CERN-TH-2023-158, EFI-23-8

Classifying the CP properties of the ggH couplingin H + 2j production
Henning Bahl∗1 , Elina Fuchs†2,3,4 , Marc Hannig‡3 , and Marco Menen§3,41

University of Chicago, Department of Physics, 5720 South Ellis Avenue, Chicago, IL 60637 USA
2
CERN, Department of Theoretical Physics, 1211 Geneve 23, Switzerland
3
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
4
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

July 18, 2024

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

∗hbahl@uchicago.edu
elina.fuchs@cern.ch‡
marc.hannig@stud.uni-hannover.de§marco.menen@itp.uni-hannover.de†1

Contents
1 Introduction

3

2 Gluon fusion with two jets in an EFT approach
2.1 Gluon fusion in association with two jets . . . . . . . . . . . . . . . . . . . .
2.2 Effective Higgs-vector boson interactions . . . . . . . . . . . . . . . . . . . .

55
6

3 Event generation

7

4 Analysis strategy
4.1 Signal-background separation . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Separating the different ggF2j contributions . . . . . . . . . . . . . . . . . .

10
11
12

5 Sensitivity to the Higgs–gluon coupling
5.1 ggF2j signal region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 VBF signal region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Highest-impact observables . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
17
17
20

6 Disentangling CP violation in the ggH and HV V couplings

23

7 Limits on the CP structure of the top-Yukawa coupling24

8 Conclusions

25

A Details of the event generation

27

B Distributions of kinematic variables

28

C Training uncertainty

30

D Likelihood evaluation

31

2

1

IntroductionMore than ten years ago, the ATLAS and CMS collaborations announced the discovery of anew particle at a mass of about 125 GeV during Run-1 of the Large Hadron Collider (LHC)[1, 2]. Since its discovery in 2012, the quantum numbers of this Higgs boson have been tested
extensively, as well as its interactions with other SM particles [3–6]. Up to now, all results
are in agreement with the predictions of the SM within the experimental and theoreticaluncertainties. However, effects from beyond the Standard Model (BSM) physics could still
be hidden in the current uncertainties and may be unveiled with the large amount of data
to be collected during LHC Run-3 and the high-luminosity phase of the LHC (HL-LHC).
A so far relatively unconstrained property of the discovered Higgs boson is its behaviour
under CP transformations. This is of particular interest given that the amount of CP violation present in the SM is — by several orders of magnitude — insufficient to explain thebaryon asymmetry of the Universe [7, 8]. The possibility of the Higgs boson being a pure
CP-odd state was already ruled out shortly after its discovery by analyzing the CP structureof its couplings to massive vector bosons [3, 4]. It is, nevertheless, still possible that thestructure of the discovered Higgs boson’s couplings is CP-mixed instead of (nearly) CP-even
as predicted by the SM.
Electric dipole moments (EDMs) are sensitive probes of CP violation beyond the SM [9,10] and their experimental upper bounds, in particular of the electron [11, 12], neutron [13]and mercury [14], place strong constraints on CP-violating Higgs interactions [15–19]. These
constraints, however, strongly depend on the assumption about the first-generation Yukawa
couplings [20], which themselves are only very weakly constrained [21–30]. It is therefore ofgreat interest to search for possible CP-violating effects also at colliders, which allow for a
distinction between different Higgs couplings.
At the LHC, CP-violating Higgs interactions can be constrained using the kinematic
information of the final state particles. Most of the observables built from these kinematics
are CP-even, but can show sensitivity towards CP violating effects via changes in theirdistributions. However, an observed deviation in a CP-even observable is only indicativeof CP violation. Dedicated CP-odd observables offer an advantage as measuring a nonzero value for them is an unambiguous sign of CP violation. On the other hand, it canbe difficult to measure CP-odd observables since it often requires measuring at least four
independent momenta associated either with the Higgs production or with its decay (seee.g. Ref. [31]). In this situation, the measurement of CP-sensitive observables can provide
valuable complementary information.
The most stringent constraints on CP-violating Higgs couplings so far have been set on
the Higgs interactions with massive vector bosons [3, 4, 32–41]. In most BSM theories, CPviolation in these interactions is expected to be loop-suppressed (given that a pseudoscalar
cannot be coupled to massive vector bosons at the tree level). CP violation in the Higgs interactions with fermions can, in contrast, occur unsuppressed. These are far less constrainedwith the existing experimental analyses targeting the Higgs interaction with tau leptons [42,343] and with top quarks [39, 44–49].
Since the top-Yukawa coupling is of special interest due to its magnitude, also many
phenomenological studies have been carried out focusing mainly on top-associated Higgs
production as a tree-level probe of the Higgs–top-quark interaction [18, 20, 50–83]. Besides
that, the CP character of the top-Yukawa coupling can also be probed by investigationof the Higgs production via gluon fusion (ggF). While ggF production alone (without theassociation of jets) is only sensitive to the CP nature of the top-Yukawa interaction via its
total rate, ggF production in association with one1 or two jets is more directly sensitive.
In particular, Higgs production in association with two jets (ggF2j) is well known to be
CP-sensitive via the difference of the azimuthal angles of the two jets that enables theconstruction of a CP-odd observable [56, 83, 85–97].
Generally, the ggF2j production does not directly probe the CP structure of the topYukawa interaction but of the Higgs–gluon interaction. Conversely, a CP-violating Higgs–
gluon interaction cannot only be induced by a CP-violating Higgs–top-quark interaction but
also by CP-violating Higgs interactions with coloured BSM states. Yet, given the increasingly
strong limits on the mass scale of coloured BSM states set by experimental searches atthe LHC [98–103], any sign for CP violation in the Higgs–gluon interaction is plausible tooriginate at least partially from the Higgs–top-quark interaction. This makes the ggF2jchannel not only interesting in its own respect, but also in the context of constraining the
CP structure of the effective Higgs-gluon interaction and the top-Yukawa coupling.
Existing collider measurements so far only put relatively weak constraints on the CP
character of the Higgs–gluon interaction [38, 39, 46] (while EDM limits assuming the absenceof CP violation in all other Higgs couplings give competitive results [104]). In this work,
we investigate how these limits could be improved. In particular, we compare the potentialof two distinct kinematic regions (a VBF-like and a ggF2j-like kinematic region) and use
well-established machine-learning techniques (boosted classifiers) to identify whether a given
event originates from a CP-even interaction, a CP-odd interaction, or from their interference.
This allows us to construct both CP-even and CP-odd observables. In the ggF2j-like region,these show a significantly better sensitivity than the difference of the azimuthal angles of thetwo jets, which is widely used in the literature. Moreover, we highlight how the separation
into two kinematic regions helps to distinguish CP violation in the Higgs–gluon interaction
from CP violation appearing in the Higgs interaction with massive vector bosons.
The paper is structured as follows. In Section 2, we review Higgs production in association with two jets and introduce the effective theory that parameterizes our BSM couplings.
Afterwards, we discuss the event generation and applied cuts in Section 3. Section 4 deals
with the training setup of our classifiers. In this section, we also define the observables thatare used to probe the CP-structure of the Higgs-gluon coupling. We present expected exclusion limits in Section 5. The interplay of the CP structure of the Higgs–gluon interaction
with the CP structure of the Higgs couplings with massive vector bosons is discussed in Sec1

For ggF production in association with one jet, jet substructure information has to be exploited toconstruct a CP-odd observable [84].

4

(a) gg-initiated(b) gq-initiated(c) qq-initiatedFigure 1: Examplary Feynman diagrams for the various ggF2j sub-channels.

(b) V H(a) VBF

Figure 2: Examplary Feynman diagrams for the considered background processes whereV ∈ [W + , W − , Z] and q ∈ [q, q̄].

tion 6. The interpretation of our limits in terms of limit on the Higgs–top-quark interaction
and a comparison to current experimental limits is shown in Section 7. Finally, Section 8
concludes our findings and gives a brief outlook into possible future studies.

2

Gluon fusion with two jets in an EFT approach

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

2.1

Gluon fusion in association with two jets

Higgs production via gluon fusion is the main production channel at the LHC with its crosssection being (at minimum) an order of magnitude higher than any other Higgs production√
mode at s = 13 TeV. Even if two or more jets associated with the parton level interaction
are required in the final state, the cross section is still higher than all other Higgs production√j≥2
modes [105] (σggF= 7.88 pb and σVBF = 3.78 pb at s = 13 TeV). In the SM, the gluon
fusion process is induced mainly by a top-quark loop, while for our work we consider an
effective point-like interaction with unknown CP state (see Section 2.2).
The ggF2j process can be classified by its initial state. Examplary diagrams for thethree initial states can be seen in Fig. 1 where effective interactions are represented as a
black dot. The individual initial states with gluons have higher cross sections due to the5larger contribution of the gluons compared to those of the quarks to the parton distributionfunctions (PDFs) of the proton. In our setup, we find for the SM case that the gg initial state
contributes ∼ 72% to the total ggF2j cross section, while the gq initial state still contributes∼ 26.5%. The contribution of the qq initial state is very small (∼ 1.5%). We furthermore
note that the interference between the qq-induced ggF2j production and Higgs production
via VBF was found to be negligible [106, 107].
Example Feynman diagrams for the considered background (BG) processes — VBF andV H production — are shown in Fig. 2.

2.2

Effective Higgs-vector boson interactions

We perform our study in an effective field theory (EFT) framework. Following Ref. [108],
we parameterize the interaction of the Higgs boson with gluons in the following form,


αs
1
αs
a
µν,a
a
µν,a
−
cg Gµν G+
c̃g Gµν G̃
H,(1)
LggH = −4v
3π2π
where H is a Higgs field, Gaµν is the gluon field strength tensor, v = 246 GeV is the vacuum expectation value and αs = gs2 /4π is the strong coupling constant. Here, we assume
that this interaction already includes the Higgs–gluon vertex induced by the top quark in
the infinite top-mass limit.2 Accordingly, cg = 1 and c̃g = 0 in the SM. A mixing angle
αHgg = tan−1 (c̃g /cg ) can be defined to parameterize the CP state of the Higgs–gluon coupling. Deviations from the SM can either be induced by a modified top-Yukawa coupling orby coloured BSM particles.
The prefactors of the operators are chosen such that modifications of the top-Yukawa
interaction directly map to the operators of Eq. (1) if we assume that no coloured BSM
particles affect the Higgs–gluon interaction. In this case, we have cg = ct and c̃g = c̃t in theinfinite top-quark mass limit, if we parameterize the top-Yukawa interaction via
y SM
Ltop-Yuk = − √t t̄(ct + iγ5 c̃t )tH,2(2)

where ytSM is the SM top-Yukawa interaction and ct = 1 as well as c̃t = 0 for the SM.
We account for effects due to the finite mass of the top quark by rescaling the total rate
and placing an upper bound on the Higgs transverse momentum (see Section 3).
The modified Higgs-gluon coupling of Eq. (1) affects ggF2j production. Its matrix element
can be separated into three pieces


2
2
2
∗
2
(3)
MggF2j = cg Meven + 2cg c̃g Re Meven Modd + c̃2g Modd .
2In other works (see e.g. Refs. [39, 46], the Higgs–gluon interaction is written in the form LggH =cgg Gaµν Gµν,a + c̃gg Gaµν G̃µν,a H without including the effect of the top-quark loop. In the infinite− αvs πtop-quark mass limit, both parameterizations are related via cg = 1 + 12π 2 cgg and c̃g = −8π 2 c̃gg .

6

The first and third terms proportional to the squared values of the coupling modifiers are
CP-even, while the second term parameterizing the interference between the two CP-states
is CP-odd. CP violation in the Higgs-gluon interaction is therefore only realised for a nonzero value of the interference term. Correspondingly, the interference term gives a non-zerocontribution only for distributions of CP-odd observables.
In addition to a modification of the Higgs–gluon interaction, in Section 6 we will also
consider a CP-violating interaction of the Higgs boson with massive vector bosons. We
parameterize this interaction as3
LΦV V = Lgauge +

cΦW
f
† f
µν
O f with OΦW
f ≡ Φ ΦWµν W ,
Λ2 ΦW

(4)

f are the SU (2)L field strength and its
where Lgauge is the SM gauge Lagrangian. W and W
dual, respectively. Φ is the Higgs doublet and Λ denotes the cut-off scale of the EFT, which
we set to 1 TeV. If not stated otherwise, we will for the main part of this work assume thatcΦW̃ = 0 such that the Higgs coupling to massive vector bosons is SM-like.

3

Event generation

As discussed in Section 1, we focus on Higgs production via gluon fusion in association
with two jets, which we refer to as ggF2j in the following. This production channel offersa unique sensitivity to probe the CP nature of the Higgs–gluon interaction, since the two
additional jets allow the construction of CP-odd observables. Moreover, in this work, weconcentrate on the Higgs decay to two photons. While this simplifies our analysis, ourgeneral analysis strategy can be straightforwardly adapted to other Higgs decay channels.
As background processes, we consider Higgs production via vector-boson fusion (VBF) and
Higgsstrahlung (V H). We leave a more in-depth study including the continuous backgroundfrom the qq → qq process with two photons from final state radiation for future work.
We generate parton-level events at leading order and scale them to NLO by using flat
K-factors from Ref. [56] with MadGraph5_aMC@NLO (version 3.4.0) [111]. The events arethen passed on to Pythia8 (version 8.306) [112] and Delphes3 (version 3.4.2) [113] forparton showering, hadronisation and detector simulation. For the detector simulation with
Delphes3, we employ the ATLAS detector card with a modified jet cone radius of ∆R = 0.4
which is more widely used in experimental analyses. A jet is then defined as a set of hadronic
particles in this cone which in total reach a minimum of pjT ≥ 20 GeV. Reconstruction of thejet is performed using the anti-kt algorithm [114]. Photons (which are used for reconstructing
the Higgs boson) are required to have a transverse momentum of pT ≥ 0.5 GeV.
Data sets are generated for Higgs production from the ggF2j, VBF and V H processes.
While the latter two do not receive any BSM contribution in our model, the event generation
3

At dimension six in the SM effective field theory (SMEFT), three distinct operators can introduce CP
violation in VBF production (see e.g. Ref. [109]). In this work, we concentrate on the OΦW
f which has thelargest impact on VBF production [110]. We expect similar results for the other operators.

7

Fraction of accepted events
Applied cut

ggF2j|Meven |2

ggF2jInterf.

ggF2j|Modd |2

VBF

VH

Initial events

100%

100%

100%

100%

100%

Nj ≥ 2; Nγ ≥ 2

48.1%

50.8 %

48.1%

62.6%

49.8%

110 GeV ≤ mγγ

47.8%

50.5%

47.9%

62.0%

49.4%

39.4%

40.9%

39.8%

50.0%

40.5%

38.6%

40.2%

38.6%

49.7%

39.9%

22.9%

21.5%

22.7%

39.8%

31.2%

18.6%

18.4%

18.3%

34.4%

26.8%

mγγ ≤ 140 GeV
pγ1
T /mγγ ≥ 0.35pγ2
T /mγγ ≥ 0.25
pj1
T ≥ 30 GeV
pj2
T ≥ 20 GeV|ηj | ≤ 2.5|ηγ | ≤ 2.5
pHT ≤ 200 GeV

Table 1: Cutflow table for the Higgs production mechanisms considered in this work.
Listed are all cuts applied after the event generation, along with the percentage of events
that survives the cut.

8

of ggF2j production is split into three terms following the parameterization in Eq. (3). We
therefore obtain separate data sets for the terms proportional to c2g (the CP-even amplitude
squared), c̃2g (the CP-odd amplitude squared), and cg c̃g (the interference between the CPeven and CP-odd amplitude). This amounts to five distinct classes which we further split
into independent data sets for training, validating and testing the classifiers, which will be
introduced in Section 4. Further details on the event generation and the employed UFO
model [115, 116] can be found in Appendix A.
Before training the classifiers, we impose a number of simple baseline cuts mimicking
the event selection typically used by ATLAS and CMS for H → γγ measurements [44, 45].
A cutflow table detailing the imposed cuts and the corresponding reduction in the event
numbers can be found in Table 1, where the percentage of acceptance after each cut isdisplayed for all signal and background processes examined in this work.
First of all, a cut on Nj and Nγ ensures that at least two jets and two photons arefound in each event, which are needed to reconstruct the Higgs boson and identify ggF2j
production. Only accepting events with 110 GeV ≤ mγγ ≤ 140 GeV defines a region in whichmost Higgs events fall. The cuts on low values of pγT and pjT suppress misidentified photons
and jets that do not originate from the parton level interaction. Finally, only events with
|ηj | ≤ 2.5 and |ηγ | ≤ 2.5 are accepted to match the pseudorapidity coverage of the ATLAS
inner detector. As an additional requirement, we place an upper limit of pH
T ≤ 200 GeV on
the Higgs transverse momentum (reconstructed out of the two photon momenta). This cut
ensures that the top-quark loop inducing ggF2j production cannot be resolved and ggF2j
production can be reliably interpreted using the effective Lagrangian given in Eq. (1) (seeRefs. [117, 118] for early calculations and e.g. the discussion in Ref. [119]).
As the cutflow in Table 1 shows, the cut on pH
T reduces the number of surviving events
by 14 − 19%. While this reduction is not a critical limitation for our analysis, in principle
one can include the events with higher pH
T beyond the infinite top-quark mass limit byemploying the FTapprox approximation [120]. This approximation combines the exact topquark mass and width in the Born, one-loop and real amplitudes with approximated andre-scaled two-loop virtual contributions, originally applied to multi-Higgs production. The
FTapprox approximation has been successfully validated against the full NLO calculation in
the SM for H + j production [121]. Under the assumption that the FTapprox approximation
yields similarly accurate results for Higgs production in association with a higher multiplicity
of jets, Ref. [121] has also improved the calculation of differential cross sections for H + 2j bycomplementing the exact real corrections with the approximated two-loop virtual corrections.
The good agreement between their NLO result in the heavy-top limit (HTL) and in the
FTapprox up to pH
T ≤ 300 GeV justifies our treatment in the HTL within the chosen cut onH
pT . The relevance of the finite top-quark mass at higher pH
T and the success of FTapprox in
H + j motivate a future study of CP properties of the Higgs boson without cutting out theHhigh-pHT events. For the scope of our work, our robust treatment with the pT -cut results inconservative limits that might be improved by a refined event generation.

9X = ggF2j, VBF
ℳeven

VBF

2

Classifier
P ℳeven 2
ggF2jClassifier Xproduction

Limits fromthe X-SR

Interf.
Classifier
P Interf.

VH

ℳodd

2

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

4

Analysis strategy

Our analysis is structured into two steps. In the first step, two distinct classifiers are trained
to separate the events originating from the different Higgs production channels. One classifieris trained to separate the signal events (from ggF2j production) from the considered Higgs
background processes, which is dubbed the ggF2j signal region (SR). A second classifier isused to define a signal region which is enriched with q q̄-induced ggF2j events. Although
the ggF2j process is our signal process throughout the entire analysis, these events share
the topology of VBF Higgs production, and we, therefore, train the classifier to identify
VBF events. This is used to define what we call the ”VBF-SR”. Both SRs offer possible
advantages in the analysis. In the ggF2j-SR, we expect an increased sensitivity as it contains
a significant number of signal events. The VBF-SR is enriched with q q̄ initial state events
which were shown to carry the most CP information after applying typical VBF-cuts in an
earlier work Ref. [91]. Events from the continuous background are expected to be enhancedin the ggF2j-SR compared to the VBF-SR, although we do not consider this background in
this study.
Both classifiers are trained on the same input data. Since the two classifiers are independent, it is possible for events to appear in both or none of the kinematic regions. For ourdata set, the percentage of ggF2j data that appear in both the ggF2j-SR and the VBF-SR
is about 8% of the total data set, while about 11% of the events do not appear in eitherof them. Subsequently, the events identified in each category are passed to two additionalclassifiers (see description below). These are used to distinguish the squared CP-odd andCP-even matrix elements and the interference term in the ggF2j production (see Eq. (3)) and10

thereby to construct CP-sensitive observables. The analysis steps for each of the kinematic
regions are summarized in Fig. 3.

4.1

Signal-background separation

As mentioned above, two classifiers are trained to define a ggF2j- and a VBF-SR with distinct
kinematics. In both cases, we reduce the separation to a binary classification problem, where
the respective signal is trained against all other types of events. The classifiers are set up
with PyTorch and the same data set is used for training, validation, and testing of bothclassifiers.
The following kinematic variables are used as input for the signal-background separation:• the energy E, the transverse momentum pT , the pseudorapidity η, and the azimuthal
angle ϕ of the Higgs boson and the two leading-pT jets,• the invariant mass of the two leading-pT jets mjj , their pseudorapidity difference ∆ηjj ,and their azimuthal angle difference ∆ϕjj (where the sign is chosen such that the ϕ of
the jet with the smaller η is subtracted from the one with larger η),• the number of jets in the event Nj , and• the respective energy E of each jet that is not leading or sub-leading in pT .
Among the most important variables for the signal/background separation are Nj , mjj ,|∆ηjj |, and ∆ϕjj (see e.g. Ref. [56]).4 It can be seen that the three processes (ggF2j, VBF
and V H) show quite distinct topologies which are important for training the NN. While theusage of higher-level observables (like |∆ηjj | and mjj ) does not provide the classifiers withnew information, they can still be useful in order to improve the training process. Suchobservables should, however, only be included if they are expected or known to impact theoutcome of the classifier, as they can slow down the training otherwise. In the present case,|∆ηjj | and mjj are well-known examples for variables allowing one to distinguish the events
stemming from VBF and ggF2j production.
The classifier creating the ggF2j-SR (VBF-SR) reaches an accuracy of about 70% (79%).
The classifier score, which gives an estimate of the probability for each event to be a signal
event, is calculated for ggF2j, VBF and V H Higgs production, respectively, and plotted inFig. 4(a) (Fig. 4(b)). The ggF2j production is additionally split up into the three possible
initial states (qq, gq and gg). We observe that the more quarks are in the initial state, themore likely it is for an event to be classified as VBF. Especially for the qq initial state, thiscan be easily understood by comparing the example Feynman diagrams in Fig. 1(c) and
Fig. 2(a), which only differ via the exchange of the gluon/heavy vector boson propagators
creating the Higgs. We additionally observe that interference events from ggF2j are more
likely to be identified as VBF-like events than events from the squared terms are.
4

For more advanced approaches using e.g. jet energy profiles or jet charge see Refs. [122, 123].

11

ggF2j - gg
ggF2j - gq
ggF2j - qq
VBF
VH

0.10

ggF2j - gg
ggF2j - gq
ggF2j - qq
VBF
VH

0.30
Occurrence (normalized)

Occurrence (normalized)

0.12

0.08
0.06
0.04

0.25
0.20
0.15
0.10
0.05

0.02
0.00
0.0

0.2

0.4
0.6
P (ggF2j)

0.8

0.00
0.0

1.0

0.2

0.4

0.6

0.8

1.0

P (VBF)(a)(b)

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

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

4.2

Separating the different ggF2j contributions

The three terms in the squared ggF2j amplitude in Eq. (3) can be probed by exploiting
rate information as well as their different kinematics. The interference term can only beprobed by CP-odd observables since its positive and negative contributions cancel out for
CP-even observables. One such observable is ∆ϕjj , which has been used as a CP observablein previous analyses (see e.g. Ref. [38]). Fig. 5 shows the contributions of the squared
and interference terms to the differential cross section, plotted against ∆ϕjj . Since ∆ϕjj isa CP-odd observable, the distribution for the interference term is antisymmetric, while the
distribution of the squared CP-even and CP-odd matrix elements are symmetric. We impose
a CP flip on the events5 before the training to ensure the symmetry of the distributions evenfor a limited number of Monte-Carlo events. The two peaks close to ∆ϕjj = 0 in the ggF2j-SR
originate from events with low mjj and vanish if a minimal mjj cut is imposed.
For the training of the CP classifiers, we use the full kinematic information of the twoleading jets and the reconstructed Higgs boson, as well as mjj , ∆ηjj , and ∆ϕjj as additional
high-level observables. The distributions of all variables used in the training can be found in
Appendix B where they are split up into the three possible contributions to the ggF2j crosssection. Compared to the signal-background separation, we dropped the information about
5This corresponds to flipping the sign of the particle momenta as well as in case of the interference term
the weight on an event-by-event basis.

12

1.0

ggF2j-SR|Meven |2Interf.

0.8|Meven |2Interf.

0.8
dσ/d∆φjj [fb/rad]|Modd |2dσ/d∆φjj [fb/rad]VBF-SR

0.6
0.4
0.2

|Modd |2

0.6
0.4
0.2
0.0

0.0
−3

−2

−1

0
∆φjj

1

2

−3

3

(a)

−2

−1

0
∆φjj

1

2

3

(b)

Figure 5: The differential cross section as a function of ∆ϕjj in the (a) ggF2j- and(b) VBF-SR, plotted for the |Meven |2 (blue), interference (orange) and |Modd |2 (green)contributions.

additional jets, since these were found to carry little to no information about the CP characterof the Higgs–gluon interaction. For the CP classification, we used the Gradient Boosting
Classifier from the scikit-learn [124] package. We train two classifiers independently
for each of the signal regions defined by the signal-background classifiers. Two observables
are constructed from their output in the following way:
• The first classifier only separates between the squared CP-even and the squared CPodd terms. The corresponding output observable is defined as P (c2g ), which can beinterpreted as a probability that a given event originates from the |Meven |2 contribution.
• The second classifier deals with the interference term. It is trained to separate positive
and negative interference events as well as to distinguish these from events originatingfrom either of the squared terms. Following Ref. [125], the observable built from theoutput is defined as P+ −P− where P+ (P− ) is the probability of an event to correspondto positive (negative) interference. This observable is CP-odd by definition.
A similar approach has been followed in Refs. [126, 127], which also connect it to the matrix
element approach.
Fig. 6 shows the output of the two CP classifiers in the ggF2j-SR. Each classifier has been
trained 100 times (see Appendix C) and here we show the one with the strongest constraints
on the Higgs-gluon coupling modifiers (see Section 5.1). For the P (c2g ) classifier (see left
panel of Fig. 6), this corresponds to the case in which the difference between the squared
terms was learned best, while the contribution from the interference term is zero. When
applying the same normalization to both of the squared terms, the distribution from the c̃2g

13

80

L = 139 fb−1

ggF2j-SR120|Modd |2
VBF + V H

100
Nevents

Nevents

60|Meven |2Interf.

40

L = 139 fb−1

ggF2j-SR|Meven |2Interf.
|Modd |2
VBF + V H

80
60
40

20

20
0
0.0

0.2

0.4

0.6

0.8

0
−1.0

1.0

P (c2g )

(a)

−0.5

0.0
P+ − P−

0.5

1.0

(b)

Figure 6: Distributions of the two CP discriminants (see text for the definition) for theclassifiers yielding the strongest limits on the coupling modifiers in the ggF2j-SR.

term shows to be higher towards low values of P (c2g ) and lower around P (c2g ) = 0.5. The
difference between the classifier with the strongest constraints and other trained classifiers
is mainly in the shape of the left outermost bin.
The P+ − P− classifier (see right panel of Fig. 6) learning the interference terms showed
only very slight fluctuations during the training processes. Since its output is a CP-odd
observable by construction, the squared ggF2j contributions, as well as BG contributions,show the expected behaviour of being symmetric around zero. The interference contribution
is asymmetric around zero. Its amplitude compared to the squared contributions is verysmall due to its small cross section, an overall lower number of interference events in the
ggF2j-SR from the signal-background separation, and misidentified events during the CP
classification.
The output of the CP classifiers in the VBF-SR is plotted in Fig. 7. Again, we show the
distributions of the classifiers yielding the strongest limits (see Section 5.2) after 100 training
iterations. The shapes of the two squared distributions show similar differences as in the
ggF2j-SR with the distribution stemming from the c̃2g contribution being higher towards
low values of P (c2g ) and lower around and above P (c2g ) = 0.5. Here, the visible statisticalfluctuations in the |Meven |2 and |Modd |2 contributions arise due to the comparably low
number of ggF2j events in this signal region.
In contrast, the asymmetry of the interference term in the P+ − P− observable is much
more visible. Even in the VBF-SR, the number of BG events (consisting of VBF and VH
events) is still lower than the number of ggF2j events, which is a consequence of the smaller
cross section.

14

L = 139 fb−1

12

VBF-SR10

L = 139 fb−1

VBF-SR|Meven |2Interf.

10.0|Modd |2
VBF + V H

Nevents

8
Nevents

12.5|Meven |2Interf.

64|Modd |2
VBF + V H

7.5
5.0
2.5

2

0.0

0
0.0

0.2

0.4

0.6

0.8

−2.5
−1.0

1.0

P (c2g )

(a)

−0.5

0.0
P+ − P−

0.5

1.0

(b)

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

5

Sensitivity to the Higgs–gluon coupling

In this section, we evaluate the sensitivity to the Higgs–gluon couplings. Assuming SM
data, we fit the coupling modifiers to the expected measurements of the two signal regions
as well as their combination resulting in expected limits. Details on the parameter fit canbe found in Appendix D. Furthermore, we evaluate the impact of individual observables on
the expected limits.

5.1

ggF2j signal region

We first present our results in the ggF2j-SR, where the contributions from background processes are strongly suppressed by the signal-background classifier. Expected limits are derived under the assumption that the data is SM-like for a luminosity of 139 fb−1 . This fit isbased on the binned distribution of the various observables. The results are then plotted inthe (cg , c̃g ) parameter plane.
First, we show the limits obtained from the ∆ϕjj distribution (see Fig. 5(a)) in Fig. 8(a).
It can be seen that the allowed parameter space is constrained to the form of an ellipse forwhich the ggF2j total rate is close to its SM value. The ∆ϕjj observable alone is not able toexclude any region of this ellipse within the 1σ region.
The limits from the two classifiers are obtained from Fig. 6(a) and Fig. 6(b), respectively.
The P (c2g ) observable provides by far the strongest constraints among the two observables
constructed from the classifiers (see Fig. 8(b)). Here, the ellipse is split up within the 3 σregion, which constrains the CP-odd Higgs–gluon coupling c̃g to the interval [−0.35, 0.35] atthe 1 σ level. While the P+ − P− observable based on the interference contribution leads toweaker constraints (see Fig. 8(c)), it still outperforms the ∆ϕjj variable. Computing the lim-

15

∆φjj

20

20

1.0 L = 139 fb−1

0.5

15

0.5

15

0.0

10

0.0

10

−0.5

5

−0.5

5

0

−1.0 ggF2j-SR−1.0 −0.5 0.0 0.5 1.0
cg0−1.0 ggF2j-SR−1.0 −0.5 0.0 0.5 1.0cg(a)P+ − P−

1.0 L = 139 fb−1

(b)

∆χ2

∆χ2

P+ − P−, P (c2g )2020

1.0 L = 139 fb−1

0.5

15

0.5

15

0.0

10

0.0

10

−0.5

5

−0.5

5

0

−1.0 ggF2j-SR−1.0 −0.5 0.0 0.5 1.0
cg0−1.0 ggF2j-SR−1.0 −0.5 0.0 0.5 1.0
cg

c̃g

c̃g

∆χ2

P (c2g )c̃gc̃g

1.0 L = 139 fb−1

∆χ2

(c)

(d)
2

Figure 8: Limits from (a) ∆ϕjj , (b) the classifier trained to distinguish Meven vs.
2
Modd , (c) the classifier trained to distinguish positive vs. negative interference, and (d)the combined limits from both classifiers. All limits are shown for the ggF2j-SR. The 1-,
2- and 3-σ contours are marked by white, grey and black dashed lines, respectively. The
SM is marked by an orange cross and the best-fit (BF) point by a red star.

16

its from a two-dimensional histogram with both CP-observables shows a small improvement
over the P (c2g ) limit (see Fig. 8(d)), limiting c̃g ∈ [−0.32, 0.32] at the 1 σ level.

5.2

VBF signal region

Since the VBF-SR contains significantly more VBF events than the ggF2j-SR, the differencein the rate between the BSM and SM scenarios, which both contain the same VBF and V H
events, is reduced compared to the ggF2j-SR. This is reflected in Fig. 9 where the ellipse is
now wider.
Just like in the ggF2j-SR, the ∆ϕjj variable in Fig. 9(a) is not able to exclude partof the ellipse. In contrast to the previous results, the P (c2g ) observable gives very similar
limits to ∆ϕjj (see Fig. 9(b)). The interference classifier (see Fig. 9(c)) again results inworse limits than the P (c2g ) observable, but now even worse limits than ∆ϕjj . This canbe understood by taking into account which information each of the observables is provided
with. While the classifiers directly use ∆ϕjj during the training process, the classifier training
2
2
Meven vs. Modd is missing the information about the interference contribution. Thiseffect is negligible in the ggF2j-SR due to the small amplitude of the interference term, but
it is non-negligible here (see Fig. 6(b) and Fig. 7(b)). On the other hand, the interference22classifier does not differentiate between the Meven and Modd contributions. Finally,
the combination of both classifiers (see Fig. 9(d)), which uses the full information about thecontributing terms to σggF2j , puts stronger constraints on the Higgs–gluon coupling modifiers
than ∆ϕjj . In contrast to the ggF2j-SR, the improvement over ∆ϕjj is, however, not sosignificant. This highlights the high sensitivity of the ∆ϕjj variable in the VBF-SR (see also
the discussion in Section 5.4). We obtain c̃g ∈ [−0.58, 0.58] for the best classifier in thissignal region.

5.3

Combination

We now evaluate how the constraints on the Higgs–gluon coupling modifiers change when we
combine the ggF2j-SR and the VBF-SR. The limits obtained from the P (c2g ) and P+ − P−classifiers can be found in Fig. 10(a) and Fig. 10(b), respectively. The limits combining
the P (c2g ) and P+ − P− classifiers are depicted in Fig. 10(c). The 1 σ (2 σ) limit is shown
as a dashed (dotted) line for the ggF2j-, the VBF- and the combined SRs.6 As the plots
display, the combination of both SRs leads to an advantage over the single ggF2j-SR only
for the interference classifier. In contrast to this, the limits from the P (c2g ) observable in the
combined region are slightly weaker than in the ggF2j-SR alone. These opposite effects of the
combined SR on the two classifiers originate from a different fraction of the ggF2j, VBF andV H events when adding the events from the VBF-SR to the ggF2j-SR: The relative numberof both VBF events and ggF2j interference events is increased, as the latter make up the
6

In the two left panels, the constraints from the ggF2j-SR and the combined SRs overlap such that thetwo curves look like a solid black line.

17

∆φjj

20

20

1.0 L = 139 fb−1

0.5

15

0.5

15

0.0

10

0.0

10

−0.5

5

−0.5

5

0

−1.0 VBF-SR−1.0 −0.5 0.0 0.5 1.0
cg0−1.0 VBF-SR−1.0 −0.5 0.0 0.5 1.0cg(a)P+ − P−

1.0 L = 139 fb−1

(b)

∆χ2

∆χ2

P+ − P−, P (c2g )2020

1.0 L = 139 fb−1

0.5

15

0.5

15

0.0

10

0.0

10

−0.5

5

−0.5

5

0

−1.0 VBF-SR−1.0 −0.5 0.0 0.5 1.0
cg0−1.0 VBF-SR−1.0 −0.5 0.0 0.5 1.0
cg

c̃g

c̃g

∆χ2

P (c2g )c̃gc̃g

1.0 L = 139 fb−1

∆χ2

(c)

(d)
2

Figure 9: Limits from (a) ∆ϕjj , (b) the classifier trained to distinguish Meven vs.
2
Modd , (c) the classifier trained to distinguish positive vs. negative interference and (d)the combined limits from both classifiers. All limits are shown for the VBF-SR. The 1-, 2and 3-σ contours are marked by white, grey and black dashed lines, respectively. The SM
is marked by an orange cross and the BF point by a red star.

18

P+ − P−P (c2g )1.0 L = 139 fb−1

0.5

0.5

0.0

0.0

c̃g

c̃g

1.0 L = 139 fb−1

−0.5
−1.0

−0.5

ggF2j-like
VBF-likecombined−1.0−1.0 −0.5 0.0 0.5 1.0
cg

ggF2j-like
VBF-likecombined−1.0 −0.5 0.0
cg

(a)

1.0

(b)

P+ − P−, P (c2g )P+ − P−, P (c2g )1.0 L = 139 fb−1

1.0 L = 139 fb−1 αHgg = 45◦

0.5

0.5

0.0

0.0

c̃g

c̃g

0.5

−0.5
−1.0

−0.5

ggF2j-like
VBF-likecombined−1.0−1.0 −0.5 0.0 0.5 1.0
cg(c)

ggF2j-like
VBF-likecombined−1.0 −0.5 0.0 0.5 1.0cg(d)Figure 10: Comparison of the limits on the coupling modifiers in the ggF2j-, VBF- andcombined SRs for the P (c2g ) (a) and P+ − P− (b) observables, as well as their combination(c) for SM data and data generated for αHgg = 45◦ (d). The 1σ regions are shown asdashed lines, while the dotted lines correspond to the 2σ confidence levels. The SM ismarked by an orange cross.

19

P+ − P−, P (c2g )

0.4

13
9f

b −1

0
30
1

−1

0 fb

c̃g

−

ggF2j+VBF-

0.0 SR @ 2σ CL

300

fb

0.2

−0.2
−0.4

0.6

0.8

1.0

1.2

|cg |
Figure 11: Comparison of the limits on the coupling modifiers in the combined SR for
projections to higher luminosities. The dotted lines correspond to the 2σ confidence levels.
The SM is marked by an orange cross.

majority of misidentified ggF2j events in the VBF-SR. The limits obtained from using bothclassifier observables are not affected when combining both signal regions, see Fig. 10(c).
So far we have only looked at exclusion limits obtained for the scenario that the measured
data is SM-like. As an addition, we show in Fig. 10(d) the limits of the combined classifierin the three signal regions, for the case that αHgg = 45◦ is realised in Nature. While, as
before, the constraints from the VBF-SR are a full ellipse and therefore allow the SM pointat both the 1σ and 2σ levels, the ggF2j-SR excludes the SM at the 1σ level. The combined
SR is able to exclude the SM at the 2σ level.
Going back to the case of SM data, the expected limits at different benchmark luminosities corresponding to Run-2 and Run-3 of the LHC and further the HL-LHC are depicted
in Fig. 11. We note that the improvementof the limits with increased luminosity is slightly√
worse than naively expected (∝ 1/ L). This is a consequence of the suppression of the|Modd |2 contribution close to the SM point as well as the small impact of the interference
term. We expect the limits at the 2σ level to improve from c̃g ∈ [−0.44, 0.44] at L = 139 fb−1
to c̃g ∈ [−0.35, 0.35] at L = 300 fb−1 and c̃g ∈ [−0.2, 0.2] at L = 3000 fb−1 , respectively. Itshould be noted, however, that the limits derived here are expected to degrade once systematic uncertainties, particularly the non-Higgs background, are taken into account. We leavea full investigation of all systematic effects for further work.

5.4

Highest-impact observables

Here, we evaluate the impact of the various observables used by the classifiers on the outputof the CP classifier. We evaluate this impact with the SHAP package [128], which explains theoutput in terms of Shapley values from cooperative game theory [129, 130]: Each classifier

20

has a fixed ”worth” determined from its separation power when all variables are used in
the training. The Shapley value of one variable xi is then determined by taking the sum
over all possible subsets of the parameter space not containing xi and evaluating the loss in”worth” compared to the full case. Taking every possible subspace into account guarantees
that possible correlations between variables do not falsify the result. For more details, werefer to Ref. [131]. In the following, the Shapley values are referred to as SHAP values tomeet the nomenclature of the SHAP package.
The SHAP plots are structured as follows:• Individual variables are plotted against their SHAP value. A point at a high absolute
SHAP value means that the variable had a large impact on deciding what label theevent has been given by the classifier. The sign of the SHAP value determines whichlabel was chosen, i.e., for the |Meven |2 vs. |Modd |2 classifier, a positive sign signals that
|Meven |2 is more likely to be chosen.
• The color of the points represents the value of the variable itself. Red points stand forhigh values of the variable, while blue ones stand for low ones.
• The variables are ranked by the mean of the absolute SHAP value. Therefore, variables
which have many events (displayed as bulks) deviating from zero are identified as themost important ones. Single points of variables that reach high absolute SHAP values
are understood as outliers and only have a minor impact.
We first focus on the classifier separating the squared ggF2j terms in the respective signal
regions in Fig. 12. In the ggF2j-SR (see Fig. 12(a)) the most important variables are pT and
E of the leading-pT jets, as well as their invariant mass, while the observables associatedwith the Higgs boson play a subordinate role. This picture is flipped in the VBF-SR (seeFig. 12(b)) where now pT and E of the Higgs boson are the most important variables,
followed by the jet kinematics. Note that the overall absolute SHAP values are relatively
low compared to their spread indicating that no single observable drives the outcome of theclassifiers. Instead, the full kinematic information is needed to separate between the squared
ggF2j terms.
In the case of the interference classifiers (see Fig. 13), ∆ϕjj is by far the most importantvariable for both signal regions. As a CP-odd observable by construction, ∆ϕjj is expected
to give a good separation between the positive and negative parts of the interference. In
the ggF2j-SR (Fig. 13(a)), other important variables are the pT of the Higgs boson, as wellas the pseudorapidity of the jets — in the form of the combination ∆ηjj as well as η j1 andη j2 alone. For the VBF-SR (Fig. 13(b)), it should be noted that the mean of the ∆ϕjj
distribution peaks at much higher SHAP values than in the ggF2j-SR. This is in agreementwith the limits obtained in Section 5.2, where the limits from the classifier using the full
event kinematics showed only a slight improvement over the limits from ∆ϕjj alone. The
next two most sensitive variables are pjT1 and pjT2 . Our findings agree with Ref. [132], in which
CP-violating effects in the Higgs coupling to W bosons were studied with a different machine21

|Meven|2 vs. |Modd|2 (VBF-like)

High

pH
T

pjT2

EH

E

j2

pjT1

E

j1

E j1

Feature value

pjT1

mjj
pH
T

∆φjj
pjT2

EH

E j2

∆ηjj

mjj

η j2

∆ηjj

Other

Other

−3

−2

−1

0

1

2

3

High

Feature value|Meven|2 vs. |Modd|2 (ggF2j-like)

Low

−3

SHAP value (impact on model output)

−2

−1

0

1

2

3

Low

SHAP value (impact on model output)

(a)

(b)

Figure 12: SHAP values for the variables used in classifiers separating the |Meven |2 and|Modd |2 terms. The classifiers are trained in (a) the ggF2j-SR and (b) the VBF-SR. Highabsolute SHAP values indicate a strong influence on the classifier score. The color ofindividual points marks the value of the specific variable. The variables are ordered bytaking the mean of the absolute SHAP value. All variables used in the training which are
not shown in the plot are summed up and marked as “Others”.

pos. interf. vs. neg. interf. (VBF-like)

High

∆φjj

pH
T

pjT1

∆ηjj

pjT2

pjT1

pH
T

η

Feature value

∆φjj

j2

η j1

η j2
E j1

H

φH

j1

ηH

φ j2

φ j2

φ
φ

Other
−8

High

Feature value

pos. interf. vs. neg. interf. (ggF2j-like)

Other
−6

−4

−2

0

2

4

6

8

Low

−8

SHAP value (impact on model output)

(a)

−6

−4

−2

0

2

4

6

8

Low

SHAP value (impact on model output)(b)

Figure 13: Same as Fig. 12, but the SHAP values for the classifiers separating the positive
and negative interference terms are shown.

22

P+ − P−, P (c2g )P+ − P−, P (c2g )20

1.0 L = 139 fb−1

∆χ2
20

0.5

15

0.5

15

0.0

10

0.0

10

−0.5

5

−0.5

5

−1.0 ggF2j-SR−1.0 −0.5 0.0 0.5 1.0
cg0−1.0 VBF-SR−1.0 −0.5 0.0 0.5 1.0
cg

0

c̃g

c̃g

1.0 L = 139 fb−1

∆χ2

(a)

(b)

Figure 14: Combined limits of the two classifiers for the case that CP violation is present
in the HV V coupling in the (a) ggF2j-SR and the (b) VBF-SR.

learning approach, identifying ∆ϕjj , pjT1 and pjT2 as the three most sensitive observables forthis coupling when using the product of all three variables.

6

Disentangling CP violation in the ggH and HV V couplings

So far, we have concentrated on CP violation in the Higgs–gluon interaction. However, BSM
physics may affect multiple Higgs couplings at once. As discussed above, VBF and V H
production are the main Higgs background processes for investigations of ggF2j production.
While, as mentioned in Section 1, a lot of effort has been put forward already to investigate
the CP structure of the Higgs couplings to massive vector bosons, we investigate in thissection how the presence of CP violation in the HV V couplings influence our limits on the
Higgs–gluon coupling in the two signal regions.
For this, we generate additional data sets for VBF and V H production in the context of
the SMEFT, where a non-zero value of the cΦW̃ Wilson coefficient introduces CP violation in
the Higgs coupling to W bosons (for details see Appendix A). These data sets are added to
the SM-like VBF and V H data sets which have been used in our analysis so far. To quantify
the effects on our Higgs–gluon coupling limits, we do not re-train our classifiers, but directlybuild new observables based on the adjusted events.
Fig. 14 depicts the combined limits of the two classifiers obtained in the ggF2j-SR andthe VBF-SR when using the data sets with CP violation in the HV V coupling. We obtainour limits in the (cg , c̃g ) parameter plane by keeping cΦW̃ ∈ [−1, 1] as a free parameter in the
fit and then profiling over it. In the ggF2j-SR, we observe no change in the limits comparedto the previous results (compare Fig. 14(a) with Fig. 8(d)). Although the classifiers are not23

retrained with the CP violating HV V samples, we find only a low number of events fromthose samples in the ggF2j-SR. Varying cΦW̃ ∈ [−1, 1]7 is therefore not enough to induce asignificant change in the limits.
The limits based on the VBF-SR with CP violation in the HV V coupling are plotted inFig. 14(b). In the VBF-SR, the amount of CP violating HV V events is enriched even without
explicitly training the classifier to recognize them. For the P (c2g ) classifier, no changes in thelimits can be seen since the interference contributions cancel in a CP-even observable. We
do observe changes in the ∆χ2 of the P+ − P− classifier, but these are hard to see by eye
due to the closed ellipse in Fig. 9(c). We therefore only show the combined limits here, inwhich a clear difference can be observed between the 1σ regions of Fig. 9(d) and Fig. 14(b).
Specifically, the 1σ limits are weakened from c̃g ∈ [−0.58, 0.58] to c̃g ∈ [−0.64, 0.64].
We conclude that the limits in the (cg , c̃g ) parameter plane get slightly weaker only in the
VBF-SR if a Wilson coefficient parameterizing CP violation in the HV V coupling is kept
free-floating in the fit. This is expected since VBF-like events are strongly suppressed in the
ggF2j-SR, which is, therefore, robust against possible CP violation in the HV V coupling.

Limits on the CP structure of the top-Yukawa coupling

7

As discussed in Section 2.2, we normalized the coefficients of the effective Higgs–gluon interaction (see Eq. (1)) such that they directly correspond to the modifiers of the top-Yukawa
coupling (see Eq. (2)) if no low-mass coloured BSM particles are present and if we neglect thecontributions of the lighter quarks. Since experimental limits on coloured BSM particles are
becoming increasingly strong [98–103], these particles can also only induce a small CP-odd
Higgs–gluon coupling. Taking as an example a coloured fermion with a mass of 1 TeV anda Yukawa-type coupling of O(1) to the Higgs boson, we expect this fermion to contribute to
the Higgs–gluon couplings at the O(v 2 /Λ2 ) ∼ 0.1.
Assuming that the contribution of coloured BSM particles is negligible, we can reinterpret
our projected limits on the Higgs–gluon coupling as projected limits on the top-Yukawa
coupling (by simply replacing cg by ct and c̃g by c̃t ). Then, our projected limits on the CP
mixing angle αHtt = tan−1 (c̃t /ct ) for a luminosity of 139 fb−1 result in
HttHtt
αggF2j∈ [−15◦ , 15◦ ], αVBF
∈ [−25◦ , 25◦ ] @ 68% CL(5)for the respective signal regions.
Besides exploiting the kinematics of ggF2j production, also the total rate information of
Higgs production via gluon fusion (without jets) can be used to constraint the CP structure of
the top-Yukawa coupling (see e.g. the recent Refs. [20, 75]). These global fits — besides othermeasurements — take into account total rate information from Higgs production via gluon
fusion and the Higgs decay to two photons, which are very sensitive, but model-dependent
constraints.
7

Current experimental limits are well within this range [41, 133].

24

(ct, c̃t) free
0.6

ggF2j
LHC fit

L = 139 fb−1

c̃t

0.3
0.0
−0.3
−0.6
0.6

0.8

1.0

1.2

ct
Figure 15: Comparison of the projected combined ggF2j limits (blue) to the current
global limits from LHC data (green). Shown are the 1 σ (dashed) and 2 σ regions (dotted)
in both cases. The SM is marked by an orange cross and the BF point of the LHC databy a red star.

To provide an up-to-date comparison to our ggF2j results, we have updated the fits
performed in Refs. [20, 75] using the new HiggsSignals which is part of HiggsTools andcontains additional recent experimental results [134–136].8 The result is shown in the (ct , c̃t )parameter plane in Fig. 15. The blue contours depict the limits from our ggF2j analysis,
and the green ones are based on the global fit to existing experimental results. As discussed
in detail in Ref. [75], the fit constraints are dominated by total rate measurements of Higgs
production via gluon fusion and the Higgs decay to two photons.9 Fig. 15 shows that a
ggF2j CP measurement has the potential to provide sensitivity to a CP-odd top-Yukawa
coupling that is competitive with a more model-dependent global fit to Higgs cross-section
and branching ratio measurements.

8

Conclusions

The CP nature of the Higgs boson is among the few Higgs properties which are still only
loosely constrained. One coupling, which is especially important for LHC physics, is the
Higgs–gluon interaction — facilitated at the loop level. The CP structure of this interaction
8

We are using version 1.1 of the HiggsSignals data set.
The observed shift of the 1 σ and 2 σ regions to lower values of ct is caused by recent experimental
measurements of top-associated Higgs production and Higgs production via gluon fusion with a lower rate
than expected in the SM [42, 137].
925

can be probed via Higgs production in association with two jets (ggF2j).
In this work, we studied how the information about the CP structure of the Higgs–gluon
interaction can be best extracted. Focusing on the Higgs decay to two photons, we firsttrained a classifier to define a ggF2j-enriched signal region differentiating it from VBF andV H production as the relevant backgrounds. In order to fully exploit the quark-initiatedggF2j channel, which is kinematically very similar to VBF production, we also defined a
VBF-enriched signal region.
For both SRs, we trained two classifiers to separate the different contributions to thesquared ggF2j amplitude: the square of the CP-even amplitude, the square of the CPodd amplitude, and the interference contribution. Out of these classifiers, we constructed
one CP-even and one CP-odd observable. Based on the distributions for these observables, we derived expected upper limits. These constrain the CP-odd Higgs–gluon coupling modifier to |c̃g | ≤ {0.35, 0.28, 0.15} at the 2 σ level for integrated luminosities of{139 fb−1 , 300 fb−1 , 3000 fb−1 }, respectively. We found these to significantly outperform
limits based only on the difference in the azimuthal angle of the two leading jets, which isan observable commonly employed in the literature. Moreover, the ggF2j-SR has a significantly higher sensitivity than the VBF-SR, which is expected by the higher number of ggF2j
events. These results suggest that significant improvements in current experimental limits
are possible with well-established techniques. We leave a more detailed study including theeffects of systematic uncertainties for future work.
We note that our limits could be further improved by including other Higgs decay channels
and removing the cut on the Higgs transverse momentum. Moreover, advanced analysis
techniques like e.g. the matrix-element approach [63, 138–146] or machine-learning-based
inference [147–151] could be used. We leave such improvements for future work.
In addition, we used interpretable machine learning (in the form of SHAP values) toinvestigate which observables have the highest impact on the classifications. While the azimuthal angle of the two leading jets is by far the most important observable for separating
the interference term from the squared terms, also other momenta like the transverse momenta of the jets and the Higgs boson play a sizeable role. For separating the squared
CP-even and CP-odd amplitudes, the transverse momenta are most decisive.
We also investigated the model dependence of our limits. In particular, we checked
whether CP-violating couplings in the VBF channel could mimic a CP-violating Higgs–gluon
interaction in our analysis. Our results show that our analysis — in particular the ggF2j-SR— is very robust to such a situation and therefore allows disentangling CP-violating Higgs
couplings to massive vector bosons and gluons.
Finally, we reinterpreted our expected limits in terms of limits on the CP character of
the top-Yukawa coupling. This reinterpretation relies on the assumption that no coloured
BSM particles contribute sizeably to Higgs production to gluon fusion, which is motivatedby the current limits from direct searches for such particles. Comparing our projections to a
global fit Higgs precision measurements using the Run-2 data of the LHC, we demonstrated
the large potential of ggF2j production as a precision probe of the CP nature of the Higgs

26

boson.

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

A

Details of the event generation

The model file used for the event generation is a custom UFO model file [115, 116] which
only contains the terms in Eq. (1), as well as an effective Higgs-photon coupling. The Hγγ
coupling is assumed to be SM-like throughout this study. It is implemented as an effective
coupling to enable the generation of the ggF2j process at the tree level. With G ≡ g, GA ≡ γ,H ≡ scalar and A ≡ pseudoscalar, there are four effective couplings: QGGH, QGGA, QGAGAH
and QGAGAA that are used to generate specific parts of the amplitude. Setting QGGH==1,
QGAGAH==1 (QGGA==1, QGAGAA==1) implies the effective interaction of a scalar (pseudoscalar)
Higgs boson with exactly one pair of gluons and one pair of photons. The parameters canbe set during the event generation, as seen in the following. All events are generated with
two hard jets at the leading order in QCD. The overall rates are scaled to the next-toleading order (NLO) results by flat K-factors from Ref. [56]. As an additional cross-checkof our results, we generated two inclusive NLO Higgs plus jets samples using the Higgs
Characterization UFO model [108] for αHgg = 0 and αHgg = 35◦ .10 Passing these samples
through the analysis workflow trained with LO samples, we find the ∆χ2 value between thetwo αHgg hypotheses to agree within ∼ 10% if comparing the LO and inclusive NLO samples(with the NLO ∆χ2 being lower). This difference is likely mainly due to the background–signal classifiers being trained with LO samples. Assuming the same selection efficiencies as
for the LO samples, the ∆χ2 values calculated using the LO samples and the inclusive NLO
samples agree within ∼ 1% indicating the robustness of our results.
The syntax for the event generation is “generate p p > a a j j QGGH==1 QGAGAH==1
2
QED=4 QCD=4” for the Meven term in ggF2j, where the restrictions on the QED and QCD
2magnitude are set to exclude di-Higgs production. The interference and Modd terms are
created by replacing “QGGH==1” by “QGGH^2==1 QGGA^2==1” and “QGGA==1”,11 respectively.
10

While this model supports NLO event generation, it does not allow to generate the interference term
separately. Therefore, we concentrate on two benchmark points for this cross-check.
11
We explicitly avoid using MadGraph’s built-in decay routines by directly simulating the γγjj final stateincluding off-shell effects. The quantum number syntax allows separating the Higgs amplitude from othercontributions. Using decay routines would not allow simulating the interference term separately.

27

For the background processes, the effective Higgs-gluon coupling has been disabled in themodel file. The VBF background is generated via “generate p p > a a j j $$ w+ w- z
QGAGAH==1” where heavy vector bosons are forbidden to appear in the s-channel to exclude
contributions from the V H background. The latter is in turn generated via “generate p p> z > a a j j QGAGAH==1; add process p p > w+ > a a j j QGAGAH==1; add process
p p > w- > a a j j QGAGAH==1”.
An additional data set introducing CP violation in the VBF channel, used for the analysis
in Section 6, is generated using the SMEFTsim 3.0 package [152]. Specifically, the U35 modelis used in the mW -scheme and we set cΦW̃ = 1 TeV−1 as the only non-zero BSM parameterof the model. Events are then generated via “generate p p > a a j j $$ w+ w- z /a
QCD=0 NP=1 NP^2==1” and we apply an additional diagram filter to ensure that only VBF
events are generated.

B

Distributions of kinematic variables

Fig. 16 shows the distributions of all variables used in the training of the classifiers (see Section 4.1), separately for the three different CP contributions to the squared ggF2j amplitude.
These include as low-level observables the energy E, transverse momentum pT and pseudorapidity η of the reconstructed final state particles, as well as some high-level observables.
An exception to this is the azimuthal angle ϕ of the aforementioned objects as this does notexhibit any separation between the three contributions due to the rotational invariance ofthe production process.

28

|Modd|2

0.15
0.10
0.05
0.00
0

200

400
EH

600

|Modd|2

0.05

0.00
0

100

200
pH
T

300

0.10
0.05

200

400
E j1

600

0.20

|Meven|2Interf.

Nevents (normalized)

Nevents (normalized)

0.15

0.10

|Modd|2

0.00
0

800

0.15

|Modd|2

0.10
0.05

100

200
pjT1

300

|Modd|2
−2

−1

0
ηH

1

2

0.125

|Meven|2Interf.

0.100|Modd|2

0.050
0.025
200

mjj

400

Nevents (normalized)

0.03
0.02

|Meven|2Interf.

0.01|Modd|2
−2

−1

0
η j1

1

0.10

0.075

0.000
0

0.04

0.00

600

200

400
E j2

600

800

|Meven|2Interf.

0.3|Modd|2

0.2

0.1

100

200
pjT2

300

0
η j2

1

400

2

|Modd|2

0.06
0.04
0.02
1

2

3
|∆ηjj |

4

0.04
0.03
0.02

5

|Meven|2Interf.

0.01
0.00|Meven|2Interf.

0.08

0.00
0

0.1

0.05

Nevents (normalized)|Meven|2Interf.

0.01
0.00

Nevents (normalized)

Nevents (normalized)

0.02

Nevents (normalized)

Nevents (normalized)

0.03|Modd|2

0.2

0.0
0

400

0.05
0.04

|Meven|2Interf.

0.3

0.0
0

800

|Meven|2Interf.

0.00
0

400

Nevents (normalized)

0.20|Meven|2Interf.

0.15

Nevents (normalized)|Meven|2Interf.

Nevents (normalized)

Nevents (normalized)

0.25|Modd|2−2−1

2|Meven|2Interf.

0.06|Modd|2

0.04

0.02

0.00

−2

0
∆φjj

2

Figure 16: Distributions of the energy (first row), pT (second row) and η (third row) of
the Higgs boson (left column), as well as the jets leading (middle column) and sub-leading(right column) in pT . The last row shows distributions of high-level observables, namely
the invariant mass of the di-jet system mjj (left), as well as the difference in pseudorapidity∆ηjj (middle) and azimuthal angle ∆ϕjj (right) of the two jets. All distributions are split
2
2
up into the Meven (blue), interference (orange) and Modd (green) contributions to
the total ggF2j cross section.

29

80

L = 139 fb−1

ggF2j-SR120|Modd |2
VBF + V H

100
Nevents

Nevents

60|Meven |2Interf.

40

L = 139 fb−1

ggF2j-SR|Meven |2Interf.
|Modd |2
VBF + V H

80
60
40

20

20
0
0.0

0.2

0.4

0.6

0.8

0
−1.0

1.0

P (c2g )−0.5

0.0
P+ − P−0.5

1.0

Figure 17: Distributions of the two CP discriminants for the mean of 100 trained classifiersin the ggF2j-like kinematic region.
L = 139 fb−1

12

|Meven |Interf.

10

2

12.5

|Modd |2
VBF + V H

L = 139 fb−1

VBF-SR|Meven |2Interf.
|Modd |2
VBF + V H

10.0
Nevents

8
Nevents

15.0

VBF-SR

6
4

7.5
5.0
2.5

2

0.0

0
0.0

0.2

0.4

0.6

0.8

−2.5
−1.0

1.0

P (c2g )−0.5

0.0
P+ − P−0.5

1.0

Figure 18: Distributions of the two CP discriminants for the mean of 100 trained classifiersin the VBF-like kinematic region.

C

Training uncertainty

In Fig. 17, the classifier outputs for the two CP discriminants in the ggF2j-like kinematic region are shown as the mean of 100 individual training processes, with the standard deviationin each bin plotted as a shaded region. This allows us to estimate the uncertainty due to theclassifier training. We observe that the largest uncertainties come from bins close to 0 and 1
in the P (c2g ) classifier. For the P+ −P− classifier, the uncertainty associated with the training
process is negligible. In the VBF-like region (see Fig. 18), the visible statistical fluctuationsin the |Meven |2 and |Modd |2 contributions arise due to the comparably low number of ggF2j
events in this kinematic region.
It should be noted that, in the present study, the uncertainty associated with the training
process does not result in an uncertainty on the extraction of the Higgs–gluon couplings.
Each trained classifier represents a slightly different observable. Consequently, the best-30performing classifier can be chosen without introducing any further uncertainty in the final
limits on the Higgs–gluon coupling.

D

Likelihood evaluation

As mentioned above, the output of the CP classifiers is used to construct different observables.
These observables are represented as histograms filled by all events from the respective
test data set. Since each bin corresponds to a number of events, which follow a Poisson
distribution, the histograms can be used to construct a likelihood
LX =e−λX λnX.
n!

(6)Here, λX is the expected number of events for a specific CP hypothesis, while n corresponds
to the observed number of events. We set n = λSM , which corresponds to a perfect agreementof the data with the SM hypothesis, in order to construct limits on the Higgs–gluon coupling.
These limits are calculated using a binned likelihood ratio Y −λi,tot ni 
λi,toteLtot
t = −2 ln= −2 ln
i
−λ
i,SM
LSM
e
λni,SM
i
 i 

X
λtot
i
i
i
= −2
λSM · ln i
− λtot + λSM ,
λSM
i


(7)

where t is the test statistic and the sum runs over all bins i. λiSM = λiE1 + λiBG is the
expected SM distribution including ggF2j with cg = 1, c̃g = 0 and the BG processes, while
λitot = c2g λiE1 + cg c̃g λiO + c̃2g λiE2 + λiBG is the total bin value in the BSM case where cg and c̃gcan be varied. Following Wilks’ theorem [153], we assume that t follows a χ2 -distribution
and can, therefore, be used to construct confidence intervals.

31

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