# Machine Learning the Higgs-Top CP Phase

Rahool Kumar Barman<EMAIL_ADDRESS>Department of Physics, Oklahoma
State University, Stillwater, OK, 74078, USA  Dorival Gonçalves
<EMAIL_ADDRESS>Department of Physics, Oklahoma State University,
Stillwater, OK, 74078, USA  Felix Kling<EMAIL_ADDRESS>SLAC
National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025,
USA  Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg,
Germany

###### Abstract

We explore the direct Higgs-top CP measurement via the $pp\to t\bar{t}h$
channel at the high-luminosity LHC. We show that a combination of machine
learning techniques and efficient kinematic reconstruction methods can boost
new physics sensitivity, effectively probing the complex $t\bar{t}h$ multi-
particle phase space. Special attention is devoted to top quark polarization
observables, uplifting the analysis from a raw rate to a polarization study.
Through a combination of hadronic, semi-leptonic, and di-leptonic top pair
final states in association with $h\to\gamma\gamma$, we obtain that the HL-LHC
can probe the Higgs-top coupling modifier and CP-phase, respectively, up to
$|\kappa_{t}|\lesssim 8\%$ and $|\alpha|\lesssim 13^{\circ}$ at $68\%$ CL.

††preprint: DESY 21-161

##  I Introduction

New sources of CP violation can be a key ingredient to explain the matter-
antimatter asymmetry of the universe Sakharov:1967dj ; Kajantie:1996mn ;
Huet:1994jb . Hence, the quest for new CP violating interactions is a clear
target for beyond the Standard Model (SM) searches, being a critical component
of the physics program of the LHC. A particularly interesting option is that
the Higgs boson couplings present these new physics sources.

From the theoretical point of view, some Higgs interactions are more inclined
to display CP violation effects than others. While the widely studied beyond
the SM CP structure for the Higgs to vector boson couplings are loop
suppressed, arising only at dimension-6 or higher Buchmuller:1985jz ;
Grzadkowski:2010es , CP violation in Higgs to fermion interactions can
manifest already at the tree-level Buckley:2015vsa , being naturally larger.
Owning to its magnitude, the top quark Yukawa coupling can play a significant
role in this context and be most sensitive to new physics.

Whereas it is possible to access the Higgs-top coupling through loop induced
processes Brod:2013cka ; Dolan:2014upa ; Englert:2012xt ; Kobakhidze:2016mfx ;
Bernlochner:2018opw ; Englert:2019xhk ; Gritsan:2020pib ; Bahl:2020wee , the
direct Higgs-top production via $pp\to t\bar{t}h$ is crucial to disentangle
possible new physics effects Ellis:2013yxa ; Boudjema:2015nda ;
Buckley:2015vsa ; Buckley:2015ctj ; Gritsan:2016hjl ; Goncalves:2016qhh ;
Mileo:2016mxg ; AmorDosSantos:2017ayi ; Azevedo:2017qiz ; Li:2017dyz ;
Goncalves:2018agy ; ATLAS:2018mme ; CMS:2018uxb ; Ren:2019xhp ;
Bortolato:2020zcg ; Cao:2020hhb ; MammenAbraham:2021ssc ; Martini:2021uey ;
Goncalves:2021dcu . This channel was observed in 2018 by both ATLAS and CMS
with significances of 6.3 $\sigma$ and 5.2 $\sigma$, respectively
ATLAS:2018mme ; CMS:2018uxb . The high-luminosity LHC (HL-LHC) studies
indicate that the Higgs-top interaction will be probed to outstanding accuracy
at the end of the LHC run, reaching $\delta y_{t}\lesssim 4\%$ when combining
the HL-LHC ATLAS and CMS data Cepeda:2019klc . The same projections indicate
that the $t\bar{t}h$ channel in the $h\to\gamma\gamma$ final state will
display dominant sensitivities. While the di-photon final state presents
limited statistics, it highly benefits from controlled backgrounds from side-
bands. Recently, ATLAS and CMS have reported the first experimental Higgs-top
CP studies, exploring the $t\bar{t}h$ channel ATLAS:2020ior ; CMS:2020cga .
Both analyses focus on the di-photon final state, $h\to\gamma\gamma$. ATLAS
and CMS exclude Higgs-top CP-mixing angles above $43^{\circ}$ and $55^{\circ}$
at 95% CL, respectively.

In the present manuscript, we perform a detailed investigation of the Higgs-
top CP sensitivity with the $pp\to t\bar{t}h$ channel at the HL-LHC,
considering the most promising decay mode, $h\to\gamma\gamma$. We explore the
complex multiparticle final state with a combination of machine learning
techniques and efficient kinematic reconstruction methods. Since distinct
Higgs-top CP-phases affect the net top and anti-top quark polarization,
propagating the spin effects to the top quark final states, we devote special
attention to include the top polarization observables in our study. In
particular, these spin effects are used to define genuine CP-observables.
After motivating and constructing the relevant kinematic observables, we
evaluate how much information can be extracted with them. The convenient
metric adopted to quantify this is given by the Fisher information. We show
that the ability of probing the $pp\to t\bar{t}h$ channel exploring the
complex multiparticle final state not only in terms of a raw rate, but as a
polarized process, can offer a crucial pathway to probe the underlying
production dynamics, accessing possible new physics effects.

The structure of this paper is as follows. In Section II, we present the
theoretical parametrization for the top Yukawa coupling. We discuss the new
physics effects to the top polarization, define the CP-sensitive observables,
and quantify how much information on the CP-phase can be extracted using
distinct observables. In Section III, we present the kinematic reconstruction
methods, which are relevant to build prominent observables to new physics and
maximally explore the $t\bar{t}h$ final state. Next, in Section IV, we move on
to the detailed analysis, where we derive the projected sensitivities for the
Higgs-top CP-phase at the HL-LHC. This study is inclusive in respect to the
top pair final states, combining the leptonic, semi-leptonic, and hadronic
channels. Finally, a summary of our key findings is delivered in Section V.

##  II CP Structure and Observables

We parametrize the top quark Yukawa coupling with the following Lagrangian

$\displaystyle\mathcal{L}=-\frac{m_{t}}{v}\kappa_{t}\bar{t}(\cos\alpha+i\gamma_{5}\sin\alpha)th\,,$
(1)

where $m_{t}$ is the mass of the top quark, $v$ is the vacuum expectation
value in the SM ($v=246$ GeV), $\kappa_{t}$ is a real number, and $\alpha$ is
the CP-phase. The interaction between the CP-even Higgs boson and the top
quark in the SM is represented by $(\kappa_{t},\alpha)=(1,0)$, while
$\alpha=\pi/2$ results in a pure CP-odd Higgs-top interaction. New physics
contributions in Eq. 1 will display effects both in the Higgs $t\bar{t}h$
production and decay, $h\rightarrow\gamma\gamma$. Whereas the Higgs decay will
more relevantly change the total signal rate Ellis:2013yxa , we will devote
special attention to probe the new physics effects in the Higgs production,
exploring the top quarks’ final state kinematics. This will be an essential
ingredient to uplift the new physics sensitivity from CP-phase effects.

Figure 1: Top panels: Distributions for the transverse momentum for the Higgs
boson $p_{Th}$ (left), invariant mass for the top pair $m_{t\bar{t}}$
(center), and the product of the projections of the top and anti-top momentum
$b_{4}=p_{t}^{z}p_{\bar{t}}^{z}/p_{t}p_{\bar{t}}$ (right). Bottom panels:
Distributions for the Collins-Soper angle $\theta^{*}$ (left), the azimuthal
angle between the two charged leptons in the top pair rest frame
$\Delta\phi_{\ell\ell}^{t\bar{t}}$ for fully leptonic $t\bar{t}h$ events
(center), and the same angle between the charged lepton and the softest light
jet in the top rest frame $\Delta\phi_{\ell j_{\text{soft}}}^{t\bar{t}}$ for
semi-leptonic $t\bar{t}h$ events (right). Each panel shows parton level
results for the $t\bar{t}h$ sample for the SM Higgs ($\alpha=0$), a CP-odd
Higgs ($\alpha=\pi/2$) and mixed hypotheses ($\alpha=\pm\pi/4$). We also
present the ratio between new physics and SM scenarios on the bottom panel of
each figure. The results are presented for the 14 TeV LHC.

###  II.1 Top Polarization

Among the observables sensitive to the structure of the top quark Yukawa
interaction, the spin correlations between the top and anti-top in $t\bar{t}h$
associated production offer a prominent pathway for precision studies Bar-
Shalom:1995quw ; Gunion:1996xu ; Atwood:2000tu ; Berge:2008wi ; Ellis:2013yxa
; Buckley:2015vsa ; Boudjema:2015nda ; Gritsan:2016hjl ; Buckley:2015ctj ;
Goncalves:2016qhh ; Mileo:2016mxg ; Goncalves:2018agy ; Aad:2020ivc ;
Goncalves:2021dcu ; Degrande:2021zpv . Owing to its short lifetime ($\sim
10^{-25}$s) ParticleDataGroup:2020ssz , the top quark is expected to decay
before hadronization occurs ($\sim 10^{-24}$s) and spin decorrelation effects
take place ($\sim 10^{-21}$s) Mahlon:2010gw . Thus, the spin-spin correlations
between $t$ and $\bar{t}$ can be traced back from the top quark decay
products. In particular, it is possible to observe correlations between any
two decay products, one from the top quark and the other from the anti-top
quark. The correlations scale with the spin analyzing power associated with
each top decay product.

More accurately, the top quark final states in the leptonic $t\to
W^{+}b\to\ell^{+}\nu b$ and hadronic $t\to W^{+}b\to\bar{d}ub$ channels are
correlated with the top quark spin axis as follows:

$\displaystyle\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\xi_{i}}=\frac{1}{2}\left(1+\beta_{i}P_{t}\cos\xi_{i}\right)\,,$
(2)

where $\Gamma$ is the partial decay width, $\xi_{i}$ is the angle between the
$i$-th decay product and the top quark spin axis in the top quark rest frame,
$P_{t}$ is the polarization of the decaying top $(-1\leq P_{t}\leq 1)$, and
$\beta_{i}$ is the spin analyzing power of the final state particle $i$
Bernreuther:2010ny . At leading order, the coefficient $\beta_{i}$ is $+1$ for
charged lepton $\ell^{+}$ and $\bar{d}$-quark, $-0.3$ for $\bar{\nu}$ and
$u$-quark, $-0.4$ for the $b$-quark, and $0.4$ for $W$-boson. The sign of the
coefficient $\beta_{i}$ is flipped for anti-top decays.

Granted by the $V\\!-\\!A$ current structure of the weak interaction, the
charged lepton will be a prominent spin analyzer, favoring studies with di-
leptonic top pairs. Exploring this phenomenology, the
$\Delta\phi_{\ell\ell}^{\text{lab}}$ observable, which is the azimuthal angle
difference between the two charged leptons in the lab frame, is a good example
of probe that has been found effective in accessing the Higgs-top CP-
properties Buckley:2015vsa ; Goncalves:2018agy . Remarkably, the sensitivity
of $\Delta\phi_{\ell\ell}^{\text{lab}}$ improves further in the boosted Higgs
regime due to the change in the net polarization for the top-pair at high
energies.

Analogously to the charged lepton, the $d$-quark also presents maximal spin
analyzing power. However, it is a challenging task to tag a $d$-quark jet in a
collider environment. An efficient solution is to select the softest of the
two light-quark jets, $j_{\text{soft}}$, in the top quark rest frame. This
choice uplifts the spin analyzing power of $j_{\text{soft}}$ to 50% of the
lepton’s Jezabek:1994qs . This approach boosts the spin correlation analyses
for the semi-leptonic and hadronic top quark pairs. Several observables can be
defined exploiting this fact, a particularly relevant example, that we will
explore in this manuscript, is the azimuthal angle difference between the
charged lepton and softest light jet in the top pair rest frame,
$\Delta\phi_{\ell j_{\text{soft}}}^{t\bar{t}}$.

###  II.2 CP-sensitive Observables

Various kinematic observables have been studied in the literature to access
the Higgs-top CP structure in the $pp\to t\bar{t}h$ channel. Some illustrative
distributions are presented in Fig. 1, such as the transverse momentum for the
Higgs boson $p_{Th}$ (top left) Demartin:2014fia ; Demartin:2015uha , the
invariant mass for the top pair $m_{tt}$ (top center), the product of
projections of top and anti-top momentum
$b_{4}=p_{t}^{z}p_{\bar{t}}^{z}/p_{t}p_{\bar{t}}$ (top right) Gunion:1996xu ,
and the angle between the top quark and the beam direction in the $t\bar{t}$
CM frame $\theta^{*}$ which is also known as Collins-Soper angle (bottom left)
Goncalves:2018agy . These observables result in distinct profiles for
different Higgs-top CP-phases. The pure CP-odd phase, $\alpha=\pm\pi/2$, leads
to a shift to higher energies in the peak of the distributions compared to the
SM scenario, $\alpha=0$. Different CP-phases interpolate between these two
profiles without sensitivity for the sign of the phase.

The variables $p_{Th}$, $m_{t\bar{t}}$, $b_{4}$, and $\theta^{*}$ are CP-even
observables, being sensitive to the squared terms: $\cos^{2}\alpha$ and
$\sin^{2}\alpha$. Thus, these probes are indifferent to the CP-even and CP-odd
Higgs-top interference terms, which are proportional to
$\cos\alpha\sin\alpha$. In particular, they are not sensitive to variations
from a relative sign-difference in the CP-phase. Genuine CP-sensitive
observables can be constructed from antisymmetric tensor products that require
four linearly independent four-momenta. Owing to the top polarization being
carried out to the decays, it is possible to construct such observable using,
for instance, the top, anti-top and their decay products Boudjema:2015nda ;
Mileo:2016mxg ; Goncalves:2018agy . In general, the antisymmetric tensor
product can be expressed as

$\displaystyle\epsilon(p_{t},p_{\bar{t}},p_{i},p_{k})\equiv\epsilon_{\mu\nu\rho\sigma}p_{t}^{\mu}p_{\bar{t}}^{\nu}p_{i}^{\rho}p_{k}^{\sigma}\,,$
(3)

where $\epsilon_{0123}=1$, and $\\{i,k\\}$ represent the final state particles
produced from the top and the anti-top decays, respectively.

In the $t\bar{t}$ CM frame, Eq. 3 can be fortuitously simplified to
$p_{t}\cdot(p_{i}\times p_{k})$. This mathematical relation can be used to
define azimuthal angle differences between the decay products, in the
$t\bar{t}$ CM frame, that are odd under CP-transformations Goncalves:2018agy :

$\displaystyle\\!\\!\\!\Delta\phi_{ik}^{t\bar{t}}\\!=\\!\text{sgn}\left[\vec{p}_{t}\\!\cdot\\!(\vec{p}_{i}\\!\times\\!\vec{p}_{k})\right]\arccos\\!\left[\frac{\vec{p}_{t}\\!\times\\!\vec{p}_{i}}{|\vec{p}_{t}\\!\times\\!\vec{p}_{i}|}\\!\cdot\\!\frac{\vec{p}_{t}\\!\times\\!\vec{p}_{k}}{|\vec{p}_{t}\\!\times\\!\vec{p}_{k}|}\right]\\!.$
(4)

For illustration, we present in Fig. 1 the azimuthal angle between the two
charged leptons $\Delta\phi_{\ell\ell}^{t\bar{t}}$ in the fully leptonic case
(bottom center) and between the charged lepton and the softest light jet in
the top rest frame $\Delta\phi_{\ell j_{\text{soft}}}^{t\bar{t}}$ in the semi-
leptonic case (bottom right). Two comments are in order. First, we notice that
$\Delta\phi_{ik}^{t\bar{t}}$ is indeed sensitive to the sign of the CP-phase,
as illustrated in a comparison between the distribution profiles for
$\alpha=\pi/4$ against $-\pi/4$. Second, in light of the spin analyzing power
of the charged lepton in relation to $j_{\text{soft}}$, the relative CP-
sensitivity of the di-leptonic against the semi-leptonic correlation follows
our expectation. Namely, the beyond the SM effects in the $\Delta\phi_{\ell
j_{soft}}^{t\bar{t}}$ observable are $\sim 50\%$ weaker in respect to
$\Delta\phi_{\ell\ell}^{t\bar{t}}$. This can be observed by comparing the
bottom panel of these plots, where we display the BSM/SM ratio.

###  II.3 Observable Information

Figure 2: Comparison of sensitivity on the CP-phase $\alpha$ arising from
different observables in terms of the Fisher Information $I$ for CP-odd
observables probing linear new physics effects (left) and in terms of the
modified Fisher Information $I^{\prime}$ for CP-even observables probing the
non-linear new physics effects (right).

Before proceeding to a full analysis, let us pause for a moment to better
understand which distributions and channels are sensitive to the CP-phase
$\alpha$. In particular, we would like to quantify and compare how much
information on the CP-phase is available using the different observables in a
parton level setup. This will provide some benchmarks and highlight the main
ingredients required for an efficient analysis strategy that will be presented
in Sec. IV.

Let us first consider the spin correlation observables
$\Delta\phi_{ik}^{t\bar{t}}$ between two decay products from the top and anti-
top, which probe the new physics effects linear in $\alpha$. A convenient
metric to quantify the sensitivity of these observables to constrain the
parameters of our model is given by the Fisher information Brehmer:2016nyr ;
Brehmer:2017lrt . Its component describing the sensitivity to the CP-phase
$\alpha$ is defined as

$\displaystyle I=\mathbb{E}\left[\frac{\partial\log
p(x|\kappa_{t},\alpha)}{d\alpha}\frac{\partial\log
p(x|\kappa_{t},\alpha)}{d\alpha}\right]\ .$ (5)

Here, $p(x|\kappa_{t},\alpha)$ is the likelihood function, which describes the
probability to observe a set of events with corresponding observables $x$ as a
function of the model parameter $\kappa_{t}$ and $\alpha$. $\mathbb{E}[\cdot]$
denotes the expectation value evaluated at the SM point,
$(\kappa_{t},\alpha)_{\rm SM}=(1,0)$. In the following we use the MadMiner
package to calculate the Fisher information Brehmer:2019xox .

In the left panel of Fig. 2, we show the Fisher information associated with
the CP-sensitive spin correlation observables for the di-leptonic (red), semi-
leptonic (gray), and hadronic (blue) channels. The bars on the left show the
full information, _i.e._ , the information that could be accessed via a
comprehensive multivariate analysis. This was estimated using the machine
learning method based on the SALLY algorithm Brehmer:2018eca ; Brehmer:2018kdj
; Brehmer:2018hga  trained with all possible spin correlation observables. The
remaining bars show the information in individual observables
$\Delta\phi_{ik}^{t\bar{t}}$, which were estimated using a histogram based
approach.

Focusing first on the di-leptonic channel, the most sensitive among these
observables is the spin correlation between the leptons,
$\Delta\phi^{t\bar{t}}_{\ell\ell}$, since the spin analyzing power for the
charged leptons are maximal. The next most sensitive observables are those
where a charged lepton has been replaced with a $b$-jet or a $W$ boson. We
observe that the corresponding Fisher information in
$\Delta\phi^{t\bar{t}}_{\ell b}$ and $\Delta\phi^{t\bar{t}}_{\ell W}$ are
suppressed compared to $\Delta\phi^{t\bar{t}}_{\ell\ell}$ by the square of the
spin analyzing power $\beta_{b/W}^{2}\sim 0.4^{2}$, as expected. The
information in the spin correlations observables between a pair of $b$-jet(s)
and/or $W$ boson(s) is further suppressed by an additional factor of
$\beta_{b/W}^{2}$.

Let us now also consider the other top decay channels. As the Fisher
information is proportional to the rate Brehmer:2016nyr , we expect it to
increase relative to the fully leptonic channel by a factor
${2\times\text{BR}_{W\\!\to\text{had}}/\text{BR}_{W\\!\to\text{lep}}\sim 6}$
for the semi-leptonic channel and
$(\text{BR}_{W\\!\to\text{had}}/\text{BR}_{W\\!\to\text{lep}})^{2}\sim 9$ for
the hadronic channel. Looking at the last three observables involving $b$-jets
and $W$-bosons, this is indeed the case. For the other observables, we notice
an additional loss of about a factor 2 in spin analyzing power, and hence a
factor 4 in the Fisher information, which is caused by probing
$j_{\text{soft}}$ instead of the $d$-quark.

Overall, we see that the different observables have distinct overall
importance in the three channels. For di-leptonic top decays, most of the
information is contained in the spin correlation between the leptons, while
the information in other observables is significantly suppressed. In contrast,
for the hadronic decay channel, all shown observables have almost similar
information. In this case, the resulting full information, that can be
obtained by combining the different spin correlation observables,
significantly exceeds the information of any individual observable. Overall,
all three channels carry a similar amount of information on the CP-phase
$\alpha$, which suggest performing a combined analysis.

Due to the limited $t\bar{t}h$ event rate at the LHC, we expect the non-linear
new physics effects to dominate over the linear ones. We therefore expect most
of the sensitivity on the CP-structure of the top Yukawa coupling to arise
from these non-linear terms, despite the fact that the corresponding
observables are not genuine CP-sensitive. To quantify the sensitivity of these
CP-even observables to the squared terms, we use modified version of the
Fisher information that was introduced in Ref. Brehmer:2019gmn . In this
approach, we simply consider the square of the coupling as our new model
parameter and define

$\displaystyle I^{\prime}=\mathbb{E}\left[\frac{\partial\log
p(x|\kappa_{t}^{2},\alpha^{2})}{d\alpha^{2}}\frac{\partial\log
p(x|\kappa_{t}^{2},\alpha^{2})}{d\alpha^{2}}\right]\ .$ (6)

The result is shown in the right panel of Fig. 2. Here, we show the
information associated with a two-dimensional distribution of two observables,
relative to the full information associated with a multivariate analysis using
all observables. As none of the presented observables relies on the top quark
final state kinematics, the results are identical for all three top quark
decay channels.

The distribution of the invariant mass of the photon pair, $m_{\gamma\gamma}$,
is only sensitive to the theory parameters through its normalization.
Correlating it with itself, we obtain the information associated with the
signal strength measurements, which accounts for $31\%$ of the information on
the CP-phase. In the absence of background, the correlation of
$m_{\gamma\gamma}$ and any other observable is equivalent to the information
in a single differential distribution of that observable. This is shown in the
bottom row. As expected, it is also identical to the information for the
correlation of an observable with itself, which are shown in the diagonal. We
can identify $\Delta\eta_{t\bar{t}}$ and $\theta^{*}$ as the two most
sensitive observables, which individually carry about $60\%$ of the full
information.

Combing two different observables further increases the information. The two
most promising combinations are $\Delta\eta_{t\bar{t}}$ vs. $p_{Th}$ as well
as $\theta^{*}$ vs. $m_{t\bar{t}h}$, which carry about $73\%$ of the full
information. Successively adding more observables further increases the
information. This shows that a multivariate analysis is vital to maximize the
sensitivity on the CP-phase $\alpha$.

##  III Kinematic Reconstruction

Most of the new physics probes discussed so far, $viz.$ $m_{tt}$,
$\theta^{*}$, $b_{4}$, and $\Delta\phi_{ik}^{t\bar{t}}$, require a full
reconstruction of the top and anti-top momenta. This is a challenging task at
the LHC due to combinatorial ambiguities and the presence of up to two
neutrinos in the $t\bar{t}(h\to\gamma\gamma)$ final state. In this section, we
discuss the strategies adopted for the kinematic reconstruction of the semi-
leptonic and hadronic channels, and the more complex di-leptonic mode.

Semi-leptonic channel: In the semi-leptonic channel, the full reconstruction
of the $t\bar{t}$ system requires the determination of the longitudinal
momentum of the missing particle $\nu$. We compute it by constraining the
invariant mass of the lepton and the neutrino to the $W$-boson mass.
Typically, either two solutions or zero solutions are obtained. Around $35\%$
of events give zero solutions, and discarding all such events would lead to a
significant reduction in event statistics. Therefore, in such events, we vary
the transverse momentum of the missing system (at most by $\pm 10\%$) while
keeping the azimuth angle of $\nu$ unchanged until physical solutions of
$p_{z,\nu}$ are obtained. Events which give zero solutions even after the
aforesaid variation are ignored. We perform the reconstruction for the top
quarks iterating over all possible partitions of light jets ($j$) and $b$-jet
forming the hadronic top ($jjb$) and leptons and $b$-jet for the leptonic top
($\ell\nu b$). The two possible neutrino solutions are separately accounted
for, forming different partitions. We select the combination that minimizes

$\displaystyle(m_{jjb}-m_{t})^{2}+(m_{\ell\nu b}-m_{t})^{2},$ (7)

where $m_{t}$ is the on-shell mass of the top quark.

Hadronic channel: We follow a similar mass minimization strategy in the
hadronic channel. We reconstruct the two top quarks, $t_{1}$ and $t_{2}$, by
iterating over all possible combinations of light jets and $b$-jets. The
combination which minimizes

$\displaystyle(m_{t_{1}}-m_{t})^{2}+(m_{t_{2}}-m_{t})^{2},$ (8)

is chosen.

Di-leptonic channel: In the more complex di-leptonic $t\bar{t}h$ channel, the
invisible system is constituted by two neutrinos. Therefore, in addition to
determining the unknown longitudinal momentum of the missing particles, it is
also indispensable to partition the four-momentum of the missing system into
the two neutrinos in order to fully reconstruct the top and the anti-top. An
additional combinatorial ambiguity arises from the tandem $b$-jet and $\ell$
pairing. The study in Ref. Goncalves:2018agy  reconstructed the $t\bar{t}(h\to
b\bar{b})$ system in di-leptonic mode using $M_{2}$ assisted reconstruction
algorithm and a boosted $h\to b\bar{b}$, with jet substructure techniques, to
suppress the additional combinatorics between the Higgs boson and top quark
decays. In contrast, the present analysis reconstructs the
$t\bar{t}(h\to\gamma\gamma)$ system following the Recursive Jigsaw
Reconstruction (RJR) algorithm presented in Ref. Jackson:2017gcy . The RJR
approach utilizes a series of jigsaw rules optimized to estimate the unknown
kinematic degrees of freedom in an event topology and resolve the
combinatorial ambiguities between/within the final state visible and invisible
objects. It results is a complete kinematic basis which can be used to define
the four-momenta of all the final state and intermediate objects in an event
decay tree.

The first step involves the resolution of combinatorial ambiguity between the
$b$-jets and the leptons by using the “Combinatorial Minimization” Jigsaw Rule
(JR) Jackson:2017gcy , identifying the ($b$-jet, $\ell$) pairs by minimizing

$(m_{b_{j}\ell^{+}}^{2}+m_{b_{k}\ell^{-}}^{2});~{}j,k=1,2;~{}j\neq k.$ (9)

After establishing the two visible hemispheres corresponding to the top and
the anti-top, we apply the “Invisible Mass” JR to estimate the invariant mass
of the invisible system ($m_{I}$) Jackson:2017gcy  defined as

$m_{I}^{2}=m_{V}^{2}-4m_{V_{a}}^{2}m_{V_{b}}^{2},$ (10)

where $m_{V}$ is the invariant mass of all the two $b$-tagged jets and the two
leptons in the final state. $m_{V_{a}}$ and $m_{V_{b}}$ correspond to the
invariant mass of the two visible hemispheres associated with the top and the
anti-top that were reconstructed in the previous step. $m_{I}$ is chosen such
that it is the smallest Lorentz invariant mass that ensures a non-tachyonic
four-momenta for the individual neutrinos upon partitioning the invisible
system. Next, we determine the longitudinal momentum of the invisible system,
$\not{p}_{z}$, using the following relation given by the “Invisible Rapidity”
JR Jackson:2017gcy :

$\not{p}_{z}=p_{z}^{V}\frac{\sqrt{{|\not{p}_{T}|}^{2}+m_{I}^{2}}}{\sqrt{{|p_{T}^{V}|}^{2}+m_{V}^{2}}}.$
(11)

Here, $p_{z}^{V}$ and $p_{T}^{V}$ represent the longitudinal and transverse
momenta, respectively, of the visible system constituted by the two $b$-jets
and the two leptons, while $\not{p}_{T}$ is the missing transverse momentum.

At this point, we have all the ingredients required to reconstruct the
$t\bar{t}$ system. However, in order to reconstruct the top and the anti-top
individually, the invisible four-momentum has to be correctly partitioned into
the two neutrinos. This is achieved by using the “Contraboost Invariant” JR
specified in Ref. Jackson:2017gcy  which estimates the four-momenta of the
neutrinos produced from top and anti-top decay in the $t\bar{t}$ CM frame
under the assumption that both $t$ and $\bar{t}$ have the same invariant mass.
The resolved four-momenta of the neutrinos along with the correctly paired
$b$-jets and leptons allows defining the $t$ and the $\bar{t}$ systems
independently. The reconstruction efficiency of this method is about $80\%$,
which is comparable with $M_{2}$ assisted reconstruction algorithm
Goncalves:2018agy .

With the fully-resolved $t\bar{t}h$ system, we can reconstruct a multitude of
CP-even and CP-odd spin correlation observables defined in the $t\bar{t}$ CM
frame and the lab frame. Several observables that do not depend on the spin-
polarization of $t\bar{t}$ pair are also considered. Our goal here is to
maximally explore the $t\bar{t}h$ multi-particle final state, augmenting the
CP-sensitivity of the $pp\to t\bar{t}h$ channel at the HL-LHC.

##  IV Analysis

###  IV.1 Simulation and Event Selection

Figure 3: Reconstructed detector level distributions for the Collins-Soper
angle $\theta^{*}$ for the di-leptonic channel (left), semi-leptonic channel
(center), and hadronic channel (right). We present for the SM $(\alpha=0)$ and
several beyond the SM Higgs-top CP-hypotheses $(\alpha=\pi/2,\pm\pi/4)$.

In this section, we explore the _direct_ Higgs-top CP measurement combining
machine learning techniques and efficient kinematic reconstruction methods. We
consider $t\bar{t}h$ signal with $h\to\gamma\gamma$ in the di-leptonic, semi-
leptonic, and hadronic top decay modes at the HL-LHC. The dominant background
to this process is given by continuum $t\bar{t}\gamma\gamma$ production. We
simulate both the signal and background event samples with
MadGraph5$\\_$aMC@NLO Alwall:2014hca  within the MadMiner framework
Brehmer:2019xox  at leading order (LO) with a center-of-mass energy of
$\sqrt{s}=14~{}\text{TeV}$. Higher order effects to the signal rate are
included via a flat next-to-leading order k-factor deFlorian:2016spz ; HXSWG .
We use NNPDF2.3QED parton distribution function Ball:2013hta . No generation-
level cuts have been applied for the signal events, while the backgrounds have
been generated in the mass window
$105~{}\text{GeV}\\!<\\!m_{\gamma\gamma}\\!<\\!145~{}\text{GeV}$. Parton
shower and hadronization effects have been included with Pythia 8
Sjostrand:2007gs  and fast detector simulation with the Delphes3 package
deFavereau:2013fsa , using the default HL-LHC detector card HLLHC_card ;
Cepeda:2019klc .

To obtain the cross section and likelihood function as a function of the
theory parameters, we use the morphing technique that is already implemented
in MadMiner. Here, we take into account the dependence of new physics theory
parameters at both $t\bar{t}h$ production and $h\to\gamma\gamma$ decay, and
therefore choose a quartic ansatz in the morphing setup, which is used to
interpolate the event weights as a function of
$\kappa_{H}=\kappa_{t}\cos{\alpha}$ and $\kappa_{A}=\kappa_{t}\sin\alpha$.

We start our analysis by selecting events consisting of two photons and at
least two $b$-tagged jets. In addition, we require the final state to contain
exactly two opposite-sign leptons for the di-leptonic channel, exactly one
lepton and at least two light jets for the semi-leptonic channel, and at least
four light jets for the hadronic channel. We demand the individual particles
to pass the following identification cuts:

$\displaystyle p_{T\ell}$ $\displaystyle\\!>\\!15~{}\text{GeV},\
|\eta_{\ell}|\\!<\\!4,\quad p_{T\gamma}\\!>\\!15~{}\text{GeV},\
|\eta_{\gamma}|\\!<\\!4,$ (12) $\displaystyle p_{Tb}$
$\displaystyle\\!>\\!25~{}\text{GeV},\ |\eta_{b}|\\!<\\!4,\quad
p_{Tj}\\!>\\!25~{}\text{GeV},\ |\eta_{j}|\\!<\\!5.$

In addition, we require the di-photon invariant mass to satisfy
$|m_{\gamma\gamma}\\!-\\!125|\\!<\\!10~{}\text{GeV}$.

We fully reconstruct the $t\bar{t}h$ system following the strategy described
in Sec. III. In particular, this allows to obtain both the lab frame and the
$t\bar{t}$ CM frame observables. As an example for an observable that requires
the top reconstruction, we present the distribution of the Collins-Soper angle
$\theta^{*}$ in Fig. 3. When comparing these detector level distributions to
the result at parton level, presented in Fig. 1, we observe the robustness of
our analysis in respect to the reconstruction strategy and detector effects.
The distributions are found to retain the CP sensitivity at the detector
level, albeit a reduction of about $20\%$ for the di-leptonic channel, $40\%$
for the semi-leptonic channel and $50\%$ for the hadronic channel, compared to
parton level.

###  IV.2 Analysis Methodology

As we have seen in Sec. II, there is no single observable that carries all the
information on the CP-structure of the top quark Yukawa. Instead, there is a
variety of sensitive observables. Hence, a multi-variate analysis is needed to
extract the maximal information on the theory parameters from the data. In the
following, we will summarize the adopted observables and the analysis
methodology.

In this analysis, we consider the following list of 80 observables to describe
the kinematics of signal and background events.

Observables: $\displaystyle\Delta\phi_{ik}^{t\bar{t}},\
\Delta\phi_{hi(k)}^{t\bar{t}},\ \Delta\phi_{ik}^{\text{lab}},$ (13)
$\displaystyle\ \Delta\phi_{ht/h\bar{t}},\ \Delta\phi_{t\bar{t}},\theta^{*},\
b_{4},\ $ $\displaystyle m_{\gamma\gamma},\ m_{t\bar{t}},\
m_{t\bar{t}h},\Delta R^{\rm 2^{nd}min}_{\gamma j},$ $\displaystyle\Delta
R_{j_{\text{soft}}j_{\text{hard}}},\ \Delta R_{\ell\nu},\Delta R_{Wb},\ \Delta
R^{\text{min}}_{\gamma j},$ $\displaystyle\Delta\eta_{t\bar{t}},\
m_{ht/h\bar{t}},\ H_{T},\not{E}_{T}/\sqrt{H_{T}},$
$\displaystyle\\{p_{T},~{}\eta\\}_{X}\text{ for }X=i,k,t,\bar{t},h.$

We include the complete set of observables used by the ATLAS collaboration in
a recent Higgs-top CP study ATLAS:2020ior  and complement this set with
additional CP-even observables that show strong sensitivity to the CP-phase
$(\theta^{*},b_{4},m_{t\bar{t}},m_{t\bar{t}h})$ together with the transverse
momentum and pseudorapidity of all final state and reconstructed objects. We
also incorporate a comprehensive list of spin correlations, as introduced in
Eq. 4, which are constructed between all possible final state pairs. We
include both observables constructed in the $t\bar{t}$ rest frame,
$\Delta\phi_{ik}^{t\bar{t}}$, and in the lab frame,
$\Delta\phi_{ik}^{\text{lab}}$. Finally, we account for the correlation
observables $\Delta\phi_{hi(k)}^{t\bar{t}}$ that arise from the tensor
products involving the Higgs boson momentum,
$\epsilon(p_{t},p_{\bar{t}},p_{h},p_{i(k)})$. The following pairs $\\{i,k\\}$
are considered for the different channels

$\displaystyle\text{di-leptonic: }i$
$\displaystyle=\ell^{+},\nu_{t},b_{t},W_{t}$ (14) $\displaystyle k$
$\displaystyle=\ell^{-},\nu_{\bar{t}},b_{\bar{t}},W_{\bar{t}}$
$\displaystyle\text{semi-leptonic: }i$
$\displaystyle={\ell,\nu,b_{\ell},W_{\ell}}$ $\displaystyle k$
$\displaystyle={j_{\text{soft}},j_{\text{hard}},b_{\text{had}},W_{\text{had}}}$
$\displaystyle\text{hadronic: }i$
$\displaystyle=j_{\text{soft}}^{t_{1}},j_{\text{hard}}^{t_{1}},b_{t_{1}},W_{t_{1}}$
$\displaystyle k$
$\displaystyle=j_{\text{soft}}^{t_{2}},j_{\text{hard}}^{t_{2}},b_{t_{2}},W_{t_{2}}$

In the semi-leptonic case, $b_{\ell}/W_{\ell}$ and
$b_{\text{had}}/W_{\text{had}}$ represent the $b$-jets/$W$-bosons produced
from the leptonically and hadronically decaying top quarks, respectively. In
events with more than two $b$-tagged jets, the hardest two are considered
while reconstructing the top and the anti-top quarks. $j_{\text{hard}}$
corresponds to the hardest light jet, from the hadronic top quark, in the top
rest frame.

To interpret the results of our analysis and obtain projected sensitivities,
we follow a likelihood-based approach. According to the Neyman-Pearson lemma,
the most powerful test statistic to discriminate between two hypotheses, in
our case a new physics model parameterized by $\theta=(\kappa_{t},\alpha)$ and
the SM with $\theta_{\text{SM}}=(1,0)$, is the likelihood ratio
$r(x|\theta;\theta_{\text{SM}})$. Here, $x$ denotes the set of reconstructed
observables introduced above.

Figure 4: Projected sensitivity in the $(\alpha,\kappa_{t})$ plane. In the
left panel we show the projected $68\%$ CL contours from direct Higgs-top
searches in the semi-leptonic (blue), di-leptonic (green), hadronic (red)
$t\bar{t}h$ channels, and their combination (black), considering all input
observables. The right panel shows the projected $68\%$ CL (dashed) and $95\%$
CL (solid) contours from the combination of the three channels considering all
input observables. The color palette illustrates the expected p-value of the
estimated log-likelihood ratio. The projections are derived for 14 TeV LHC
assuming an integrated luminosity of $3~{}\text{ab}^{-1}$.

Whereas the likelihood ratio involving detector level observables is
intractable, meaning that it cannot be computed directly, it can be estimated
using simulations. To address this issue, we use the machine learning based
technique introduced in Refs. Brehmer:2018eca ; Brehmer:2018kdj ;
Brehmer:2018hga ; Stoye:2018ovl ; Brehmer:2019bvj ; Brehmer:2020ako ;
Brehmer:2020zwh , which has been implemented in the MadMiner tool
Brehmer:2019xox . This approach uses both reconstructed observables and
matrix-element information, which are then used to train neural network models
that estimate the likelihood ratio. It therefore accounts for the effects of
parton shower, hadronization, and detector effects, while the matrix-element
information helps to significantly improve the performance of the neural
network training. Using the estimated likelihood ratio function
$r(x|\theta;\theta_{\text{SM}})$, which describes both the linear and non-
linear new physics effects, we then perform a likelihood ratio test to obtain
our projected sensitivities.

We simulate $10^{6}$ signal and $10^{6}$ background events before event
selection. Using MadMiner, we train neural networks to estimate the likelihood
ratio using the ALICES algorithm with its hyperparameter set to unity
Stoye:2018ovl . We use fully connected neural networks with three hidden
layers, each containing 100 nodes and $\tanh$ activation function. The neural
network training is performed over 100 epochs using the Adam optimizer. To
avoid overtraining, we evaluate the loss function on an independent validation
set and employ an early stopping procedure. We use a batch size of 128, and an
exponentially decaying learning rate (from $10^{-4}$ to $10^{-5}$). The limit
setting is performed with MadMiner’s Likelihood class.

###  IV.3 Results

Let us now turn to the results of our study. In Fig. 4 we show the projected
sensitivity on the top Yukawa coupling in terms of $\kappa_{t}$ and $\alpha$
using the $t\bar{t}(h\to\gamma\gamma)$ measurement. In the left panel, we
present the $68\%$ CL contours for the individual top decay channels as
colored dashed lines. A combination of all channels is shown in the black
solid line. The studied channels can be organized in ascending order of
sensitivity as: di-leptonic, semi-leptonic, and hadronic modes. Since the
leading observables display efficient reconstruction for all channels, as
illustrated in Fig. 3, the order of sensitivity among the final state modes
follow their correspondent event rate.

In the right panel, we show the $68\%$ and $95\%$ CL contours as dashed and
solid lines, respectively. The p-values in the $({\kappa_{t},\alpha})$
parameter space are presented through the color palette. We observe that
$|\kappa_{t}|$ can be constrained within $\mathcal{O}(8\%)$ of the SM value at
$68\%$ CL through a combination of direct searches in the
$t\bar{t}(h\to\gamma\gamma)$ channel at the HL-LHC. Assuming $\kappa_{t}=1$,
the combined search would be able to probe the Higgs-top CP phase up to
$|\alpha|\lesssim 13^{\circ}$ at $68\%$ CL.

We also perform a separate analysis in which we train a neural network
exclusively with the CP-even observables shown in the right panel of Fig. 2.
We observe that the projected sensitivity of such an analysis, using this
smaller set of CP-even observables that are most-sensitive to the non-linear
new physics effects, is almost comparable to the projected sensitivity of the
combination study performed using the full set of observables. Overall, almost
all the sensitivity to the Higgs-top CP-structure is provided by the non-
linear terms in $\alpha$. The limited $t\bar{t}(h\to\gamma\gamma)$ event
statistics renders sub-leading sensitivity to the observables which probe the
linear terms.

###  IV.4 Systematic Effects

In this section, we explore the implications from systematic uncertainties on
the projected sensitivity of $\kappa_{t}$ and $\alpha$. In particular, we will
consider two sources of uncertainty associated with the normalization of both
signal and background.

In the statistical analysis, these uncertainties are parameterized through
nuisance parameters $\nu_{s}$ and $\nu_{b}$ for the signal and background
normalization, respectively. These nuisance parameters encode theoretical and
experimental uncertainties on the normalization of distributions, neglecting
possible shape uncertainties. As before, we train a neural network using the
ALICES method in MadMiner to estimate the likelihood ratio
$r\left(x|\theta,\nu;\theta_{SM},\nu_{SM}\right)$. This is now a function of
both the model parameters $\theta=(\kappa_{t},\alpha)$ and the nuisance
parameters $\nu=(\nu_{s},\nu_{b})$ which have a nominal value
$\nu_{SM}=(0,0)$. Before setting limits, a constraint term describing our
prior knowledge on the nuisance parameter is added. Adopting a conservative
approach, we assume a prior constraint of $20\%$ and $50\%$ in the
$t\bar{t}(h\to\gamma\gamma)$ signal and the $t\bar{t}\gamma\gamma$ background,
respectively. Finally, we profile over the nuisance parameters following the
procedure described in Ref. Brehmer:2018eca .

Before turning to the sensitivity contours, let us remind ourselves that the
presented results are based on a multi-variate analysis. In particular, this
includes the invariant mass of the di-photon pair. The considered range,
$115~{}\text{GeV}<m_{\gamma\gamma}<135~{}\text{GeV}$, was chosen sufficiently
wide to contain both a signal dominated region at the Higgs resonance and a
background dominated region around it. MadMiner uses this background dominated
region to constrain the nuisance parameter associated with the background
normalization $\nu_{b}$, and therefore effectively performs a data-driven
side-band analysis. As we will see in a moment, the effective uncertainty of
the background normalization is therefore significantly smaller than the 50%
which we assumed as a prior.

In the following, we analyze three scenarios to study the impact of systematic
uncertainties on the projected sensitivity in the
$\left(\alpha,\kappa_{t}\right)$ plane. In the first scenario, we study the
impact associated with only the uncertainty on the background normalization.
To do so, we fix the $\nu_{s}=0$ in the estimated likelihood ratio and profile
over $\nu_{b}$. Similarly, in a second scenario, we fix $\nu_{b}=0$ and we
profile over $\nu_{s}$ to study the impact of the signal uncertainty. Finally,
in a third scenario, we obtain the limits after profiling the likelihood ratio
over both $\nu_{s}$ and $\nu_{b}$. In Fig. 5 we present the projected
sensitivity on $\alpha$ and $\kappa_{t}$ for all scenarios. The blue, green,
and red contours correspond to the first, second, and third scenarios,
respectively. The black contour represents the sensitivity assuming no
systematic uncertainty and corresponds to the black-solid contours in Fig. 4.

Figure 5: Projected $68\%$ CL contours on $(\alpha,\kappa_{t})$ from the
combination of the three channels considering all input observables, profiled
over background (B) uncertainties (blue), signal (S) uncertainties (green),
and both signal and background systematic uncertainties (red). The projections
are derived for 14 TeV LHC assuming $\mathcal{L}=3~{}\text{ab}^{-1}$.

At first, we observe that the sensitivity in $\alpha$ remains unaffected from
systematic uncertainties Goncalves:2021dcu . This stems from the reason that
at $\kappa_{t}=1$, the sensitivity in $\alpha$ is dominantly controlled by the
shape information from kinematic distributions and is largely independent of
the event rate due to the combination of two competing effects. On the one
hand, the signal cross section
$\sigma_{t\bar{t}\left(h\to\gamma\gamma\right)}$ decreases with $\alpha$: for
example at $\kappa_{t}=1$ the cross section
$\sigma_{t\bar{t}\left(h\to\gamma\gamma\right)}$ falls by $\mathcal{O}(25\%)$
from $\alpha=0$ until $\alpha\sim\pi/3$ and then remains roughly unchanged
until $\alpha=\pi/2$. On the other hand, the signal efficiency also improves
with $\alpha$. These two effects roughly offset any overall dependence on the
event rate, thereby leading to unchanged projection contours in the direction
of $\alpha$ even after profiling over the nuisance parameters.

The situation is qualitatively different in the $\kappa_{t}$ direction. When
$\alpha=0$, the measurement is purely based on a rate information, implying
that the Higgs coupling strength $\kappa_{t}$ and the signal normalization, as
parameterized by $\nu_{s}$, are essentially degenerate. Therefore, our prior
uncertainty of the signal normalization will directly propagate into a
systematic uncertainty on $\kappa_{t}$. The new physics effects in the Higgs-
top coupling manifest as $\sim\kappa_{t}^{2}$ at the $t\bar{t}h$ production
level and as $\sim\left(1.28-0.28\kappa_{t}\right)^{2}$ in $h\to\gamma\gamma$
decay Brod:2013cka . After combining these two factors, an uncertainty of
$20\%$ in the $pp\to t\bar{t}(h\to\gamma\gamma)$ cross section translates to
roughly $12\%$ uncertainty in $\kappa_{t}$. We observe this effect in Fig. 5:
for $\alpha=0$ the projected sensitivity falls from $|\kappa_{t}|\lesssim 8\%$
in the absence of systematic uncertainties to $|\kappa_{t}|\lesssim 13\%$ on
profiling over $\nu_{s}$. We observe that despite a prior $50\%$ uncertainty
in the background normalization compared to $20\%$ in the signal, its impact
on the projection contours in the $\left(\kappa_{t},\alpha\right)$ plane is
milder. As discussed above, this is a consequence of the side-band measurement
and illustrates the robustness of our multi-variate analysis.

##  V Summary

In this study, we derived the prospects to _direct_ measure the Higgs-top CP-
structure in $t\bar{t}(h\to\gamma\gamma)$ channel at the HL-LHC. We show that
a combination of machine learning techniques and efficient kinematic
reconstruction methods can boost new physics sensitivity, effectively
exploring the complex $t\bar{t}h$ multi-particle phase space.

Among the several probes included in our machine learning analysis, this study
encompass a comprehensive set of spin correlation observables. Beyond the SM
CP-phases steer the spin-polarization of the top pair, and the spin
correlations are carried forward by their decay products. We harness the
potential of the spin correlation observables via the full reconstruction of
the top and anti-top, evaluating these particular observables in the
$t\bar{t}$ CM frame, where the correlations are maximal. In the hadronic and
semi-leptonic $t\bar{t}h$ channels, we used mass minimization to fully
reconstruct the $t\bar{t}h$ system. In the more complex di-leptonic channel,
we employed the Recursive Jigsaw Reconstruction technique to resolve the
combinatorial ambiguities and determine the unknown degrees of freedom. In all
channels, the effects of parton showering, hadronization, and detector
resolution were included.

Exploring the intricate $t\bar{t}h$ multi-particle phase space with CP-odd and
even observables defined in the laboratory frame and the $t\bar{t}$ CM frame,
we obtain strong projections for the Higgs-top CP-phase. Through a combined
semi-leptonic, hadronic, and di-leptonic $t\bar{t}(h\to\gamma\gamma)$ search,
the HL-LHC can directly probe the Higgs-top coupling modifier and CP-phase
respectively up to $|\kappa_{t}|\lesssim 8\%$ and $|\alpha|\lesssim
13^{\circ}$ at $68\%$ CL.

Possible improvements can be expected by including other relevant channels,
such as $t\bar{t}(h\to b\bar{b})$ Buckley:2015vsa ; AmorDosSantos:2017ayi ;
Goncalves:2018agy ; Goncalves:2021dcu . While this channel displays the bulk
of the Higgs decay, $\mathcal{BR}(h\to b\bar{b})\sim 58\%$, it results in sub-
leading limits in comparison to $t\bar{t}(h\to\gamma\gamma)$ as it endures a
substantial QCD background that is associated with sizable uncertainties
ATLAS:2017fak ; CMS:2018hnq . Fast-moving precision calculations Jezo:2018yaf
; Denner:2020orv ; Bevilacqua:2021cit  and possible combination of side-band
analysis with $t\bar{t}h/t\bar{t}Z$ ratios Mangano:2015aow ; Goncalves:2021dcu
may change this scenario, controlling the respective uncertainties, and
pushing further forward the sensitivity with this extra channel in the near
future.

###### Acknowledgements.

We thank Johann Brehmer and Sam Homiller for helpful discussions. RKB and DG
thank the U.S. Department of Energy for the financial support, under grant
number DE-SC 0016013. The work of FK is supported by the U.S. Department of
Energy under Grant No. DE-AC02-76SF00515 and by the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy - EXC 2121 Quantum
Universe - 390833306. Part of this work was performed at the Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-1607611.
Some computing for this project was performed at the High Performance
Computing Center at Oklahoma State University, supported in part through the
National Science Foundation grant OAC-1531128.

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