††institutetext:  Institute for Theoretical Particle Physics and Cosmology,
RWTH Aachen University,
D-52056 Aachen, Germany

#  ${\cal CP}$ structure of the top-quark Yukawa interaction: NLO QCD
corrections and off-shell effects

Jonathan Hermann,  Daniel Stremmer  and Malgorzata Worek
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Since its discovery at the Large Hadron Collider in 2012 the Higgs boson has
arguably become the most famous of the Standard Model particles and many
measurements have been performed in order to assess its properties. Among
others, these include measurements of the Higgs boson’s ${\cal CP}$ state
which is predicted to be ${\cal CP}$-even. Even though a pure ${\cal CP}$-odd
state has been ruled out, a possible admixture of a ${\cal CP}$-odd Higgs
state has yet to be excluded. In this work we present predictions for the
associated production of a leptonically decaying top quark pair and a stable
Higgs boson $pp\to e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ with
possible mixing between ${\cal CP}$-even and ${\cal CP}$-odd states at NLO in
QCD for the LHC with $\sqrt{s}=13$ TeV. Finite top-quark and gauge boson width
effects as well as all double-, single- and non-resonant Feynman diagrams
including their interference effects are taken into account. We compare the
behaviour of the ${\cal CP}$-even, -odd and -mixed scenarios for the
integrated fiducial cross-sections as well as several key differential
distributions. In addition, we show that both NLO corrections and off-shell
effects play an important role even at the level of integrated fiducial cross-
sections and that these are further enhanced in differential distributions.
Even though we focus here on the Standard Model Higgs boson, the calculations
could be straightforwardly applied to models that have an extended Higgs-boson
sector and predict the existence of ${\cal CP}$-odd Higgs-like particles, such
as the two-Higgs-doublet model.

###### Keywords:

Higher Order Perturbative Calculations, Top Quark, Higgs Production, Higgs
Properties

††dedication: TTK-22-19, P3H-22-054

##  1 Introduction

The discovery of the Higgs boson with a mass of $125$ GeV by the ATLAS and CMS
collaborations in 2012 ATLAS:2012yve ; CMS:2012jmk  has launched an extensive
research program to establish the true origin of this particle and probe all
its properties. Determination of its couplings to the Standard Model (SM)
particles and verification of its quantum numbers are among the most important
tasks for current and future runs of the LHC and, in fact, for any future
collider. Accurate measurements of the Higgs boson’s properties are critical
for validating the SM or potentially discovering new physical effects by
detecting deviations from the SM predictions. The ${\cal CP}$ nature of the
Higgs boson has been extensively studied at the LHC. As a matter of fact, LHC
measurements have already ruled out the hypothesis of the pure ${\cal CP}$-odd
Higgs boson CMS:2014nkk ; ATLAS:2015zhl . However, only rather weak constrains
exist on a possible admixture between a ${\cal CP}$-even and ${\cal CP}$-odd
component. Moreover, most existing experimental analysis concentrate on the
coupling of the Higgs boson to two massive gauge bosons $HVV$, where $V=Z,W$.
In particular, both ATLAS and CMS have investigated Higgs boson production via
vector-boson fusion as well as the following decays $H\to ZZ^{*}\to 4\ell$ and
$H\to WW^{*}\to 2\ell 2\nu$, see e.g. Refs. ATLAS:2016ifi ; CMS:2017len ;
CMS:2019ekd ; CMS:2019jdw ; ATLAS:2020evk ; CMS:2021nnc . It seems that so far
the properties of the discovered Higgs boson agree with the predictions of the
SM ATLAS:2013dos ; CMS:2014fzn ; ATLAS:2016neq . It remains an intriguing
possibility, however, that the observed Higgs boson might just be one member
of an extended Higgs boson sector. A very good motivation for such a beyond
the SM Higgs boson sector is the fact that it allows for a new source of
${\cal CP}$ violation. The latter is required to explain the matter–antimatter
asymmetry problem in the observed universe Sakharov_1991 . Several competing
hypotheses exist to explain the imbalance of matter and antimatter that
resulted in baryogenesis. As of yet, there is no consensus theory explaining
this phenomenon. However, it is clear that ${\cal CP}$ violation in the SM is
insufficient for this purpose Kuzmin:1985mm . Thus, we are forced to look
elsewhere and, in particular, beyond the well-known SM. Among the simplest
Higgs boson extensions are the Two-Higgs-Doublet models (2HDMs) where the SM
is extended by one extra Higgs boson doublet with the same quantum numbers as
the SM one Gunion:1989we ; Gunion:2002zf ; Branco:2011iw ; Grzadkowski:2014ada
. These 2HDMs, both in their ${\cal CP}$-conserving and ${\cal CP}$-violating
versions, are usually used as benchmark models to search for new scalars at
the LHC. The 2HDM has four extra degrees of freedom with two extra neutral
scalars and two charged scalars. In the ${\cal CP}$-conserving version of a
generic 2HDM the two neutral states are ${\cal CP}$-even (often denoted as
$h,H$) and one is ${\cal CP}$-odd (often denoted as $A$). Moreover, in this
case there are no tree-level couplings between the pseudoscalar Higgs boson
and vector bosons as the bosonic sectors of most 2HDM extensions conserve
parity. Thus, $AZZ$ and $AWW$ couplings must be induced trough fermion loops.
The branching ratios for $A\to ZZ$ and $A\to WW$ decays are usually expected
to be small, however, Higgs-fermion couplings can be enhanced by large fermion
masses and other model parameters. On the other hand, in the ${\cal
CP}$-violating version of a generic 2HDM, the three neutral states are a
mixture of ${\cal CP}$-even and ${\cal CP}$-odd states. In addition, one of
the three states is identified as the $125$ GeV Higgs boson and all of them
have non-zero $HZZ$ and $HWW$ type interactions. Consequently, one must also
concentrate on investigating the couplings of the observed Higgs boson to the
SM fermions. Such studies are of the utmost importance to settle the question
of the true nature of this recently discovered Higgs boson. Of all the SM
fermions, the top quark plays a special role due to its large mass of
$m_{t}=172.5$ GeV. The top-quark Yukawa coupling $(Y_{t})$ is the only
coupling with the magnitude of the order of one in the SM. Any hint of the
${\cal CP}$-odd or ${\cal CP}$-mixed component would first manifest itself in
the connection with the top quarks, for example in the $t\bar{t}H$ production
process. Such a modified coupling would be a direct indication of the presence
of new physics. Furthermore, it would impact not only integrated cross-
sections but also various differential cross-section distributions, see e.g.
Ellis:2013yxa ; Boudjema:2015nda ; Buckley:2015vsa ; AmorDosSantos:2017ayi .

Indirect measurements of the Yukawa coupling between the Higgs boson and the
top quark have been carried out at the LHC using the gluon fusion process,
very often with $H\to\gamma\gamma$ decays CMS:2017dib ; CMS:2018piu ;
ATLAS:2018doi ; CMS:2018uag . In such a case, the production and, depending on
the final state, the decay occur via virtual loops CMS:2017dib ; CMS:2018piu ;
ATLAS:2018doi ; CMS:2018uag . Therefore, these measurements have been largely
based on the assumption that no unknown particles are allowed in these loops.
A direct test of the $Y_{t}$ coupling can be performed through the production
of the Higgs boson in association with a top quark pair. In 2018 this process
was observed by the ATLAS and CMS collaborations ATLAS:2018mme ; CMS:2018uxb .
Recently, ATLAS and CMS have reported first experimental Higgs-top ${\cal CP}$
studies, exploring the $t\bar{t}H$ process with various Higgs boson decay
channels ATLAS:2020ior ; CMS:2020cga ; CMS:2022def . Even though the
measurements favor the SM Higgs-top coupling, the ${\cal CP}$-violating
coupling has not yet been excluded. Using the Higgs boson decays to two
photons, the ATLAS collaboration has been able to exclude the purely ${\cal
CP}$-odd hypothesis with $3.9$ standard deviations and establish a $95\%$ C.L.
exclusion limit for the mixing angle of $43^{\circ}$. On the other hand, the
CMS collaboration has performed the first measurement of the ${\cal CP}$
mixing angle for the $\tau$ lepton Yukawa coupling CMS-PAS-HIG-20-006  which
has been found to be $4^{\circ}\pm 17^{\circ}$. This has allowed them to set
an exclusion limit on the mixing angle of $36^{\circ}$. Of course for current
and future comparisons with increasingly accurate experimental measurements by
ATLAS and CMS, equally precise theoretical predictions must also be provided.

The effective Lagrangian for the spin ${\cal J}=0$ state can be written down
using the so-called Higgs characterisation model Artoisenet:2013puc . In this
parametrisation, the SM case, i.e. the ${\cal CP}$-even state that is
invariant under charge-parity (${\cal CP}$) inversion and described by ${\cal
J}^{\cal CP}=0^{++}$, can easily be recovered. Furthermore, a ${\cal CP}$-odd
state as described by ${\cal J}^{\cal CP}=0^{+-}$, which is typical for a
generic 2HDM, is included. In addition, ${\cal CP}$ mixing between the ${\cal
CP}$-even and ${\cal CP}$-odd state is allowed and it is parameterised in
terms of the mixing angle $\alpha_{CP}$. Finally, within this framework the
${\cal CP}$-odd, ${\cal CP}$-even and ${\cal CP}$-mixed states can all couple
to SM particles. In practice, the effective Lagrangian that describes the top
Yukawa coupling can be written as a superposition of a ${\cal CP}$-even and a
${\cal CP}$-odd phase. Any deviation from the SM value for the coupling would
indicate ${\cal CP}$ violation in the top-Higgs sector and could be
immediately translated into new physics effects beyond the SM.

The Higgs characterisation model has been studied extensively in the
literature. These studies include, among other processes, the Higgs boson
production in association with top quarks, see e.g. Refs. Hou:2018uvr ;
Bahl:2020wee ; Azevedo:2020fdl ; Bortolato:2020zcg ; Martini:2021uey ;
Bahl:2021dnc . The impact of higher-order corrections on ${\cal CP}$-sensitive
observables has been investigated e.g. in Refs. Frederix:2011zi ;
Garzelli:2011vp ; Artoisenet:2012st ; Biswas:2014hwa ; Demartin:2014fia ;
Hartanto:2015uka ; Demartin:2015uha ; Demartin:2016axk . Specifically, $pp\to
t\bar{t}H$ production, the combined $t$-channel $tH+\bar{t}H$ production
process and $tWH$ have been studied including parton shower effects. For all
three processes next-to-leading-order (NLO) QCD corrections have only been
taken into account for the production stage while top-quark decays have been
treated using parton shower approximations. In some cases spin-correlated LO
top-quark decays have been included as well. In practise, however, such
simulations are always based on the on-shell top-quark approximation.
Moreover, for the computation of NLO QCD corrections to the $tWH$ process the
isolation of the $t\bar{t}H$ process and elimination of its double resonant
contributions is required. In order to achieve this, the authors have used
subtractions that are fully local in the phase space. Specifically, they have
applied the two main schemes that are present in the literature and are known
under the name of diagram removal and diagram subtraction Frixione:2008yi .
However, the only way to perform theoretically consistent simulations that
comprise both $t\bar{t}H$ and $tW(b)H$ is to compute the $pp\to
W^{+}b\,W^{-}\bar{b}\,H$ process using a complex top-quark mass. We point out
here that the $tWH$ contribution in the final state with one $b$-jet can be
obtained with massive bottom quarks using the four-flavour scheme (see e.g.
Jezo:2016ujg  for a simpler case of $t\bar{t}$ and $tW$). Instead, for $tWbH$
in the final state with two $b$-jets, massless bottom quarks and the five
flavour-scheme can be employed. A simulation of the $pp\to
W^{+}b\,W^{-}\bar{b}\,H$ process would include all contributions that are, on
the one hand, gauge invariant and, on the other hand, include interference and
finite top-quark width effects. It would also contain contributions from the
amplitude without any resonant top-quark propagator and, in turn, interference
effects between single-top and single-antitop contributions. The latter are
absent in $tWbH$ simulations. A particular final state that we are interested
in can be enhanced and probed by means of kinematic cuts, closely following
what is being done in the ATLAS and CMS experiments. We note that such full
NLO QCD predictions for the $t\bar{t}H$ process with leptonic top-quark decays
exist but only for the ${\cal CP}$-even case Denner:2015yca ; Stremmer:2021bnk
. Later also electroweak corrections have been calculated for this process
Denner:2016wet . Very recently even various Higgs boson decays have been added
albeit in the narrow-width-approximation (NWA) Stremmer:2021bnk . Taking into
account how crucial the measurements of the top-quark Yukawa coupling are, it
is of utmost importance to provide the full NLO QCD calculation for the
$t\bar{t}H$ process allowing beyond the SM top-Higgs couplings and taking into
account all effects whilst consistently avoiding any approximations.

The purpose of the article is, therefore, manifold. Firstly, we provide state-
of-the-art NLO QCD predictions for the $pp\to t\bar{t}H$ process in the di-
lepton top-quark decay channel with an extended top-Higgs Yukawa coupling
within the Higgs characterisation framework. Specifically, we calculate NLO
QCD corrections to the $e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$
final state, consistently taking into account double-, single and non-resonant
top-quark contributions together with all interference effects. Moreover, in
the computation off-shell, top quarks are treated in the complex mass scheme
with a physical top-quark width $\Gamma_{t}$. Similarly, non-resonant and off-
shell effects due to the finite $W$ boson width are also incorporated in the
calculation. Schematically, our calculations for the
$e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ matrix element squared can
be written down as $|{\cal M}_{t\bar{t}H}|^{2}=|{\cal M}_{2tH}+{\cal
M}_{1tH}+{\cal M}_{1\bar{t}H}+{\cal M}_{0tH}|^{2}$. Secondly, we investigate
the sensitivity of the $t\bar{t}H$ process to beyond the SM physics in the
Higgs boson sector and identify observables that are particularly useful in
distinguishing the three different Higgs boson ${\cal CP}$ states. Thirdly, we
assess the impact of full off-shell effects on the $t\bar{t}H$ process with
the extended top-Higgs coupling both at the integrated and differential
fiducial cross-section level. To perform such a study, a second computation
based on the NWA is carried out for the $pp\to t\bar{t}H$ process. In this
case the $e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ final state is
generated in the $pp\to t\bar{t}H\to W^{+}b\,W^{-}\bar{b}\,H\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ decay chain where the top
quarks and $W$ gauge bosons are always kept on-shell. On the one hand, we
compare our results from the full off-shell calculation to the predictions
based on the full NWA. In that case NLO QCD corrections are calculated both to
the production stage and the top-quark decays. On the other hand, we also
provide NLO QCD predictions in the NWA but with leading order (LO) top-quark
decays (abbreviated as ${\rm NWA}_{\rm LOdec}$). By employing the ${\rm NWA}$
and ${\rm NWA}_{\rm LOdec}$ results, we are able to estimate the effects of
NLO QCD corrections to the top-quark decays.

The paper is organised as follows. In Section 2 we summarise the framework of
our calculation. The theoretical setup for LO and NLO QCD results is given in
Section 3. Results for the integrated fiducial cross-sections for the SM
(${\cal CP}$-even), ${\cal CP}$-mixed and ${\cal CP}$-odd Higgs boson are
presented in Section 4. In Section 5 predictions for differential fiducial
cross-sections are given and we investigate how the different ${\cal CP}$
configurations are affected by the higher-order effects in various phase-space
regions. Additionally, we present observables that can be used to distinguish
the ${\cal CP}$ states. Finally, in Section 5 we study the impact of off-shell
effects on differential fiducial cross sections and examine the size of NLO
QCD corrections to top-quark decays at the differential level. We summarise
the results and outline our conclusions in Section 6.

##  2 Description of the calculation and Higgs parametrisation

Figure 1:  Representative LO Feynman diagrams for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process involving two (top
left), only one (top right and bottom left) or no (bottom right) top-quark
resonances. The Feynman diagrams were created with the help of the FeynGame
Harlander:2020cyh program.

In this work we consider the production of a stable ${\cal CP}$-even, ${\cal
CP}$-odd and ${\cal CP}$-mixed Higgs boson in association with a leptonically
decaying top-quark pair at order
$\mathcal{O}\left(\alpha_{s}^{3}\alpha^{5}\right)$ in perturbation theory.
Specifically, we study the process $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H+X$, i.e. we calculate NLO
QCD corrections to the LO process $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ which is of order
$\mathcal{O}\left(\alpha_{s}^{2}\alpha^{5}\right)$. Following the Higgs
characterisation framework Artoisenet:2013puc , we allow for a possible
admixture of a ${\cal CP}$-odd Higgs state and extend the top Yukawa
interaction according to

$\mathcal{L}_{t\bar{t}H}=-\bar{\psi}_{t}\frac{Y_{t}}{\sqrt{2}}\left(\kappa_{Ht\bar{t}}\,\cos(\alpha_{CP})+i\kappa_{At\bar{t}}\,\sin(\alpha_{CP})\gamma_{5}\right)\psi_{t}\,H,$
(1)

where $\psi_{t}$ and $H$ are the top and Higgs fields, respectively, whereas
$\alpha_{CP}$ is the ${\cal CP}$-mixing angle. Furthermore,
$Y_{t}=\sqrt{2}m_{t}/v$ is the top quark Yukawa coupling with the Higgs vacuum
expectation value $v\approx 246$ GeV and the top quark mass $m_{t}$. In this
parametrisation:

  * •

$\alpha_{CP}=0$ corresponds to the ${\cal CP}$-even case,

  * •

$\alpha_{CP}=\pi/2$ corresponds to the ${\cal CP}$-odd case,

  * •

$\alpha_{CP}=\pi/4$ corresponds to the ${\cal CP}$-mixed case.

In practice, a ${\cal CP}$-mixed scenario could be realised by choosing any
value of $\alpha_{CP}$ from the range $\alpha_{CP}\in\left(0,\pi/2\right)$ but
we focus here on the $\alpha_{CP}=\pi/4$ case for which
$\cos(\alpha_{CP})=\sin(\alpha_{CP})$. Different choices for the real-valued
couplings $\kappa_{Ht\bar{t}}$ and $\kappa_{At\bar{t}}$ will be discussed in
Section 4. In addition to changing the Yukawa interaction, we also introduce
an additional parameter $\kappa_{HVV}$ in the $HVV$ ($V=W^{\pm},Z$)
interaction term

$\mathcal{L}_{HVV}=\kappa_{HVV}\left(\frac{1}{2}\,g_{HZZ}Z_{\mu}Z^{\mu}+g_{HWW}W^{+}_{\mu}W^{-\mu}\right)H,$
(2)

where $g_{HZZ}=2m_{Z}^{2}/v$ and $g_{HWW}=2m_{W}^{2}/v$ are the $HVV$
couplings with $m_{Z,W}$ being the masses of the electroweak (EW) gauge
bosons. We do not take into account higher dimensional $HVV$ couplings and
neglect any contribution from loop-induced couplings of the Higgs boson to
gluons or photons. The above described $HVV$ couplings are only of importance
in full off-shell calculations such as the one presented here since on-shell
top quarks are not heavy enough to produce both a Higgs and a $W$ boson.

In order to incorporate full off-shell effects, we consistently take into
account all Feynman diagrams with the
$e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ final state at order
$\mathcal{O}\left(\alpha_{s}^{2}\alpha^{5}\right)$ (for the LO computation)
and interference effects, irrespective of whether a $t\bar{t}$ pair actually
occurs as an intermediate state. The same applies to the $W$ bosons appearing
in the calculation. Examples for such diagrams with two, only one or no top-
quark resonances, to which we will be referring as double-, single- and non-
resonant contributions in the following, are depicted in Figure 1. Assuming a
diagonal Cabibbo-Kobayashi-Maskawa (CKM) matrix and using the $5$-flavor
scheme, i.e. setting both the mass and the Yukawa coupling of the bottom quark
to zero, we obtain $174$ diagrams for the $gg$ and $76$ for the $q\bar{q}$
initial state at LO where $q\in\left\\{u,d,c,s\right\\}$. On the other hand,
$b\bar{b}$ initial states are neglected as they are at the per-mille level
Stremmer:2021bnk and thus well within NLO scale uncertainties. The numbers of
diagrams are independent of the ${\cal CP}$-structure of the Higgs boson as
long as $\kappa_{HVV}\,,\kappa_{At\bar{t}}\,,\kappa_{Ht\bar{t}}\neq 0$. To
include Breit-Wigner propagators for the top quark and gauge bosons in a gauge
invariant way, we use the complex-mass scheme Denner:1999gp ; Denner:2005fg ;
Bevilacqua:2010qb ; Denner:2012yc . In addition to the full off-shell case we
also provide results in the NWA, i.e. the limit in which the unstable
intermediate top quarks and $W$ gauge bosons are put on-shell. As only double-
resonant diagrams (see e.g. top-left diagram in Figure 1) are included when
employing the NWA, the numbers of considered diagrams for the $gg$ and
$q\bar{q}$ initial states are reduced to $8$ and $2$, respectively, at LO.

All of the results presented in this work were obtained using the Helac-NLO
Bevilacqua:2011xh ; Bevilacqua:2019quz MC program which consists of Helac-
Dipoles Czakon:2009ss and Helac-1Loop vanHameren:2009dr . Amplitudes are
constructed in a recursive manner based on recursive Dyson-Schwinger equations
Draggiotis:1998gr ; Draggiotis:2002hm ; Papadopoulos:2005ky and the phase-
space integration is performed with the help of Parni vanHameren:2007pt and
Kaleu vanHameren:2010gg . NLO corrections to the Born level process can be
divided into virtual and real corrections which are computed in Helac-1Loop
and Helac-Dipoles, respectively. Note that from the point of view of QCD
corrections, this process is essentially identical to $t\bar{t}\gamma$,
$t\bar{t}W^{\pm}$ or $t\bar{t}Z$ production as presented in Refs.
Bevilacqua:2018woc ; Bevilacqua:2019cvp ; Bevilacqua:2020pzy . For the virtual
corrections we use the CutTools Ossola:2007ax implementation of the OPP
reduction Ossola:2006us in order to reduce the occurring one loop amplitudes
to scalar integrals at the integrand level. The resulting scalar integrals are
then evaluated using OneLOop vanHameren:2010cp . The real corrections receive
contributions from $2\to 8$ partonic processes. In our case we need to
consider the following subprocesses:

$\displaystyle\begin{split}gg&\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,Hg\\\ q\bar{q}&\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,Hg\\\ gq&\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,Hq\\\ g\bar{q}&\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H\bar{q}\end{split}$ (3)

Their respective contributions are computed using Helac-Dipoles. It implements
two independent subtraction schemes, the Catani-Seymour dipole formalism
Catani:1996vz ; Catani:2002hc and the Nagy-Soper subtraction scheme
Bevilacqua:2013iha . We use the Nagy-Soper scheme for the full off-shell
calculations while those done in the NWA are performed in the Catani-Seymour
formalism and its extension to top-quark decays Campbell:2004ch ;
Bevilacqua:2019quz .

Our full off-shell results are stored as (partially) unweighted events
Bevilacqua:2016jfk in (modified) Les Houches Event Files (LHEFs) Alwall:2006yp
and Root Ntuples Antcheva:2009zz . This allows us to re-weight our results in
order to efficiently accommodate different parton distribution functions
(PDFs) as well as renormalisation and factorisation scale settings.

The implementation of the modified $HVV$ and $Ht\bar{t}$ interactions in
Helac-NLO according to equations (1) and (2) have been cross-checked with
MadGraph_aMC@NLO Alwall:2014hca and MadLoop Hirschi:2011pa . Furthermore,
various tests have been performed within the Helac-NLO code. Specifically, we
have compared the LO full off-shell squared amplitudes for $gg\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ and $u\bar{u}\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ with MadGraph_aMC@NLO for a
few random phase-space points. The latter have been generated with the help of
the Rambo Kleiss:1985gy MC phase-space generator. In addition, we have checked
the finite parts along with the coefficients of the poles in $\epsilon$, i.e.
$1/\epsilon$ and $1/\epsilon^{2}$, of the virtual amplitudes for stable
$t\bar{t}H$ production for $gg$ and $u\bar{u}$ initial states for all three
${\cal CP}$ states. For the real corrections we have ensured that the poles in
$\epsilon$ occurring in the $\mathbf{I}$-operator and those appearing in the
virtual part cancel. Finally, we have confirmed that the real emission part is
independent from any phase-space restriction imposed by the unphysical
$\alpha_{max}$ parameter Nagy:1998bb ; Nagy:2003tz ; Bevilacqua:2009zn ;
Czakon:2015cla .

##  3 LHC setup

We consider the ${\cal CP}$-even, ${\cal CP}$-odd and ${\cal CP}$-mixed Higgs
production in association with top quarks at the LHC with $\sqrt{s}=13$ TeV
and calculate higher-order corrections to the
$e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ final state at ${\cal
O}(\alpha_{s}^{3}\alpha^{5})$. We work in the 5-flavour scheme ($N_{f}=5$) and
neglect the contribution from bottom quarks in the initial state. The LHAPDF
interface Buckley:2014ana is used to provide an access to parton density
functions and we employ the NNPDF31-nlo-as-0118 PDF set Ball:2017nwa with
$\alpha_{s}(m_{Z})=0.118$ at LO and NLO in QCD. The running of the strong
coupling constant is performed with two-loop accuracy. The $G_{\mu}-$scheme is
used for the electroweak input parameters and we adopt the same values as in
Ref. Bevilacqua:2020pzy . Thus, the electromagnetic coupling constant $\alpha$
is given by

$\alpha=\frac{\sqrt{2}}{\pi}\,G_{\mu}\,m_{W}^{2}\,\left(1-\frac{m_{W}^{2}}{m_{Z}^{2}}\right)\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ G_{\mu}=1.166378\cdot
10^{-5}\textrm{ GeV}^{-2}\,.$ (4)

We use the following masses and widths

$\begin{array}[]{lll}m_{t}=172.5\leavevmode\nobreak\ {\rm
GeV}\,,&&m_{H}=125\leavevmode\nobreak\ {\rm GeV}\,,\vspace{0.2cm}\\\
m_{W}=80.385\leavevmode\nobreak\ {\rm
GeV}\,,&&\Gamma_{W}=2.09767\leavevmode\nobreak\ {\rm GeV}\,,\vspace{0.2cm}\\\
m_{Z}=91.1876\leavevmode\nobreak\ {\rm
GeV}\,,&&\Gamma_{Z}=2.50775\leavevmode\nobreak\ {\rm GeV}\,,\end{array}$ (5)

and neglect the masses of all other particles. From these input parameters we
calculate the top-quark width according to the formulae derived in Refs.
Denner:2012yc ; Jezabek:1988iv . For the values provided above, the LO and NLO
top-quark widths are given by

$\Gamma_{t}^{\rm LO}=1.45759\leavevmode\nobreak\ {\rm
GeV}\,,\quad\quad\quad\Gamma_{t}^{\rm NLO}=1.33247\leavevmode\nobreak\ {\rm
GeV}\,$ (6)

in the full off-shell case and by

$\Gamma_{t,{\rm NWA}}^{\rm LO}=1.48063\leavevmode\nobreak\ {\rm
GeV}\,,\quad\quad\quad\Gamma_{t,{\rm NWA}}^{\rm
NLO}=1.35355\leavevmode\nobreak\ {\rm GeV}\,$ (7)

in the NWA. The top-quark width is treated as a fixed parameter and we use
$\alpha_{s}(\mu_{R}=m_{t})$ when calculating $\Gamma^{\rm NLO}_{t}$. Since the
Higgs boson is stable, its width is set to $\Gamma_{H}=0$. IR-safety is
ensured by the anti-$k_{T}$ jet algorithm Cacciari:2008gp which is used to
cluster final state partons with pseudo-rapidity $|\eta|<5$ into jets with the
jet-resolution parameter $R=0.4$. We require exactly two charged leptons, two
$b$-jets and one stable Higgs boson. All final states have to fulfil the
following experimental cuts

$\begin{array}[]{ll l l }p_{T,\,\ell}>25\leavevmode\nobreak\ {\rm
GeV}\,,&&p_{T,\,b}>25\leavevmode\nobreak\ {\rm GeV}\,,\vspace{0.2cm}\\\
|y_{\ell}|<2.5\,,&&|y_{b}|<2.5\,,\end{array}$ (8)

where $\ell=\mu^{-},e^{+}$. No restrictions on the extra light jet and missing
transverse momentum, denoted as
$p_{T,\,miss}=|\vec{p}_{T,\,\nu_{e}}+\vec{p}_{T,\,\bar{\nu}_{\mu}}|$, are
applied. We use a dynamical scale setting $\mu_{0}=\mu_{R}=\mu_{F}=H_{T}/2$ as
our central scale, where $H_{T}$ is defined according to

$H_{T}=p_{T,\,b_{1}}+p_{T,\,b_{2}}+p_{T,\,e^{+}}+p_{T,\,\mu^{-}}+p_{T,\,miss}+p_{T,\,H}\,,$
(9)

where $b_{1}$ and $b_{2}$ stand for the first and second hardest $b$-jet in
$p_{T}$. A detailed discussion and comparison with other scale choices is
given for the SM case ($\alpha_{CP}=0$) in Ref. Stremmer:2021bnk . In order to
estimate the theoretical uncertainties arising from neglected higher-order
terms in the perturbative expansion, we use the $7$-point scale variation in
which the factorisation and renormalisation scales are varied independently in
the range

$\frac{1}{2}\,\mu_{0}\leq\mu_{R}\,,\mu_{F}\leq
2\,\mu_{0}\,,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\frac{1}{2}\leq\frac{\mu_{R}}{\mu_{F}}\leq
2\,.$ (10)

This leads to the following seven pairs

$\left(\frac{\mu_{R}}{\mu_{0}}\,,\frac{\mu_{F}}{\mu_{0}}\right)=\Big{\\{}\left(2,1\right),\left(0.5,1\right),\left(1,2\right),(1,1),(1,0.5),(2,2),(0.5,0.5)\Big{\\}}\,,$
(11)

which are taken into account for the error estimation. We do not include PDF
uncertainties into the theoretical error estimation, since these are
significantly smaller than the scale uncertainties Stremmer:2021bnk ,
especially for the chosen PDF set.

##  4 Integrated fiducial cross-sections

Figure 2:  LO integrated fiducial cross-section depending on $\alpha_{CP}$ for
the $pp\to e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the
LHC with $\sqrt{s}=13\textrm{ TeV}$. The red curve indicates the results
computed in the NWA for $\kappa_{At\bar{t}}=2/3$ whereas the blue, orange and
green curves correspond to full off-shell results for different choices of the
coupling constants. In all cases we have $\kappa_{Ht\bar{t}}=1$. In the lower
panel the ratio to ${\alpha_{CP,\,max}}$ is displayed for each plotted case.

In this first part of our analysis we will discuss the results for integrated
fiducial cross-sections. The two coupling parameters $\kappa_{Ht\bar{t}}$ and
$\kappa_{At\bar{t}}$ appearing in Eq. (1) are chosen as follows. We take
$\kappa_{Ht\bar{t}}=1$ everywhere to recover the SM results for
$\alpha_{CP}=0$. For $\kappa_{At\bar{t}}$ we consider two different case,
$\kappa_{At\bar{t}}=1$ and $\kappa_{At\bar{t}}=2/3$. The first choice is
simply designed to have the same coupling for both the scalar and pseudoscalar
Higgs boson. The second one is motivated by the measurements of Higgs boson
production in gluon-fusion (GF) process which indicate good agreement with the
SM predictions ATLAS-CONF-2021-014 ; CMS-PAS-HIG-18-029 . Assuming that the
top-loop induced terms dominate the GF we find

$\frac{\sigma_{gg\to H}}{\sigma_{gg\to
H}^{SM}}=\cos^{2}\left(\alpha_{CP}\right)\kappa_{Ht\bar{t}}^{2}+\sin^{2}\left(\alpha_{CP}\right)\kappa_{At\bar{t}}^{2}\,\frac{g_{Agg}^{2}}{g_{Hgg}^{2}}$
(12)

for the ratio between the GF in our model and the SM Demartin:2014fia ;
Demartin:2015uha . Here $g_{Agg}=\alpha_{s}/\left(2\pi v\right)$ and
$g_{Hgg}=-\alpha_{s}/\left(3\pi v\right)$ are the effective loop-induced
gluon-Higgs couplings for a ${\cal CP}$-even and ${\cal CP}$-odd Higgs boson,
respectively. Hence, to recover the SM results for any value of $\alpha_{CP}$
one must set $\kappa_{At\bar{t}}=2/3$ and $\kappa_{Ht\bar{t}}=1$.

For the remaining free parameter $\kappa_{HVV}$ we also analyse two different
cases, specifically $\kappa_{HVV}=\cos\left(\alpha_{CP}\right)$ and
$\kappa_{HVV}=1$. Note that in the first case $\kappa_{HVV}$ actually becomes
dependent on the mixing angle $\alpha_{CP}$. This choice is relevant for SM
extensions such as the ${\cal CP}$-conserving $2$HDM where only ${\cal
CP}$-even Higgs-like particles couple to the EW gauge bosons, see e.g. Refs.
Aoki:2009ha ; Branco:2011iw . Thus, the corresponding couplings receive the
same scaling as the scalar Higgs-top coupling. However, when considering a
possible non-zero $\alpha_{CP}$ for the SM Higgs boson it is more sensible to
choose $\kappa_{HVV}=1$ in order to recover results from Higgs boson
production through vector boson fusion (VBF) which, just as the GF
measurements, indicate agreement with the SM CMS:2018uag . Let us mention that
in the NWA, the choice of $\kappa_{HVV}$ is irrelevant since in this case the
Higgs boson can only be radiated off top quarks in the production stage.

In Figure 2 we present the LO integrated fiducial cross-sections depending on
the mixing angle $\alpha_{CP}$ for several different combinations of the
parameter choices outlined above. The blue, orange and green curves indicate
results for the full off-shell case with different choices for the couplings
and the red one describes the NWA with $\kappa_{At\bar{t}}=2/3$. To create
these curves one can make use of the fact that at LO the cross-section can be
split into six parts according to

$\displaystyle\begin{split}\sigma\left(\alpha_{CP}\right)&=\cos^{2}\left(\alpha_{CP}\right)\kappa_{Ht\bar{t}}^{2}\,\,\sigma_{1}+\sin^{2}\left(\alpha_{CP}\right)\kappa_{At\bar{t}}^{2}\,\,\sigma_{2}+\cos\left(\alpha_{CP}\right)\sin\left(\alpha_{CP}\right)\kappa_{Ht\bar{t}}\,\kappa_{At\bar{t}}\,\,\sigma_{3}\\\
&+\cos\left(\alpha_{CP}\right)\kappa_{Ht\bar{t}}\,\kappa_{HVV}\left(\alpha_{CP}\right)\,\sigma_{4}+\sin\left(\alpha_{CP}\right)\kappa_{At\bar{t}}\,\kappa_{HVV}\left(\alpha_{CP}\right)\,\sigma_{5}+\kappa_{HVV}^{2}\left(\alpha_{CP}\right)\,\sigma_{6}\,,\end{split}$
(13)

where the $\sigma_{i}$ coefficients are independent of the mixing angle,
$\kappa_{Ht\bar{t}}$, $\kappa_{At\bar{t}}$ and $\kappa_{HVV}$. Thus, it is
enough to compute six integrated cross-sections with different sets of input
parameters
$\\{\alpha_{CP},\kappa_{Ht\bar{t}},\kappa_{At\bar{t}},\kappa_{HVV}\\}$. From
these, the cross-section contributions $\sigma_{i}$ in Eq. (13) can then be
calculated by solving a system of equations. The final results for
$\sigma(\alpha_{CP})$ with all required sets of parameters
$\\{\alpha_{CP},\kappa_{Ht\bar{t}},\kappa_{At\bar{t}},\kappa_{HVV}\\}$ can
then easily be assembled from Eq. (13). In case of the NWA it is even
sufficient to only evaluate the first three contributions as no $HVV$
couplings can occur. Therefore, we have $\sigma_{4,5,6}^{\rm NWA}=0$ both at
LO and NLO in QCD. The numerical values for the cross-section coefficients
$\sigma_{i}$ are listed in Table 1. Although there is a non-zero contribution
from interference effects between the ${\cal CP}$-odd coupling and the SM
contribution at the matrix element level, any hadron collider observable is
averaged over charge conjugate processes. As a result, interference effects
vanish in typical hadronic (differential) cross-sections when integrated over
a ${\cal CP}$-symmetric part of the phase space. This observation still holds
for fiducial cross-sections $(\sigma_{3},\sigma_{5})$ when ${\cal
CP}$-symmetric selection criteria are applied to all final states, see e.g.
Ref. Plehn:2001nj .

Table 1: Cross-section coefficients, $\sigma_{i}$, for the splitting of the LO integrated fiducial cross-sections according to Eq. (13) for the full off-shell and narrow-width treatments. For the latter the final three coefficients do not contribute as no diagrams with couplings between the Higgs boson and the EW gauge bosons occur in the NWA. The values in brackets indicate the respective Monte Carlo integration error on the last digit.  |  Off-shell |  NWA
---|---|---
$\sigma_{1}$ [fb] | $\leavevmode\nobreak\ \leavevmode\nobreak\ 2.0643(4)$ |  $\leavevmode\nobreak\ \leavevmode\nobreak\ 2.0388(2)$
$\sigma_{2}$ [fb] | $\leavevmode\nobreak\ \leavevmode\nobreak\ 0.7800(1)$ |  $\leavevmode\nobreak\ \leavevmode\nobreak\ 0.74583(7)$
$\sigma_{3}$ [fb] | $-0.0002(8)$ |  $-0.0001(3)$
$\sigma_{4}$ [fb] | $-0.0693(8)$ |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ -$
$\sigma_{5}$ [fb] | $-0.0001(9)$ |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ -$
$\sigma_{6}$ [fb] | $\leavevmode\nobreak\ \leavevmode\nobreak\ 0.0363(9)$ |  $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ -$

We note that Eq. (13) is also instructive in understanding and interpreting
the different behaviours of the curves depicted in Figure 2. First, one should
note that the production of pseudoscalar particles in association with top
quarks is suppressed compared to scalar ones for masses below $\sim 200$ GeV
if the two couplings $\kappa_{Ht\bar{t}}$ and $\kappa_{At\bar{t}}$ are equal
Haisch:2016gry . This difference can be understood when looking at the $t\to
t+H/A$ fragmentation functions Dittmaier:2000tc ; Dawson:1997im

$\displaystyle\begin{split}f_{t\to
t+H}(x)&=\frac{\kappa^{2}_{Ht\bar{t}}}{(4\pi)^{2}}\left[\frac{4(1-x)}{x}+x\ln\left(\frac{s}{m_{t}^{2}}\right)\right]\,,\\\
f_{t\to
t+A}(x)&=\frac{\kappa^{2}_{At\bar{t}}}{(4\pi)^{2}}\left[x\ln\left(\frac{s}{m_{t}^{2}}\right)\right]\,,\end{split}$
(14)

where $x$ is the momentum fraction that the Higgs boson carries. The scalar
fragmentation function has an additional $1/x$ term which enhances the
production of soft scalar particles compared to pseudoscalar particles and
thus results in a larger integrated fiducial cross-section for a scalar Higgs
boson. This can clearly be seen in Figure 2 as all depicted curves have a
minimum around $\alpha_{CP}=\pi/2$ which corresponds to the production of a
${\cal CP}$-odd Higgs boson. Reducing $\kappa_{At\bar{t}}$ from $1$ to $2/3$
only enhances this effect which is why around its minimum the orange curve
$(\kappa_{At\bar{t}}=1)$ is about twice as large as the green one
$(\kappa_{At\bar{t}}=2/3)$. For the latter choice of $\kappa_{At\bar{t}}$, the
production cross-section of $2.03$ fb for a ${\cal CP}$-even Higgs boson is
more than five times as large as the $0.38$ fb for the ${\cal CP}$-odd case.
The mixed case falls almost exactly in the middle with $1.19$ fb.

One can also observe that the cross-sections computed in the NWA (red curve)
are symmetric with respect to $\alpha_{CP}\rightarrow\pi-\alpha_{CP}$ which is
equivalent to changing the sign of the top-quark Yukawa coupling $Y_{t}$. As
we have $\sigma_{4,5,6}^{\rm NWA}=0$ in the NWA, we only need to consider the
terms in the first line of Eq. (13) in order to understand this phenomenon.
For the first two terms the symmetry is inherent in the explicit $\alpha_{CP}$
dependence since
$\cos^{2}\left(\pi-\alpha_{CP}\right)=\cos^{2}\left(\alpha_{CP}\right)$ and
$\sin^{2}\left(\pi-\alpha_{CP}\right)=\sin^{2}\left(\alpha_{CP}\right)$. In
contrast, for the third term we have
$\cos\left(\pi-\alpha_{CP}\right)\sin\left(\pi-\alpha_{CP}\right)=-\cos\left(\alpha_{CP}\right)\sin\left(\alpha_{CP}\right)$
which means that $\sigma_{3}$ must vanish in order to explain the observed
behaviour which is in line with our numerical results in Table 1. From this we
can conclude that there is no mixing between the production of ${\cal
CP}$-even and ${\cal CP}$-odd Higgs bosons which is known for stable top
quarks, see e.g. Ref. Bahl:2020wee .

In the full off-shell case we only observe the same symmetry if we set
$\kappa_{HVV}\left(\alpha_{CP}\right)=\cos\left(\alpha_{CP}\right)$. With this
choice the fourth and sixth term in (13) have the same
$\cos^{2}\left(\alpha_{CP}\right)$ dependence as the first one and the fifth
term becomes proportional to
$\cos\left(\alpha_{CP}\right)\sin\left(\alpha_{CP}\right)$. As the symmetry of
the cross-section under the replacement
$\alpha_{CP}\rightarrow\pi-\alpha_{CP}$ is clearly visible in the blue curve
of Figure 2, we conclude that both $\sigma_{3}$ and $\sigma_{5}$ must either
vanish or cancel, which is again in line with our numerical results. This is
broken if we take the other choice for our $HVV$ coupling, $\kappa_{HVV}=1$,
which corresponds to the green and orange curves in Figure 2. In this case the
fourth term becomes proportional to just $\cos\left(\alpha_{CP}\right)$. As
the corresponding coefficient is negative, we find that the cross-section for
$\alpha_{CP}=\pi$ is actually larger than the one for $\alpha_{CP}=0$ by about
$0.14$ fb which translates to an increase of $7\%$.

Table 2: Comparison of LO and NLO QCD integrated fiducial cross-sections as calculated in the NWA, NWA with LO top-quark decays and full off-shell approach for $\alpha_{CP}=0,\pi/4$ and $\pi/2$. Also shown are their respective scale uncertainties and Monte Carlo integration errors. All values are given for the $pp\to e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with $\sqrt{s}=13\textrm{ TeV}$. $\alpha_{CP}$ |  | Off-shell | NWA |  Off-shell effects
---|---|---|---|---
$0$ (SM) |  $\sigma_{\text{LO}}$ [fb] | $2.0313(2)^{+0.6275\,(31\%)}_{-0.4471\,(22\%)}$ |  $2.0388(2)^{+0.6290\,(31\%)}_{-0.4483\,(22\%)}$ | $-0.37\%$
$\sigma_{\text{NLO}}$ [fb] | $2.466(2)^{+0.027\,(1.1\%)}_{-0.112\,(4.5\%)}$ |  $2.475(1)^{+0.027\,(1.1\%)}_{-0.113\,(4.6\%)}$ | $-0.36\%$
$\sigma_{\text{NLO}_{\text{LOdec}}}$ [fb] | $-$ |  $2.592(1)^{+0.161\,(6.2\%)}_{-0.242\,(9.3\%)}$ |
|  ${\cal K}=\sigma_{\text{NLO}}/\sigma_{\text{LO}}$ | $1.21$ |  $1.21$ (LOdec: $1.27$) |
$\pi/4$ |  $\sigma_{\text{LO}}$ [fb] | $1.1930(2)^{+0.3742\,(31\%)}_{-0.2656\,(22\%)}$ |  $1.1851(1)^{+0.3707\,(31\%)}_{-0.2633\,(22\%)}$ | $0.66\%$
$\sigma_{\text{NLO}}$ [fb] | $1.465(2)^{+0.016\,(1.1\%)}_{-0.071\,(4.8\%)}$ |  $1.452(1)^{+0.015\,(1.0\%)}_{-0.069\,(4.8\%)}$ | $0.89\%$
$\sigma_{\text{NLO}_{\text{LOdec}}}$ [fb] | $-$ |  $1.517(1)^{+0.097\,(6.4\%)}_{-0.144\,(9.5\%)}$ |
|  ${\cal K}=\sigma_{\text{NLO}}/\sigma_{\text{LO}}$ | $1.23$ |  $1.23$ (LOdec: $1.28$) |
$\pi/2$ |  $\sigma_{\text{LO}}$ [fb] | $0.38277(6)^{+0.13123\,(34\%)}_{-0.09121\,(24\%)}$ |  $0.33148(3)^{+0.11240\,(34\%)}_{-0.07835\,(24\%)}$ | $13.4\%$
$\sigma_{\text{NLO}}$ [fb] | $0.5018(3)^{+0.0083\,(1.2\%)}_{-0.0337\,(6.7\%)}$ |  $0.4301(2)^{+0.0035\,(0.8\%)}_{-0.0264\,(6.1\%)}$ | $14.3\%$
$\sigma_{\text{NLO}_{\text{LOdec}}}$ [fb] | $-$ |  $0.4433(2)^{+0.0323\,(7.3\%)}_{-0.0470\,(11\%)}$ |
|  ${\cal K}=\sigma_{\text{NLO}}/\sigma_{\text{LO}}$ | $1.31$ |  $1.30$ (LOdec: $1.34$) |

Let us also mention that the final term in equation (13) becomes independent
of $\alpha_{CP}$ when choosing $\kappa_{HVV}=1$. This means that in this case,
the breaking of the symmetry around $\alpha_{CP}=\pi/2$ in the full off-shell
case is only a result of the fourth term. This contribution corresponds to the
interference between diagrams where the Higgs boson is radiated off a $W$ or
$Z$ boson ( single- and non-resonant top-quark contributions) with diagrams
where the Higgs boson is produced from one of the top quarks (double- and
single-resonant top-quark contributions), see Figure 1. As single- and non-
resonant Feynman diagrams are absent in the NWA, the dependence on
$\alpha_{CP}$ is symmetric in this case. The very different behaviors in the
full off-shell treatment and the NWA lead us to conclude that the inclusion of
off-shell effects is vital, in particular when considering large mixing
angles. Since the discrepancy between the two approaches is mostly driven by
the $\sigma_{4}$ term, their difference increases with increasing mixing
angle.

This observation also holds at NLO, as can be seen from the integrated
fiducial cross-sections listed in Table 2. Here, we compare full off-shell and
NWA as well as LO and NLO cross-sections for $\alpha_{CP}=0,\pi/4$ and
$\pi/2$. We do this comparison only for a few values of $\alpha_{CP}$ and
fixed $\kappa_{HVV}$ and $\kappa_{At\bar{t}}$ since equation (13) does not
hold at one-loop level for the full off-shell calculation, making a simple
interpolation impossible at NLO. Following the argumentation of Ref.
Demartin:2016axk , we will henceforth use $\kappa_{HVV}=1$ and
$\kappa_{At\bar{t}}=2/3$ in order to be consistent with ATLAS and CMS
measurements since we focus here on a potential non-zero mixing angle
$\alpha_{CP}$ for the SM Higgs boson.

Just like at LO, the off-shell effects at NLO increase for larger values of
$\alpha_{CP}$. For the SM case we find that the full off-shell results at LO
and NLO in QCD are slightly smaller than those obtained using the NWA.
Specifically, the full top-quark and $W$ boson off-shell effects change the
integrated LO and NLO fiducial cross-section by about $0.4\%$. The finding is
consistent with the expected uncertainty of the NWA Fadin:1993kt , which for
sufficiently inclusive observables is of the order of ${\cal
O}(\Gamma_{t}/m_{t})\sim 0.8\%$. It is the other way around for ${\cal
CP}$-mixed and ${\cal CP}$-odd Higgs boson production. In these two cases the
NWA results are smaller than the full off-shell ones. Moreover, the full off-
shell effects are of the order of $0.7\%-0.9\%$ ($13\%-14\%$ ) for the ${\cal
CP}$-mixed (${\cal CP}$-odd) case. Even at the level of integrated fiducial
cross-sections, off-shell effects for the pure ${\cal CP}$-odd case are
substantial and therefore easily distinguishable from the other two cases. A
similar pattern will be visible at the differential cross-section level.

Concerning the NLO QCD corrections we find that these are positive for all
three values of $\alpha_{CP}$ and consistent between the NWA and full off-
shell results. Similarly to the off-shell effects, they increase with larger
mixing angle, albeit not as drastically. For the ${\cal CP}$-even and -mixed
scenarios we find corrections of $21\%-23\%$ while they are at the level of
$30\%-31\%$ for the ${\cal CP}$-odd case. Scale uncertainties for the ${\cal
CP}$-even and -mixed cases, taken as the maximum of the lower and upper
bounds, amount to $31\%$ at LO. After the inclusion of NLO QCD corrections,
they are reduced substantially to $5\%$. For the ${\cal CP}$-odd case
theoretical uncertainties arising from the scale dependence are of the order
of $34\%$ at LO and $6\%-7\%$ at NLO. Thus, by including higher-order effects
in $\alpha_{s}$ we have reduced the theoretical error by a factor of $6$.

In Table 2 we additionally provide results for the NWA${}_{\text{LOdec}}$,
i.e. for the NWA with NLO QCD corrections to the production stage and with LO
top-quark decays (denoted as $\sigma_{\text{NLO}_{\text{LOdec}}}$ in Table 2).
In line with previous findings for top-quark pair associated production with
$\gamma$, $H$, $W^{\pm}$ and $Z$ bosons Hermann:2021xvs ; Stremmer:2021bnk ;
Bevilacqua:2019quz ; Bevilacqua:2020pzy ; Bevilacqua:2022nrm we observe that
NWA${}_{\text{LOdec}}$ predictions are higher compared to the NLO QCD results
in the full NWA. Specifically, for the process at hand we have an increase of
$3\%-5\%$. The difference is largest for the $\cal CP$-even and smallest for
the $\cal CP$-odd Higgs boson. The corresponding $\cal{K}$-factors, defined as
${\cal K}=\sigma_{\text{NLO}_{\text{LOdec}}}/\sigma_{\text{LO}}$, are included
in Table 2 in parenthesis. We observe that not only the size of higher-order
corrections is amplified in this case, but also the associated scale
uncertainties are larger. Due to the lack of higher-order effects in top-quark
decays the latter amount to $9\%-11\%$.

Finally, let us also mention that, as expected from $t\bar{t}H$ production
with stable top quarks, we find that the mixing term $\sigma_{3}$ in Eq. (13)
between the ${\cal CP}$-even and ${\cal CP}$-odd states vanishes also for the
NLO QCD predictions in the full NWA as well as for the case with LO top-quark
decays. Indeed, we can write

$\sigma(\alpha_{CP}=\pi/4)=\frac{1}{2}\left\\{\sigma(\alpha_{CP}=0)+\sigma(\alpha_{CP}=\pi/2)\right\\}\,,$
(15)

which is fulfilled within statistical uncertainties for both cases (see Table
2).

##  5 Differential fiducial cross-sections

In this section we perform the comparison between the ${\cal CP}$-even (SM),
${\cal CP}$-mixed and ${\cal CP}$-odd Higgs boson at the differential level.
In the first step, we analyse the impact of NLO QCD corrections on various
differential cross-section distributions for the three ${\cal CP}$ cases.
Subsequently, the size of off-shell and higher-order effects in top-quark
decays is discussed.

Figure 3:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,\pi/4,\pi/2$ for the observables $p_{T,\,H}$, $p_{T,\,miss}$,
$p_{T,\,b_{1}}$ and $p_{T,\,\mu^{-}}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13\textrm{ TeV}$. The lower panels show the differential
$\mathcal{K}\textrm{-factor}$. Scale uncertainties are given for
$\alpha_{CP}=0$ in the upper panel while Monte Carlo integration errors are
displayed in both panels.

Figure 4:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,\pi/4,\pi/2$ for the observables $M_{b_{1}b_{2}}$ and
$M(be^{+})_{min}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13\textrm{ TeV}$. The lower panels show the differential
$\mathcal{K}\textrm{-factor}$. Scale uncertainties are given for
$\alpha_{CP}=0$ in the upper panel while Monte Carlo integration errors are
displayed in both panels.

In Figures 3 and 4 we present differential distributions for dimensionful
observables, specifically the transverse momentum of the Higgs boson
($p_{T,\,H}$), the total missing transverse momentum ($p_{T,\,miss}$), the
transverse momentum of the hardest $b$-jet ($p_{T,\,b_{1}}$), the transverse
momentum of the muon ($p_{T,\,\mu^{-}}$), the invariant mass of the two
$b$-jets ($M_{b_{1}b_{2}}$) and the minimum mass of the positron and a $b$-jet
($M(be^{+})_{min}$). In the upper panels of each plot the differential cross-
section distributions are given for the SM, ${\cal CP}$-mixed and ${\cal
CP}$-odd Higgs boson. Moreover, as reference, we display the theoretical
uncertainties obtained by scale variation for the SM Higgs boson. The lower
panels show the differential $\mathcal{K}\textrm{-factor}$ for all three
$\cal{CP}$ states. For $p_{T,\,H}$ we observe that the relative QCD
corrections are very similar for the SM, ${\cal CP}$-mixed and ${\cal CP}$-odd
Higgs boson. They are about $9\%-12\%$ at the beginning of the spectrum,
whereas towards the tails they increase up to $45\%$. In contrast, the QCD
corrections for $p_{T,\,miss}$ behave quiet differently for the ${\cal
CP}$-odd Higgs boson and increase from $28\%$ to $45\%$ for increasing
transverse momenta. For the other two ${\cal CP}$ configurations, the QCD
corrections rise more drastically from $15\%$ at the beginning up to $60\%$ in
the tail. Generally, in the case of $t\bar{t}$ production, $p_{T}$ observables
constructed from the decay products of both top quarks, like for example
$p_{T\,miss}$, $p_{T\,b_{1}b_{2}}$ and $p_{T\,e^{+}\mu^{-}}$, are known to
have huge ${\cal K}$-factors in the tails, see e.g. Denner:2012yc ;
Czakon:2020qbd . At LO these observables exhibit a strong suppression of the
$t\bar{t}$ cross-section above $150$ GeV. In the case of the dominant double-
resonant top-quark contributions, top-quark decay products are boosted via
their parents, which have opposite momenta. Even though the transverse
momentum of the particular decay products can be substantial, the $p_{T}$ of
the $bb$, $e^{+}\mu^{-}$ and $\nu_{e}\bar{\nu}_{\mu}$ systems can only acquire
rather small values. At NLO the $t\bar{t}$ system can attain large transverse
momentum by recoiling against extra jet radiation. Thus, the kinematical
constraint is partially lifted resulting in an enhancement of the NLO cross-
section and a huge ${\cal K}$-factor of $2-4$. For the $t\bar{t}H$ production,
where radiation of the Higgs boson off top quarks is already taking place at
LO, smaller NLO corrections are foreseen. Indeed, the latter generate ${\cal
K}$-factors in the range of $1.6-1.8$ Denner:2015yca ; Stremmer:2021bnk .
Additionally, the harder Higgs boson spectrum in the ${\cal CP}$-odd case
leads to a relaxation of the kinematical suppression already at LO for
$p_{T,\,miss}$. Similar higher-order effects are observed for
$p_{T,\,b_{1}b_{2}}$ and $p_{T,\,e^{+}\mu^{-}}$. For $p_{T,\,b_{1}}$ we again
find an enhancement of the NLO QCD corrections at the beginning of the
spectrum for the ${\cal CP}$-odd Higgs boson leading to effects up to $52\%$
compared to $40\%$ for the other ${\cal CP}$ configurations. These differences
decrease towards the high $p_{T}$-region and we observe
$\mathcal{O}(\alpha_{s})$ corrections up to $40\%$ for all three cases. For
$p_{T,\,\mu^{-}}$ we find NLO QCD corrections of about $20\%-25\%$ for the SM
and ${\cal CP}$-mixed Higgs boson while they increase up to $25\%-35\%$ for
the ${\cal CP}$-odd one. For all three ${\cal CP}$ configurations, the higher-
order effects mainly impact the normalisation and we do not observe
significant shape distortions. The QCD corrections for the differential
distribution $M_{b_{1}b_{2}}$ are very similar for all three ${\cal CP}$
configurations at the beginning of the spectrum and amount to $25\%-28\%$.
Towards the tail they increase up to $45\%$ for the ${\cal CP}$-odd Higgs
boson and up to $30\%-35\%$ for the other two ${\cal CP}$ configurations.

In the case of on-shell top quarks and $W$ gauge bosons, the observable
$M(be^{+})_{min}$ is characterised by a sharp upper bound,
$\sqrt{m_{t}^{2}-m_{W}^{2}}\approx 153$ GeV, which renders it very sensitive
to the top-quark mass. Due to additional radiation, this kinematical edge is
smeared at NLO in QCD for the NWA. For the off-shell prediction, small but
non-negligible single- and non-resonant contributions that elude the
kinematical bound around $153$ GeV are already present at LO. Thus, this
observable is particularly interesting for investigating the top-quark
modeling. At NLO this feature becomes more pronounced because of QCD radiation
that enters the $b$-jet without being emitted from its parent $b$ quark. As a
result, we find very significant higher-order effects both below and above
this kinematical bound. Additionally, we observe that the QCD corrections are
very similar for small invariant masses while towards the kinematical edge
around $153$ GeV the ${\cal CP}$-odd Higgs boson leads to $12\%-15\%$ larger
QCD corrections. The QCD corrections increase significantly for all three
${\cal CP}$ configurations for invariant masses above this edge but their size
differs substantially. We find the smallest QCD corrections in this region for
the ${\cal CP}$-odd Higgs boson at abound $50\%$ while they increase even
further to $100\%$ for the ${\cal CP}$-mixed case and to $130\%$ for the SM
Higgs boson which means that the enhancement due to real radiation above this
kinematical edge is significantly reduced in the ${\cal CP}$-odd case. As we
have already seen at the level of integrated cross-sections (Table 2), the
relative size of full off-shell effects and, in turn, the contribution from
$HVV$ couplings, are significantly larger than for the other two ${\cal CP}$
states. Also here the single- and non-resonant contributions play a crucial
role since they are less affected by this kinematical edge and thus typically
receive smaller QCD corrections than the double-resonant part in this phase-
space region.

Figure 5:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,{\pi}/{4},{\pi}/{2}$ for the observables
$\cos\theta^{*}_{e^{+}\mu^{-}}$, $\cos\theta_{e^{+}\mu^{-}}$,
$\Delta\phi_{e^{+}\mu^{-}}$ and $\Delta R_{e^{+}\mu^{-}}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13\textrm{ TeV}$. The lower panels show the differential
$\mathcal{K}\textrm{-factor}$. Scale uncertainties are given for
$\alpha_{CP}=0$ in the upper panel while Monte Carlo integration errors are
displayed in both panels.

Figure 6:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,{\pi}/{4},{\pi}/{2}$ for the observables $y_{\mu^{-}}$ and
$y_{H}$ for the $pp\to e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$
process at the LHC with $\sqrt{s}=13\textrm{ TeV}$. The lower panels show the
differential $\mathcal{K}\textrm{-factor}$. Scale uncertainties are given for
$\alpha_{CP}=0$ in the upper panel while Monte Carlo integration errors are
displayed in both panels.

In Figures 5 and 6 we show the differential distributions and differential
$\mathcal{K}\textrm{-factors}$ for the dimensionless observables
$\cos\theta^{*}_{e^{+}\mu^{-}}$, the cosine of the opening angle between the
two charged leptons ($\cos\theta_{e^{+}\mu^{-}}$), the azimuthal angle between
the two charged leptons ($\Delta\phi_{e^{+}\mu^{-}}$), the distance in the
azimuthal angle rapidity plane between two charged leptons ($\Delta
R_{e^{+}\mu^{-}}$), the rapidity of the muon ($y_{\mu^{-}}$) and the rapidity
of the Higgs boson ($y_{H}$). The observable $\cos\theta^{*}_{e^{+}\mu^{-}}$
has been introduced in Ref. Barr:2005dz and is defined as

$\cos\theta^{*}_{e^{+}\mu^{-}}=\textrm{tanh}\left(\frac{y_{e^{+}}-y_{\mu^{-}}}{2}\right)\,.$
(16)

It should be distinguished from
$\cos\theta_{e^{+}\mu^{-}}=\hat{p}_{e^{+}}\cdot\,\hat{p}_{\mu^{-}}$ which we
also use in the following. We note that $\cos\theta^{*}_{e^{+}\mu^{-}}$ can be
used to determine the ${\cal CP}$ nature of spin-0 mediators in associated
production of dark matter and $t\bar{t}$ pairs, see e.g. Haisch:2016gry ;
Hermann:2021xvs . For this observable we find that the NLO QCD corrections for
the SM and ${\cal CP}$-mixed Higgs boson again behave rather similarly. They
amount to about $20\%$ for $\cos\theta^{*}_{e^{+}\mu^{-}}\approx 0$ and
increase towards $\cos\theta^{*}_{e^{+}\mu^{-}}\approx\pm 1$ up to $40\%$ for
the SM and $45\%$ for the ${\cal CP}$-mixed case. We find the same general
behaviour for the ${\cal CP}$-odd Higgs boson. However, in this case the
higher-order corrections can even reach $55\%$. The differences between LO and
NLO distributions for $\cos\theta_{e^{+}\mu^{-}}$ are very similar for all
three ${\cal CP}$ configurations for small opening angles between the two
charged leptons and amount to $30\%-35\%$ for these configurations. In the
case of the SM and ${\cal CP}$-mixed Higgs boson they decrease for increasing
opening angle down to $15\%$ while they remain almost constant for the ${\cal
CP}$-odd case. The rather similar behaviour of the NLO QCD corrections in the
${\cal CP}$-even and ${\cal CP}$-mixed Higgs boson cases is also present in
the $\Delta\phi_{e^{+}\mu^{-}}$ distributions. Here, the corrections decrease
with rising opening angle from about $40\%-45\%$ around
$\Delta\phi_{e^{+}\mu^{-}}\approx 0$ to $10\%$ for
$\Delta\phi_{e^{+}\mu^{-}}\approx\pi$. For the ${\cal CP}$-odd Higgs boson,
higher-order corrections lead to similar differences between the LO and NLO
distributions for small angles, but these effects decrease less substantially
and only down to $20\%$ for larger angles. Thus, shape distortions are less
pronounced for the pure pseudo-scalar Higgs boson case. For $\Delta
R_{e^{+}\mu^{-}}$, the $\mathcal{O}(\alpha_{s})$ corrections for the ${\cal
CP}$-odd Higgs boson are rather flat up to about $\Delta
R_{e^{+}\mu^{-}}\approx 3$, whereas for $\Delta R_{e^{+}\mu^{-}}>3$ they
increase from $30\%$ to $55\%$. On the other hand, we observe significant
shape distortions over the whole range for the SM and ${\cal CP}$-mixed Higgs
boson where the NLO QCD corrections vary between $10\%$ and $45\%-55\%$.
Finally, for both rapidity distributions $y_{\mu^{-}}$ and $y_{H}$, the
higher-order QCD effects for the ${\cal CP}$-odd case exceed the ones for the
SM and ${\cal CP}$-mixed Higgs boson over the entire range by up to $20\%$ for
$y_{\mu^{-}}$ and $10\%$ for $y_{H}$. The SM and ${\cal CP}$-mixed cases
obtain higher-order corrections of about $20\%-25\%$ for $y_{\mu^{-}}$ while
they vary between $10\%$ and $25\%$ for $y_{H}$.

Concluding, the NLO QCD corrections for the SM (${\cal CP}$-even) and ${\cal
CP}$-mixed case are comparable for most of the observables we have examined. A
similar behaviour has already been found at the integrated fiducial level for
these two cases. Just like at the integrated level, the
$\mathcal{O}(\alpha_{s})$ corrections for the ${\cal CP}$-odd Higgs are larger
than for the other two cases. Indeed, the harder Higgs-boson radiation for the
pure pseudo-scalar case as compared to the SM Higgs boson has an important
impact on the size and shape of the higher-order QCD corrections for
dimensionful and dimensionless observables. This is especially visible for
observables which obtain large NLO QCD corrections due to additional
radiation.

Figure 7:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,{\pi}/{4},{\pi}/{2}$ for the observables $H_{T}$, $p_{T,H}$,
$M(be^{+})_{min}$ and $M_{T2,t}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13\textrm{ TeV}$. The upper panels show full off-shell and NWA
(dashed lines) results. The middle panels provide the ratio to $\alpha_{CP}=0$
of the normalised differential distributions for the full off-shell case. The
ratio NWA/off-shell is given in the lower panels. Scale uncertainties are
shown for $\alpha_{CP}=0$ in the upper and middle panels while Monte Carlo
integration errors are displayed in all panels.

In the next step we discuss the shape distortions due to the different Higgs
boson ${\cal CP}$ configurations and the size of full off-shell effects. The
latter analysis is performed by comparing the full off-shell calculation to
the NWA predictions. In this way we can estimate the impact of off-shell
effects on dimensionful and dimensionless observables for the three different
${\cal CP}$ states. We start this discussion with the four dimensionful
observables $H_{T}$, $p_{T,\,H}$, $M(be^{+})_{min}$ and $M_{T2,\,t}$ shown in
Figure 7. In the upper panels we give the differential distributions for the
SM (${\cal CP}$-even), ${\cal CP}$-mixed and ${\cal CP}$-odd Higgs boson in
the full off-shell calculation (solid lines) and in the NWA (dashed lines).
The middle panels provide the ratio to the $\alpha_{CP}=0$ case for the
normalised differential distributions in the full off-shell approach. The
ratio $\textrm{NWA}/\textrm{off-shell}$ for each $\alpha_{CP}$ value is
displayed in the lower panels. The scale uncertainties are provided as
reference for the SM case in the two upper panels while Monte Carlo errors are
reported in all three panels. For $H_{T}$, as defined in Eq. (9), we find that
compared to the SM Higgs boson, the spectra for the ${\cal CP}$-mixed and
${\cal CP}$-odd case are harder towards the tails. Specifically, we find an
increase of only about $10\%$ for the ${\cal CP}$-mixed case while for the
${\cal CP}$-odd case the increase is very significant with more than $100\%$.
In both cases these effects are larger than the corresponding scale
uncertainties for the ${\cal CP}$-even case which only amount to up to $6\%$.
The full off-shell effects for the SM and ${\cal CP}$-mixed Higgs boson cases
are very similar to each other and amount to about $5\%$ in the tails which is
comparable to the size to the scale uncertainties. On the other hand, we
obtain large off-shell effects of about $45\%$ for the ${\cal CP}$-odd Higgs
boson. The latter exceed the full off-shell effects at the integrated level by
more than a factor of three. Since these off-shell effects rise towards the
tails, they also have a significant impact on the normalised ratios shown in
the middle panels. Had we taken the NWA predictions instead of the full off-
shell ones, the differences between the ${\cal CP}$ states would only be up to
$60\%$ in the tails instead of about $120\%$. The shape differences in the
tails of $p_{T,\,H}$ are even larger than for $H_{T}$. In particular, we
observe differences with respect to the SM case at the level of $30\%$ for the
${\cal CP}$-mixed and effects larger than $200\%$ for the ${\cal CP}$-odd
Higgs boson. Thus, these differences are significant when compared to the
scale uncertainties of about $10\%$ for the SM Higgs boson case. Consequently,
the $p_{T,\,H}$ observable is an excellent example of a dimensionful
differential cross-section distribution that might be employed to decipher the
${\cal CP}$ quantum numbers of the Higgs boson and look for any hints of new
physics in the Higgs-top sector. In addition, the off-shell effects are of the
order of $5\%$ for the SM and ${\cal CP}$-mixed case and thus would not have
an impact on this comparison. For the ${\cal CP}$-odd case, on the other hand,
the full off-shell effects increase up to $35\%$ in the tails of the
$p_{T,\,H}$ distribution rendering them indispensable in studies of the ${\cal
CP}$ state of the Higgs boson. If we had set
$\kappa_{Ht\bar{t}}=\kappa_{At\bar{t}}=1$ instead, then the absolute
differential distributions for $p_{T,\,H}$ would coincide in the tails for all
three ${\cal CP}$ cases in the NWA, see e.g. Refs. Frederix:2011zi ;
Demartin:2014fia . Taking into account the fragmentation functions in Eq.
(14), this behaviour becomes clear since the soft enhancement for the scalar
Higgs boson is absent in the tails of this distribution. On the other hand, in
the full off-shell calculation we would not obtain such a behaviour due to the
large effects coming from single- and non-resonant top-quark contributions as
well as their interference effects with double-resonant ones. For
$M(be^{+})_{min}$ the SM and ${\cal CP}$-mixed Higgs boson cases lead to the
exact same normalised distributions below the kinematical edge around
$M(be^{+})_{min}\approx 153$ GeV. Above this edge the difference between the
two scenarios is in the $20\%-25\%$ range. Nevertheless, it is still
comparable to the scale uncertainties of the ${\cal CP}$-even case, which are
of the order of $20\%$ in this phase-space region. In the ${\cal CP}$-odd case
the ratio of the normalised differential distribution to the SM distribution
is larger than four. Even the absolute differential cross-section distribution
for the ${\cal CP}$-odd case is larger than the ${\cal CP}$-mixed one in this
phase-space region, even though the overall normalisation is almost three
times smaller. This behaviour is closely related to the size of the full off-
shell effects, which reach up to $90\%$ for the ${\cal CP}$-odd Higgs boson
but only $50\%$ for the SM and $60\%$ for the ${\cal CP}$-mixed Higgs boson.
This is a result of a larger relative contribution from single- and non-
resonant diagrams for the ${\cal CP}$-odd Higgs boson which is partially
caused by the non-vanishing $HVV$ coupling. A similar phenomenon can be
observed in the stransverse mass $M_{T2,t}$ Barr:2003rg ; Lester:1999tx ;
Lester:2014yga which is a generalisation of the transverse mass in case of
pairs of unstable particles and is defined as

$M^{2}_{T2,t}=\textrm{min}_{\mathbf{p}^{\nu_{1}}_{T}+\mathbf{p}^{\nu_{1}}_{T}=\mathbf{p}_{T,miss}}\left[\textrm{max}\left\\{M_{T}^{2}\left(\mathbf{p}_{T}^{(lb)_{1}},\mathbf{p}_{T}^{\nu_{1}}\right),M_{T}^{2}\left(\mathbf{p}_{T}^{(lb)_{2}},\mathbf{p}_{T}^{\nu_{2}}\right)\right\\}\right],$
(17)

where the transverse mass of the lepton+$b$-jet system in presence of a
missing transverse momentum $\mathbf{p}_{T}^{\nu_{i}}$ is given by

$M_{T}^{2}\left(\mathbf{p}_{T}^{(lb)_{i}},\mathbf{p}_{T}^{\nu_{i}}\right)=M^{2}_{(lb)_{i}}+2\left(E_{T}^{(lb)_{i}}E_{T}^{\nu_{i}}-\mathbf{p}_{T}^{(lb)_{i}}\cdot\mathbf{p}_{T}^{\nu_{i}}\right).$
(18)

We choose the lepton+$b$-jet pairs by minimising the value
$Q=M_{e^{+}b_{i}}+M_{\mu^{-}b_{j}}$ so that we do not take into account the
charge of the $b$-jets. Similarly to $M(be^{+})_{min}$, the stransverse mass
has a kinematical edge around the top-quark mass ($m_{t}=172.5$ GeV). For
$M_{T2,t}>m_{t}$, large suppression of double-resonant top-quark contributions
occurs. Thus, the absolute value of the cross-section is significantly reduced
above this edge and the full off-shell effects, which are driven by the
single-resonant top-quark contributions, increase. Following the same
reasoning as before, this reduction is suppressed for the ${\cal CP}$-odd
case. When examining the absolute differential cross-section distributions we
can observe that the ${\cal CP}$-odd case exceeds even the SM results in these
phase-space regions. The difference between the normalised $M_{T2,t}$
distributions can reach a factor of $10$. A less significant enhancement
occurs for the ${\cal CP}$-mixed Higgs boson so that the SM and ${\cal
CP}$-mixed cases are very similar towards the end of the absolute $M_{T2,t}$
spectrum. This is due to the large shape differences of about $50\%-60\%$
which are substantial compared to the scale uncertainties. The latter are in
the $15\%-18\%$ range for the ${\cal CP}$-even case. Compared to the
$M(be^{+})_{min}$ differential cross-section distribution the full off-shell
effects are even larger here. Indeed, they are of the order of $70\%$, $80\%$
and $98\%$ for the ${\cal CP}$-even, ${\cal CP}$-mixed and ${\cal CP}$-odd
case, respectively. Both $M(be^{+})_{min}$ and $M_{T2,t}$ are very sensitive
to the various Higgs boson ${\cal CP}$ numbers. However, they are also subject
to large off-shell effects. Not only the finite top-quark width effects should
be properly taken into account for these observables but also single-resonant
top-quark contributions and interference effects must be incorporated to
adequately describe the phase-space regions above these kinematical edges.

Figure 8:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,{\pi}/{4},{\pi}/{2}$ for the observables
$\cos\theta^{*}_{e^{+}\mu^{-}}$, $\cos\theta_{e^{+}\mu^{-}}$,
$\cos\theta^{*}_{b_{1}b_{2}}$ and $\cos\theta_{b_{1}b_{2}}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13\textrm{ TeV}$. The upper panels show full off-shell and NWA
(dashed lines) results. The middle panels provide the ratio to $\alpha_{CP}=0$
of the normalised differential distributions for the full off-shell case. The
ratio NWA/off-shell is given in the lower panels. Scale uncertainties are
shown for $\alpha_{CP}=0$ in the upper and middle panels while Monte Carlo
integration errors are displayed in all panels.

In the next step we discuss the shape differences between the ${\cal
CP}$-even, ${\cal CP}$-mixed and ${\cal CP}$-odd Higgs boson as well as the
respective size of off-shell effects in several angular observables. In Figure
8 we present the differential cross-section distributions for
$\cos\theta^{*}_{e^{+}\mu^{-}}$, $\cos\theta_{e^{+}\mu^{-}}$,
$\cos\theta^{*}_{b_{1}b_{2}}$ and $\cos\theta_{b_{1}b_{2}}$. In the case of
$\cos\theta^{*}_{e^{+}\mu^{-}}$ we find that the prediction for the central
value of the scale is smaller by about $7\%$ for the ${\cal CP}$-mixed and
$35\%$ for ${\cal CP}$-odd Higgs boson compared to the SM case. These
differences increase significantly towards
$\cos\theta^{*}_{e^{+}\mu^{-}}\approx\pm 1$ to about $50\%$ for the ${\cal
CP}$-mixed and more than $200\%$ for the ${\cal CP}$-odd case. Thus, the two
leptons tend to be more separated in the rapidity plane in the case of the
${\cal CP}$-mixed and ${\cal CP}$-odd Higgs boson. Furthermore, the
theoretical uncertainties due to the scale dependence are not only very small
but also very similar for all the four observables. More specifically, we have
theoretical uncertainties at the level of $5\%$ in the central regions for the
SM Higgs boson case, whereas towards $\cos\theta^{*}_{e^{+}\mu^{-}}\approx\pm
1$ they increase only slightly up to $7\%-9\%$. This means that the shape
differences outlined above are much larger than the scale uncertainties. As a
result, we can conclude that the $\cos\theta^{*}_{e^{+}\mu^{-}}$ angular
differential cross-section distribution is a promising ${\cal CP}$-sensitive
observable to be measured at the LHC. This is further underlined when
examining the full off-shell effects for the SM and ${\cal CP}$-mixed Higgs
boson. We can observe that these effects are less than $2\%$ in both cases.
Thus, they are even comparable in size to the full off-shell top-quark effects
at the integrated fiducial cross-section level. However, the situation is
rather different for the ${\cal CP}$-odd Higgs boson for which the off-shell
effects lead to a wrong normalisation as well as additional shape distortions
and we find that the off-shell effects are reduced from the centre of the plot
towards the edges from $20\%$ to $5\%$. Thus, even for angular distributions,
single- and non-resonant contributions can significantly affect the shape,
especially in the $\cos\theta^{*}_{e^{+}\mu^{-}}\approx 0$ phase-space region
where theoretical uncertainties are four times smaller than the size of the
full off-shell effects. Consequently, the latter should be included in
theoretical predictions when comparing to experimental LHC data. For
$\cos\theta_{e^{+}\mu^{-}}$ we can see the largest differences between the SM
and the ${\cal CP}$-mixed Higgs boson around
$\cos\theta_{e^{+}\mu^{-}}\approx\pm 1$. At about $7\%$, they are comparable
in size to the scale uncertainties in those phase-space regions. The situation
is again quite different for the ${\cal CP}$-odd Higgs boson. In this case,
differences of up to $40\%-50\%$ are obtained for
$\cos\theta_{e^{+}\mu^{-}}\approx\pm 1$. The off-shell effects for the SM and
${\cal CP}$-mixed Higgs boson cases are again quite small and rather similar
in size while for the ${\cal CP}$-odd one they significantly affect the
normalisation and shape of the distribution. In particular, the off-shell
effects reduce towards large opening angles from $20\%$ to $10\%$.

The observables $\cos\theta^{*}_{b_{1}b_{2}}$ and $\cos\theta_{b_{1}b_{2}}$,
which are the hadronic counterparts of the two previously discussed
observables, are important for leptonic Higgs boson decays since in those
cases we have a larger lepton multiplicity which makes a precise measurement
of leptonic observables more challenging. All in all, the shape differences
between the three ${\cal CP}$ states are similar to those of the leptonic
observables. This is especially true for $\cos\theta_{b_{1}b_{2}}$ for which
the normalised ratios are essentially the same as for
$\cos\theta_{e^{+}\mu^{-}}$. For the ${\cal CP}$-odd Higgs boson, on the other
hand, both the $\cos\theta^{*}_{b_{1}b_{2}}\approx 0$ and
$\cos\theta^{*}_{b_{1}b_{2}}\approx\pm 1$ phase-space regions receive
additional enhancements. Compared to the corresponding scale uncertainties the
off-shell effects of the ${\cal CP}$-even and ${\cal CP}$-mixed Higgs boson
are again negligible for both observables but sizable for the ${\cal CP}$-odd
case. In the latter case, they amount to $10\%-30\%$ for
$\cos\theta^{*}_{b_{1}b_{2}}$ and $10\%-20\%$ for $\cos\theta_{b_{1}b_{2}}$
which substantially exceed the corresponding theoretical uncertainties and
lead to significant shape distortions.

In the following, we investigate the contribution of double-, single- and non-
resonant fiducial phase-space regions to the integrated and differential
cross-sections for the full off-shell case. This should give us a qualitative
picture of the importance of the individual contributions, as well as their
general distribution for a given value of the ${\cal CP}$-mixing angle of the
Higgs boson. To identify these contributions, we use the method introduced in
Ref. Kauer:2001sp . This approach was further modified and employed in various
studies, see e.g. Liebler:2015ipp ; Baskakov:2018huw ; Bevilacqua:2019quz ;
Bevilacqua:2020srb . Specifically, at NLO in QCD we have the following three
resonance histories

$\begin{array}[]{llll}(\mathrm{i})&t=W^{+}(\to
e^{+}\nu_{e})\,b&\quad\quad\quad\mathrm{and}&\bar{t}=W^{-}(\to\mu^{-}\bar{\nu}_{\mu})\,\bar{b}\,,\\\\[5.69046pt]
(\mathrm{ii})&t=W^{+}(\to
e^{+}\nu_{e})\,bj&\quad\quad\quad\mathrm{and}&\bar{t}=W^{-}(\to\mu^{-}\bar{\nu}_{\mu})\,\bar{b}\,,\\\\[5.69046pt]
(\mathrm{iii})&t=W^{+}(\to
e^{+}\nu_{e})\,b&\quad\quad\quad\mathrm{and}&\bar{t}=W^{-}(\to\mu^{-}\bar{\nu}_{\mu})\,\bar{b}j\,,\\\
\end{array}$ (19)

where the light jet (if resolved by the anti-kT jet algorithm with $R=0.4$) is
added to the list only if it passes the following cuts: $|y_{j}|<2.5$ and
$p_{T,\,j}>15$ GeV. In this way, we closely mimic what is done on the
experimental side for the light jet to be observed in the ATLAS or CMS
detector. For each history we compute the following quantity

${\cal Q}=|M_{t}-m_{t}|+|M_{\bar{t}}-m_{t}|\,,$ (20)

where $m_{t}=172.5$ GeV and $M_{t}$ ($M_{\bar{t}}$) is the invariant mass of
the reconstructed top quark (anti-top quark). For each phase-space point, one
history is selected by minimising the value of ${\cal Q}$. The boundaries of
the fiducial double-resonant (DR), single-resonant (SR) and non-resonant (NR)
regions are expressed in terms of $\Gamma_{t}^{\rm NLO}$ and the parameter
$n=15$. The value of the latter is rather arbitrary. The DR phase-space region
is defined according to

$|M_{t}-m_{t}|<n\,\Gamma_{t}^{\rm NLO}\,,\quad\quad\quad{\rm
and}\quad\quad\quad|M_{\bar{t}}-m_{t}|<n\,\Gamma_{t}^{\rm NLO}\,.$ (21)

There are two SR regions given by

$|M_{t}-m_{t}|<n\,\Gamma_{t}^{\rm NLO}\,,\quad\quad\quad{\rm
and}\quad\quad\quad|M_{\bar{t}}-m_{t}|>n\,\Gamma_{t}^{\rm NLO}\,,$ (22)

or

$|M_{t}-m_{t}|>n\,\Gamma_{t}^{\rm NLO}\,,\quad\quad\quad{\rm
and}\quad\quad\quad|M_{\bar{t}}-m_{t}|<n\,\Gamma_{t}^{\rm NLO}\,.$ (23)

The NR region is determined via

$|M_{t}-m_{t}|>n\,\Gamma_{t}^{\rm NLO}\,,\quad\quad\quad{\rm
and}\quad\quad\quad|M_{\bar{t}}-m_{t}|>n\,\Gamma_{t}^{\rm NLO}\,.$ (24)

For the above outlined procedure, the contributions at the integrated fiducial
cross-section level for these three regions for the ${\cal CP}$-even
($\alpha_{CP}=0$), ${\cal CP}$-mixed ($\alpha_{CP}=\pi/4$) and ${\cal CP}$-odd
($\alpha_{CP}=\pi/2$) case are given in Table 3. The integrated fiducial
cross-sections for all three cases are dominated by the DR contributions. For
the SM and ${\cal CP}$-mixed Higgs boson, more than $90\%$ of $\sigma^{\rm
NLO}_{\rm Off-shell}$ comes from the DR contribution. For the ${\cal CP}$-odd
Higgs boson, however, the DR contribution is reduced to $80\%$. A similar
behaviour can be observed for the remaining fiducial phase-space regions. For
both $\alpha_{CP}=0$ and $\alpha_{CP}=\pi/4$, the SR region is of the order of
$9\%$, while the NR regions of the phase space are completely negligible.
Again, for $\alpha_{CP}=\pi/2$, the SR and NR contributions are larger at the
level of $19\%$ and $1\%$, respectively. Hence, in the case of the ${\cal
CP}$-odd Higgs boson, the SR regions of the phase space are more substantial,
which is reflected in the larger size of the full off-shell effects as we have
already seen in Table 2.

Table 3: NLO QCD integrated fiducial cross-sections as calculated in the full off-shell approach together with predictions for DR, SR and NR fiducial phase-space regions for $\alpha_{CP}=0,\pi/4$ and $\pi/2$. Also shown are their respective Monte Carlo integration errors. All values are given for the $pp\to e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with $\sqrt{s}=13$ TeV. $\alpha_{CP}$ | Off-shell | Double-resonant | Single-resonant | Non-resonant
---|---|---|---|---
| [fb] | [fb] | [fb] | [fb]
$0$ (SM) | 2.466(2) | 2.246(2) | 0.2138(5) | 0.00594(7)
$\pi/4$ | 1.465(2) | 1.325(2) | 0.136(1) | 0.0039(1)
$\pi/2$ | 0.5018(3) | 0.4037(9) | 0.0948(4) | 0.00346(5)

Figure 9:  Differential distributions at NLO in QCD for $\alpha_{CP}=0$ and
$\pi/2$ for the observables $M(be^{+})_{min}$ and $p_{T,\,H}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13$ TeV. The upper panels show full off-shell and NWA results as
well as predictions for DR, SR and NR fiducial phase-space regions. The ratios
of all contributions to the full off-shell result are shown in the lower
panels. Monte Carlo integration errors are displayed in both panels.

We have confirmed these findings also at the differential level. In Figure 9
we display the differential cross-section distribution as a function of
$M(be^{+})_{min}$ and $p_{T,\,H}$ for $\alpha_{CP}=0$ and $\pi/2$. The ${\cal
CP}$-mixed case is omitted as it follows the SM Higgs boson very closely. For
each plot, we show the full off-shell and NWA predictions as well as the
results for the DR, SR and NR fiducial phase-space regions. The lower panels
display the ratio of all contributions to the full off-shell result. For the
$M(be^{+})_{min}$ distribution, we observe that in the case of the ${\cal
CP}$-even Higgs boson, the SR part rapidly increases in the region that starts
to be sensitive to off-shell effects, i.e. for $M(be^{+})_{min}\approx 153$
GeV, and even starts to dominate the full off-shell result towards the end of
the spectrum. This effect is enhanced for the ${\cal CP}$-odd Higgs boson. As
a matter of fact, we have the following distribution of DR, SR and NR phase-
space regions for $M(be^{+})_{min}$ around $153$ GeV; DR $65\%$ ($42\%$), SR
$35\%$ ($53\%$) and NR $1\%$ ($4\%$) for $\alpha_{CP}=0$
($\alpha_{CP}=\pi/2$). Even for $M(be^{+})_{min}\leq 153$ GeV, the SR
contribution for $\alpha_{CP}=\pi/2$ is twice as large as for the
$\alpha_{CP}=0$ case. A similar effect can be observed for $p_{T,\,H}$. Even
though in the high $p_{T}$ tail of the differential cross-section distribution
the SR part does not dominate the full off-shell result, its contribution
increases substantially from $10\%$ for the ${\cal CP}$-even Higgs boson to
$34\%$ for the ${\cal CP}$-odd Higgs boson. Again, even at the beginning of
the spectrum, the SR contribution is of the order of $16\%$ for
$\alpha_{CP}=\pi/2$, while for the SM Higgs boson case we have a rather
constant contribution of the order of $10\%$ for the whole plotted range.
Consequently, the DR part is reduced from around $90\%$ for the
$\alpha_{CP}=0$ case down to $84\%$ for the $\alpha_{CP}=\pi/2$ one.

Figure 10:  Differential distributions at NLO in QCD for
$\alpha_{CP}=0,{\pi}/{4},{\pi}/{2}$ for the observables $p_{T,b_{1}}$,
$p_{T,H}$, $M(be^{+})_{min}$ and $\cos\theta^{*}_{b_{1}b_{2}}$ for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13\textrm{ TeV}$. The upper panels show NWA and
$\textrm{NWA}_{\textrm{LOdec}}$ (dashed lines) results. The lower panels
provide the ratio NWA/$\textrm{NWA}_{\textrm{LOdec}}$ of the normalised
distributions. Scale uncertainties are given for $\alpha_{CP}=0$ in the upper
panel while Monte Carlo integration errors are displayed in both panels.

Finally, we investigate the effects of the inclusion of higher-order QCD
corrections to top-quark decays in an NLO calculation. As an example, we
display the observables $p_{T,\,b_{1}}$, $p_{T,\,H}$, $M(be^{+})_{min}$ and
$\cos\theta^{*}_{b_{1}b_{2}}$ in Figure 10. In each case we plot the full NWA
predictions (solid lines) and the $\textrm{NWA}_{\textrm{LOdec}}$ ones (dashed
lines). The lower panels display the ratio NWA/$\textrm{NWA}_{\textrm{LOdec}}$
of the normalised differential cross-section distributions. For
$p_{T,\,b_{1}}$ we find that the NLO QCD corrections to the top-quark decays
amount to about $20\%$ for small transverse momenta and about $10\%$ for
larger values of $p_{T,\,b_{1}}$. They are very similar for all three ${\cal
CP}$ configurations as the corresponding normalized ratios differ by less than
$2\%$ amongst themselves. For $p_{T,H}$ the shape distortions due to QCD
corrections in the decay stage range from $-5\%$ to $10\%$ for all three
${\cal CP}$ configurations. Thus again, the higher-order effects do not depend
on the ${\cal CP}$ nature of the Higgs boson. Even for $M(be^{+})_{min}$, for
which we have found huge differences between the three different ${\cal CP}$
states in the full off-shell calculation, the inclusion of corrections in top-
quark decays is similar in all three cases. Still, we find a large enhancement
of QCD corrections for $M(be^{+})_{min}>153$ GeV of the order of $30\%$. Even
for $M(be^{+})_{min}<153$ GeV the $\mathcal{O}(\alpha_{s})$ corrections to the
top-quark decays lead to substantial shape changes ranging from about $-20\%$
up to $+10\%$. Therefore, in the vicinity of the kinematical edge, the overall
shape distortions reach up to $50\%$. The dimensionless observables are less
affected by higher-order effects in the top-quark decays. For angular
distributions, like $\cos\theta^{*}_{b_{1}b_{2}}$, the shape distortions in
the normalised ratio do not exceed $4\%$ for all three ${\cal CP}$
configurations. Furthermore, these shape distortions do not differ by more
than $4\%$ amongst the different ${\cal CP}$ states.

Thus, as expected and already discussed at the integrated fiducial cross-
section level, the NLO QCD corrections to top-quark decays are similar in size
and shape for all three considered ${\cal CP}$ states of the Higgs boson in
differential cross-section distributions. Accordingly, on their own they do
not help in deciphering the ${\cal CP}$ nature of the Higgs boson.
Nevertheless, higher-order corrections to top-quark decays cannot simply be
omitted. Not only do they range from a few percent up to $30\%$ depending on
the phase-space region, introducing vast distortions in the shape of various
observables, but they also significantly impact the final theoretical
uncertainties. Therefore, they should not be ignored in high-precision
theoretical predictions for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process.

In summary, when considering high precision theoretical predictions for the
associated production of the Higgs boson and top-quark pairs various effects
have to be taken into account. We have seen that NLO QCD corrections to top-
quark production and decays are indispensable when studying the ${\cal CP}$
property of the Higgs boson. Furthermore, the non-factorisable higher-order
corrections, which introduce the cross-talk between the production and decay
stage, play a crucial role. The latter cannot simply be approximated by some
overall factor. Indeed, they are vastly different for various observables and
their size depends on the examined phase-space region.

##  6 Summary

In this paper we have presented NLO QCD calculations for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13$ TeV for the ${\cal CP}$-even and ${\cal CP}$-odd state of the
Higgs boson as well as for possible mixing between these two states. We have
placed particular emphasis on how the different ${\cal CP}$ states are
affected by higher-order corrections and full off-shell top-quark and $W$
gauge boson effects. To this end, we have compared integrated fiducial cross-
sections and differential distributions including full off-shell effects to
those computed in the narrow-width approximation for the ${\cal CP}$-even
($\alpha_{CP}=0$), ${\cal CP}$-odd ($\alpha_{CP}=\pi/2$) and ${\cal CP}$-mixed
($\alpha_{CP}=\pi/4$) states. For the full off-shell calculations we have
taken into account all double-, single- and non-resonant diagrams as well as
interference effects between those and introduced Breit-Wigner distributions
for the unstable top quarks and gauge bosons through the complex-mass scheme
to describe their invariant mass distributions.

As a first step, we have demonstrated that even at the level of integrated
fiducial cross-sections, off-shell effects can significantly exceed the
typical size of $\mathcal{O}(\Gamma/m)$. While these effects are below one
percent for the ${\cal CP}$-even and -mixed scenarios and thus still well
within scale uncertainties, they reach $13\%-14\%$ for the pseudoscalar Higgs
boson. This increase for the $\alpha_{CP}=\pi/2$ case is driven by a
substantially larger contribution from the single-resonant top-quark region of
the phase space. In addition, we have investigated how the
$\alpha_{CP}$-dependence of the integrated cross-section changes between the
NWA and the full off-shell treatment. Due to the absence of any $HVV$
couplings in the NWA, the corresponding integrated cross-section is symmetric
around $\alpha_{CP}=\pi/2$. However, this symmetry is broken if full off-shell
effects are included. This is a result of interference effects between
diagrams in which the Higgs boson is radiated of a gauge boson and those in
which it is emitted from a top quark. These contributions are proportional to
$\cos\left(\alpha_{CP}\right)$ instead of $\cos^{2}\left(\alpha_{CP}\right)$
or $\sin^{2}\left(\alpha_{CP}\right)$ and thus not symmetric around
$\alpha_{CP}=\pi/2$. Consequently, the integrated fiducial cross-section is
about $7\%$ larger for $\alpha_{CP}=\pi$ than for the SM case.

Just like the off-shell effects, NLO corrections are also largest for the
${\cal CP}$-odd scenario. We have shown that the $\mathcal{K}-$factor is of
the order of $1.21-1.23$ for the SM and ${\cal CP}$-mixed cases and $1.31$ for
the ${\cal CP}$-odd one. Furthermore, theoretical scale uncertainties are
substantially reduced when higher-order effects are incorporated.
Specifically, they are at the level of $31\%-34\%$ at LO and in the range of
$5\%-6\%$ for the NLO predictions. When only considering NLO QCD corrections
to the production stage, we have observed that the size of higher-order
effects increases together with the corresponding scale uncertainties. Indeed,
not only the $\mathcal{K}-$factor is now in the range of $1.27-1.28$ for the
${\cal CP}$-even and ${\cal CP}$-mixed Higgs boson and up to ${\cal K}=1.34$
for the ${\cal CP}$-odd case, but also theoretical uncertainties due to the
scale dependence increase up to $11\%$. Therefore, they are almost twice as
large as in the full NWA or when full off-shell effects are included.
Independently of the modeling, the scale dependence is roughly the same when
comparing the different Higgs boson ${\cal CP}$ states.

Next to the overall normalisation, the NLO QCD corrections also change the
shape of several differential cross-section distributions. In contrast to
their effects on the normalisation at the fiducial cross-section level, the
shape distortions are generally larger for the SM case. This is particularly
apparent in the case of the $p_{T,\,\text{miss}}$ and $M(be^{+})_{min}$
distributions. For these observables the smaller shape distortions for the
pseudoscalar Higgs boson can mainly be ascribed to the generally harder Higgs
boson radiation and the larger relative size of single- and non-resonant
contributions. In general, the $\mathcal{K}-$factors are larger in the
high-$p_{T}$ and -mass regions, irrespective of the Higgs bosons ${\cal CP}$
state. Just like in the case of $t\bar{t}$ production and for other
$t\bar{t}$-associated production processes, NLO QCD corrections are especially
large for the $M(be^{+})_{min}$ observable above the kinematical edge around
$153$ GeV. In this phase-space region higher-order effects can even exceed
$100\%$ in the ${\cal CP}$-even and ${\cal CP}$-mixed cases.

In the second part of our discussion on differential cross-section
distributions we have investigated the impact of full off-shell effects on the
distribution shapes. Just as for the integrated fiducial cross-section, the
effects are fairly small for the SM and ${\cal CP}$-mixed cases apart from the
high-$M(be^{+})_{\text{min}}$ and -$M_{T2,t}$ regions. Both of these
observables have kinematical edges which are related to the masses of unstable
intermediate particles. For these two observables full off-shell effects can
exceed $50\%$ above these edges. For the pseudoscalar Higgs boson, on the
other hand, all observables are significantly affected by off-shell effects.
Again, these effects are largest for $M(be^{+})_{\text{min}}$ and $M_{T2,t}$
at up to $90\%$. However, even in observables without kinematic edges the
differences can reach $45\%$ in dimensionful and $20\%$ in dimensionless
observables. In general, the full off-shell effects are larger in the
high-$p_{T}$ and -mass phase-space regions and for small angles between the
decay products coming from two different top quarks.

These changes in the behaviour of differential distributions also have a
significant impact on how well one can distinguish the three Higgs boson
${\cal CP}$ states from each other at the differential fiducial cross-section
level. We have shown that while the normalised distributions for the ${\cal
CP}$-even and ${\cal CP}$-mixed cases are generally quite similar to each
other, the ${\cal CP}$-odd case leads to much more pronounced distribution
tails in transverse momentum and invariant mass distributions. Again, this is
mostly a result of the harder radiation of a pseudoscalar Higgs boson compared
to a scalar one. For $M_{T2,t}$ this effect is so large that the ${\cal
CP}$-odd distribution actually exceeds the one for the SM case above $240$ GeV
even though the integrated cross-section is about five times as large for the
latter. Although not quite as drastic, the shape differences in $H_{T}$,
$p_{T,\,H}$ and $M(be^{+})_{\text{min}}$ still result in ratios of normalised
distributions that reach a factor $2$ or more in the tails of the respective
distributions. Amongst dimensionless cross-section distributions we have found
that the $\cos\theta^{*}_{e^{+}\mu^{-}}$ observable is the most promising
candidate for distinguishing the different ${\cal CP}$ states of the Higgs
boson.

In addition, we have discussed the effects of omitting NLO QCD corrections in
top-quark decays at the differential fiducial cross-section level by comparing
the full NWA with the $\text{NWA}_{\text{LOdec}}$ predictions. In contrast to
the full NLO QCD corrections, the shape distortions coming from NLO QCD
corrections to top-quark decays are largely independent of the ${\cal CP}$
state, even above kinematic edges. Nevertheless, they should be taken into
account as they can reach up to $30\%$ depending on the considered observable
and phase-space region. Furthermore, they introduce vast shape distortions and
significantly impact the size of scale uncertainties.

In summary, both NLO QCD corrections and full off-shell effects of the top
quark and $W$ gauge boson have a significant impact on the normalisation and
shape of various differential cross-section distributions for the $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,b\bar{b}\,H$ process at the LHC with
$\sqrt{s}=13$ TeV for all three Higgs boson ${\cal CP}$ states. However, the
most affected is the ${\cal CP}$-odd Higgs boson. For the ${\cal CP}$-even and
-mixed cases higher-order corrections play a crucial role but full off-shell
effects come into play mostly in the case of observables with kinematic edges
such as $M(be^{+})_{\text{min}}$ and $M_{T2,t}$. Due to these kinematic edges,
these observables are also the ones that are most sensitive to the ${\cal CP}$
state of the Higgs boson.

Finally, let us mention again that the calculations presented in this paper
and the Helac-NLO framework constructed for this work could easily be altered
in order to obtain results for a general beyond the SM Higgs-boson sector and
are thus not limited to predictions for an SM-like Higgs boson.

###### Acknowledgements.

The work was supported by the Deutsche Forschungsgemeinschaft (DFG) under
grant 396021762 $-$ TRR 257: P3H - Particle Physics Phenomenology after the
Higgs Discovery and by the DFG under grant 400140256 - GRK 2497: The physics
of the heaviest particles at the LHC. Support by a grant of the
Bundesministerium für Bildung und Forschung (BMBF) is additionally
acknowledged. The authors gratefully acknowledge the computing time granted by
the Resource Allocation Board and provided on the supercomputer CLAIX at RWTH
Aachen University as part of the NHR infrastructure. The calculations for this
research were conducted with computing resources under the projects rwth0414
and rwth0871.

## References

  * (1) ATLAS collaboration, _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 716 (2012) 1 [1207.7214].
  * (2) CMS collaboration, _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , _Phys. Lett. B_ 716 (2012) 30 [1207.7235].
  * (3) CMS collaboration, _Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton collisions at 7 and 8 TeV_ , _Phys. Rev. D_ 92 (2015) 012004 [1411.3441].
  * (4) ATLAS collaboration, _Study of the spin and parity of the Higgs boson in diboson decays with the ATLAS detector_ , _Eur. Phys. J. C_ 75 (2015) 476 [1506.05669].
  * (5) ATLAS collaboration, _Test of CP Invariance in vector-boson fusion production of the Higgs boson using the Optimal Observable method in the ditau decay channel with the ATLAS detector_ , _Eur. Phys. J. C_ 76 (2016) 658 [1602.04516].
  * (6) CMS collaboration, _Constraints on anomalous Higgs boson couplings using production and decay information in the four-lepton final state_ , _Phys. Lett. B_ 775 (2017) 1 [1707.00541].
  * (7) CMS collaboration, _Measurements of the Higgs boson width and anomalous $HVV$ couplings from on-shell and off-shell production in the four-lepton final state_, _Phys. Rev. D_ 99 (2019) 112003 [1901.00174].
  * (8) CMS collaboration, _Constraints on anomalous $HVV$ couplings from the production of Higgs bosons decaying to $\tau$ lepton pairs_, _Phys. Rev. D_ 100 (2019) 112002 [1903.06973].
  * (9) ATLAS collaboration, _Test of CP invariance in vector-boson fusion production of the Higgs boson in the $H\to\tau^{+}\tau^{-}$ channel in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector_, _Phys. Lett. B_ 805 (2020) 135426 [2002.05315].
  * (10) CMS collaboration, _Constraints on anomalous Higgs boson couplings to vector bosons and fermions in its production and decay using the four-lepton final state_ , _Phys. Rev. D_ 104 (2021) 052004 [2104.12152].
  * (11) ATLAS collaboration, _Measurements of Higgs boson production heis and couplings in diboson final states with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 726 (2013) 88 [1307.1427].
  * (12) CMS collaboration, _Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8 TeV_, _Eur. Phys. J. C_ 75 (2015) 212 [1412.8662].
  * (13) ATLAS, CMS collaboration, _Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at $\sqrt{s}=7$ and 8 TeV_, _JHEP_ 08 (2016) 045 [1606.02266].
  * (14) A. D. Sakharov, _Violation of cp invariance, c asymmetry, and baryon asymmetry of the universe_ , _Soviet Physics Uspekhi_ 34 (1991) 392.
  * (15) V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, _On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe_ , _Phys. Lett. B_ 155 (1985) 36.
  * (16) J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, _The Higgs Hunter’s Guide_ , vol. 80. 2000.
  * (17) J. F. Gunion and H. E. Haber, _The CP conserving two Higgs doublet model: The Approach to the decoupling limit_ , _Phys. Rev. D_ 67 (2003) 075019 [hep-ph/0207010].
  * (18) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, _Theory and phenomenology of two-Higgs-doublet models_ , _Phys. Rept._ 516 (2012) 1 [1106.0034].
  * (19) B. Grzadkowski, O. M. Ogreid and P. Osland, _Measuring CP violation in Two-Higgs-Doublet models in light of the LHC Higgs data_ , _JHEP_ 11 (2014) 084 [1409.7265].
  * (20) J. Ellis, D. S. Hwang, K. Sakurai and M. Takeuchi, _Disentangling Higgs-Top Couplings in Associated Production_ , _JHEP_ 04 (2014) 004 [1312.5736].
  * (21) F. Boudjema, R. M. Godbole, D. Guadagnoli and K. A. Mohan, _Lab-frame observables for probing the top-Higgs interaction_ , _Phys. Rev. D_ 92 (2015) 015019 [1501.03157].
  * (22) M. R. Buckley and D. Goncalves, _Boosting the Direct CP Measurement of the Higgs-Top Coupling_ , _Phys. Rev. Lett._ 116 (2016) 091801 [1507.07926].
  * (23) S. Amor Dos Santos et al., _Probing the CP nature of the Higgs coupling in $t{\bar{t}}h$ events at the LHC_, _Phys. Rev. D_ 96 (2017) 013004 [1704.03565].
  * (24) CMS collaboration, _Measurements of properties of the Higgs boson decaying into the four-lepton final state in pp collisions at $\sqrt{s}=13$ TeV_, _JHEP_ 11 (2017) 047 [1706.09936].
  * (25) CMS collaboration, _Measurements of Higgs boson properties in the diphoton decay channel in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _JHEP_ 11 (2018) 185 [1804.02716].
  * (26) ATLAS collaboration, _Combined measurements of Higgs boson production and decay using up to 80 fb -1 of proton–proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment, [ATLAS-CONF-2018-031]_, .
  * (27) CMS collaboration, _Combined measurements of Higgs boson couplings in proton–proton collisions at $\sqrt{s}=13\,\text{Te}\text{V}$_, _Eur. Phys. J. C_ 79 (2019) 421 [1809.10733].
  * (28) ATLAS collaboration, _Observation of Higgs boson production in association with a top quark pair at the LHC with the ATLAS detector_ , _Phys. Lett. B_ 784 (2018) 173 [1806.00425].
  * (29) CMS collaboration, _Observation of $t\overline{t}H$ production_, _Phys. Rev. Lett._ 120 (2018) 231801 [1804.02610].
  * (30) ATLAS collaboration, _$CP$ Properties of Higgs Boson Interactions with Top Quarks in the $t\bar{t}H$ and $tH$ Processes Using $H\rightarrow\gamma\gamma$ with the ATLAS Detector_, _Phys. Rev. Lett._ 125 (2020) 061802 [2004.04545].
  * (31) CMS collaboration, _Measurements of ${t\bar{t}}H$ Production and the CP Structure of the Yukawa Interaction between the Higgs Boson and Top Quark in the Diphoton Decay Channel_, _Phys. Rev. Lett._ 125 (2020) 061801 [2003.10866].
  * (32) CMS collaboration, _Search for CP violation in $t\bar{t}H$ and $tH$ production in multilepton channels at $\sqrt{s}=13\leavevmode\nobreak\ \mathrm{TeV}$, [CMS-PAS-HIG-21-006]_, .
  * (33) CMS collaboration, _Analysis of the CP structure of the Yukawa coupling between the Higgs boson and $\tau$ leptons in proton-proton collisions at $\sqrt{s}=13\leavevmode\nobreak\ \mathrm{TeV}$, [CMS-PAS-HIG-20-006]_, tech. rep., CERN, Geneva, 2020.
  * (34) P. Artoisenet et al., _A framework for Higgs characterisation_ , _JHEP_ 11 (2013) 043 [1306.6464].
  * (35) W.-S. Hou, M. Kohda and T. Modak, _Probing for extra top Yukawa couplings in light of $t\bar{t}h(125)$ observation_, _Phys. Rev. D_ 98 (2018) 075007 [1806.06018].
  * (36) H. Bahl, P. Bechtle, S. Heinemeyer, J. Katzy, T. Klingl, K. Peters et al., _Indirect $\mathcal{CP}$ probes of the Higgs-top-quark interaction: current LHC constraints and future opportunities_, _JHEP_ 11 (2020) 127 [2007.08542].
  * (37) D. Azevedo, R. Capucha, E. Gouveia, A. Onofre and R. Santos, _Light Higgs searches in $t\overline{t}\phi$ production at the LHC_, _JHEP_ 04 (2021) 077 [2012.10730].
  * (38) B. Bortolato, J. F. Kamenik, N. Košnik and A. Smolkovič, _Optimized probes of $CP$ -odd effects in the $t\bar{t}h$ process at hadron colliders_, _Nucl. Phys. B_ 964 (2021) 115328 [2006.13110].
  * (39) T. Martini, R.-Q. Pan, M. Schulze and M. Xiao, _Probing the CP structure of the top quark Yukawa coupling: Loop sensitivity versus on-shell sensitivity_ , _Phys. Rev. D_ 104 (2021) 055045 [2104.04277].
  * (40) H. Bahl and S. Brass, _Constraining $\mathcal{CP}$-violation in the Higgs-top-quark interaction using machine-learning-based inference_, _JHEP_ 03 (2022) 017 [2110.10177].
  * (41) R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, R. Pittau and P. Torrielli, _Scalar and pseudoscalar Higgs production in association with a top–antitop pair_ , _Phys. Lett. B_ 701 (2011) 427 [1104.5613].
  * (42) M. V. Garzelli, A. Kardos, C. G. Papadopoulos and Z. Trocsanyi, _Standard Model Higgs boson production in association with a top anti-top pair at NLO with parton showering_ , _EPL_ 96 (2011) 11001 [1108.0387].
  * (43) P. Artoisenet, R. Frederix, O. Mattelaer and R. Rietkerk, _Automatic spin-entangled decays of heavy resonances in Monte Carlo simulations_ , _JHEP_ 03 (2013) 015 [1212.3460].
  * (44) S. Biswas, R. Frederix, E. Gabrielli and B. Mele, _Enhancing the $t\bar{t}H$ signal through top-quark spin polarization effects at the LHC_, _JHEP_ 07 (2014) 020 [1403.1790].
  * (45) F. Demartin, F. Maltoni, K. Mawatari, B. Page and M. Zaro, _Higgs characterisation at NLO in QCD: CP properties of the top-quark Yukawa interaction_ , _Eur. Phys. J. C_ 74 (2014) 3065 [1407.5089].
  * (46) H. B. Hartanto, B. Jager, L. Reina and D. Wackeroth, _Higgs boson production in association with top quarks in the POWHEG BOX_ , _Phys. Rev. D_ 91 (2015) 094003 [1501.04498].
  * (47) F. Demartin, F. Maltoni, K. Mawatari and M. Zaro, _Higgs production in association with a single top quark at the LHC_ , _Eur. Phys. J. C_ 75 (2015) 267 [1504.00611].
  * (48) F. Demartin, B. Maier, F. Maltoni, K. Mawatari and M. Zaro, _tWH associated production at the LHC_ , _Eur. Phys. J. C_ 77 (2017) 34 [1607.05862].
  * (49) S. Frixione, E. Laenen, P. Motylinski, B. R. Webber and C. D. White, _Single-top hadroproduction in association with a W boson_ , _JHEP_ 07 (2008) 029 [0805.3067].
  * (50) T. Jezo, J. M. Lindert, P. Nason, C. Oleari and S. Pozzorini, _An NLO+PS generator for $t\bar{t}$ and $Wt$ production and decay including non-resonant and interference effects_, _Eur. Phys. J. C_ 76 (2016) no.12, 691 [1607.04538].
  * (51) A. Denner and R. Feger, _NLO QCD corrections to off-shell top-antitop production with leptonic decays in association with a Higgs boson at the LHC_ , _JHEP_ 11 (2015) 209 [1506.07448].
  * (52) D. Stremmer and M. Worek, _Production and decay of the Higgs boson in association with top quarks_ , _JHEP_ 02 (2022) 196 [2111.01427].
  * (53) A. Denner, J.-N. Lang, M. Pellen and S. Uccirati, _Higgs production in association with off-shell top-antitop pairs at NLO EW and QCD at the LHC_ , _JHEP_ 02 (2017) 053 [1612.07138].
  * (54) R. V. Harlander, S. Y. Klein and M. Lipp, _FeynGame_ , _Comput. Phys. Commun._ 256 (2020) 107465 [2003.00896].
  * (55) A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, _Predictions for all processes $e^{+}e^{-}\to$ 4 fermions $+$ $\gamma$_, _Nucl. Phys. B_ 560 (1999) 33 [hep-ph/9904472].
  * (56) A. Denner, S. Dittmaier, M. Roth and L. H. Wieders, _Electroweak corrections to charged-current $e^{+}e^{-}\to 4$ fermion processes: Technical details and further results_, _Nucl. Phys. B_ 724 (2005) 247 [hep-ph/0505042].
  * (57) G. Bevilacqua, M. Czakon, A. van Hameren, C. G. Papadopoulos and M. Worek, _Complete off-shell effects in top quark pair hadroproduction with leptonic decay at next-to-leading order_ , _JHEP_ 02 (2011) 083 [1012.4230].
  * (58) A. Denner, S. Dittmaier, S. Kallweit and S. Pozzorini, _NLO QCD corrections to off-shell top-antitop production with leptonic decays at hadron colliders_ , _JHEP_ 10 (2012) 110 [1207.5018].
  * (59) G. Bevilacqua, M. Czakon, M. V. Garzelli, A. van Hameren, A. Kardos, C. G. Papadopoulos et al., _HELAC-NLO_ , _Comput. Phys. Commun._ 184 (2013) 986 [1110.1499].
  * (60) G. Bevilacqua, H. B. Hartanto, M. Kraus, T. Weber and M. Worek, _Off-shell vs on-shell modelling of top quarks in photon associated production_ , _JHEP_ 03 (2020) 154 [1912.09999].
  * (61) M. Czakon, C. G. Papadopoulos and M. Worek, _Polarizing the Dipoles_ , _JHEP_ 08 (2009) 085 [0905.0883].
  * (62) A. van Hameren, C. G. Papadopoulos and R. Pittau, _Automated one-loop calculations: A Proof of concept_ , _JHEP_ 09 (2009) 106 [0903.4665].
  * (63) P. Draggiotis, R. H. P. Kleiss and C. G. Papadopoulos, _On the computation of multigluon amplitudes_ , _Phys. Lett. B_ 439 (1998) 157 [hep-ph/9807207].
  * (64) P. D. Draggiotis, R. H. P. Kleiss and C. G. Papadopoulos, _Multijet production in hadron collisions_ , _Eur. Phys. J. C_ 24 (2002) 447 [hep-ph/0202201].
  * (65) C. G. Papadopoulos and M. Worek, _Multi-parton cross sections at hadron colliders_ , _Eur. Phys. J. C_ 50 (2007) 843 [hep-ph/0512150].
  * (66) A. van Hameren, _PARNI for importance sampling and density estimation_ , _Acta Phys. Polon. B_ 40 (2009) 259 [0710.2448].
  * (67) A. van Hameren, _Kaleu: A General-Purpose Parton-Level Phase Space Generator_ , 1003.4953.
  * (68) G. Bevilacqua, H. B. Hartanto, M. Kraus, T. Weber and M. Worek, _Hard Photons in Hadroproduction of Top Quarks with Realistic Final States_ , _JHEP_ 10 (2018) 158 [1803.09916].
  * (69) G. Bevilacqua, H. B. Hartanto, M. Kraus, T. Weber and M. Worek, _Towards constraining Dark Matter at the LHC: Higher order QCD predictions for $t\bar{t}+Z(Z\to\nu_{\ell}\bar{\nu}_{\ell})$_, _JHEP_ 11 (2019) 001 [1907.09359].
  * (70) G. Bevilacqua, H.-Y. Bi, H. B. Hartanto, M. Kraus and M. Worek, _The simplest of them all: $t\bar{t}W^{\pm}$ at NLO accuracy in QCD_, _JHEP_ 08 (2020) 043 [2005.09427].
  * (71) G. Ossola, C. G. Papadopoulos and R. Pittau, _CutTools: A Program implementing the OPP reduction method to compute one-loop amplitudes_ , _JHEP_ 03 (2008) 042 [0711.3596].
  * (72) G. Ossola, C. G. Papadopoulos and R. Pittau, _Reducing full one-loop amplitudes to scalar integrals at the integrand level_ , _Nucl. Phys. B_ 763 (2007) 147 [hep-ph/0609007].
  * (73) A. van Hameren, _OneLOop: For the evaluation of one-loop scalar functions_ , _Comput. Phys. Commun._ 182 (2011) 2427 [1007.4716].
  * (74) S. Catani and M. H. Seymour, _A General algorithm for calculating jet cross-sections in NLO QCD_ , _Nucl. Phys. B_ 485 (1997) 291 [hep-ph/9605323].
  * (75) S. Catani, S. Dittmaier, M. H. Seymour and Z. Trocsanyi, _The Dipole formalism for next-to-leading order QCD calculations with massive partons_ , _Nucl. Phys. B_ 627 (2002) 189 [hep-ph/0201036].
  * (76) G. Bevilacqua, M. Czakon, M. Kubocz and M. Worek, _Complete Nagy-Soper subtraction for next-to-leading order calculations in QCD_ , _JHEP_ 10 (2013) 204 [1308.5605].
  * (77) J. M. Campbell, R. K. Ellis and F. Tramontano, _Single top production and decay at next-to-leading order_ , _Phys. Rev. D_ 70 (2004) 094012 [hep-ph/0408158].
  * (78) G. Bevilacqua, H. B. Hartanto, M. Kraus and M. Worek, _Off-shell Top Quarks with One Jet at the LHC: A comprehensive analysis at NLO QCD_ , _JHEP_ 11 (2016) 098 [1609.01659].
  * (79) J. Alwall et al., _A Standard format for Les Houches event files_ , _Comput. Phys. Commun._ 176 (2007) 300 [hep-ph/0609017].
  * (80) I. Antcheva et al., _ROOT: A C++ framework for petabyte data storage, statistical analysis and visualization_ , _Comput. Phys. Commun._ 180 (2009) 2499 [1508.07749].
  * (81) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079 [1405.0301].
  * (82) V. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli, F. Maltoni and R. Pittau, _Automation of one-loop QCD corrections_ , _JHEP_ 05 (2011) 044 [1103.0621].
  * (83) R. Kleiss, W. J. Stirling and S. D. Ellis, _A New Monte Carlo Treatment of Multiparticle Phase Space at High-energies_ , _Comput. Phys. Commun._ 40 (1986) 359.
  * (84) Z. Nagy and Z. Trocsanyi, _Next-to-leading order calculation of four jet observables in electron positron annihilation_ , _Phys. Rev. D_ 59 (1999) 014020 [hep-ph/9806317].
  * (85) Z. Nagy, _Next-to-leading order calculation of three jet observables in hadron hadron collision_ , _Phys. Rev. D_ 68 (2003) 094002 [hep-ph/0307268].
  * (86) G. Bevilacqua, M. Czakon, C. G. Papadopoulos, R. Pittau and M. Worek, _Assault on the NLO Wishlist: $pp\to t\bar{t}b\bar{b}$_, _JHEP_ 09 (2009) 109 [0907.4723].
  * (87) M. Czakon, H. B. Hartanto, M. Kraus and M. Worek, _Matching the Nagy-Soper parton shower at next-to-leading order_ , _JHEP_ 06 (2015) 033 [1502.00925].
  * (88) A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht et al., _LHAPDF6: parton density access in the LHC precision era_ , _Eur. Phys. J. C_ 75 (2015) 132 [1412.7420].
  * (89) NNPDF collaboration, _Parton distributions from high-precision collider data_ , _Eur. Phys. J. C_ 77 (2017) 663 [1706.00428].
  * (90) M. Jezabek and J. H. Kuhn, _QCD Corrections to Semileptonic Decays of Heavy Quarks_ , _Nucl. Phys. B_ 314 (1989) 1.
  * (91) M. Cacciari, G. P. Salam and G. Soyez, _The anti- $k_{t}$ jet clustering algorithm_, _JHEP_ 04 (2008) 063 [0802.1189].
  * (92) ATLAS collaboration, _Measurements of gluon fusion and vector-boson-fusion production of the Higgs boson in $H\rightarrow WW^{*}\rightarrow e\nu\mu\nu$ decays using $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, [ATLAS-CONF-2021-014]_, tech. rep., CERN, Geneva, Mar, 2021.
  * (93) CMS collaboration, _Measurements of Higgs boson production via gluon fusion and vector boson fusion in the diphoton decay channel at $\sqrt{s}=13$ TeV, [CMS-PAS-HIG-18-029]_, tech. rep., CERN, Geneva, 2019\.
  * (94) M. Aoki, S. Kanemura, K. Tsumura and K. Yagyu, _Models of Yukawa interaction in the two Higgs doublet model, and their collider phenomenology_ , _Phys. Rev. D_ 80 (2009) 015017 [0902.4665].
  * (95) T. Plehn, D. L. Rainwater and D. Zeppenfeld, _Determining the Structure of Higgs Couplings at the LHC_ ,  _Phys. Rev. Lett._ 88 (2002), 051801 [hep-ph/0105325].
  * (96) U. Haisch, P. Pani and G. Polesello, _Determining the CP nature of spin-0 mediators in associated production of dark matter and $t\overline{t}$ pairs_, _JHEP_ 02 (2017) 131 [1611.09841].
  * (97) S. Dittmaier, M. Kramer, Y. Liao, M. Spira and P. M. Zerwas, _Higgs radiation off quarks in supersymmetric theories at $e^{+}e^{-}$ colliders_, _Phys. Lett. B_ 478 (2000) 247 [hep-ph/0002035].
  * (98) S. Dawson and L. Reina, _QCD corrections to associated Higgs boson production_ , _Phys. Rev. D_ 57 (1998) 5851 [hep-ph/9712400].
  * (99) V. S. Fadin, V. A. Khoze and A. D. Martin, _How suppressed are the radiative interference effects in heavy instable particle production?_ , _Phys. Lett. B_ 320 (1994) 141 [hep-ph/9309234].
  * (100) J. Hermann and M. Worek, _The impact of top-quark modelling on the exclusion limits in ${t\bar{t}}+{DM}$ searches at the LHC_, _Eur. Phys. J. C_ 81 (2021) 1029 [2108.01089].
  * (101) G. Bevilacqua, H. B. Hartanto, M. Kraus, J. Nasufi and M. Worek, _NLO QCD corrections to full off-shell production of $t\bar{t}Z$ including leptonic decays_, _JHEP_ 08 (2022) 060 [2203.15688].
  * (102) M. Czakon, A. Mitov and R. Poncelet, _NNLO QCD corrections to leptonic observables in top-quark pair production and decay_ , _JHEP_ 05 (2021) 212 [2008.11133].
  * (103) A. J. Barr, _Measuring slepton spin at the LHC_ , _JHEP_ 02 (2006) 042 [hep-ph/0511115].
  * (104) A. Barr, C. Lester and P. Stephens, _m(T2): The Truth behind the glamour_ , _J. Phys. G_ 29 (2003) 2343 [hep-ph/0304226].
  * (105) C. G. Lester and D. J. Summers, _Measuring masses of semiinvisibly decaying particles pair produced at hadron colliders_ , _Phys. Lett. B_ 463 (1999) 99 [hep-ph/9906349].
  * (106) C. G. Lester and B. Nachman, _Bisection-based asymmetric M T2 computation: a higher precision calculator than existing symmetric methods_, _JHEP_ 03 (2015) 100 [1411.4312].
  * (107) N. Kauer and D. Zeppenfeld, _Finite width effects in top quark production at hadron colliders_ ,  _Phys. Rev._ D 65 (2002) 014021 [hep-ph/0107181].
  * (108) S. Liebler, G. Moortgat-Pick and A. S. Papanastasiou, _Probing the top-quark width through ratios of resonance contributions of $e^{+}e^{-}\rightarrow W^{+}W^{-}b\bar{b}$_,  _JHEP_ 03 (2016) 099 [1511.02350].
  * (109) A. Baskakov, E. Boos and L. Dudko, _Model independent top quark width measurement using a combination of resonant and nonresonant cross sections_ ,  _Phys. Rev._ D 98 (2018) no.11, 116011 [1807.11193].
  * (110) G. Bevilacqua, H. Y. Bi, H. B. Hartanto, M. Kraus, J. Nasufi and M. Worek, _NLO QCD corrections to off-shell ${t{\bar{t}}W^{\pm}}$ production at the LHC: correlations and asymmetries_,  _Eur. Phys. J._ C 81 (2021) no.7, 675 [2012.01363].