BONN-TH-2020-04, DESY 20-102, IFT--UAM/CSIC--20-090

Indirect ${\cal CP}$ probes of the Higgs–top-quark interaction:
current LHC constraints and future opportunities

Henning Bahl100footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Philip Bechtle2, Sven Heinemeyer3,4,5, Judith Katzy1,
Tobias Klingl2,

Krisztian Peters1, Matthias Saimpert6, Tim Stefaniak1, Georg Weiglein1

1Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
2Physikalisches Institut, Universität Bonn, Nußallee 12, 53115 Bonn, Germany
3Instituto de Física Teórica, (UAM/CSIC), Universidad Autónoma de Madrid,
Cantoblanco, E-28049 Madrid, Spain
4Campus of International Excellence UAM+CSIC, Cantoblanco, E-28049, Madrid,
Spain
5Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain
6CERN, Geneva, Switzerland

###### Abstract

The ${\cal CP}$ structure of the Higgs boson in its coupling to the particles
of the Standard Model is amongst the most important Higgs boson properties
which have not yet been constrained with high precision. In this study, all
relevant inclusive and differential Higgs boson measurements from the ATLAS
and CMS experiments are used to constrain the ${\cal CP}$-nature of the top-
Yukawa interaction. The model dependence of the constraints is studied by
successively allowing for new physics contributions to the couplings of the
Higgs boson to massive vector bosons, to photons, and to gluons. In the most
general case, we find that the current data still permits a significant ${\cal
CP}$-odd component in the top-Yukawa coupling. Furthermore, we explore the
prospects to further constrain the ${\cal CP}$ properties of this coupling
with future LHC data by determining $tH$ production rates independently from
possible accompanying variations of the $t\bar{t}H$ rate. This is achieved via
a careful selection of discriminating observables. At the HL-LHC, we find that
evidence for $tH$ production at the Standard Model rate can be achieved in the
Higgs to diphoton decay channel alone.

##  1 Introduction

In 2012 the ATLAS and CMS collaborations have discovered a new particle that –
within current theoretical and experimental uncertainties – is consistent with
the predictions of a Standard-Model (SM) Higgs boson at a mass of $\sim
125\,\,\mathrm{GeV}$ [1, 2]. No conclusive signs of physics beyond the SM have
been found so far at the LHC. However, the measurements of Higgs-boson
couplings, which are known experimentally to a precision of roughly $\sim
20\%$, leave room for Beyond Standard-Model (BSM) interpretations.
Consequently, one of the main tasks of the LHC Run 3 and the high-luminosity
LHC (HL-LHC) will be to determine the Higgs-boson coupling structures and
quantum numbers with highest precision.

This experimental program has a direct link to cosmology. One of the most
important questions connecting the two fields is related to the fact that the
Cabbibo-Kobayashi-Maskawa (CKM) matrix, the only source of ${\cal CP}$
violation in the SM, cannot explain the observed baryon asymmetry of the
universe (BAU), as the SM prediction is off by many orders of magnitude [3,
4]. Consequently, additional sources of ${\cal CP}$ violation must exist in
nature. The main idea of this work is to investigate to which extent ${\cal
CP}$ violation beyond the CKM matrix can be present in the interactions of the
detected Higgs boson. Specifically we investigate here the interaction between
the Higgs boson and the top quark.

While the hypothesis that the Higgs boson at 125 GeV is a pure ${\cal CP}$-odd
state was ruled out based on LHC Run 1 data [5, 6], only rather weak bounds
exist on a possible admixture between a ${\cal CP}$-even and a ${\cal CP}$-odd
component. The analyses so far were mainly based on observables involving the
coupling of the observed Higgs boson, $H$, to two gauge bosons, $HVV$, where
$V=W,Z$, in particular $H\to ZZ^{*}\to 4\ell$, $H\to WW^{*}\to\ell\nu\ell\nu$,
and Higgs production in weak vector boson fusion (VBF). Since in many BSM
models only a small loop-induced coupling of a ${\cal CP}$-odd component to
gauge bosons, $H^{{\cal CP}-{\rm odd}}VV$, is generated, the effects of the
${\cal CP}$-odd component are heavily suppressed compared to the tree-level
contribution of $H^{{\cal CP}-{\rm even}}VV$, even if the ${\cal CP}$-odd
component itself is large. Consequently, the couplings of the observed Higgs
boson to fermions (in particular to the top quark) are crucial for
investigating the ${\cal CP}$-nature of the observed state, since a ${\cal
CP}$-odd component in the $Hf\bar{f}$ coupling may be unsuppressed and could
be of similar magnitude as the ${\cal CP}$-even component. In addition to
being a spectacular discovery by itself, the establishment of a non-zero
${\cal CP}$-odd component of the top-Yukawa coupling would also be a strong
hint for an extended BSM Higgs sector.

Our investigations make use of the so-called “Higgs characterization model”, a
framework based on an effective field theory (EFT) approach. This framework
allows one to take into account ${\cal CP}$-violating effects in the couplings
and to perform studies in a consistent, systematic and accurate way, see e.g.
Ref. [7]. The top-Yukawa coupling between the Higgs boson and the top quark in
this approach is parametrized in terms of rescaling parameters for the ${\cal
CP}$-even (SM-like) coupling and a ${\cal CP}$-odd (BSM) coupling.
Alternatively, these can be expressed in terms of a magnitude and a ${\cal
CP}$-violating phase. Such types of approaches are the basis of most published
data analyses sensitive to ${\cal CP}$-violation and ${\cal CP}$-admixtures in
the Higgs sector, briefly reviewed below.

${\cal CP}$ violation in the interactions between the Higgs boson and the top
quark can be probed experimentally within this framework using two different
strategies: In a _direct_ approach, a pure ${\cal CP}$-odd observable is
constructed, typically from angular distributions of Higgs boson production
[8, 9] or decay [6, 10, 11]. Such a measurement can be used to constrain all
models where both the ${\cal CP}$-odd and the ${\cal CP}$-even components of
the Higgs boson couple with significant strength to the particle from which
the ${\cal CP}$-odd observable is measured. For the experimental establishment
of ${\cal CP}$ violation in the Higgs boson sector, the measurement of a
${\cal CP}$-odd observable with a non-zero result would be crucial. For the
Higgs–top-quark coupling, no experimental analysis of a pure ${\cal CP}$-odd
observable [12, 13] exists yet. In an _indirect_ approach, the model
parameters are fitted to ${\cal CP}$-even observables such as decay rates and
kinematic distributions (e.g. $p_{T}$ spectra), and constraints on the ${\cal
CP}$ nature of the Higgs boson are derived indirectly from the constraints on
the model parameters. This latter approach is the one followed in this paper.
While the indirect approach is a very powerful test of possible deviations
from the SM, in case a significant deviation is found it is not guaranteed
that its origin can uniquely be associated with the presence of ${\cal
CP}$-violating effects. Thus, in order to determine the ${\cal CP}$ nature of
the Higgs sector, the two approaches are complementary.

The properties of the Higgs boson under the pure ${\cal CP}$-even hypothesis
and its coupling to top quarks are studied by ATLAS and CMS in great detail.
An overview of early LHC Run 2 results on the constraints on the ${\cal
CP}$-even top-Yukawa coupling $y_{t}$ can be found in Refs. [14, 15]. Early
fits to a possible ${\cal CP}$ admixture to the observed Higgs state have been
performed using Run 1 and partially early Run 2 data, either investigating all
Higgs-boson couplings [16, 17], or focusing on the Higgs–top-quark interaction
[18, 19]. These analyses could set only very weak bounds on possible ${\cal
CP}$ violation in the Higgs-boson sector. Projections for future bounds on
possible ${\cal CP}$-admixtures of the discovered Higgs-boson based on $gg\to
H+2$ jets data were obtained in Ref. [20, 21, 22], focusing on current and
future Run 3 data. Experimental bounds on the top-Yukawa coupling under the
${\cal CP}$-even coupling assumption have been obtained more recently using an
integrated luminosity of $36\,\,\mathrm{fb}^{-1}$ in the $t\bar{t}\,t\bar{t}$
channel [23] and $t\bar{t}$ kinematic distributions [24].

Up to now, the most stringent experimental bounds on the ${\cal CP}$ nature of
the top-Yukawa coupling have been obtained by dedicated CMS [25] and ATLAS
[26] analyses using full Run 2 data. Both analyses combine the direct and the
indirect approach by fitting the rate of the $t\bar{t}H$, $tH$ and $tWH$
processes to data in the Higgs to di-photon decay channel in certain kinematic
configurations. Under the assumptions of the considered model, a pure ${\cal
CP}$-odd top-Yukawa coupling is excluded by $3.2\,\sigma$ (CMS) and
$3.9\,\sigma$ (ATLAS), and, assuming a pure ${\cal CP}$-even coupling, the
$t\bar{t}H$ signal is observed at the $5\,\sigma$ level. Combining the direct
and the indirect approaches leads to the best sensitivity for the targeted
model. Presenting the results of such a combined study in a fully model-
independent format, however, is very challenging and would entail the public
release of the likelihood as a function of all relevant parameters (including
the relevant Higgs coupling scale factors). As this has not been done for the
present analyses [25, 26] we could not include these results as input for the
${\cal CP}$ study carried out in the present paper, which is performed for
several “Higgs characterization models” of different complexities. We will
discuss the results of Refs. [25, 26] further in Sec. 5.5.

Another type of observables that are sensitive to ${\cal CP}$-violating
interactions are electric dipole moment (EDM) measurements. These were used in
Ref. [27] to set upper bounds on the ${\cal CP}$-odd Higgs–top-quark coupling
of ${\cal O}(0.5)$. The most recent EDM measurements (see e.g. Refs. [28, 29])
which were not taken into account in Ref. [27] may lead to stronger
constraints (see Refs. [30, 31, 32], in which the complementarity between
collider and BAU constraints is also discussed). In the analysis of these
constraints, however, it is assumed that the Higgs couples only to third-
generation fermions. While assuming SM values also for the first- and second-
generation fermions leads to substantially stricter bounds, non-zero ${\cal
CP}$-odd components in the couplings of the Higgs boson to leptons or first-
and second-generation quarks, which are hardly experimentally accessible so
far, could actually loosen the constraints on the ${\cal CP}$-odd component of
the top-Yukawa interaction (see Refs. [33, 34, 35]).

Besides the experimental publications, ${\cal CP}$ violation in the Higgs–top-
quark coupling has also been analysed in various phenomenological works
focussing on $t\bar{t}H$, $tWH$ and $tH$ production. Tree-level analyses using
Run 1 data can be found in Ref. [36]. Corresponding studies at the next-to-
leading (NLO) QCD level have been performed for $t\bar{t}H$ [37], $tH$ [38]
and $tWH$ production [39]. In these studies the different dependence of
$t\bar{t}H$, $tWH$ and $tH$ production on a ${\cal CP}$-admixture of the Higgs
boson was investigated. A study of the Higgs–top-quark interaction including
partial Run 2 data has been presented in Ref. [40]. In Ref. [41], future
constraints from $t\bar{t}H$, $tH$ and $t\bar{t}\,t\bar{t}$ total rate
measurements at the LHC Run 3 and the HL-LHC were investigated. Assuming that
no deviations from the SM will be detected, prospective bounds on the ${\cal
CP}$-admixture of the Higgs boson were derived. ${\cal CP}$-violating
asymmetries involving $t\bar{t}H$ production were analyzed in Refs. [42, 43,
44, 45, 46]. These analyses correlated the top-quark spin and the four-momenta
to obtain information about a possible ${\cal CP}$-admixture of the Higgs that
is involved in the $t\bar{t}H$ vertex. However, no results based on the
available data were presented.

In this work we will use all relevant inclusive and differential Higgs boson
rate measurements (based on ${\cal CP}$-even observables) that are presently
available to derive bounds on a possible ${\cal CP}$-odd coupling (or the
aforementioned ${\cal CP}$-violating phase) using the _indirect_ approach
discussed above. Based on the results of our analysis, we develop a possible
strategy to measure the single top quark associated Higgs production mode with
upcoming LHC data independently from possible ${\cal CP}$-structure-related
effects in the $t\bar{t}H$ rate. This measurement would significantly enhance
the sensitivity of the fit to the ${\cal CP}$ nature of the top-Yukawa
coupling in the future.

Our paper is organized as follows. The various LHC processes which depend on
the top-Yukawa coupling at leading-order (LO) are briefly reviewed in Sec. 2.
The effective model description we are using, i.e. the employed “Higgs
characterization model”, is defined in Sec. 3. These definitions are applied
to the various Higgs-boson production and decay modes in Sec. 4, which also
demonstrates the numerical dependences of the various Higgs-boson production
modes on the ${\cal CP}$-even and ${\cal CP}$-odd couplings. In Sec. 5, we
define four parametrizations with increasing complexity, ranging from two free
parameters in the Higgs–top-quark coupling to five free parameters, including
also additional Higgs-boson production modes and the relevant Higgs-boson
decay modes. The main results, using all relevant inclusive and differential
Higgs boson rate measurements that are presently available, are obtained for
these parametrizations. In Sec. 6, we present a possible strategy to measure
separately $tH$ and $t\bar{t}H+tWH$ production with more data and discuss the
additional constraints on the ${\cal CP}$-structure that one would obtain from
this new measurement. Our conclusions can be found in Sec. 7.

##  2 Higgs–top-quark interaction at hadron colliders

At hadron colliders the Higgs top-Yukawa coupling appears in multiple
processes. For the present discussion we restrict ourselves to those processes
in which the top-Yukawa coupling appears at the leading-order (LO). All of
these processes are sensitive to the properties of the top-Yukawa coupling via
the total rate and potentially via kinematic observables.

Figure 1: Exemplary Feynman diagrams for $gg\rightarrow H$ and
$H\rightarrow\gamma\gamma$.

First, we discuss loop-induced processes which are mediated among others by a
top-quark loop. One example is gluon fusion, the dominant LHC Higgs production
mechanism for a SM-like Higgs boson. Its dominant leading-order contribution
is mediated by a top-quark loop, shown in the left panel of Fig. 1. One
phenomenologically relevant Higgs decay mode – the decay into two photons – is
also mediated by a top-quark loop, shown in the right panel of Fig. 1, besides
a dominantly contributing $W$-boson loop (for the case of a SM-like Higgs
boson) and other subdominant contributions.

Figure 2: Exemplary Feynman diagrams for $qq\rightarrow ZH$ and $gg\rightarrow
ZH$.

Another production mechanism sensitive to the top-Yukawa coupling is $Z$-boson
associated production. While the dominant contribution to $ZH$ production
involving a quark and an anti-quark in the initial state (see left diagram of
Fig. 2) does not depend on the top-Yukawa coupling at leading order, the
subdominant gluon-induced channel has two LO contributions of which one
involves the top-Yukawa coupling (see exemplary diagrams in the middle and
right panel of Fig. 2). In addition to the total rate, also the $p_{T}$-shape
of the Higgs boson produced via $gg\rightarrow ZH$ is sensitive to the ${\cal
CP}$-nature of the top-Yukawa coupling (alongside other kinematic
distributions, see e.g. [22]). This distribution can be studied in the
simplified template cross-section (STXS) framework [47]. In contrast, the
$p_{T}$-shape of the Higgs boson produced via $gg\rightarrow H$ is not
sensitive to the ${\cal CP}$-nature of the top-Yukawa coupling (see e.g. Ref.
[37]). If the Higgs is produced in association with two jets, the azimuthal
correlations between the jets, however, offer sensitivity to the ${\cal
CP}$-nature [37].

Figure 3: Exemplary Feynman diagrams for $t\bar{t}H$, $tH$ and $tWH$
production.

So far we discussed only loop-induced processes, where the top-Yukawa coupling
dependence enters due to a virtual top quark appearing among other particles
in the loop. At the LHC, however, we can also study channels which are
sensitive to the top-Yukawa coupling already at the tree-level. These are
Higgs production in association with one or two top-quarks. While all
contributions to $t\bar{t}H$ production are proportional to the top-Yukawa
coupling (see exemplary diagram in the upper left panel of Fig. 3), $tH$
production receives contributions proportional to the top-Yukawa coupling (see
upper middle diagram of Fig. 3) and proportional to the electroweak gauge
couplings (see upper right diagram of Fig. 3).111In addition to the
$t$-channel $tH$ contributions, shown in Fig. 3, there is also a $s$-channel
contribution mediated by a $W$ boson. The $s$-channel contribution is an order
of magnitude smaller than the $t$-channel contribution [38]. Therefore, we
neglect it in the present study. Similar to $tH$ production, also $tWH$
production receives contributions proportional to the top-Yukawa coupling and
to the electroweak gauge couplings (see bottom diagrams of Fig. 3).
Experimentally, $tWH$ is challenging to distinguish from $t\bar{t}H$
production. At next-to-leading order in the five-flavor scheme or at leading-
order in the four-flavor scheme, $tWH$ and $t\bar{t}H$ production even
interfere with each other (see [39] for a detailed discussion). The
distributions of the Higgs transverse momentum in $tH$, $t\bar{t}H$ and $tWH$
production offer additional sensitivity to the ${\cal CP}$-nature of the top-
Yukawa coupling. Measurements of these shapes are not yet possible but are
expected to become feasible in the future. STXS bins for the $t\bar{t}H$ Higgs
$p_{T}$-shape have been defined already [48].

In addition to the processes discussed above, also the Higgs decay mode into a
photon and a $Z$ boson, four leptons, as well as four-top-quark production
[49, 41] can be used to constrain the ${\cal CP}$-nature of the top-Yukawa
coupling. With the current experimental precision, these processes are,
however, not competitive to the processes discussed above (but may become
relevant after the high-luminosity upgrade of the LHC). Therefore, we do not
include them into our analysis.

##  3 Effective model description

For our analysis, we use a model similar to the Higgs-characterization model
defined in Refs. [7, 50, 37]. The top-Yukawa part of the Lagrangian is
modified with respect to the SM,

$\displaystyle\mathcal{L}_{\text{yuk}}=-\frac{y_{t}^{\text{SM}}}{\sqrt{2}}\bar{t}\left(c_{t}+i\gamma_{5}\tilde{c}_{t}\right)tH,$
(1)

where $y_{t}^{\text{SM}}$ is the SM top-Yukawa coupling, $H$ is used to denote
the Higgs boson field and $t$ to denote the top quark field. The parameter
$c_{t}$ rescales the ${\cal CP}$-even coupling with respect to the SM
prediction ($c_{t}=1$). The ${\cal CP}$-odd coupling of the Higgs boson to top
quarks is parametrized by $\tilde{c}_{t}$ (in the SM, $\tilde{c}_{t}=0$). In
an EFT framework (e.g. SMEFT [38]), this modification would be generated by
dimension-six operators. For simplicity, we do not allow for any modification
of the other Yukawa couplings in this work. The coupling rescaling parameters
$c_{t}$ and $\tilde{c}_{t}$ in Eq. 1 can furthermore be rewritten in terms of
an absolute value $|g_{t}|$ and a ${\cal CP}$-violating phase $\alpha$,
defined as

$\displaystyle|g_{t}|\equiv\sqrt{c_{t}^{2}+\tilde{c}_{t}^{2}},\qquad\qquad\tan\alpha=\frac{\tilde{c}_{t}}{c_{t}}.$
(2)

Note that $|g_{t}|$ corresponds to the $\kappa_{t}$ parameter used in Ref.
[26].

In addition to the modification of the top-Yukawa sector we shall consider an
$SU(2)_{L}$ preserving modification of the Higgs-gauge-boson coupling
strength,

$\displaystyle\mathcal{L}_{V}=c_{V}H\left(\frac{M_{Z}^{2}}{v}Z_{\mu}Z^{\mu}+2\frac{M_{W}^{2}}{v}W_{\mu}^{+}W^{-\mu}\right),$
(3)

where $Z$ and $W$ are the massive vector boson fields ($M_{Z}$ and $M_{W}$ are
their respective masses). The SM Higgs vacuum expectation value, $v$, is
$\simeq 246\,\,\mathrm{GeV}$. The parameter $c_{V}$ rescales the SM-Higgs
interaction strength with the massive vector bosons. In addition to rescaling
the SM-Higgs–gauge interaction, we could also include additional non-SM like
operators of the form $Z_{\mu\nu}Z^{\mu\nu}H$ or
$Z_{\mu\nu}\widetilde{Z}^{\mu\nu}H$ (and analogously for the $W$ boson), where
$Z_{\mu\nu}$ and $\widetilde{Z}_{\mu\nu}$ are the $Z$ boson’s field strength
and its dual, respectively. Since the focus of this study lies mainly on the
top-Yukawa interaction, we do not take into account these additional
couplings, which are not expected to have a significant impact on our
analysis.

We furthermore include a set of operators that couple the Higgs boson directly
to photons and gluons,

$\displaystyle\mathcal{L}_{Hgg,H\gamma\gamma}={}$
$\displaystyle-\frac{1}{4v}H\left(-\frac{\alpha_{s}}{3\pi}c_{g}G_{\mu\nu}^{a}G^{a,\mu\nu}+\frac{\alpha_{s}}{2\pi}\tilde{c}_{g}G_{\mu\nu}^{a}\widetilde{G}^{a,\mu\nu}\right)$
$\displaystyle-\frac{1}{4v}H\left(\frac{47\alpha}{18\pi}c_{\gamma}A_{\mu\nu}A^{\mu\nu}+\frac{4\alpha}{3\pi}\tilde{c}_{\gamma}A_{\mu\nu}\widetilde{A}^{\mu\nu}\right),$
(4)

where $\alpha_{s}=g_{3}^{2}/(4\pi)$, with $g_{3}$ being the strong gauge
coupling, and $\alpha=e^{2}/(4\pi)$, with $e$ being the elementary electric
charge. $A_{\mu\nu}$ and $G^{a}_{\mu\nu}$ (with $a$ being the color index) are
the field strengths of the photon and the gluons. The couplings $c_{g}$,
$\tilde{c}_{g}$, $c_{\gamma}$ and $\tilde{c}_{\gamma}$ can be generated by
heavy undiscovered BSM particle(s), while the SM limit corresponds to
$c_{g}=\tilde{c}_{g}=c_{\gamma}=\tilde{c}_{\gamma}=0$. The prefactors are
chosen as in the SM for the case where the top quark and the $W$-boson are
integrated out. These additional couplings take into account the possibility
that new-physics contributions could decorrelate the tight constraints from
Higgs production via gluon fusion and the Higgs decay to photons from the
Higgs–top-quark interactions. In the analysis, we will always assume that
$\tilde{c}_{g}=\tilde{c}_{\gamma}=0$.

The parametrization of the adopted effective Lagrangian with up to five free
parameters, $c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $c_{g}$ and $c_{\gamma}$, has
been chosen for demonstrating the impact of the experimental constraints on
possible ${\cal CP}$-violating effects in the top quark sector. More general
parametrizations allowing additional sources of ${\cal CP}$ violation or
additional free couplings would in general lead to somewhat increased allowed
ranges of the top quark Yukawa coupling parameters $c_{t}$ and
$\tilde{c}_{t}$.

##  4 Coupling dependence of cross-sections and decays

In order to constrain the effective model described in Sec. 3, we need to
obtain theoretical predictions for the relevant Higgs production and decay
channels in dependence of the various coupling parameters. In this section we
derive these theory predictions. Following the structure of Sec. 2, we first
discuss Higgs production via gluon fusion and the Higgs to di-photon decay,
and then turn to $Z$\- and top-associated Higgs production. The cross section
of processes involving a Higgs–vector-boson interaction but not the Higgs top-
Yukawa coupling (e.g. weak-vector-boson fusion) can be obtained by multiplying
the corresponding SM cross section by $c_{V}^{2}$ (according to Eq. 3).

###  4.1 Cross-section calculation and event generation

We calculate the cross sections for different parameter choices in order to
derive numerical fit formulas which we will use as input for the global fit in
Sec. 5. While for gluon fusion and the Higgs decay into two photons such fit
formulas are available at leading-order (LO) in analytic form including the
full dependence on the generalized top-Yukawa coupling, Eq. 1, this is not the
case for Higgs boson production in association with one or two top-quarks or a
$Z$ boson.

For calculating the cross-sections for these processes, we use
MadGraph5_aMC@NLO 2.7.0 [51] with Pythia 8.244 [52] as parton shower (PS)
employing the A14 set of tuned parameters [53]. As model file, we use the
“Higgs charaterization model” [7, 50, 37]. We derive the cross-sections at LO
in the five-flavor scheme. For the parton-distribution functions (PDF) we use
the MSTW2008LO [54] fit evaluated through the LHAPDF interface [55]. The
derived results are then rescaled to the state-of-the-art SM predictions
reported in Ref. [47].222In the case of Higgs production via gluon fusion,
this procedure could be improved by taking into account the full NLO
predictions available for scalar and pseudoscalar production. However, we
cross-checked our predictions with the ones of the code SusHi [56, 57],
finding differences which are negligible within the theoretical and
experimental uncertainties.

Note that the “Higgs characterization model” allows the derivation of results
at next-to-leading order in the strong coupling constant only if the
Higgs–gluon–gluon BSM operators of Sec. 3 are zero (or in case of the infinite
top-mass limit). Therefore, we include the BSM effects only at the LO and then
rescale the predictions by the ratio of the state-of-the-art SM predictions to
the SM LO predictions. Since the contributions of the Higgs–gluon–gluon
operators and the explicit loop contributions from top quarks that are coupled
to the external Higgs boson should largely compensate each other if
$\kappa_{g}\sim 1$, we expect that neglecting BSM NLO corrections leads to a
negligible effect in the phenomenologically relevant parameter region. For the
analysis of the kinematic properties of $ZH$ and Higgs plus jets production,
we use Rivet [58] to distribute the generated events into bins (using the STXS
analysis script provided in Ref. [59]). In the case of $ZH$ production, we
cross-checked the results obtained using Rivet against an independent analysis
script written in the MadAnalysis framework [60, 61, 62, 63].

###  4.2 Higgs boson production in gluon fusion

Following the notation of Ref. [16], the modification of the SM gluon fusion
production cross-section at leading order (LO) is given by

$\displaystyle\kappa_{g}^{2}\equiv\frac{\sigma_{gg\rightarrow
H}}{\sigma_{gg\rightarrow
H}^{\text{SM}}}=\frac{\left|c_{t}H_{1/2}(\tau_{t})+\frac{2}{3}c_{g}+\ldots\right|^{2}+\left|\tilde{c}_{t}A_{1/2}(\tau_{t})+\tilde{c}_{g}\right|^{2}}{\left|H_{1/2}(\tau_{t})+\ldots\right|^{2}},$
(5)

where $\tau_{t}=M_{H}^{2}/(4M_{t}^{2})$. The ellipses (i.e., the “$+\dots$”)
denotes the contributions involving the remaining SM quarks. They are included
with their SM values in our numerical analysis, but their numerical effect is
negligible (and we omit those contributions in the following discussions). The
loop functions $H_{1/2}$ (${\cal CP}$-even scalar attached to fermion loop)
and $A_{1/2}$ (${\cal CP}$-odd scalar attached to fermion loop) are given by

$\displaystyle H_{1/2}(\tau)=\frac{(\tau-1)f(\tau)+\tau}{\tau^{2}},\hskip
14.22636ptA_{1/2}(\tau)=\frac{f(\tau)}{\tau}$ (6)

with

$\displaystyle f(\tau)=\begin{cases}\arcsin^{2}(\sqrt{\tau})&\text{for
}\tau\leq 1,\\\
-\frac{1}{4}\left(\log\frac{1+\sqrt{1-1/\tau}}{1-\sqrt{1-1/\tau}}-i\pi\right)^{2}&\text{for
}\tau>1.\end{cases}$ (7)

In the heavy top-quark mass limit the dependence simplifies to333For our
numerical results, we employ Eq. 5 including the full top-quark mass
dependence.

$\displaystyle\frac{\sigma_{gg\rightarrow H}}{\sigma_{gg\rightarrow
H}^{\text{SM}}}\bigg{|}_{M_{t}\rightarrow\infty}=(c_{g}+c_{t})^{2}+\frac{9}{4}(\tilde{c}_{g}+\tilde{c}_{t})^{2},$
(8)

in correspondence to the equation employed in Ref. [38]. According to Refs.
[37, 38], also in the total rates for Higgs plus up to three jets no
interference effects between the scalar and pseudoscalar components appear.
Therefore, we employ Eq. 5 also for the numerical evaluation of gluon fusion
in association with jets.

In addition, we investigated the azimuthal correlations between the jets in
$gg\rightarrow H+2j$, $\Delta\phi(j_{1},j_{2})$, as a function of $c_{t}$ and
$\tilde{c}_{t}$. This observable is known to have some sensitivity to the
${\cal CP}$ character of the Higgs boson [64, 65, 21, 38]. However,
numerically, we find modifications due to the ${\cal CP}$-properties at the
sub-percent level, which cannot be resolved with the current experimental and
theoretical precision. Nevertheless, we include $\Delta\phi(j_{1},j_{2})$
observables and their ${\cal CP}$-dependence in our fit (more details are
given in App. A). As mentioned in Sec. 2, the $p_{T}$-shape of Higgs bosons
produced via gluon fusion is not sensitive to the ${\cal CP}$-nature of the
top-Yukawa coupling. Therefore, we do not include corresponding observables
into our fit.

###  4.3 Higgs decay to two photons

The decay width of the Higgs boson into photons is modified as follows,

$\displaystyle\kappa_{\gamma}^{2}\equiv\frac{\Gamma_{H\rightarrow\gamma\gamma}}{\Gamma_{H\rightarrow\gamma\gamma}^{\text{SM}}}=\frac{\left|\frac{4}{3}c_{t}H_{1/2}(\tau_{t})-c_{V}H_{1}(\tau_{W})-\frac{47}{18}c_{\gamma}+\ldots\right|^{2}+\left|\frac{4}{3}\tilde{c}_{t}A_{1/2}(\tau_{t})+\frac{4}{3}\tilde{c}_{\gamma}\right|^{2}}{\left|\frac{4}{3}H_{1/2}(\tau_{t})-H_{1}(\tau_{W})+\ldots\right|^{2}},$
(9)

where $\tau_{W}=M_{H}^{2}/(4M_{W}^{2})$. The ellipses denotes subdominant SM
contributions (e.g. from $c$ and $b$ quarks and $\tau$ leptons), which we take
into account in our numerical analysis with their SM values. The loop function
$H_{1}$ (${\cal CP}$-even scalar attached to vector boson loop) is defined as

$\displaystyle
H_{1}(\tau)=\frac{3(2\tau-1)f(\tau)+3\tau+2\tau^{2}}{2\tau^{2}}.$ (10)

With the exception of $H\rightarrow gg$, which is modified in accordance with
$gg\rightarrow H$ as outlined in Sec. 4.2, the partial widths of all other
Higgs decay channels are unmodified with respect to the SM or trivially
rescaled by $c_{V}^{2}$ (see explanation above).

###  4.4 $Z$-boson associated Higgs production

Figure 4: Ratio of the $gg\rightarrow ZH$ cross-section over the SM cross-
section in dependence of $c_{t}$ and $\tilde{c}_{t}$ for $c_{V}=1$. Red
contours: $\kappa_{g}=1.0\pm 0.2$ for $c_{g}=0$. White contours:
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}=1.0\pm
0.2$ for $c_{\gamma}=0$. The parameter point $(c_{t}=1,\tilde{c}_{t}=0)$
corresponding to the SM case is marked by an orange cross, while the SM value
for the plotted observable is indicated by the gray curve.

The largest contribution to $Z$-boson associated Higgs production is the
$q\bar{q}$-initiated process. The $gg$-initiated process plays a subdominant
role ($\sigma_{\text{SM}}(q\bar{q}\to ZH)=761$ fb and
$\sigma_{\text{SM}}(gg\to ZH)=123$ fb at $13\,\,\mathrm{TeV}$).

The $q\bar{q}$-initiated process scales trivially with $c_{V}^{2}$. For the
signal strength of gluon-initiated $Z$-boson associated Higgs production, we
obtain

$\displaystyle\mu_{gg\rightarrow ZH}$
$\displaystyle\equiv\kappa_{ggZH}^{2}\equiv\frac{\sigma_{gg\rightarrow
ZH}}{\sigma_{gg\rightarrow
ZH}^{\text{SM}}}=0.45c_{t}^{2}+0.50\tilde{c}_{t}^{2}+2.44c_{V}^{2}-1.89c_{V}c_{t}$
(11)

with $\sigma$ and $\sigma^{\text{SM}}$ denoting the $13\,\,\mathrm{TeV}$
inclusive cross sections at the parameter point and for the SM, respectively,
for the given process. We cross-checked this formula against the corresponding
fit function implemented in HiggsBounds-5 [66] (based on vh@nnlo-v2 [67]) and
found very good agreement. Note that there can be contributions from
additional operators affecting the $gg\rightarrow ZH$ channel. We do not
include the effect of such operators in the parametrization given in Eq. 11.
However, in order to assess the possible impact of these operators, we treat
$\kappa_{ggZH}$ as a free parameter instead of using Eq. 11 in some of the
fits presented in Sec. 5.

Fig. 4 shows the numerical dependence of the $gg\rightarrow ZH$ signal
strength, as obtained via Eq. 11, on $c_{t}$ and $\tilde{c}_{t}$ for
$c_{V}=1$. As additional contour lines, we display $\kappa_{g}^{2}$ (red
solid: $\kappa_{g}^{2}=1$; red dashed: $\kappa_{g}^{2}=0.8,1.2$) and
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}$
(white solid:
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}=1.0$;
white dashed:
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}=0.8,1.2$)
for illustration setting $c_{g}=c_{\gamma}=0$. Furthermore the contour
corresponding to $\mu_{gg\rightarrow ZH}=1$ is indicated in gray. It forms an
ellipse that is roughly centered around the point
$(c_{t},\tilde{c}_{t})\simeq(2,0)$, while the contours for constant
$\kappa_{g}$ are centered at the origin, $(c_{t},\tilde{c}_{t})=(0,0)$.

Figure 5: Left: $Z$-boson $p_{T}$-shape for $gg\rightarrow ZH$ and
$q\bar{q}\rightarrow ZH$. Right: $Z$-boson $p_{T}$-shape for $pp\rightarrow
ZH$. The gray vertical lines mark the boundaries of the bins used in Ref.
[68].

As additional input, we take into account modifications of the Higgs-boson
$p_{T}$-distribution in $Z$-associated Higgs production. Experimentally,
though, measurements are performed in $p_{T}$ bins of the leptonically
decaying $Z$-boson which recoils against the Higgs boson. The $Z$-boson
$p_{T}$ is strongly correlated to the Higgs boson $p_{T}$ and provides an
experimentally cleaner observable. The transverse momentum distribution of
$gg\rightarrow ZH$ is sensitive to the ${\cal CP}$-structure of the top-Yukawa
coupling while it is insensitive for the $q\bar{q}\rightarrow ZH$ process.
Exemplary $Z$-boson $p_{T}$-shapes are shown in the left panel of Fig. 5. The
$p_{T}$-distribution for $gg\to ZH$ is harder in the pure ${\cal CP}$-odd
case. As mentioned above, the $q\bar{q}$\- and $gg$-initiated processes are
difficult to distinguish experimentally. The $p_{T}$-shape of the $Z$ boson
produced in the combined $pp\rightarrow ZH$ channel is shown in the right
panel of Fig. 5, which features a visible increase in the high-$p_{T}$ range
in the case of a large ${\cal CP}$-odd coupling due to the behavior of the
$gg\to ZH$ subprocess.

Figure 6: Fraction of $gg\rightarrow ZH$ events in the specific $p_{T}$ bins
in dependence of $c_{t}$ and $\tilde{c}_{t}$ for $c_{V}=1$. Red contours:
$\kappa_{g}=1.0\pm 0.2$ for $c_{g}=0$. White contours:
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}=1.0\pm
0.2$ for $c_{\gamma}=0$. The parameter point $(c_{t}=1,\tilde{c}_{t}=0)$
corresponding to the SM case is marked by an orange cross, while the SM value
for the plotted observable is indicated by the gray curve.

Measurements of the kinematic shape of $ZH$ production employ the STXS
framework, in which the $p_{T}$-bins are defined. The current ATLAS analysis
[68, 69] uses the reduced $VH,V\to\text{leptons}$ stage-1.2 STXS region
scheme, given by [70, 71, 48]:

  * •

bin 1: $75\,\,\mathrm{GeV}<p_{T}^{Z}\leq 150\,\,\mathrm{GeV}$,

  * •

bin 2: $150\,\,\mathrm{GeV}<p_{T}^{Z}\leq 250\,\,\mathrm{GeV}$,

  * •

bin 3: $250\,\,\mathrm{GeV}<p_{T}^{Z}$,

and illustrated in the right panel of Fig. 5. For the ${\cal CP}$-dependent
predictions of the signal strength in these $p_{T}$-bins we obtain

$\displaystyle\mu_{gg\rightarrow ZH}^{\text{bin-1}}$
$\displaystyle=0.20c_{t}^{2}+0.22\tilde{c}_{t}^{2}+1.96c_{V}^{2}-1.13c_{V}c_{t},$
(12) $\displaystyle\mu_{gg\rightarrow ZH}^{\text{bin-2}}$
$\displaystyle=0.53c_{t}^{2}+0.61\tilde{c}_{t}^{2}+2.76c_{V}^{2}-2.29c_{V}c_{t},$
(13) $\displaystyle\mu_{gg\rightarrow ZH}^{\text{bin-3}}$
$\displaystyle=2.02c_{t}^{2}+2.22\tilde{c}_{t}^{2}+5.12c_{V}^{2}-6.14c_{V}c_{t}.$
(14)

In order to confirm the results shown in Fig. 5, the fraction of events, $f$,
falling into the respective bins is plotted in Fig. 6. As in Fig. 5, the
$Z$-boson $p_{T}$-shape becomes harder if the ${\cal CP}$-odd component of the
top-Yukawa coupling is increased.

In addition to the analysis performed in Refs. [68, 69], a specialized
analysis targeting the high $p_{T}$-range was presented in Ref. [72]. Since a
significant statistical correlation between these analyses is expected (and
the corresponding correlations are not available), we only take into account
the most sensitive analyses – i.e., Refs. [68, 69] – in our global fit (see
Sec. 5).

###  4.5 Top-quark associated Higgs production

For the signal strengths for inclusive top-quark associated Higgs production,
we obtain the following fit formulas,

$\displaystyle\mu_{t\bar{t}H}$ $\displaystyle\equiv\frac{\sigma_{pp\rightarrow
t\bar{t}H}}{\sigma_{pp\rightarrow
t\bar{t}H}^{\text{SM}}}=1.00c_{t}^{2}+0.42\tilde{c}_{t}^{2},$ (15)
$\displaystyle\mu_{tH}$ $\displaystyle\equiv\frac{\sigma_{pp\rightarrow
tH}}{\sigma_{pp\rightarrow
tH}^{\text{SM}}}=3.28c_{t}^{2}+1.00\tilde{c}_{t}^{2}+3.82c_{V}^{2}-6.10c_{V}c_{t},$
(16) $\displaystyle\mu_{tWH}$ $\displaystyle\equiv\frac{\sigma_{pp\rightarrow
tWH}}{\sigma_{pp\rightarrow
tWH}^{\text{SM}}}=2.73c_{t}^{2}+2.07\tilde{c}_{t}^{2}+2.01c_{V}^{2}-3.74c_{V}c_{t}.$
(17)

The $tH$, $tWH$ and $t\bar{t}H$ production, channels are difficult to
disentangle experimentally. Therefore, so far an inclusive quantity is
measured, with possible contributions from all three processes.
Correspondingly, we define the combined signal strength for $t\bar{t}H$, $tWH$
and $tH$ production as

$\displaystyle\mu_{t\bar{t}H+tWH+tH}\equiv\frac{\sigma(pp\to
t\bar{t}H)+\sigma(pp\to tH)+\sigma(pp\to tWH)}{\sigma_{\text{SM}}(pp\to
t\bar{t}H)+\sigma_{\text{SM}}(pp\to tH)+\sigma_{\text{SM}}(pp\to tWH))}.$ (18)

Since a distinction of $tH$ production from $ttH$ and $tWH$ production may be
feasible at the LHC in the future (see Sec. 6), we in addition define the SM-
normalized $tH$ over $t\bar{t}H+tWH$ cross section ratio as

$\displaystyle\mu_{tH/(t\bar{t}H+tWH)}\equiv\frac{\sigma(pp\to
tH)/\left[\sigma(pp\to t\bar{t}H)+\sigma(pp\to
tWH)\right]}{\sigma_{\text{SM}}(pp\to tH)/\left[\sigma_{\text{SM}}(pp\to
t\bar{t}H)+\sigma_{\text{SM}}(pp\to tWH)\right]}$ (19)

and the combined signal strength for $t\bar{t}H$ and $tWH$ as

$\displaystyle\mu_{t\bar{t}H+tWH}\equiv\frac{\sigma(pp\to
t\bar{t}H)+\sigma(pp\to tWH)}{\sigma_{\text{SM}}(pp\to
t\bar{t}H)+\sigma_{\text{SM}}(pp\to tWH)}.$ (20)

Figure 7: Signal strengths of $tH$, $ttH$ and $tWH$ (and combinations) in
dependence of $c_{t}$ and $\tilde{c}_{t}$ for $c_{V}=1$. Red contours:
$\kappa_{g}=1.0\pm 0.2$ for $c_{g}=0$. White contours:
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}=1.0\pm
0.2$ for $c_{\gamma}=0$. The parameter point $(c_{t}=1,\tilde{c}_{t}=0)$
corresponding to the SM case is marked by an orange cross, while the gray
curve marks $\mu=1$.

We plot $\mu_{t\bar{t}H+tWH}$ and $\mu_{tH}$ in dependence of $c_{t}$ and
$\tilde{c}_{t}$ (for $c_{V}=1$) in the upper left and right panel of Fig. 7,
respectively. The contours of constant $\mu_{t\bar{t}H+tWH}$ are centered
around the origin, $(c_{t},\tilde{c}_{t})=(0,0)$. This result originates from
the fact that $\mu_{t\bar{t}H+tWH}$ is dominated by $t\bar{t}H$ production (in
the SM case, $\sigma_{\text{SM}}(pp\to t\bar{t}H)=506$ fb and
$\sigma_{\text{SM}}(pp\to tWH)=15$ fb). This is different for $tH$ production.
The interference of the top-Yukawa contribution with the $W$-boson
contribution shifts the center of the contours for constant $\mu_{tH}$ from
the origin towards positive $c_{t}$ values.

We show the signal strength for the sum of the three production channels, Eq.
18, in the lower left plot of Fig. 7. Since $t\bar{t}H$ has the highest cross-
section in most parts of the considered parameter space
($\sigma_{\text{SM}}(pp\to tH)=74$ fb), $t\bar{t}H$ production is the dominant
contribution to the combined signal strength.

The lower right panel of Fig. 7 shows the ratio of the signal strengths of
$tH$ and the combined $t\bar{t}H$ and $tWH$ production, Eq. 19. The comparably
large variation of this ratio along the contours of constant $\kappa_{g}$
indicates that this ratio is a promising future observable for disentangling
the effects of $c_{t}$ and $\tilde{c}_{t}$ (see Sec. 6).

##  5 Constraints from current LHC measurements

In this section we discuss the fit of the model presented in Sec. 3 to
experimental data, including all relevant inclusive and differential Higgs
boson rate measurements available, and using the theory predictions outlined
in Sec. 4.444The fitting code is publicly available at
https://gitlab.com/timstefaniak/higgs_cp_fitting_code.

###  5.1 Fit setup

In order to study the impact of current Higgs boson rate measurements from the
LHC on the ${\cal CP}$-nature of the Higgs–top-quark interaction we performed
numerical scans of the parameter space using four different model
parametrizations:555Note that we do not use the parameters $c_{g}$,
$\tilde{c}_{g}$, $c_{\gamma}$ and $\tilde{c}_{\gamma}$ directly, but instead
we fit the coupling modifiers $\kappa_{g}$ and $\kappa_{\gamma}$, see Sec. 4.

  1. 1.

($c_{t}$, $\tilde{c}_{t}$) free [_2D parametrization_]

Only the ${\cal CP}$-even and ${\cal CP}$-odd components of the Higgs–top-
quark coupling are allowed to vary freely; all remaining tree-level Higgs
couplings (including $c_{V}$) are fixed to their SM value; the loop-induced
coupling scale factors $\kappa_{g}$ and $\kappa_{\gamma}$ are derived in terms
of $c_{t}$ and $\tilde{c}_{t}$, i.e. we assume $c_{g}=c_{\gamma}=0$ (see Sec.
4);

  2. 2.

($c_{t}$, $\tilde{c}_{t}$, $c_{V}$) free [_3D parametrization_]

In addition to the ${\cal CP}$-even and ${\cal CP}$-odd components of the
Higgs–top-quark coupling, we allow a freedom in the Higgs coupling to weak
vector bosons ($V=W^{\pm},Z$) via $c_{V}$; again, $\kappa_{g}$ and
$\kappa_{\gamma}$ are derived in terms of $c_{t}$ and $\tilde{c}_{t}$, i.e. we
assume $c_{g}=c_{\gamma}=0$ (see Sec. 4);

  3. 3.

($c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $\kappa_{\gamma}$) free [_4D
parametrization_]

In addition to the three free parameters of the previous setup, the loop-
induced Higgs coupling to photons, $\kappa_{\gamma}$, is allowed to vary
freely, parametrizing our ignorance about possible contributions to the
$H\to\gamma\gamma$ decay from new, undiscovered color-neutral but electrically
charged particles; the loop-induced Higgs coupling to gluons, parametrized by
$\kappa_{g}$, is still derived in terms of $c_{t}$ and $\tilde{c}_{t}$, i.e.
we assume $c_{g}=0$ (see Sec. 4);

  4. 4.

($c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $\kappa_{\gamma}$, $\kappa_{g}$) free [_5D
parametrization_]

Both loop-induced Higgs coupling parameters – $\kappa_{\gamma}$ and
$\kappa_{g}$ – are allowed to vary freely, accounting for the possibility of
additional contributions to gluon fusion and the ${H\to\gamma\gamma}$ decay
from undiscovered particles carrying color or electric charge. This five-
dimensional fit setup is the most general parametrization that we consider.

For brevity, we shall refer to these parametrizations by stating the number of
dimensions (D), i.e. the number of free fit parameters, as denoted above. In
the baseline fits for these four parametrizations, $\kappa_{ggZH}$ is derived
from the other coupling modifiers following Eq. 11. However additional fits
are performed in order to address specific issues in the analysis.
Specifically, we performed fits where the 5D parametrization is extended by
$\kappa_{ggZH}$ as an additional free fit parameter so that a fit with six
free parameters is performed, as well as fits to a reduced set of observables
(without including the dedicated $t\bar{t}H$ analyses) and/or without taking
Higgs-$p_{T}$-shape effects due to ${\cal CP}$-violation into account. More
details on those fits are given below in the discussion of the results.

Technically, we sample the parameter space in each fit setup randomly with
$\sim\mathcal{O}(10^{7}-10^{8})$ points. We use uniform priors for $c_{t}$,
$\tilde{c}_{t}$, $c_{V}$ (and $\kappa_{ggZH}$ when applicable), and Gaussian
priors with mean value at the SM prediction for $\kappa_{g}$ and
$\kappa_{\gamma}$. Note that the details of the sampling technique do not
influence the fit results, but are only chosen to ensure a sufficiently dense
sampling of the relevant parameter space.

At each parameter point we evaluate the predicted cross sections and decay
rates, as detailed in Sec. 4, and include them into the
HiggsBounds/HiggsSignals framework [73, 74, 75, 76, 77, 66]. HiggsSignals
(version 2.5.0) [75, 76, 77, 78] incorporates the latest available Higgs rate
measurements from ATLAS and CMS from Run 2 [79, 80, 81, 82, 83, 84, 85, 86,
69, 25, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97], as well as the combined
measurements from Run 1 [98]. In total, HiggsSignals includes 81 Run-2
measurements and 20 Run-1 measurements. A detailed list of the included
observables is given in App. B. Note, in particular, that we take into account
the shape modification of the Higgs transverse momentum distribution expected
in the presence of ${\cal CP}$-violation for the available $p_{T}$-binned STXS
measurements of $pp\to ZH$, $H\to b\bar{b}$ by ATLAS [68, 69], as well as in
the $\Delta\Phi_{j_{1}j_{2}}$ distribution for the $pp\to
H+jj,H\to\gamma\gamma$ channel, for which ATLAS provided measurements in four
$\Delta\Phi_{j_{1}j_{2}}$ bins [84].

Based on this large collection of measurements, HiggsSignals is used to
calculate a total $\chi^{2}$ value, which includes various information on
correlations of systematic uncertainties as provided by the experiments. In
the $\chi^{2}$ evaluation, we assume the theoretical uncertainties on Higgs
boson production and decay rates, as well as their correlations, to be the
same as in the SM across the parameter space. We then determine the favored
regions in parameter space by calculating the $\chi^{2}$ difference,
$\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{\text{min}}$, with
$\chi^{2}_{\text{min}}$ being the minimal $\chi^{2}$ value found within the
parameter scan at the so-called best-fit point. In the following we mainly
present the fit results projected onto binned two-dimensional parameter
planes. In each bin we display the minimal $\chi^{2}$ for the subset of scan
points within the bin. In the frequentist approach, this corresponds to the
profile likelihood for the two-dimensional plane of the parameters of
interest. We then derive the $1\,\sigma$, $2\,\sigma$ and $3\,\sigma$
confidence regions from the profile $\Delta\chi^{2}$, corresponding (in a two-
dimensional projection of the parameter space) to $\Delta\chi^{2}<2.3,6.18$
and $11.83$, respectively, assuming the Gaussian limit is approximately
realized. In the case of a profile likelihood in a one-dimensional parameter
space (used for the ${\cal CP}$-violating phase interpretation) the
$1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ corresponding $\Delta\chi^{2}$ values
are $1$, $4$ and $9$, respectively.

In all four parametrizations we find the best-fit point to be remarkably close
to the SM. The minimal $\chi^{2}$ improves only insignificantly with respect
to the SM $\chi^{2}$, despite the up to five additional free parameters. A
more quantitative discussion about how well the models describe the data, as
well as implications about which model may be favored, can be found in App. C.

###  5.2 ${\cal CP}$-violating Higgs–top-quark interactions

First, we investigate the two-dimensional plane of the ${\cal CP}$-even and
${\cal CP}$-odd Higgs–top-quark coupling parameters, $c_{t}$ and
$\tilde{c}_{t}$, respectively. The corresponding fit results for all four
model parametrizations are shown in Fig. 8. The profile $\Delta\chi^{2}$
distribution is shown in color (the maximal value includes all values of
$\Delta\chi^{2}\geq 20$) and the $1\,\sigma$, $2\,\sigma$ and $3\,\sigma$
confidence regions as defined above are indicated by the white, light-gray and
dark-gray dashed contours, respectively.

Figure 8: Fit results in the ($c_{t}$, $\tilde{c}_{t}$) parameter plane for
the four considered models (see header labels), using all currently available
measurements from the LHC. The color corresponds to the profile
$\Delta\chi^{2}$ of the global fit, and the $1\,\sigma$, $2\,\sigma$ and
$3\,\sigma$ confidence regions are shown as white, light-gray and dark-gray
dashed contours, respectively. The best-fit point and the SM case are marked
by a white star and an orange cross, respectively. Note the larger scale
required to display the bottom plots.

In the first three parametrizations (see above) the gluon fusion cross section
(or equivalently $\kappa_{g}$) is derived from $c_{t}$ and $\tilde{c}_{t}$.
The fact that the experimental results are well compatible with a gluon fusion
cross section that is close to the SM value gives rise to the feature that the
favored region of parameter space is located around the ellipse in the
($c_{t}$, $\tilde{c}_{t}$) parameter space where $\kappa_{g}\approx 1$ (see
also Figs. 7 and 4). However, the favored region is restricted to the part
where $c_{t}$ is around $+1$. In the 2D and 3D parametrizations – ($c_{t}$,
$\tilde{c}_{t}$) free and ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$) free – this
constraint originates from the fact that the preferred value for the
$H\to\gamma\gamma$ partial width modifier, $\kappa_{\gamma}$, which is derived
in these parametrizations from the fit parameters, is close to $+1$ (see also
the discussion in Sec. 4). As the 3D parametrization has additional freedom in
the Higgs-vector boson coupling, $c_{V}$, which influences the
$H\to\gamma\gamma$ partial width, the allowed region in the ($c_{t}$,
$\tilde{c}_{t}$) parameter space is slightly larger than in the 2D
parametrization (which has fixed $c_{V}=1$). In contrast, the 4D
parametrization treats $\kappa_{\gamma}$ as a free parameter, hence the lower
bound on $c_{t}$ is weakened here, as shown in the _bottom left panel_ of Fig.
8 (note the different scale of the $c_{t}$ axis in comparison to the previous
plots in the figure). In fact, two local minima of the likelihood appear in
the negative $c_{t}$ region, at $(c_{t},\tilde{c}_{t})\sim(-0.3,0.6)$ and
$(-0.3,-0.6)$, which are still allowed at the $3\,\sigma$ level. In these
regions, the predictions for Higgs production in gluon fusion, $gg\to H$, as
well as the combined top-quark associated Higgs production, $pp\to
t\bar{t}H,tWH$ and $tH$, are SM-like [see the intersection of the gray and red
solid contour in Fig. 7 (_bottom left panel_)]. However, the process $gg\to
ZH$ is strongly enhanced here, which excludes this region at the $2\,\sigma$
level. Overall, the interplay of Higgs measurements in channels containing the
top quark associated and $gg\to ZH$ Higgs production modes restricts the
$2\,\sigma$ favored region to the positive $c_{t}$ range, as will be discussed
in more detail in the following subsections.

The shape of the favored region in the 5D parametrization is different, as
$\kappa_{g}$ and $\kappa_{\gamma}$ are both treated as free parameters. Here,
the favored region in the ($c_{t}$, $\tilde{c}_{t}$) parameter space extends
to much larger values of $\tilde{c}_{t}$ than in the previous
parametrizations. It should be noted that the scale on the
$\tilde{c}_{t}$-axis has been enlarged by a factor two compared to the other
plots in this figure, while the scale on the $c_{t}$-axis has been much
enlarged and displays also negative values. As Higgs production in gluon
fusion is now governed by an individual fit parameter, the fit does not
constrain the allowed parameter space to the region near the $\kappa_{g}=1$
ellipse derived from $c_{t}$ and $\tilde{c}_{t}$ (see Sec. 4). Instead, again,
the Higgs measurements in channels containing the top quark associated and
$gg\to ZH$ Higgs production modes become important and impose the constraints
on the ($c_{t}$, $\tilde{c}_{t}$) parameter space observed in Fig. 8 (bottom
right). In particular, the allowed region aligns with the ellipse in the
parameter space where SM-like rates are obtained for inclusive top quark
associated Higgs production, see Fig. 7, and allow substantially smaller
$c_{t}$ values as compared to the other parametrizations. However, they are
constrained at the $2\,\sigma$ level to the region $c_{t}>0$ by measurements
that include $gg\to ZH$ production, see Fig. 4, as well as by dedicated top-
quark associated Higgs measurements that feature different signal acceptances
among the various top quark associated Higgs production modes (see App. B and
discussion below).

Figure 9: Fit results for the ${\cal CP}$-violating phase $\alpha$ in the four
different model parametrizations.

The constraints derived in the various parametrizations can also be
interpreted in terms of the ${\cal CP}$-violating phase $\alpha$, defined in
Eq. 2. The fit results for $\alpha$ are shown in Fig. 9, where we again
profile over the other parameters. Since $\tilde{c}_{t}$ enters the
theoretical predictions only in squared form (see Sec. 4), the constraints on
the ${\cal CP}$-violating phase $\alpha$ are symmetric around zero. The
constraints are similar in the 2D, 3D and 4D parametrizations, with $|\alpha|$
being bounded to be $\lesssim 22.5-27$ degrees at the $2\,\sigma$ level. In
the 4D parametrization, a second minimum appears around $\alpha\sim 110$
degrees, which corresponds to the local minima in the negative $c_{t}$ region
discussed above [see Fig. 8 (_bottom left panel_)]. In the 5D parametrization,
the $2\,\sigma$ upper limit on the ${\cal CP}$-violating phase is $\lesssim
72$ degrees.

###  5.3 $pp\to ZH$ subprocesses

We now study the behavior of the rate of Higgs production in association with
a $Z$ boson in our fit. In particular, we focus on the gluon-initiated
subprocess, $gg\to ZH$, which depends non-trivially on the ${\cal CP}$-nature
of the Higgs–top-quark interaction, as discussed in Sec. 4. In contrast, the
quark-initiated subprocess, $q\bar{q}\to ZH$, depends dominantly on the
Higgs–$Z$-boson interaction, i.e. on the coupling modifier $c_{V}$, which is
either set to the SM prediction (in the 2D parametrization) or treated as an
individual fit parameter (in the other parametrizations). For the latter, in
an inclusive measurement of $pp\to ZH$, the $c_{V}$-dominated $q\bar{q}\to ZH$
process can at least partially compensate a non-trivial modification of the
$gg\to ZH$ rate from the ${\cal CP}$-violating Higgs–top-quark interaction.
Consequently, non-inclusive measurements, where the subprocesses $q\bar{q}\to
ZH$ and $gg\to ZH$ can be separated, as well as Higgs transverse momentum
($p_{T}$) dependent measurements are crucial to increase the sensitivity to
the ${\cal CP}$-nature of the Higgs–top-quark interaction in $Z$-boson
associated Higgs production.

Figure 10: Predicted signal strength of gluon-initiated $ZH$ production,
$\mu_{gg\to ZH}$, in dependence of $c_{t}$ (_top row_) and $\tilde{c}_{t}$
(_middle row_) and its correlation with the $q\bar{q}\to ZH$ signal strength,
$\mu_{q\bar{q}\to ZH}$, (_bottom row_) for the 3D parametrization (_left
panels_) and the 5D parametrization (_right panels_). The color corresponds to
the profile $\Delta\chi^{2}$ of the global fit, and the $1\,\sigma$,
$2\,\sigma$ and $3\,\sigma$ confidence regions are shown as white, light-gray
and dark-gray dashed contours, respectively. The best-fit point and the SM
case are marked by a white star and an orange cross, respectively. Note the
larger scale required to display the plots in the right panels.

We display in Fig. 10 the inclusive cross section of the $gg\to ZH$ process
normalized to the SM prediction, $\mu_{gg\to ZH}$, in dependence of $c_{t}$
(_top panels_) and $\tilde{c}_{t}$ (_bottom panels_) for the 3D
parametrization – ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$) free – (_left panels_)
and the 5D parametrization – ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$,
$\kappa_{\gamma}$) free – (_right panels, note the larger scale required to
display the plots_). In both models, this rate is derived via Eq. 11 from the
fit parameters $c_{t}$, $\tilde{c}_{t}$ and $c_{V}$. The differences in the
allowed $\mu_{gg\to ZH}$ ranges in the two models therefore entirely originate
from differences in the allowed ranges of the fit parameters $c_{t}$,
$\tilde{c}_{t}$ and $c_{V}$.666The corresponding results for the 2D and 4D
parametrization not shown here are qualitatively very similar to those for the
3D parametrization. We find the following allowed ranges for $\mu_{gg\to ZH}$
at the $2\,\sigma$ level:

$\displaystyle\mu_{gg\to ZH}\in[0.70,1.36]$
$\displaystyle\qquad\text{in~{}the~{}3D parametrization -- ($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$) free},$ (21) $\displaystyle\mu_{gg\to
ZH}\in[0.44,2.90]$ $\displaystyle\qquad\text{in~{}the~{}5D parametrization --
($c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$, $\kappa_{\gamma}$) free}.$
(22)

The upper limits of the allowed ranges are reached only for a non-zero ${\cal
CP}$-odd component of the Higgs–top-quark interaction, i.e. $\tilde{c}_{t}\neq
0$. This feature is particularly pronounced in the 5D parametrization, see
Fig. 10 (_bottom right panel_), but is also visible in the 3D parametrization
(_bottom left panel_). From the upper panels in Fig. 10 we again observe how
$\mu_{gg\to ZH}$ increases for decreasing $c_{t}$, as shown in Fig. 4 in Sec.
4, thus limiting the allowed $c_{t}$ range in the 5D parametrization to
positive values at the $2\,\sigma$ level.

Concerning the $q\bar{q}\to ZH$ subprocess, we found in all parametrizations
with $c_{V}$ as a fit parameter that the $q\bar{q}\to ZH$ cross section is
restricted at the $2\,\sigma$ level to be within $20\%$ of the SM prediction,
as can be seen in the bottom panels of Fig. 10 for the 3D and 5D
parametrization. While we observe a mild linear correlation of the $gg\to ZH$
and $q\bar{q}\to ZH$ signal strengths in the 3D and 4D (not shown)
parametrization, the signal strengths are rather uncorrelated in the 5D
parametrization. Hence, observing a strongly enhanced $gg\to ZH$ component in
Higgs channels targeting $ZH$ production may indicate a ${\cal CP}$-violating
Higgs–top-quark interaction.

We now analyze to what extent measurements of differential distributions
already help to constrain the parameter space. As noted earlier, the only
measurements that allow us to account for kinematic modifications due to the
${\cal CP}$-odd component of the Higgs–top-quark coupling are the STXS
measurements in three Higgs-$p_{T}$ bins of the $pp\to ZH$, $H\to b\bar{b}$
channel, as well as four $\Delta\Phi_{j_{1}j_{2}}$ bins of the $pp\to H+2j$,
$H\to\gamma\gamma$ channel, both provided by ATLAS. The modifications of the
$\Delta\Phi_{j_{1}j_{2}}$ distribution in $pp\to H+2j$ are very small compared
to current experimental and theoretical uncertainties, see Sec. 4 and App. A.
The impact of these effects on the fit to current data is therefore
negligible.

Figure 11: Fit results in the ($c_{t}$, $\tilde{c}_{t}$) plane if
Higgs-$p_{T}$-shape modifications in $gg\to ZH$ due to a modified top-Yukawa
coupling are ignored. Results are shown for the 5D parametrization (_left
panel_) and for a parametrization where the $gg\to ZH$ cross section is
additionally modified by an individual fit parameter, $\kappa_{ggZH}$ (_right
panel_). The color corresponds to the profile $\Delta\chi^{2}$ of the global
fit, and the $1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ confidence regions are
shown as white, light-gray and dark-gray dashed contours, respectively. The
best-fit point and the SM case are marked by a white star and an orange cross,
respectively.

In order to assess the impact of the Higgs-$p_{T}$-shape modifications in the
$pp\to ZH$, $H\to b\bar{b}$ channel, we repeated the fit in the 5D
parametrization – ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$,
$\kappa_{\gamma}$) free – while ignoring the modifications of the kinematic
Higgs boson properties that could be induced by ${\cal CP}$-violating
effects.777Technically, the predicted signal strength in each $pp\to ZH,H\to
b\bar{b}$ $p_{T}$-bin is set to the signal strength of the inclusive $pp\to
ZH$ rate. In Fig. 11 we show the fit result in the ($c_{t}$, $\tilde{c}_{t}$)
plane using the full observable set without these kinematic shape
modifications (_left panel_). Comparing these results with the result when
taking these effects into account, see Fig. 8 (_bottom right panel_), we find
the impact of the additional information provided through the current
kinematic shape measurements to be moderate, but clearly visible. The impact
is most pronounced in the regions of large $|\tilde{c}_{t}|$ and small
$c_{t}$, as well as in the negative $c_{t}$ region at small $\tilde{c}_{t}$
values. This can be seen, for instance, by comparing the $2\,\sigma$ and
$3\,\sigma$ contours. In particular, if the shape modifications are neglected,
the $3\,\sigma$ allowed region closes in the negative $c_{t}$ parameter space.
We conclude that accounting for the Higgs-$p_{T}$ modifications in the $gg\to
ZH$ process, even at the current stage, already helps to constrain the
parameter space. We therefore strongly encourage the ATLAS and CMS
collaborations to (continue to) provide such measurements, e.g. by employing
the STXS framework.

So far, the discussion and results were based on the assumption that the
$gg\to ZH$ cross section can be derived entirely from the modified Higgs–top-
quark and Higgs–vector-boson interactions, i.e. from the parameters $c_{t}$,
$\tilde{c}_{t}$ and $c_{V}$. In other words, genuine contributions to $gg\to
ZH$ from higher-dimensional operators were assumed to be absent. If we relax
this assumption, both the inclusive and Higgs-$p_{T}$ differential cross
sections can be altered by new operators. While a detailed study of these
effects would require to consider all relevant SMEFT operators888At dimension-
six, three operators directly contribute to $gg\to ZH$ production [99]. In
addition, there are three dimension-six operators modifying the Higgs
$Z$-boson coupling. which is beyond the scope of the present work, we
nevertheless assess the possible effects of additional loop contributions to
$gg\to ZH$ production using a single scale factor ($\kappa$) parametrization.
For this purpose, we consider higher-dimensional operator contributions that —
in combination with the modifications from our model parameters $c_{t}$,
$\tilde{c}_{t}$ and $c_{V}$ — change the inclusive $gg\to ZH$ cross section by
$\kappa_{ggZH}^{2}$ with respect to the SM prediction, see Eq. 11. We can then
treat $\kappa_{ggZH}$ as an additional free floating parameter, which
decorrelates the $gg\to ZH$ cross section from the other fit parameters.
However in this approach we have to neglect Higgs-$p_{T}$-shape modifications
in $gg\to ZH$, as this would require knowledge of all underlying relevant
operators in the Lagrangian.

The resulting constraints in the ($c_{t}$, $\tilde{c}_{t}$) parameter plane
for this fit are displayed in the right panel of Fig. 11. We find these to be
slightly weaker than those from the previous parametrization, Fig. 11 (_left
panel_), i.e. when performing the fit without the additional freedom in $gg\to
ZH$ and without accounting for the Higgs-$p_{T}$-shape modifications. From
this we conclude that, in the model parametrization(s) considered here, both
the inclusive and Higgs-$p_{T}$ differential rate measurements of the $gg\to
ZH$ process have a sizable impact on the allowed parameter space.999As noted
earlier, in an inclusive $ZH$ rate measurement, the $gg\to ZH$ rate
modification can be partially compensated by modifications of the $q\bar{q}\to
ZH$ rate if $c_{V}$ is treated as a free parameter in the fit. In contrast, if
$c_{V}$ were fixed, the inclusive $gg\to ZH$ measurement would have an even
larger impact.

###  5.4 Top quark associated Higgs production

We now discuss the rates for Higgs production in association with top quarks.
In the SM the dominant process is $pp\to t\bar{t}H$ with a
$13\,\,\mathrm{TeV}$ cross section of $\sim 0.5~{}\mathrm{pb}$, while the
rates for Higgs production in association with a single top quark, $pp\to tH$
and $pp\to tWH$, are smaller by roughly a factor of $7$ and $33$,
respectively. However this picture can be strongly affected by the presence of
a non-zero ${\cal CP}$-odd component. Experimental measurements of top-quark
and Higgs associated production have been so far inclusive in the Higgs boson
$p_{T}$, as well as in all three processes, due to the limited size of the LHC
dataset. The measured rate is then composed of $t\bar{t}H$, $tH$ and $tWH$
events. The $tH$ contribution is sometimes subtracted assuming the SM
prediction for the $tH$ rate. Unfortunately, information about the acceptance
of $t\bar{t}H$, $tH$ and $tWH$ events is rarely provided by the experiments.
If no such information is given for a measurement, we have to make assumptions
about the relative contributions of these processes. In these cases, we
assumed equal acceptances for $t\bar{t}H$ and $tWH$ events. The acceptance of
$tH$ events is also assumed to be equal to the $t\bar{t}H$ acceptance, except
in measurements requiring two leptons originating from the Higgs production
process, where the acceptance of $tH$ events is set to zero. Furthermore, we
assume all acceptances to be independent of the ${\cal CP}$-nature of the
Higgs–top-quark coupling. More details and explanations are given in App. B.

While detailed information on the $tH$ and $tWH$ signal acceptance may appear
irrelevant for the SM, it is very relevant when testing models where these
processes can be strongly enhanced. The impact of our assumptions on the
signal acceptances on the fit is demonstrated in App. B, where we show fit
results obtained for a variation of the $tH$ acceptance in all Higgs
measurements. The associated uncertainty of this assumption on the fit is
sizable in parameter regions of enhanced $tH$ production. We therefore
strongly encourage the experimental collaborations to publicly provide more
detailed information about the $t\bar{t}H$, $tWH$ and $tH$ signal acceptances
in future measurements. In fact, in case experimental analyses feature
significantly different acceptances for these processes, it should be possible
to determine the rate of each process separately in a global fit including
several measurements of top quark associated Higgs production, which in turn
can constrain the ${\cal CP}$-character of the Higgs–top-quark interaction.

Figure 12: Predicted signal strength for combined $t\bar{t}H+tH+tWH$
production, $\mu_{t\bar{t}H+tH+tWH}$, in dependence of $c_{t}$ (_top panels_)
and $\tilde{c}_{t}$ (_bottom panels_) for the 3D parametrization (_left
panels_) and the 5D parametrization (_right panels_). The color corresponds to
the profile $\Delta\chi^{2}$ of the global fit, and the $1\,\sigma$,
$2\,\sigma$ and $3\,\sigma$ confidence regions are shown as white, light-gray
and dark-gray dashed contours, respectively. The best-fit point and the SM
case are marked by a white star and an orange cross, respectively. The green
band in the bottom left panel indicates the parameter points with $c_{V}$ and
$c_{t}$ being within $2\%$ of the SM prediction. Note the larger scale
required to display the plots in the right panels.

The combined $t\bar{t}H+tH+tWH$ signal strength is shown in Fig. 12 in
dependence of $c_{t}$ (_top panels_) and $\tilde{c}_{t}$ (_bottom panels_) for
the 3D parametrization – ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$) free – (_left
panels_) and 5D parametrization – ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$,
$\kappa_{g}$, $\kappa_{\gamma}$) free – (_right panels, note the larger scale
required to display the plots_). We find the $2\,\sigma$ allowed ranges to be

$\displaystyle\mu_{t\bar{t}H+tH+tWH}\in[0.76,1.19]$
$\displaystyle\qquad\text{in~{}the~{}3D parametrization -- ($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$) free},$ (23)
$\displaystyle\mu_{t\bar{t}H+tH+tWH}\in[0.64,1.56]$
$\displaystyle\qquad\text{in~{}the~{}5D parametrization -- ($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$, $\kappa_{\gamma}$) free}.$ (24)

In the 3D parametrization101010The phenomenology of the $t\bar{t}H$, $tH$ and
$tWH$ production modes in the 2D and 4D parametrization (not shown here) is
qualitatively very similar to the behavior found in the 3D parametrization.
the constraints on $\mu_{t\bar{t}H+tH+tWH}$ are mainly induced through model
correlations. Measurements of Higgs production in gluon fusion and the
$H\to\gamma\gamma$ decay rate strongly constrain the model parameters $c_{t}$
and $\tilde{c}_{t}$, which, in turn, restricts the allowed range for
$\mu_{t\bar{t}H+tH+tWH}$. From Fig. 12 (_top left panel_) we see that, in this
parametrization, the allowed values of the combined signal strength when
$c_{t}\geq 1$ can be approximated by $\mu_{t\bar{t}H+tH+tWH}\approx
c_{t}^{2}$, as $\tilde{c}_{t}$ is small and a strong $tH$ enhancement is not
possible here. For $c_{t}<1$, larger values of $\tilde{c}_{t}$ are allowed
(compare with Fig. 8). This can also be observed by comparing the areas above
($c_{t}\gtrsim 1$) and below ($c_{t}\lesssim 1$) the green band in the _bottom
left panel_ of Fig. 12. Thus an enhancement in $tH$ production leads to
$\mu_{t\bar{t}H+tH+tWH}$ values slightly above the naive $c_{t}^{2}$
expectation. Still, in the $c_{t}<1$ region, the combined signal strength
cannot significantly exceed the SM expectation ($\mu_{t\bar{t}H+tH+tWH}=1$).

In contrast, in the 5D parametrization, where the gluon fusion cross section
and the $H\to\gamma\gamma$ decay rate are governed by individual fit
parameters, the signal strength is much less constrained (we stress again that
the scale in the right panels is enlarged as compared to the left panels), due
to the absence of correlations with the sensitive gluon fusion and
$H\to\gamma\gamma$ observables. Here, dedicated measurements of top quark
associated Higgs production directly restrict the allowed range of
$\mu_{t\bar{t}H+tH+tWH}$. This, in turn, yields constraints on the underlying
model parameters $c_{t}$ and $\tilde{c}_{t}$, in an important interplay with
measurements sensitive to $gg\to ZH$, as explained in Sec. 5.2. It is
interesting to note that the correlation between $c_{t}$ and
$\mu_{t\bar{t}H+tH+tWH}$ is much weaker in the 5D parametrization, Fig. 12
(_upper right panel_), than in the 3D parametrization (_upper left panel_).
Here, even for $c_{t}<1$, an enhancement of the $t\bar{t}H+tH+tWH$ inclusive
cross section by more than $+50\%$ is possible at the $2\,\sigma$ level.

Figure 13: Predicted ratio between the SM-normalized rates of $tH$ and
$t\bar{t}H$ production, $\mu_{tH}/\mu_{t\bar{t}H}$, in dependence of $c_{t}$
(_top panel_) and $\tilde{c}_{t}$ (_bottom panel_) for the 3D parametrization
(_left panels_) and the 5D parametrization (_right panels_). The color
corresponds to the profile $\Delta\chi^{2}$ of the global fit, and the
$1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ confidence regions are shown as
white, light-gray and dark-gray dashed contours, respectively. The best-fit
point and the SM case are marked by a white star and an orange cross,
respectively. The green band in the bottom left panel indicates the parameter
points with $c_{V}$ and $c_{t}$ being within $2\%$ of the SM prediction. Note
the larger scale required to display the plots in the right panels.

Nevertheless, from the lower panels in Fig. 12 we can conclude that,
independently of the specific model parametrization, the measurement of the
combined inclusive signal strength of top quark associated production does not
feature sensitive discrimination power regarding the ${\cal CP}$-odd
Higgs–top-quark coupling $\tilde{c}_{t}$. Instead, one may aim at a separate
determination of the $t\bar{t}H+tWH$ and $tH$ cross sections. We will discuss
some strategies for this in Sec. 6. In order to illustrate the possible impact
of separate experimental information on the $t\bar{t}H+tWH$ and the $tH$ cross
section, we show in Fig. 13 the corresponding plots where the SM-normalized
ratio of the $tH$ rate over the $t\bar{t}H+tWH$ rate,
$\mu_{tH/(t\bar{t}H+tWH)}$, is displayed (see Eq. 19). This quantity is shown
in Fig. 13 in dependence of $c_{t}$ (_top panels_) and $\tilde{c}_{t}$
(_bottom panels_), for the same two model parametrizations as above (note the
larger scale required to display the plots in the right panels). We find that
the cross section ratio can be significantly enhanced with respect to the SM.
Specifically, within the $2\,\sigma$ region, we find

$\displaystyle\mu_{tH/(t\bar{t}H+tWH)}\in[0.8,1.7]$
$\displaystyle\qquad\text{in~{}the~{}3D parametrization -- ($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$) free},$ (25)
$\displaystyle\mu_{tH/(t\bar{t}H+tWH)}\in[0.8,6.7]$
$\displaystyle\qquad\text{in~{}the~{}5D parametrization -- ($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$, $\kappa_{\gamma}$) free}.$ (26)

As can be seen in Fig. 13, this enhancement increases with decreasing $c_{t}$
and increasing $|\tilde{c}_{t}|$, and is therefore a sensitive probe of the
${\cal CP}$-nature of the Higgs–top-quark coupling.

In order to assess the impact of the dedicated experimental analyses targeting
top quark associated Higgs production on the parameter space, we repeated the
fit in the 5D parametrization – ($c_{t}$, $\tilde{c}_{t}$, $c_{V}$,
$\kappa_{g}$, $\kappa_{\gamma}$) free – with a reduced observable set in
HiggsSignals which excludes all specific $t\bar{t}H$ analyses (see App. B for
details).111111In this fit we furthermore ignored Higgs-$p_{T}$-shape
modifications in $pp\to ZH$, as done in Sec. 5.3. The fit result for the
($c_{t}$, $\tilde{c}_{t}$) parameter plane is given in the left panel of Fig.
14. The impact of the $t\bar{t}H$ measurements can be clearly seen in the
comparison with the left panel of Fig. 11, which includes these measurements.
Obviously, leaving out the information from the dedicated $t\bar{t}H$
measurements largely opens up the parameter space in the ($c_{t}$,
$\tilde{c}_{t}$) plane for this model parametrization. In the fit without the
$t\bar{t}H$ measurements the favored region still aligns with the contour
ellipses of constant $\mu_{t\bar{t}H+tH+tWH}$, see Fig. 7 (_bottom left
panel_), where — in contrast to the fit with all observables included, Fig. 11
(_left)_ — the $2\,\sigma$ allowed parameter space covers the whole ellipse,
i.e. it closes in the negative $c_{t}$ region. Within the $2\,\sigma$ region
we find the combined rate and the ratio of $tH$ and $t\bar{t}H$ production to
be

$\displaystyle\mu_{t\bar{t}H+tH+tWH}\in[0.3,3.5]\qquad\mbox{and}\qquad\mu_{tH/(t\bar{t}H+tWH)}\in[0.8,38.0].$
(27)

The fit still favors positive values of $c_{t}$ at the $1\,\sigma$ level. This
constraint originates from $ZH$-sensitive measurements, as the $gg\to ZH$ rate
is enhanced for decreasing $c_{t}$ in this parametrization, see Fig. 4.

Figure 14: Fit results in the ($c_{t}$, $\tilde{c}_{t}$) plane if
Higgs-$p_{T}$-shape modifications in $gg\to ZH$ due to ${\cal CP}$-violation
are ignored and measurements from dedicated $t\bar{t}H$ analyses are excluded
from the fit. Results are shown for the 5D parametrization (_left panel_) and
for a parametrization where the $gg\to ZH$ cross section is additionally
modified by an individual fit parameter, $\kappa_{ggZH}$ (_right panel_). The
color corresponds to the profile $\Delta\chi^{2}$ of the global fit, and the
$1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ confidence regions are shown as
white, light-gray and dark-gray dashed contours, respectively. The best-fit
point and the SM case are marked by a white star and an orange cross,
respectively.

In order to explicitly prove this influence of the $gg\to ZH$ process, we
repeated the fit to the same observable set (i.e. without dedicated
$t\bar{t}H$ analyses) in the six-dimensional parametrization from the
preceding subsection, treating $c_{t}$, $\tilde{c}_{t}$, $c_{V}$,
$\kappa_{g}$, $\kappa_{\gamma}$ and $\kappa_{ggZH}$ as free parameters, i.e.
allowing the $gg\to ZH$ rate to be freely adjusted by $\kappa_{ggZH}^{2}$. The
result in the ($c_{t}$, $\tilde{c}_{t}$) parameter plane is shown in Fig. 14
(_right panel_). Here, there is no sign preference in $c_{t}$, and the
likelihood is completely flat along the entire ellipse of constant
$\mu_{t\bar{t}H+tH+tWH}$, as anticipated. The fact that
$\mu_{t\bar{t}H+tH+tWH}\simeq 1$ is still preferred — albeit at a very low
significance — is due to inclusive, non-$t\bar{t}H$ specific, Higgs rate
measurements included in the fit, as well as the inclusion of the LHC Run-1
ATLAS and CMS combined measurements. Both types of measurements maintain some
sensitivity to top quark associated Higgs production processes.

###  5.5 Discussion

As shown in Sec. 5.4, top-quark associated Higgs production plays a key role
in constraining the ${\cal CP}$-nature of the Higgs–top-quark interaction. The
corresponding constraints can be tightened by taking into account not only
rate information, but also kinematic information which partly incorporate
${\cal CP}$-odd observables, as done in the recent ATLAS and CMS analyses [25,
26]. Due to differences in the used models, a comparison of these studies to
our results is difficult. The CMS study constrains the ${\cal CP}$-violating
phase to be lower than 55 degrees at the $2\,\sigma$ level under the
assumption that all Higgs production modes (apart from top-quark associated
Higgs production) are constrained to their SM predictions. In our 5D
parametrization, this would imply that $c_{V}=\kappa_{g}=\kappa_{\gamma}=1$ is
assumed. For comparison, in our results in the 5D parametrization where
$c_{V}$, $\kappa_{g}$ and $\kappa_{\gamma}$ are treated as free parameters, we
obtain an upper limit of 72 degrees (see Sec. 5.2). The ATLAS study finds a
limit of 43 degrees regardless whether $\kappa_{g}$ and $\kappa_{\gamma}$ are
treated as free parameters or not. If $\kappa_{g}$ and $\kappa_{\gamma}$ are
calculated as a function of the ${\cal CP}$-violating phase, the result could
be compared to our result of a $22.5$ degree upper limit in the 2D
parameterization. The comparability of the results is less obvious for the
case that $\kappa_{g}$ and $\kappa_{\gamma}$ are treated as free parameters.
Since the Monte-Carlo samples used in Ref. [26] are generated at the NLO level
employing the “Higgs Characterization Model”, top-quark associated Higgs
production cannot be regarded as being independent of Higgs production via
gluon fusion, since top-quark associated Higgs production depends on the
Higgs–gluon–gluon operator at NLO (see the discussion in Sec. 4.1 for more
details). Therefore, the results of Ref. [26] can only be compared to our
result in an approximate way.

As discussed in Sec. 1, results as the ones presented in Ref. [25, 26] are in
any case difficult to integrate into global fits without further experimental
information, since the interpretation of these results depends strongly on the
exact model used for deriving the constraints. Also, the resulting limit on
${\cal CP}$-odd couplings from such analyses cannot be directly applied to
other models than the model studied in the experimental analyses.

Unfolded differential cross section or STXS measurements can be used to lower
the model-dependence. But also for such measurements, the sensitivity of the
kinematic distributions to the ${\cal CP}$-character of the Higgs boson can
affect the efficiency of the event selection, thus introducing some model-
dependence (see Sec. 6). In the particular case of the measurement of
$t\bar{t}H$ kinematic shapes for ${\cal CP}$ tests, the subtraction of the
$tH$ and $tWH$ background in a model-independent way is delicate, due to the
strong dependence of the $tH$ and $tWH$ cross sections on the ${\cal CP}$
properties of the Higgs boson.

Total rate measurements are, however, largely model-independent. While total
rate measurements for combined top-quark associated Higgs production have
already been performed and first shape measurements with the full Run 2 LHC
data are on-going, few attempts have been made so far to disentangle $tH$ and
$t\bar{t}H+tWH$ production. However, as discussed above and shown in Fig. 13,
separate measurements of $tH$ and $t\bar{t}H+tWH$ production have the
potential to constrain further the $(c_{t},\tilde{c}_{t})$ parameter space.
While such measurements may not be feasible with current data, they may become
possible with upcoming data, as discussed in the next Section.

##  6 Future sensitivity to $tH$ production and constraints on ${\cal CP}$

The aim of this section is to evaluate the possibility for separately
measuring $tH$ and $t\bar{t}H+tWH$ production at the LHC with 300 fb-1 and at
the HL-LHC with 3000 fb-1 of data and to discuss the additional constraints
brought by this new measurement on the Higgs-${\cal CP}$ nature. For this
purpose, we evaluate the feasibility of an analysis for the separate
measurement of $tH$ and $t\bar{t}H+tWH$ production that is designed to remain
independent of the ${\cal CP}$-character of the top-Yukawa coupling. This
allows us to directly compare the measured rates to the theoretical
predictions, e.g. from the model described in Sec. 4, and enables their
inclusion in a global fit such as the one presented earlier.

The analysis is implemented in Delphes [100] and employs a simplified analysis
strategy. A corresponding analysis by ATLAS and CMS thus should yield an
improvement over the results presented here. The inclusion of the $tH$ signal
strength, as proposed here, in a global study of the ${\cal CP}$-properties,
along with various other inclusive and differential measurements, would yield
the highest sensitivity.

###  6.1 Analysis strategy

Only $H\to\gamma\gamma$ decays are considered in the study, due to their sharp
two photon invariant mass ($m_{\gamma\gamma}$) peak signature around the Higgs
boson mass ($m_{H}\sim 125$ GeV) at the LHC and equally at the HL-LHC. This
allows the subtraction of the background originating from non-Higgs processes,
which is typically smoothly falling as a function of $m_{\gamma\gamma}$, with
a rather simple fit of the $m_{\gamma\gamma}$ distribution in the experimental
data, including a fixed Higgs boson mass. In this work, we assume that the
non-Higgs background is subtracted by such a fit. The $H\to\gamma\gamma$ decay
channel also benefits from a relatively high decay branching ratio ($\sim
10^{-3}$), e.g. with respect to the four-lepton decay channel ($\sim
10^{-4}$), which allows to apply tighter cuts to select $tH$ events while
retaining a sufficient number of events. These properties make
$H\to\gamma\gamma$ decays particularly relevant in the context of this
phenomenological work, however we encourage the experiments to explore all
relevant Higgs boson decay channels.

We consider the splitting of data events in two categories. The first category
(1-lepton category) targets leptonic $W$ boson decays in $tH$ events and so
requires exactly one electron or muon in the final state. The second category
(2-lepton category) targets $t\bar{t}H$ and $tWH$ events in which both $W$
bosons decay leptonically, and thus requires exactly two electrons or muons
with opposite sign in the final state. The two categories are orthogonal to
each other due to the different lepton multiplicity requirement. The purpose
of the 2-lepton category, which is expected to be free of any $tH$ events, is
to allow an independent measurement of the $t\bar{t}H+tWH$ production. By
fitting the $t\bar{t}H+tWH$ event yields in the 1-lepton and 2-lepton
categories simultaneously, the number of $t\bar{t}H+tWH$ events entering the
1-lepton category, which originates mostly from events including one $W$ boson
decaying to leptons and the other one to hadrons, can then be predicted
accurately.

A particular attention will be given to the robustness of the analysis with
respect to the Higgs boson ${\cal CP}$ properties, which is an essential
feature to be able to include the results in a global ${\cal CP}$ fit such as
the one presented earlier in this paper.

###  6.2 Technical setup

Only Higgs boson production modes predicted by the SM which include at least
one ‘prompt’ lepton originating from a $Z$ or $W$ boson decay, i.e.
$t\bar{t}H$, $tWH$, $tH$ (neglecting the $tH$ $s$-channel contribution) and
$VH$ ($V=Z,W$, including the $gg\to ZH$ contribution) production, are
considered in the following. This is justified by the fact that the ATLAS and
CMS detectors allow for a very strong rejection of leptons which have a non-
prompt origin (e.g. bottom or charm hadron decays, electrons arising from a
photon conversion, jets misidentified as electrons, or muons produced from in-
flight pion or kaon decays) such that the resulting background can usually be
neglected in data analyses after requiring exactly one electron or muon (as in
the 1-lepton category) or exactly two electrons or muons with opposite sign
(as in the 2-lepton category) in the final state.

$t\bar{t}H$, $tWH$, $tH$ and $VH$ events in $\sqrt{s}=13$ TeV proton-proton
collisions are simulated using the same setup as described in Sec. 4.1. Monte
Carlo (MC) events are produced under four different ${\cal CP}$ hypotheses for
the Higgs–top-quark coupling, following the notation introduced in Sec. 3:

  * •

SM Higgs boson: $c_{t}=c_{V}=1$, $\tilde{c}_{t}=0$,

  * •

pure ${\cal CP}$-odd state: $\tilde{c}_{t}=c_{V}=1$, $c_{t}=0$,

  * •

${\cal CP}$-mixed state, benchmark 1: $c_{t}=\tilde{c}_{t}=0.8$, $c_{V}=1$,

  * •

${\cal CP}$-mixed state, benchmark 2: $c_{t}=0.5$, $\tilde{c}_{t}=1.2$,
$c_{V}=0.95$.

We assume the $H\to\gamma\gamma$ decay branching ratio to equal the SM
prediction for all four ${\cal CP}$ hypotheses ($\kappa_{\gamma}=1$), i.e.
allowing the existence of higher-dimensional BSM operators that couple the
Higgs boson directly to photons for the non-SM ${\cal CP}$ cases (see Sec. 3).
Since only $H\to\gamma\gamma$ decays are considered in this work and the
impact of the BSM operators on the $H\to\gamma\gamma$ decay kinematics
relevant for the present analysis are known to be small, a different
$H\to\gamma\gamma$ decay branching ratio is however simply equivalent to a
rescaling of the integrated luminosity at first order. In addition, the Higgs
production rate via gluon fusion is assumed to be equal to the SM prediction
for all ${\cal CP}$ hypotheses ($\kappa_{g}=1$). These assumptions are
necessary to relax the tight constraints from diagrams where the Higgs–top-
quark couplings intervene at the loop-level, see Sec. 5. They require in
general the presence of corresponding BSM operators and some non-trivial,
accidental cancellations (see Sec. 3), with $c_{g}$ and $\tilde{c}_{g}$ of the
order of $\mathcal{O}(\lesssim 0.5)$ (see Eq. 5).

The analyses presented in Refs. [25, 26] exclude a pure ${\cal CP}$-odd top-
Yukawa coupling by $3.9\,\sigma$ (ATLAS) and $3.2\,\sigma$ (CMS). The first
benchmark of a ${\cal CP}$-mixed state defined above lies within the
$1\,\sigma$ region of the five parameter fit presented in Sec. 5 [($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$, $\kappa_{\gamma}$, $\kappa_{g}$) free]. It
corresponds to a ${\cal CP}$-violating phase of $45^{\circ}$ and an absolute
rescaling factor for the top-Yukawa coupling of 0.8 with respect to the SM.
The second benchmark of a ${\cal CP}$-mixed state is chosen to be further away
from the SM point. It lies within the $2\,\sigma$ region of the five parameter
fit presented in Sec. 5.

The SM $tH$ and $t\bar{t}H$ samples are also generated at the next-to-leading
order (NLO) in QCD with the MSTW2018NLO [54] PDF set and Pythia 8 for
comparison (see discussion of Fig. 15 below). It should be kept in mind that
the SM Higgs boson scenario is the only one out of the four Higgs boson ${\cal
CP}$ hypotheses considered here which can be generated at NLO+PS with the
current implementation of the “Higgs charaterization model” when $\kappa_{g}$
is fixed to unity (see discussion in Sec. 4.1).

The simulated events are reconstructed with Delphes 3.4.2 [100] using either
the ATLAS or the HL-LHC configuration card provided in the Delphes repository.
The Delphes software allows for an approximate, very fast simulation of the
current ATLAS detector [101] response (ATLAS card) or of the future ATLAS or
CMS HL-LHC detector response (HL-LHC card) [102, 103], reconstructing typical
objects and quantities used in data analyses and accounting for the limited
energy resolution, reconstruction efficiency and acceptance of these objects
at the LHC and at the HL-LHC. Additional proton-proton interactions occurring
during the same bunch crossing (pileup) are assumed to be subtracted from
experimental data by dedicated algorithms and are not included in the MC
generation. In all the results which follow, the ATLAS card has been modified
to reconstruct hadronic jets using the anti-$k_{t}$ algorithm [104] with
radius parameter $R=0.4$ instead of $R=0.6$, which is the default value in the
configuration card. The value $R=0.4$ is more widely used in recent ATLAS data
analyses and has the additional advantage to harmonise the jet definition
between the ATLAS and the HL-LHC card. All the other parameters of the ATLAS
card are left unchanged. The HL-LHC card was used in a recent review of the
opportunities for Higgs physics at the HL-LHC [102, 103] so that it is left
unchanged. The same proton-proton center of mass energy of $\sqrt{s}=13$ TeV
is used in the LHC and HL-LHC scenario and an integrated luminosity of 300
fb-1 (3000 fb-1) is assumed at the LHC (HL-LHC). The center of mass energy may
be increased to $\sqrt{s}=14$ TeV at the HL-LHC [102, 103], which would
enhance Higgs production cross sections by about 20% (10%) for $t\bar{t}H$,
$tWH$ and $tH$ ($VH$). The choice to use $\sqrt{s}=13$ TeV for all scenarios
is therefore conservative.

###  6.3 Event selection

We start by implementing a simplified version of the typical event
preselection used in the latest $t\bar{t}H,~{}H\to\gamma\gamma$ ATLAS and CMS
measurements [25, 26]: all events must include at least two photons with
$|\eta|<2.5$ and $p_{T}^{\gamma}$ larger than 35 GeV and 25 GeV. In case of
the presence of more than two photons in the event, the two photons with
highest $p_{T}$ are selected, forming the Higgs boson candidate. The invariant
mass of the two selected photons, $m_{\gamma\gamma}$, must be within the range
[105–160] GeV, and the leading (subleading) photon in $p_{T}$ has to satisfy
$p_{T}^{\gamma}/m_{\gamma\gamma}>0.35~{}(0.25)$. At least one hadronic jet
identified as originating from the hadronization of a $b$-quark ($b$-jet) is
required in both categories, and must satisfy $p_{T}^{b}>25$ GeV and
$|\eta|<2.0$, and the missing transverse momentum $p_{T}^{miss}$ (as defined
in Delphes) must be higher than 25 GeV. The $t\bar{t}H$ event yield obtained
after applying this selection to events reconstructed with the ATLAS card, and
requiring in addition at least one electron or muon with $p_{T}^{\ell}>15$
GeV, was compared with a recent ATLAS measurement [105]. A good agreement
between the two numbers (within $10\%$) was observed after rescaling the
$t\bar{t}H$ event yield obtained in this work to the signal strength fitted by
ATLAS ($\mu_{t\bar{t}H}=1.4$).

We then split the events according to the lepton multiplicity: the 1-lepton
(2-lepton) category includes at least one (two) electron(s) or muon(s) with
$p_{T}^{\ell}>15$ GeV and no additional lepton in the full acceptance. The
two-lepton category requires in addition that the two leptons have opposite
signs and, in case they have the same flavor ($ee$ or $\mu\mu$), that their
invariant mass ($m_{\ell\ell}$) falls outside the $Z$ boson mass range,
defined as [80–100] GeV, preventing potential contamination from $ZH$ events.
The selections described above will be referred to as the _1-lepton
preselection_ and the _2-lepton selection_ in the following.

In the 1-lepton category, further discriminating variables are considered to
enhance the fraction of $tH$ events. In addition to the jet multiplicity
($N_{jet}$) and the $b$-jet multiplicity ($N_{bjet}$), which are both expected
to be larger in $t\bar{t}H$ due to the presence of a second top quark, the
highest $p_{T}$ $b$-jet is associated to the lepton and the $p_{T}^{miss}$
vector to form the top quark candidate, whose transverse mass is defined as

$\displaystyle
m_{T}^{top}=\sqrt{2p_{T}^{b+\ell}p_{T}^{miss}[1-\cos\Delta\phi(p_{T}^{b+l},p_{T}^{miss})]},$
(28)

where $p_{T}^{b+l}$ corresponds to the vector sum of the $b$-jet and the
lepton transverse momenta, and $\Delta\phi(p_{T}^{b+l},p_{T}^{miss})$
corresponds to the difference in azimuthal angle between the missing
transverse energy vector and $p_{T}^{b+l}$. The shapes of $N_{jet}$,
$N_{bjet}$, and $m_{T}^{top}$ obtained at LO+PS and NLO+PS for SM $t\bar{t}H$
and $tH$ events passing the _1-lepton preselection_ are shown in Fig. 15 (top,
middle). We observe a good consistency between the LO+PS and NLO+PS shapes
except for $N_{jet}$ in $tH$ events, where the LO+PS calculation predicts a
higher jet multiplicity. This feature has already been discussed in Ref. [38]
and also appears in the case of single top production alone. It is related to
the presence of a $b$ quark in the initial state in the five-flavor scheme
(instead of a gluon splitting to a $b\bar{b}$ pair in the four-flavor scheme),
which leads to a mismodeling of QCD radiation at leading-order. It was shown
in Ref. [38] to be mitigated once higher-order corrections are included
(NLO+PS). Since the jet multiplicity is not expected to strongly depend on the
${\cal CP}$ properties of the Higgs boson, the LO+PS $tH$ sample is reweighted
to the $N_{jet}$ shape obtained at NLO+PS in the SM for all four ${\cal CP}$
hypotheses in the following.121212A similar effect may also affect $tWH$
production. However, since the $tWH$ total rate is much smaller than the $tH$
and $t\bar{t}H$ rates, the effect is neglected and no corrections are
included. The impact of the reweighting on the $N_{bjet}$ and $m_{T}^{top}$
shapes is shown in Fig. 15 (top right and middle), and found to be small. The
discrimination power between the different production modes of $N_{jet}$,
$N_{bjet}$ and $m_{T}^{top}$ is found to be largely independent from the Higgs
${\cal CP}$ properties, and is shown in Fig. 16 (top, middle) in the SM. The
following requirements are applied to enhance the fraction of $tH$ events in
the 1-lepton category: $N_{jet}=2$, $N_{bjet}=1$ and $m_{T}^{top}<200$ GeV.

Figure 15: Comparison of the shape of (top, left) $N_{jet}$, (top, right)
$N_{bjet}$, (middle) $m_{T}^{top}$, (bottom, left) $|\Delta y^{bj}|$ and
(bottom, right) $y^{j}\oplus y^{\gamma\gamma}$ at (solid line) LO+PS and
(coarse dashed line) NLO+PS in SM (red) $tH$ and (green) $t\bar{t}H$ events
passing the _1-lepton preselection_. The shape of $N_{bjet}$, $m_{T}^{top}$,
$|\Delta y^{bj}|$ and $y^{j}\oplus y^{\gamma\gamma}$ are also shown for $tH$
events at LO+PS after reweighting the $N_{jet}$ shape to the one observed at
NLO+PS (thin dashed line). Events are reconstructed with the Delphes software
using the ATLAS card described in the text, similar results are obtained when
using the HL-LHC card.

Figure 16: Comparison of the shape of (top, left) $N_{jet}$, (top, right)
$N_{bjet}$, (middle) $m_{T}^{top}$, (bottom, left) $|\Delta y^{bj}|$ and
(bottom, right) $y^{j}\oplus y^{\gamma\gamma}$ at LO+PS in SM $tH$,
$t\bar{t}H$, $tWH$ and $VH$ events passing the _1-lepton preselection_. $tH$
events are reweighted to match the NLO+PS $N_{jet}$ shape. Events are
reconstructed with the Delphes software using the ATLAS card described in the
text, similar results are obtained when using the HL-LHC card.

Figure 17: Comparison of the shape of (left) $|\Delta y^{bj}|$ and (right)
$y^{j}\oplus y^{\gamma\gamma}$ in (top) $tH$ and (bottom) $t\bar{t}H$
production for the four ${\cal CP}$ hypotheses considered in the study: SM,
pure ${\cal CP}$-odd Higgs boson (solid lines), and two benchmarks of ${\cal
CP}$-mixed states described in the text (dashed lines). Events are
reconstructed with the Delphes software using the modified ATLAS card
described in the text, similar results are obtained when using the HL-LHC
card.

$tH$ events tend to include a high $p_{T}$ jet at large rapidity in the final
state, similarly to single top production alone, which is well separated from
the selected (central) $b$-jet. To exploit this feature, a first observable
that comes to mind is the absolute value of the difference in
rapidity131313The pseudo-rapidity ($\eta$) is a good approximation of the
rapidity ($y$) for the selected $b$-jet and the next highest $p_{T}$ jet due
to the low jet mass, i.e. $y^{b}\sim\eta^{b}$ and $y^{j}\sim\eta^{j}$, but not
for the Higgs boson candidate ($y^{\gamma\gamma}\neq\eta^{\gamma\gamma}$),
where the effect from the Higgs boson mass is sizable. between the selected
$b$-jet and the next highest $p_{T}$ jet ($|\Delta y^{bj}|$). The shape of
$|\Delta y^{bj}|$ is found to be stable after the $N_{jet}$ reweighting of
$tH$, as shown in Fig. 15 (bottom, left), and its discrimination power between
the different production modes is found to be large in the SM, as shown in
Fig. 16 (bottom, left). However, when the Higgs boson includes a ${\cal
CP}$-odd component, the $b$-jet tends to get closer (further away) in rapidity
from that jet in $tH$ ($t\bar{t}H$) events, which reduces significantly the
discrimination power and makes it dependent on the ${\cal CP}$ properties (see
Fig. 17, left). The full selection efficiency for $t\bar{t}H$ events if
requiring e.g. $|\Delta y^{bj}|>2$ varies by $\sim 40\%$ depending on the
${\cal CP}$-character of the top-Yukawa coupling. In Fig. 18, we evaluate the
use of the Higgs rapidity, $y^{\gamma\gamma}$, as an alternative to $y^{b}$.
While $t\bar{t}H$ events are clustered around the origin in the
$(y^{j},y^{\gamma\gamma})$ plane, only a few $tH$ events lie close to the
origin. Moreover, by comparing the left (SM) and the right plots (pure ${\cal
CP}$-odd), we observe that the presence of a ${\cal CP}$-odd component in the
Higgs–top-quark interaction leads to a transformation similar to a rotation in
the $(y^{j},y^{\gamma\gamma})$ plane. Consequently we introduce as an
alternative observable to $|\Delta y^{bj}|$ the sum in quadrature of $y^{j}$
and $y^{\gamma\gamma}$, defined as

$\displaystyle y^{j}\oplus
y^{\gamma\gamma}=\sqrt{(y^{j})^{2}+(y^{\gamma\gamma})^{2}}.$ (29)

The shape of $y^{j}\oplus y^{\gamma\gamma}$ is found to be stable after the
$N_{jet}$ reweighting of $tH$, as shown in Fig. 15 (bottom, right), and allows
for a good discrimination power between the different production modes in the
SM, as shown in Fig. 16 (bottom, right). The shape of $y^{j}\oplus
y^{\gamma\gamma}$ is stable with respect to the Higgs boson ${\cal CP}$
properties (see Fig. 17, bottom right), and the variation of the full
selection efficiency for $t\bar{t}H$ events if requiring e.g. $y^{j}\oplus
y^{\gamma\gamma}>2$ is within $\lesssim 2\%$ in the considered parameter
space. Due to this important property, a requirement of $y^{j}\oplus
y^{\gamma\gamma}>2$ is included in the 1-lepton selection but no selection on
$|\Delta y_{bj}|$ is considered. A summary of the event selection is presented
in Tab. 1.

Figure 18: Comparison of the correlations between $y^{j}$ and
$y^{\gamma\gamma}$ in (top) $tH$ and (bottom) $t\bar{t}H$ production in (left)
the SM and (right) for a pure ${\cal CP}$-odd Higgs boson. Events are
reconstructed with the Delphes software using the modified ATLAS card
described in the text, similar results are obtained when using the HL-LHC
card.

We showed in this section that the discriminating observables used to
disentangle $tH$ from $t\bar{t}H$ production need to be chosen with care. In
particular, the stability of their shape with respect to the ${\cal CP}$
properties of the Higgs boson is an essential ingredient making it possible to
probe $tH$ production in a ${\cal CP}$-independent way. We demonstrated that
this property can be achieved and proposed a new observable, $y^{j}\oplus
y^{\gamma\gamma}$, which fulfills the conditions.

Observable / Selection | 1-lepton selection | 2-lepton selection
---|---|---
$N_{\gamma}$ | $\geq 2$
$m_{\gamma\gamma}$ |  $[105-160]$ GeV
$(p_{T,1}^{\gamma},p_{T,2}^{\gamma})$ |  $\geq(35,25)$ GeV
$(p_{T,1}^{\gamma}/m_{\gamma\gamma},p_{T,2}^{\gamma}/m_{\gamma\gamma})$ | $\geq(0.35,0.25)$
$N_{bjet}$ | $\geq 1$
$p_{T}^{miss}$ |  $\geq 25$ GeV
$N_{\ell}$ |  exactly $1$ | exactly 2 with opposite sign
$m_{\ell\ell}$ | – |  $[80,100]$ GeV vetoed if same flavour
$N_{jet}$ | exactly 2 | –
$N_{bjet}$ | exactly 1 | –
$m_{T}^{top}$ |  $<200$ GeV | –
$y^{j}\oplus y^{\gamma\gamma}$ | $>2$ | –
Table 1: Summary of the 1-lepton and 2-lepton selection. The _1-lepton
preselection_ , not detailed in the table, corresponds to the 1-lepton
selection dropping the requirements on $N_{jet}$, $N_{bjet}$, $m_{T}^{top}$
and $y^{j}\oplus y^{\gamma\gamma}$. The object definition follows the one
written in the Delphes cards unless mentioned otherwise in the text.

###  6.4 Results

Less than two events in total are expected at the 300 fb-1 LHC in the 2-lepton
category in all four ${\cal CP}$ scenarios, indicating that additional data
from the HL-LHC is required to successfully implement the strategy being
followed in this work. We will therefore focus on the results for the 3000
fb-1 HL-LHC in the following. The $gg\to ZH$ contribution is found to be
negligible in all scenarios and is not considered further. The $t\bar{t}H$,
$tH$, $tWH$, $WH$ and $qq\to ZH$ event yields obtained after applying the
different selections described in the previous section are shown in Tab. 2 for
the SM and the pure ${\cal CP}$-odd Higgs–top-quark interaction scenarios (the
corresponding event yields for the ${\cal CP}$-mixed benchmark scenarios lead
to results in between).

Selection | 1-lepton | 1-lepton | 2-lepton
---|---|---|---
Process | preselection | selection | selection
SM $ttH$ | 300.4 | 7.2 | 29.3
SM $tH$ | 19.0 | 6.7 | $<0.1$
SM $tWH$ | 8.0 | 0.4 | 0.8
SM $WH$ | 3.6 | 0.4 | $<0.1$
SM $qqZH$ | 0.8 | $<0.1$ | 0.1
SM total | 331.8 | 14.7 | 30.2
${\cal CP}$-odd, $ttH$ | 140.2 | 3.3 | 13.9
${\cal CP}$-odd, $tH$ | 120.7 | 42.7 | 0.1
${\cal CP}$-odd, $tWH$ | 37.8 | 1.4 | 3.9
${\cal CP}$-odd total (incl. SM $WH$ \+ $qqZH$) | 303.1 | 47.8 | 18.0
Table 2: Expected number of events at the 3000 fb-1 HL-LHC passing the
different event selections defined in the text.

As expected the fraction of $tH$ ($t\bar{t}H$) events entering the 1-lepton
category is enhanced (depleted) in the pure ${\cal CP}$-odd scenario with
respect to the SM. $tWH$ production is not negligible at the HL-LHC in the
pure ${\cal CP}$-odd scenario, with about four $tWH$ events out of eighteen in
total expected in the 2-lepton category, even if it is still largely
subdominant with respect to $t\bar{t}H$. For convenience, we will refer in the
following to the merged $t\bar{t}H$ and $tWH$ contribution, labeled
$t\bar{t}H+tWH$. The $VH$ contribution is very small ($<5\%$) and independent
of the ${\cal CP}$ properties of the Higgs–top-quark interaction since the
$gg\to ZH$ contribution is negligible in all categories and scenarios.

In order to evaluate the sensitivity of the prospect analysis to measure $tH$
production, the total number of events in the 1-lepton and 2-lepton category
is included in a dedicated profile likelihood fit defined within the
HistFitter software[106]. Systematic uncertainties are neglected in this fit.
Compared to the low count rates, their effect is expected to be subdominant.
The $t\bar{t}H+tWH$ and $tH$ signal strengths, labeled as
$\mu_{t\bar{t}H+tWH}$ and $\mu_{tH}$, respectively, are free-floating in the
fit. The $VH$ signal strength, $\mu_{VH}$, is fixed to unity. Due to the very
limited size of the $VH$ contamination in both the 1-lepton and 2-lepton
category, $\mu_{VH}$ can be varied by large values without impacting the
results. The fit is repeated four times, considering each time an observed
dataset built from one of the four different Higgs ${\cal CP}$ hypotheses
introduced earlier and rounded to the closest integer number. The results are
summarised in Tab. 3. The selection efficiencies entering the fit results can
be considered as independent of the ${\cal CP}$ properties of the Higgs–top-
quark interaction assumed during the hypothesis testing within the statistical
uncertainty.

Observed data | $\mu_{tH}$ | $\mu_{t\bar{t}H+tWH}$ | $\mu_{tH/(t\bar{t}H+tWH)}$
---|---|---|---
SM | $1.06\pm 0.6$ | $0.99\pm 0.18$ | $1.07^{+0.82}_{-0.62}$
${\cal CP}$-odd | $6.41\pm 1.05$ | $0.60\pm 0.14$ | $10.78^{+3.96}_{-2.82}$
${\cal CP}$-mixed 1 | $2.29\pm 0.74$ | $0.96\pm 0.18$ | $2.39^{+1.14}_{-0.87}$
${\cal CP}$-mixed 2 | $3.56\pm 0.87$ | $1.03\pm 0.18$ | $3.47^{+1.33}_{-1.03}$
Table 3: Results of the prospect measurement with 3000 fb-1 at the HL-LHC
described in the text. The observed data sets used for the prospective
measurements are built from each of the four ${\cal CP}$ Higgs scenarios
defined earlier in the text. The selection efficiencies entering the fit
results can be considered as independent of the ${\cal CP}$ properties of the
Higgs–top-quark interaction within the statistical uncertainty. They deviate
insignificantly from unity in the SM scenario due to the rounding to the
closest integer number required by Poisson statistics for the observed data.

If the data is SM-like, an evidence for $tH$ production is obtained with about
$2\,\sigma$ significance (corresponding to a $p$-value for the null hypothesis
of no $tH$ contribution of about 0.025) while $t\bar{t}H$ is observed with
$5\,\sigma$ significance. This translates to an expected upper limit of
$\mu_{tH}<2.21$ at the 95% confidence level (CL) using the $CL_{s}$
prescription [107] if the data is SM-like including the $tH$ signal, and an
upper bound of $\mu_{tH}<1.16$ if there is no $tH$ signal. This result for
discriminating the $tH$ channel from the $t\bar{t}H+tWH$ channel is robust
with respect to the ${\cal CP}$ properties of the Higgs–top-quark interaction
assumed during the hypothesis testing owing to the careful selection of
observables used to define the 1-lepton and 2-lepton category discussed in the
previous section. Our projected result improves on the current strongest limit
[26] by about a factor of 5 ($\mu_{tH}<12$ at the 95% CL), which was set
recently by the ATLAS collaboration with the full Run 2 dataset of 139 fb-1
using $H\to\gamma\gamma$ decays, under the assumption of a pure ${\cal
CP}$-even coupling. The projected exclusion limit obtained in this work is
also stronger than the most optimistic projected limit on $tH$ production at
the 3000 fb-1 HL-LHC documented in Refs. [102, 103]. This illustrates the
potential of the proposed strategy to probe $tH$ production at the HL-LHC.

We recall that, unlike in the references quoted above, the results derived
from the prospect measurement presented in this work do not consider
systematic uncertainties. The ATLAS analysis reporting the current strongest
limit on $tH$ production [26] mentions that systematic uncertainties are
negligible. They are expected to still play a minor role at the HL-LHC [102,
103]. We report in addition in Tab. 3 the results for the measurement of the
$tH/(t\bar{t}H+tWH)$ signal strength, $\mu_{tH/(t\bar{t}H+tWH)}$, which is
obtained by reparametrizing the free-floating parameters in the fit to
$\mu_{tH/(t\bar{t}H+tWH)}$ and $\mu_{t\bar{t}H}$. The
$\mu_{tH/(t\bar{t}H+tWH)}$ measurement suffers from a larger statistical
uncertainty than the $tH$ measurement but is expected to be more robust with
respect to e.g. the uncertainty in the photon energy resolution and scale,
which should cancel out in the ratio to a large extent and was reported as the
leading experimental systematic uncertainties ($\sim 7\%$) in Ref. [105].
Current theory uncertainties are expected to be at the $\sim$ 5 to 10% level
for $tH$, $tWH$ and $t\bar{t}H$ production and therefore should be subdominant
with respect to the data statistical uncertainty quoted in Tab. 3. We found
that $\mu_{tH/\\{t\bar{t}H+tWH\\}}>2.57$ is expected to be excluded at the 95%
level if the data is SM-like including $\mu_{tH/(t\bar{t}H+tWH)}=1$, and
$\mu_{tH/(t\bar{t}H+tWH)}>1.14$ if there is no $tH$ signal.

When the observed dataset is built from other ${\cal CP}$ scenarios,
$\mu_{tH}=1$, which is the value expected in the SM, deviates from the best
fit $\mu_{tH}$ value by about $5.2\,\sigma$, $1.7\,\sigma$ and $2.9\,\sigma$
in the pure ${\cal CP}$-odd, the first and the second benchmark of a ${\cal
CP}$-mixed state scenario, respectively. It should be noted that the
hypothesis test between various ${\cal CP}$ scenarios depends on many other
features not explored in this analysis. It should be stressed that our
analysis is meant to be included in a global study of the ${\cal
CP}$-properties rather than being used alone.

Figure 19: Impact of the prospective $\mu_{tH/(t\bar{t}H+tWH)}$ determination
of the proposed $tH$ analysis with $3~{}\mathrm{ab}^{-1}$ of data [the green
areas indicate the $1\,\sigma$ precision] on the currently allowed ranges of
$c_{t}$ (_top panels_), $\tilde{c}_{t}$ (_middle panels_), as well as on the
($c_{t}$, $\tilde{c}_{t}$) parameter plane (_bottom panels_). The figures
contain the fit results of the 5D parametrization shown in Fig. 13 (_right
panels_) and Fig. 8 (_bottom right panel_), respectively. We assume the future
$\mu_{tH/(t\bar{t}H+tWH)}$ measurement to be consistent either with the SM
(_left panels_) or the ${\cal CP}$-mixed 2 benchmark scenario (_right
panels_).

An independent $tH$ measurement will in any case provide valuable information
for determining the nature of BSM physics if a deviation from the SM
predictions is observed. Here, we concentrate on illustrating the impact of a
prospective HL-LHC determination of $\mu_{tH/(t\bar{t}H+tWH)}$ on the
currently allowed ranges in the 5D model parametrization – ($c_{t}$,
$\tilde{c}_{t}$, $c_{V}$, $\kappa_{g}$, $\kappa_{\gamma}$) free – discussed in
Sec. 5. For this purpose, we overlay the anticipated $1\,\sigma$ range of
$\mu_{tH/(t\bar{t}H+tWH)}$ from our proposed analysis, Tab. 3, assuming either
a measurement consistent with the SM hypothesis or with the ${\cal CP}$-mixed
2 benchmark scenario, on the fit results obtained in Sec. 5. Figure 19
displays this comparison in dependence of $c_{t}$ (_top panels_),
$\tilde{c}_{t}$ (_middle panels_), as well as in the ($c_{t}$,$\tilde{c}_{t}$)
parameter plane (_bottom panels_), where the green band indicates the
prospective determination of $\mu_{tH/(t\bar{t}H+tWH)}$, assuming the SM
hypothesis (_left panels_) or the ${\cal CP}$-mixed 2 benchmark scenario
hypothesis (_right panels_). The fit results are the ones that have been shown
above in Fig. 13 (_right panels_) and Fig. 8 (_bottom right panel_),
respectively.

Assuming a SM-consistent measurement at the HL-LHC, the
$\mu_{tH/(t\bar{t}H+tWH)}$ determination will yield a lower limit on $c_{t}$
while the overall impact on $\tilde{c}_{t}$ will be rather weak. In contrast,
if $\mu_{tH/(t\bar{t}H+tWH)}$ is found to be consistent with the ${\cal
CP}$-mixed 2 benchmark scenario at the HL-LHC, a preference for $c_{t}$ values
below 1 and non-zero $\tilde{c}_{t}$ will be observed. From the bottom panels
in Fig. 19 one can see that a SM-consistent future measurement would further
constrain the allowed range to positive values of $c_{t}$, while a measurement
consistent with the ${\cal CP}$-mixed 2 benchmark scenario would result in a
preference for small or even negative $c_{t}$ values. In the latter case, the
combination of the $\mu_{tH/(t\bar{t}H+tWH)}$ measurement with the fit result
obtained above based on all other available inclusive and differential Higgs
rate measurements (which will be further improved with the accuracies at the
HL-LHC) would yield a preference for a parameter region with
$\tilde{c}_{t}\neq 0$, i.e. hinting towards ${\cal CP}$-violation in the top-
Yukawa sector.

##  7 Conclusions

Probing the top-Yukawa coupling is crucial to constrain the ${\cal CP}$ nature
of the Higgs boson discovered at the LHC. In this paper, we used all relevant
inclusive and differential Higgs boson rate measurements to derive bounds on a
possible ${\cal CP}$-odd component of the top-Yukawa coupling.

We performed this study in the “Higgs characterization model”, an EFT
framework which allows one to study ${\cal CP}$-violating Higgs couplings in a
consistent, systematic and accurate way. Within this model, we derived theory
predictions for all relevant Higgs production and decay channels. We evaluated
the experimental constraints in four different model parameterizations with
the parameter space spanning two, three, four or five dimensions. The 2D
parameter space, consisting of the ${\cal CP}$-even, $c_{t}$, and the ${\cal
CP}$-odd, $\tilde{c}_{t}$, top-Yukawa coupling, is successively enlarged by
allowing for new physics contributions to the couplings of the Higgs boson to
massive vector bosons, to photons, and to gluons. We found the best-fit points
in all four parameterizations to be close to the SM.

We could constrain $\tilde{c}_{t}$ in the 2D parametrization to the the
interval $[-0.3,0.3]$ at the $1\,\sigma$ level. In this parametrization, the
most constraining measurements arise from Higgs production via gluon fusion
and the Higgs decay to di-photons, which are sensitive to the top-Yukawa
coupling at the loop level. Allowing for additional freedom in the Higgs
coupling to massive vector bosons (i.e. not setting $c_{V}$ to unity) only
slightly weakens the constraints. Allowing also $\kappa_{\gamma}$ to vary
freely results in a $1\,\sigma$ interval of $[-0.4,0.4]$ for $\tilde{c}_{t}$.
In this parameterization, the strongest constraints originate from Higgs
production via gluon fusion. The constraints are significantly weakened if
also the Higgs coupling to gluons, $\kappa_{g}$, is allowed to vary freely. In
this model, $\tilde{c}_{t}$ can lie in the range $[-1.1,1.1]$ at the
$1\,\sigma$ level. Measurements of top-associated Higgs production, sensitive
to the top-Yukawa coupling at the tree-level, yield the strongest constraints
in case of a positive $c_{t}$. Moreover, Higgs production in association with
a $Z$ boson, using total rate and kinematic information, also constraints the
parameter space in particular for negative $c_{t}$. We encourage future
measurements of kinematic shapes (also for top-quark-associated production) to
further constrain the parameter space.

The constraints can also be interpreted in terms of a ${\cal CP}$-violating
phase which we find to be $\lesssim 22.5-27$ degrees for the first three
parametrizations and $\lesssim 72$ degrees for the 5D parameterization at the
$2\,\sigma$ level. Similar constraints have been derived in the latest studies
reported by ATLAS and CMS [25, 26]. Such results nevertheless cannot be
directly applied to other models than the model studied in the experimental
analyses, whereas the STXS and total rate measurements included in our global
fit are largely model-independent.

The existing constraints place only weak bounds on the possible values of the
$tH$ production cross section, so that there is large room for deviations from
the SM in this channel. The measurement of an enhanced rate of Higgs
production in association with a single top-quark with respect to the SM would
hint at a non-zero ${\cal CP}$-odd top-Yukawa coupling and a lower ${\cal
CP}$-even top-Yukawa coupling than predicted by the SM. At the current level
of experimental precision, it is, however, hard to disentangle Higgs
production in association with a single top-quark and with two top-quarks.
Therefore, we proposed a novel analysis strategy for disentangling $tH$ and
$t\bar{t}H+tWH$ production at the HL-LHC, where we compare the number of
events in 1- and 2-lepton categories designed such that the selection
efficiencies can be considered as independent of the ${\cal CP}$ properties of
the Higgs–top-quark interaction within the statistical uncertainty. Focusing
on the Higgs decay to photons, we obtained an expected upper limit of
$\mu_{tH}<2.21$ at the 95% CL if the data is SM-like including the $tH$
signal, and $\mu_{tH}<1.16$ if there is no $tH$ signal. These prospective
limits may be regarded as a conservative estimate since a dedicated effort
including additional decay channels will most likely increase the sensitivity.

If a deviation from the SM in top-associated Higgs production is observed in
the future, an independent $tH$ measurement will be important to unravel the
nature of this deviation. The prospective impact on the ${\cal CP}$ properties
of the Higgs–top-quark interaction depends on the measured value of the signal
strength. In case of a large deviation, the measurement of the $tH$ production
cross section will be a crucial ingredient for determining the ${\cal CP}$
properties of the $t\bar{t}H$ coupling. On the other hand, if the $tH$ cross
section turns out to be SM-like, according to the current projections there
will still be significant scope for non-standard ${\cal CP}$ properties of the
Higgs–top-quark interaction. Further experimental input and more precise
measurements will be needed in this case to narrow down these possibilities.

## Acknowledgments

We thank Lisa Biermann and Martin Mosny for collaboration in an early stage of
the project, Marco Zaro for clarifications regarding the “Higgs
charaterization model” as well as Robert Harlander, Lukas Simon and Jonas
Wittbrodt for useful discussion regarding vh@nnlo. H.B., J.K., K.P., T.S. and
G.W. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306. The work of S.H. is supported in part by the Spanish
Agencia Estatal de Investigación (AEI) and the EU Fondo Europeo de Desarrollo
Regional (FEDER) through the project FPA2016-78645-P, in part by the MEINCOP
Spain under contract FPA2016-78022-P, in part by the “Spanish Red Consolider
MultiDark” FPA2017-90566-REDC and in part by the AEI through the grant IFT
Centro de Excelencia Severo Ochoa SEV-2016-0597.

## Appendix

##  Appendix A CP sensitivity in $gg\rightarrow H+2j$

Figure 20: Fraction of $gg\rightarrow H+2j$ events falling into the
$\Delta\phi(j_{1},j_{2})$ bin. Red: $\kappa_{g}=1.0\pm 0.2$. White:
$\text{BR}_{H\rightarrow\gamma\gamma}/\text{BR}^{\text{SM}}_{H\rightarrow\gamma\gamma}=1.0\pm
0.2$. The SM is marked by an orange cross.

In [84], the $\Delta\phi(j_{1},j_{2})$ distribution in $gg\rightarrow H+2j$
has been split into four bins. Since the theoretical prediction for the
$\Delta\phi_{j_{1},j_{2}}$ distribution in the model considered is symmetric
around zero, we merge the four bins used in [84] into two bins:

  * •

bin 1: $\pi/2<|\Delta\phi(j_{1},j_{2})|\leq\pi$,

  * •

bin 2: $-\pi/2\leq\Delta\phi(j_{1},j_{2})\leq\pi/2$.

The corresponding fit formulas are given by

$\displaystyle\mu_{gg\rightarrow H+2j}^{\text{bin 1}}$
$\displaystyle=(c_{t}+c_{g})^{2}+0.001(c_{t}+c_{g})(\tilde{c}_{t}+\tilde{c}_{g})+2.17(\tilde{c}_{t}+\tilde{c}_{g})^{2},$
(30) $\displaystyle\mu_{gg\rightarrow H+2j}^{\text{bin 2}}$
$\displaystyle=(c_{t}+c_{g})^{2}+0.013(c_{t}+c_{g})(\tilde{c}_{t}+\tilde{c}_{g})+2.25(\tilde{c}_{t}+\tilde{c}_{g})^{2}.$
(31)

The numerical results are presented in Fig. 20. While the fraction of events
falling into the region $\pi/2<|\Delta\phi(j_{1},j_{2})|\leq\pi$ is maximized
for $\tilde{c}_{t}\approx 0$ (see Fig. 20, left), the fraction of events in
the complementary region of $-\pi/2\leq\Delta\phi(j_{1},j_{2})\leq\pi/2$
reaches its maximal value for $c_{t}\approx 0$. The overall variation is,
however, in the sub-percent level and therefore not resolvable within the
current experimental and theoretical precision.

##  Appendix B Experimental input

Tabs. 4 and 5 list the LHC Run-2 Higgs signal measurements from ATLAS and CMS,
respectively, which are implemented in HiggsSignals-2.5.0 and included in our
fit. In addition, we include the combined LHC Run-1 results from ATLAS and CMS
[98] in the fit via HiggsSignals.

Channel |  Luminosity [$\text{fb}^{-1}$] |  Signal strength $\mu$ | Reference
---|---|---|---
VBF, $H\to b\bar{b}$ | $30.6$ | $3.0\begin{subarray}{c}+1.7\\\ -1.8\end{subarray}$ | [79]
$t\bar{t}H$, $H\to b\bar{b}$ ($1\ell$) † | $36.1$ | $0.67\begin{subarray}{c}+0.61\\\ -0.69\end{subarray}$ | [80]
$t\bar{t}H$, $H\to b\bar{b}$ ($2\ell$) † | $36.1$ | $0.11\begin{subarray}{c}+1.36\\\ -1.41\end{subarray}$ | [80]
$t\bar{t}H$, multilepton ($2\ell ss$) † | $79.9$ | $0.38\begin{subarray}{c}+0.57\\\ -0.54\end{subarray}$ | [81]
$t\bar{t}H$, multilepton ($3\ell$) † | $79.9$ | $0.93\begin{subarray}{c}+0.58\\\ -0.52\end{subarray}$ | [81]
$t\bar{t}H$, multilepton ($4\ell$) † | $79.9$ | $0.52\begin{subarray}{c}+0.93\\\ -0.72\end{subarray}$ | [81]
$t\bar{t}H$, multilepton ($1\ell+2\tau_{h}$) † | $79.9$ | $0.30\begin{subarray}{c}+1.01\\\ -0.90\end{subarray}$ | [81]
$t\bar{t}H$, multilepton ($2\ell+1\tau_{h}$) † | $79.9$ | $0.49\begin{subarray}{c}+0.94\\\ -0.82\end{subarray}$ | [81]
$t\bar{t}H$, multilepton ($3\ell+1\tau_{h}$) † | $79.9$ | $0.43\begin{subarray}{c}+1.10\\\ -0.85\end{subarray}$ | [81]
|  |  $\sigma_{\text{obs}}$ [pb] |  $\sigma_{\text{SM}}$ [pb] |
$gg\to H$, $H\to W^{+}W^{-}$ | $36.1$ | $11.4\begin{subarray}{c}+2.2\\\ -2.1\end{subarray}$ | $10.4\pm 0.6$ | [82]
VBF, $H\to W^{+}W^{-}$ | $36.1$ | $0.50\begin{subarray}{c}+0.29\\\ -0.28\end{subarray}$ | $0.81\pm 0.02$ | [82]
VBF, $H\to ZZ$ ($p_{T,H}$ high) | $139.0$ | $0.0005\begin{subarray}{c}+0.0079\\\ -0.0048\end{subarray}$ | $0.00420\pm 0.00018$ | [83]
VBF, $H\to ZZ$ ($p_{T,H}$ low) | $139.0$ | $0.150\begin{subarray}{c}+0.064\\\ -0.052\end{subarray}$ | $0.1076\begin{subarray}{c}+0.034\\\ -0.0035\end{subarray}$ | [83]
$V(\text{had})H$, $H\to ZZ$ | $139.0$ | $0.021\pm 0.035$ | $0.0138\begin{subarray}{c}+0.0004\\\ -0.0006\end{subarray}$ | [83]
$V(\text{lep})H$, $H\to ZZ$ | $139.0$ | $0.22\begin{subarray}{c}+0.028\\\ -0.017\end{subarray}$ | $0.0165\begin{subarray}{c}+0.0010\\\ -0.0016\end{subarray}$ | [83]
$gg\to H$, $H\to ZZ$ ($p_{T,H}$ high) | $139.0$ | $0.038\begin{subarray}{c}+0.021\\\ -0.016\end{subarray}$ | $0.015\pm 0.004$ | [83]
$gg\to H$, $H\to ZZ$ ($0j,~{}p_{T,H}$ high) | $139.0$ | $0.63\pm 0.11$ | $0.55\pm 0.04$ | [83]
$gg\to H$, $H\to ZZ$ ($0j,~{}p_{T,H}$ low) | $139.0$ | $0.170\pm 0.055$ | $0.176\pm 0.025$ | [83]
$gg\to H$, $H\to ZZ$ ($1j,~{}p_{T,H}$ high) | $139.0$ | $0.009\begin{subarray}{c}+0.016\\\ -0.012\end{subarray}$ | $0.020\pm 0.004$ | [83]
$gg\to H$, $H\to ZZ$ ($1j,~{}p_{T,H}$ low) | $139.0$ | $0.05\pm 0.08$ | $0.172\pm 0.025$ | [83]
$gg\to H$, $H\to ZZ$ ($1j,~{}p_{T,H}$ med.) | $139.0$ | $0.170\pm 0.050$ | $0.119\pm 0.018$ | [83]
$gg\to H$, $H\to ZZ$ ($2j$) | $139.0$ | $0.040\pm 0.075$ | $0.127\pm 0.027$ | [83]
$t\bar{t}H$, $H\to ZZ$ † | $139.0$ | $0.025\begin{subarray}{c}+0.026\\\ -0.017\end{subarray}$ | $0.0154\begin{subarray}{c}+0.0010\\\ -0.0013\end{subarray}$ | [83]
$gg\to H$, $H\to\gamma\gamma$ ($0j$) | $139.0$ | $0.039\pm 0.006$ | $0.0382\begin{subarray}{c}+0.0019\\\ -0.0018\end{subarray}$ | [84]
$gg\to H$, $H\to\gamma\gamma$ ($1j$) | $139.0$ | $0.0162\begin{subarray}{c}+0.0031\\\ -0.0022\end{subarray}$ | $0.0194\begin{subarray}{c}+0.0018\\\ -0.0019\end{subarray}$ | [84]
$gg\to H$, $H\to\gamma\gamma$ ($2j$, $\Delta\Phi_{jj}\in[-\pi,-\tfrac{\pi}{2}]$) | $139.0$ | $0.0023\pm 0.0007$ | $0.0024\pm 0.0002$ | [84]
$gg\to H$, $H\to\gamma\gamma$ ($2j$, $\Delta\Phi_{jj}\in[-\tfrac{\pi}{2},0]$) | $139.0$ | $0.0011\pm 0.0004$ | $0.0020\pm 0.0002$ | [84]
$gg\to H$, $H\to\gamma\gamma$ ($2j$, $\Delta\Phi_{jj}\in[0,\tfrac{\pi}{2}]$) | $139.0$ | $0.0014\pm 0.0004$ | $0.0020\pm 0.0002$ | [84]
$gg\to H$, $H\to\gamma\gamma$ ($2j$, $\Delta\Phi_{jj}\in[\tfrac{\pi}{2},\pi]$) | $139.0$ | $0.0021\pm 0.0007$ | $0.0024\pm 0.0002$ | [84]
$t\bar{t}H$, $H\to\gamma\gamma$ † | $139.0$ | $1.59\begin{subarray}{c}+0.43\\\ -0.39\end{subarray}$ | $1.15\begin{subarray}{c}+0.09\\\ -0.12\end{subarray}$ | [84]
VBF, $H\to\tau^{+}\tau^{-}$ | $36.1$ | $0.28\begin{subarray}{c}+0.14\\\ -0.13\end{subarray}$ | $0.237\pm 0.006$ | [85]
$gg\to H$, $H\to\tau^{+}\tau^{-}$ | $36.1$ | $3.10\begin{subarray}{c}+1.90\\\ -1.60\end{subarray}$ | $3.05\pm 0.13$ | [85]
$WH$, $H\to W^{+}W^{-}$ | $36.1$ | $0.67\begin{subarray}{c}+0.36\\\ -0.30\end{subarray}$ | $0.293\pm 0.007$ | [86]
$ZH$, $H\to W^{+}W^{-}$ | $36.1$ | $0.54\begin{subarray}{c}+0.34\\\ -0.25\end{subarray}$ | $0.189\pm 0.007$ | [86]
$WH$, $H\to b\bar{b}$ ($p_{T,V}\in[150,250]~{}\mathrm{GeV}$) | $139.0$ | $0.019\pm 0.0121$ | $0.0240\pm 0.0011$ | [69]
$WH$, $H\to b\bar{b}$ ($p_{T,V}\geq 250~{}\mathrm{GeV}$) | $139.0$ | $0.0072\pm 0.0022$ | $0.0071\pm 0.0030$ | [69]
$ZH$, $H\to b\bar{b}$ ($p_{T,V}\in[75,150]~{}\mathrm{GeV}$) | $139.0$ | $0.0425\pm 0.0359$ | $0.0506\pm 0.0041$ | [69]
$ZH$, $H\to b\bar{b}$ ($p_{T,V}\in[150,250]~{}\mathrm{GeV}$) | $139.0$ | $0.0205\pm 0.0062$ | $0.0188\pm 0.0024$ | [69]
$ZH$, $H\to b\bar{b}$ ($p_{T,V}\geq 250~{}\mathrm{GeV}$) | $139.0$ | $0.054\pm 0.0017$ | $0.0049\pm 0.0005$ | [69]
Table 4: ATLAS Higgs rate measurements from LHC Run-2 included in the fit via HiggsSignals. Measurements from dedicated $t\bar{t}H+tH$ analyses are marked with † and are not included in a specific fit that we performed for comparison (see Sec. 5.4). Channel |  Luminosity [$\text{fb}^{-1}$] |  Signal strength $\mu$ | Reference
---|---|---|---
$pp\to H$, $H\to\mu^{+}\mu^{-}$ | $35.9$ | $1.0\begin{subarray}{c}+1.1\\\ -1.1\end{subarray}$ | [87]
$WH$, $H\to b\bar{b}$ | $35.9$ | $1.7\begin{subarray}{c}+0.7\\\ -0.7\end{subarray}$ | [88]
$ZH$, $H\to b\bar{b}$ | $35.9$ | $0.9\begin{subarray}{c}+0.5\\\ -0.5\end{subarray}$ | [88]
$pp\to H$ (boosted), $H\to b\bar{b}$ | $35.9$ | $2.3\begin{subarray}{c}+1.8\\\ -1.6\end{subarray}$ | [89]
$t\bar{t}H$, $H\to b\bar{b}$ ($1\ell$) † | $35.9\oplus 41.5$ | $0.84\begin{subarray}{c}+0.52\\\ -0.50\end{subarray}\oplus 1.84\begin{subarray}{c}+0.62\\\ -0.56\end{subarray}$ | [91, 90]
$t\bar{t}H$, $H\to b\bar{b}$ ($2\ell$) † | $35.9\oplus 41.5$ | $-0.24\begin{subarray}{c}+1.21\\\ -1.12\end{subarray}\oplus 1.62\begin{subarray}{c}+0.90\\\ -0.85\end{subarray}$ | [91, 90]
$t\bar{t}H$, $H\to b\bar{b}$ (hadr.) † | $41.5$ | $-1.69\pm 1.43$ | [90]
$t\bar{t}H$, multilepton ($1\ell+2\tau_{h}$) † | $35.9\oplus 41.5$ | $-1.52\begin{subarray}{c}+1.76\\\ -1.72\end{subarray}\oplus 1.4\begin{subarray}{c}+1.24\\\ -1.14\end{subarray}$ | [92, 93]
$t\bar{t}H$, multilepton ($2\ell ss+1\tau_{h}$) † | $35.9\oplus 41.5$ | $0.94\begin{subarray}{c}+0.80\\\ -0.67\end{subarray}\oplus 1.13\begin{subarray}{c}+1.03\\\ -1.11\end{subarray}$ | [92, 93]
$t\bar{t}H$, multilepton ($2\ell ss$) † | $35.9\oplus 41.5$ | $1.61\begin{subarray}{c}+0.58\\\ -0.51\end{subarray}\oplus 0.87\begin{subarray}{c}+0.62\\\ -0.55\end{subarray}$ | [92, 93]
$t\bar{t}H$, multilepton ($3\ell+1\tau_{h}$) † | $35.9\oplus 41.5$ | $1.34\begin{subarray}{c}+1.42\\\ -1.07\end{subarray}\oplus-0.96\begin{subarray}{c}+1.96\\\ -1.33\end{subarray}$ | [92, 93]