Prospect study for measurement of $Hb\bar{b}$ coupling at the LHC and FCC-hh

Zahra Abdy1 and Mojtaba Mohammadi Najafabadi2,3

1 AGH University of Science and Technology, Faculty of Physics and Applied
Computer Science, Al. Mickiewicza 30, 30-055 Krakow, Poland
2Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland
3 School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran

###### Abstract

This paper employs the $H+b+\text{jets}$ signature in proton-proton collisions
to explore the structure of the $Hb\bar{b}$ couplings.The focus of the
analysis lies in the decay of the Higgs boson into a photon pair, taking into
account both reducible and irreducible backgrounds and a realistic simulation
of the detector effects. To enhance the discrimination between signal and
background, a multivariate analysis is employed to analyse the kinematic
variables and optimise the signal-to-background ratio. The results indicate
that the $H+b+\text{jets}$ process can significantly contribute to the precise
measurement of CP-even and CP-odd couplings between the bottom quark and the
Higgs boson at the LHC and FCC-hh. Finally, a novel asymmetry is introduced
for the purpose of probing CP violation within the $Hb\bar{b}$ coupling,
formulated exclusively based on lab-frame momenta.

Keywords: Higgs boson, bottom quark, CP violation, hadron colliders.

##  1 Introduction

The observation of the $H\rightarrow b\bar{b}$ decay mode by the ATLAS and CMS
experiments has been made through the Higgs boson production in association
with a massive vector boson $V$ ($V=W,Z$) processes [1, 2]. Both the ATLAS and
CMS analyses rely on the leptonic decays of the vector boson ($W^{\pm}$ and
$Z$) for triggering the events and to suppress the QCD multi-jet background.
The most recent measurements for observation of the $H\rightarrow b\bar{b}$
via $VH$ processes are presented using the whole Run 2 data collected by the
ATLAS experiment in proton-proton collisions with an integrated luminosity of
139 fb-1 at $\sqrt{s}=13$ TeV. For a Higgs boson with 125 GeV produced in
either $HZ$ or $HW$ channel, an observed significance of 6.7 standard
deviations is obtained from the ATLAS experiment and the measured signal
strength relative to the prediction of SM is found to be [3]:

$\displaystyle\mu_{bb}=1.02\pm 0.12~{}\text{(stat.)}\pm
0.14~{}\text{(syst.)},$ (1)

Similar measurement by the CMS experiment is available with a combination of
Run 1 data (7 TeV and 8 TeV ) and part of Run 2 (2017 data corresponding to an
integrated luminosity of 41.3 fb-1). The observed significance of 5.6 standard
deviations and the measured signal strength from the combination $ZH$ and $WH$
processes is [2]:

$\displaystyle\mu_{bb}=1.04\pm 0.2~{}\text{(stat.}\oplus\text{syst.)},$ (2)

As can be seen, the overall uncertainty is around $20\%$ in both measurements
from the ATLAS and CMS experiments and the results are in agreement with the
Standard Model (SM) prediction within the uncertainties. In the future, at the
HL-LHC, an improvement of $10\%$ in tagging the b-quark jets efficiency is
expected which leads to a relative improvement in the uncertainty of signal
strength of up to $4.7\%$ [4].
A global fit of the Higgs boson couplings has been conducted in Ref.[5]
utilizing the comprehensive Higgs datasets collected at the LHC, encompassing
integrated luminosities per experiment of approximately 5 fb-1 at 7 TeV, 20
fb-1 at 8 TeV, and up to 139 fb-1 at 13 TeV. This analysis included the
exploration of CP-even and CP-odd couplings of the Higgs boson to bottom
quarks.
To probe the Higgs boson coupling with the SM particles and find any deviation
from the SM predictions, the $\kappa-$framework is used [6]. In this
framework, possible deviation for the Higgs-bottom quark coupling is defined
by $\kappa_{b}^{2}=\Gamma_{H\rightarrow b\bar{b}}/\Gamma^{\rm
SM}_{H\rightarrow b\bar{b}}$. The current measurement of $\kappa_{b}$ at
$68\%$ confidence level (CL), obtained from a general fit to the Higgs boson
couplings, is $0.99^{+0.17}_{-0.16}$ [7]. At HL-LHC, assuming similar
systematic uncertainties to the Run 2 of the LHC, $\kappa_{b}$ is expected to
be measured with an uncertainty of $\pm 0.042$ at $68\%$ CL [8].
Except for some anomalies which are listed in Ref.[9] and references therein,
all the experimental results obtained by the LHC experiments are consistent
with the SM expectations within the uncertainties. In particular, the measured
Higgs boson couplings with the SM fields are in agreement within the total
uncertainties [7]. Therefore, any new degree of freedom could be foreseen to
be well separated in mass from the SM particles [10, 11]. As there are several
beyond SM theories, with similar experimental signatures in some cases, an
easy and efficient way to search for new physics effects is to rely on a model
independent way. In a model independent approach, the impacts of new physics
could show up in an Effective Field Theory (EFT) extension of the SM which
consists of an infinite series of effective operators with higher dimensions
[12, 13, 14, 15, 16, 17, 18, 19]. The leading contributions to the effective
extension of the SM originate from the dimension 6 operators that is based on
a non-redundant and complete operator basis [20, 21]. Not only these operators
can modify the signal strengths but also the differential distributions may be
affected because of the presence of new vertex structures. Several effective
operators are contributing to the Higgs boson couplings with the SM fields
which persuade us to pay attention to all possible Higgs involved processes at
the colliders. Probing processes where Higgs boson is present such as Higgs
boson associated production, H+jets, and various processes where Higgs boson
is off-shell provides useful information of the new higher order effective
couplings. There are already several studies for investigating the EFT
extension of the SM in the Higgs boson sector [22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].
Higgs boson production in association with a pair of $b\bar{b}$ at hadron
colliders ($pp\rightarrow H+b+\bar{b}$), has received attention as a direct
way to probe both the CP-even and -odd structures of the bottom quark Yukawa
coupling [44, 45]. In Ref.[44], using the kinematic shapes of $\gamma\gamma
b\bar{b}$ final state in Boosted Decision Trees, a specific way to achieve a
strong sensitivity to bottom quark Yukawa coupling is presented. The analysis
has been performed at HL-LHC and Future Circular Collider proton-proton (FCC-
hh) considering the main sources of background processes. In conjunction with
the HL-LHC, anticipated to operate at a center-of-mass energy of 14 TeV and an
integrated luminosity of 3 ab-1, the analysis of Ref.[44] has been extended to
the forthcoming FCC-hh facility. This future endeavor is poised to operate at
an even higher center-of-mass energy, 100 TeV, coupled with a substantially
increased integrated luminosity of 15 ab-1 [46]. The inclusion of FCC-hh as
the succeeding flagship hadron collider initiative for the CERN, as indicated
in the updated European strategy report, underscores its pivotal role in
advancing the frontier of particle physics research [47].
In Ref. [45], both a cut-based analysis and a gradient boost algorithm have
been exploited to probe bottom quark Yukawa coupling through $Hb\bar{b}$
channel with $4b$ final state. This analysis only focuses on probing the CP-
even structure of the $Hb\bar{b}$ coupling.
The primary objective of this research paper is to investigate the
$H+b+(\text{jet})$ process and its potential to extract the CP-even and -odd
components of the bottom quark Yukawa coupling within the framework of
Effective Field Theory (EFT). The approach employed for this purpose involves
utilizing Machine Learning techniques. The $H+b+(\text{jet})$ process under
study allows for the inclusion of either a light flavour jet or a b-quark jet.
The rationale behind exploring this process is two-fold. Firstly, by examining
the capabilities of the $H+b+(\text{jet})$ process to probe the structure of
$Hb\bar{b}$ couplings, a comparative analysis can be performed in relation to
other relevant processes. Secondly, the research delves into the efficacy of a
multivariate technique that leverages the shapes of the final state to
discriminate between the signal and the main SM background processes. Through
this investigation, we aim to gain valuable insights into the underlying
physics of the $H+b+(\text{jet})$ process and its significance in probing both
CP-odd and CP-even $Hb\bar{b}$ couplings. Additionally, we seek to assess the
strength of the Machine Learning-based multivariate approach in effectively
distinguishing the signal from the SM background processes.
This paper is organised as follows. The theoretical framework of the
$Hb\bar{b}$ effective coupling is described in section 2. In section 3, we
discuss the Higgs production in association with a bottom quark and additional
jet. In section 4, the analysis strategy and details of event simulation are
explained. The results of the analysis, exemplified for the HL-LHC and for the
FCC-hh, are given in section 5. An asymmetry-like observable is introduced in
section 6 to explore the CP-violating term of the $Hb\bar{b}$ coupling.
Finally, we summarise and conclude in section 7.

##  2 Theoretical framework

As referenced in the preceding section, this study has been conducted within
the framework of the EFT, where manifestations of new physics are anticipated
to emerge through novel interactions among the SM fields. In this context, the
introduction of new couplings is suppressed by inverse powers of $\Lambda$,
which serves as the characteristic scale representing physics beyond the SM.
In the EFT, the effects of heavy new degrees of freedom are integrated out and
the SM gauge symmetries, Lorentz invariance and lepton and baryon number
conservation are respected. The new physics effects are parameterised by
higher dimension operators with not-known Wilson coefficients and the main
contributions to the observable come from dimension-six operators. In this
work we rely on the Higgs characterisation model [49] based on an EFT approach
where the $Hf\bar{f}$ interaction has the following form:

$\displaystyle\mathcal{L}_{Hf\bar{f}}=-\sum_{f=e,\mu,\tau,u,d,c,s,b,t}\frac{y_{f}^{\rm
SM}}{\sqrt{2}}\bar{f}(c_{f}+i\tilde{c}_{f}\gamma_{5})fH,$ (3)

where the Higgs boson field is denoted by $H$ and $f$ is the fermion field.
The SM Yukawa coupling of a fermion $f$ is shown by coupling $y_{f}^{\rm SM}$.
Modifications from the dimension six operators to the $Hf\bar{f}$ CP-even and
CP-odd couplings show up in $c_{f}$ and $\tilde{c}_{f}$ parameters,
respectively. In the SM, $c_{f}=1.0$ and $\tilde{c}_{f}=0.0$ and $y_{f}^{\rm
SM}/\sqrt{2}=m_{f}/v$, where $v$ is the vacuum expectation value. Both $c_{f}$
and $\tilde{c}_{f}$ for the top and bottom quarks can be indirectly studied
through the measured Higgs boson production cross section via gluon-gluon
fusion ($\sigma(gg\rightarrow H)$) and the decay width of the Higgs boson into
two photons ($\Gamma(H\rightarrow\gamma\gamma)$). The modifications that
$\Gamma(H\rightarrow\gamma\gamma)$ and $\sigma(gg\rightarrow H)$ receive only
from third quark generation are [23]:

$\displaystyle\kappa^{2}_{\gamma}$ $\displaystyle=$ $\displaystyle
0.08c^{2}_{t}+0.18\tilde{c}^{2}_{t}+4\times
10^{-5}\times(c^{2}_{b}+\tilde{c}^{2}_{b})-0.002\times
c_{t}c_{b}-0.004\tilde{c}_{t}\tilde{c}_{b},$ (4) $\displaystyle\kappa^{2}_{g}$
$\displaystyle=$ $\displaystyle
1.11c^{2}_{t}+2.56\tilde{c}^{2}_{t}+0.01\times(c^{2}_{b}+\tilde{c}^{2}_{b})-0.12\times
c_{t}c_{b}-0.2\tilde{c}_{t}\tilde{c}_{b},$

where $\kappa_{g}=\sigma(gg\rightarrow H)/\sigma_{\rm SM}(gg\rightarrow H)$
and $\kappa_{\gamma}=\Gamma(H\rightarrow\gamma\gamma)/\Gamma_{\rm
SM}(H\rightarrow\gamma\gamma)$. As can be seen, the gluon fusion cross section
can deviate significantly from its SM prediction even with minor deviations of
$c_{t}$ and $\tilde{c}_{t}$ from their SM values. However, the impact of
$c_{t}$ and $\tilde{c}_{t}$ on $\kappa_{\gamma}$ is less pronounced. When
compared to the top quark coupling, variations in the coupling modifiers of
the bottom quark ($c_{b}$ and $\tilde{c}_{b}$) exhibit much smaller magnitudes
in $\kappa_{g}$ and $\kappa_{\gamma}$. As discussed in Ref.[23], from Eq.4,
the gluon fusion cross section exhibits an enhancement of approximately $26\%$
when considering a negative value for $c_{b}$ parameter falling within
$2\sigma$ allowed region, specifically $-1.23\leq c_{b}\leq-1.08$. Meanwhile,
the di-photon decay width experiences a reduction of around $3\%$ within this
specified region. On the other hand, the impact of $\tilde{c}_{b}$ on
$\kappa_{g}$ and $\kappa_{\gamma}$ remains at the sub-percent level and loose
limits on $\tilde{c}_{b}$ could be derived from $\kappa_{g}$ and
$\kappa_{\gamma}$ [23].

In addition to the gluon fusion Higgs boson cross section and di-photon decay
width of the Higgs boson, the CP-odd component of the $Hb\bar{b}$ coupling,
$\tilde{c}_{b}$, can be constrained using the electron electric dipole moment
(EDM). This contribution to the EDM of the electron, $d_{e}$, occurs through
loop processes. Consequently, the value of $\tilde{c}_{b}$ can be indirectly
constrained by considering the existing experimental limit on $d_{e}$ [43].
The ACME collaboration has established an experimental limit on the electron
EDM at a $90\%$ confidence level to be $|d_{e}|\leq 1.1\times 10^{-29}$ e.cm
[48]. Using this bound on the electron EDM, the constraint on $\tilde{c}_{b}$
at a $90\%$ CL is found to be $\tilde{c}_{b}<0.26$. It is important to note
that this limit is derived under the assumption that there are no deviations
from the SM in the $He\bar{e}$ coupling and that no cancellations occur with
other contributing mechanisms.

As a direct way to probe the $Hb\bar{b}$ interaction, the analysis of
$H+b+\bar{b}$ production at the LHC and FCC-hh is expected to provide the
following constraints at $1\sigma$ [44]:

$\displaystyle c_{b}\in[-0.99,-0.82]\cup[0.84,1.14],\text{ at HL-LHC with 6
ab}^{-1}$ (5) $\displaystyle c_{b}\in[0.99,1.01],\text{at FCC-hh with of 30
ab}^{-1}.$

where the result of HL-LHC and FCC-hh are obtained with an integrated
luminosity of 6 ab-1 and 30 ab-1, respectively.

##  3 Higgs boson production associated with a bottom quark jet and an
additional jet

In this section, we describe the production of a Higgs boson associated with a
b-quark jet and additional jet in proton-proton collisions which occur at the
LHC and FCC-hh. In the SM, the $H+b+$(jet) production proceeds through the
gluon-$b$ quark interactions, quark-(anti)quark annihilation, and gluon-gluon
fusion. The representative Feynman diagrams are shown in Fig. 1. The filled
circles represent the vertices in the diagram that undergo modifications due
to the effective interaction of $Hb\bar{b}$ introduced in Eq.3.

Figure 1:  Representative Feynman diagrams for production of Higgs boson in
association with a bottom quark and an additional parton at leading order. The
filled red circle presents the effective coupling of the $Hb\bar{b}$.

The cross section for $H+b$ processes, involving the production of a Higgs
boson and a bottom quark, is of the order of
$\mathcal{O}(\alpha_{s}(y_{b}^{\rm SM})^{2})$, where $y_{b}^{\rm SM}$
represents the Yukawa coupling strength of the b-quark and $\alpha_{s}$
denotes the strong coupling constant. On the other hand, the higher-order
processes that include an additional jet have a cross section of
$\mathcal{O}(\alpha_{s}^{2}(y_{b}^{\rm SM})^{2})$. These processes involve the
emission of an extra jet alongside the Higgs boson and bottom quark, resulting
in a higher suppression factor due to the additional power of the strong
coupling constant.

To gain insights into the impact of varying the coupling modifiers at the LHC,
we examine the changes in the signal cross section concerning the SM when
$c_{b}$ is altered by approximately $10\%$ from their SM values. For
$c_{b}=1.1$, the relative change in the cross section is approximately $18\%$,
whereas for $c_{b}=0.9$, $|\Delta\sigma|/\sigma_{\rm SM}=17\%$. Additionally,
the effect of changing $\tilde{c}_{b}$ within $\pm 20\%$ and $\pm 40\%$ of the
SM value, results in relative changes of approximately $3\%$ and $21\%$,
respectively.

To assess the effects of the $Hb\bar{b}$ effective couplings on $Hb(H\bar{b})$
productions in association with a jet, the MadGraph5_aMC@NLO package (version
3.5.1) [50, 51, 52] is utilised. This package allows for the calculation of
cross-sections and event generation for various processes. In this case, the
effective Lagrangian, as introduced in Eq.3, is implemented in the FeynRule
program. The resulting model, known as the Universal FeynRules Output (UFO)
model [53], is then provided as input to the MadGraph5_aMC@NLO program. The
production of $Hb/H\bar{b}$ with an additional parton in the final state is
considered at leading order, employing relevant matrix elements. The events
with zero and one additional parton are combined using the MLM matching scheme
[55], which ensures a consistent description of the production process.

In this study, we aim to determine a realistic sensitivity of the $H+b$+jet
process to general $Hb\bar{b}$ coupling, specifically focusing on the $c_{b}$
and $\tilde{c}_{b}$. To achieve this, we carry out the analysis using the
Higgs decay channel to two photons, which is known for its well-reconstructed
and rather clean signature. This choice allows us to well estimate the
background processes and provides valuable insights into the impact of
different contributions on $c_{b}$ and $\tilde{c}_{b}$. With Higgs boson
decaying into two photons, the final state consists of two photons and at
least one jet from which at least one is originating from the hadronization of
a b-quark.

The analysis considers several dominant background processes that contribute
to the signal. These background processes, which need to be carefully taken
into account, include:

  * •

SM production of $H+b$+jet (merged $H+b$ and $H+b$+jet using MLM
prescription), where Higgs boson decays into $\gamma+\gamma$ and jet can be a
light or a heavy flavour jet.

  * •

Higgs boson production from gluon-gluon fusion process: $pp\rightarrow
H\rightarrow\gamma+\gamma$.

  * •

Higgs boson production in association with a vector boson: $pp\rightarrow
H+V\rightarrow\gamma+\gamma+jets$, where $V=W^{\pm}$ and $Z$.

  * •

Di-photon production associated with b-quark jets
($\text{b-jets}+\gamma\gamma+\text{jets}$).

  * •

Di-photon production associated with c-quark jets
($\text{c-jets}+\gamma\gamma+\text{jets}$). This background arises from the
production of a pair of photon, not coming from Higgs boson, accompanied by
c-quark jets, where the c-jets are mis-tagged as b-jets.

  * •

Di-photon production associated with light flavour jets where light flavour
jets are misidentified as b-jets.

To ensure sufficient statistics and obtain a more precise estimate of their
contributions, the di-photon production associated with different types of
jets, including b-quark jets, c-quark jets, and light flavour (non-bottom,
non-charm) quark jets, is generated separately in the analysis. By generating
these processes independently, it becomes possible to have an adequate number
of events for each specific jet flavour and accurately estimate their
individual contributions to the di-photon final state. This approach allows
for a more comprehensive understanding of the impact of different jet flavors
on the di-photon signal and helps in properly accounting for their respective
backgrounds in the analysis.

##  4 Simulation and Analysis Strategy

The SM background processes and signal events are generated using the
MadGraph5_aMC@NLO event generator. Specifically, the Higgs boson production in
association with a b-quark and a jet sample is obtained by merging the Higgs
boson plus b-quark ($H+b$) sample and the Higgs boson plus b-quark plus jet
($H+b$+jet) sample using the MLM merging prescription. By merging the samples,
a comprehensive description of the Higgs boson production in association with
a b-quark and a jet is achieved, allowing for accurate modelling and analysis
of the signal and background processes.

After generating the samples, they undergo further simulation and modelling to
account for various effects. First, the samples are passed through PYTHIA 8.3
[56], which handles parton showering, hadronisation, and the decay of unstable
particles. To account for the effects of the detector, the Delphes 3.5.0
package [57] is utilised, which simulates both CMS detector phase II card and
the FCC-hh card. Delphes takes the generated particles as input and applies
realistic detector response and resolution effects to the particles. For jet
reconstruction, the anti-$k_{\rm t}$ algorithm [58] with a cone size parameter
$R=0.4$, implemented in the FastJet package [59], is employed. This algorithm
clusters particles into jets based on their proximity in the detector. To
identify and tag jets originating from b-quarks, b-tagging efficiency and
misidentification rates are considered. The efficiency of b-tagging for a jet
is $p_{\rm T}$ and $\eta$ dependent. Additionally, misidentification rates for
charm-jets and light-flavour jets are taken into account. For example, for the
HL-LHC with CMS Phase II Delphes card, the b-tagging efficiency for a jet with
$p_{\rm T}\in[40,50]$ and $|\eta|\leq 1.8$ is set to $66.6\%$. The
misidentification rate for the charm-jets is $p_{\rm T}-\eta$ dependent and
for the light-flavour jets is flat. For a charm-jet with $p_{\rm T}\in[40,50]$
and $|\eta|\leq 1.8$ the misidentification rate is $18.8\%$ while it is
$1.0\%$ for light-flavour jets independent of $p_{\rm T}$ and $\eta$. These
rates reflect the likelihood of mistakenly tagging a charm-jet or a light-
flavour jet as a b-jet. By incorporating these simulation and modelling steps,
the analysis aims to realistically account for detector effects, jet
reconstruction, and the identification of b-jets, charm-jets, and light-
flavour jets, thereby providing a more accurate description of the
experimental observables.

###  4.1 Events Selection

To identify signal events, specific criteria are applied. The event selection
requires the presence of exactly two isolated photons. These photons must have
a transverse momentum greater than or equal to 20 GeV and a pseudorapidity
within the range of $|\eta_{\gamma_{1,2}}|\leq 3.0$. An isolated photon is
defined as one that exhibits minimal activity in its vicinity, reducing the
probability of originating from a jet. A small value of the isolation variable
indicates a high degree of isolation, implying that the photon is more likely
to be a genuine rather than originating from a jet. The isolation of a photon,
$I_{\gamma}$ , is quantified using an isolation variable, which is calculated
as the ratio of the sum of transverse momenta ($p_{\rm T}$) of other particles
around the photon to the transverse momentum of the photon itself. For both
photons the isolation variable is required to be less than $0.15$.

Additionally, each event must contain at least one jet, out of which at least
one must be identified as b-jets using a b-tagging criterion. The jets
(b-jets) are required to have a $p_{\rm T}$ greater than or equal to 30 GeV
and a pseudorapidity within the range of $|\eta|\leq 4.0~{}(3.0)$.

To ensure that the selected objects are well-isolated, an angular separation
criterion is applied. The angular separation between any two objects (photons
or jets) is quantified using the variable $\Delta
R_{i,j}=\sqrt{(\eta_{i}-\eta_{j})^{2}+(\phi_{i}-\phi_{j})^{2}}$, where $\phi$
represents the azimuthal angle and $\eta$ represents the pseudorapidity. The
requirement is set at $\Delta R_{i,j}\geq 0.4$ for all possible combinations
of objects (photon-photon, photon-jet, and jet-jet). This criterion helps to
ensure that the selected objects are sufficiently separated from each other in
the detector.

By applying these selection criteria, the analysis aims to isolate signal
events that satisfy the specific kinematic and isolation requirements,
ensuring the quality and reliability of the data used for further analysis.
Table 1 displays the efficiencies of the signal when the coupling $c_{b}$ is
set to 1.05 and the coupling $\tilde{c}_{b}$ is set to 0.0. It also includes
the efficiencies of the main background processes after applying the cuts. The
efficiencies represent the fraction of events that pass the selection
criteria, for each process.

| Signal | SM $H+b+\text{jets}$ | b-jets+$\gamma\gamma$ | c-jets+$\gamma\gamma$ | light-jets+$\gamma\gamma$
---|---|---|---|---|---
HL-LHC | $3.6\%$ | $3.4\%$ | $5.1\%$ | $0.25\%$ | $0.05\%$
FCC-hh | $4.3\%$ | $4.1\%$ | $2.7\%$ | $0.58\%$ | $0.09\%$
Table 1:  The efficiencies of the signal with $c_{b}=1.05,\tilde{c}_{b}=0.0$
and for the main SM background processes after applying the selection
criteria.

The cross sections of other background processes such as $H,HZ$, and $HW$
after the selection are found to be negligible in both HL-LHC and FCC-hh. At
the end of this section, we address the potential impact of a background
arising from the misidentification of jets as photons in multijet and
jets+$\gamma$ production. This occurs when jets contain neutral pions that
decay into two photons, resulting in overlapping photon showers that appear as
a single photon in the detector. To mitigate the QCD multijet background, it
is crucial to develop methods to identify and reject jets mimicking photons.
This background process has a significantly higher cross section compared to
other backgrounds, with a difference spanning several orders of magnitude.
However, by applying kinematic requirements (excluding criteria related to
photons) and ensuring the presence of only b-jets, the cross section of the
multijet and jets+$\gamma$ background is reduced to approximately $10^{4}$ pb.
The probability of a jet being misidentified as a photon depends on the
transverse momentum of the fake photon, typically ranging from $10^{-5}$ to
$10^{-3}$ depending on fake photons $p_{\rm T}$. By imposing a requirement of
two photons, the contribution of this background is effectively minimised to a
low level. While this analysis neglects the multijet background, it is
important to emphasise that a dedicated and more realistic detector simulation
is necessary to accurately estimate its potential contribution. Such a
simulation should consider the specific characteristics of the experimental
setup and incorporate a detailed representation of the detector response.
Future studies should address this aspect to provide a more comprehensive
assessment of the multijet background’s impact.

###  4.2 Multivariate analysis

In this analysis, the applied cuts on individual variables are generally
loose, meaning they do not significantly suppress a substantial fraction of
background events while also reducing the signal events. To achieve a better
discrimination between the signal and background processes a multivariate
analysis is used [60, 61, 62, 63, 64]. In particular a Boosted Decision Tree
(BDT) is trained to increase the sensitivity. All the backgrounds are taken
into account during the BDT training, with each background process weighted
accordingly. This helps in obtaining an effective separation of signal events
from the background events. To enhance sensitivity, separate analyses are
conducted for the CP-even and CP-odd signals using distinct BDT test and
training sets. This approach allows for a more targeted investigation of each
signal, optimizing the discrimination power and enhancing the overall
sensitivity of the analysis. Distinct sets of variables for CP-even ($c_{b}$)
and CP-odd ($\tilde{c}_{b}$) signal processes are employed in the BDT models
for analyses at both the HL-LHC and FCC-hh. The included variables are as
follows:

  * •

$p_{T,\gamma_{1}},p_{T,\gamma_{2}}:$ transverse momenta of first and second
photon.

  * •

$p_{T,\rm b-jet}:$ transverse momentum of the most energetic b-jet.

  * •

$p_{T,\gamma_{1}\gamma_{2}}=|\vec{p}_{T,\gamma_{1}}+\vec{p}_{T,\gamma_{2}}|$:
the magnitude of the vector sum of the transverse momenta of the two photons.

  * •

$M_{\gamma_{1}\gamma_{2}}$ : invariant mass of two photons (reconstructed
Higgs boson mass).

  * •

$M_{\gamma_{1},\gamma_{2},\rm b-jet}$: the invariant mass of two photons and
the leading b-jet system.

  * •

$M_{\gamma_{1},\rm b-jet}$: the invariant mass of leading photon and leading
b-jet system.

  * •

$M_{\gamma_{2},\rm b-jet}$: the invariant mass of second photon and first
leading b-jet system.

  * •

$\Delta R(\gamma_{1},\gamma_{2})$: the angular distance between the first and
second photon.

  * •

$\cos(\gamma_{1},\gamma_{2})$: the cosine of angle between the two photons.

  * •

$\Delta\phi(\gamma_{1,2},\rm b-jet)$: the azimuthal angle between the photons
and the b-jet. These azimuthal angle variables are expected to be sensitive to
the CP-violating coupling [65, 66, 67, 68].

  * •

mean IP of the b-tagged jet defined as:

$\displaystyle\frac{\sum_{\rm
i=trk}^{N_{trk}}\sqrt{d_{0,i}^{2}+d_{z,i}^{2}}}{N_{trk}},$ (6)

where $d_{0,i}$ and $d_{z},i$ are the transverse impact parameter and the
longitudinal impact parameter values of $i$th track inside the b-tagged jet
cone with $\Delta R=0.4$. The total number of tracks inside the jet cone is
$N_{trk}$.

  * •

b-jet charge (${\mathbf{Q}_{\rm b-jet}}$):

$\displaystyle\frac{\sum_{i}q_{i}\times p_{T,i}}{p_{T,\rm b-jet}},$ (7)

where the sum is over the particles inside the reconstructed b-jet cone,
$q_{i}$ is the integer charge value of the observed color-neutral object,
$p_{\rm T,i}$ is the magnitude of its transverse momentum w.r.t the beam axis,
and the total transverse momentum of the b-jet is denoted by $p_{\rm
T,b-jet}$. More details of the jet charge definition is found Ref.[69].

Figure 2 displays the normalised distributions of some the input variables for
the CP-even signal. The distributions for signal are for the case
$c_{b}=1.05,\tilde{c}_{b}=0.0$. In particular, first photon $p_{T}$, invariant
mass of diphoton, cosine of the angle between two photons
($\cos(\gamma_{1},\gamma_{2})$), and the angular distance between the di-
photon system ($\Delta R(\gamma_{1},\gamma_{2})$) are depicted in Fig.2. The
normalised distributions of some the input variables for the CP-odd case with
$c_{b}=1.0,\tilde{c}_{b}=0.1$ as for signal events are presented in Fig.3. The
invariant mass of the leading photon and the leading jet system, highest
$p_{T}$ b-jet , and the difference between the azimuthal angles of
$\gamma_{1,2}$ and the $p_{T}$ of the highest $p_{T}$ b-jet are shown as
examples of the input variables for the CP-odd BDT.

Figure 2: Normalised distributions for the signal with
$c_{b}=1.05,\tilde{c}_{b}=0.0$ and the main background processes of some of
the input variables at the HL-LHC used in the BDT after the selection
criteria. In particular, first photon $p_{T}$, invariant mass of diphoton,
cosine of the angle between two photons ($\cos(\gamma_{1},\gamma_{2})$) , and
the angular distance between the di-photon system ($\Delta
R(\gamma_{1},\gamma_{2})$).

Figure 3: Normalised distributions for the CP-odd signal events with
$c_{b}=1.0,\tilde{c}_{b}=0.1$ and the main background processes of some of the
input variables at the HL-LHC used in the BDT after the selection criteria.
The invariant mass of the leading photon and the leading jet system
($M_{\gamma_{1},\rm b-jet}$), leading b-jet $p_{T}$ , and the difference
between the azimuthal angles of $\gamma_{1,2}$ and the leading b-jet
($\Delta\phi(\gamma_{1,2},\rm b-jet)$).

There is a substantial contribution from the c-jet+$\gamma\gamma$+jets
background, primarily arising from the elevated misidentification rate of
c-quark jets as b-jets. This misattribution is rooted in the reliance of
heavy-flavor jet identification algorithms on variables linked to the
characteristics of heavy-flavor hadrons, such as their lifetimes. Notably,
heavy-flavor hadrons containing b-quarks exhibit a lifetime on the order of
1.5 ps, whereas c-hadrons have a lifetime of 1 ps or less. Consequently, b
hadrons typically display displacements ranging from a few millimeters to one
centimeter, depending on their momentum-values that can align with those of
energetic jets containing c-hadrons. To discern between backgrounds featuring
jets originating from c quarks, we utilize the mean impact parameter and the
charge of the identified b-jet as input variables for the BDT. The
distribution of these two variables for the
b-jet+$H(\rightarrow\gamma\gamma)$+jets and the c-jet+$\gamma\gamma$+jets
backgrounds is illustrated in Fig.4 for comparative analysis.

Figure 4: Normalised mean IP and $Q_{\rm b-jet}$ distributions for the
b-jet+$H(\rightarrow\gamma\gamma)$+jets and the c-jet+$\gamma\gamma$+jets
background. The definitions of mean IP and $Q_{\rm b-jet}$ are given in Eq.6
and Eq.7, respectively.

It is notable that there is always room for improvement in the analysis,
especially by selecting a more optimal set of variables. However, the
variables utilised in our study have proven to be effective discriminators, as
evidenced below. We employ the relative importance to filter out the most
crucial variables while maintaining the accuracy of the BDT. By utilising the
feature importance, we identify and retain the variables that have the highest
significance without compromising the overall accuracy of the BDT model. The
mean IP, b-jet charge, and the invariant mass of the diphoton system, and the
cosine between the two photons angle emerge as the most significant in
distinguishing the signal from the backgrounds for both the HL-LHC and FCC-hh.
In Figure 5, we present the relative importance of each observable based on
the separation between the signal and backgrounds. The ranking is shown for
both the HL-LHC and the FCC-hh. The top panel illustrates the relevance of
kinematic variables as inputs for the BDT in distinguishing CP-even signals,
while the bottom panel focuses on CP-odd signals from the main background
processes. The left subplot depicts variable importance for the LHC, and the
right subplot presents the corresponding analysis for the FCC-hh.

Figure 5:  Input kinematic variables for the Boosted Decision Tree (BDT) and
their significance in distinguishing the CP-even signal (top) and CP-odd
signal (bottom). On the left, variable importance is illustrated for the LHC,
while the right side plot showcases the same analysis for the FCC-hh.

Figure 6: The normalized BDT output distributions for signal and background at
HL-LHC (top) and FCC-hh (bottom). The distributions are presented for the CP-
even scenario with $(c_{b}=1.05,\tilde{c}_{b}=0.0)$ (left) and the CP-odd case
with $(c_{b}=1.0,\tilde{c}_{b}=0.1)$ (right). The Kolmogorov-Smirnov
probabilities are given in the plots as well.

We proceed with the implementation of the BDT algorithm using the following
methodology. The datasets containing independent event samples for both the
signal and background are randomly divided into two equal parts. One part is
used to train the BDT algorithm, while the other serves as a validation set
for both signal and background events.

As mentioned earlier, we employ twelve parameters to train the BDT algorithm
separately for CP-even and CP-odd. To ensure optimal performance and minimise
the risk of overtraining, we take necessary precautions. We have taken
explicit measures to prevent overtraining by verifying that the Kolmogorov-
Smirnov probability, which measures the similarity between distributions.
Figure 6 displays that BDT output distributions for both HL-LHC (top) and FCC-
hh (bottom) and illustrates the Kolmogorov-Smirnov probability for both the
training and testing samples, demonstrating that neither the signal nor the
background samples are overtrained. To optimise sensitivity, we apply an
appropriate cut on the BDT response. This cut is determined to maximise the
ability to discriminate between the signal and background events. By applying
this cut, we obtain the corresponding numbers of signal (${\mathbf{N}_{\rm
S}}$) and background (${\mathbf{N}_{\rm B}}$) events. Using these event
counts, we calculate the sensitivity for parameters $c_{b}$ and
$\tilde{c}_{b}$.

##  5 Results

To determine the statistical significance, denoted as $\mathcal{S}$, we employ
the following formula. Given a number of signal events (${\mathbf{N}_{\rm
S}}$) and background events (${\mathbf{N}_{\rm B}}$) at a specific luminosity
($\mathcal{L}$), considering an uncertainty of $\Delta_{\rm B}$ on background,
the significance ($\mathcal{S}$) is calculated as [70, 71]:

$\displaystyle\mathcal{S}=\left[2\times\left((\mathbf{N}_{\rm
S}+\mathbf{N}_{\rm B})\ln\left[\frac{(\mathbf{N}_{\rm S}+\mathbf{N}_{\rm
B})(\mathbf{N}_{\rm B}+\Delta^{2}_{\rm B})}{\mathbf{N}^{2}_{\rm
B}+(\mathbf{N}_{\rm S}+\mathbf{N}_{\rm B})\Delta^{2}_{\rm
B}}\right]-\frac{\mathbf{N}^{2}_{\rm B}}{\Delta^{2}_{\rm
B}}\ln\left[1+\frac{\Delta^{2}_{\rm B}\times\mathbf{N}_{\rm
S}}{\mathbf{N}_{\rm B}(\mathbf{N}_{\rm B}+\Delta^{2}_{\rm
B})}\right]\right)\right]^{1/2}.$ (8)

In case that $\Delta_{\rm B}=0$, $\mathcal{S}$ is reduced to the following
formula:

$\displaystyle\mathcal{S}=\sqrt{2\times\left[(\mathbf{N}_{\rm
S}+\mathbf{N}_{\rm B})\ln(1+\frac{\mathbf{N}_{\rm S}}{\mathbf{N}_{\rm
B}})-\mathbf{N}_{\rm S}\right]}.$ (9)

In the limit of large number of background events with respect to signal,
$\mathcal{S}=\mathbf{N}_{\rm S}/\sqrt{\mathbf{N}_{\rm B}}$. Now, we present
the sensitivity reach for both the HL-LHC and the FCC-hh. In Fig.7, the
$1\sigma$ and $2\sigma$ allowed regions for $c_{b}$ and $\tilde{c}_{b}$ are
displayed in Table 2 and Fig.7. The integrated luminosity considered for the
HL-LHC is 3 ab-1, while for the FCC-hh, it is taken as 15 ab-1. The regions
allowed within $1\sigma$ and $2\sigma$ are also displayed, accounting for a
systematic uncertainty of $25\%$ for background.

Figure 7: The $1\sigma$ and $2\sigma$ regions for $c_{b}$ and $\tilde{c}_{b}$ by considering only statistical uncertainty and statistical uncertainty plus $25\%$ overall systematic uncertainty at the HL-LHC and FCC-hh.  $c_{b}/\tilde{c}_{b}$ |  $1\sigma/2\sigma$ |  Uncertainty |  HL-LHC |  FCC-hh
---|---|---|---|---
$c_{b}\;(\tilde{c}_{b}=0)$ |  1$\sigma$ |  Stat. |  $[-1.057,-0.948]$ $\cup\;[0.949,1.057]$ |  $[-1.0045,-0.995]$ $\cup\;[0.995,1.0045]$
Stat.+Exp. |  $[-1.063,-0.940]$ $\cup\;[0.941,1.064]$ |  $[-1.0048,-0.996]$ $\cup\;[0.996,1.0048]$
2$\sigma$ |  Stat. |  $[-1.107,-0.889]$ $\cup\;[0.889,1.108]$ |  $[-1.009,-0.990]$ $\cup\;[0.990,1.009]$
Stat.+Exp. |  $[-1.120,-0.873]$ $\cup\;[0.873,1.120]$ |  $[-1.010,-0.991]$ $\cup\;[0.991,1.010]$
$\tilde{c}_{b}\;(c_{b}=1)$ |  1$\sigma$ |  Stat. |  $[-0.33,0.33]$ |  $[-0.069,0.069]$
Stat.+Exp. |  $[-0.35,0.35]$ |  $[-0.072,0.072]$
2$\sigma$ |  Stat. |  $[-0.46,0.46]$ |  $[-0.097,0.097]$
Stat.+Exp. |  $[-0.49,0.49]$ |  $[-0.101,0.101]$
Table 2: The expected $1\sigma$ and $2\sigma$ limits on $c_{b}$ and
$\tilde{c}_{b}$ couplings considering only statistical uncertainty and
statistical uncertainty plus $25\%$ overall systematic uncertainty at the HL-
LHC and FCC-hh with the integrated luminosity of 3 ab-1 and 15 ab-1.

###  5.1 Bounds on $(c_{b},\tilde{c}_{b})$ space

Figure 8 illustrates the anticipated $1\sigma$ exclusion regions within the
$c_{b}-\tilde{c}_{b}$ parameter space for both the HL-LHC and FCC-hh
colliders, taking into account integrated luminosities of 3 ab-1 and 15 ab-1,
respectively. These exclusion limits were calculated under the assumption that
the kinematics of the signal events remain independent of the specific values
of the $c_{b}$ and $\tilde{c}_{b}$ couplings. Additionally, the presented
limits incorporate an overall systematic uncertainty of $25\%$. Upon
comparison, it is evident that the limits experience slight enhancements with
the increase in center-of-mass energy from the HL-LHC to the FCC-hh. The
parameter space region obtained for the FCC-hh exhibits a circular shape,
similar to that for the HL-LHC. However, due to the compactness of the region
delineated for the FCC-hh, we opted to zoom in on a restricted area
surrounding the SM value. Consequently, this adjustment resulted in an
elliptical appearance, facilitating a more refined visualization of the
parameter space vicinity to the SM point.

Figure 8:  Expected $2\sigma$ regions in the $c_{b}-\tilde{c}_{b}$ plane
obtained for the HL-LHC and FCC-hh with the integrated luminosities of 3 ab-1
and 15 ab-1, respectively. The limits are derived assuming an overall
uncertainty of $25\%$ on the background estimation. The red circle indicates
the SM.

##  6 Exploration of the $Hb\bar{b}$ CP-odd coupling

In this section, we introduce an asymmetry observable that is sensitive to the
magnitude of the pseudo-scalar coupling ($\tilde{c}_{b}$). To evaluate this
observable, we apply the simulation chain and selection criteria described in
Section 4 for various values of the $\tilde{c}_{b}$ coupling.

The differential production cross section associated to any CP-mixed case of
the $Hb\bar{b}$ coupling in H+b+jet signal can be parameterized according to
the following:

$\displaystyle d\sigma=c_{b}^{2}\times d\sigma_{\rm CP-
even}+\tilde{c}_{b}^{2}\times d\sigma_{\rm CP-
odd}+c_{b}\times\tilde{c}_{b}\times d\sigma_{\rm int.},$ (10)

where $d\sigma_{\rm CP-even}$, $d\sigma_{\rm CP-odd}$, and $d\sigma_{\rm
int.}$ are corresponding to the signal differential cross sections for the CP-
even, CP-odd couplings and interference terms, respectively. The integration
of the interference term in Eq. 10 over the whole phase space disappears
because when a CP-even amplitude and a CP-odd amplitude interfere, the
resulting interference term oscillates in sign across different regions of
phase space and the integral of an odd function over a symmetric interval
vanishes. Consequently, the interference term doesn’t add anything to the
overall rate or to CP-even measurements like transverse momenta and invariant
masses distributions. Instead, it only affects observables designed
specifically to measure CP-odd phenomena. We construct an asymmetry observable
from the azimuthal angular distributions of the final state objects in
$pp\rightarrow H(\rightarrow\gamma\gamma)+b+j$ process. This observable,
$\mathcal{O}_{\phi}$, is sensitive to the CP-violating $\tilde{c}_{b}$
coupling of the $Hb\bar{b}$ interaction. We define the angular asymmetry
$\mathcal{O}_{\phi}$ with respect to the azimuthal angle as:

$\displaystyle\mathcal{O}_{\phi}=\frac{N^{+}-N^{-}}{N^{+}+N^{-}},$ (11)

where

$\displaystyle
N^{+}=\int_{0}^{\pi}\frac{dN}{d\Delta\phi(H,b)},~{}~{}\text{and}~{}~{}N^{-}=\int_{-\pi}^{0}\frac{dN}{d\Delta\phi(H,b)},$
(12)

where $\Delta\phi(H,b)$ is defined as the azimuthal angle between the Higgs
boson and the highest $p_{\rm T}$ b-quark in the event, where the Higgs boson
is reconstructed from the two photons. According to the general expression
provided in Eq.10, we anticipate the following functional form for the
asymmetry:

$\displaystyle\mathcal{O}_{\phi}(c_{b},\tilde{c}_{b})=\frac{A\times
c_{b}^{2}+B\times\tilde{c}_{b}^{2}+C\times c_{b}\tilde{c}_{b}}{D\times
c_{b}^{2}+E\times\tilde{c}_{b}^{2}},$ (13)

where $\mathcal{O}_{\phi}(c_{b}=1.0,\tilde{c}_{b}=0.0)$ is the SM case. We
note that the denominator represents the total cross section and thus does not
contain an interference term. Assuming $c_{b}=1.0$, the deviation of
$\mathcal{O}_{\phi}$ due to the CP-odd coupling from the SM has the following
form:

$\displaystyle\delta\mathcal{O}_{\phi}(\tilde{c}_{b})=\mathcal{O}_{\phi}(c_{b}=1.0,\tilde{c}_{b})-\mathcal{O}_{\phi}(c_{b}=1.0,\tilde{c}_{b}=0.0)=\frac{B^{\prime}\times\tilde{c}_{b}^{2}+C^{\prime}\times\tilde{c}_{b}}{1.0+E^{\prime}\times\tilde{c}_{b}^{2}},$
(14)

where parameters $B^{\prime}\equiv(BD-AE)/D^{2}$, $C^{\prime}\equiv C/D$, and
$E^{\prime}\equiv E/D$. To illustrate the sensitivity of the asymmetry, we
explore the relationship between the $\delta\mathcal{O}_{\phi}$ and
$\tilde{c}_{b}$ coupling. Several Monte-Carlo simulated samples consisting of
500K events is analyzed to discern the degree to which the asymmetry is
influenced by the presence of the $\tilde{c}_{b}$ coupling. The difference
between the asymmetry and its SM value, $\delta\mathcal{O}_{\phi}$, is plotted
against $\tilde{c}_{b}$ at the LHC, as illustrated in Figure 9. In the plot,
the value of $c_{b}$ is fixed at the SM value of $1.0$ and the uncertainty
depicted is purely statistical. By conducting a fit, we derive the following
result: $B^{\prime}=-0.09\pm 0.003,C^{\prime}=0.0006\pm 0.0005$, and
$E^{\prime}=2.9\pm 0.3$. It is evident from the plot that the
$\mathcal{O}_{\phi}$ exhibits sensitivity to the magnitude of CP-odd coupling.
As $|\tilde{c}_{b}|$ increases, $\delta\mathcal{O}_{\phi}$ falls below the SM
value. It is noteworthy that $\delta\mathcal{O}_{\phi}$ receives a minor
impact from the interference term because of the very small coefficient of
linear term of $\tilde{c}_{b}$ with respect to the term $\tilde{c}_{b}^{2}$,
i.e. $|C^{\prime}|\ll|B^{\prime}|$. As a result, the bounds which will be
derived on $\tilde{c}_{b}$ using $\mathcal{O}_{\phi}$ are expected to exhibit
only a minimal degree of asymmetry.

Figure 9:  The variation of the asymmetry from its SM prediction is examined
by plotting the difference $\delta\mathcal{O}_{\phi}$ as defined in Eq.14.
This variation is analyzed as a function of $\tilde{c}_{b}$ at the LHC. The
uncertainty is pure statistical.

We also investigate the sensitivity of the asymmetry as a probe of CP-
violating couplings at the HL-LHC and FCC-hh. To quantify this, all selection
criteria presented in section 4.1 are followed. We assess the statistical
significance in the measurement of the asymmetry $\mathcal{O}_{\phi}$ as
$S_{\mathcal{O}_{\phi}}=\mathcal{O}_{\phi}/\Delta\mathcal{O}_{\phi}$, where
$\Delta\mathcal{O}_{\phi}$ is:

$\displaystyle\Delta\mathcal{O}_{\phi}=\sqrt{\frac{1-\mathcal{O}^{2}_{\phi}}{\sigma_{\rm
SM}\times\mathcal{L}}},$ (15)

where $\sigma_{\rm SM}$ is the SM cross section and $\mathcal{L}$ is the
integrated luminosity. The signal significance $S_{\mathcal{O}_{\phi}}$ is
dependent on $c_{b}$ and $\tilde{c}_{b}$ and has the following form:

$\displaystyle
S_{\mathcal{O}_{\phi}}(c_{b},\tilde{c}_{b})=\frac{\mathcal{O}_{\phi}-\mathcal{O}_{\phi,\rm
SM}}{\sqrt{1-\mathcal{O}^{2}_{\phi,\rm SM}}}\times\sqrt{\sigma_{\rm
SM}\times\mathcal{L}}.$ (16)

As our focus lies on the CP-odd coupling, $c_{b}$ is set to its SM value and
we concentrate on $\tilde{c}_{b}$. In Figure 10, the $1\sigma$ and $2\sigma$
regions of $\tilde{c}_{b}$ are depicted versus the integrated luminosity for
both the HL-LHC and FCC-hh.

Figure 10: The $1\sigma$ and $2\sigma$ regions versus the integrated
luminosity for $\tilde{c}_{b}$ for the HL-LHC (left) and FCC-hh (right). The
regions are obtained considering only statistical uncertainty.

As observed, the $1\sigma$ ($2\sigma$) region of $\tilde{c}_{b}$ is accessible
down to $-0.40\leq\tilde{c}_{b}\leq 0.38~{}(-0.62\leq\tilde{c}_{b}\leq 0.61)$
with an integrated luminosity of 3000 fb-1 at the HL-LHC and down to
$(-0.16\leq\tilde{c}_{b}\leq 0.15~{}(-0.23\leq\tilde{c}_{b}\leq 0.22))$ with
an integrated luminosity of 15000 fb-1 at the FCC-hh.

##  7 Summary and conclusions

This study focused on investigating the sensitivity of the $H+b+\text{jets}$
process at the HL-LHC and FCC-hh colliders in order to explore the CP-even and
CP-odd couplings of $Hb\bar{b}$. The analysis involved constraining the new
physics couplings, $c_{b}$ and $\tilde{c}_{b}$, by conducting a search through
the $H+b+\text{jets}$ channel and studying the subsequent decay of the Higgs
boson into two photons. Monte Carlo event generation was employed to simulate
the signal and relevant background processes, and detector effects were taken
into account. To distinguish the signal from the background, a carefully
selected set of discriminating variables was analysed using a multivariate
technique, in particular, by employing Boosted Decision Trees (BDTs). The
expected $1\sigma$ and $2\sigma$ limits on $c_{b}$ and $\tilde{c}_{b}$ were
obtained, and the exclusion region in the $c_{b}$-$\tilde{c}_{b}$ plane were
determined. These limits correspond to integrated luminosities of 3 ab-1 and
15 ab-1 for the HL-LHC and FCC-hh, respectively.

Comparing the obtained limits with the current experimental bounds, it was
found that a significant portion of the unexplored $c_{b}$-$\tilde{c}_{b}$
parameter space could be accessed through this analysis. By assuming one non-
zero coupling at a time for comparison, the $c_{b}$ coupling was found to
range from $[-1.2,-1.17]\cup[0.88,1.12]$ based on the LHC Higgs signal
strength data at $90\%$ confidence level (CL). The direct predicted ranges
from this study were $[-1.12,-0.873]\cup[0.873,1.12]$ for the HL-LHC and
$[-1.010,-0.991]\cup[0.991,1.010]$ for the FCC-hh.

Regarding the $\tilde{c}_{b}$ parameter, the current limits were within the
range $[-0.5,0.5]$ based on the LHC Higgs signal strength data and
$[-0.26,0.26]$ from electron EDM at $90\%$ CL. The direct measurement from
this analysis excluded the range $[-0.49,0.49]$ for the HL-LHC and
$[-0.10,0.10]$ for the FCC-hh at the $2\sigma$ level. Both the HL-LHC and FCC-
hh demonstrated the potential to provide limits on $c_{b}$ and $\tilde{c}_{b}$
at the same order of magnitude as or even better than those obtained from
indirect bounds.

We also introduced a novel asymmetry tailored to investigate CP violation
within the $Hb\bar{b}$ coupling, applying exclusively lab-frame momenta. Our
analysis underscored the effectiveness of this asymmetry in constraining the
CP-odd couplings of the bottom quark Yukawa couplings.

Upon contrasting the findings of this study, it becomes apparent that the
$H+b+\text{jets}$ process emerges as a powerful and direct avenue for
investigating both the CP-even and CP-odd aspects of the $Hb\bar{b}$
interactions at proton-proton colliders. This efficacy is primarily attributed
to the incorporation of a more extensive final state. There are potential
avenues to improve the results. Firstly, incorporating complete next-to-
leading order predictions for the $H+b+\text{jet}$ process, including loop
level diagrams, can yield more accurate and reliable outcomes. Secondly, to
enhance sensitivity and statistical significance, the inclusion of additional
decay modes of the Higgs boson, such as $WW$ and $ZZ$ needs to be considered.

Acknowledgement

The authors would like to thank M. Ebrahimi for her fruitful discussions in
multivariate analysis algorithms and M. Zaro for his valuable guidances in
generating events with MadGraph.

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unactuated hinge. Sparse rewards.

  * •

walker-stand ($\mathbf{a}\in\mathbb{R}^{6}$). A bipedal planar walker that is
rewarded for standing with an upright torso at a constant minimum height.
Dense rewards.

  * •

walker-walk ($\mathbf{a}\in\mathbb{R}^{6}$). A bipedal planar walker that is
rewarded for walking forward at a certain target velocity. Dense rewards.

X

###  F.2. Robotic experiments

We train MaDi and three other baselines on the UR-VisualReacher task (Yuan and
Mahmood, 2022; Wang et al., 2023) shown in Figure 21, and test the
generalization performance on a novel experiment that we designed, dubbed UR-
VisualReacher-VideoBackgrounds. The task of the robotic arm is the same in
both environments: it gets rewarded for positioning its camera as close as
possible to the red dot on the screen. In the testing environment, one of the
videos shown in Figure 22 is playing in the background. We take videos from
the DeepMind Control Generalization Benchmark (Hansen and Wang, 2021).

For the reward function: we adjust the RGB thresholds used to determine
whether pixels are part of the red dot or not, such that it is robust against
the video backgrounds. We selected videos that do not have any strong red
colors in them, so that the rewards are well calibrated. It would be an
interesting direction for future work to verify whether MaDi can detect the
red dot even if other task-irrelevant pixels are red too. This would require a
different setup of the reward function.

(a) Initial state

(b) Reaching towards the goal

Figure 21. The UR5 Robotic Arm setup used in our experiments. The agent is
rewarded for getting its camera as close as possible to the red dot randomly
located on the screen.

This figure presents two images displaying the UR5 Robotic Arm setup from a
side view. In the first image, labeled ”(a) Initial state,” there’s a computer
monitor on the left showing a white screen with a red dot about 3 cm wide (in
real life). To the right of the monitor, the UR5 robotic arm with 6 joints
(only 4 are clearly visible) is in a resting position. It has a webcam camera
attached to the tip of the arm. The second image, ”(b) Reaching towards the
goal,” depicts the same desk with monitor and robotic arm setup. However, in
this image, the UR5 robotic arm has extended towards the monitor, aiming to
approach the red dot. The robotic arm is sophisticated, with joints and
sections allowing for flexibility in movement. Its arm section are made of
silver-colored pipes. The joints between sections are colored light blue.

(a)

(b)

(c)

(d)

(e)

Figure 22. Examples of frames from each video shown on the screen in the
UR5-VisualReacher task to test generalization.

This figure shows five distinct rectangular frames, each depicting a different
visual environment with a prominent red dot located in different random
positions in the frame. These frames were displayed on the screen during the
UR5-VisualReacher-VideoBackgrounds task to test generalization: (a) A serene
tropical setting, featuring a beach with many swaying palm trees, calm azure
waters, and a clear blue sky. (b) A more minimalistic and uniform blue
gradient background, showing a starry night. (c) An image of a tranquil
forested lakeside, where calm waters reflect the dense trees and foliage of
the surrounding forest. (d) A majestic scene of icy mountains towering in the
background, bathed in the light of a setting sun. The foreground showcases a
calm lake with some ice floating around. (e) A beach environment seen from the
top where waves flow towards the beach. Most of the image is dark blue-green
from the ocean. Beach is at the top.