dimensional significance tables for the $g_{t}=1$,
$\alpha_{t}=35^{\circ}$ sample are shown in Fig. 17 for the
$t\bar{t}H(\to\gamma\gamma)$ channel, in Fig. 18 for the
$t\bar{t}H$(multilep.) channel, and in Fig. 19 for the $t\bar{t}H(\to bb)$
channel. The combined significances including the three channels are depicted
in Fig. 20.

Figure 17: Significances of the $g_{t}=1$, $\alpha_{t}=35^{\circ}$ signal for
the $t\bar{t}H(\to\gamma\gamma)$ channel. The diagonal matrix elements show
the significances obtained from single observables with default binning, while
the off-diagonal matrix elements show the signifiances obtained from the
combination of two observables. Only half of the histogram is filled to avoid
repeating the same information. Figure 18: Significances of the $g_{t}=1$,
$\alpha_{t}=35^{\circ}$ signal for the $t\bar{t}H$(multilep.) channel. The
diagonal matrix elements show the significances obtained from single
observables with default binning, while the off-diagonal matrix elements show
the signifiances obtained from the combination of two observables. Only half
of the histogram is filled to avoid repeating the same information. Figure 19:
Significances of the $g_{t}=1$, $\alpha_{t}=35^{\circ}$ signal for the
$t\bar{t}H(\to bb)$ channel. The diagonal matrix elements show the
significances obtained from single observables with default binning, while the
off-diagonal matrix elements show the signifiances obtained from the
combination of two observables. Only half of the histogram is filled to avoid
repeating the same information. Figure 20: Combined significances of the
$g_{t}=1$, $\alpha_{t}=35^{\circ}$ signal including the
$t\bar{t}H(\to\gamma\gamma)$, $t\bar{t}H$(multilep.) and ${t\bar{t}H(\to bb)}$
channels. The diagonal matrix elements show the significances obtained from
single observables with default binning, while the off-diagonal matrix
elements show the signifiances obtained from the combination of two
observables. Only half of the histogram is filled to avoid repeating the same
information.

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