# Jets and QCD measurements at high energy colliders

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## Introduction

Quantum Chromodynamics (QCD) is the quantum field theory of the strong nuclear force. It is based upon an SU(3) gauge symmetry, and couples together particles which carry the color charge quantum number. Amongst the fundamental fermions of the Standard Model, this means only the quarks experience the strong force. Each flavour of quark comes with three different possible colour states. The gauge boson associated with QCD is the gluon, which comes in eight different color states (combinations of colors and anti-colors).

Quarks and gluons, the fundamental degrees of freedom in QCD, are not observable as isolated, stable particles in nature. The coupling constant of QCD ($$\alpha_s$$) evolves with distance, becoming large at large distances and leading to the confinement of quarks and gluons within color-neutral composite particles – hadrons. Conversely, at short distances the coupling becomes small, and quarks exhibit asymptotic freedom, behaving as quasi-free particles. The large momentum transfers present in so-called "hard scattering" at high-energy colliders make such short distances accessible, and in regions where the coupling is much less than one, perturbative techniques may be used to make calculations that may be compared to the measurements. When scattered at such high energies, quarks and gluons give rise to collimated sprays of particles known as jets.

It is these properties that have made rigorous investigation of QCD challenging, but possible, at such colliders.

This article describes key collider results relevant to QCD, starting with those processes with the largest cross sections and moving to smaller cross sections. The internal structure of hadrons, the production and features of hadronic jets, the production of heavy quarks and of electroweak bosons will be addressed along the way.

## Total Cross Sections and Diffractive Processes

While the annihilation cross section for electron-positron collisions falls rapidly with increasing centre-of-mass energy ($$\sqrt{s}$$), the total cross section for hadron-hadron scattering rises. The photon-photon cross section also rises with $$\sqrt{s}$$. This is understood as being due to the fact that the photon can undergo fluctuations into quark-antiquark pairs long-lived enough to develop a complex hadronic ‘structure’, effectively making high-energy photon-photon collisions similar in behaviour to vector meson collisions.

Both the $$e^+e^-$$ annihilation cross section, and the cross section for deep inelastic scattering (DIS) in lepton-hadron collisions are dominated by electroweak interactions. However, since components of both the $$e^+e^-$$ and lepton-hadron cross sections involve the radiation and subsequent interaction of almost-on-shell photons, asymptotically all cross sections at high-energy colliders should exhibit hadronic behaviour.

The rise in hadronic cross sections is understood within scattering theory as being due to a virtual exchange of vacuum quantum numbers, known within Regge theory as a Pomeron. Pomeron exchange is eikonalised (that is, copied – multiple Pomeron exchanges are included) at very high energies, avoiding the violation of unitarity. The component of the cross section mediated by Pomeron exchange is commonly referred to as the diffractive component. This may be elastic or quasi-elastic - that is, one or both hadrons may disassociate and fragment. However, the fact that vacuum quantum numbers (that is, no charge or color) are exchanged leads to the presence of regions in rapidity that are unpopulated by particles, even in the cases where the hadrons have fragmented. Particle production in processes in which color is exchanged populate rapidity uniformly, leading to the exponential suppression of rapidity gaps. The presence of "non-exponentially suppressed rapidity gaps”, has been widely used as a working final-state definition of diffraction, as suggested by Bjorken.

Measurements of diffractive processes have been made at $$ep, pp$$ and $$p\bar{p}$$ colliders. The presence of a hard (short distance) scale $$Q^2$$, which may be ensured for example by requiring a large four-momentum transfer ($$Q^2$$) in DIS or high transverse momentum ($$p_T$$) hadronic jets in hadronic collisions, allows some connection to be made between Regge theory and QCD. This will be discussed in subsequent sections.

Hadrons are extended objects consisting of quarks, bound together by gluons. Quarks and gluons are collectively referred to as partons, and the distribution of partons, and their momenta, inside the hadron is a critical factor in physics at hadron colliders.

Under the assumption of factorisation, cross sections at an ep collider are given by$d\sigma_{lh \rightarrow X} = \int_x f_{a/h}(x,\mu_F) \int_{\Phi_X}|M_{la \rightarrow X}(\Phi_X,\mu_F,\mu_R)|d\Phi_X dx$

Where $$\Phi$$ is the phase space, $$|M|$$ is the matrix element, $$f_{a/h}$$ represents the probability of resolving the parton $$a$$ inside hadron $$h$$ carrying a fraction $$x$$ of the hadron's momentum (i.e. the PDF), and $$\mu_F$$ and $$\mu_R$$ are the factorization and renormalisation scales respectively, usually taken to be the magnitude of the exchanged four-momentum. In this equation and the discussion below, $$X$$ indicates inclusivity; that is it stands for all possible final states.

At hadron colliders, this expression becomes similarly$d\sigma_{h_1h_2 \rightarrow X} = \int_{x_1} \int_{x_2} f_{a/h_1}(x_1,\mu_F) f_{b/h_2}(x_2,\mu_F) \int_{\Phi_X}| M_{ab\rightarrow X}(\Phi_X,\mu_F,\mu_R)| d\Phi_X dx_1 dx_2$

Thus, knowledge of the PDFs is needed in order to predict cross sections, and conversely, precise measurements of cross sections can be used to constrain PDFs.

Several groups perform global PDF fits, including data from fixed target experiments, the HERA $$ep$$ collider, the Tevatron $$p\bar{p}$$ collider and, most recently, the first run of the LHC; the most precise and comprehensive information comes from HERA. DIS ($$ep \rightarrow e X$$ and $$ep \rightarrow \nu X$$) as studied at HERA involves colliding electrons and protons; as electrons do not interact with gluons, these measurements only directly probe the quark content of the proton. However, the collinear radiation of gluons from quarks and the splitting of gluons into quark-antiquark pairs at high scales is predicted in QCD and described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations. The interdependence thus introduced allows measurements of quark densities to be translated into information on the gluon content of the proton, principally via scaling violations. Measurements from HERA are of such precision and coverage that the gluon momentum density is highly constrained over a significant range of $$x$$. Further constraints on PDFs come from:

1. More exclusive DIS measurements, where for instance the fragmented proton (in $$X$$) contains charm quarks, or high transverse momentum ($$p_T$$) hadronic jets (see below). Such tags enhance "boson-gluon fusion" processes, where a quark-antiquark pair couples the electroweak boson to a gluon from the proton.
2. the production of jets, charged-leptons and neutrinos (so-called Drell-Yan events) and of prompt photons at the Tevatron and LHC, as well as potentially other processes such as top pair production

The structure of the hadronic fluctuations of the photon (see previous section) has also been studied, using $$e \gamma$$ DIS collisions between electrons and almost-on-shell photons, and $$\gamma \gamma$$ collisions, at LEP (principally LEP2), as well as photoproduction ($$\gamma p$$) collisions at HERA.

In DIS events containing a large, empty gap in the hadronic final state (in $$X$$), the gap can be used to define a separation between the proton, or a relatively low-mass disassociated proton system, and the rest of the event. Events tagged this way can be used to extract so-called diffractive PDFs of the proton. This gives information on the QCD structure of the colour singlet (Pomeron) exchange discussed in the previous section. However, although diffractive processes are also measured at hadron-hadron colliders, the applicability of such diffractive PDFs is complicated by the presence of secondary interactions between the proton remnants - i.e. those parts of the proton not directly participating in the hard scattering process.

## Particle multiplicities and "Soft" QCD

Soft QCD forms the final stage of any high-energy collision involving strongly interacting particles in the initial or final state.

At lepton colliders, precise measurements of particle multiplicities in hadronic final states provide stringent constraints on models for hadronization. These models then form the starting point for understanding particle production in the more complex environment of hadronic collisions.

The vast majority of hadron collisions can be characterised as peripheral scatters between hadrons in which no particular hard energy scale exists. Such events give no insight into short-distance physics, but they can shed light on the process of hadronization. They also need to be measured and understood, at least phenomenologically, because they form a background to all other studies at hadron colliders, often appearing in conjunction with events containing a harder scatter in the same, or adjacent, time window of the experiment (an effect known as 'pile up').

These events are often called 'minimum bias', since they generally pass even the loosest selection criteria of the experiment. Studying short and long-range correlations in such events constrains models of particle-production implemented in Monte Carlo simulations. As well as being useful input for other physics studies at colliders, such modelling is useful for comparisons with studies of quark-gluon matter in heavy-ion collisions, and for understanding high-energy cosmic ray air showers. Measurements of identical-particle production have shown evidence for Bose-Einstein correlations, which are generally insensitive to the short-distance hard scattering and can give information on the radial distance from the collision at which hadronization occurs.

At the mid-to-high transverse momentum end of the 'minimum-bias' spectrum, scatters involve momentum transfers of a few GeV, sufficient to resolve partons inside the hadrons. A naive convolution of the parton densities with the cross sections for parton-parton scattering shows that at high hadron-hadron centre-of-mass energies and low parton-parton centre-of-mass energy the parton-parton cross section becomes bigger than the hadron-hadron cross section. This implies that each hadron-hadron collision may contain more than one parton-parton scatter, a phenomenon known as multiple parton interactions. These additional partonic scatters may also accompany a much harder parton scattering processes, in which case they form part of the so-called 'underlying event'. Double parton scattering has been measured at several experiments, and all models capable of providing a reasonable description of minimum bias and/or the underlying event incorporate a multiple parton interaction model of some form.

## Partons and Jets

The most common method for mapping final-state particle measurements to partonic physics is to use algorithmically defined event-shape variables, designed to be insensitive to soft, long distance physics but to preserve those features of an event which reflect the momenta of high energy quarks and gluons. Global variables such as thrust or sphericity have been used, especially in leptonic collisions. However, the dominance of soft QCD radiation at angles close to the direction of an initial parton direction motivates the most commonly-used event-shape concept: a "jet", or a more-or-less collimated spray of particles and energy. Each of these sprays of particles can be combined into a composite object (the "jet"), whose kinematic properties reflect those of an initiating (unmeasurable) parton.

Figure 1: A collision recorded by the ATLAS experiment, showing the production of two jets of hadrons.

The reconstruction of jets from the hadrons produced in a typical collision is carried out using a recombination or geometric jet algorithm, and any such algorithm must meet a number of requirements. Formally, to enable them to be implemented to arbitrary order in QCD calculation, jet algorithms must be insensitive to very soft QCD radiation and to collinear splitting of partons (infrared and collinear safety) producing additional soft particles. Important experimental properties include speed and geometrical regularity, as well as a well-defined relationship between the reconstructed jet and input particles, allowing the resulting jets to be calibrated. Accurate measurement of jets presents particular l challenges for detector design, principally because in general the response of a detector to hadrons will differ from that to photons, and jets typically contain a statistically-fluctuating sample of both, since photons are produced in the decay of neutral pions.

Many jet algorithms have been proposed and used in experiments, and take as input typically a list of energy or momentum measurements from a particle detector, or a list of simulated or reconstructed final-state particles. Successive elements in this list are the compared, and if they meet a given criteria, are merged; this process is repeated until a stable list composite objects, the jets, is obtained. The main differences between algorithms lie in the criteria for merging inputs, and the most commonly used jet algorithms today base this decision on a distance separation $$d_{ij}$$ between inputs i and j, and $$d_{iB}$$ between input i and the beam, defined by$d_{ij} = {\rm min}(k_{T i}^{2p},k_{T j}^{2p})\Delta^2_{ij}/R^2\\ d_{iB}=k_{T i}^{2p},$

where $$\Delta^2_{ij}=(y_i - y_j)^2 + (\phi_i-\phi_j)^2$$, and $$k_{T i}, y_i$$ and $$\phi_i$$ are the transverse momentum, rapidity and azimuthal angle of an input particle respectively. "R" is a radius parameter which controls the size (in $$y,\phi$$ space) of the resulting jets, and "p" controls the relative importance of energy and geometrical scales. When looping through inputs, if $$d_{ij}<d_{iB}$$ the two inputs are combined; otherwise, the input is labelled a jet and removed from the list. Setting p=-1 results in the anti-$$k_T$$ algorithm, the most commonly used jet algorithm at the LHC, which essentially clusters the hardest inputs first, then successively softer inputs, all based on angular separation. The angular cut-off, or distance parameter, can be adjusted to yield larger or smaller jets, with 0.4 being a typical value. Setting p=1 gives the $$k_T$$ algorithm (put simply, this clusters softest to hardest), and p=0 gives the purely geometric Cambridge/Aachen algorithm, both commonly used in the jet substructure studies detailed below.

Jets have been an essential tool in high energy physics since measurements of three-jet production in $$e^+e^-$$ annihilation at centre-of-mass energies of around 10 GeV and above at the PETRA collider, DESY, Hamburg, provided the first compelling evidence for the existence of gluons in the final state - the events are interpreted as being due to the $$e^+e^- \rightarrow q\bar{q}g$$ hard process. Jet production in $$e^+e^-, ep, p\bar{p}$$ and $$pp$$ collisions has been used to test QCD calculations over many orders of magnitude in cross-section, to constrain proton structure, to measure the strong coupling, and to search for physics beyond the Standard Model.

As colliding beam energies have increased, the increasing phase space for jet production increases sensitivity to QCD effects and mandates calculations of increasing sophistication. Higher order calculations in the strong coupling are needed both to describe high jet multiplicities and to obtain precise predictions of cross sections. In some regions of phase space, large kinematic factors can enter into the calculations, usually in the form of logarithms of ratios of momentum scales. These terms can disrupt the convergence of a perturbative expansion in the strong coupling; practically this is often manifested as high multiplicity gluon radiation. Such contributions can often be recast as an exponential series and resummed, incorporating a portion of the calculation to infinite order in the strong coupling.

Increasingly, such effects are implemented in Monte Carlo event generators, in which a fixed (and often next-to-leading order) matrix element is matched to a logarithmic parton shower (which models high-multiplicity QCD radiation) and in which models for soft physics such as hadronization and underlying event are also incorporated, giving a complete prediction for the final state.

Measurements of jets at colliders have demonstrated the accuracy of QCD as a theory of the strong interaction, including the above effects, and demonstrating key features of QCD, including coherent radiation, the spins of the quark and gluon, and the color factors involved in their couplings.

## Charm, Bottom and Top

Moving up the generations of the Standard Model, quark masses increase. The three heaviest quarks (charm, bottom, top) all have masses above the typical baryon binding energy (~1 GeV), breaking the approximate flavour symmetry among the three lightest quarks (up, down, strange) and providing the dominant contribution to the masses of hadrons containing these quarks.

Figure 2: The dimuon mass spectrum measured in early CMS data, showing the $$J/\psi$$ and $$\Upsilon$$ resonances, as well as several others.

The existence of the charm and bottom was proved through the discovery of the quarkonia states: the $$J/\psi$$ ($$c\bar{c}$$) with a mass of 3.1 GeV in 1974, and $$\Upsilon$$ ($$b\bar{b}$$) with a mass of 9.4 GeV in 1977. While quarkonia decays proceed through quark-antiquark annihilation, in general these massive quarks decay via the electroweak force, emitting a W boson. Mixing of the quark generations, parameterised by the CKM matrix, allows decays across generations, such as $$b \rightarrow W c$$. As the charm and bottom quark masses are far below the W boson mass (80 GeV) these decays are suppressed, leading to lifetimes of order $$10^{-12}$$ s for most b- and c-hadrons. As a result, when these hadrons are produced in high-energy collisions, the hadrons can travel millimeters before decaying - an observable distance. Identifying such displaced decays forms the basis for most "tagging" algorithms, which attempt to assign a flavour (bottom, charm or light) to a jet. Production of charm and bottom in proton-proton collisions proceeds through $$g\rightarrow c\bar{c}$$ or $$g\rightarrow b\bar{b}$$, and following the factorisation above, can occur from an initial or final state gluon in the matrix element, or be folded into the PDF. The charm or bottom PDFs are generated perturbatively from the gluon PDF, and the quark masses are set to zero.

Figure 3: The relative proportions of different $$t\bar{t}$$ decay modes.

The top quark is the heaviest particle in the Standard Model of particle physics, with a mass of approximately 173 GeV. Its large mass relative to both the W boson and b-quark means there is no suppression of the weak process $$t\rightarrow Wb$$, and the top decays before forming QCD bound states. Therefore there is no "toponia" state, and the top quark is measured through its decay products. There is almost no mixing of the top with other quarks in the CKM matrix, and it decays $$t \rightarrow Wb$$ with a branching fraction of 99.9%. Events containing $$t\bar{t}$$ are characterised by the decay modes of the W: all hadronic ($$W^{+}(\rightarrow qq')b + W^{-}(\rightarrow qq')\bar{b}$$), single lepton ($$W^{+}(\rightarrow l^{+}\nu)b + W^{-}(\rightarrow qq')\bar{b}$$ or $$W^{+}(\rightarrow qq')b + W^{-}(\rightarrow l^{-}\bar{\nu})\bar{b}$$)) or dilepton ($$W^{+}(\rightarrow l^{+}\nu)b + W^{-}(\rightarrow l^{-}\bar{\nu})\bar{b}$$), with the relative proportions therefore determined by the branching fraction of the W boson. The single- and di-lepton mode, while being less common than the all-hadronic, provide a more distinctive experimental signature and are used in the majority of top quark measurements including the discovery of the top quark at the Tevatron in 1995. Single top quark production is rarer and experimentally harder to identify, but can occur through the $$qq'\rightarrow W\rightarrow tb$$, or $$b\rightarrow Wt$$.

Heavy quarks remain active areas of research. The CKM matrix contains a complex phase, allowing CP violation in both in the decays and mixing of b- and c-hadrons, a possible contribution to the matter-antimatter asymmetry in the universe. The Higgs boson should decay to $$b\bar{b}$$ at a rate of 56%, and a definitive observation of this signal is a key aim for the LHC Run 2. The top quark Yukawa coupling to the Higgs field is approximately 1, which is often taken as a suggestion that a deeper understanding of the top quark properties will shed further light on the Higgs mechanism itself. The explanation of the large top mass may also lie outside the Standard Model, and many theories for beyond-the-Standard-Model physics predict enhanced couplings between some new particles and top or bottom quarks, so those signals are also used extensively in searches at the LHC.

## Jets in association with photons and weak bosons

The production of jets in association with vector bosons ("V") provides a rich testing ground for the study of QCD at hadron colliders. Photons, and the leptonic (specifically electron and muon) decay modes of the W and Z, are relatively clear experimental signals at hadron colliders. Triggering on, and reconstructing these signals with low backgrounds allows an unbiased study of any jets present in these events. Essentially, the colourless photon or weak boson can then be used as a probe of the underlying QCD process. The high collision energy at the LHC allows for the production of multiple hard jets (typically selected with a minimum of 20-30 GeV of momentum transverse to the beam direction) in addition to the vector boson. These "V+jets" signatures are relatively common at the LHC, and provide an excellent testing ground for the understanding and modelling of high energy, high multiplicity final states.

Figure 4: Inclusive jet multiplicity in dilepton events, from an ATLAS analysis of 13 TeV proton proton collisions. While Z+jets dominates, contributions from other sources are visible.

In perturbative QCD calculations, inclusive vector boson production has been available at next-to-next-to-leading Order (NNLO) for some time. However, this fixed order calculation includes up to two real parton emissions, corresponding (approximately) to up to two high energy jets in association with the vector boson. In order to model the higher jet multiplicities frequently seen at the LHC, different predictions are merged together. This was initially achieved for the separate "tree level" (leading order, LO) matrix-element calculations of V+0, V+1, V+2, ... V+5 partons, which were merged to form a complete, tree-level V+{0-5} parton sample. To convert this parton level calculation into a full particle level prediction, it must be matched to a parton shower and hadronisation model, as described above. But now this matching introduces possible double-counting between samples: as an example, the parton shower may add an additional emission to a V+1 parton event, turning it into a V+2 parton event. But the V+2 parton events are already being modelled by a dedicated V+2 sample, so there are now too many V+2 parton events. To remove this overlap, merging cuts are used. As the parton shower best describes softer parton emission, partons below a certain scale are allowed to come only from the parton shower, and above a certain scale must come from a matrix element. In this way, the different multiplicity samples can still be merged to produce a full sample with no double-counting. These merged matrix-element + parton-shower (ME+PS) samples are the only way to model the high jet multiplicities seen at the LHC, and while the exact definition of the scale and the cut value used varies, there are several implementations of this idea.

Recent developments have allowed the merging of NLO calculations of V+[0-3] jets with LO calculations of higher multiplicities. In this case, there is an overlap between the matrix element calculations (for example, V+1 parton at NLO contains the LO V+2 parton prediction) so additional merging techniques have been developed. While this increases the complication of the merging process, it comes with the benefit that the NLO matrix elements have smaller theoretical uncertainties than the LO equivalent, so have better predictive power. These predictions will be used extensively for the LHC Run 2 data analysis.

Extensive measurements at both the Tevatron and LHC have proved the success of these merging techniques in describing real data. Accurate modelling of V+jets across a huge kinematic range is of key importance across the physics programme at any hadron collider: from low to high momentum and jet multiplicity, V+jets signals are used to test the understanding of basic QCD, and frequently form the dominant background to rare signals, such as associated Higgs boson production (VH, with $$H \rightarrow b\bar{b}$$), or new phenomena which may appear in the high energy tails of distributions.

## Jet Substructure

Jets are reconstructed from the collimated sprays of particles produced in high energy QCD interactions, and it is generally the properties of the resulting composite jet that are of interest: the total energy, direction, and so on. However, the distribution of particles and energy within a jet can also reveal information about both the production mechanism and underlying physics. The study of jet substructure is an area that has seen significant interest and activity at the LHC, as well as in the theory and phenomenology communities.

Given that soft QCD takes over somewhere around the 1 GeV region, and that typical energy scales involved in jet production extend far above this, much of the evolution of jet substructure takes places within the perturbative regime. The substructure of jets is to a large extent dictated by harder QCD radiation, and much of it is therefore calculable in perturbative QCD. Measurements of jet substructure including jet shapes, masses, particle multiplicities, as well as the presence of narrower subjets within the jet, have indeed been successfully compared to such calculations, and consistent values for the strong couplings constant have been extracted, as well as information on the subprocess mixture and the initiating parton (quark or gluon).

Figure 5: The decay products fo a boosted top quark may be reconstructed in a single jet.

However, jet substructure has obtained wider relevance at the LHC, since the electroweak scale lies squarely between the soft QCD scale and the kinematic limit for jet production, extending into the multi-TeV range. This has the practical consequence that particles with masses around the electroweak scale ($$W, Z$$ and Higgs bosons and the top quark) may be produced with high Lorentz boost, and thus their decay products will be highly collimated and, in the case of hadronic decays, may be reconstructed as a single jet. Such configurations have the advantage that combinatorial and other backgrounds can be suppressed, but they mandate the use of jet mass and other jet substructure variables in order to identify and reconstruct the decaying massive particle. Such techniques have been widely employed in measurements at the LHC (for example of the top quark), in searches both for the SM Higgs decaying to b-quarks, and especially in searches for BSM physics, since they become increasingly relevant at the highest transverse momenta, the area of most interest for such searches.

Well-understood jet substructure variables are also widely used at the LHC to reduce sensitivity to soft physics and to energy from pile-up (see above). Various methods for removing soft particles and energy flow are used, collectively referred to as 'jet grooming' techniques. These can improve the energy and mass resolution of jets significantly and dramatically reduce sensitivity to pile-up.

## Summary and Outlook

Jet measurements and the comparison to QCD calculations at high energy colliders have been critical in establishing QCD as the precision theory of the strong interaction, in probing the structure of the proton and in measuring the strong coupling. They have also played an important role in studying the physics of electroweak symmetry-breaking and in searching for physics beyond the Standard Model. This role is set to continue at the LHC and future colliders, as the production of jets in association with gauge bosons, top quarks and the Higgs boson become common and highly interesting processes. Most recently, jet substructure has become established as an essential component of the tool-kit for experiments above the electroweak symmetry-breaking scale. Jet measurements at colliders, matched by sophisticated calculations, will for the foreseeable future be crucial in extending our knowledge of fundamental physics at the energy frontier.

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