STUDY OF HIGGS BOSON PRODUCTION AT HIGH TRANSVERSE MOMENTUM IN THE B-QUARK PAIR DECAY MODE By José Gabriel Reyes Rivera A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics—Doctor of Philosophy 2023 ABSTRACT This document presents constraints on Higgs boson production at high transverse mo- √ mentum using the bb̄ channel. The study is based on data collected at s = 13 TeV with the ATLAS detector, corresponding to an integrated luminosity of 136 fb−1 . The events of interest consist of two large radius jets recoilling against each other. The Higgs boson decaying to b-quarks is identified using b-tagging techniques, exploiting the experimental signature of b-hadron decays while the other jet is a fully hadronic system. Z → bb̄ events are used to validate experimental techniques. Upper limits at the 95% confidence level on the Higgs boson production cross section are established for transverse momenta above 450 GeV and above 1 TeV. Studies related to possible improvements of these results, by reducing the uncertainties are also discussed, such as the use of modern jet definitions like UFO jets and the development of jet substructure taggers using machine learning techniques. ACKNOWLEDGMENTS I would like to start by thanking my advisor Joey Huston, who has always been a source of guidance and support in all the avenues of research I wanted to explore. Thank you for the valuable feedback as well as the great stories. I would like to also thank the MSU ATLAS team of professors and postdocs for always providing me with feedback and suggestions on how to proceed with my projects. I am also thankful of the many friends I’ve made throughout my life. Appreciate all the support and thank you for the good times. Termino agradeciendo a mis seres queridos. A toda mi familia, todo lo que soy y he logrado se lo debo a ustedes. Por último, a Marı́a, sin tu apoyo durante este proceso, completar esta meta no hubiese sido posible. Me inspiras a ser un mejor ser humano, a creer en mi, y a no rendirme nunca. iii TABLE OF CONTENTS Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Relativistic Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Higgs Boson Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Simulation of proton-proton Collisions . . . . . . . . . . . . . . . . . . . . . 28 Chapter 3 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 ATLAS Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Muon system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7 Magnet system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8 Trigger system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 4 Object Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1 Track and Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 b-hadron Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Chapter 5 Boosted H → bb̄ Analysis . . . . . . . . . . . . . . . . . . . . . . . 73 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3 Object Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.5 Higgs Boson Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.6 Background Process Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.7 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 6 Boosted H → bb̄ Results . . . . . . . . . . . . . . . . . . . . . . . 107 6.1 Inclusive Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2 Fiducial Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Differential Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 iv Chapter 7 Unified Flow Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 Jet Substructure Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.3 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.4 Binary Taggers for Boosted UFO jets . . . . . . . . . . . . . . . . . . . . . . 135 7.5 High pT Scale Factor Extrapolation . . . . . . . . . . . . . . . . . . . . . . . 138 7.6 Multiclass Tagger for Boosted UFO jets . . . . . . . . . . . . . . . . . . . . . 157 Chapter 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 APPENDIX A: H → bb̄ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 193 APPENDIX B: Unified Flow Objects . . . . . . . . . . . . . . . . . . . . . . . . 207 v Chapter 1 Introduction Humanity’s quest to understand our place in the universe has lead us to the study of what is the most fundamental representation of our reality, particles. Even though the idea of elementary particles has existed for millenia, it wasn’t until the 19th century that a “modern” view of particles was defined with the discovery of atoms. Atoms are the basic particle of chemical elements and it didn’t take phycisists too long to discover that atoms are in fact composite particles themselves made of protons, neutrons and electrons. With the developments of quantum physics to explain nuclear phenomena coupled with technogical advancements in acceleration physics and particle colliders, we soon found ourselves within a “particle zoo” of supossedly elementary particles during the 1950’s. It wasn’t until the 1960’s, when physicists formulated what we call today the Standard Model (SM), that the origin of so many particles was explained as combinations of a smaller amount of true fundamental particles. High energy physics (also known as particle physics) attempts to create a robust mathe- matical framework that models all the fundamental interactions observed in nature through experimental observations. The SM is constantly being tested and re-tested by continuous analysis of particle collisions produced on the largest and most complex machines ever built by humanity. Teams of scientists and engineers perform a multitude of studies to confirm with greater accuracy the established SM and test theories beyond the Standard Model 1 (BSM). This document presents one of those measurements in one of those experiments for one specific particle, the Higgs boson. The Higgs boson plays a fundamental role in the SM, as it is the particle responsible for the generation of the W, Z and fermion masses. The discovery of a particle with the properties of the Higgs boson in 2012 by ATLAS [1] and CMS [2] concluded one of the main goals of the Large Hadron Collider (LHC) program [3]. In subsequent years, with a larger dataset, more studies have been made putting the measured resonance on more solid grounds [4][5]. One of those was the measurement of H → bb̄ published in 2018 [6]. This measurement uses final states that limit a specific Higgs boson production channel: gluon-gluon fusion (ggF), which in itself is a window to BSM physics effects [7][8]. This document explores a fully inclusive Higgs boson production using the H → bb̄ decay mode at very high energies. A boosted all-hadronic H → bb̄ search requires an extra hadronic jet. Therefore, the analysis will focus on H(→ bb̄) + j where both jets must have a boosted topology. It is the first study in the ATLAS collaboration that targets Higgs boson production cross-sections with transverse momenta above 1 TeV. An analysis in a particle collision experiment requires multiple steps as a setup. You need a collider to create the collisions, a detector to collect signals and procedures to turn those signals into representations of physical objects. At the same time you cannot extract much information without simulating first what process is being produced and how it inter- acts with the detector. Only after that we can then define the scope of the measurement and complete an analysis that measures a physical observable. Figure 1.1 presents a diagra- matic representation of the steps required to perform a collider experiment analysis by the combination of collider data and simulations. 2 Figure 1.1: Diagram that shows the steps required to perform an analysis in a collider experiment [9]. This thesis is divided into three parts. The first part, starts with the theoretical back- ground behind the SM, the Higgs boson, jets and hadronic collisions in Chapter 2. Then, a description of the experimental apparatus: the Large Hadron Collider and the ATLAS detector is explored in Chapter 3. Chapter 4 describes the reconstruction algorithms used to define the physics objects used within the analysis, mainly Large-R jets and the techniques to identify b-hadrons (b-tagging). The second part of this thesis, composed of Chapters 5 and 6, presents the boosted all- hadronic H → bb̄ analysis and the results obtained, where the author contributed to various studies, including uncertainties of the signal modeling, multijet background modeling and combination studies. The analysis and the results shown on this document were published on the paper: “Constraints on Higgs boson production with large transverse momentum using H → bb̄ decays in the ATLAS detector” by the ATLAS Collaboration [10]. 3 Finally, the third part of this thesis, consisting of Chapter 7, presents the studies per- formed by the author as a member of the ATLAS Jet Tagging and Scale Factor Derivation group. This chapter explores modern jet definitions and tagging techniques using Unified Flow Objects [11]. These projects support ATLAS efforts to reduce the systematic uncer- tainties associated with jet reconstruction as well as the development of tools to identify boosted hadronic jets, which will be useful not just for the next iterations of the analysis presented in this thesis, but for any analysis that aims to make measurements using hadronic jets in the boosted regime. 4 Chapter 2 Theoretical Background To give a theoretical description of the Higgs boson, the subject of this thesis, we must first explore the underlying framework that is used to describe fundamental interactions. The framework is based on relativistic quantum mechanics and quantum field theories (QFT) that respect certain symmetry transformations. The introduction given here for these subjects is based on “The Quantum Theory of Fields” by Steven Weinberg [12], for a more detailed approach refer to the original source. After the introduction, a summary of the QFTs that compose the Standard Model is presented. The specifics of the Electroweak (EW) interaction, where the introduction of the Higgs field becomes a neccesity to explain experimental observations of certain physical procesess, is explored. To finalize, Quantum Chromodynamics is discussed to be able to understand the origin of hadronic jets and how we model hadronic collisions. 2.1 Relativistic Quantum Mechanics Any physical state is represented by rays in a finite complex vector space known as Hilbert space [13]. A ray R is a set of normalized state vectors where two states Ψ,Ψ0 belong to the same ray if Ψ0 = ξΨ with ξ being an arbitrary complex number that satisfies |ξ| = 1. An observable is represented by a Hermitian operator that satisfies the reality condition A† = A. They represent mappings Ψ → AΨ of Hilbert space into itself. An observable 5 represented by A acting on state represented by ray R must have a definitie value α for the observable if its vectors are eigenvectors of A with eigenvalue α: AΨ = αΨ for Ψ in R. (2.1) If a system is in a state represented by R and an experiment is performed to test if it is in one of the different states represented by mutually orthogonal rays Rn then the probability of finding it is given by P (R → Rn ) = |(Ψ, Ψn )|2 (2.2) Symmetries A symmetry transformation is a change of point of view that does not change the results of an experiment [12]. That is, two observers O, O0 , looking at the same system represented by rays R, R0 must find the same probabilities P (R → Rn ) = P (R0 → R0n ). (2.3) Any transformation (R → R0 ) is defined by an operator U on Hilbert space such that if a state Ψ is in ray R then U Ψ is in ray R0 . The operator U must satisfy U † = U −1 . Symmetry transformations have certain properties that define them as mathematical groups [14]. A group is a set and an operation such that any two elements of the set produce a third element of the same set. The operation must be associative, the set must have an identity element and every element of the group has an inverse. For particle physics the symmetry groups of interest for this thesis are the Lie groups. In particular, SU(n), the Lie 6 group of n × n unitary matrices with determinant 1, and the Poincaré group, the Lie group of Minkowski spacetime isometries. Lie Group A Lie group is a group of transformations T (θ) that can be described by a finite set of real continuous parameters θa . On the Hilbert space the unitary operator U (T (θ)) can be represented by a power series 1 U T (θ) = 1 + iθa ta + θb θc tbc ...,  (2.4) 2 where ta , tbc are Hermitian operators. The operator ta is known as the generator of the group. Higher order terms of the expansion are related to the generator by the equation tbc = −tb tc − ifbc a t . (2.5) a It is required that the generators satisfy a set of commutation relations known as the Lie algebra: a t [ta , tc ] = iCbc (2.6) a where Cbca ≡ −f a + f a are known as structure constants. For the special case where the bc cb generators commute, the group goes from non-abelian to abelian. The unitary operator can be expressed as simply U T (θ) = exp(ita θa ).  (2.7) 7 Poincaré Group The Poincaré group, also known as the inhomogeneous Lorentz group is a 10-dimensional non-Abelian Lie group represented by the set of transformations with the form T (Λ, a). A Lorentz transformation connects coordinate systems in different intertial frames in a linear form: µ x0µ = Λν xν + aµ . (2.8) The constant matrix Λ satisfies µ ηµν Λρ Λνσ = ηρσ (2.9) where ηµν is a Minkowski metric tensor. In Hilbert space, for an infinitesimal Lorentz transformations, the unitary operator U (Λ, a) can be expanded in the form 1 U (1 + ω, ) = 1 + iωρσ J ρσ − iρ P ρ . (2.10) 2 Commutation relations between the combinations of J µν and P µ with themselves and each other define the Lie algebra of the Poincaré group. The Hamiltonian operator is given by P 0 and is the generator of time translations. The momentum three-vector P = {P 1 , P 2 , P 3 } is the generator of space translations. These form a subgroup of the Poincaré group. In Hilbert space pure translations are represented by U (1, a) = exp(−iP µ aµ ), (2.11) In the same fashion, the angular-momentum three-vector J = {J 23 , J 31 , J 12 } is the generator 8 of rotations U (Rθ , 0) = exp(iJ · θ). (2.12) Finally the other generators form the boost three-vector K = {J 10 , J 20 , J 30 }. Particles A general one particle state Ψp,σ , with momentum p, and degrees of freedom σ, under any Lorentz transformation Λ is given by N (p) X U (Λ)Ψp,σ = D 0 (W (Λ, p))ΨΛp,σ0 , (2.13) N (Λp) 0 σ ,σ σ where N is a normalization factor, W is a Lorentz transformation that leaves the momentum invariant (known as the little group) and D(W ) are the coefficients that form a representation of the little group. Finding irreducible representations of the little group is how we classify physical states and thus how we define the different types of particles. For example, for particles with (j) positive-definite mass, an irreducible representation D of dimensionality 2j + 1 with j = σσ 0 0, 1/2, 1, · · · , can be built using the standard rotation matrices. For this case σ runs over the values j, j − 1, · · · , −j for a particle with spin j. Experimental Observables The Hamiltonian (H) can be divided into two terms, a free-particle Hamiltonian H0 and an interaction term V : H = H0 + V. (2.14) 9 The free-particle Hamiltonian has eigenstates Φα , with eigenvalue Eα and the full Hamilto- nian has eigenstates Ψ±α with the same eigenvalue as the free-particle Hamiltonian. These are known as “in”(+) and “out” (−) states and can be written in terms of the free-particle eigenstates: Ψ±α = Ω(∓∞)Φα , (2.15) where Ω(τ ) = exp(+iHτ )exp(−iH0 τ ). (2.16) The in and out states contain the particles described by the label α if observations are made at τ → ±∞. The probability amplitude for a transition of states α → β is governed by the scalar product of the “in”(+) and “out”(-) states known as the S-matrix: Sβα = (Ψ− + β , Ψα ) ≡ (Φβ , SΦα ). (2.17) Where S is the S-operator defined as S = Ω(∞)† Ω(−∞). (2.18) The master formula to interpret calculations of S-matrix elements in terms of predictions for actual experiments is Sβα ≡ −2πiδ 4 (pβ − pα )Mβα . (2.19) Here the delta function ensures the conservation of total energy and momentum and Mβα represents the non-trivial scattering matrix elements. 10 Decay Rate The decay rate for a single particle state α into a general multi-particle state β is given by dΓ(α → β) = 2π|Mβα |2 δ 4 (pβ − pα )dβ. (2.20) When multiple decay modes are available to a specific particle the total decay rate will be the sum of all of the individual modes X n Γtotal = Γi . (2.21) i=1 Then, it is useful to define the branching fractions to quantify the probability of each specific decay mode. The branching fraction of mode i is given by Γi Bi = . (2.22) Γtotal Cross-Section When α is a two particle state, we can calculate the transition rate per flux, known as the differential cross section, using the decay rate dσ(α → β) = dΓ(α → β)/Φα = (2π)4 u−1 2 4 α |Mβα | δ (pβ − pα )dβ, (2.23) where Φα is defined as the product of the density and the relative velocity uα between the two particles in state α. When the differential cross section is integrated over all the possible 11 configurations i we call it the total or inclusive cross-section X n σtotal = σi . (2.24) i=1 Interactions To consider the interaction term we rewrite the S-operator as a Dyson series for the time ordered interaction Hamiltonian density H(x) defined as: ∞ (−i)n Z d4 x1 · · · d4 xn T {H(x1 ) · · · H(xn )}, X S =1+ (2.25) n! n=1 where Z V (τ ) ≡ exp(H0 τ )V exp(−iH0 τ ) = d3 xH(x, t). (2.26) Then it is possible to write an asymptotic expansion of the S-operator in whatever cou- pling constant factors appear in the inveraction terms of the Hamiltonian density. This technique is known as perturbation theory. 2.2 Quantum Field Theory For the Hamiltonian density to satisfy both Lorentz invariance and the cluster decomposi- tion principle, the Hamiltonian density must be constructed as a function of creation and annihilation fields. A creation field is defined to have a creation operator (ac† (p, σ)) that adds a particle to the list of particles in a physical state. The annihilation field contains the annihilation operator (a(p, σ)) and does the opposite, it removes a particle from any state in which it acts. A general quantum field in the irreducible (A, B) representation of the ho- 12 mogeneous Lorentz group, where A and B are the spins, is defined by a linear combination of creation and annihilation fields: XZ (2π)−3/2 d3 p uab (p, σ)a(p, σ)eip·x + (−)2B vab (p, σ)ac† e−ip·x .   ψab (x) = (2.27) σ where the coefficients a, b are integers or half-integers running over the values a = −A, −A + 1, · · · , +A and b = −B, −B + 1, · · · , +B. (2.28) A field according to the (A, B) representation has components that rotate like objects of spin j with j = A + B, A + B − 1, · · · , |A − B|. (2.29) Fields with integer values for the spin commute with each other and are classified as bosons. Bosons do not obey the Pauli exclusion principle and thus are described by Bose-Einstein statistics. On the other hand, half-integer spin fields, known as fermions, anticommute with each other. Fermions obey the Pauli exclusion principle and therefore, a system of fermions follow Fermi-Dirac statistics. Lagrangians In practice it is preferable to work with Langrangians (L) instead of the Hamiltonians (H). These two quantities are related to each other by taking the Legendre transformation: XZ H= d3 x Πl (x, t)Ψ̇l (x, t) − L[Ψ(t), Ψ̇(x)], (2.30) l 13 where Ψ is a set of generic fields, Π are the conjugate fields and the dotted variables represent time-derivatives. The conjugate field is defined using variational derivatives: δL[Ψ(t), Ψ̇(t)] Πl (x, t) = . (2.31) δ Ψ̇l (x, t) These are knwon as the Euler-Lagrange equations, and their time derivatives Π̇l (x, t) are the equations of motion. Defining the Lagrangian density L, Z ˙ = L[Ψ(t), Ψ(t)] d3 x L(Ψ(x, t), ∇Ψ(x, t), Ψ̇(x, t)) (2.32) we can express the Euler-Lagrange equations in their usual form ∂L ∂L ∂µ l = . (2.33) ∂(∂µ Ψ ) ∂Ψl Scalar Fields A scalar field is a field of type (0,0) in the irreducible representation of the homogeneous Lorentz group and therefore are spin 0 fields. A general Langrangian density L for a massive free scalar field Φ is given by 1 m2 2 L = − ∂µ Φ∂ µ Φ − Φ . (2.34) 2 2 The Euler-Lagrange equation is then (∂µ ∂ µ − m2 )Φ = 0, (2.35) 14 which is the usual Klein-Gordon equation. Vector Fields A vector field is a field of type ( 12 , 21 ) in the irreducible representation of the homogenous Lorentz group. Therefore they can be spin 0 or spin 1. The spin 0 vector field is just the derivative of a spin 0 scalar particle (∂µ Φ). For a massive spin 1 vector field Aµ and no external currents the Langrangian density L is 1 L = − Fµν F µν − m2 Aµ Aµ , (2.36) 4 where Fµν ≡ ∂µ Aν − ∂ν Aµ is the field strength tensor. In cojunction with ∂µ Aµ = 0, the Euler-Lagrange equation takes the form: (∂µ ∂ µ − m2 )Aµ = 0, (2.37) which is known as the Proca equation. This implies that each component of the field fulfils the Klein-Gordon equation. Dirac Fields Dirac fields represent particles of spin 1/2 and are of the type ( 12 , 0) (0, 21 ) in the irreducible L representation of the homegeneous Lorentz group. A general Langrangian density for Dirac fields is of the form L = −ψ̄(γ µ ∂µ + m)ψ, (2.38) 15 where ψ̄ = ψ † γ 0 is the Dirac adjoint. The Euler-Lagrange equations for ψ is known as the Dirac equation (iγ µ ∂µ + m)ψ(x) = 0. (2.39) Taking the hermitian conjugate of the Dirac equation and multiplying on the right by γ 0 , the adjoint Dirac equation can be derived. When both solutions are combined we arrive at (∂µ ∂ µ − m2 )ψ µ = 0. (2.40) Therefore, each component of the Dirac field satisfies the Klein-Gordon equation. 2.3 Standard Model The Standard Model (SM) is the collection of quantum field theories (QFTs) that describe three of the four fundamental forces of the universe and classifies all the elementary par- ticles currently known. The three forces it describes are: the strong interaction, the weak interaction and electromagnetism. The strong interaction is described by quantum chro- modynamics (QCD). Electromagnetism (EM) and the weak interaction are unified into the same theory, called the electroweak interaction (EW). On its entirety the SM respects the symmetry under the non-abelian SU(3)C × SU(2)L × U(1)Y gauge group. The SM contains both types of particles, fermions and bosons. The fermions can be divided into quarks and leptons and they exist in three generations; each one with increasing mass. There are six types of quarks: up, down, strange, charm, bottom and top. There are also six leptons: electron, muon, tau and their respective neutrinos. The bosons are divided into the spin 1 (vector) force carriers and the spin 0 (scalar) Higgs boson. The force carriers 16 are the photon γ for the electromagnetic interaction, the gluon g for the strong interaction and the W ± , Z bosons that mediate the weak interaction. Figure 2.1 summarizes all the SM fundamental particles. Figure 2.1: Standard Model of elementary particles showing the twelve fundamental fermions and five fundamental bosons [15]. 2.4 Quantum Chromodynamics Quantum Chromodynamics (QCD) is the non-abelian gauge field theory that describes the strong interaction, a force only felt by quarks and gluons. Quarks are in the fundamental representation of the SU(3) color group. They are represented by quark field spinors ψi , where i the color-index that goes from 1 to Nc = 3 (the number of colors). The gluon is a vector field, Aaµ , where a runs from 1 to Nc2 − 1 = 8. Gluons transform under the adjoint 17 representation of SU(3) color group. The eight 3 × 3 matrices taij are the generators of SU(3), which can be represented explicitly by the Gell-Mann matrices (λa ) given by ta = λa /2. The QCD Lagragian is given by 1 L = ψ̄i (iγ µ ∂µ δij − gs γ µ taij Aaµ − mδij )ψj − Gaµν Gaµν . (2.41) 4 The QCD coupling constant is αs = gs2 /4π. The gluon field strength tensor Gaµν is given by Gaµν = ∂µ Aaν − ∂ν Aaµ − gs fabc Abµ Acν (2.42) where fabc are the structure constants of the SU(3) group. The coupling constant αs is a function of the scale at which the process happens. Figure 2.2 shows different experimental measurements of the coupling constant αs as a function of energy scale Q. Quarks and gluons cannot be isolated, only color-singlet (color neutral) combinations of them can be observed as free particles. Given that the coupling is really strong at low energies, it leads to the confinement of quarks and gluons into hadrons, a non-perturbative process called hadronization. On the other hand, for hard processes the strong coupling is weak and the theory becomes suitable to perturbative theory techniques, a phenomenon known as asymptotic freedom. Before hadronization occurs, a hard scattering event involving a QCD interaction starts with the interacting particles radiating more gluons and quarks (parton showering) until the parton energy gets to the hadronization scale (Λ). This leads to the formation of collimated sprays of energetic hadrons, which we call jets [17]. In Chapter 4 we will discuss more about the particular type of jets used in this analysis, how we define them, the rules to group 18 Figure 2.2: Measurements of αs as a function of the energy scale Q. The degree of QCD perturbation theory used in the extraction of αs is indicated in parentheses [16]. particles and how to determine their momenta. 2.5 Electroweak Theory The standard model of electroweak interactions is based on the gauge field theory that respects the symmetry of the SU(2)L × U(1)Y product group, with gauge bosons Wµi , i = 1, 2, 3 and Bµ , and their corresponding gauge coupling constants g and g 0 . The fundamental quantities are the SU(2)L weak isospin and the U(1)Y weak hypercharge. The right-handed fields are singlets in SU(2). On the other hand, the left-handed fermion fields transform as 19 doublets      vi  ui  Ψi =   and   (2.43) li− d0i where d0i = P j Vij dj and V is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [18]. The CKM matrix is a unitary 3 × 3 matrix and it describes the quark flavor mixing in weak interactions. In one of the standard parametrizations it can be expressed as     1 0 0   c13 0 s13 e−iδ   c12 s12 0     VCKM = 0 c23 s23   0 (2.44)     1 0   −s12 c12 0         0 −s23 c23 −s13 e iδ 0 c13 0 0 1 where sij = sin θij , cij = cos θij and δ is the phase responsible for CP-violating phenomena in flavor-changing processes. Higgs Mechanism For the EW theory to be consistent with observations we require a mechanism that makes the W and Z bosons massive to render the weak interaction short range. This can be achieved by the introduction of a scalar field Φ, called the Higgs field, that causes a spontaneous breaking of the electroweak gauge symmetry (EWSB) [16]. The Higgs potential is of the form: V (Φ) = µ2 Φ† Φ + λ(Φ† Φ)2 (2.45) 20 The Higgs field is a self-interacting SU(2) complex doublet with a weak hypercharge Y = 1 normalized such that it has a neutral charge (Q = T3L + Y /2): √ +   1  2φ  Φ= √   (2.46) 2 0 0 φ + ia where φ0 and a0 are the CP-even and CP-odd neutral components and φ+ is the complex charged component. If the quadratic term of V (Φ) is negative the neutral component of the scalar doublet acquires a non zero vacuum expectation value (VEV)   1 0 hΦi = √   (2.47) 2 v with φ0 = H + hφ0 i and hφ0 i = v, inducing the spontanueos breaking of the gauge symmetry SU(3)C ×SU(2)L ×U(1)Y into SU(3)C ×U(1)em . Three of the four generators of the SU(2)L × U(1)Y are spontaneously broken; this imples the existence of three massless Goldstone bosons which can be identified as three of the four Higgs field degrees of freedom. Figure 2.3 illustrates the fact that the Higgs field VEV is not a single state with an energy of 0 and instead it has degenerate vacua with a VEV of v. The kinetic term of the Higgs Lagrangian shows how the Higgs field couples to the Wµ and Bµ gauge fields of the SU(2)L × U(1)Y local symmetry: LHiggs = (Dµ Φ)† (Dµ Φ) − V (Φ) (2.48) 21 Figure 2.3: Illustration of the Higgs potential V (Φ). After EWSB, the Higgs field VEV is not a single state with an energy of 0 (represented by A), it has degenerate vacua with a VEV of v (represented by B) [19]. The covariant derivative is: Dµ = ∂µ + igσ a Wµa /2 + ig 0 Y Bµ /2 (2.49) where g and g 0 are the SU(2)L and U(1)Y couplings and σ a are the Pauli matrices. Expanding the kinetic term and rearranging then we can see that the presence of the Higgs field gives mass to the gauge bosons. Examining the mass term of the Lagrangian: g2v2 v2 µ Lm = (W12 + W22 ) + (gW3 − g 0 B µ ) (2.50) 8 8 we can see that the physical massive bosons are combinations of the original gauge bosons. Rewriting in terms of the physical fields: 1 1 Lm = m2W Wµ+ W −µ + m2Z Zµ Z µ (2.51) 2 2 22 where Wµ1 ∓ iWµ2 1 Wµ± = √ , Zµ = p (gWµ3 − g 0 Bµ ) (2.52) 2 2 g +g 02 and g2v2 (g 02 + g 2 )v 2 m2W = , m2Z = (2.53) 4 4 There is one combination of W3 and B, orthogonal to Z, that is not present in the mass Lagrangian; it corresponds to the photon: 1 µ Aµ = p (g 0 W3 + gB µ ) (2.54) 2 g +g 02 Of the initial four degrees of freedom of the Higgs field, three of them were absorbed by the W ± and Z bosons and the remaining degree of freedom, H, becomes the physical Higgs boson. The Higgs boson is a CP-even spin 0 (scalar) particle with a mass given by mH = √ 2λv, where λ is the self coupling parameter. The Higgs field expectation value is fixed by √ the Fermi coupling costant GF : v = ( 2GF )−1/2 ≈ 246 GeV. Fermions acquire mass through interactions with the Higgs field, also known as Yukawa interactions. Yukawa couplings respect all the symmetries of the SM but generates fermion masses after the EWSB occurs. LYukawa = −ĥdij q̄Li ΦdRj − ĥuij q̄Li iσ2 Φ∗ uRj − ĥdij ¯lLi ΦeRj + h.c. (2.55) After the Higgs field acquires a VEV the fermions acquire a mass in the form: mfi = √ hfi v/ 2, where hfi is the Yukawa coupling and i = 1, 2, 3 refer to the three families of the up-quark, down-quark or charged lepton sectors. 23 The coupling of the Higgs boson to other fundamental particles is dictated by how massive the particle is. The interaction is strongest with particles such as the W/Z bosons and to top quarks. For fermions the coupling is linearly proportional to the fermion mass (gHf f¯ = mf /v) and for bosons it is proportional to the square of the boson masses (gHV V = 2m2V /v). 2.6 Higgs Boson Phenomenology This section explores the Higgs boson production modes and the branching ratios for all of its decay channels [20]. We will finish the section by exploring how a boosted Higgs boson can be used as a probe for beyond the Standard Model (BSM) physics. Production and Decays Experimentally, the Higgs boson mass is measured to be mH = 125.25 ± 0.17 GeV. To produce a Higgs boson we require a collider experiment with a large center-of-mass (CoM) energy such as the Fermilab Tevatron [21] or the CERN LHC [3]. Given that we are studying the Higgs boson at the LHC, we need to first understand the production mechanisms in hadron (on this case: proton-proton (pp)) collisions. The principal production mode at the LHC is the gluon-gluon fusion (ggF) process, followed by weak-boson (vector-boson) fusion (VBF). Other production modes include associated production with a gauge boson (VH), associated production with tt̄ quark pair (tt̄H) or associated production with a single top quark (tHq). Figure 2.4 illustrates the leading order Feynman diagrams for some of the Higgs boson production modes at the LHC. Figure 2.5 shows the Higgs boson production cross section for pp collisions at a CoM √ energy of s = 13 TeV as a function of Higgs boson mass. For a Higgs boson mass of mH = 24 Figure 2.4: Leading order Feynman diagrams that contribute to Higgs boson production in (a) ggF, (b) VBF, (c) Higgs-strahlung, (d) associated production with a gauge boson and (e) associated production with a pair of top-quarks [16]. 125 GeV, the total production cross section is 55.1 pb. The production mode breakdown as a percentage is as follows: ggF is the largest contribution with 88% of the total production cross section, VBF accounts for around 7%, VH (WH and ZH combined) sum to 4% and tt̄H is close to 1%. Detecting the Higgs boson requires an understanding of all the relevant decay channels. The Higgs boson has a natural width of 4 MeV, meaning it has a lifetime of the order of 10−22 seconds. Figure 2.6 shows the Higgs boson decay branching ratios as a function of Higgs boson mass. The dominant decay mode is H → bb̄, with a branching fraction of about 58%. This decay mode is the focus of the study presented in this thesis. Even though it is the most common decay mode, the channel suffers from large backgrounds, primarily from bb̄ production. To measure its mass the two high mass-resolution sensitive channels are H → γγ and H → ZZ → 4l, which despite having low branching ratios, have clean signals. These two channels were used for the original discovery of the Higgs boson in 2012 [23][24]. H → bb̄ is a promising channel to study the Higgs field coupling to quarks. For the direct observation of the Higgs boson decaying to a pair of b-quarks, the production mode used in the original studies was the VH channel. The presence of a vector boson reduces the 25 LHC HIGGS XS WG 2016 102 s= 13 TeV σ(pp → H+X) [pb] pp → H (N3LO QCD + NLO EW) 10 pp → qqH (NNLO QCD + NLO EW) pp → WH (NNLO QCD + NLO EW) pp → ZH (NNLO QCD + NLO EW) 1 pp → ttH (NLO QCD + NLO EW) pp → bbH (NNLO QCD in 5FS, NLO QCD in 4FS) 10−1 pp → tH (NLO QCD) 120 122 124 126 128 130 MH [GeV] √ Figure 2.5: Standard Model Higgs boson production cross sections at s = 13 TeV as a function of Higgs boson mass for pp collisions [22]. relative background because the leptonic decay of the W and Z enable efficient triggering and a significant reduction of the multijet background. The Higgs candidate was reconstructed from two b-tagged jets in the event. Both ATLAS and CMS observed a significance of the excesses greater than 5σ when combining Run 1 and Run 2 data [26][27]. Sensitivity for an inclusive search for H → bb̄ in the ggF production mode is limited because of the large amount of background from the inclusive production of pp → bb̄ + X. From the Run 1 dataset, no meaningful results exist. The analysis presented in this document is the first ever performed by the ATLAS collaboration that attempts to do this with the full Run 2 data, with the sensitivity increased by focusing on Higgs boson production at high transverse momentum. 26 Branching Ratio LHC HIGGS XS WG 2016 1 bb WW gg 10-1 ττ cc ZZ 10-2 γγ 10-3 Zγ µµ 10-4 120 121 122 123 124 125 126 127 128 129 130 MH [GeV] Figure 2.6: Standard Model Higgs boson decay branching ratios as a function of Higgs boson mass [25]. Boosted Higgs boson To boost a Higgs boson to high momenta, it is required to have an extra jet in the event for the Higgs boson to recoil against. Figure 2.7 contains a couple examples of diagrams that contribute to the H + j production cross section. A Higgs boson with high transverse momentum can be used to set constraints for beyond the Standard Model (BSM) [29][30]. The inclusion of a set of dimension-six operators [31] in the SM lagrangian that describe physics at a scale Λ above the EW scale, modify the Yukawa operator, provide a contact interaction of the Higgs boson with gluons and introduce the chromomagnetic dipole moment operator [32]. All of these interactions have an impact in the Higgs boson pT distribution. In particular, when considering the chromomagnetic dipole 27 Figure 2.7: Examples of gluon-gluon fusion Feynman diagrams that contribute to the H + j process [28]. moment operator in the case of single Higgs production, it has been shown to have a large impact at high pT [33]. Figure 2.8 illustrates these results. The extra term related to the chromomagnetic dipole moment in this context is of the form c3 gs mt 2 O3 = c3 3 (v + h)GA µν A µν (ψ̄L σ t ψR + h.c.) (2.56) Λ 2v where c3 is the Wilson coefficient, σ µν are the Pauli matrices and ψ is the spinor representing the top quarks. 2.7 Simulation of proton-proton Collisions Any cross section that involves QCD interactions of initial-state hadrons is inherently not calculable in perturbative QCD. Structure functions are needed to describe these complex objects. The structure functions are given in terms of non-perturbative parton distribution functions (PDFs) . A PDF fq/p (x) represents the number density of quarks of type q inside a hadron that carry a fraction x of its longitudinal momentum. A typical hadron-hadron 28 Figure 2.8: Impact of the chromomagnetic operator on the pT spectrum of the Higgs boson. The bottom panel shows the ratio with respect to the SM prediction [33]. (h1 ,h2 ) collision cross-section is of the form ∞ XZ (n) αsn (µ2R ) dx1 dx2 fi/h (x1 , µ2F )fj,h2 (x2 , µ2F ) × σ̂ij→X X σ(h1 , h2 → X) = (2.57) 1 n=0 i,j where s is the squared center-of-mass energy of the collision, µR is the renormalization scale and µF is the factorization scale, the scale at which emissions with transverse momenta below it are accounted for within the PDFs. The parton level cross-section σ̂ij→X (x1 x2 s, µ2R , µ2F ) can be calculated using perturbative QCD. PDFs are determined empirically by fitting a large number of cross section data points from many experiments, including Deep Inelastic Scattering experiments (DIS) and hadron collider experiments. To evolve those functions to different energy scales, the Dokshitzer- 29 Gribov-Lipatov-Altarelli-Parisi (DGLAP) [34] equation is employed. Usually the default choice of the scales is µR = µF = Q. Figure 2.9 shows the CT18 parton distribution functions at different energy scales. Figure 2.9: The CT18 parton distribution functions at Q = 2 GeV and Q = 100 GeV for ¯ s = s̄, and g [35]. u, ū, d, d, The parton-hadron transition is non-perturbative, so it is not possible to calculate quan- tities like the energy spectrum of hadrons in high-energy collisions. Nevertheless it is possible to factorize the perturbative and non-perturbative behaviours using the concept of fragmen- tation functions. Similarly to PDFs, they depend on a factorization scale and satisfy the DGLAP evolution equation. To create simulations of this entire process, we use parton-shower Monte Carlo (MC) event generators such as Pythia [36], Herwig [37] and Sherpa [38]. They provide a full simulation of QCD events at the level of measurable particles. Figure 2.10 shows a sketch of a pp collision as simulated by a MC generator. There are MC generators that only produce the matrix elements, such as MadGraph5 aMC@NLO [39], which are then passed to a shower/hadronization program such as Pythia. The parton shower MC programs model the gluon emissions and gluon splittings simulating a cascade of particles. Each emission is generated at a lower scale, with the emissions stopping at a scale of the order of 1 GeV. 30 At this point a hadronization model is used to combine the resulting particles into hadrons. There are different hadronization/shower models which might have slight differences in the end result. In practice multiple programs are considered when generating MC predictions for an analysis and the differences are quantified as a source of uncertainty. The remnants of hadron collisions also have to be modeled; this is refered to as the underlying event (UE). The UE is usually implemented by introducing multiple parton in- teractions (MPI) at a scale of a few GeV. Similarly, pile-up also has to be simulated. Pile-up refers to any other pp collisions in addition to the collision of interest. Figure 2.10: Sketch of a proton-proton collision as simulated by a multi-purpose Monte Carlo event generator [40]. As a last step all of the particles/jets generated and their kinematic variables, at “truth- level”, are subjected to a detector simulation. All of the particle interactions with the different detector modules are done with Geant4 [41]. Geant4 is a toolkit for simulating 31 the passsage of particles through matter. After the detector simulation is completed all the kinematic variables modified are then refered to as being at “reconstructed-level”. 32 Chapter 3 Experimental Apparatus 3.1 Large Hadron Collider To study physics at small scales it is necesary to accelerate particles to high energies and have them interact, that is, make them collide. This is done by the Large Hadron Collider (LHC) [3] at the European Organization for Nuclear Research, known as CERN located at the French-Swiss border of Geneva. The LHC is the largest and most powerful parti- cle accelerator ever built and is part of the CERN accelerator complex shown in Figure 3.1. The process starts with a cylinder of hydrogen gas. The hydrogen atoms are ionized to obtain protons. These protons are then accelerated in bunches by using a series of ac- celerators, first a linear accelerator (LINAC), then the proton synchrotron (PS), the super proton synchrotron (SPS) and finally the LHC. The collider itself consists of two rings with a circumference of approximately 26.7 km, where the two counter-rotating proton beams are accelerated to a momentum of 6.5 TeV per beam, leading to a center of mass energy of √ s = 13 TeV. To maintain the beams along the trajectory, the LHC uses superconducting dipole magnets which are cooled to a temperature below 2 K using superfluid helium. The superconducting magnets produce magnetic fields with a strength of about 8 T. Quadrupole magnets are used to squeeze the beams as they enter the interaction points. The LHC is designed to run with 2808 bunches per beam separated by a 25 ns gap with each bunch con- 33 taining 100 billion (1011 ) protons. This translates to a crossing rate of 40 MHz with typicaly 50 collisions per crossing. There are four distinct interaction points where the beams cross and the protons collide. On these sites the main detectors are placed: ALICE (A Large Ion Collider Experiment) [42], LHCb (LHC-beauty) [43], CMS (Compact Muon Solenoid) [44] and ATLAS (A Toroidal Large ApparatuS) [45]. Figure 3.1: Illustration of the CERN accelerator complex. The LHC is the last ring in a complex chain of particle accelerators [46]. The number of events per second generated in the LHC can be described by the equation: Nevent = Lσevent (3.1) where σevent is the cross section for a certain process under study and L is the machine luminosity. Given that the luminosity depends only on the beam parameters it can be used 34 as a measure of the performance of the collider. The full definition for a gaussian beam distribution is: Nb2 nb frev γ L= F (3.2) 4πn β where Nb is the number of particles per bunch, nb the number of bunches per beam, frev the revolution frequency, γ is the relativistic gamma factor, n the normalized transverse beam emittance (area occupied by the beam), β the beta function (function of the transverse size of the beam) at the collision point, and F the geometric luminosity reduction factor due to the crossing angle at the interaction point. Integrating (with respect to time) the luminosity over the different runs would then give us a measure of the amount of data that was delivered by the LHC. Figure 3.2 shows the total integrated luminosity for Run 2 (2015-2018) of the LHC, as well as the data recorded by the ATLAS detector that was deemed good for physics. Figure 3.2: Total Integrated Luminosity and Data Quality of the LHC during Run 2 (2015- 2018) [47]. 35 3.2 ATLAS Detector ATLAS [45] is a multi-purpose particle detector of 25 meters in height and 44 meters in length that weighs about 7000 tonnes. It consists of various layers that perform specific measurements of the particles from the collision. ATLAS is located 100 meters below the surface at the CERN LHC Point 1. The detector was designed to have forward-backward symmetry along the beam pipe with a large azimuthal angle coverage. It contains a super- conductiong solenoid that surrounds the inner detector, inmersing it in a 2 T solenoid field. ATLAS also contains electromagnetic and hadronic calorimeters that are surrounded by su- perconducting air-core toroids arranged with an azimuthal symmetry. A muon spectrometer is located within the toroids. Figure 3.3 shows an ATLAS schematic of the different detector modules. Figure 3.3: Schematic of the ATLAS detector showing its main sub-components. The people in the diagram indicate the scale of the detector [48]. 36 3.3 Coordinate System To describe the ATLAS detector in detail, we must first describe the conventions regarding the coordinate system used. The clockwise direction of the beam defines the z-axis while the x-y plane lies transversal to the beam direction. The positive x-axis points towards the center of the LHC ring and the positive y-axis points upwards. The azimuthal angle φ is defined around the beam axis and the polar angle θ is the angle from the beam axis. Rapidity then is defined as y = 1/2 ln (E + pz )/(E − pz ), which for massless particles becomes the pseudorapidity η = − ln tan(θ/2). With these quantities the distance in the pseudorapidity- p azimuthal angle space can be defined as ∆R = ∆η 2 + ∆φ2 . Other quantities of interest are the kinematic variables defined on the transverse (x-y) plane, the transverse momentum pT , transverse energy ET and missing transverse energy ET miss . 3.4 Tracking Because of the large number of particles that emerge from the collision point, the inner detector (ID) must have fine granularity in order to make high precision measurements. It also is designed to provide hermetic and robust pattern recognition. The ID achieves this with its 3 sub-detectors: the pixel detector [49], the silicon microstrip trackers also known as the semiconductor Tracker (SCT) and the transition radiation tracker (TRT). The ID covers the region η < 2.5, extends to 1.15 m radially and has a length of 6.2 m. It is contained in a solenoid that inmerses its 3 sub-detectors in a 2 T magnetic field which allows charge and momentum measurements. With its track reconstruction capabilities, the ID is the main system used to construct primary and secondary interaction vertices as well as identifying heavy-flavor jets (i.e. b-tagging). Figure 3.4 shows the overall layout of the inner detector. 37 A schematic view of the overrall path of a charged particle in the inner detector is shown in Figure 3.5. Figure 3.4: Schematic diagram of the ATLAS inner detector showing its different sub- components [50]. Pixel detector The innermost part of the ID is the pixel detector [49]. The pixel detector provides the highest granularity around the vertex region with a total of 1744 modules. Each module is composed of oxygenated silicon sensors, front-end electronics and flex-hybrids with control circuits. The silicon sensors are the sensitive part of the pixel detector and function as a solid-state ionization chambers. The pixel sensor is an array of bipolar diodes placed on a silicon wafer. The p-n junctions operate under a reverse bias. Ionizing particles passing through the active volume create drifting electron-hole pairs that produce electrical signal that can be measured. The bulk contains oxygen impurities to increase tolerance of the silicon against damage caused by charged hadrons [51]. 38 The modules are arranged in three concentric cylinders around the beam axis and as three disks in the end-cap regions. Three layers allow an effective reconstruction of tracks by requiring a minimum of 3 hits. An Insertable B-Layer (IBL) [52] was installed between the beam pipe and the pixel detector during the 2016 LHC shutdown to maintain robust tracking in the presence of increased pileup and radiation, while also providing improved precision for vertexing and tagging. Semiconductor Tracker After the pixel detector we have the semiconductor tracker (SCT), arranged in four concentric cylinders around the beam axis and nine disks in the end-cap regions. Instead of pixels the SCT contains silicon strip sensors. Each module is composed of two sensors glued together. Eight strip layers are crossed by each track. Small-angle stereo strips consisting of two 6.4 cm long daisy-chained sensors measure both coordinates in the barrel region . In the end-cap region, the strips run radially with a set of stereo strips at an angle of 40 mrad. The SCT has a resolution of 16 µm in φ and 580 µm in z. Transition Radiation Tracker At a larger radius, the straw tubes of the Transition Radiation Tracker (TRT) provide infor- mation on particle tracking and identification. The TRT consists of gas-filled (Xe,C02 ,O2 ) drift tubes with a gold plated tungsten wire inside. In the barrel region, these straws are parallel to the beam axis, while in the end-cap region, they are arranged radially in wheels. In the barrel region the TRT achieves a resolution of 130 µm while in the end-cap region it provides an accuracy of 30-50 µm. Each of the 3 cylidrincal layers contains 32 modules, 39 and each module is composed of a carbon-fiber laminated shell with an internal array of the straws embedded in a matrix of polypropylene fibers that serve as the transition radiation material. Figure 3.5: Schematic diagram of the structural elements traversed by a charged particle in the barrel inner detector [53]. 3.5 Calorimetry Calorimeters are used to measure the energy of particles. Due to their segmented nature they also can provide directional information about energetic charged leptons and hadrons and even neutral particles that don’t interact with the trackers. When particles enter the calorimeter they initiate a particle shower. These lower energy particles are absorbed by the 40 calorimeters and consequently produce a signal that allows the measurement of the energy deposited. The ATLAS calorimeters cover the range |η| < 4.9. They can be divided in two main categories: the EM calorimeters and hadronic calorimeters. A diagram of the calorime- ter system is shown in Figure 3.6. Both calorimeter systems must provide good containment for electromagnetic and hadronic showers and also limit the punch-through into the muon system. The EM calorimeter has a finer granularity that is suited for precision measurements of electrons and photons. The hadronic calorimeter has coarser granularity and is sufficient to satisfy the physics requirements for jet reconstruction and ETmiss measurements. Figure 3.6: Schematic diagram of the ATLAS calorimeter system showing all of its sub- components [54]. The energy resolution of a sampling calorimeter is parametrized as σ a b = √ ⊕ ⊕ c, (3.3) E E E where a is the stochastic term, b is the noise term and c corresponds to a constant term. The stochastic term represents the random nature of the showering process and is dependent 41 of the active and absorber materials in the calorimeter as well as the number of layers and their thickness. The noise term describes the electronic noise of the readout system. The constant term reflects local non-uniformities in the response of the calorimeter. The EM calorimeter was measured [55] to have an energy resolution of σ 2.8% 0.12 GeV = √ ⊕ ⊕ 0.3%. (3.4) E E E For the hadronic calorimeter the electronic noise was found to be negligible and thus not included. The energy resolution measured [55] was σ 52.9% = √ ⊕ 5.7%. (3.5) E E LAr Electromagnetic Calorimeter The EM calorimeter has a barrel part (|η| < 1.475) and two end-cap components (1.375 < |η| < 3.2). It is a liquid argon (LAr) detector with accordion-shaped electrodes and lead absorber plates. This geometry provides a full φ symmetry without azimuthal cracks and lead to an uniform performance in terms of linearity and resolution as a function of φ. The absorbers have two stainless-steel sheets glued on either side using a resin-impregnated glass- fiber fabric to provide mechanical strength. The readout electrodes, consisting of conductive copper layers separated by insulating polyminide sheets, are located in the gaps between the absorbers. The barrel EM calorimeter is composed of two half-barrels, each with a length of 3.2m and a weight of 57 tonnes. One half-barrel consists of 1024 accordion-shaped absorbers interleaved with readout electrodes. For the EM calorimeter, one parameter of interest is 42 the radiation length X0 , defined as the mean distance a particle can travel before its energy is reduced by a factor of 1/e. Each half-barrel is divided in 16 modules, each with a total thickness of a minimum of 22 X0 and cover ∆φ = 22.5◦ . These modules have three layers of depth. The front layer is read out at the low-radius side of the electrode while the middle and back layers are read out at the high-radius side of the electrode. A sketch of the different layers of the EM barrel module is provided in Figure 3.7. Figure 3.7: Schematic diagram showing the different layers of the EM calorimeter barrel module [45]. Hadronic Calorimeter The hadronic calorimeter system consists of the tile calorimeter (TileCal), the hadronic end-cap calorimeter (HEC) and the forward calorimeter (FCal). 43 TileCal is located outside the EM calorimeter envelope. The barrel covers |η| < 1.0 and the extended barrels cover the range 0.8 < |η| < 1.7. It is a sampling calorimeter that uses scintillating tiles as the active medium and steel as the aborber. The tile calorimeter extendeds radially from an inner radius of 2.28 m to an outer radius of 4.25 m. Azimuthally, TileCal is divided into 64 modules. It is segmented in three longitudinal layers with different interaction lengths. The interaction length λ is defined as the mean free path of a hadronic particle before undergoing an inelastic nuclear interaction. The three segments have 1.5, 4.1 and 1.8 λ for the barrel, and 1.5, 2.6 and 3.3 λ for the extended barrel. The scintillating tiles are read out by wavelength shifting fibers into two photomultiplier tubes (PMT). When an ionising particle crosses the tiles, they induce the production of blue scintillation light that is then converted to green light by the wavelength-shifting fluors in the fibers. A schematic drawing of a TileCal module with its components is shown in Figure 3.8. The HEC module is a copper/liquid-argon sampling calorimeter with a flat-plate design that covers the range 1.5 < |η| < 3.2. It consists of two cylindrical wheels, each with two longitudinal sections. HEC shares the end-cap cryostats with FCal and the electromagnetic end-cap calorimeter. Each of the HEC wheels is constructed of 32 identical wedge-shaped modules. The modules of the front wheels have 24 copper plates with a thickness of 25 mm. For the rear wheels, the modules are made of 16 copper plates with a thickness of 50 mm making its sampling fraction coarser. Figure 3.9 depicts the HEC module views from different angles. Seven stainless-steel tie-rods provide the structural strength of the modules. Honeycomb sheets are used to fill the space between three electrodes that divide the gaps into four separate LAr drift zones. Each of these drift zones is supplied with a high voltage. The middle electrode serves as the readout electrode and the other two carry surfaces of high resistivity to which high voltage is applied, forming an electrostatic transformer. 44 Figure 3.8: Schematic diagram of a TileCal module, showing the slots in the steel for scin- tillating tiles and the method of light collection by wavelength-shifting fibers to PMTs [56]. The FCal system provides coverage over 3.1 < |η| < 4.9. The FCal modules are located at a distance of 4.7 m from the interaction point and are exposed to high particle fluxes. To avoid ion build-up problems it is designed with very small liquid-argon gaps. These gaps are constructed by using an electrode structure of small-diameter rods centered in tubes that are oriented parallel to the beam direction. Three modules make up the FCal: an electromagnetic module (FCal1) and two hadronic modules (FCal2, FCal3). Figure 3.10 provides a schematic diagram of the FCal modules. FCal1 uses copper as an absorber to optimize resolution and heat removal. FCal2 and FCal2, on the other hand, use mainly tungsten. Extra shielding behind FCal3 is employed to reduce backgrounds in the end-cap muon system. 45 Figure 3.9: Schematic of the R-φ (left) and R-z (right) views of the hadronic end-cap calorimeter (HEC) module [57]. Figure 3.10: Schematic diagram showing the three forward calorimeter (FCal) modules [58]. 46 3.6 Muon system The outer part of the ATLAS detector is the muon spectrometer. It is designed to detect muons exiting the barrel and end-cap calorimeters in the range |η| < 2.7. Most muons pass through the inner detector without much interaction and thus it is neccesary to have a dedicated system for them. The muon spectrometer is based on the magnetic deflection of the muon tracks in the large superconducting air-core toroid magnets. Magnetic bending is provided by the large barrel toroid for |η| < 1.4. Two smaller end-cap magnets inserted at the ends of the barrel toroid provide the track bending for 1.6 < |η| < 2.7. A combinantion of these two fields provide the magnetic deflection in the transisiton region 1.4 < |η| < 1.6. In the barrel region, located between the eight coils of the superconducting barrel toroid magnet, there are eight precision-tracking chambers. In the end-cap, they are in front and behind the two end-cap toroid magnets. Each octant is divided in the azimuthal direction in two sectors (a large and a small sector). The chambers are arranged in three concentric cylindrical shells around the beam axis at a radius of 5, 7.5 and 10 m. In the two end-cap regions, the muon chambers form large wheels that are perpendicular to the z-axis at a distance of |z| = 7.4, 10.8, 14 and 21.5 m. The momentum measurement is performed by the Monitored Drift Tube chambers (MDT) that cover the range |η| < 2.7. The chambers consist of three to eight layers of drift tubes with an average resolution of 80 µm per tube (35 µm per chamber). In the forward region (2 < |η| < 2.7), the Cathode-Strip Chambers (CSC) are used due to their higher rate capa- bility and time resolution. The CSCs are multiwire chambers with cathode planes segmented into strips in orthogonal directions. This configurations allows the measurement of both co- ordinates using the induced-charge distribution. The resolution of these chambers is 40 µm 47 in the bending plane and 5 mm in the transverse plane. The muon system also has the capability to trigger on muon tracks. The precision- tracking chambers have a system of fast trigger chambers capable of delivering track infor- mation in nanoseconds after the passage of a particle. Resistive Plate Chambers (RPC) and Thin Gap Chambers (TGC) were chosen for this, in the barrel and end-cap respectively. Both chamber types deliver signals with a spread below 25 ns, thus they provide the ability to tag the beam crossing. They also measure both coordinates of the track, one in the bend- ing (η) plane and one in the non-bending (φ) plane. Muons can be measured in the inner detector and in the muon system. Figure 3.11 shows the elements of the muon system as they are arranged in the ATLAS detector. Figure 3.11: Schematic view of the ATLAS muon spectrometer system showing its sub- components [59]. 48 3.7 Magnet system ATLAS features a unique system of four large superconducting magnets. This system con- sists of a solenoid, a barrel toroid and two end-cap toroids. Figure 3.12 depicts the magnet system layout in the detector. The powerful magnetic fields produced enable the momentum measurement of electrically charged particles generated in the collisions. The solenoid is aligned on the beam axis and provides a 2 T magnetic field for the inner detector. It was designed to to keep the material thickness in front of the calorimeter as low as possible. It has an inner diameter of 2.46 m, an outer diameter of 2.56 m and an axial length of 5.8 m. The material used is Al-stabilised NbTi conductor, which achieves a high field with a reduced thickness. Figure 3.12: Schematic representation of the ATLAS magnet system, showing the central solenoid and the toroids [60]. The barrel toroid system produces a magnetic field that fills the cylindrical volume sur- rounding the calorimeter and both end-cap toroids. It consists of eight coils encased indi- vidually in racetrack-shaped stainless-steel vacuum vessels. It has a length of 25.3 m with 49 an inner and outer diameters of 9.4 m and 20.1 m respectively. The techology used for the all the toroid system is based on using a conductor of pure Al-stabilised Nb/Ti/Cu reshaped into “pancakes” followed by vacuum impregnation. 3.8 Trigger system Only a fraction of all the events that ATLAS detects contain interesting and useful infor- mation. For this reason a system is needed to ensure a proper selection of events for study. The Trigger and Data Acquisition (TDAQ) systems, the timing and trigger-control logic and the Detector Control System (DCS) achieves this goal [61]. The trigger system [62][63] has three distict levels: L1, L2 and the event filter. Each one refines the decisions the previous trigger made by applying additional selection criteria. The LHC has a collision rate of 40 MHz and ATLAS collects about 60 TB/s of data. The TDAQ and DCS systems reduce the rate of events to the order of 1 kHz and saves to permanent storage around 1.5 GB/s. The L1 trigger searches for high transverse momentum muons, photons and jets. Its selection is based on information from a subset of detectors. Muons are identified using trigger chambers in the barrel and end-cap regions of the muon spectrometer. Calorimeter selections are based on reduced-granularity information from all the calorimeters. L1 also identifies Regions-of-Interest (RoI) where the detector has identified interesting features. This first level of triggers makes a decision in less than 2.5 µs and reduces the rate from 40 MHz to 75 kHz. Events passing the L1 selection are transferred to the next stages of the detector-specific electronics and to the data acquisition. The L2 selection uses all the available detector data within the RoI’s. It is designed to reduce the trigger rate to approximately 3.5 kHz, processing an event in about 40 ms. The final stage is the event 50 selection carried out by the event filter. The event filter reduces the event rate to about 200 Hz and it is implemented offline, with an average event processing time of four seconds. 51 Chapter 4 Object Reconstruction Before we can perform any type of analysis we have to transform the electrical signals recorded by the TDAQ system from particle interactions with the detector to actual physical objects. This thesis is focused in the identification of high momentum H → bb̄ decays. To study this specific process we have to discuss how we reconstruct hadronic jets and how we identify b-hadrons. Muon reconstruction will also be discussed as muons are used in the analysis to perform corrections to the mass spectrum of the Higgs as well as triggers to fill a control region for tt̄ events. 4.1 Track and Vertex Reconstruction Tracking is performed by the inner detector, except for muons where the outer detector may also be involved. Track reconstruction using the ID covers two sequences, a main inside-out track reconstruction and a consecutive outside-in track reconstruction [64]. The pattern recognition sequence (inside-out) starts with the formation of a seed from at least 3 hits in the inner silicon tracker. This is done with the creation of three-dimensinal representations of the silicon detector measurements. From these, track seeds are built. Then, through a window search, using the seed direction, the track candidates are built. Kalman filtering [65] and smoothing are applied to the nearby hits from the detector elements to decide if they are added or rejected to the track candidates. There is a dedicated module for resolving and 52 cleaning the initial track collection to avoid ambiguity due to the presence of fake tracks or overlapping track segments with shared hits. The ambiguity solving module is based on a scoring algorithm that is optimised for each sub-detector. After this, two modules perform a track extension from the silicon detectors to the TRT. The extension to the TRT improves momentum resolution and particle identification. The final fit of the track is done using a maximum likelihood approach that involves minimizing a global χ2 . Not all tracks can be found using an inside-out procedure. Some ambiguos hits survive the ambiguity solving process and also tracks coming from secondary decay vertices may not have any silicon hits for the inside-out sequence to proceed. This could occur due to kaons (Ks ) decaying or from photon conversions. The outside-in procedure starts with the identification of tracks in the TRT using a Hough transform mechanism [66]. An association tool prevents double counting of hits that have been assigned already to tracks in the inside- out procedure. The TRT segments are then traced back into the silicon detector, which allows one to find small track segments that were missed in the initial inside-out stage. Figure 4.1 provides an example of an event showing the two track reconstruction methods. The primary vertex is reconstructed by using an iterative vertex finding algorithm [67]. Looking at the reconstructed tracks, vertex seeds are obtained. A χ2 fit is made using the seeds and nearby tracks. Each possible track gets a weight associated with it which quantifies the compatibility with the fitted vertex. Any track that has a displacement larger than 7σ from the vertex is used to seed a new vertex. The algorithm is iterated until no more vertices are found. 53 Figure 4.1: Example of an event showing the two possible TRT hit associations. Red shows extensions using the inside-out method and black shows extensions using the outside-in method [64]. 4.2 Jets Jets are a collimated spray of particles coming from a single hard interaction. A jet, in an experimental context, will be detected by its interaction with the different detector modules, creating tracks in the inner detector and depositing energy in the calorimeters. To define a jet, we need a set of rules for grouping these particles and the calorimeter deposits. Jet clustering algorithms are the main way of performing this task. For ATLAS, and this thesis, the clustering is done by using the anti-kt algorithm [68] an algorithm in the same family as the kt and Cambridge/Aachen sequential recombination algorithms. To define the clustering algorithms, we consider two distances: dij beween entities i and j, and diB between entity i and the beam B. Entities refer to particles, energy deposits or 54 pseudo-jets. The distance metrics are defined as 2 2p 2p ∆ij dij = min(kti , ktj ) 2 , (4.1) R 2p diB = kti , (4.2) where ∆2ij = (yi − yj )2 + (φi − φj )2 , kt is the transverse momentum, yi is the rapidity, φ is the azimuthal angle, R is the jet radius and p is a parameter that governs the relative power of the energy. A value of p = −1 results in the anti-kt algorithm, while p = 1 is the usual kt algorithm and p = 0 corresponds to the Cambridge/Aachen algorithm. The algorithm proceeds by identifying the shortest between the distance measures. If it is dij , then i and j get combined into one pseudo-jet. If the shortest distance is diB , then the entity i is classified as a jet and removed from the list. This procedure ends when every entity has been combined and eventually classified as a jet. Soft particles will cluster with hard particles before they cluster among themselves. A hard particle that doesn’t have another hard particle close to it, will just accumulate all the soft particles within the radius R, resulting in a conical jet. Figure 4.2 presents how a particular event is clustered into jets with four different jet algorithms. From these algorithms, only the anti-kt algorithm is simple, yet Infrared-Collinear (IRC) safe, and soft-resilient in terms of shape. Infrared safety is used to describe algorithms that are robust under the addition of soft radiation. Collinear satefy describes the fact that the result from the algorithm is not changed irregardless if the the particles are collinear (moving together in the same direction). The impact of the underlying event (UE) and pile- up on the momentum resolution for jets is close to zero, which is crucial for high luminosity experiments, like at the LHC. This can be observed by looking at the average jet area at a 55 Figure 4.2: Sample parton-level event clustered with four different jet algorithms with a radius parameter value of R = 1 [68]. given pT for dijet events clustered using different algorithms when including the underlying event and pile-up. When the ratio of the jet area and πR2 is calculated as a function of pT using different jet clustering algorithms, only the anti-kt clustered jets stay close to 1 [68]. Topological Clustering Jets deposit their energies into the calorimeters. Before appyling the anti-kt algorithm we must first find the energy clusters deposited in the detector. There are many algorithms that have been used to construct the clusters. The fixed-sized sliding window algorithm [69] was used in the early years of the ATLAS experiment but currently a more complex dynamical topological cell clustering approach is employed [70]. Topological clustering consists of finding topologically connected calorimeter signals due to a specific collision event in an attempt to extract a significant signal from a noisy back- ground. The metric used for the formation of topo-clusters is the cell signal significance σcell , 56 defined as the ratio of the cell signal Ecell to the average noise in the cell σnoise,cell : Ecell σcell = . (4.3) σnoise,cell Topo-clusters are then formed starting from a calorimeter cell with a highly significant seed signal. Three parameters (S,N,P) control how the algorithm evolves and define signal thresh- olds for seeding, growth and boundary features of the topological clustering. To begin, proto- cluster seeds from calorimeter cells with σcell > S are selected. Then all the neighboring cells satisfying σcell > N around the seeds are added. Finally the neighboring cells with σcell > P are also added to the cluster. The optimised configuration for ATLAS is: (S=4,N=2,P=0) making the resulting clusters 4-2-0 topo-clusters. Figure 4.3 shows an example of the stages of topo-cluster formation. Large Radius Jets When highly boosted massive particles decay, their decay products tend to become colli- mated, resulting in high levels of overlap between them. For a quasi-collinear splitting [17] into two objects i and j, the total mass is given by m2 ' pTi pTj ∆Rij 2 . Defining the total momentum pT = pTi + pTj and z = pTj /pT , then m2 ' z(1 − z)p2T ∆Rij 2. (4.4) 57 (a) (b) (c) Figure 4.3: Stages of topo-cluster formation in the first module of the FCAL calorimeter for a simulated dijet event. Shown in (a) are the cells with signal significance σcell > 4 that can seed topo-clusters, in (b) cells with σcell > 2 controlling the topo-cluster growth, and in (c) all clustered cells and the outline of topo-clusters in this module [71]. 58 In the case of a Higgs boson decaying to a b-quark pair, the momentum fraction is uniform (z = 0.5). Therefore the angular separation of its decay products is approximately 2mH ∆R ' . (4.5) pT For massive particles with high pT , the ability to resolve individual hadronic decay products using standard narrow-radius jets begins to degrade. The b-quark pair coming from a Higgs boson at a pT ' 250 GeV would be separated by approximately ∆R ' 1. Reconstructing these objects in a single large-radius (large-R) jet is advantageous in order to maximize efficiency [72]. Figure 4.4 contains an illustration of the degree of collimation of the decay products of a massive Z 0 boson when the pT increases. Figure 4.4: Diagram showing the degree of collimation of the decay products of massive particle decaying as pT increases [73]. A single jet containing all the decay products of a massive particle has different properties than a jet originating from a light quark. These large-R jets are rich with multi-pronged substructure, properties that are absent in jets formed from gluons and light quarks. In ATLAS, large-R jets are reconstructed using the anti-kt clustering algorithm with a radius 59 parameter R = 1.0. Jet Trimming When a hard scattering event occurs, the detector records more than just the final states. Initial state radiation (ISR), multiple parton interactions (MPI), underlying event (UE) remnants and pile-up all contribute to the final state. This complicates the jet definition as it is often important to discriminate between these types of energy and the jet of interest. In the case of large-R jets, the subtle substructure differences of jets formed from a massive particle decay products and jets coming from quarks and gluons can be resolved more clearly by removing soft QCD radiation from them [72]. The process of removing soft radiation during the jet reconstruction is referred to as jet grooming. One of these grooming procedures is known as jet trimming [74]. The trimming algo- rithm starts by clustering cells into jets with any clustering algorithm, for example, the anti-kt algorithm, and calling them seed jets. For each seed jet, all of its constituents are then reclustered using another jet algorithm into subjects with a characteristic radius Rsub . Subjets from the original seed jet are discarded if they have pTi < fcut ·Λhard , where fcut is a fixed dimensionless parameter and Λhard is a hard scale chosen depending on the kinematics of the event. Finally, the remaining subjets are assembled into the new trimmed jet. Figure 4.5 contains a diagramatic representation of how the trimming procedure is performed. The analysis presented in this document uses trimmed large-R jets with parameters Rsub = 0.2 and fcut = 0.05. The scale Λhard chosen is the original jet pT , and therefore the subjets with a pT of less than 5% of the original jet pT are removed. 60 Figure 4.5: Diagram that depicts the jet trimming procedure employed in this analysis [75]. Jet Calibration Before applying a jet clustering algorithm, cell clusters need calibration to correct for the effects of a non-compensating calorimeter response to hadrons, to accidental signal losses and to energy lost in the inactive material. The calibration strategy is referred to as “lo- cal hadronic cell weighting” (LCW) [70]. Topo-clusters calibrated using this method are transformed to be at the LCW scale. After the calibration a large-R jet is defined with the topo-clusters using the anti-kt algorithm and subsequently trimmed. The energy, pseudorapidity and mass calibration of the LCW jets are corrected for resid- ual detector effects, using energy and pseudorapidity dependent calibration factors derived from simulation [76]. The correction restores the average reconstructed calorimeter jet en- ergy scale (JES) to that of particle-level jets. These scale factors are applied as multiplicative weights that correct the distributions to the proper scales. The reconstructed large-R jet energy, mass, pseudorapidity and transverse momentum become Ereco = cJES E0 , mreco = cJES m0 , (4.6) ηreco = η0 + ∆η, preco T = cJES |~ p0 | cosh (η0 + ∆η), (4.7) where E0 , m0 , η0 , p~0 are the jet properties before any calibration. The correction factors 61 cJES and ∆η are smooth functions of the large-R jet kinematics. The JES factor cJES is parametrized by a Gaussian fit of the average jet energy response RE = hEreco /Etruth i. An extra jet mass scale (JMS) calibration is performed after the energy scale calibra- tion. The correction factor cJMS is applied as a function of Ereco , η and log (mreco /Ereco ). The definition of the correction factor is determined using the same procedure as the jet energy calibration but using the jet mass response, Rm = hmreco /mtruth i, instead. The reconstructed kinematic variables corrected are then q mreco = cJES cJMS m0 , preco T = cJES E02 − c2JMS m20 / cosh (η0 + ∆η). (4.8) The final step of the calibration is the in situ calibration method [77] to bring data to agreement with MC using response measurements in pp collision data of well known objects, such as dijet events, that work as a reference. Scale factors are derived in the same fashion as the JES and JMS calibration but the response is defined by the ratio of jet properties of data and MC. At the end the groomed jets should be at the proper jet energy scale (JES) and jet mass scale (JMS). Uncertainties are also derived through this process. An overview of all the calibration steps is shown in Figure 4.6. Figure 4.6: Overview of the reconstruction and calibration procedure for large-R jets [77]. 62 Jet Mass One of the most powerful tools to distinguish jets that contain the decay products of massive particles from the multijet background is its mass. Jet mass is considered one of the most important jet substructure (JSS) variables for large-R jets. The jet mass used in the analysis is referred to as combined jet mass mcomb [77]. It is called the combined jet mass because it is a smooth interpolation between two other mass definitions: a calorimeter-based jet mass mcalo and track-assisted jet mass mTA [78]. For a large-R jet J with calorimeter-cell cluster constituents i the mcalo is defined as: v !2 !2 u u X mcalo = t X Ei − p~i . (4.9) i∈J i∈J Given that the angular spread of the decay products of a boosted massive particle scales as 1/pT , the spread is comparable with the calorimeter granularity at high values of pT . It is possible to include tracking information to maintain performance at high pT . The track-assisted mass is defined as: pcalo mTA = trackT × mtrack (4.10) pT where pcalo T is the transverse momentum of the large-R calorimeter jet, pT track is the transverse momentum of the four-vector sum of tracks associated to the large-radius calorimeter jet, and mtrack is the invariant mass of the four-vector sum. This mass measurement has a better resolution for high-pT jets with low values of m/pT . The combined jet mass smoothly interpolates between mcalo at low pT and mTA at high 63 pT . A weighted least-squares combination is performed to define mcomb : mcomb = wcalo mcalo + wTA mTA (4.11) where the weights are determined by the mass resolutions σcalo , σTA of the calorimeter and track-assisted measurements. These are derived using the jet mass response distribution in dijet events. They are defined as −2 σcalo −2 σTA wcalo = −2 −2 wTA = −2 −2 (4.12) σcalo + σTA σcalo + σTA with the constraint wcalo + wTA = 1. Variable Radius Track Jets Track jets are formed by applying jet clustering on tracks in the inner detector from charged particles originating from the hard scattering vertex. They are crucial for finding b-hadrons and are used in this analysis to integrate b-tagging methods with large-R groomed jets. Using a fixed radius size approach for reconstructing track jets from highly boosted massive particles presents problems in the identification of more than one charged jet given the high degree of collimation. A variable radius (VR) approach is neccesary to maintain acceptable levels of efficiency. A VR jet is a jet that has been reconstructed with the use of a pT dependent effective radius Reff [79]. It requires the defnition of a parameter ρ that controls how the radius 64 changes as a function of pT , " # ρ Reff (pT ) = min , Rmax . (4.13) pT In this analysis the VR track jets are reconstructed using the anti-kt algorithm, with ρ = 30 GeV and the lower and upper bounds of the track-jet radius being Rmin = 0.02 and Rmax = 0.4. The value of Rmin is dictated by the detector resolution. These provide the optimal performance for high-pT Higgs jets decaying to b-quarks [80]. Figure 4.7 shows the efficiency of subjet double b-labelling of track jets associated with a Higgs jet (in MC) using different track jet clustering algorithms. The efficiency using the standard track jets with R = 0.2 degrades sharply for Higgs jets with pT > 1 TeV. Figure 4.7: Efficiency of subjet double b-labelling at the truth level of a Higgs jet as a function of pT using VR track jets with Rmin = 0.02 and Rmax = 0.4 for different values of ρ [80]. Track jets are matched to large-R jets using a process called ghost-association [81]. This procedure consists of treating track jets as infinitely soft particles by setting their pT to 1 eV. 65 This is done to not affect the reconstruction of the calorimeter jets. The jets are added to the list of inputs of the jet finding algorithm which makes it possible to identify which tracks were clustered in which subjets. This technique facilitates the measurement of the ghost area, the effective area of a jet. Instead of identifying tracks associated with the resulting jets, the number of ghost particles present in the jet after recontruction defines the effective area of that jet. 4.3 b-hadron Identification ATLAS uses various tagging algorithms to identify b-jets [82]. These are referred to as b-tagging algorithms [83], and they exploit the long lifetime of b-hadrons as well as the properties of the b-quark fragmentation. Measureable b-hadrons have a significant mean flight length in the detector before decaying. This leads to an extra vertex displaced from the hard-scatter collision point. An illustration of a track jet with a displaced secondary vertex is shown in Figure 4.8. b-tagging Taggers can be divided into two main categories, low-level taggers and high-level taggers. Low level taggers are traditional track-based impact parameter taggers. Examples of these include IP2D and IP3D, SV1 [85] and JetFitter [86]. They are based on a log-likelihood ratio (LLR) discriminant that separates tracks associated to jets according to their compati- bility to the primary vertex. IP2D and IP3D use the transverse impact parameter significance as discriminating variables. The other low-level taggers, SV1 and JetFitter, are secondary vertex-based b-tagging algorithms. All the discriminating variables produced by the low-level 66 Figure 4.8: Diagram of a track jet with displaced secondary vertex from the primary vertex [84]. taggers are used as inputs for the high-level algorithms. The high-level taggers used in this thesis are MV2 [87] and DL1 [87]. MV2 consists of a boosted decision tree (BDT) algorithm. It is trained using the ROOT Toolkit for Multivariate Data Analysis (TMVA) [88] on a hybrid tt̄ + Z 0 sample. The kine- matic properties of the jets are included in the training in order to exploit the correlations with the other input variables. However, for b-jets and c-jets, pT and |η| are reweighted to match the spectrum of the light-flavor jets. DL1 is based on an Artificial Deep Neural Network (DNN) trained using Keras [89] with a Theano [90] backend. DL1 has a multidimensional output corresponding to the probabilities for a jet to be a b, c or light-flavor jets. Similar to MV2, a reweighting of pT 67 (a) (b) Figure 4.9: Distribution of the output discriminant for (a) MV2 and (b) DL1 b-tagging algorithms [83] . and |η| is performed. DL1’s final discriminant is defined as:   pb DDL1 = ln (4.14) fc · pc + (1 − fc ) · plight where pb , pc , plight and fc are the b-jet, c-jet, light-flavor jet probabilities, and the effective c-jet fraction in the background training sample. Figure 4.9 the values of the MV2 and DL1 discriminant is shown for the different types of jets. The b-tagging algorithms are calibrated in terms of their efficiency working points (WP) by making a cut on the discriminant values. The b-tagging efficiency is defined as Nbtagged bjets btagging = . (4.15) Ntotal bjets A cut on the corresponding discriminant is done such that the overall efficiency WP at a desired kinematic range stays constant. The WP chosen for the study shown in this 68 (a) (b) Figure 4.10: The (a) light-flavor jet (b) c-jet rejections versus the b-tagging efficiency [83] . thesis is the 77% efficiency WP. For every efficiency, there is also a background rejection rate associated with it. The background rejection is calculated as the inverse of the b-mistag rate (1/bkg ). The light-flavor and c-flavor rejections are shown in Figure 4.10 as a function of b-tagging efficiency for multiple b-taggers. Double b-tagging Given that we are looking for H → bb̄ we require the identification of 2 b-jets. When tagging 2 b-jets, multiple schemes have been developed [91]. The benchmarks are: double, asymmetric, single and leading single b-tagging. Double b-tagging takes the two highest-pT track jets that pass the b-tagging requirement. For asymmetric b-tagging, the track jet that is more consistent with the interpretation of being a b-jet must pass a fixed efficiency working point, while the b-tagging requirement on the second track jet is varied. In single b-tagging 69 (a) (b) Figure 4.11: The multijet rejection as a function of the Higgs tagging efficiency for large-R jets with pT above (a) 250 GeV and above (b) 1000 GeV [91] . at least one of the two highest-pT track jets must pass the b-tagging requirement, while on leading single b-tagging only the highest-pT track jet must pass the b-tagging requirement. The scheme adopted by the study performed in this thesis is double b-tagging as it is the scheme with the largest multijet and top-jet rejection across a large range of Higgs efficiencies. This can be seen in Figure 4.11. 4.4 Muons Muon reconstruction [92] is performed both in the Inner Detector (ID) and in the Muon Spectrometer (MS). This information is then combined to form the muon tracks that are used in the analyses. The Monitored Drift Tubes (MDT) segments are reconstructed with a straight-line fit to the hits of each layer. The Resistive Plate Chambers (RPC) and Thin Gap Chambers (TGC) are used to measure the coordinate orthogonal to the bending plane. For segments in the Cathode Strip Chambers (CSC), a separate combinatorial search in the η, φ detector planes is performed. Muon track candidates are then built by combining together all the segments in the different layers. 70 There are four muon types defined depending on which subdetectors are used in their reconstruction. They are: Combined (CB) muons, Segment-tagger (ST) muons, Calorimeter- tagger (CT) muons and Extrapolated (ME) muons. Combinations of these are used to select and identify muons. Muon identification selection is divided into four categories: Medium, Loose, Tight and High-pT . In this analysis, Medium muons are used, the default ATLAS selection. This selection uses only tracks from CB muons and ME muons and it minimizes the systematic uncertainties associated with reconstruction and calibration. For CB muons, reconstruction follows an outside-in pattern recognition; the muons are first reconstructed in the MS and then are extrapolated inward by matching to an ID track. The independent tracks from the ID and MS are combined using a global fit. ME muon reconstruction is based only on the MS track and a requirement on compatiblity with the interaction point (IP). ME muons are mainly used to extend the acceptance into the region not covered by the ID (2.5 < η < 2.7). Muon isolation criteria optimized for different analyses are also defined [92]. Two vari- topocone20 ables are used to make the isolation cuts, pvarcone30 T and ET . The track-based variable, pvarcone30 T , is the scalar sum of the transverse momenta of the tracks with pT > 1 µ GeV in a cone size ∆R = min(10 GeV/pT , 0.3) around the muon of transverse momentum µ topocone pT . The calorimeter-based isolation variable, ET , is the sum of the transverse en- ergy of the topological clusters in a cone size ∆R = 0.2 around the muon direction. The contribution of the muon itself to the energy deposits is subtracted. The isolation working point (WP) used in this analysis is referred to as the FixedCutLoose 71 WP. The cuts on the discriminating variables are µ topocone20 µ pvarcone30 T /pT < 0.15 and ET /pT < 0.30. (4.16) Figure 4.12 shows the efficiency for the FixedCutLoose muon isolation working point as a function of pT . Figure 4.12: Isolation efficiency for the FixedCutLoose muon isolation working point [92]. 72 Chapter 5 Boosted H → bb̄ Analysis 5.1 Introduction After describing how the different physics objects are constructed from collisions in the de- tectors we can now perform the analysis. The analysis has the goal of extracting information by examining the events from real data gathered by the ATLAS detector using guidance from MC based collision simulations. For this particular analysis, the goal is to measure the Higgs boson production cross section inclusively, in the fiducial volume and differentially (in pT ), using its H → bb̄ decay. The measurement of the Higgs boson cross section for pT > 1 TeV is of particular interest and is the first of its kind made by the ATLAS Collaboration. Before this study, the measurement of the Higgs boson cross section using its H → bb̄ decay exploited the semileptonic decays of vector bosons in the V H production modes by requiring additional triggers involving muons and missing energy [26], or in the VBF produc- tion mode by requiring additional jets [93]. In this analysis, we do not impose any restrictions on the production mode and simply require the presence of an extra jet accompanying the Higgs. This selection allows for access to a large Higgs boson pT range using triggering with large-R jets. Without this requirement, the large QCD backgrounds would require single b-jet triggering which limits the mass range for which the two jets can be reconstructed [94], eliminating the possibility of measuring the 125 GeV Higgs boson. 73 Signal Measurement The measurements presented in this thesis uses the signal strength µH extracted through a binned likelihood fit. The signal strength is defined as the ratio of the observed yield of the signal (NH ) over the Standard Model prediction (NSM ): NH µH = . (5.1) NSM The measurement of the signal strength µH is then used to calculate the Higgs boson production cross-sections for different phase space regions. The measured cross-section is given by NH σH = , (5.2) c×L where c is a correction factor, L is the integrated luminosity and NH = µH NSM is the yield of signal events. The correction factor is defined as c = A where A is the fiducial acceptance and  is the selection efficiency. By this definition, the signal strength can also be defined as the ratio between the measured and the expected signal cross sections µH = σH /σSM . Maximum Likelihood Method Suppose you have a set of random variables x that are distributed following a probability density function (p.d.f.) f (x; Θ) with unknown parameters Θ = (Θ1 , . . . , Θm ). By choosing a specific functional form of f (x; Θ) we can use the maximum likelihood method to estimate the values of the parameters given a finite sample of data. The likelihood function is defined as Yn L(Θ) = f (xi ; Θ) (5.3) i=1 74 where xi are the different outcomes of repeated measurements. Then the estimators for the parameters are those that maximize the likelihood function, ∂L = 0, i = 1, . . . , m. (5.4) ∂Θi In our analysis the random variable xi is the number of events in a specific mass bin given the signal and background models. Therefore xi ≡ xi (µ, θ) where µ are the set of all the signal strengths for signal or scale factors for the backgrounds and θ are nuisance parameters used for uncertainty estimation. In each bin, the expected number of events is given by X X xi (µ, θ) = µs xi,s (θ) + µb xi,b (θ). (5.5) s∈sig b∈bkg The quantities xi,s and xi,b are taken from the Asimov datasets constructed using MC mass templates of the resonances, or in the case of the QCD background, a parametric function. When a maximum likelihood fit is performed on an Asimov dataset, the results are the expected signal strengths and their expected uncertainties µ̂H = 1 ± σ̂(µH ). The hat on the parameters is used to identify them as estimators (parameter values for which the likelihood function is a maximum) of the real parameters. The specifics of the MC templates for the signal and backgrounds are explored in the modeling sections 5.5 and 5.6. The functional form of the p.d.f. is a Poisson distribution. The Poisson distribution expresses the probability of a given number of events ocurring in a fixed interval. The binned likelihood function therefore takes the following form: Y (xi (µ, θ))Ni L(µ, θ) = exi (µ,θ) , (5.6) Ni ! i∈bins 75 where Ni is the number of data events in bin i. The nuisance parameters θ = (θ1 , . . . , θm ) encode the dependence on systematic uncer- tainties. The prior knowledge on these parameters is used to make the displacement from the nominal value disfavored. They enter the equation as multiplicative factors in the likelihood function and are Gaussian distributions centered at 0 and a standard deviation of 1: 1 2 g(θm ) = √ e−θm /2 . (5.7) 2π Confidence Levels To test the compatibility of a background only model with the data or derive confidence limits on our observations we use a test statistic. According to the Neyman-Pearson lemma, we can consider the function constructed from a ratio of likelihood functions for a set of parameters that describe the null hypothesis H0 and a set of parameters that describe the alternate hypothesis H1 as the best hypothesis discriminator L(H0 ) q = −2 ln . (5.8) L(H1 ) This reduces to a χ2 distribution in the large sample limit. The fit result is obtained by maximizing the log-likelihood function with respect to all the parameters. For a discovery, the null hypothesis H0 is the likelihood function where µ = 0 (no signal). More specifically in our case, where the observed signal is small, we can build confidence limits using the CLs [95] method based on the test statistic ˆ L(µ, θ̂(µ)) qµ = −2 ln , (5.9) L(µ̂, θ̂) 76 where µ̂, θ̂ are the parameters that maximize the constrained (0 ≤ µ̂ ≤ µ) likelihood and ˆ θ̂(µ) are the nuisance parameters which maximize the likelihood function for a given value of µ. The variance of µ̂, can be directly calculated using the test statistic (µ − µ̂)2 σµ̂2 = . (5.10) qµ The p-value of an hypothesized µ and the corresponding significance is given by √ √ pµ = 1 − Φ( qµ ) and Zµ = qµ , (5.11) where Φ is the cumulative distribution function for the standard Gaussian. Finally the upper limit of an estimator at a 1 − α confidence level is given by µup = µ̂ + σµ̂ Φ−1 (1 − α). (5.12) Therefore, for a 95% upper limit, α would be 0.05. 5.2 Samples Data √ The data used was collected by the ATLAS detector at s = 13 TeV during Run 2 (2015- 2018) of the LHC. All the events come from the Good Runs Lists (GRL) [96]. The events in the GRL have collisions with bad detector performance removed. Table 5.1 lists all the GRL datasets used. The GRL list of events sums up to an ingrated luminosity of 139 fb−1 [97]. This quantity 77 Table 5.1: Summary of the Good Run List (GRL) datasets used in this analysis. The data was collected during Run 2 of the LHC. Year Dataset Name 2015 data15 13TeV.periodAllYear DetStatus-v89-pro21-02 Unknow PHYS StandardGRL All Good 25ns.xml 2016 data16 13TeV.periodAllYear DetStatus-v89-pro21-01 DQDefects-00-02-04 PHYS StandardGRL All Good 25ns.xml 2017 data17 13TeV.periodAllYear DetStatus-v99-pro22-01 Unknown PHYS StandardGRL All Good 25ns Triggerno17e33prim.xml 2018 data18 13TeV.periodAllYear DetStatus-v102-pro22-04 Unknown PHYS StandardGRL All Good 25ns Triggerno17e33prim.xml is less than the total integrated luminosity (156 fb−1 ) that was delivered by the LHC due to the quality requirements and trigger efficiencies. Table 5.2 summarizes the integrated luminosity and its uncertainties for each year of data taking period included in the Run 2 dataset. The uncertainty sources can be correlated between all years, correlated between a subset of years or completely uncorrelated. Therefore the relative error on the total is not a weighted sum of the relative errors on the individual years. It considers the covariance matrix of the absolute luminosity uncertainties for the differrent years where correlated sources are represented by terms with non-zero off-diagonal entries. Table 5.2: √ Summary of the integrated luminosities and uncertainties for the Run 2 pp data sample at s = 13 TeV [97]. Data sample Luminosity Uncertainty 2015-2016 36.2 fb−1 2.1 % 2017 44.3 fb−1 2.4 % 2018 58.5 fb−1 2.0 % Total 139.0 fb−1 1.7 % In this analysis, the data amounts to an integrated luminosity of 136 fb−1 due to the specific Large-R jet and muon trigger requirements. Each event satisfies a trigger that requires a large-R (R = 1.0) jet reconstructed using the anti-kt algorithm. For each year of data taking, the jet pT and mass thresholds for triggers differ due to changes in luminosity profiles, inclusion of new techniques [98] and generally different beam conditions. The jet pT thresholds go from 360 to 460 GeV and the mass trigger is either 0, 30 or 35 GeV. The 78 additional muon triggering is used to fill a control region for top quarks that requires a muon with pT > 50 GeV [99]. Table 5.3 summarizes the triggers used in this analysis. Plots for the efficiency curves for the triggers used in this analysis are presented in Appendix B. Table 5.3: Summary of the triggers used in this analysis. Large-R jet triggers are used to identify candidate jets, while the muon trigger is used to fill the tt̄ control region. Year Trigger Treshhold Luminosity [fb−1 ] Large-R jet Triggers 2015 HLT j360 a10 lcw sub L1J100 pT > 360 GeV 3.2 2016 HLT j420 a10 lcw L1J100 pT > 420 GeV 33.0 2017 HLT j390 a10t lcw jes 30smcINF L1J100 pT > 390 GeV, mJ > 30 GeV 41.0 HLT j440 a10t lcw jes L1J100 pT > 440 GeV 41.2 HLT j420 a10t lcw jes 35smcINF L1J100 pT > 420 GeV, mJ > 35 GeV 58.5 2018 HLT 420 a10t lcw jes 35smcINF L1SC111 pT > 420 GeV, mJ > 35 GeV 55.4 HLT 460 a10t lcw jes LJ100 pT > 460 GeV 58.5 Muon triggers µ All years HLT mu50 L1 MU20 pT > 50 GeV 139 fb−1 Simulated Samples Monte Carlo (MC) programs are used to simulate events which then are used to model the signal and backgrounds pertinent to this analysis. The signal consists of the Higgs production processes ggF, VBF, VH and tt̄H. The background samples include W +jets, Z+jets, top quark production and dijets events. Table 5.4 has a summary of all the simulated samples used in the analysis. Signal Samples Higgs production through ggF is simulated at NLO QCD accuracy including, the finite top mass effects with the Hj-MINLO (Multi-Scale Improved NLO) [100] prescription using 79 Powheg Box v2 [101] [102]. In a similar manner, NLO accuracy in QCD is achieved for VBF and tt̄H production [103] [104]. gg → V H production at LO accuracy is calculated using Powheg Box v2 as calculations at NLO required new developments [105] not pub- lished at the time this analysis was performed. qq → V H [106] is calculated at NLO accuracy using GoSam [107]. Electroweak (EW) NLO corrections are also applied as a function of the Higgs boson momentum for the VBF, VH and tt̄H production modes using HAWK [108] (see Appendix B). Finally the branching ratios are calculated using Hdecay [109] and Prophecy4F [110]. Background Samples Vector boson + jets was simulated using Sherpa [111] with NLO QCD accuracy. NLO EW approximate corrections were applied which reduced the predicted yield by 10 − 20%. The NNLOJET group provided NNLO QCD custom corrections as a function of the generated vector-boson momentum (pVT ) that are then applied on top of the NLO EW corrections. Every top quark production mode was modeled using Powheg Box v2 at NLO QCD. Top quark pair production, tW and single-top t-channel and s-channel are all included [112] [113]. QCD multijet events were modeled using a parametric model. The MC used to study the model was generated by Pythia 8.230 [114]. After generation, hadronization and showering every event is put through an ATLAS detector simulation that is based on Geant4 [41]. Pileup and multi-particle interactions were also modeled using Pythia 8.186 with the A3 tuning [115] and were also fed through the same ATLAS detector simulation. 80 Table 5.4: Summary of the simulated samples for the signal and background processes [10]. Process ME generator ME PDF PS and hadronization UE model tune Cross-section order Higgs Boson gg → H → bb̄ Powheg Box v2 + MINLO NNPDF3.0N N LO Pythia 8.212 AZNLO NLO(QCD) + LO(EW) qq → H → q 0 q 0 bb̄ Powheg Box v2 NNPDF3.0N LO Pythia 8.230 AZNLO NLO(QCD) + NLO(EW) qq → W H → q 0 q 0 bb̄ Powheg Box v2 + MINLO + GoSam NNPDF3.0N LO Pythia 8.240 AZNLO NNLO(QCD) + NLO(EW) qq → W H → lνbb̄ Powheg Box v2 + MINLO + GoSam NNPDF3.0N LO Pythia8.212 AZNLO NNLO(QCD) + NLO(EW) qq → ZH → q q̄bb̄ Powheg Box v2 + MINLO + GoSam NNPDF3.0N LO Pythia 8.240 AZNLO NNLO(QCD) + NLO(EW) qq → ZH → ννbb̄ Powheg Box v2 + MINLO + GoSam NNPDF3.0N LO Pythia 8.212 AZNLO NNLO(QCD) + NLO(EW) qq → ZH → llbb̄ PowhegBox v2 + MINLO + GoSam NNPDF3.0N LO Pythia 8.212 AZNLO NNLO(QCD) + NLO(EW) gg → ZH → q q̄bb̄ Powheg Box v2 NNPDF3.0N LO Pythia 8.240 AZNLO LO +NLL(QCD) gg → ZH → ννbb̄ Powheg Box v2 NNPDF3.0N LO Pythia 8.212 AZNLO LO +NLL(QCD) gg → ZH → llbb̄ Powheg Box v2 NNPDF3.0N LO Pythia 8.212 AZNLO LO +NLL(QCD) gg → tt̄H → all Powheg Box v2 NNPDF3.0N LO Pythia 8.230 AZNLO NLO(QCD) + NLO(EW) gg → tt̄H → all Powheg Box v2 NNPDF3.0N LO Pythia 8.230 AZNLO NLO(QCD) + NLO(EW) Vector Boson + jets W → q q̄ Sherpa 2.2.8 NNPDF3.0N N LO Sherpa 2.2.8 Default NNLO(QCD) + approx NLO(EW) Z → q q̄ Sherpa 2.2.8 NNPDF3.0N N LO Sherpa 2.2.8 Default NNLO(QCD) + approx NLO(EW) Top quark tt̄ →all Powheg Box v2 NNPDF3.0N LO Pythia 8.230 A14 NNLO + NNLL tW Powheg Box v2 NNPDF3.0N LO Pythia 8.230 A14 NLO t t-channel Powheg Box v2 NNPDF3.0N LO Pythia 8.230 A14 NLO t s-channel Powheg Box v2 NNPDF3.0N LO Pythia 8.230 A14 NLO Multijet Dijets Pythia 8.230 NNPDF2.3.LO Pythia 8.230 A14 LO 5.3 Object Definition Object Reconstruction A Lorentz boosted Higgs boson event has a topology of the form: pp → H(→ bb̄) + j. Therefore the events of interest are better described by two large-R (R = 1.0) jets in which one of them contains the decay products of two b-hadrons. The large-R jets are defined by applying the anti-kt algorithm, using the software package Fastjet [116], to topological clusters of calorimeter energy deposits. A jet trimming procedure is employed with param- subjet jet eters R = 0.2 and pT /pT < 0.05. The jet mass mJ is defined as the combined mass, a weighted combination of the calorimeter based mass and the track-assisted jet mass. Variable radius (VR) track jets are formed using the anti-kt algorithm with parameters Reff = ρ/pT where ρ = 30 GeV and have an upper bound of Rmax = 0.4. Ghost association is used to match large-R jets (before trimming) to VR track jets. For simulation events, 81 track jets are labeled as having b,c or light (u,d,c and g) flavor by truth matching hadrons with pT > 5 GeV within ∆R = 0.3 of the jet axis [87]. The b-tagger MV2 is used to tag VR track jets containing a b-hadron decay. Track jets must have pT > 10 GeV and |η| < 2.5. At least two track jets per event are considered. The working point is tuned to have an average b-tagging efficiency of 77% for b-jets in simulated tt̄ events. The misidentification efficiencies are 0.9% for light-jets and 25% for c-jets. To prevent overlap between VR track jets, if the ∆R between any two track jets with pT > 5 GeV associated with a large-R jet [91] is less than their respective radii, then the jet is not considered for b-tagging. µ Muons satisfy the Medium quality criterion. Muons have to satisfy |η| < 2.5 and pT > 10 GeV. Isolated muons also have to satisfy loose track and calorimeter based isolation conditions [92]. For a more detailed view of the object reconstruction, algorithms, calibrations and ref- erences used refer to Chapter 4. Analysis Object Definitions Reconstructed jets that have the properties compatible with a H → bb̄ decay are labeled candidate jets. Theoretically the Higgs boson and the hadronic recoil system both have the same pT . In practice, the reconstructed jet pT is affected by final state radiation, jet resolution and any other activity outside the jet cone like pile-up. From simulation it is estimated that roughly 50% of the Higgs jets are leading jets (jet with the highest pT ) and 47% are subleading jets (jet with the second largest pT ). Figure 5.1 illustrates this phenomenon using simulated ggF events. For this reason, candidate jets are defined as either the leading or sub-leading jet that has a pT > 250 GeV, has |η| < 2.0 and satisfies the 82 boosted condition: 2mJ /pT < 1. Each candidate jet must contain at least two track jets. The candidate jets are classified as double-tagged if its two leading track jets are b-tagged or anti-tagged if neither of them are b-tagged. Figure 5.1: Difference between the pT of the Higgs matched large-R jet and the recoil jet. Higgs jets are leading pT if the difference is positive (right of the dashed line) or subleading if the difference is negative (left of the dashed line). The presence of semileptonic b-hadron decays motivates the application of a correction to candidate jets. The ‘muon in jet’ correction uses the leading-pT muon found within µ ∆R = min(0.4, 0.4 + 10/pT ) of a b-tagged VR track jet. The correction consists of removing the energy deposited by the muon in the calorimeter and adding its four-momenta to the trimmed large-R jet. The corrrection is of the order of 13% for leading Higgs jets and 33% for subleading Higgs jets in simulated ggF events. The mJ width is reduced by 5% and 12% respectively. The uncorrected momentum and mass in candidate jets are denoted as p0T and m0J respectively. 83 5.4 Event Selection Events are classified into three separate regions: a signal region (SR), a control region (CRtt̄ ) and a validation region (VR). The SR is used to calculate the signal strength. The CRtt̄ is used as a control region to study top quark events. The VR is used to test the multijet and V +jets models. For both the SR and the VR at least one jet with p0T > 450 GeV and m0J > 60 GeV is required. The second jet required has to have a p0T > 200 GeV. From these two jets at least one of them have to satisfy the candidate jet criteria. The SR and VR are subdivided into subregions according to the pT ordering of the candidate jets. Figure 5.2 summarizes the regions used for event categorization in the analysis. Leading Large-R Jet Not s ta g Cand. b- Not Cand. VRL Subleading VRS SRL Large-R Jet SRS Figure 5.2: Diagram showing the event categorization criteria. Candidate jets are categorized by the number of ghost-associated b-tagged track jets as well as its pT ordering in the event. 84 Signal Region For an event to be assigned to the SR it must be a double-tagged candidate jet. If the event is the leading jet then it will populate the leading jet signal region (SRL). If the double-tagged jet is not the leading jet, but the subleading jet, it will populate the subleading-jet signal region (SRS). Figure 5.3 shows the mass distributions for all the processes that contribute to the signal region. (a) (b) Figure 5.3: Jet mass distributions for the Higgs boson, Z+jets, W+jets, and top quark contributions from the SM prediction as well as the multijet jet mass distribution extracted from data in the signal region (SR) defined by the leading (a) and subleading (b) jets [10]. SR Configurations The signal is extracted in three different SR configurations, providing three measurements of the Higgs cross-section. First, for the inclusive measurement, the Higgs boson signal strength µH is extracted from the signal region containing candidate jets with pT > 250 GeV. Second, a fiducial measurement is performed on the fiducial volume where the candidate jets have pT > 450 GeV and |yH | < 2 defined by the acceptance cuts of this analysis. Finally, the 85 differential measurement, where the signal strength extraction is performed for candidate jets in the pT ranges 250-450 GeV, 450-650 GeV, 650-1000 GeV and > 1 TeV. The bin with 250 < pT < 450 GeV is populated only by candidate jets from the subleading signal region (SRS). Table 5.5: Summary of the candidate jet pT requirements for the three Signal Region con- figurations [10]. Candidate jet pT (GeV) Region SRL SRS Inclusive > 450 > 250 Fiducial > 450 > 450 450 − 650, 250 − 450, Differential 650 − 1000, 450 − 650, > 1000 650 − 1000 Validation Region Similarly to the SR region the VR is subdivided into the leading jet validation region (VRL) and the subleading jet validation region (VRS). The main difference lies in the b-tagging requirements. Every event of the VR must be an anti-tagged candidate jet. Control Region To constrain the tt̄ background a dedicated control region CRtt̄ was defined. The high purity of top quark pair events is achieved using the muon-trigger to choose events in which one of the tops decays semiletopnically (t → lνb) while the other decays hadronically (t → qq 0 b). Each large-R jet in this region must have at least one b-tagged VR track-jet associated to it. The large-R jet that has a close isolated muon with pT > 52.5 GeV is labeled as Jb and is associated to the semileptonically decaying top quark. On the other hand, for the 86 hadronically decaying top, the large-R jet Jt requires at least 3 associated VR track-jets. These two jets (Jb , Jt ) must have an angular separation of at least ∆φ > 2π/3 to ensure a back-to-back topology. Table 5.6 summarizes the selection criteria used for the tt̄ control region CRtt̄ . Table 5.6: Summary of the CRtt̄ selection criteria for the semileptonically decaying top quark (Jb ) and the hadronically decaying top quark (Jt ) [10]. Jet N track-jets N b-tags Angular selection Jet mass [GeV] µ Jb ≥1 1 0.04 + 10/pT < ∆R(µ, J b ) < 1.5 − Jt ≥3 1 ∆φ(J b , J t ) > 2π/3 140 − 200 5.5 Higgs Boson Modeling The selection criteria chosen in this analysis provides an inclusive view of the Higgs boson in terms of its four main production modes. Therefore the production modes considered are ggF, V H, VBF and tt̄H. Considering Higgs bosons near the mass peak (105 < mJ < 140 GeV) and pT > 450 GeV, the largest contribution for Higgs production comes from the ggF process. On the other hand, in the SRS (pT < 450 GeV) the largest production mode is tt̄H. In this bin, highly energetic hadronic tops can satisfy the trigger requirements in events where the Higgs boson has low pT due to the nature of three body decays. Table 5.7 shows the relative contribution of the main production modes in the SR. The Higgs boson also contributes to the VR to a lesser extent. The breakdown of the contribution of the Higgs boson to the SR and VR as a function of mass is shown in Figure 5.4 and as a function of pT in Figure 5.5. Multiple modeling systematics were considered for the Higgs boson. These are: factor- ization and renormalization scale variations, cross-section and acceptance, PDF uncertainty, 87 Table 5.7: Fractional contribution of each production mode to the SR configurations around the Higgs boson mass peak (105 < mJ < 140 GeV) [10]. Jet pT range (GeV) Process 250 − 450 450 − 650 650 − 1000 > 1000 SRL ggF − 0.56 0.50 0.39 VBF − 0.17 0.16 0.17 VH − 0.14 0.18 0.25 tt̄H − 0.13 0.16 0.19 SRS ggF 0.28 0.46 0.43 − VBF 0.07 0.19 0.21 − VH 0.26 0.24 0.26 − tt̄H 0.39 0.11 0.10 − jet shower systematics and EW correction uncertainties. For the factorization and renormal- ization scale variations a 7-point scale variation on µR/F was performed. The variation was found to show a flat effect in the cross section of 2% for ggF, 0.5% for VBF, 5% for V H and 13% for tt̄H. For the shower systematics, Pythia and Herwig samples at the truth level were compared. From these samples, pT and mJ dependent reweighting maps were constructed, applied to the recontructed level samples and used to estimate the uncertainty. The shower systematics were found to be negligible, as no substantial differences were seen in the mJ shape in the resonance peak region. Appendix A contains plots related to the studies of the jet parton showers as well as the EW correction systematics [117]. Higgs Boson Resolution The pT and mass resolution of the Higgs jet is studied by truth matching candidate jets to a Higgs boson. A Gaussian is then fitted to the difference of the reconstructed jet and the truth jet values. The Gaussian resolution is therefore taken as the standard deviation of the 88 (a) (b) (c) (d) Figure 5.4: Breakdown of the Higgs boson contributions to the mass peak for the different production modes for signal (a,b) and validation regions (c,d). The plots on the left (a,c) show the contribution to the leading regions and on the right (b,d) to the subleading regions. fitted Gaussian. The impact of pile-up (PU) on the mass resolution was also studied and shown to be small, given that the large-R jets are subject to a trimming procedure. The pT resolution is shown in Table 5.8 and the mass resolution is shown in Table 5.9. 5.6 Background Process Modeling The dominant background process is QCD multijet production which presents itself as a non-resonant monotonically decreasing spectrum. The resonant backgrounds, the V +jets process and the top quark, peak outside the Higgs boson signal window but still contribute 89 (a) (b) Figure 5.5: Breakdown of the contributions for the different Higgs boson production modes for the (a) SRL and (b) SRS as a function of pT . Table 5.8: Momentum resolution of the candidate jets truth matched to a Higgs boson for ggF events. pT Resolution [GeV] pT [GeV] Leading Subleading 250 < pT < 450 − 38.3 ± 1.2 450 < pT < 650 29.3 ± 0.3 39.9 ± 1.2 650 < pT < 1000 38.3 ± 0.9 56.1 ± 2.1 in a minor way. Within the Higgs mass peak (105 < mj < 140 GeV), the V +jets process, represents approximately 1% of the total background, top quarks represent about 3%, and the rest of the background is due to QCD multijets. Figure 5.6 shows the mass distributions of the expected MC estimates (Asimov datasets) of the signal and backgrounds for the signal regions. Table 5.9: Mass resolution of the candidate jets truth matched to a Higgs boson for ggF events. Mass Resolution [GeV] pT [GeV] Leading Subleading 250 < pT < 450 − 17.7 ± 0.7 450 < pT < 650 11.3 ± 0.1 13.4 ± 0.2 650 < pT < 1000 10.8 ± 0.3 13.5 ± 0.4 90 (a) (b) Figure 5.6: Mass distributions in the (a) SRL and (b) SRS for the MC estimates of the signal and backgrounds [10]. Top Quark Modeling A top quark event is characterized by the presence of a b-quark and two hadronic decay products of a W boson. The tt̄ control region CRtt̄ was defined to estimate the top quark contributions to the signal regions. The jet mass distributions in both the CRtt̄ and the SR are comparable given that both regions probe a similar phase space. Figure 5.7 shows the breakdown of the tt̄ contribution to the candidate jet mass distribution in the signal and control regions. The shape of the spectrum is taken from MC but the normalization is extracted from data. Adjustments performed on the CRtt̄ are directly applied to the SR by including it in the global likelihood fit. To extract the scale factor, a simultaneous fit is performed on the CRtt̄ and SR together. Given that the CRtt̄ has a tt̄ purity of 97% (with similar levels in the fiducial and differential regions), the normalization was determined from data with better than or equal to 10% precision. In the SR, single top events contribute between 2-3% (tW ) and 1-5% (t-channel) of the candidate jets relative to the total tt̄ yield. These events have a candidate jet mass distribution similar to that of the tt̄ events. The s-channel contribution was found to be negligible. To account for this contribution to the likelihood fits the tt̄ MC was scaled 91 (a) (b) (c) Figure 5.7: Breakdown of the tt̄ contribution to the candidate jet mass for the inclusive (a) SRL, (b) SRS and (c) CRtt̄ . accordingly to match number of events in tW and t-channel MC samples for each pT bin. A 50% normalization uncertainty was applied to the estimated number of single top quark events due to comparisons between diagram subtraction and diagram removal schemes [118] in tW events. Standalone fits were also performed for the CRtt̄ using Asimov datasets. The measure- ment of the scale factor had greater uncertainty in the high pT region due to the lower number of events. It was found that the scale factor between the data and the MC was about 0.8. For the global fit of the SR with the control region, the data and simulation seem to agree. Figure 5.8 shows the results of the fit for the differential analysis regions. 92 Systematic uncertainty estimates were calculated using simulated samples for alternative parton shower models (Powheg vs Herwig) finding 6-19% difference in yield across the analysis regions. Similarly, uncertainties due to matrix element calculations (Madgraph5 vs Powheg Box v2) were performed and found to have a 1-19% difference in yield. Weight variations on the nominal sample associated with initial and final state radiation (ISR and FSR) produced uncertainties between 1-7%. Renormalization and factorization scale varia- tions were found to be negligible. The two largest uncertainties on the tt̄ normalization were from b-tagging efficiency and JMS. (a) (b) (c) (d) Figure 5.8: The post-fit CRtt̄ Jt mass distribution in the four pT regions used in the global likelihood of the differential fit. The W (lv) contribution is flat in jet mass and for events with pT < 1 TeV it is estimated to be 1-3% of the total. The pT > 1 TeV region is shown in 10 GeV jet mass bins. The ratio of the data to the background prediction is shown in the lower panel. The shaded areas indicate the 68% CL for all background processes [10]. 93 V + jets Modeling Vector boson production offers a unique opportunity to validate the signal measurement procedure for the Higgs given that their decay structure, mass peak and resolution are similar. The Z boson, especially, given that it populates the signal region with around 20 times more events than the Higgs boson, it can be used to study experimental effects that would not be apparent with the statistically limited Higgs boson measurement. Therefore, to have a proper measurement of H, a well understood Z background is neccesary. In the SR, the number of Z+jets events is more than 3 times that of W +jets because of the large branching ratio of its b-quark pair decay (Z → bb̄) coupled with our b-tagging selection criteria. For approximately 90% of the candidate jets in Z+jets events, the decay products of the Z bosons are fully contained within the jet. On the other hand, only 40% of the candidate jets in W +jets events contain its decay products. This is due to the low misidentification rate for b-tagging. The remaining candidates from W +jets events come from the recoiling hadronic system resulting in a broader mJ distribution in the SR. In the VR, due to the requirement that the candidate jets must be anti-tagged, W +jets are three times larger than Z events. They both have comparable acceptance but the W has a larger cross-section. The decay products of the vector bosons are reconstructed within the candidate jets only in 60% of events. This results in a non-resonant mass distribution, with a shape that is similar to the QCD multijet background. Given that the Z+jets normalization is directly extracted from the data with the global likelihood fit, the systematic uncertainties of the modeling are limited to changes in accep- tance in the different regions and to the mass distribution shape. For the W +jets cross section, a 10% uncertainty in the signal region is assigned [119]. The semi-leptonic W +jets 94 decays (W → lν) contribution in the CRtt̄ has a total uncertainty of 30%. Systematic un- certainties due to renormalization and factorization scale variation represent a 3-20% error to the acceptance across the different regions. Other variations were studied, but found to have a negligible impact. These include an alternative PDF (MMHT2015NLO [120]), αs variations in the nominal PDF and alternative cluster fragmentation modeling (Lund string model [121]). For the normalization, the largest experimental uncertainties are associated with the JMR and JMS. Agreement between simulation and data in the leading jet VR is shown in Figure 5.10. V + jets Resolution It was found that the fitted Z+jets normalization in the SR had a correlation with the re- constructed mass resolution. This is due to the flexibility of the Z+jets template and the multijet model (discussed in the next section). In some cases, the best value of the JMR parameter broadened the Z+jets peak, which corresponds to a increase of Z+jets normal- ization and a decrease of the contribution of multijets compared to the expected values. A dedicated control region rich in large-R jets containing W bosons from the decay products of semileptonic tt̄ decays was created to constrain the JMR systematic in conjunction with the VRL. This control region is denoted as the WCRtt̄ . The WCRtt̄ requires the presence of two top quarks in different hemispheres where one top quark decays leptonically while the other top quark decays hadronically. The decay products of the W boson from the hadronically decaying top quark must be isolated in the large-R jet. This region provides a high purity reconstructed W peak with pT from 200-600 GeV. Similar to the CRtt̄ , an isolated medium quality muon is used. The selection requires at least one large-R jet (leading will be the W candidate) with pT > 200 GeV and 95 at least 2 VR track jets with pT > 10 GeV. Both VR track jets must pass the b-tagging requirements. One of the b-tagged VR track jets has to be close to the muon by satisfying: µ 0.04 + 10/pT < ∆Rbtag1,muon < 1.5 and to also have a pT > 25 GeV. This b-tagged jet must be well separated from the W candidate (∆Rbtag1,Wcand > 2.0). The second b-tagged VR track jet is required within 1.0 < ∆Rbtag2,Wcand < 1.5 of the W candidate. Figure 5.9 shows a diagram of the topology of the WCRtt̄ events. The mass and pT distributions of the inclusive WCRtt̄ can be seen on Figure 5.11. (a) Figure 5.9: Diagram depicting the topology of a WCRtt̄ event. The hadronically decaying W boson must be isolated in the large-R jet. The WCRtt̄ provides a good source of W bosons in the pT range below 600 GeV. The VRL provides a clear peak in pT ranges above 450 GeV but with more multijet background. The jet mass width of the W and Z resonances show a slow evolution from low pT in the WCRtt̄ to high pT in the VRL. This can be seen in Figure 5.12. The results from of the jet mass width using the WCRtt̄ have around 1/5 of the original JMR uncertainty after the contraint transfer to the Z → bb̄ dominated V +jets sample in the SR. The correlation 96 between the Z+jets normalization and the JMR is reduced when included in the global likelihood fit. In the inclusive signal region, the correlation reduces from around 90% to 30% when this auxiliary mass measurement is considered. 97 (a) (b) (c) Figure 5.10: Post-fit leading-jet invariant mass distributions after the multijet background was subtracted in the validation region for data and the V +jets (W +Z) and top quark components for (a) 450 < pT < 650 GeV, (b) 650 < pT < 1000 GeV, and (c) pT > 1 GeV shown in wider 10 GeV jet mass bins. The V +jets contribution is split into five generator ‘truth’ pVT volumes labeled p0T –p4T for pVT < 300 GeV, 300–450 GeV, 450–650 GeV, 650–1000 GeV, and > 1000 GeV, respectively. The tt̄ normalization and its uncertainty are set to the corresponding values from the CRtt̄ . The mJ range has been extended down to 60 GeV for only this fit to show the level of agreement along the rising edge of the V +jets mJ distribution. The ratio of the data to the background prediction is shown in the lower panel. The shaded areas indicate the 68% CL for all background processes [10]. 98 (a) (b) Figure 5.11: Inclusive WCRtt̄ (a) mass distribution and (b) pT distribution of the W candi- dates [122]. Figure 5.12: A summary of the Z and W resonance peak reconstructed-width measurements as a function of the jet pT using a resolved W boson in top quark decays in the WCRtt̄ region and the combined W and Z boson mass distribution in the validation region. The continuous black curve is a fit to the measurements with resultant errors shown as a cyan band [122]. 99 Multijet Modeling The QCD multijet background has a monotonically decreasing mass spectrum. It is modeled using an exponential function of a polynomial of degree N with the form: X N  fN (x|Θ) = Θ0 exp i Θi x , (5.13) i=1 where Θi are the parameters of the fit and x = (mJ − 140)/70 GeV. The parameters are simultaneously determined during the signal extraction fit independently for each region. The number of events and the shape of the spectrum has an impact on the optimal degree of the polynomial of the model. Small values of N make the function too rigid and therefore prone to bias in the resonant process yields. On the other hand, large values of N decrease the statistical significance of the resonant process models, due to the increased correlation which can create or absorb the resonances. Modified VR (which we call hybrid VR) ensembles are used to study the optimal values for N given that they contain more than 50 times the amount of data than the SR. Ten of these ensemble (VR slices), with roughly the same amount of data as the SR, are used to find the optimal parameters of the fit by taking an average for each of the regions. The hybrid VR is constructed by replacing the VR resonance peaks with the SM pre- diction in the SR while correcting the mass spectrum to match the SR. A shape correction factor, defined as the ratio of the SR multijet estimate (MJSR ) with the VR model (MJVR ), is applied to the VR slices. The values of MJSR are obtained from the likelihood fit of the SR and CRtt̄ while including all the systematic uncertainties. The MJVR is taken as the average of likelihood fits of 10 random orthogonal subsets of the VR while including all sys- tematic uncertainties. The resonant peak estimates for V + jets and Top (VVR and TopVR ) 100 are extracted from the average post-fit contributions of the same 10 VR fits. Each hybrid VR (VRihyb ) is defined as MJSR VRihyb = (VRi − VVR − TopVR ) × + VSR + tt̄SR + HSR , (5.14) MJVR where VRi is the mass distribution of the data events in the VR slice and the variables with subscript SR are the nominal MC predictions for the resonant sources in the SR. The log-likelihood ratio (LLR) is used to test the results between different values of the polynomial degree N in each VRhyb but without the injected resonances. The null hypothesis is defined as the fit using a polynomial of degree N , while the alternate hypothesis is the fit using a polynomial of degree N + 1. Wilk’s theorem [123] relates the log-likelihood ratio to a χ2 distribution with N + 1 − N = 1 degrees of freedom (d.o.f.). The smallest value of N that yields a uniform distribution of p-values is selected as the optimal model. A uniform distribution is represented by a linear increase in the corresponding cumulative distribution function (CDF). Figure 5.13 shows the CDF as a function of p-value for the VR. (a) (b) Figure 5.13: Cumulative distribution function (CDF) of the p-values of the log-likelihood ratio of the exponential polynomial of degrees N and N + 1. Plots correspond to the (a) VRL and (b) VRS [122]. 101 To look for local effects due to the resonances, tests were performed by including a free normalization parameter µVR ± σstat VR for either the Z + jets process or the Higgs boson in a fit for the VRhyb by doing an artificial signal injection. The quantity F2σ was used to estimate the probability that the multijet model allows artificial excesses or deficits. F2σ is VR : defined as the fraction of fitted Z and H signal in excess of twice its error σstat |µVR − 1| > 2σstat VR . (5.15) The average ratio µ/σ = (µVR − 1)/σstat VR quantifies the bias in the signal strength determi- nation and can be used to estimate the spurious signal systematic uncertainty when applied to VRhyb without any signal. The value of N is chosen so that F2σ is compatible with a value of 0.05 and µ/σ is stable for both Z + jets and Higgs production. The spurious signal systematic uncertainties range from 0.01-0.33 for H and 0.15-0.65 for Z. Figure 5.14 shows the values of F2σ for the Higgs and the Z bosons. (a) (b) Figure 5.14: Fraction of fitted signal in excess of twice its error for (a) H and (b) Z as a function of the exponential polynomial degree N [122]. 102 The optimal values for N were found to be N = 5 in the inclusive region and between 4 and 5 for the differential pT bins. The results are summarized in Table 5.10, where the differential bins are labeled as p0T (250 < pT < 450 GeV), p1T (450 < pT < 650 GeV), p2T (650 < pT < 1000 GeV) and p3T (pT > 1 TeV). A comparison of the QCD multijet fits for all the pT binned analysis regions is shown on Figure 5.15. Table 5.10: Optimal degree N of the exponential polynomial used to model the QCD multijet background for all the analysis regions Inclusive Differential Candidate jet p0T p1T p2T p3T Leading 5 − 5 4 4 Subleading 5 5 4 5 4 103 (a) (b) (c) (d) (e) (f) Figure 5.15: Multijet jet-mass distribution from the different pT -binned analysis regions. The solid lines show the multijet function after a fit to the SR data (gray) and VR data (blue). The solid points are the data from VR slices with the same number of events as the SR after the SM resonances are subtracted. The bottom panel shows the ratio of the SR data fit to the VR data fit [10]. 104 5.7 Statistical Analysis The signal extraction is performed using the maximum likelihood method. In practice, this is achieved through the minimization of the negative log likelihood function L(µ, θ) using the RooStats framework [124] and the RooFit library [125]. The likelihood function is defined as a product of Poisson probability density functions as described in Sec 5.1. One of these terms is defined for each mJ bin of the SRL, SRS and CRtt̄ . The bin width for the mass distribution was chosen to be 5 GeV. A recent RooFit extension [126] was needed to remove an existing bias for wide binned datasets. The nuisance parameters θ represent the systematic uncertainties and are constrained with Gaussian or log-normal probability density functions. The V +jets JMR contraints obtained from the WCRtt̄ and VRL are implemented as Gaussian p.d.f priors. The normalization of the MC templates is controlled by free parameters for each pT region or the truth-based volume common to the SRL, SRS and CRtt̄ . For the multijet model, both the normalization and the polynomial coefficients are treated as free parameters independent from each jet mass distribution. The yield of each of the signals is given by the signal strength µ. The signal stregnth µ is defined as ratio between the fitted number of signal events and the corresponding SM prediction. Upper limits on the Higgs boson signal strength µH and production cross section σH are obtained using the CLs method, where the expected limits are determined by assuming no Higgs boson contribution. Systematic Uncertainties Uncertainties related to a low number of events in MC samples for the background predictions were parametrized with the Beeston-Barlow technique [127]. A smoothing procedure was 105 also used to remove large variations with a threshold for pruning of only 2% [128]. Table 5.11 summarizes all the systematic uncertainties considered in the likelihood fit for the H and Z signal strength extraction. Table 5.11: Summary of the systematic uncertainties included in the proifle likehood fit for the signal strength extaction. The second column states the processes for which an independent nuisance parameter is considered. The third column indicates the regions for which the systematic uncertainty is correlated. The fourth column describes the effect of the systematic uncertainty induced by the parameter: N denotes a normalization change and S represents an impact to the shape. (*) tt̄ and V +jets events have two extra minor components only applied to them. (?) This uncertainty only covers relative acceptance across regions instead of the absolute cross section uncertainty. (•) Only applied to Z+jets when the signal extraction performed on truth-based volumes is tested using the SR [10]. Description Processes Category Effect Reconstructed object systematic uncertainties JMR tt̄, V + jets, H pT N+S JMS (dominant) tt̄, V + jets, H pT N+S JMS (rest) tt̄,V + jets +H all N+S Jet Energy Scale all(∗) all N+S Jet Energy Resolution all all N+S b-tag efficiency for b-jets all all N+S b-tag efficiency for c-jets all all N+S b-tag efficiency for light-jets all all N+S Process modeling systematic uncertainties Renormalization and factorization scale V + jets all N+S Cross section W + jets all N Cross section and acceptance W (lν) all N Parton shower model tt̄ all N+S Matrix element calculation tt̄ all N+S Initial and final state radiation tt̄ all N+S Cross section and acceptance t all N Cross section and acceptance(?) H all N VBF +V H+tt̄H all N NLO EW corrections H pH T bins × LS N Spurious signal Z+ jets(•) pZT bins × LS N 106 Chapter 6 Boosted H → bb̄ Results Three different configurations are used to study Higgs boson production at high pT . The inclusive region is used to measure the H signal strength, the fiducial region is used to measure the fiducial cross section and the differential regions are used to measure the cross section for four different pT bins. The signal strength extraction in the fiducial region considers the events on the fiducial volume defined by the requirements applied to the truth Higgs boson transverse momentum pH T and its rapidity yH . The same truth information is used for the differential regions. The pT -y volume bins are based on the simplified template cross-section (STXS) framework [129][130] for ggF production, with the modification of a tighter yH requirement and the inclusion of all production modes. The STXS framework was developed to maximize the sensitivity of the Higgs boson measurements, while at the same time minimizing the theory dependence of their determination. The same pT boundaries are used for the V +jets pro- duction cross section measurements in the VRL, and for Z+jets production in the SR, and are used to validate the method. The fiducial and STXS volumes for these are defined by requirements on the generator truth vector boson transverse momentum pVT . The summary of the fiducial and differential region volumes is given in Table 6.1. 107 Table 6.1: Summary of the fiducial and STXS volumes used to determine the signal events considered for the signal strength measurement [10]. Volume pHT |yH | Fiducial > 450 <2 300 − 450, 450 − 650, STXS <2 650 − 1000, > 1000 6.1 Inclusive Region The inclusive region is the signal region containing candidate jets with pT > 250 GeV. The extraction of the Higgs boson signal strength in the inclusive region yields a value of µH = 0.8 ± 3.2 when combining the SRL, SRS and CRtt̄ information. The breakdown of the uncertainty was found to be ± 3.2 (total) = ± 3 (stat) ± 1.1 (syst). The measurement is dominated by the statistical uncertainty (size of the data sample) which limits the sensitivity of the signal. The observed signal strength corresponds to a sensitivity of 0.29σ (0.36σ was expected). The largest contributions to the systematic uncertainty are from the jet mass resolution (JMR) and the jet mass scale (JES). For the tt̄ contribution, the value was found to be µtt̄ = 0.80 ± 0.06. The poor modeling seen for tt̄ agrees with previous published results where the boosted top-quark pair differential cross-section in the l+jets channel was measured [131]. The Z+jets process had a signal strength value of µZ = 1.29 ± 0.22. These results are summarized in Table 6.2. The yields in the three regions pertinent to the Higgs boson signal extraction are presented in Table 6.3. 108 (a) (b) Figure 6.1: The ratios of the measured fiducial phase-space absolute differential cross-sections to the predictions obtained to (a) the Powheg+Pythia8 MC generator and to (b) the NNLO predicitions, in the resolved and boosted topologies as a function of the top quark pT [131]. Table 6.2: Expected and observed values of the signal strengths for the H, Z, and tt̄ com- ponents in the inclusive fit [10]. Result µH µZ µtt̄ Expected 1.0 ± 3.2 1.00 ± 0.17 1.00 ± 0.07 Observed 0.8 ± 3.2 1.29 ± 0.22 0.80 ± 0.06 Table 6.3: Event yields and associated uncertainties after the global likelihood fit in the inclusive region [10]. Process SRL SRS CRtt̄ Multijet 590 700 ± 4200 529 300 ± 3500 − Z+ jets 16 100 ± 2800 12 000 ± 2100 − W + jets 3050 ± 720 2510 ± 500 − Top 16 300 ± 1900 15 900 ± 2000 3737 ± 68 W (lν) − − 53 ± 16 H 400 ± 1500 300 ± 1300 − Total 626 530 ± 820 560 090 ± 770 3790 ± 66 Data 626 532 560 083 3791 109 (a) (b) Figure 6.2: Post-fit jet mass distributions for the various components in the inclusive SRL (left) and SRS (right) regions. In the middle panels the shaded areas indicate the 68% CL for the multijet background from the fitted parameters and normalizations of the exponentiated polynomials. In the lower panels the shaded areas indicate the 68% CL for all background processes [10]. 110 6.2 Fiducial Region In the fiducial region, the Higgs boson yield is determined using the fiducial volume defined by the Higgs boson transverse momentum (pH T > 450) GeV and rapidity (|yH | < 2.0). The transverse momentum cut was chosen to ensure an unbiased truth spectrum due to the trigger turn on. Therefore, this measurement doesn’t include the SRS region below 450 GeV. The signal acceptance times efficiency in the fiducial volume is presented in Table 6.4. Table 6.4: Signal acceptance times efficiency within the fiducial volume used in the fiducial region [10]. pHT > 450 GeV Process |yH | < 2 All 0.24 ggF 0.26 VBF 0.22 VH 0.27 tt̄H 0.20 The signal outside the fiducial region is set to the SM value within uncertainties. There- fore the fit considers two Higgs boson mass templates in each SR. The component from the fiducial volume accounts for more than 80% of the Higgs boson signal. The component from outside the fiducial volume has a broader mass spectrum shifted to higher values. This procedure was tested with W → qq 0 and Z → q q̄ in the VR and with Z → bb̄ in the SR. By fixing the Higgs signal to the SM values, in both in and out the fiducial region, the signal strength for V +jets was found to be µV = 1.01 ± 0.09. Similarly, for Z events in the SR the signal strength was found to be µZ = 1.35 ± 0.25, both being in agreement with the SM. For the Higgs boson signal strength, the likelihood fit yields a value of µH = −0.1 ± 3.5. The results are summarized in Table 6.5. Post-fit mass distributions are shown in Figure 6.3. No signal of the Higgs boson is shown given that the signal strength was found to be 111 below 0. The SM prediction for the Higgs boson production with pH T > 450 GeV is 18.4 fb. Our measurement corresponds to a 95% CL upper limit on the observed (expected) Higgs boson production cross section of σH (pH T > 450 GeV) < 115 (128) fb. (6.1) Table 6.5: Expected and observed values of the signal strengths for the H, Z and tt̄ compo- nents in the fiducial fits [10]. Result µH µZ µtt̄ Expected 1.0 ± 3.4 1.00 ± 0.18 1.00 ± 0.08 Observed −0.1 ± 3.5 1.30 ± 0.22 0.75 ± 0.06 The statistical uncertainty is the largest contributor to the total uncertainty of the signal strength, with the systematic uncertainty being somewhat smaller. The largest component of the systematic uncertainty, with almost a 80% contribution, is the jet systematics driven by the jet mass scale (JMS) effects. The JMS uncertainty comes from both background (V +jets and tt̄ contribute 50%) and the reconstructed Higgs bosons (which contributes the other 50%). The breakdown is shown in Table 6.6. Table 6.6: Contributions to the systematic uncertainties for the measurement of the fiducial volume signal strength [10]. Uncertainty Contribution pH T > 450 GeV Total 3.5 Statistical 2.6 Systematic 2.3 Jet systematic uncertainties 2.2 Modeling and theory systs. 0.8 Flavor-tagging systs. 0.2 112 (a) (b) Figure 6.3: Post-fit jet mass distributions for the various components in the fiducial SRL (left) and SRS (right) regions. In the middle panels the shaded areas indicate the 68% CL for the multijet background from the fitted parameters and normalizations of the exponentiated polynomials. In the lower panels the shaded areas indicate the 68% CL for all background processes [10]. 113 6.3 Differential Regions The differential region measurements aims to extract the Higgs boson transverse momentum spectrum in four pH T volumes. These are based o the STXS template and consist of the pH T volumes with 300-450 GeV, 450-650 GeV, 650-1000 GeV and above 1 TeV. The same procedure established for the Higgs boson measurement for the fiducial region is employed. The Higgs boson mass template for each pH T volume is used within each pT region in the global likelihood. Only the SRL and CRtt̄ regions are included for measurements above 1 TeV, given that the SRS expected sensitivity in this region is marginal. Outside the volumes (pH T < 300 GeV) the components were fixed to their SM expectations. The signal acceptance times the efficiency for the STXS volumes is shown in Table 6.7. The expected yield and percentage contributions of the Higgs boson subprocesses is shown in Figure 6.4. Table 6.7: Signal acceptances times efficiency for the STXS volumes in the differential mea- surement [10]. Process 300 < pH T < 450 GeV 450 < pH H T < 650 GeV 650 < pT < 1000 GeV pH T > 1 TeV All 1.3 × 10−2 0.23 0.31 0.23 ggF 0.7 × 10−2 0.25 0.35 0.28 VBF 0.4 × 10−2 0.21 0.32 0.25 VH 1.7 × 10−2 0.26 0.30 0.20 tt̄H 4.7 × 10−2 0.19 0.24 0.19 Similar to what was done to the fiducial measurement, the signal determination method was tested with W → qq 0 and Z → q q̄ in the VR, and with Z → bb̄ in the SR. The VRL is divided into 5 slices with the fit being performed independently in each slice. The results are then averaged. The Z fit is performed in the SRL, SRS, CRtt̄ regions with the Higgs boson contribution fixed to the SM prediction. The results of the differential fit signal strengths for V +jets in the VRL and Z+jets in the SR are shown in Figure 6.5. 114 (a) (b) Figure 6.4: For each of the pH T differential volumes (x-axis), the expected signal event yield for all Higgs boson events (left) and the fraction of signal in percent (right) in each reconstructed jet pT region (y-axis) is shown. The leading jet pT in the SRL is denoted by pL T and the S subleading jet pT in the SRS is denoted by pT [10]. (a) (b) Figure 6.5: Comparison of differential fit signal strengths for (a) V +jets in the VRL and (b) Z+jets in the SR. The signal strength within the STXS volumes is calculated relative to the prediction at NLO QCD and LO EW accuracy. They are compared with the NLO EW correction provided by SHERPA, the NNLO QCD correction provided by the NNLOJET group, and their product. The points are located at the weighted center of the bin considering the underlying pVT or pHT spectrum [10]. 115 The Higgs boson signal strengths in the STXS volumes is extracted by fitting simulta- neously the ten differential SR and CR regions defined in Tables 5.5 and 6.1. The results are summarized in Tables 6.8 and 6.9. The four Higgs boson signal strengths are compatible with the SM and have a p-value of 0.53. Table 6.8: Expected and observed values of the signal strengths for the H component in the differential fits [10]. µH pHT [GeV] Exp. Obs. 300 − 450 1.0 ± 18 −6 ± 18 450 − 600 1.0 ± 3.3 −3 ± 5 650 − 1000 1.0 ± 6 5±7 > 1000 1.0 ± 30 18 ± 32 Table 6.9: Expected and observed values of the signal strengths for the Z and tt̄ components in the differential fits [10]. µZ µtt̄ Jet pT [GeV] Exp. Obs. Exp. Obs. 300 − 450 1.0 ± 1.1 1.8 ± 1.1 1.0 ± 0.07 0.85 ± 0.06 450 − 600 1.0 ± 0.17 1.28 ± 0.22 1.0 ± 0.07 0.76 ± 0.06 650 − 1000 1.0 ± 0.33 1.4 ± 0.4 1.0 ± 0.09 0.74 ± 0.08 > 1000 1.0 ± 1.6 2.4 ± 1.7 1.0 ± 0.22 0.57 ± 0.18 The Higgs boson production cross section for pH T > 1 TeV was found to be σH (pHT > 1 TeV) = 2.3 ± 3.9 (stat) ± 1.3 (syst) ± 0.5 (theory) fb. (6.2) The SM prediction for this quantity is 0.13 fb. Because the sensitivity was low, upper limits were calculated. The 95% CL upper limits on the Higgs boson differential production cross 116 section were found to be σH (300 < pH T < 450 GeV) < 2.9 (3.1) pb, σH (450 < pH T < 650 GeV) < 89 (102) fb, (6.3) σH (650 < pHT < 1000 GeV) < 39 (34) fb, σH (pH T > 1000 GeV) < 9.6 (7.4) fb. These results are shown in Figure 6.6. As for the first two results of this analysis, the largest source of uncertainty is of statistical nature given that the sample is small. Table 6.10 summarizes the breakdown of the uncertainties associated with the measurement. The largest contribution to the systematic uncertainties is the JMS uncertainty. Table 6.10: Contributions to the systematic uncertainties for the differential measurements of the signal strength [10]. Uncertainty Contribution 300 < pH T < 450 GeV 450 < pH T < 650 GeV 650 < pH T < 1000 GeV pH T > 1 TeV Total 18 5.0 6.5 32 Statistical 16 3.0 5.5 30 Systematic 7 3.9 3.4 10 Jet systematic uncertainties 6 3.8 3.4 9.5 Modeling and theory systs. 4 0.7 0.7 2 Flavor-tagging systs. 0.2 0.4 0.4 2 The correlations between the differential Higgs boson signal strength measurements in pHT bins were found to be small. This implies a low number of events migrating from the analysis bins and the STXS truth Higgs bins. Figure 6.7 shows the correlations of the µH and µZ measurements. A post-fit mass distribution of all the components in the differential leading jet signal region is shown in Figure 6.8. 117 (a) Figure 6.6: Summary of the STXS volume signal strengths measured using the differential signal regions. Within the same kinematic regimes, measurements of the Z → bb̄ process agree with the Standard Model predictions, validating the methods. The points are located at the weighted center of the bin considering the underlying pH T spectrum [10]. 118 (a) (b) Figure 6.7: Correlations among the four Higgs boson signal strengths, and between the four Higgs boson and Z+jets signal strengths. The Higgs boson signal strengths µH are labeled with the corresponding pH T range as a superscript. The Z+jets signal strengths µZ are labeled with the corresponding jet pT range as a superscript [10]. 119 (a) (b) (c) Figure 6.8: Post-fit jet mass distributions of the various components in the differential leading-jet signal region defined by the selected candidate jet with (a) 450 < pT < 650 GeV, (b) 650 < pT < 1000 GeV, and (c) pT > 1000 GeV shown in wider 10 GeV bins [10]. 120 Chapter 7 Unified Flow Objects 7.1 Introduction From the results presented in Chapter 6, it is evident that the optimization of large-radius jet definitions could result in considerable gains for our measurement in terms of preci- sion. Approximately 90% of the systematic uncertainties result from the jet definitions, in particular the jet mass resolution and jet mass scale. In the analysis presented, large-R jet reconstruction was based on topological cluster inputs reconstructed using calorimeter-based energy measurements. A trimming procedure was performed and a combined mass scheme was employed. Even though a good energy resolution is achieved, in the high pT regime, the resulting showers are so collimated that the calorimeter’s granularity is not sufficient to spatially resolve individual particles in the jet. For this reason, the use of jet substruc- ture variables (JSS) is limited with these type of jet definitions. As a step to reduce these limitations during Run 2, particle-flow (PFlow) [134] algorithms were implemented to im- prove performance at low pT . For high pT , on the other hand, Track-CaloClusters (TCCs) [135] were designed in order to reconstruct jet substructure (JSS) variables. A new type of jet input, called “unified flow object” (UFO) [11] was then developed using both particle- flow objects (PFOs) and TCCs. This object combines calorimeter and inner detector based signals in order to achieve optimal performance across a wide kinematic range. This new 121 definition, combined with better pile-up mitigation techniques, such as Constituent Subtrac- tion (CS) [136] and SoftKiller (SK) [137], as well as grooming algorithms, such as Soft-Drop [138], motivated the re-optimization of the large-R jet definitions used by ATLAS. The AT- LAS Jet Tagging and Scale Factor Derivation group has been developing UFO reconstructed large-R jet dedicated taggers for hadronically decaying boosted objects. In this chapter we explore the reconstruction algorithm behind UFO jets, the different dedicated taggers devel- oped and the main ideas behind their development, such as jet substructure variables and machine learning. The strategy used to extrapolate their scale factors to higher momenta and the manner in which we estimate the uncertainties associated with the extrapolation procedure, will be presented. The chapter contains final results of the extrapolation un- certainties for the already calibrated UFO taggers supported by the ATLAS Collaboration. Finally, studies regarding a multiclass tagger (MCT) that includes Higgs boson identification will also be explored. These projects include work that started as the author’s “qualification task” (authorship project) and the subsequent collaboration with the ATLAS Jet Tagging and Scale Factor Derivation group. Particle-flow Objects Particle-flow (PFlow) [134] reconstruction combines both track and calorimeter based mea- surements. Particle-flow objects (PFOs) themselves improve pile-up stability relative to jets reconstructed from topo-clusters. The PFlow algorithm matches each selected track to a single topo-cluster. For each track/topo-cluster system the probability that the particle’s energy was deposited in more than one topo-cluster is evaluated. Then, the expected en- ergy deposited in the calorimeter by the particle that produced the track is subtracted. Any topo-cluster that is not matched to a track is considered to be produced by a neutral particle 122 and is left unmodified. The subtraction is gradually disabled for tracks with pT < 100 GeV if the energy deposited (Eclus ) in a cone of size ∆R = 0.15 by the extrapolated track satisfies Eclus − hEdep i > 33.2 × log10 (40 GeV/ptrk T ), (7.1) σ(Edep ) where Edep is the expected energy deposition. Any charged PF0 that is not matched to the primary vertex is removed to reduce the contribution from pile-up. This procedure is known as “Charged Hadron Subtraction” [139]. Track-CaloClusters Track-CaloClusters (TCCs) [135] are optimized to perform jet substructure reconstruction for very high pT jets. TCCs use energy scale information from topo-clusters and angular information from tracks. The algorithm matches a loose track in a particular event to topo- clusters that have been calibrated to the local hadronic scale. When a track is matched to a topo-cluster, the pT is determined using the pT associated with the topo-cluster while its angular coordinates (η,φ) are taken from the track. If a topo-cluster is not matched to any track then the TCC is created only using the topo-clusters 4-vector directly. Similarly, for an independent track that is not matched to any topo-cluster, the track information is used directly to create the TCC. If multiple tracks are matched to a single topo-cluster, each TCC object is given a fraction of the total pT of the topocluster. The momentum fraction is determined using the ratios of momenta of the matched tracks. Any unmatched topo-clusters are included as unmodified neutral objects. 123 Pile-Up Mitigation and Grooming Algorithms Before jet reconstruction, pile-up mitigation techniques are employed. For topo-clusters, the techniques are applied to the entire set of inputs. On the other hand, for charged PFOs, only the CHS method is employed, while neutral PFOs and TCCs are subject to the same preprocessing techniques used for topo-clusters. During reconstruction, grooming algorithms are applied to reduce contamination from soft radiation originating from the underlying event (UE), pile-up and initial state radiation (ISR). Constituent Subtraction Constituent Subtraction (CS) [136] is a method that performs area subtraction on jet input objects. Therefore, it is a local subtraction of pile-up at the level of individual jet con- stituents. Each input is defined using ghost association, a process where massless particles (ghosts) with low momentum are overlaid uniformly in the event. Ghosts have to satisfy g pT = Ag × ρ, (7.2) g where Ag is the area of the ghosts and pT is the expected contribution from pile-up radiation in a small angular area. The pile-up energy density ρ is assumed to have a weak dependence in azimuthal angle φ and rapidity y. ρ is estimated as the median of pT /A distribution of R = 0.4 kt jets in each event. The distance ∆Ri,k between a cluster i and ghost k is given by q ∆Ri,k = (ηi − ηk )2 + (φi − φk )2 . (7.3) 124 The algorithm proceeds iteratively through each cluster-ghost pair, after sorting in order of ascending ∆Ri,k , by modifying the pT of each pair as follows if pT,i ≥ pT,k : pT,i → pT,i − pT,k , pT,k → 0; (7.4) otherwise : pT,k → pT,k − pT,i , pT,i → 0. The procedure continues until ∆Ri,k > ∆Rmax , where ∆Rmax is a free parameter. The particles with zero transverse momentum are then discarded. In the original formulation of this procedure, a similar modification was performed for the mass, but this is ignored in the latest ATLAS implementation given that all neutral ATLAS jet inputs are defined to be massless. SoftKiller SoftKiller (SK) [137] is an algorithm that applies a pT cut to input objects. The pT cutoff, pcut T , is chosen such that the value of ρ is approximately zero. The event is broken into square patches in the rapidity-azimuth plane. The parameter ρ is the event-wide estimate of the transverse momentum density in an area patch pT,i   ρ = median . (7.5) i∈patches Ai The cut is determined such that when the cut is applied, half of the grid spaces are empty. 125 Computationally, this is given by the next formula pcut T = median {pT,i }, max (7.6) i∈patches where pmax T,i is the pT of the hardest particle in patch i. This implies that half the patches will contain only particles that have pT < pcut T . Therefore, after applying the cut, the value of ρ will be zero. The best performance is achieved when SK is applied after the CS algorithm. Soft-Drop Soft-Drop (SD) [138] is a grooming technique that removes soft and wide-angle radiation from jets. For this procedure, the constituents of the large-R jets are reclustered using the C/A algorithm. Using the angle-ordered jet clustering history, determined from the C/A algorithm, the clustering sequence is traversed from the widest-angled radiation iterating to the smallest-angle radiation. The condition min(pT,1 , pT,2 ) ∆R12 β   < zcut , (7.7) pT,1 + pT,2 R is evaluated for each splitting, where 1 and 2 represent the harder and softer branches of the splitting respectively. The parameters zcut and β dictate how aggresive the removal of soft and wide-angle radiation is and its dependence on the distance parameter. When a splitting fails the condition, the lower pT branch is removed. If the condition is satisfied, the process ends and the consituents remaining form the groomed jet. The use of the SD grooming algorithm allows the calculation of certain jet substructure observables to beyond leading-logarithm (LL) accuracy, while other trimming algorithms do not [140]. 126 Unified Flow Objects The fact that no single jet definition is optimal according to all metrics motivated the development of a new jet input that combines all the desirable aspects of both particle- flow objects (PFOs) and Track-CaloClusters (TCCs) reconstruction. TCCs improve tagging performance at high pT but their performance is worse than with baseline trimmed topo- cluster based jets at low pT , as they are more sensitive to pile-up. PFOs on the other hand can improve on the baseline definition for the entire pT range but their tagging performance is worse than with TCCs at high pT . Combining both approaches by defining a new jet input object, called Unified Flow Object (UFO) [11], we can achieve optimal performance across the full kinematic range. Figure 7.1 contains an illustration of how the UFO reconstruction algorithm is performed. The process starts by the application of the standard particle-flow (PFlow) algorithm. Any charged PFO that is matched to a pile-up vertex is removed. The remaining PFOs are then divided into three categories: neutral PFOs, charged PFOs that were used to subtract energy from a topo-cluster, and the charged PFOs that were not used for energy subtraction. At this point jet input pile-up mitigation algorithms (i.e. CS+SK) are applied. Then a modified TCC splitting algorithm is applied. Tracks that have been used for the PFlow subtraction are not considered as they have already been subtracted from the energy of the topo-clusters. The TCC algorithm proceeds with the remaining collection of tracks to split neutral and unsubtracted charged PFOs. The use of UFOs improves jet mass resolution (JMR) relative to topo-cluster-based jets, by 40% for high pT hadronically decaying W bosons and by 26% for hadronically decaying high pT top quarks. Figure 7.2 shows the JMR relative performance of different UFO 127 Figure 7.1: Illustration of the UFO reconstruction algorithm. The procedure starts with the identification of particle-flow objects (PFOs) and inner-detector tracks [11]. definitions compared to the ATLAS baseline jet definitions. For tagging, UFOs bring significant improvements over the usual topo-cluster and TCC definitions. Some studies show an increase of 120% in background rejection at a fixed signal effifiency of 50% at high pT . One example of the performance of UFOs for tagging is shown in Figure 7.3. In this example, using UFOs for hadronically decaying top quarks at high pT , improves the background rejection by 135% when compared with the baseline jet definitions. The pT resolution is degraded for large-R jets coming from UFOs compared to the baseline topo-cluster and TCC definitions, but given the improvements in jet mass resolution and jet tagging at high pT , it is worthwhile to proceed with the development and study of these jet definitions and to define taggers for future ATLAS analyses. 7.2 Jet Substructure Variables The jet substructure techniques [141] can be summarized as a set of tools to exploit the radiation pattern inside hadronic jets. These correlations are quantified by a set of variables 128 (a) (b) Figure 7.2: Jet mass resolution for (a) W boson jets and (b) top quark jets as a function of pT . The relative performance of the studied UFO definitions compared to the current ATLAS baseline topo-cluster and TCC jets is shown [11]. Background Rejection Background Rejection ATLAS Simulation Preliminary ATLAS Simulation Preliminary s = 13 TeV, t → qqb s = 13 TeV, t → qqb 500 GeV ≤ p true < 1000 GeV 1000 GeV ≤ p true < 1500 GeV 103 ηtrue < 1.2 T ηtrue < 1.2 T JES+JMS JES+JMS 102 102 LC Topo Trimming LC Topo Trimming TCC Trimming TCC Trimming CS+SK UFO Trimming CS+SK UFO Trimming CS+SK UFO Soft Drop CS+SK UFO Soft Drop 10 10 CS+SK UFO Recursive SD CS+SK UFO Recursive SD CS+SK UFO Bottom-up SD CS+SK UFO Bottom-up SD 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Top-tagging efficiency Top-tagging efficiency (a) (b) Figure 7.3: Background rejection as a function of signal efficiency for a tagger using the jet mass and τ32 for top quark jets at (a) low pT and (b) high pT . The relative performance of different UFO definitions are compared with the current ATLAS baseline topo-cluster and TCC jets [11]. 129 which in principle are infrared and collinear (IRC) safe [142], i.e. they are insensitive to infinitesimally soft or collinear emissions, the presence of which presents difficulties for higher order QCD predictions. Angularities The generalized angularities [143] are a family of jet shapes defined as ∆Ri,J β   aκβ ziκ X = , (7.8) R i∈J where zi is the jet transverse momentum fraction carried by the constituent i of jet J, and ∆Ri,J its distance to the jet axis. Only angularities with κ = 1 are IRC safe. Angularities can be seen as a measure of QCD radiation around the jet axis. N-subjettiness N-subjettiness [144] is an angularity-type observable. Within a jet, N -subjettiness identifies N subjet axes, calculates the jet thrust about each and sums all of them together. For a jet J, with transverse momentum pTJ and particles i each with pTi , the N-subjettiness is given by (β) 1 X τN = pTi min{Ri,1 , Ri,2 , · · · Ri,N }β , (7.9) pTJ i∈J where Ri,n is the distance between particle i and the closest subjet axis in the η-φ plane. The angular exponent β controls the sensitivity to collinear radiation. By taking the ratio of multiple N-subjettiness variables, new dimensionless quantities can be derived, which aid 130 in the discrimination of multi-pronged objects. For example, the ratio (β) (β) τN τN,N −1 = , (7.10) (β) τN −1 can be used to identify the presence of N subjets within a jet. Generalized Energy Correlation Functions The N -point energy correlation function (ECF) [145] is defined as n ! n−1 n !β X Y Y Y ECF(n, β) = pTia Rib ic , (7.11) i1