# Improved Precision in $Vh(\rightarrow b\bar{b})$
via Boosted Decision Trees

Philipp Englert<EMAIL_ADDRESS>
###### Abstract

Extracting bounds on BSM operators at hadron colliders can be a highly non-
trivial task. It can be useful or, depending on the complexity of the event
structure, even essential to employ modern analysis techniques in order to
measure New-Physics effects. A particular class of such modern methods are
Machine-Learning algorithms, which are becoming more and more popular in
particle physics. We attempt to gauge their potential in the study of
$Vh(\rightarrow b\bar{b})$ production processes, focusing on the leptonic
decay channels of the vector bosons. Specifically, we employ boosted decision
trees using the kinematical information of a given event to discriminate
between signal and background. Based on this analysis strategy, we derive
bounds on four dimension-6 SMEFT operators and subsequently compare them with
the ones obtained from a conventional cut-and-count analysis. We find a mild
improvement of $\mathcal{O}(\mathrm{few}\,\%)$ across the different operators.

###### Keywords:

##  1 Introduction

Machine-Learning (ML) algorithms are a set of very powerful tools that really
shine in situations where we have access to large amounts of data. Data is
something that is usually abundant in hadron-collider experiments, so it seems
natural to apply these kinds of algorithms in their study. One of their main
applications in this context is the classification of events into signal and
background, which can be quite hard to do well with traditional cut-and-count-
based analyses or can simply be improved upon, depending on the process at
hand. We would like to note, however, that there are many other important
applications of ML techniques in particle physics, a detailed discussion of
which is beyond the scope of this paper. For more in-depth information about
the current applications of ML in high-energy physics and its advantages, we
would like to refer the reader to, e.g., refs. Guest:2018yhq ;
Bourilkov:2019yoi ; Schwartz:2021ftp ; Karagiorgi:2021ngt . Furthermore, ref.
Feickert:2021ajf  attempts to curate a steadily up-to-date list of both
general and more specialised reviews and recent research at the interface
between ML and high-energy physics.
This paper accompanies our earlier study Bishara:2022vsc0  of the
$Vh(\rightarrow b\bar{b})$ production processes at hadron colliders, where we
derived bounds on the couplings of four dimension-6 operators within the
framework of the Standard Model Effective Field Theory (SMEFT). The goal of
the present study is to leverage the strengths of Machine-Learning techniques
to improve the sensitivity to New-Physics effects with respect to the
conventional cut-and-count approach we took in said previous analysis, and to
compare the performance of the two methods.
Diboson production channels like this one allow EW precision measurements at
hadron colliders and can be used to probe the dynamics of the Higgs boson at
high energies Farina:2016rws ; deBlas:2013qqa ; Franceschini:2017xkh . Because
of this, they are a viable way to test for a large class of New Physics
models. For further theoretical details on $Vh$-diboson-production channels in
the context of EW precision measurements, we would like to refer the reader to
our previously mentioned companion paper Bishara:2022vsc0  and our closely
related earlier papers, refs. Bishara:2020vix0 ; Bishara:2020pfx0 , on the
corresponding diphoton channels $Wh(\rightarrow\gamma\gamma)$ and
$Zh(\rightarrow\gamma\gamma)$. Previous studies on precision measurements in
the $Vh$ production channel, where a boosted Higgs decaying into two
$b$-quarks was considered can be found in refs. Banerjee:2018bio ; Liu:2018pkg
; Banerjee:2019pks ; Banerjee:2019twi ; Banerjee:2021efl . Furthermore, the
ATLAS collaboration published a comprehensive study using LHC Run 2 data in
refs. ATLAS:2020fcp ; ATLAS:2020jwz .
The reason we decided to study the potential of ML techniques in the
$h\rightarrow b\bar{b}$ channel instead of the $h\rightarrow\gamma\gamma$
channel is that, as explained in refs. Bishara:2020vix0 ; Bishara:2020pfx0 ,
it is possible to essentially render the $h\rightarrow\gamma\gamma$ channel
background-free by applying cuts on the kinematical variables. Therefore,
there is not much room for improvement with an ML analysis of the diphoton
channel. The situation is different for the $h\rightarrow b\bar{b}$ channel,
though, where the backgrounds are large and more difficult to separate from
the signal.
We decided to use BDTs for this analysis because we only have at most
$\mathcal{O}(10)$ kinematical variables that are potentially useful for
discriminating signal and background, and BDTs are relatively easy to optimise
compared to, e.g., NNs.
We opted for performing this analysis using LHC simulations because the FCC-hh
data we simulated for our previous study in ref. Bishara:2022vsc0  is limited
and as a consequence, we estimate that the statistical uncertainties are of
the order of the performance difference between the BDT analysis and the
conventional cut-and-count analysis. However, we want to be able to
quantitatively compare the two methods with each other. We have access to
better statistics for the LHC, since we performed more extensive simulations
for this collider. We choose to derive the bounds assuming a luminosity of
$3\,\mathrm{ab^{-1}}$, which corresponds to the HL-LHC.
ML algorithms have numerous advantages over more traditional algorithms. For
example, they are sometimes able to recognise data patterns that are not
immediately obvious using more conventional methods and hence could not be
leveraged by the latter. Or they might be much simpler to set up for some
types of problems where a more traditional solution would require very complex
code or a lot of hand-tuning or even be outright impossible to find. However,
there are arguably also some disadvantages to using an ML solution to a given
problem. A common point of criticism is that it is sometimes difficult to
understand in detail what the ML model is doing and why it is performing a
certain way for a given data set. But there are ways of mitigating this issue
by making the results more interpretable. One way to do this is the use of
SHAP values, as suggested in ref. Grojean:2022mef .
The first part of section 2 is dedicated to a concise recapitulation of the
general setup of the physics analysis that has been described in much more
detail in ref. Bishara:2022vsc0 . Since these prerequisites are identical to
what we discussed in said companion paper, we refrain from going into much
detail here. In the second part of this section, we briefly introduce the
concept of the aforementioned SHAP values. In section 3, we describe the setup
of the ML analysis we performed. Finally, we report our results in section 4.
On the one hand, we present the bounds derived from our ML-based analysis and
compare them to the bounds derived from our previous cut-and-count analysis.
On the other hand, we attempt to explain and interpret the workings of the
BDTs we trained. For this, we present the aforementioned SHAP values and we
visualise how the BDTs act to separate signal from background by comparing the
distributions of the kinematical variables after the ML analysis with the
distributions after the cut-and-count analysis and with the distributions
before the application of either of the two.

##  2 Theoretical background, physical and technical prerequisites

In this section, we briefly discuss the theoretical fundamentals related to
our analysis and the physical and technical concepts we adopt from our
companion paper Bishara:2022vsc0 . In subsection 2.1, we concisely summarise
the general setup of our analysis from a physical perspective. A more detailed
discussion can be found in our companion paper. In section 2.2, we introduce
the concept of SHAP values, which we will later use to explain the output of
our ML model.

###  2.1 General setup of the physics analysis

Following the analysis strategy explained in ref. Bishara:2022vsc0 , we focus
on the four operators

$\displaystyle\mathcal{O}_{\varphi q}^{(1)}=$
$\displaystyle\left(\overline{Q}_{L}\gamma^{\mu}Q_{L}\right)\left(iH^{\dagger}\overset{\leftrightarrow}{D}_{\mu}H\right)\,,$
(1) $\displaystyle\mathcal{O}_{\varphi q}^{(3)}=$
$\displaystyle\left(\overline{Q}_{L}\sigma^{a}\gamma^{\mu}Q_{L}\right)\left(iH^{\dagger}\sigma^{a}\overset{\leftrightarrow}{D}_{\mu}H\right)\,,$
(2) $\displaystyle\mathcal{O}_{\varphi u}=$
$\displaystyle\left(\overline{u}_{R}\gamma^{\mu}u_{R}\right)\left(iH^{\dagger}\overset{\leftrightarrow}{D}_{\mu}H\right)\,,$
(3) $\displaystyle\mathcal{O}_{\varphi d}=$
$\displaystyle\left(\overline{d}_{R}\gamma^{\mu}d_{R}\right)\left(iH^{\dagger}\overset{\leftrightarrow}{D}_{\mu}H\right)\,.$
(4)

They are responsible for the leading contributions to the energy growth of BSM
corrections to the $Vh$-production processes, assuming Minimal Flavour
Violation (MFV). These operators are elements of the Warsaw basis, see ref.
Grzadkowski:2010es . In figure 1, we display representative Feynman diagrams
contributing at leading order (LO) to said processes.

Figure 1: Tree-level Feynman diagrams contributing to the $qq^{\prime}\to Vh$
production processes. The gray circles represent an insertion of one of the
BSM operators in equations (1)–(4).

The $Vh(\rightarrow b\bar{b})$ production processes can be divided in the
following three categories depending on the number of charged leptons in the
final state:

  * •

_Zero-lepton category_ : The biggest contribution to the signal in this
category comes from $pp\rightarrow Z(\rightarrow\nu\bar{\nu})h(\rightarrow
b\bar{b})$, but the signal also receives a smaller additional contribution
from $pp\rightarrow W(\rightarrow\ell\nu)h(\rightarrow b\bar{b})$, whenever
the charged lepton is not detected. Here, we collectively denote
$\ell=e,\,\mu,\,\tau$ and $\nu=\nu_{e},\,\nu_{\mu},\,\nu_{\tau}$. The dominant
contribution to the $Wh$ signal in this category is the one where
$W\rightarrow\tau\nu_{\tau}$. The backgrounds we consider in this analysis are
$t\bar{t}$-, $Wb\bar{b}$\- and $Zb\bar{b}$ production with the same $W$\- and
$Z$ decay modes as in the signal.

  * •

_One-lepton category_ : The signal process in this category is ${pp\rightarrow
W(\rightarrow\ell\nu)h(\rightarrow b\bar{b})}$, where we denote
$\nu=\nu_{e},\,\nu_{\mu}$ and the charged lepton $\ell=e,\,\mu$ is detected.
The backgrounds we include in our analysis are $Wb\bar{b}$ and $t\bar{t}$.

  * •

_Two-lepton category_ : In this category, the signal is the $pp\rightarrow
Z(\rightarrow\ell^{+}\ell^{-})h(\rightarrow b\bar{b})$ process, where
$\ell=e,\,\mu$ and both leptons are detected and the only background we
consider is the $Zb\bar{b}$ process.

As discussed in refs. Bishara:2020pfx0 ; Bishara:2022vsc0 ; Bishara:2019iwh ,
the phase-space region that yields the highest sensitivity to the BSM
operators is the high-energy tail. The Higgs boson is usually boosted in this
region, which entails characteristic kinematical properties that can be tested
for in order to distinguish signal- from background events. A possible way to
test for such a boosted Higgs is the mass-drop-tagging technique
Butterworth:2008iy . In order to make use of the available events in the most
efficient way, we divide them into two classes: events that contain such a
boosted Higgs candidate, which we call ‘boosted events’, and ‘resolved
events’, which contain a pair of resolved $b$-jets instead. We explain this
classification procedure in detail in ref. Bishara:2022vsc0 . Since in our
analysis, we combine these two classes of events, the tagging procedure we
employ is a scale invariant one Gouzevitch:2013qca . The concrete tagging
strategy we follow in our studies is heavily inspired by the one implemented
in ref. Bishara:2016kjn . As a result, we can divide all the events into six
categories depending on whether the final state contains 0, 1 or 2 charged
leptons and depending on whether the event is boosted or resolved.
In our companion study, we optimised cuts on the kinematical variables of each
of these six categories separately in order to distinguish signal- from
background events. In the study described here, we simply use BDTs for this
discrimination instead.
The parton-level events were generated using MadGraph5_aMC@NLO v.2.7.3
Alwall:2014hca  together with the NNPDF23 parton-distribution functions
Ball:2013hta . The subsequent parton showers and the Higgs decay were modelled
via Pythia8.24 Sjostrand:2014zea . The SMEFT operators were defined and
integrated into our analysis using the SMEFTatNLO UFO model Degrande:2020evl ;
Degrande:2011ua . More technical details about the simulations are explained
in ref. Bishara:2022vsc0 .

###  2.2 SHAP values

In order to make the predictions of our BDTs more transparent, we make use of
the so-called SHAP (SHapley Additive exPlanations) values NIPS2017_7062 ,
which are based on the game-theoretic concept of Shapley values
shapley1951notes .
Both Shapley values and SHAP values are feature-importance measures, i.e.
their goal is to quantify the influence of a certain input feature on the
outcome of the prediction for a given model.
We can translate the game-theoretic philosophy behind Shapley values into the
context of ML in the following way: the importance of an input feature can be
defined as the effect of including that feature as an input on the prediction
of the model. There are several ways to technically implement a quantitative
measure for this effect and perhaps the most intuitive one is the approach of
Shapley regression values Lipovetsky2001 . The idea behind this is to train a
model with the feature present in the input data and another model with the
feature absent and compare the predictions of the two models.
There is one caveat to this idea, though, which is that the effect of
including versus withholding the feature in the input data depends on which
other features are included in the input data. If, for example, there is
another input feature that is highly correlated with the feature whose
importance we are trying to determine, then the importance is shared between
those two input features. Since this should be reflected by a quantitative
importance measure, Shapley values are computed in the following way.
Let $i$ be the input feature whose importance we want to measure and $F$ be
the set of all the input features we are using. We call $S$ a subset of the
set of all the input features excluding the one whose importance we want to
measure, i.e. $S\subseteq F\backslash\\{i\\}$. Now we train a model
$f_{S\cup\\{i\\}}(x_{S\cup\\{i\\}})$ and a model $f_{S}(x_{S})$ for each
possible subset $S$, including the empty set $\O$. Here, $x_{S\cup\\{i\\}}$
and $x_{S}$ are the input feature values of a specific sample $x$, restricted
to the subset $S$ of the features with and without the feature $i$,
respectively. The Shapley regression value $\phi_{i}(x)$ for the sample $x$ of
the input feature $i$ is defined as

$\displaystyle\phi_{i}(x)=\sum\limits_{S\subseteq
F\backslash\\{i\\}}\frac{|S|!(|F|-|S|-1)!}{|F|!}[f_{S\cup\\{i\\}}(x_{S\cup\\{i\\}})-f_{S}(x_{S})]\,.$
(5)

The normalisation factor that weights the contributions is closely related to
the inverse number of subsets of input features of size $|S|$ that can be
chosen from the set of all the input features excluding $i$, which is of size
$|F|-1$. For example, the contribution from the model where $S$ is the full
set, $S=F\backslash\\{i\\}$, is much more important than the contribution from
one of the models where half of the input features are in $S$. This is to
account for the fact that if we did not apply the weights in that way, the
effect of, e.g., a strong correlation of $i$ with another input variable on
the Shapley value would be diluted if there was a large number of other input
variables, even if they did not contribute much to the prediction at all.
To give a more applied example, imagine the problem of discriminating between
the ${Zh(\rightarrow b\bar{b})}$ and ${Wh(\rightarrow b\bar{b})}$ signal
events of the resolved 0-lepton category and the background consisting of
$Zb\bar{b}$-, $Wb\bar{b}$\- and $t\bar{t}$ events. Let the input variables be
the mass $m_{bb}$ of the $b\bar{b}$ pair, the missing transverse momentum
$E_{T}^{\mathrm{miss}}$ and the angular distance between the two $b$-jets
$\Delta R_{bb}$. Of course, this is not an optimal selection of input
variables, but it serves us well to illustrate the point.
To assess the importance of $m_{bb}$ for the prediction of a specific event,
we train a model with only $\Delta R_{bb}$ as an input variable and compare
its prediction for this event with the one of the model with $\Delta R_{bb}$
and $m_{bb}$ as an input variable. Multiplying this difference with the
according combinatorial weight gives us the contribution of $S=\\{\Delta
R_{bb}\\}$ to the Shapley value. Now, we do the same with
$S=\\{E_{T}^{\mathrm{miss}}\\}$, $S=\\{E_{T}^{\mathrm{miss}},\,\Delta
R_{bb}\\}$ and $S=\\{\O\\}$, weight them accordingly and add up the
contributions. The result is the Shapley value of $m_{bb}$ for the given
event.111Note that for $S=\\{\O\\}$, we need to define the output of the model
with no input variable $f_{\O}(x_{\O})$. The only information available to
train this model is the distribution of the true labels of the training set,
so we define $f_{\O}(x_{\O})$ to be the expectation value of the true label
over the training set. Since, the discriminating power of $m_{bb}$ in this
process is large, an ML model that performs well would likely yield a large
Shapley value for $m_{bb}$. The same can be done to compute the Shapley values
of $\Delta R_{bb}$ and $E_{T}^{\mathrm{miss}}$.
This procedure, however, is very computationally expensive because a new model
needs to be trained for every possible subset of the input variables. In fact,
this problem is even NP-hard.
A different approach, the so-called SHAP (SHapley Additive exPlanations) value
NIPS2017_7062 , unifies the ideas of several methods that approximate the
Shapley regression values with the ideas of several other feature-importance
measures and circumvents this issue of having to train a model for every
possible subset of the input variables. Instead, one measures the effect of
removing a variable from the model by averaging the predictions of the model
when drawing values for the variable to be removed from random samples of the
data set. I.e., one uses the expectation function of the model conditional on
the variables that have not been removed.
Since even SHAP values are still computationally very expensive, one can apply
different approximations, e.g. one can also sample from the feature subsets
instead of taking into account all of them, which is known as Shapley sampling
values Strumbelj2013 . Alternatively, if one focuses on specific types of
models, e.g. decision trees, one can make use of their properties to use
algorithms that are able to exactly compute the classic Shapley regression
values in polynomial time. One of these algorithms is the TreeExplainer
lundberg2019explainable ; 2018arXiv180203888L , which is used in this work to
measure the feature importance in BDTs.

##  3 Setup of the analysis

In this section, we present the methodology of the ML analysis replacing the
conventional cut-and-count analysis described in ref. Bishara:2022vsc0 .
We use one BDT per bin, according to the definitions in table 1, for each of
the 0-, 1- and 2-lepton categories and separately for both boosted and
resolved events. Furthermore, we treat the extracted observables to be
independent in the context of the computation of the $\chi^{2}$ function. We
decided to perform this bin-by-bin analysis for the following reason. The
sensitivity to the New-Physics operators varies significantly between the
different bins, and this approach avoids the potential problem that the
performance of the ML model could be optimised for a relatively unimportant
bin at the cost of the performance in a bin with larger sensitivity.

Categories | Variable | bin boundaries
---|---|---
| boosted | $p_{T,\mathrm{min}}\,\mathrm{[GeV]}$ | $\\{0,300,350,\infty\\}$
0-lepton | resolved | $\\{0,160,200,250,\infty\\}$
| boosted | $p_{T}^{h}\,\mathrm{[GeV]}$ | $\\{0,175,250,300,\infty\\}$
1-lepton | resolved | $\\{0,175,250,\infty\\}$
| boosted | $p_{T,\mathrm{min}}\,\mathrm{[GeV]}$ | $\\{250,\infty\\}$
2-lepton | resolved | $\\{175,200,\infty\\}$

Table 1: Bin definitions used in the different categories of the analyses for
the HL-LHC, adopted from ref. Bishara:2022vsc0 .

We preselect the events that are used for each of the individual BDT analyses
according to the number of charged leptons that pass the acceptance cuts and
whether the event contains a boosted Higgs candidate or two resolved $b$-jets.
Furthermore, we do not accept events where the Higgs candidate does not have
two $b$-tags and for all but the boosted 2-lepton category we veto events with
untagged jets inside the acceptance region. The identical preselection cuts
are implemented in the conventional cut-and-count analysis and only events
that pass these criteria are used for the respective ML analyses.
As a loss function, we used the binary cross-entropy, which can be defined via
Mehta:2018dln

$\displaystyle
H_{p}(x)=-\frac{1}{N}\sum_{i=1}^{N}w_{i}\bigg{[}y_{i}\cdot\log\big{(}p\left(x_{i}\right)\big{)}+(1-y_{i})\cdot\log\big{(}1-p\left(x_{i}\right)\big{)}\bigg{]}\,,$
(6)

where $N$ is the number of samples being used for the computation of the loss
function, $y_{i}\in\\{0,1\\}$ is the true class label (signal or background)
of the $i$-th sample and $x_{i}$ are the features of the $i$-th sample, which,
in the case of this study, are the kinematical variables the predictions are
based on. The prediction the model in question makes for the $i$-th sample is
denoted as $p(x_{i})$. Each sample can be assigned an individual weight
$w_{i}$ that quantifies its importance for the performance of the model. In
order to account for the differential cross-section $\sigma_{i}$ associated to
each event, we define the weights to be

$\displaystyle
w_{i}=\frac{\tilde{w}_{i}\cdot\sigma_{i}}{\langle\tilde{w}_{j}\cdot\sigma_{j}\rangle\Big{|}_{j=1}^{N}}\,,$
(7)

where $\langle.\rangle\Big{|}_{i=1}^{N}$ denotes the mean value over the $N$
samples and

$\displaystyle\tilde{w}_{i}=\begin{cases}w_{\mathrm{sig}}\,&\text{ for
}y_{i}=1\\\ 1\,&\text{ for }y_{i}=0\end{cases}$ (8)

assigns a relative weight $w_{\mathrm{sig}}$ to signal events compared to
background events in order to be able to maximise

$\displaystyle\frac{s}{\sqrt{b}}:=\frac{\mathrm{signal}}{\sqrt{\mathrm{background}}}\,,$
(9)

instead of just $s/b$. For this, the signal weight $w_{\mathrm{sig}}$ is to be
tuned during the stage of hyperparameter optimisation.
The hyperparameters signal weight, maximum depth of the individual decision
trees and learning rate were optimised via $5$-fold cross-validation (CV) for
each individual BDT used in the analysis. For each bin, $n$-lepton- and
boosted/resolved-category, we performed an exhaustive scan of the grid defined
by the following hyperparameter values:

$\displaystyle\begin{split}\text{maximum depth}&=\\{5,\,10,\,15,\,20\\}\\\
\text{learning rate}&=\\{10^{-3},\,10^{-4}\\}\\\ \text{signal
weight}&=\\{0.5,\,1,\,5,\,10,\,30,\,50,\,100,\,500,\,1000\\}\,.\end{split}$
(10)

After each of these grid searches, we picked the configuration with the
largest value of $s/\sqrt{b}$ averaged over the five iterations of CV under
the side condition that the variance be moderate.
During the grid search, the maximum number of trees for each BDT was set to
$10^{5}$. An optimisation of this parameter was not necessary because we used
early-stopping as a regularisation method, which automatically truncates the
number of BDTs at the optimal value. For the BDT of the boosted 2-lepton
category and the last bin of the resolved 2-lepton category, however, the
early stopping condition was not met within those $10^{5}$ training steps, so
for those two exceptions, we decided to set the maximum number of trees to
$10^{6}$ during the final training with the optimised hyperparameters.
However, due to limited computational resources, we could not afford to do a
full grid search with this higher maximum number of trees during the
hyperparameter-optimisation stage.
We randomly split the total amount of data into training set (60%), test set
(30%) and validation set (10%). This random assignment of events into
training-, test- and validation set was performed using stratified sampling,
i.e., the random sampling was subject to the side condition that each physical
process (e.g. $Wh$, $Wb\bar{b}$ and $t\bar{t}$ for the 1-lepton category)
should be represented in equal proportions in each of the of the three data
sets.

##  4 Results

In this section, we present the results we obtained from our final analysis
using the BDTs with the optimised sets of hyperparameters displayed in table
2.

Category |  $p_{T}$ bin [GeV]  | max. depth | learning rate | signal weight
---|---|---|---|---
$\begin{aligned} &\text{0 - lepton}\\\ &\text{resolved}\end{aligned}$ |  $[0,\,160]$ |  15 | $10^{-3}$ | 100
$[160,\,200]$ |  5 | $10^{-3}$ | 30
$[200,\,250]$ |  5 | $10^{-3}$ | 50
|  $[250,\,\infty]$ |  10 | $10^{-2}$ | 500
$\begin{aligned} &\text{0 - lepton}\\\ &\text{boosted}\end{aligned}$ |  $[0,\,300]$ |  15 | $10^{-3}$ | 500
$[300,\,350]$ |  5 | $10^{-2}$ | 10
$[350,\,\infty]$ |  5 | $10^{-2}$ | 10
$\begin{aligned} &\text{1 - lepton}\\\ &\text{resolved}\end{aligned}$ |  $[0,\,175]$ |  5 | $10^{-3}$ | 100
$[175,\,250]$ |  5 | $10^{-2}$ | 30
$[250,\,\infty]$ |  5 | $10^{-3}$ | 1000
$\begin{aligned} &\text{1 - lepton}\\\ &\text{boosted}\end{aligned}$ |  $[0,\,175]$ |  5 | $10^{-3}$ | 1000
$[175,\,250]$ |  10 | $10^{-3}$ | 100
$[250,\,300]$ |  10 | $10^{-2}$ | 100
|  $[300,\,\infty]$ |  15 | $10^{-3}$ | 50
$\begin{aligned} &\text{2 - lepton}\\\ &\text{resolved}\end{aligned}$ |  $[175,\,250]$ |  10 | $10^{-3}$ | 500
$[250,\,\infty]$ |  5 | $10^{-3}$ | 100
$\begin{aligned} &\text{2 - lepton}\\\ &\text{boosted}\end{aligned}$ |  $[250,\,\infty]$ |  10 | $10^{-3}$ | 30

Table 2: Optimal sets of hyperparameters for the BDTs given bin-by-bin for
each category, determined by performing a grid search defined by the
hyperparameter values given in equation (10). The sets of hyperparameters we
consider to be optimal are the ones that yield the largest average value of
$s/\sqrt{b}$ over five CV folds under the side condition of not having an
excessively large variance.

On the one hand, we compute the bounds derived from the BDT-based analysis of
the $Vh(\rightarrow b\bar{b})$ process, which we will do in section 4.1.
On the other hand we would like to shine some light on how the BDTs use the
kinematical variables of the events to discriminate between signal and
background. For this purpose, we make use of the previously introduced SHAP
values and to complement this, we show some of the kinematical distributions
of both signal and background events before and after both the BDT-based
analysis and the cut-and-count analysis. These aspects of explainability and
interpretability of the ML model are explored in section 4.2.
Lastly, we draw conclusions from these results and attempt to use them to
highlight the advantages and disadvantages of an ML-based approach over a
conventional cut-and-count approach in section 5.

###  4.1 Presentation of the bounds

Like in the conventional cut-and-count analysis discussed in our previous
paper Bishara:2022vsc0 , we generate histograms using the events that have
been accepted by the BDTs according to the bin definitions given in table 1.
For the signal, we fit one-dimensional quadratic functions of the Wilson
coefficients $c_{\varphi q}^{(3)}$, $c_{\varphi q}^{(1)}$, $c_{\varphi u}$ and
$c_{\varphi d}$ to this data in a bin-by-bin fashion. From these bin-by-bin
fits and the background histograms, we construct one-dimensional $\chi^{2}$
functions of the Wilson coefficients in order to derive the $95\%$ CL bounds
at $\chi^{2}=3.84$. The results of these fits and the background histograms
are given in tables 5 \- 10 in appendix A.
In table 3, we present the bounds we derived for the three benchmarks of
$1\%$, $5\%$ and $10\%$ systematic uncertainty. All these bounds represent an
improvement over the bounds obtained via the conventional cut-and-count
analysis discussed in our companion paper. The relative improvements of both
the lower and upper bounds are also given in table 3 and they are mostly of
the order $\mathcal{O}(\mathrm{few}\,\%)$.
We find the largest relative improvements in the lower bound on the
$c_{\varphi q}^{(1)}$ coefficient and the weakest improvements are found for
the lower bound on the $c_{\varphi d}$ coefficient. For most of the
coefficients, the relative improvement grows with increasing systematic
uncertainties. The exception to this is the $c_{\varphi q}^{(3)}$ operator,
where we find the largest improvement for the $1\%$ systematic-uncertainty
benchmark.

Coefficient | Bounds [TeV-2] |  Relative improvement
---|---|---
Lower bound | Upper bound
$c_{\varphi q}^{(3)}\,$ |  |  $[-1.1,\,1.0]\times 10^{-2}$ |  $1\%$ syst.
---|---
$[-1.7,\,1.5]\times 10^{-2}$ |  $5\%$ syst.
$[-2.9,\,2.2]\times 10^{-2}$ |  $10\%$ syst.
|  $5\%$
---
$1.5\%$
$1.5\%$
|  5%
---
$2\%$
$3\%$
$c_{\varphi q}^{(1)}\,$ |  |  $[-4.2,\,6.2]\times 10^{-2}$ |  $1\%$ syst.
---|---
$[-5.0,\,7.0]\times 10^{-2}$ |  $5\%$ syst.
$[-6.4,\,8.3]\times 10^{-2}$ |  $10\%$ syst.
|  $7\%$
---
$7\%$
$7\%$
|  $2\%$
---
$3\%$
$4\%$
$c_{\varphi u}\,$ |  |  $[-10.7,\,3.8]\times 10^{-2}$ |  $1\%$ syst.
---|---
$[-11.9,\,4.8]\times 10^{-2}$ |  $5\%$ syst.
$[-13.6,\,6.5]\times 10^{-2}$ |  $10\%$ syst.
|  $4\%$
---
$5\%$
$5\%$
|  $2\%$
---
$3\%$
$4\%$
$c_{\varphi d}\,$ |  |  $[-6.5,\,10.2]\times 10^{-2}$ |  $1\%$ syst.
---|---
$[-7.9,\,11.5]\times 10^{-2}$ |  $5\%$ syst.
$[-10.1,\,13.7]\times 10^{-2}$ |  $10\%$ syst.
|  $0.9\%$
---
$1.6\%$
$2.6\%$
|  $4\%$
---
$4\%$
$5\%$
Table 3:  Bounds at $95\%$ CL ($\chi^{2}=3.84$) on the coefficients of the
$\mathcal{O}_{\varphi q}^{(3)}$, $\mathcal{O}_{\varphi q}^{(1)}$,
$\mathcal{O}_{\varphi u}$ and $\mathcal{O}_{\varphi d}$ operators for 14 TeV
HL-LHC with integrated luminosity of $3\,\mathrm{ab}^{-1}$ using the BDT-based
analysis. In the third and fourth column, we present the relative improvements
of these bounds with respect to the corresponding results derived from the
cut-and-count approach.

###  4.2 Explaining and interpreting the results

In this section, we would like to tackle a point of criticism that is commonly
direct towards ML-based approaches to particle-physics analyses. Such ML-based
analyses are often regarded as black boxes whose internal workings are
difficult to understand and therefore hard to learn from.
However, we would like to highlight how these algorithms can be demystified by
exemplarily studying the impact of our BDT-based analysis on the kinematical
distributions of the boosted 0-lepton $Vh(\rightarrow b\bar{b})$ category and
comparing it with the effects of the corresponding cut-and-count-based
approach. Furthermore, we will analyse the mean absolute SHAP values of the
different input features used in the BDT-based analysis in order to get some
insight into how this particular ML algorithm decides whether an event should
be accepted or rejected.
In figures 2 \- 4, we present the kinematical distributions of the BDTs’ input
features in the boosted 0-lepton category in the different bins. Each plot
shows six distributions. The input features are $\eta^{H_{\mathrm{cand}}}$,
$\Delta\phi(E_{T}^{\mathrm{miss}},H_{\mathrm{cand}})$, $m_{H}$ and
$p_{T}^{Z}$. For both signal- (blue) and background events (green), we display
the number of unweighted Monte-Carlo events just after the preselection cuts
as solid lines, after additionally applying the BDT analysis as dotted lines
and after instead applying the rest of the cuts used in the cut-and-count
analysis as dash-dotted lines.

Figure 2: Kinematical distributions of both signal- and background events in
the ${p_{T,\mathrm{min}}\in[0,\,300]\,}$GeV bin of the boosted 0-lepton
category. The numbers of events are given as unweighted Monte-Carlo events.
The signal distributions are displayed in blue, while the background
distributions are displayed in green. The solid lines represent the
distributions after the application of preselection cuts like the jet veto and
the charged lepton veto, but before applying any other cuts or BDT analysis.
The dotted lines display the distributions after the BDTs have been applied to
reject events. The dash-dotted lines are associated to the distributions after
the conventional cuts used in the analysis discussed in ref. Bishara:2022vsc0
. In the top-left panel, we present the distribution of
$\eta^{H_{\mathrm{cand}}}$, in the top-right panel
$\Delta\phi(E_{T}^{\mathrm{miss}},H_{\mathrm{cand}})$, in the bottom-left
panel $m_{H}$ and in the bottom-right panel $p_{T}^{Z}$. Figure 3: Kinematical
distributions of both signal- and background events in the
${p_{T,\mathrm{min}}\in[300,\,350]\,}$GeV bin of the boosted 0-lepton
category. The numbers of events are given as unweighted Monte-Carlo events.
The signal distributions are displayed in blue, while the background
distributions are displayed in green. The solid lines represent the
distributions after the application of preselection cuts like the jet veto and
the charged lepton veto, but before applying any other cuts or BDT analysis.
The dotted lines display the distributions after the BDTs have been applied to
reject events. The dash-dotted lines are associated to the distributions after
the conventional cuts used in the analysis discussed in ref. Bishara:2022vsc0
. In the top-left panel, we present the distribution of
$\eta^{H_{\mathrm{cand}}}$, in the top-right panel
$\Delta\phi(E_{T}^{\mathrm{miss}},H_{\mathrm{cand}})$, in the bottom-left
panel $m_{H}$ and in the bottom-right panel $p_{T}^{Z}$. Figure 4: Kinematical
distributions of both signal- and background events in the
${p_{T,\mathrm{min}}\in[350,\,\infty]\,}$GeV bin of the boosted 0-lepton
category. The numbers of events are given as unweighted Monte-Carlo events.
The signal distributions are displayed in blue, while the background
distributions are displayed in green. The solid lines represent the
distributions after the application of preselection cuts like the jet veto and
the charged lepton veto, but before applying any other cuts or BDT analysis.
The dotted lines display the distributions after the BDTs have been applied to
reject events. The dash-dotted lines are associated to the distributions after
the conventional cuts in the analysis discussed in ref. Bishara:2022vsc0 . In
the top-left panel, we present the distribution of $\eta^{H_{\mathrm{cand}}}$,
in the top-right panel $\Delta\phi(E_{T}^{\mathrm{miss}},H_{\mathrm{cand}})$,
in the bottom-left panel $m_{H}$ and in the bottom-right panel $p_{T}^{Z}$.

A general observation that can be made for all three bins is that while the
cut-and-count analysis typically manifests itself as sharp cuts in the
kinematical distributions, the BDT-based analysis does not do this. Instead,
we find that the distributions go to zero smoothly, more similarly to how the
initial distributions behave. This behaviour is most pronounced in the
distributions in the lowest bin displayed in figure 2, especially in the
$m_{H}$ and the $p_{T}^{Z}$ distributions. This difference in effects on those
distributions can be explained by the fact that a given BDT is modelling a
function of the input features that is supposed to represent the probability
of a given event to be a signal event. The modelled probability distribution
can very well be smooth in the input features and this is what we are
observing. The advantage of this strategy over the conventional cut-and-count
analysis is that it is potentially able to make better use of the available
statistics, since sharp cuts on the distributions typically remove a lot of
events from an analysis.
We furthermore find that in the last bin, apart from the aforementioned sharp
cuts, the distributions sculpted by the BDT analysis are very similar to the
ones sculpted by the cut-and-count analysis. In the first two bins however,
they differ more. The largest differences can be found in the lowest bin,
where the overall number of accepted events is much lower for the cut-and-
count analysis, while especially the signal distributions after the BDT
analysis follow the initial distributions very closely. This observation is
mirrored by the numbers of events expected in the lowest bin at the HL-LHC
from the ML-based analysis and the cut-and-count-based analysis reported in
tables 6 and 18, respectively. The number of expected events for the SM is at
least an order of magnitude larger after the BDT-based analysis than after the
cut-and-count analysis for both signal and background. In the second bin, the
overall number of events is larger for the cut-and-count analysis. The shapes
of the distributions sculpted by the two different approaches, however, are
rather similar in this bin.
We will now discuss figures 5 \- 10, which display the mean absolute values of
the SHAP values introduced in section 2.2. We compare these results to the
impact of the different cuts in the corresponding cut-and-count analysis given
in tables 11 \- 16. The comparison shows that the results of the feature-
importance analysis for the BDTs mostly agrees with our expectations from the
cut-flow tables.

Figure 5: Averages of the absolute SHAP values of the different input features
used by the BDTs in the resolved 0-lepton category. The top-left panel
corresponds to the $[0,\,160]\,\mathrm{GeV}$ bin, the top-right panel to the
$[160,\,200]\,\mathrm{GeV}$ bin, the bottom-left panel to the
$[200,\,250]\,\mathrm{GeV}$ bin and the bottom-right panel to the
$[250,\,\infty]\,\mathrm{GeV}$ bin.

Figure 6: Averages of the absolute SHAP values of the different input features
used by the BDTs in the boosted 0-lepton category. The top-left panel
corresponds to the $[0,\,300]\,\mathrm{GeV}$ bin, the top-right panel to the
$[300,\,350]\,\mathrm{GeV}$ bin and the bottom panel to the
$[350,\,\infty]\,\mathrm{GeV}$ bin.

Figure 7: Averages of the absolute SHAP values of the different input features
used by the BDTs in the resolved 1-lepton category. The top-left panel
corresponds to the $[0,\,175]\,\mathrm{GeV}$ bin, the top-right panel to the
$[175,\,250]\,\mathrm{GeV}$ bin and the bottom panel to the
$[250,\,\infty]\,\mathrm{GeV}$ bin.

Figure 8: Averages of the absolute SHAP values of the different input features
used by the BDTs in the boosted 1-lepton category. The top-left panel
corresponds to the $[0,\,175]\,\mathrm{GeV}$ bin, the top-right panel to the
$[175,\,250]\,\mathrm{GeV}$ bin, the bottom-left panel to the
$[250,\,300]\,\mathrm{GeV}$ bin and the bottom-right panel to the
$[300,\,\infty]\,\mathrm{GeV}$ bin.

Figure 9: Averages of the absolute SHAP values of the different input features
used by the BDTs in the resolved 2-lepton category. The left panel corresponds
to the $[175,\,200]\,\mathrm{GeV}$ bin and the right panel to the
$[200,\,\infty]\,\mathrm{GeV}$ bin. Figure 10: Averages of the absolute SHAP
values of the different input features used by the BDT in the boosted 2-lepton
category.

With the exception of the lowest bin in the resolved 0-lepton category,
$m_{H_{\mathrm{cand}}}$ is by far the most important kinematical variable for
the discrimination between signal and background. In the lowest bin, the
missing transverse momentum surpasses the mass of the Higgs candidate in
importance. However, this bin corresponds to the region in phase space, where
the cut-and-count analysis has not been optimised and the cut-flow table does
not include this low-energy region, since the sensitivity coming from it is
very limited. This is the reason why the large discriminatory impact of
$E_{T}^{\mathrm{miss}}$ seen in the SHAP values for the lowest bin of the
resolved 0-lepton category is not reflected by the cut-flow tables.
Studying the importance of the other input features, we find that none of them
stands out in particular. This is in accord with the cut-flow tables, where we
can read off that the impact of most cuts is rather mild. The cut-flow tables
show that some cuts, like $\Delta R_{bb}$ in the resolved 2-lepton category,
or the $p_{T}^{\ell}$ imbalance in the boosted 2-lepton category do exhibit a
small discriminatory power, which agrees with their moderate mean absolute
SHAP values.
This comparison of the cut-flow tables with the mean SHAP values shows that
SHAP values are able to provide an understanding about the decision process of
a BDT similar to the one a cut-flow table can provide about a cut-and-count
analysis. Of course, the SHAP-value analysis we provide here does not give any
information about the importance of the different kinematic variables for the
rejection or acceptance of events of a specific process like $Zb\bar{b}$,
whereas the cut-flow tables 11 \- 16 provide this kind of information.
However, this lost information can be restored by computing the mean absolute
SHAP values for the events of each process individually.

##  5 Summary and conclusions

With this study, we showcased how an ML algorithm can be used as an
alternative to a conventional cut-and-count analysis by the example of the
$V(h\rightarrow b\bar{b})$ process. We demonstrated that it is possible to
quantify the importance of the different kinematical variables for the
predictions using the concept of SHAP values. This technique helps alleviating
a common point of criticism directed towards ML applications in the field of
particle physics, claiming that it is difficult to interpret and explain their
decision processes.
To emphasise the point that ML algorithms do not have to be regarded as
dubious black boxes, we show that more insight can be gained by, e.g.,
studying their effects on kinematical distributions. Our findings highlight
that, on the one hand, the effects of, e.g., BDTs on the kinematical
distributions usually do not manifest as sharp cuts on the distributions as
opposed to conventional cut-and-count analyses. This can be advantageous
because it can enable a more efficient usage of the available statistics. On
the other hand, our study shows that in general, the shapes of the
distributions sculpted by the BDTs are not necessarily vastly different from
the shapes sculpted by a conventional cut-and-count analysis. This further
emphasises the fact that such ML analyses can in fact be made intuitive and
interpretable.
In terms of quantitative performance, it is difficult to perform a direct
comparison between the conventional cut-and-count analysis and the BDT
analysis we performed because the BDTs were optimised on a bin-by-bin basis
and the cut-and-count analysis was only optimised in detail in the phase-space
region of $p_{T}^{V}>200\,$GeV. We find a relative improvement of the bounds
of the order $\mathcal{O}(\mathrm{few}\,\%)$ using the BDT-based analysis
compared to the cut-and-count analysis, and this advantage could very well be
due to the aforementioned difference in methodology.
However, we would like to argue that the comparison of tables 5 \- 10,
containing the number of expected signal- and background events per bin from
the ML-based analysis, to the corresponding tables 17 \- 22 from the cut-and-
count-based analysis indicates that even in the phase-space region of
$p_{T}^{V}>200\,$GeV, where the cut-and-count analysis has been carefully
optimised, the BDTs are performing better.
We highlight this by computing the two quantities $s/b$ and $s/\sqrt{b}$,
where $s$ denotes the number of signal events at the HL-LHC assuming a
luminosity of $3\,\mathrm{ab}^{-1}$ and $b$ denotes the number of background
events. The $\chi^{2}$ function depends on these two quantities such that
larger values of both those quantities yield a larger value of $\chi^{2}$ and
consequently stronger bounds. The quantity $s/b$ is associated with the term
that depends on the systematic uncertainty and is therefore more important
when the bounds are dominated by systematic uncertainties. The quantity
$s/\sqrt{b}$ on the other hand appears in the term that is associated to the
statistical uncertainty. We can therefore compute $s/b$ and $s/\sqrt{b}$ bin-
by-bin for each category and consider them to be metrics for the performance
of the two analysis approaches. We present these quantities for both the cut-
and-count analysis and the BDT-based analysis in table 4.

Category |  $p_{T}$ bin  |  $s/\sqrt{b}$ |  $s/b$
---|---|---|---
[GeV]  | BDTs | cut-and-count |  BDTs | cut-and-count
$\begin{aligned} &\text{0 - lepton}\\\ &\text{resolved}\end{aligned}$ |  $[0,\,160]$ |  25.7 | 3.3 | 0.030 | 0.037
$[160,\,200]$ |  14.0 | 9.5 | 0.26 | 0.13
$[200,\,250]$ |  5.6 | 5.3 | 0.18 | 0.15
|  $[250,\,\infty]$ |  2.2 | 2.0 | 0.19 | 0.12
$\begin{aligned} &\text{0 - lepton}\\\ &\text{boosted}\end{aligned}$ |  $[0,\,300]$ |  9.3 | 5.3 | 0.076 | 0.24
$[300,\,350]$ |  5.1 | 5.3 | 0.26 | 0.24
$[350,\,\infty]$ |  8.3 | 7.1 | 0.59 | 0.46
$\begin{aligned} &\text{1 - lepton}\\\ &\text{resolved}\end{aligned}$ |  $[0,\,175]$ |  32.2 | 13.4 | 0.036 | 0.035
$[175,\,250]$ |  13.9 | 9.6 | 0.21 | 0.12
$[250,\,\infty]$ |  3.6 | 2.9 | 0.27 | 0.20
$\begin{aligned} &\text{1 - lepton}\\\ &\text{boosted}\end{aligned}$ |  $[0,\,175]$ |  1.3 | 0.6 | 0.0054 | 0.016
$[175,\,250]$ |  8.4 | 7.4 | 0.080 | 0.098
$[250,\,300]$ |  7.9 | 6.6 | 0.20 | 0.20
|  $[300,\,\infty]$ |  4.2 | 3.8 | 0.21 | 0.19
$\begin{aligned} &\text{2 - lepton}\\\ &\text{resolved}\end{aligned}$ |  $[175,\,250]$ |  3.9 | 3.0 | 0.17 | 0.16
$[250,\,\infty]$ |  2.7 | 2.8 | 0.15 | 0.16
$\begin{aligned} &\text{2 - lepton}\\\ &\text{boosted}\end{aligned}$ |  $[250,\,\infty]$ |  5.1 | 5.4 | 0.29 | 0.28

Table 4: Comparison of $s/b$\- and $s/\sqrt{b}$ ratios from the ML analysis
with the ones from the cut-and-count analysis in each bin and for the
different categories. Here, $s$ denotes the number of signal events at the HL-
LHC assuming an integrated luminosity of $3\,\mathrm{ab}^{-1}$ and $b$ denotes
the corresponding number of background events.

This shows that the BDTs yield both larger values for $s/\sqrt{b}$ and for
$s/b$ than the conventional cut-and-count analysis in the majority of bins for
the different categories. Since this better performance by the BDTs is not
exclusive to the phase-space region $p_{T}^{V}>200\,$GeV where the cut-and-
count analysis has not been carefully optimised, we argue that this can be
considered as an indication of the superiority of the BDT approach in terms of
quantitative performance in the $Vh(\rightarrow b\bar{b})$ analysis overall.
Note that the improvement on the bounds is only of the order
$\mathcal{O}(\mathrm{few}\,\%)$ despite the large improvements on $s/\sqrt{b}$
and $s/b$ in some of the bins. This can be explained by the fact that the most
important bins for the determination of the bounds are the ones exhibiting
only a moderate BDT enhancement of $s/\sqrt{b}$ and $s/b$. This is due to the
fact that the cut-and-count analysis in ref. Bishara:2022vsc0  has been
optimised exactly for these most important bins.
On this note, we would like to point out that the grid search we performed to
optimise the hyperparameters was by no means exhaustive. One could both scan
over a more fine-grained grid or a grid that covers a larger hyperparameter
space overall, in order to fully exploit the potential performance gain of the
ML approach compared to the conventional cut-and-count approach. However,
increasing the hyperparameter space is usually accompanied with an increased
computation time. In particular for the learning rate, we found it technically
difficult to include significantly smaller values into the grid search. This
is because smaller learning rates usually require an increased number of
training steps, making the grid search even more computationally expensive. A
similar effect can potentially be observed for the maximum depth of the trees.
However, in our analysis, we found the computation time to be less sensitive
to the variations in maximum depth than to the variations in learning rate we
applied.
Another possibility for a potential performance increase of an ML-based
approach to this study of $Vh(\rightarrow b\bar{b})$ is the following. In the
analysis presented in this paper, we only attempted to improve the efficiency
of the discrimination between signal and background after the stage of the
boosted-Higgs tagging. In principle, one could however attempt to even replace
this step by using lower-level kinematical variables as input for an ML
algorithm. However, we would like to point out that due to the increased
complexity of the input variables, the hyperparameter-optimisation- and
training stage in such an analysis is likely much more involved than the one
presented in this chapter. It is conceivable that in order to achieve good
sensitivity, one would even have to resort to a different ML algorithm
altogether, like e.g. a deep-learning algorithm.
Apart from the potential advantage in quantitative performance, another
benefit of ML-based approaches is the fact that the work-intensive process of
cut-optimisation is replaced by the process of training the ML model. This
step is not free of subtleties, e.g. depending on the process to be analysed
and depending on the ML algorithm, the hyperparameter optimisation can be more
or less intricate. However, we believe that there is much more potential for
automatisation compared to the cut-and-count analysis, allowing for human
effort to be replaced with computer effort.
Even in the study of the $Vh(\rightarrow b\bar{b})$ process in particular,
there is still more potential for the use of our BDT-based approach. As a
proof of concept, we only computed the one-dimensional bounds on the four
Wilson coefficients $c_{\varphi q}^{(3)}$, $c_{\varphi q}^{(1)}$, $c_{\varphi
u}$ and $c_{\varphi d}$. However, it could be interesting to also investigate
the impact of an ML-based approach on a global analysis turning on several
operators at the same time, as we did in the course of the corresponding cut-
and-count analysis in ref. Bishara:2022vsc0 . We leave this study for future
work.

##  Appendix A Signal- and background cross-sections in the ML-analysis

In this appendix, we present the fits for the expected number of signal events
and the expected number of background events derived from the BDT-based
analysis of $Vh(\rightarrow b\bar{b})$ at the HL-LHC assuming an integrated
luminosity of $3\,\mathrm{ab}^{-1}$. We give the results for both the resolved
and boosted versions of the different $n$-lepton categories.

0-lepton channel, resolved, HL-LHC
---
$p_{T,\mathrm{min}}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-160]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}2.3&\times 10^{4}\,+6.3\times 10^{5}\,c_{\varphi q}^{(3)}+(4\pm 4)\times 10^{3}\,c_{\varphi q}^{(1)}+8.9\times 10^{4}\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-(3.6\pm 0.3)\times 10^{4}\,c_{\varphi d}+6.1\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}+4.6\times 10^{6}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+(2.37\pm 0.14)\times 10^{6}\,\left(c_{\varphi u}\right)^{2}+2.0\times 10^{6}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $(7.3\pm 0.9)\times 10^{5}$
$[160-200]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}730\,&+(5.34\pm 0.03)\times 10^{4}\,c_{\varphi q}^{(3)}-(1.9\pm 0.4)\times 10^{3}\,c_{\varphi q}^{(1)}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+(9.8\pm 1.3)\times 10^{3}\,c_{\varphi u}-(4.6\pm 0.6)\times 10^{3}\,c_{\varphi d}+1.2\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+9.3\times 10^{5}\,\left(c_{\varphi q}^{(1)}\right)^{2}+(5.4\pm 0.4)\times 10^{5}\,\left(c_{\varphi u}\right)^{2}+4.0\times 10^{5}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $2800\pm 300$
$[200-250]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}170\,&+1.7\times 10^{4}\,c_{\varphi q}^{(3)}-(9\pm 3)\times 10^{2}\,c_{\varphi q}^{(1)}+3600\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-(1.35\pm 0.15)\times 10^{3}\,c_{\varphi d}+5.0\times 10^{5}\,\left(c_{\varphi q}^{(3)}\right)^{2}+4.5\times 10^{5}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+2.4\times 10^{5}\,\left(c_{\varphi u}\right)^{2}+1.8\times 10^{5}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $980\pm 160$
$[250-\infty]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}27\,&+5.3\times 10^{4}\,c_{\varphi q}^{(3)}-(3.8\pm 1.0)\times 10^{2}\,c_{\varphi q}^{(1)}+(9.4\pm 0.9)\times 10^{2}\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-(5.4\pm 0.7)\times 10^{2}\,c_{\varphi d}+2.6\times 10^{5}\,\left(c_{\varphi q}^{(3)}\right)^{2}+2.7\times 10^{6}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+1.5\times 10^{5}\,\left(c_{\varphi u}\right)^{2}+9.5\times 10^{4}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $140\pm 70$

Table 5: Number of expected signal events as a function of the Wilson
coefficients (in units of TeV-2) and of total background events in the
$Zh\rightarrow\nu\bar{\nu}b\bar{b}$ channel, resolved category, at HL-LHC
based on the ML analysis. The Monte-Carlo errors on the fitted coefficients,
when not explicitly specified, are $\lesssim\mathrm{few}\,\%$. Note that as
opposed to the corresponding tables 17 \- 22, the functions given for the
signal do not include the $c_{\varphi q}^{(3)}$-$c_{\varphi q}^{(1)}$ mixed
terms, since they are just a linear combination of 1D fits.

0-lepton channel, boosted, HL-LHC
---
$p_{T,\mathrm{min}}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-300]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}1130\,&+1.4\times 10^{5}\,c_{\varphi q}^{(3)}-(6.5\pm 0.8)\times 10^{3}\,c_{\varphi q}^{(1)}+2.7\times 10^{4}\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-9900\,c_{\varphi d}+5.1\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}+4.4\times 10^{6}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+2.5\times 10^{6}\,\left(c_{\varphi u}\right)^{2}+1.8\times 10^{6}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $1.5\times 10^{4}$
$[300-350]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}100\,&+2.0\times 10^{4}\,c_{\varphi q}^{(3)}-(2.1\pm 0.2)\times 10^{3}\,c_{\varphi q}^{(1)}+4100\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-1100\,c_{\varphi d}+1.2\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}+1.1\times 10^{6}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+6.2\times 10^{5}\,\left(c_{\varphi u}\right)^{2}+4.3\times 10^{5}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $400\pm 70$
$[350-\infty]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}120\,&+3.7\times 10^{4}\,c_{\varphi q}^{(3)}-(3.8\pm 0.3)\times 10^{2}\,c_{\varphi q}^{(1)}+8400\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-2700\,c_{\varphi d}+3.7\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}+3.4\times 10^{6}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+2.1\times 10^{6}\,\left(c_{\varphi u}\right)^{2}+1.3\times 10^{6}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $200\pm 20$

Table 6: Number of expected signal events as a function of the Wilson
coefficients (in units of TeV-2) and of total background events in the
$Zh\rightarrow\nu\bar{\nu}b\bar{b}$ channel, boosted category, at HL-LHC based
on the ML analysis. The Monte-Carlo errors on the fitted coefficients, when
not explicitly specified, are $\lesssim\mathrm{few}\,\%$. Note that as opposed
to the corresponding tables 17 \- 22, the functions given for the signal do
not include the $c_{\varphi q}^{(3)}$-$c_{\varphi q}^{(1)}$ mixed terms, since
they are just a linear combination of 1D fits.

1-lepton channel, resolved, HL-LHC
---
$p_{T}^{h}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-175]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}2.9\times 10^{4}\,+(8.6\pm 4.7)\times 10^{5}\,c_{\varphi q}^{(3)}+8.8\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $(8.0\pm 0.7)\times 10^{5}$
$[175-250]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}780\,+(7.3\pm 0.2)\times 10^{4}\,c_{\varphi q}^{(3)}+1.8\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $4500\pm 300$
$[250-\infty]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}49\,+74005\,c_{\varphi q}^{(3)}+3.0\times 10^{5}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $180\pm 30$

Table 7: Number of expected signal events as a function of the Wilson
coefficients (in units of TeV-2) and of total background events in the
$Wh\rightarrow\nu\ell b\bar{b}$ channel, resolved category, at HL-LHC based on
the ML analysis. The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$. Note that as opposed to
the corresponding tables 17 \- 22, the functions given for the signal do not
include the $c_{\varphi q}^{(3)}$-$c_{\varphi q}^{(1)}$ mixed terms, since
they are just a linear combination of 1D fits.

1-lepton channel, boosted, HL-LHC
---
$p_{T}^{h}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-175]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}247\,+(2.2\pm 0.7)\times 10^{4}\,c_{\varphi q}^{(3)}+5.5\times 10^{5}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $(6.0\pm 1.3)\times 10^{4}$
$[175-250]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}885\,+1.0\times 10^{5}\,c_{\varphi q}^{(3)}+3.0\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $1.1\times 10^{4}$
$[250-300]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}305\,+4.6\times 10^{4}\,c_{\varphi q}^{(3)}+1.8\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $1500\pm 100$
$[300-\infty]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}82\,+1.9\times 10^{4}\,c_{\varphi q}^{(3)}+1.3\times 10^{6}\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $390\pm 40$

Table 8: Number of expected signal events as a function of the Wilson
coefficients (in units of TeV-2) and of total background events in the
$Wh\rightarrow\nu\ell b\bar{b}$ channel, boosted category, at HL-LHC based on
the ML analysis. The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$. Note that as opposed to
the corresponding tables 17 \- 22, the functions given for the signal do not
include the $c_{\varphi q}^{(3)}$-$c_{\varphi q}^{(1)}$ mixed terms, since
they are just a linear combination of 1D fits.

2-lepton channel, resolved, HL-LHC
---
$p_{T,\mathrm{min}}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-160]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}86\,&+6500\,c_{\varphi q}^{(3)}+(260\pm 50)\,c_{\varphi q}^{(1)}+(1400\pm 200)\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-(530\pm 60)\,c_{\varphi d}+1.5\times 10^{5}\,\left(c_{\varphi q}^{(3)}\right)^{2}+1.4\times 10^{5}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+(8.3\pm 0.9)\times 10^{4}\,\left(c_{\varphi u}\right)^{2}+7.3\times 10^{4}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $510\pm 50$
$[250-\infty]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}49\,&+5100\,c_{\varphi q}^{(3)}-(34\pm 16)\,c_{\varphi q}^{(1)}+(1100\pm 100)\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-(400\pm 90)\,c_{\varphi d}+1.6\times 10^{3}\,\left(c_{\varphi q}^{(3)}\right)^{2}+(1.3\pm 0.2)\times 10^{5}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+(9.0\pm 0.6)\times 10^{4}\,\left(c_{\varphi u}\right)^{2}+6.9\times 10^{4}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $330\pm 40$

Table 9: Number of expected signal events as a function of the Wilson
coefficients (in units of TeV-2) and of total background events in the
$Zh\rightarrow\ell^{+}\ell^{-}b\bar{b}$ channel, resolved category, at HL-LHC
based on the ML analysis. The Monte-Carlo errors on the fitted coefficients,
when not explicitly specified, are $\lesssim\mathrm{few}\,\%$. Note that as
opposed to the corresponding tables 17 \- 22, the functions given for the
signal do not include the $c_{\varphi q}^{(3)}$-$c_{\varphi q}^{(1)}$ mixed
terms, since they are just a linear combination of 1D fits.

2-lepton channel, boosted, HL-LHC
---
$p_{T,\mathrm{min}}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[250-\infty]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}88\,&+1.5\times 10^{4}\,c_{\varphi q}^{(3)}-(1.4\pm 0.3)\times 10^{3}\,c_{\varphi q}^{(1)}+3700\,c_{\varphi u}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&-(1.21\pm 0.12)\times 10^{3}\,c_{\varphi d}+8.0\times 10^{5}\,\left(c_{\varphi q}^{(3)}\right)^{2}+9.2\times 10^{5}\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-9.00002pt]{0.0pt}{9.00002pt}&+4.9\times 10^{5}\,\left(c_{\varphi u}\right)^{2}+3.3\times 10^{6}\,\left(c_{\varphi d}\right)^{2}\end{aligned}$ | $300\pm 30$

Table 10: Number of expected signal events as a function of the Wilson
coefficients (in units of TeV-2) and of total background events in the
$Zh\rightarrow\ell^{+}\ell^{-}b\bar{b}$ channel, boosted category, at HL-LHC
based on the ML analysis. The Monte-Carlo errors on the fitted coefficients,
when not explicitly specified, are $\lesssim\mathrm{few}\,\%$. Note that as
opposed to the corresponding tables 17 \- 22, the functions given for the
signal do not include the $c_{\varphi q}^{(3)}$-$c_{\varphi q}^{(1)}$ mixed
terms, since they are just a linear combination of 1D fits.

##  Appendix B Tables for reference from our companion study

In this appendix, we list a collection of tables from our companion paper,
ref. Bishara:2022vsc0 , for reference. Tables 11-16 display the results of the
cut-flow analyses for the different categories. In tables 17-22, we show the
expected numbers of signal- and background events at the HL-LHC based on the
cut-and-count analysis.

Cuts / Eff. | $Zh$ | $Wh$ | $Wb\bar{b}$ | $Zb\bar{b}$ | $t\bar{t}$
---|---|---|---|---|---
$0$ $\ell^{\pm}$ | $1$ | $0.32$ | $0.34$ | $0.78$ | $0.98$
0 UT jets | $0.37$ | $0.036$ | $0.02$ | $0.12$ | $0.011$
1 MDT DBT jet | $0.29$ | $0.026$ | $0.014$ | $0.048$ | $0.0018$
$\eta_{\max}^{H_{\mathrm{cand}}}$ | $0.26$ | $0.022$ | $0.012$ | $0.044$ | $0.0016$
$\Delta\phi(E_{T}^{\mathrm{miss}},H_{\mathrm{cand}})$ | $0.26$ | $0.022$ | $0.012$ | $0.044$ | $0.0016$
$E_{T}^{\mathrm{miss}}$ | $0.12$ | $0.007$ | $0.003$ | $0.013$ | $0.0005$
$m_{H_{\mathrm{cand}}}$ | $0.12$ | $0.007$ | $0.0008$ | $0.003$ | $4\cdot 10^{-5}$
Table 11: Cut-flow for the boosted events in the 0-lepton category at HL-LHC taken from ref. Bishara:2022vsc0 . The acceptance regions for charged leptons and jets at the different colliders are defined in the text therein. UT, MDT and DBT stand for untagged, mass-drop-tagged and doubly-b-tagged respectively. Cuts / Eff. | $Zh$ | $Wh$ | $Wb\bar{b}$ | $Zb\bar{b}$ | $t\bar{t}$
---|---|---|---|---|---
$0$ $\ell^{\pm}$ | $1$ | $0.32$ | $0.34$ | $0.78$ | $0.98$
0 UT jets | $0.37$ | $0.036$ | $0.020$ | $0.12$ | $0.011$
2 res. $b$-jets | $0.028$ | $0.0027$ | $0.0016$ | $0.015$ | $6\cdot 10^{-5}$
$\Delta R_{bb}$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$H_{T}$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$p_{T,\min}^{b,\mathrm{leading}}$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$\Delta\phi(E_{T}^{\mathrm{miss}},H_{\mathrm{cand}})$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$\Delta\phi(b_{1},b_{2})$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$\Delta\phi(E_{T}^{\mathrm{miss}},b\mathrm{-jets})$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$E_{T}^{\mathrm{miss}}$ | $0.027$ | $0.0024$ | $0.0006$ | $0.0035$ | $1\cdot 10^{-5}$
$m_{H_{\mathrm{cand}}}$ | $0.027$ | $0.0024$ | $3\cdot 10^{-5}$ | $10^{-4}$ | $<10^{-5}$
Table 12: Cut-flow for the resolved events in the 0-lepton category at the HL-LHC taken from ref. Bishara:2022vsc0 . UT stands for untagged.  Cuts / Eff. | $Wh$ | $Wb\bar{b}$ | $t\bar{t}$
---|---|---|---
$1$ $\ell^{\pm}$ | $0.66$ | $0.59$ | $0.88$
0 UT jets | $0.25$ | $0.075$ | $0.021$
1 MDT DBT jet | $0.18$ | $0.051$ | $0.010$
$E_{T}^{\mathrm{miss}}$ | $0.16$ | $0.043$ | $0.0097$
$\eta_{\max}^{H_{\mathrm{cand}}}$ | $0.14$ | $0.038$ | $0.0089$
$\Delta y(W,H_{\mathrm{cand}})_{\mathrm{max}}$ | $0.13$ | $0.030$ | $0.0072$
$m_{H_{\mathrm{cand}}}$ | $0.13$ | $0.007$ | $0.0005$
Table 13: Cut-flow for the boosted events in the 1-lepton category at the HL-LHC taken from ref. Bishara:2022vsc0 . UT, MDT and DBT stand for untagged, mass-drop-tagged and doubly-b-tagged respectively. Cuts / Eff. | $Wh$ | $Wb\bar{b}$ | $t\bar{t}$
---|---|---|---
$1$ $\ell^{\pm}$ | $0.66$ | $0.59$ | $0.88$
0 UT jets | $0.25$ | $0.075$ | $0.021$
2 res. b-jets | $0.025$ | $0.006$ | $0.002$
$\Delta R_{bb}$ | $0.025$ | $0.004$ | $0.0017$
$E_{T}^{\mathrm{miss}}$ | $0.024$ | $0.003$ | $0.0016$
$p_{T,\min}^{b,\mathrm{leading}}$ | $0.024$ | $0.003$ | $0.0016$
$m_{H_{\mathrm{cand}}}$ | $0.024$ | $7\cdot 10^{-5}$ | $<5\cdot 10^{-6}$
Table 14: Cut-flow for the resolved events in the 1-lepton category at the HL-LHC taken from ref. Bishara:2022vsc0 . UT stands for untagged. Cuts / Eff. | $Zh$ | $Zb\bar{b}$
---|---|---
$2$ $\ell^{\pm}$ | $0.48$ | $0.71$
1 MDT DBT jet | $0.21$ | $0.18$
Leptons | $0.21$ | $0.18$
$\eta_{\max}^{H_{\mathrm{cand}}}$ | $0.19$ | $0.17$
$\Delta y(Z,H_{\mathrm{cand}})_{\mathrm{max}}$ | $0.16$ | $0.10$
$m_{\ell\ell}$ | $0.15$ | $0.10$
max. $p_{T}^{\ell}$ imbalance | $0.14$ | $0.078$
$p_{T,\mathrm{min}}^{Z,E_{T}^{\mathrm{miss}}}$ | $0.14$ | $0.078$
$p_{T,\mathrm{min}}^{Z}$ | $0.14$ | $0.074$
$m_{H_{\mathrm{cand}}}$ | $0.14$ | $0.020$
Table 15: Cut-flow for the boosted events in the 2-lepton category at the HL-LHC taken from ref. Bishara:2022vsc0 . MDT and DBT stand for mass-drop-tagged and doubly-b-tagged, respectively. Cuts / Eff. | $Zh$ | $Zb\bar{b}$
---|---|---
$2$ $\ell^{\pm}$ | $0.48$ | $0.71$
2 res. b-jets | $0.061$ | $0.057$
$\Delta R_{bb}$ | $0.052$ | $0.014$
0 UT jets | $0.033$ | $0.0051$
Leptons | $0.033$ | $0.0051$
$p_{T,\min}^{b,\mathrm{leading}}$ | $0.033$ | $0.0051$
$m_{\ell\ell}$ | $0.031$ | $0.0047$
$m_{H_{\mathrm{cand}}}$ | $0.031$ | $0.0002$
Table 16: Cut-flow for the resolved events in the 2-lepton category at the HL-
LHC taken from ref. Bishara:2022vsc0 . UT stands untagged.

0-lepton channel, resolved, HL-LHC
---
$p_{T,\mathrm{min}}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-160]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}300\,&+1120\,c_{\varphi q}^{(3)}-(39\pm 47)\,c_{\varphi q}^{(1)}+(155\pm 39)\,c_{\varphi u}-(80\pm 30)\,c_{\varphi d}+1250\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-7.00002pt]{0.0pt}{7.00002pt}&+1010\,\left(c_{\varphi q}^{(1)}\right)^{2}+(550\pm 75)\,\left(c_{\varphi u}\right)^{2}+(400\pm 47)\,\left(c_{\varphi d}\right)^{2}-(190\pm 250)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $8200\pm 3700$
$[160-200]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}708\,&+3230\,c_{\varphi q}^{(3)}-(160\pm 60)\,c_{\varphi q}^{(1)}+(596\pm 49)\,c_{\varphi u}-(263\pm 39)\,c_{\varphi d}+4460\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-7.00002pt]{0.0pt}{7.00002pt}&+3340\,\left(c_{\varphi q}^{(1)}\right)^{2}+1920\,\left(c_{\varphi u}\right)^{2}+1510\,\left(c_{\varphi d}\right)^{2}-(1400\pm 340)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $5500\pm 1400$
$[200-250]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}195\,&+1160\,c_{\varphi q}^{(3)}-(55\pm 22)\,c_{\varphi q}^{(1)}+(223\pm 15)\,c_{\varphi u}-(89\pm 13)\,c_{\varphi d}+2075\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-7.00002pt]{0.0pt}{7.00002pt}&+1750\,\left(c_{\varphi q}^{(1)}\right)^{2}+955\,\left(c_{\varphi u}\right)^{2}+698\,\left(c_{\varphi d}\right)^{2}-(430\pm 150)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $1340\pm 90$
$[250-\infty]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}33\,&+312\,c_{\varphi q}^{(3)}-(32\pm 10)\,c_{\varphi q}^{(1)}+(66\pm 7)\,c_{\varphi u}-(26\pm 6)\,c_{\varphi d}+1020\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-7.00002pt]{0.0pt}{7.00002pt}&+907\,\left(c_{\varphi q}^{(1)}\right)^{2}+517\,\left(c_{\varphi u}\right)^{2}+351\,\left(c_{\varphi d}\right)^{2}-(360\pm 85)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $265\pm 37$

Table 17: Expected number of signal events as a function of the Wilson
coefficients (in units of TeV-2) and total background events in the resolved
0-lepton category, at the HL-LHC based on the cut-and-count analysis in ref.
Bishara:2022vsc0 . The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$.

0-lepton channel, boosted, HL-LHC
---
$p_{T,\mathrm{min}}$ bin $[$GeV$]$ | Number of expected events
Signal | Background
$[0-300]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}118\,&+1175\,c_{\varphi q}^{(3)}-(17\pm 20)\,c_{\varphi q}^{(1)}+248\,c_{\varphi u}-(123\pm 13)\,c_{\varphi d}+3479\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ &+3064\,\left(c_{\varphi q}^{(1)}\right)^{2}\rule[-7.00002pt]{0.0pt}{7.00002pt}+1724\,\left(c_{\varphi u}\right)^{2}+1300\,\left(c_{\varphi d}\right)^{2}-(1190\pm 171)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $492\pm 50$
$[300-350]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}117\,&+1423\,c_{\varphi q}^{(3)}-(123\pm 21)\,c_{\varphi q}^{(1)}+272\,c_{\varphi u}-(77\pm 13)\,c_{\varphi d}+5222\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-7.00002pt]{0.0pt}{7.00002pt}&+4643\,\left(c_{\varphi q}^{(1)}\right)^{2}+2670\,\left(c_{\varphi u}\right)^{2}+1810\,\left(c_{\varphi d}\right)^{2}-(1670\pm 191)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $492\pm 43$
$[350-\infty]$ | $\begin{aligned} \phantom{\frac{C^{Q}}{1}}111\,&+2115\,c_{\varphi q}^{(3)}-(217\pm 15)\,c_{\varphi q}^{(1)}+489\,c_{\varphi u}-(162\pm 9)\,c_{\varphi d}+12820\,\left(c_{\varphi q}^{(3)}\right)^{2}\\\ \rule[-7.00002pt]{0.0pt}{7.00002pt}&+11790\,\left(c_{\varphi q}^{(1)}\right)^{2}+7060\,\left(c_{\varphi u}\right)^{2}+4650\,\left(c_{\varphi d}\right)^{2}-4700\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $243\pm 16$

Table 18: Expected number of signal events as a function of the Wilson
coefficients (in units of TeV-2) and total background events in the boosted
0-lepton category, at the HL-LHC based on the cut-and-count analysis in ref.
Bishara:2022vsc0 . The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$.  1-lepton channel,
resolved, HL-LHC
---
$p_{T}^{h}$ bin [GeV] | Number of expected events
Signal | Background
$[0-175]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}5100\,+14900\,c_{\varphi q}^{(3)}+12800\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $144000\pm 9800$
$[175-250]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}780\,+4400\,c_{\varphi q}^{(3)}+6600\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $6550$
$[250-\infty]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}41\,+380\,c_{\varphi q}^{(3)}+950\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $203\pm 35$
Table 19: Expected number of signal events as a function of the Wilson
coefficients (in units of TeV-2) and total background events in the resolved
1-lepton category, at the HL-LHC based on the cut-and-count analysis in ref.
Bishara:2022vsc0 . The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$.  1-lepton channel,
boosted, HL-LHC
---
$p_{T}^{h}$ bin [GeV] | Number of expected events
Signal | Background
$[0-175]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}(26\pm 6)\,+(154\pm 19)\,c_{\varphi q}^{(3)}+(221\pm 20)\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $1630$
$[175-250]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}560\,+3770\,c_{\varphi q}^{(3)}+6650\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $5690$
$[250-300]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}214\,+1920\,c_{\varphi q}^{(3)}+4530\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $1046$
$[300-\infty]$ | $\rule[-7.65005pt]{0.0pt}{21.15009pt}(79\pm 5)\,+1150\,c_{\varphi q}^{(3)}+4700\,\left(c_{\varphi q}^{(3)}\right)^{2}$ | $425\pm 25$
Table 20: Expected number of signal events as a function of the Wilson
coefficients (in units of TeV-2) and total background events in the boosted
1-lepton category, at the HL-LHC based on the cut-and-count analysis in ref.
Bishara:2022vsc0 . The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$.  2-lepton channel,
resolved, HL-LHC
---
$p_{T,\mathrm{min}}$ bin [GeV] | Number of expected events
Signal | Background
$[175-200]$ | $\begin{aligned} \rule{0.0pt}{11.25002pt}57\,&+277\,c_{\varphi q}^{(3)}-(3\pm 7)\,c_{\varphi q}^{(1)}+(73\pm 5)\,c_{\varphi u}-(19\pm 4)\,c_{\varphi d}\\\ &+402\,\left(c_{\varphi q}^{(3)}\right)^{2}+403\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-5.84995pt]{0.0pt}{9.00002pt}&+238\,\left(c_{\varphi u}\right)^{2}+172\,\left(c_{\varphi d}\right)^{2}-(141\pm 47)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $361\pm 21$
$[200-\infty]$ | $\begin{aligned} \rule{0.0pt}{11.25002pt}48\,&+299\,c_{\varphi q}^{(3)}-(5\pm 6)\,c_{\varphi q}^{(1)}+(65\pm 5)\,c_{\varphi u}-(25\pm 4)\,c_{\varphi d}\\\ &+580\,\left(c_{\varphi q}^{(3)}\right)^{2}+560\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-5.84995pt]{0.0pt}{9.00002pt}&+324\,\left(c_{\varphi u}\right)^{2}+256\,\left(c_{\varphi d}\right)^{2}-(110\pm 49)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $296\pm 19$
Table 21:  Expected number of signal events as a function of the Wilson
coefficients (in units of TeV-2) and total background events in the resolved
2-lepton category, at the HL-LHC based on the cut-and-count analysis in ref.
Bishara:2022vsc0 . The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$. 2-lepton channel,
boosted, HL-LHC
---
$p_{T,\mathrm{min}}$ bin [GeV] | Number of expected events
Signal | Background
$[250-\infty]$ | $\begin{aligned} \rule{0.0pt}{11.25002pt}103\,&+974\,c_{\varphi q}^{(3)}-(53\pm 11)\,c_{\varphi q}^{(1)}+231\,c_{\varphi u}-(79\pm 7)\,c_{\varphi d}\\\ &+2800\,\left(c_{\varphi q}^{(3)}\right)^{2}+2850\,\left(c_{\varphi q}^{(1)}\right)^{2}\\\ \rule[-5.84995pt]{0.0pt}{9.00002pt}&+1660\,\left(c_{\varphi u}\right)^{2}+1150\,\left(c_{\varphi d}\right)^{2}-(1070\pm 93)\,c_{\varphi q}^{(3)}\,c_{\varphi q}^{(1)}\end{aligned}$ | $370\pm 21$
Table 22:  Expected number of signal events as a function of the Wilson
coefficients (in units of TeV-2) and total background events in the boosted
2-lepton category, at the HL-LHC based on the cut-and-count analysis in ref.
Bishara:2022vsc0 . The Monte-Carlo errors on the fitted coefficients, when not
explicitly specified, are $\lesssim\mathrm{few}\,\%$.

## References

  * (1) D. Guest, K. Cranmer and D. Whiteson, _Deep Learning and its Application to LHC Physics_ , _Ann. Rev. Nucl. Part. Sci._ 68 (2018) 161 [1806.11484].
  * (2) D. Bourilkov, _Machine and Deep Learning Applications in Particle Physics_ , _Int. J. Mod. Phys. A_ 34 (2020) 1930019 [1912.08245].
  * (3) M. D. Schwartz, _Modern Machine Learning and Particle Physics_ , 2103.12226.
  * (4) G. Karagiorgi, G. Kasieczka, S. Kravitz, B. Nachman and D. Shih, _Machine Learning in the Search for New Fundamental Physics_ , 2112.03769.
  * (5) M. Feickert and B. Nachman, _A Living Review of Machine Learning for Particle Physics_ , 2102.02770.
  * (6) F. Bishara, P. Englert, C. Grojean, G. Panico and A. N. Rossia, _Revisiting $\mathbf{Vh(\rightarrow b\bar{b})}$ at the LHC and FCC-hh_, 2208.11134.
  * (7) M. Farina, G. Panico, D. Pappadopulo, J. T. Ruderman, R. Torre and A. Wulzer, _Energy helps accuracy: electroweak precision tests at hadron colliders_ , _Phys. Lett._ B772 (2017) 210 [1609.08157].
  * (8) J. de Blas, M. Chala and J. Santiago, _Global Constraints on Lepton-Quark Contact Interactions_ , _Phys. Rev._ D88 (2013) 095011 [1307.5068].
  * (9) R. Franceschini, G. Panico, A. Pomarol, F. Riva and A. Wulzer, _Electroweak Precision Tests in High-Energy Diboson Processes_ , _JHEP_ 02 (2018) 111 [1712.01310].
  * (10) F. Bishara, P. Englert, C. Grojean, M. Montull, G. Panico and A. N. Rossia, _A New Precision Process at FCC-hh: the diphoton leptonic Wh channel_ , _JHEP_ 07 (2020) 075 [2004.06122].
  * (11) F. Bishara, S. De Curtis, L. Delle Rose, P. Englert, C. Grojean, M. Montull et al., _Precision from the diphoton Zh channel at FCC-hh_ , _JHEP_ 04 (2021) 154 [2011.13941].
  * (12) S. Banerjee, C. Englert, R. S. Gupta and M. Spannowsky, _Probing Electroweak Precision Physics via boosted Higgs-strahlung at the LHC_ , _Phys. Rev. D_ 98 (2018) 095012 [1807.01796].
  * (13) D. Liu and L.-T. Wang, _Prospects for precision measurement of diboson processes in the semileptonic decay channel in future LHC runs_ , _Phys. Rev. D_ 99 (2019) 055001 [1804.08688].
  * (14) S. Banerjee, R. S. Gupta, J. Y. Reiness and M. Spannowsky, _Resolving the tensor structure of the Higgs coupling to $Z$-bosons via Higgs-strahlung_, _Phys. Rev._ D100 (2019) 115004 [1905.02728].
  * (15) S. Banerjee, R. S. Gupta, J. Y. Reiness, S. Seth and M. Spannowsky, _Towards the ultimate differential SMEFT analysis_ , _JHEP_ 09 (2020) 170 [1912.07628].
  * (16) A. Banerjee, S. Dasgupta and T. S. Ray, _Probing composite Higgs boson substructure at the HL-LHC_ , _Phys. Rev. D_ 104 (2021) 095021 [2105.01093].
  * (17) ATLAS collaboration, _Measurements of $WH$ and $ZH$ production in the $H\rightarrow b\bar{b}$ decay channel in $pp$ collisions at 13 TeV with the ATLAS detector_, _Eur. Phys. J. C_ 81 (2021) 178 [2007.02873].
  * (18) ATLAS collaboration, _Measurement of the associated production of a Higgs boson decaying into $b$-quarks with a vector boson at high transverse momentum in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector_, _Phys. Lett. B_ 816 (2021) 136204 [2008.02508].
  * (19) C. Grojean, A. Paul, Z. Qian and I. Strümke, _Lessons on interpretable machine learning from particle physics_ , _Nature Rev. Phys._ 4 (2022) 284 [2203.08021].
  * (20) B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, _Dimension-Six Terms in the Standard Model Lagrangian_ , _JHEP_ 10 (2010) 085 [1008.4884].
  * (21) F. Bishara and M. Montull, _(Machine) Learning amplitudes for faster event generation_ , 1912.11055.
  * (22) J. M. Butterworth, A. R. Davison, M. Rubin and G. P. Salam, _Jet substructure as a new Higgs search channel at the LHC_ , _Phys. Rev. Lett._ 100 (2008) 242001 [0802.2470].
  * (23) M. Gouzevitch, A. Oliveira, J. Rojo, R. Rosenfeld, G. P. Salam and V. Sanz, _Scale-invariant resonance tagging in multijet events and new physics in Higgs pair production_ , _JHEP_ 07 (2013) 148 [1303.6636].
  * (24) F. Bishara, R. Contino and J. Rojo, _Higgs pair production in vector-boson fusion at the LHC and beyond_ , _Eur. Phys. J. C_ 77 (2017) 481 [1611.03860].
  * (25) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079 [1405.0301].
  * (26) NNPDF collaboration, _Parton distributions with QED corrections_ , _Nucl. Phys. B_ 877 (2013) 290 [1308.0598].
  * (27) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten et al., _An introduction to PYTHIA 8.2_ , _Comput. Phys. Commun._ 191 (2015) 159 [1410.3012].
  * (28) C. Degrande, G. Durieux, F. Maltoni, K. Mimasu, E. Vryonidou and C. Zhang, _Automated one-loop computations in the SMEFT_ , 2008.11743.
  * (29) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, _UFO - The Universal FeynRules Output_ , _Comput. Phys. Commun._ 183 (2012) 1201 [1108.2040].
  * (30) S. M. Lundberg and S.-I. Lee, _A unified approach to interpreting model predictions_ , in _Advances in Neural Information Processing Systems 30_ , I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan et al., eds., pp. 4765–4774, Curran Associates, Inc., (2017), http://papers.nips.cc/paper/7062-a-unified-approach-to-interpreting-model-predictions.pdf.
  * (31) L. S. Shapley, _Notes on the n-Person Game-II: The Value of an n-Person Game_ , _Rand Corporation_ (1951) .
  * (32) S. Lipovetsky and M. Conklin, _Analysis of regression in game theory approach_ , _Applied Stochastic Models in Business and Industry_ 17 (2001) 319 .
  * (33) E. Štrumbelj and I. Kononenko, _Explaining prediction models and individual predictions with feature contributions_ , _Knowledge and Information Systems_ 41 (2013) 647.
  * (34) S. M. Lundberg, G. Erion, H. Chen, A. DeGrave, J. M. Prutkin, B. Nair et al., _From local explanations to global understanding with explainable AI for trees_ , _Nature Machine Intelligence_ 2 (2020) 56.
  * (35) S. M. Lundberg, G. G. Erion and S.-I. Lee, _Consistent Individualized Feature Attribution for Tree Ensembles_ , _arXiv e-prints_ (2018) [1802.03888].
  * (36) P. Mehta, M. Bukov, C.-H. Wang, A. G. R. Day, C. Richardson, C. K. Fisher et al., _A high-bias, low-variance introduction to Machine Learning for physicists_ , _Phys. Rept._ 810 (2019) 1 [1803.08823].