List of Figures

1 The timeline and evolution of forces in the early universe.
1.1 Particles and their classifications in the SM, reproduced from Ref. [1].
3.1 Feynman diagrams for meson decay (left) and nucleon-antinucleon annihilation (right).
3.2 The two lowest order nucleon-antinucleon scattering diagrams.
3.3 Cross section for

e^{+} e^{-} \to

hadron scattering as a function of

\sqrt{s}

with a clear resonance at the

Z

boson mass, reproduced from Ref. [2].
3.4 An example of a higher-order scattering diagram with a “loop”.
3.5 Feynman rules unique to non-abelian Yang-Mills theories, reproduced from Ref. [3].
3.6 The running of the inverse strength of the SM coupling constants, with the strong coupling constant (

SU (3)

) in green, weak (

SU (2)

) in red, and electromagnetic (

U (1)

) in black, reproduced from Ref. [4].
3.7 The “sombrero” potential for the Higgs field, reproduced from Ref. [5]. An initial state and a ground state breaking the

U (1)

symmetry are represented by the green balls at the top and bottom of the potential, respectively.
4.1 A graphical summary of the SM, reproduced from Ref. [6].
4.2 The quarks in the SM, reproduced from Ref. [1].
4.3 The theoretically predicted running of the strong coupling

α_{s} = \frac{g_{s}^{2}}{4 π}

as a function of the energy scale along with experimental measurements, reproduced from Ref. [7].
4.4 “Flux tubes” between a quark and anti-pair inside a meson (left) and three quarks in a proton (right), reproduced from Refs. [8, 9].
4.5 A table of what were considered to be elementary particles in 1964, reproduced from Ref. [10].
4.6 Baryons in the octet (left) and decuplet (right) representations of

SU (3)

, reproduced from Ref. [11].
4.7 Feynman diagram for deep inelastic scattering, reproduced from Ref. [12] (left) and an illustrative example of proton-proton collisions reproduced from Ref. [13] (right).
4.8 PDFs for the proton at

Q^{2} = 10 GeV

, reproduced from Ref. [14].
4.9 LO, NLO, and NNLO predictions and uncertainties for

pp

to Z boson production, differential in rapidity

Y

at the LHC, reproduced from Ref. [15].
4.10 The splitting functions for quarks and gluons, reproduced from Ref. [16].
4.11 PDF measurements at different energy scales

Q^{2}

and momentum fraction

x

by the H1 collaboration in DIS experiments, reproduced from Ref. [17].
4.12 A cartoon of a jet, reproduced from Ref. [18].
4.13 An example of real jets in an event collected by CMS and identified in the search described in Chapter 14 [19, 20]. An interactive version of this event display is available at https://cms3d.web.cern.ch/HIG-23-012/.
4.14 An illustration of double-counting when combining matrix element predictions (in black) with parton showering algorithms (in red) for

Z +

parton and

Z +

2-parton events, reproduced from Ref. [21].
4.15 An illustration of the Lund string model of hadronization, reproduced from Ref. [22].
4.16 Feynman diagrams for beta decay (left) and muon decay (right) in Fermi’s theory.
4.17 Single Higgs boson production modes and decay channels at the LHC, reproduced from Ref. [23].
4.18 Higgs branching fractions predicted in the SM as a function of

m_{H}

(reproduced from Refs. [24, 25]).
4.19 Constraints on Higgs to fermion and vector boson couplings, reproduced from Ref. [23] (left) and measurements of the Higgs mass, reproduced from Ref. [26] (right) by the CMS experiment.
4.20 Cartoon of the Higgs potential in the SM and potential deviations due to BSM physics.
4.21 Leading-order diagrams for nonresonant

H H

production via gluon gluon fusion.
4.22 Leading-order diagrams for nonresonant

H H

production via vector boson fusion. In this chapter, we refer to the left-most VBF diagram as the

{(H V V)}^{2}

and the right-most as the

H H V V

diagram.
4.23 Differential cross section at

13 TeV

center of mass for VBF

H H

production as a function of the invariant mass of the

H H

system (

m_{H H}

) for different diagrams and couplings.
4.24

H H

decays and their respective branching fractions (reproduced from Ref. [27]).
4.25 Distribution of events in the high-

m_{H H}

ggF category of the Run 2 CMS

H H \to b \bar{b} b \bar{b}

resolved analysis [28].
4.26 Combination of bins of all postfit distributions of the Run 2 CMS

H H \to b \bar{b} ττ

analysis [29], ordered according to the expected signal-to-square-root-background ratio, separately for the

τ_{h} τ_{e}

(left), the

τ_{h} τ_{μ}

(center), and

τ_{h} τ_{h}

(right) channels.
4.27 Invariant two-photon mass distribution of the Run 2 CMS

H H \to b \bar{b} γγ

analysis [30].
4.28 Distribution of events in the resolved

1 b

, resolved

\geq 2 b

, and boosted signal categories of the Run 2 CMS semi-leptonic

H H \to b \bar{b} W W

analysis [31].
4.29 The expected and observed limits on the ratio of experimentally estimated production cross section and the expectation from the SM in searches using different final states and their combination, reproduced from Ref. [32].
4.30 The expected and observed limits on the ratio of experimentally estimated production cross section and the theory expectation.
4.31

X \to H Y

production in the symmetric (left) and asymmetric (right) cases.
4.32 The fraction of

Y \to V V

jets containing three or four generator-level quarks for the resonant

X \to (H \to b \bar{b}) (Y \to V V)

r

z

plane, reproduced from Ref. [43]. In green are the pixel detector layers, in red the single-sided strip modules, and in blue the double-sided strip modules.
6.8 Total thickness

t

of the tracker material traversed by a particle produced at the nominal interaction point.
6.9 Layout of the CMS ECAL, reproduced from Ref. [44], with one of the 36 barrel regions highlighted in yellow, the preshower in pink, and the endcap regions in green.
6.10 PbWO

_{4}

crystals and photodiodes used in the CMS ECAL.
6.11 Layout of the CMS detector in the

r

z

plane with the four HCAL sections labeled, reproduced from Ref. [45].
6.12 Simulation of the distribution of measured

∕

incident energy for pions with incident energies of

200 GeV

η = 0

, reproduced from Ref. [46].
6.13 Layout of the CMS detector in the

r

z

plane, with the muon system highlighted and the steel return yoke in dark grey, reproduced from Ref. [47].
6.14 Response and resolution of single neutral hadron energies in the barrel as a function of the true energy, before and after calibration, reproduced from Ref. [48].
6.15 Layout of the CMS HGCAL, reproduced from Ref. [49].
6.16 (Left) layers of the individual HGCAL modules and (right) their layout in the all-silicon and mixed layers, reproduced from Ref. [49].
7.1 A “nomological net” of ML applications in HEP, reproduced from Ref. [50].
7.2 Illustration of gradient descent in a 2D parameter space of

(𝜃_{0}, 𝜃_{1})

.
7.3 (Left) a single perceptron and (right) a neural network built using multiple layers of perceptrons (MLPs).
7.4 Schematic of a convolutional neural network, reproduced from Ref. [51].
7.5 Schematic of a message passing graph neural network.
7.6 Schematic of set self-attention.
7.7 Examples of a jet (left) and calorimeter shower (right) represented as 2D and 3D images, respectively.
7.8 Schematic of a steerable CNN, reproduced from Ref. [52].
7.9 Schematic of the

E (3)

-equivariant neural network architecture used for predicting phonon density of states, reproduced from Ref. [53].
7.10 Schematic of a Lorentz group-equivariant network layer, reproduced from Ref. [54].
7.11 Diagram of an image autoencoder, reproduced from Ref. [55].
7.12 Summary of popular generative models, reproduced from Ref. [56].
7.13 Sample point clouds from the ShapeNet dataset, reproduced from Ref. [57].
8.1 Sample 2D Poisson distributions.
8.2 The profile likelihood ratio

λ (s)

(left) and the

t_{s}

test statistic (right) for our one-bin Poisson model.
8.3 The profile likelihood ratio

λ (s)

(left) and the

t_{s}

test statistic (right) with

b = m

, demonstrating the effect of decreasing uncertainties on our nuisance parameters.
8.4 Comparing the nominal vs alternative test statistic.
8.5 Estimating

p ({\tilde{t}}_{s} | s)

through toys.
8.6 Asymptotic form of

p ({\tilde{t}}_{s} | s)

.
8.7 Testing

H_{s}

in Example 8.2.1.
8.8 Relationship between significance

Z

and the

p

-value, reproduced from Ref. [58].
8.9 Testing the background-only hypothesis in Example 8.2.1.
8.10 Demonstration of the Neyman construction for a 95% confidence interval for the experiment in Example 8.2.1 (

n_{obs} = 20, m_{obs} = 5

).
8.11 Comparing

{\tilde{t}}_{s}

and

{\tilde{q}}_{s}

.
8.12 Comparing

p ({\tilde{t}}_{s} | s)

and

p ({\tilde{q}}_{s} | s)

p ({\tilde{q}}_{s} | s)

asymptotically follows a half-

χ^{2}

distribution (green).
8.13 Extending the Neyman construction to an upper limit on

s

.
8.14 Demonstration of CL

_{s}

criterion for Examples 8.2.1 (left) and 8.2.2 (right).
8.15 Left: Distributions of

{\tilde{t}}_{0}

under the background-only and background + signal hypotheses using 30,000 toys each.
8.16 Gaussian quantiles, reproduced from Ref. [59].
8.17 Calculating the median expected

p_{μ = 1}^{'}

-value with respect to the signal + background hypothesis, for test statistics

{\tilde{q}}_{μ}

sampled under the background-only hypothesis.
8.18 Left: The expected median and

\pm 1 σ, \pm 2 σ

quantiles of

p_{μ}^{'}

for different

μ

’s. The intersection of these with

p_{μ}^{'} = 0.05

(gray) corresponds to the expected exclusion limits. Right: The median and

\pm 1 σ, \pm 2 σ

expected limits at 95% CL

_{s}

μ

.
8.19 Expected and observed 95% CL

_{s}

upper limits for the SM Higgs by ATLAS in 2012, for different hypothetical Higgs masses [60].
8.20 Distribution of the MLE of

μ

for different

s

and

b

produced using 30,000 toy experiments each. (Note the x-axis range is becoming narrower from the left-most to the right-most plot.)
8.21 Gaussian fits to distributions of

\hat{μ}

for different

s

and

b

from Figure 8.20.
8.22 The Fisher information

I_{μμ} (μ, b)

for different

μ

and

s

, as a function of the expected background

b

.
8.23 Asymptotic (dotted lines) and toy-based (solid lines) distributions, using 30,000 toys each, of the MLE of

μ

for different

s

b

, and true signal strengths

μ^{'}

.
8.24 Error between the sampled toy distributions, using 50,000 toys each, and the asymptotic distributions of the MLE of

μ

for different

s

and

b

(blue), with

1 ∕ \sqrt{N}

fits in red.
8.25 Central

χ_{k}^{2}

and non-central

χ_{k}^{' 2} (Λ)

distributions for

Λ

between

1 - 30

(left) and

30 - 300

(right).
8.26 Comparing the distribution

p (t_{μ} | μ^{'})

(solid) with non-central

χ_{1}^{' 2} (Λ)

distributions (dotted) for a range of

s, b, μ, μ^{'}

values, with

σ_{\hat{μ}}^{2}

estimated using the inverse of the Fisher information matrix.
8.27 Comparing the sampling distribution

p (t_{μ} | μ^{'})

with non-central

χ_{1}^{' 2} (Λ)

distributions for a range of

s, b, μ, μ^{'}

values, with the Asimov sigma estimation for

σ_{\hat{μ}}^{2}

.
8.28 Comparing the sampling distribution

p (t_{μ} | μ^{'})

with non-central

χ_{1}^{' 2} (Λ)

distributions for different

s, b \leq 10

, showing the break-down of the

σ_{A}

approximation for

σ_{\hat{μ}}^{2}

at low statistics.
8.29 Comparing the significances, as a function of the signal strength

μ

of the hypothesis being tested, for simple counting experiments (Eq. 8.2.3) with different

s, n_{obs}, m_{obs}

’s.
9.1 The three jet classes we simulate.
10.1 Top: The MP generator uses message passing to generate a particle cloud.
10.2 Samples from our sparse MNIST dataset (far left) compared to samples from MPGAN (center left). Samples from the MNIST superpixels dataset (center right) compared to samples from MPGAN (far right).
10.3 Comparison of real and generated distributions for a subset of jet and particle features.
10.4 Random samples of discretized images in the

η^{rel}

-

ϕ^{rel}

plane, with pixel intensities equal to particle

p_{T}^{rel}

, of real and generated gluon jets (left), and an average over 10,000 such sample images (right).
10.5 Random samples of discretized images in the

η^{rel}

-

ϕ^{rel}

plane, with pixel intensities equal to particle

p_{T}^{rel}

, of real and generated light quark jets (left), and an average over 10,000 such sample images (right).
10.6 Random samples of discretized images in the

η^{rel}

-

ϕ^{rel}

plane, with pixel intensities equal to particle

p_{T}^{rel}

, of real and generated top quark jets (left), and an average over 10,000 such sample images (right).
10.7 Correlation plots between pairs of evaluation metrics, evaluated on 400 separate batches of 50,000 MPGAN generated top quark jets.
10.8 Diagram of the iGAPT generator and discriminator networks.
10.9 Illustration of an induced particle attention block (IPAB).
10.10 Low-level particle feature distributions (far left and center left) and high-level jet feature distributions (center right and far right).
11.1 Samples of (mixtures of) Gaussian distributions used for testing evaluation metrics.
11.2 Scores of each metric on samples from the true distribution for varying sample sizes.
11.3 The probability, in arbitrary units (A.U.), of the relative jet mass for truth and distorted gluon jet distributions. On the left are distribution-level distortions, and on the right particle-level.
11.4 Correlations between FPD and FPND, KPD, and

W_{1}^{M}

on 400 separate batches of 50,000 GAPT-generated jets.
13.1 Soft drop (left) and regressed (right) mass distributions for the

b \bar{b}

and

V V

-candidate AK8 jets for 2018 data and simulated samples following a loose pre-selection for boosted jets.
13.2 Full set of training jet classes for GloParT.
13.3 Receiver operating characteristic (ROC) curve for the

T_{HVV}

discriminator on

V V

-candidate jets passing the AK8 online and offline selections for a subset of nonresonant and resonant signals versus QCD and

t \bar{t}

backgrounds.
13.4 Regions of the primary Lund plane (left) and data versus MC Lund plane ratios in

W \to q \bar{q}

jets, binned in subjet

p_{T}

(right), reproduced from Refs. [61] and [62], respectively.
13.5 Distributions of the GloParT

T_{HVV}

discriminant before and after the Lund plane reweighting of top matched jets.
14.1 The boosted

H H \to b \bar{b} (V V \to 4 q)

(left) and

X \to (H \to b \bar{b}) (Y \to V V \to 4 q)

(right) processes.
14.2 Trigger efficiencies for the 2018 dataset measured in bins of the AK8 jet

p_{T}

, soft drop mass (MassSD) and

T_{Xbb}

score.
14.3 Illustration of the signal and fail nonresonant analysis region selections in terms of the

T_{Xbb}^{bb}

and BDT scores.
14.4 Post-background-only-fit distributions of the

b \bar{b}

-candidate jet regressed mass (

m_{reg}^{bb}

) in the ggF (left) and VBF (right) signal regions. The data is not shown in the Higgs mass window.
14.5 Observed and expected exclusion limits at 95% CL for the

H H \to b \bar{b} V V

signal SM cross section (top) and cross section at

κ_{2 V} = 0

(bottom).
14.6 1D upper limits scans on the inclusive HH cross section as a function of

κ_{2 V}

.
14.7 Post-background-only-fit distributions in the fully-merged category of the

V V

-candidate jet regressed mass (

m_{reg}^{VV}

).
14.8 Post-background-only-fit distributions in the fully-merged category of the dijet mass (

m^{jj}

).
14.9 Median expected exclusion limits in the fully-merged category for resonant

X \to (H \to b \bar{b}) (Y \to V V \to 4 q)

signals for different

m_{X}

and

m_{Y}

.
15.1 The JetNet logo.
16.1 Individual Lorentz group equivariant message passing (LMP) layers are shown on the left, and the LGAE architecture is built out of LMPs on the right.
16.2 Jet image reconstructions.
16.3 Particle and jet feature reconstruction by the LGAE, GNNAE, and CNNAE models.
16.4 Anomaly detection ROC curves for the LGAE, GNNAE, and CNNAE models.
16.5 The correlations between the total momentum of the imaginary components in the

τ_{(1 ∕ 2, 1 ∕ 2)} = 2

LGAE-Mix model and the target jet momenta. The Pearson correlation coefficient

r

is listed above.
16.6 Distributions of the invariant mass squared of the latent 4-vectors and jet momenta of the LGAE models.
16.7 Median magnitude of relative errors of jet mass reconstruction by LGAE and CNNAE models at trained on different fractions of the training data.
B.1 Examples of a disconnected (left) and an un-amputated (right) Feynman diagram.
B.2 The two lowest order nucleon scattering diagrams.
B.3 Tree-level Feynman diagram for meson decay via a Yukawa interaction.
C.1 Comparison of real and PCGAN-generated distributions for a subset of jet and particle features. Top: gluon jet features, Middle: light quark jets, Bottom: top quark jets.
C.2 Particle

p_{T}^{rel}

and relative jet mass distributions of real jets and those generated by MPGAN without our masking strategy.
C.3 The four alternative masking strategies which we test.
C.4 Loss curve of a training on light quark jets with masking strategy 3, typical of loss curves with all four strategies.
C.5 Low-level particle feature distributions and high-level jet feature distributions for 150-particle gluon jets.
D.1 Example real and generated jet mass distributions used to illustrate the benefit of IPMs in Appendix D.1, based on Refs. [63, 64].
D.2 Time taken per each metric on Gaussian-distributed datasets as described in Section 11.2.
D.3 Scores of each metric on Gaussian-distributed datasets as described in Section 11.2.
D.4 Scores of each metric on Gaussian-distributed datasets as described in Section 11.2.
D.5 The probability, in arbitrary units (A.U.), of the particle

p_{T}^{rel}

, a sample

d = 3

EFP, and a sample

d = 4

EFP for truth and distorted gluon jet distributions. On the left are distribution-level distortions, and on the right particle-level.
E.1 An MPNN layer in the GNNAE. Here,

EdgeNet

and

NodeNet

are feed-forward neural networks.
E.2 The relative deviations, as defined in Eq. E.3.1, of the output 4-momenta

p^{μ}

to boosts along the

z

-axis (left) and rotations around the

z

-axis (right).

⭠ ⭢