Search

We compute the average Lund multiplicity of high-energy QCD jets. This extends an earlier calculation, done for event-wide multiplicity in $e^+e^-$ collisions [arxiv:2205.02861], to the large energy range available at the LHC. Our calculation achieves next-to-next-to-double logarithmic (NNDL) accuracy. Our results are split into a universal collinear piece, common to the $e^+e^-$ calculation, and a non-universal large-angle contribution. The latter amounts to 10-15% of the total multiplicity. We provide accurate LHC predictions by matching our resummed calculation to fixed-order NLO results and by incorporating non-perturbative corrections via Monte Carlo simulations. Including NNDL terms leads to a 50% reduction of the theoretical uncertainty, with non-perturbative corrections remaining below 5% down to transverse momentum scales of a few GeV. This proves the suitability of Lund multiplicities for robust theory-to-data comparisons at the LHC., Comment: 37 pages, 9 figures

Jet physics : theory overview

Discriminating quark and gluon jets is a long-standing topic in collider phenomenology. In this paper, we address this question using the Lund jet plane substructure technique introduced in recent years. We present two complementary approaches: one where the quark/gluon likelihood ratio is computed analytically, to single-logarithmic accuracy, in perturbative QCD, and one where the Lund declusterings are used to train a neural network. For both approaches, we either consider only the primary Lund plane or the full clustering tree. The analytic and machine-learning discriminants are shown to be equivalent on a toy event sample resumming exactly leading collinear single logarithms, where the analytic calculation corresponds to the exact likelihood ratio. On a full Monte Carlo event sample, both approaches show a good discriminating power, with the machine-learning models usually being superior. We carry on a study in the asymptotic limit of large logarithm, allowing us to gain confidence that this superior performance comes from effects that are subleading in our analytic approach. We then compare our approach to other quark-gluon discriminants in the literature. Finally, we study the resilience of our quark-gluon discriminants against the details of the event sample and observe that the analytic and machine-learning approaches show similar behaviour., Comment: 40 pages, 11 figures

Standard dipole parton showers are known to yield incorrect subleading-colour contributions to the leading (double) logarithmic terms for a variety of observables. In this work, concentrating on final-state showers, we present two simple, computationally efficient prescriptions to correct this problem, exploiting a Lund-diagram type classification of emission regions. We study the resulting effective multiple-emission matrix elements generated by the shower, and discuss their impact on subleading colour contributions to leading and next-to-leading logarithms (NLL) for a range of observables. In particular we show that the new schemes give the correct full colour NLL terms for global observables and multiplicities. Subleading colour issues remain at NLL (single logarithms) for non-global observables, though one of our two schemes reproduces the correct full-colour matrix-element for any number of energy-ordered commensurate-angle pairs of emissions. While we carry out our tests within the PanScales shower framework, the schemes are sufficiently simple that it should be straightforward to implement them also in other shower frameworks., 54 pages, 17 figures

In a series of previous papers, we have presented a new approach, based on perturbative QCD, for the evolution of a jet in a dense quark-gluon plasma. In the original formulation, the plasma was assumed to be homogeneous and static. In this work, we extend our description and its Monte Carlo implementation to a plasma obeying Bjorken longitudinal expansion. Our key observation is that the factorisation between vacuum-like and medium-induced emissions, derived in the static case, still holds for an expanding medium, albeit with modified rates for medium-induced emissions and transverse momentum broadening, and with a modified phase-space for vacuum-like emissions. We highlight a scaling relation valid for the energy spectrum of medium-induced emissions, through which the case of an expanding medium is mapped onto an effective static medium. We find that scaling violations due to vacuum-like emissions and transverse momentum broadening are numerically small. Our new predictions for the nuclear modification factor for jets $R_{AA}$, the in-medium fragmentation functions, and substructure distributions are very similar to our previous estimates for a static medium, maintaining the overall good qualitative agreement with existing LHC measurements. In the case of $R_{AA}$, we find that the agreement with the data is significantly improved at large transverse momenta $p_T\gtrsim 500$ GeV after including the effects of the nuclear parton distribution functions., 35 pages, 7 figures

The Lund-jet plane has recently been proposed as a powerful jet substructure tool with a broad range of applications. In this paper, we provide an all-order single logarithmic calculation of the primary Lund-plane density in Quantum Chromodynamics, including contributions from the running of the coupling, collinear effects for the leading parton, and soft logarithms that account for large-angle and clustering effects. We also identify a new source of clustering logarithms close to the boundary of the jet, deferring their resummation to future work. We then match our all-order results to exact next-to-leading order predictions. For phenomenological applications, we supplement our perturbative calculation with a Monte Carlo estimate of non-perturbative corrections. The precision of our final predictions for the Lund-plane density is 5-7% at high transverse momenta, worsening to about 20% at the lower edge of the perturbative region, corresponding to transverse momenta of about 5 GeV. We compare our results to a recent measurement by the ATLAS collaboration at the Large-Hadron Collider, revealing good agreement across the perturbative domain, i.e. down to about 5 GeV., 44 pages, 17 figures

Using a recently-developed perturbative-QCD approach for jet evolution in a dense quark-gluon plasma, we study the nuclear modification factor for the jet fragmentation function. The qualitative behaviour that we find is in agreement with the respective experimental observations in Pb+Pb collisions at the LHC: a pronounced nuclear enhancement at both ends of the spectrum. Our Monte Carlo simulations are supplemented with analytic estimates which clarify the physical interpretation of the results. The main source of theoretical uncertainty is the sensitivity of our calculations to a low-momentum cutoff which mimics confinement. To reduce this sensitivity, we propose a new observable, which describes the jet fragmentation into subjets and is infrared-and-collinear safe by construction. We present Monte Carlo predictions for the associated nuclear modification factor together with their physical interpretation., Comment: 37 pages, 11 figures, to appear in JHEP

The past few years have seen a rapid development of machine-learning algorithms. While surely augmenting performance, these complex tools are often treated as black-boxes and may impair our understanding of the physical processes under study. The aim of this paper is to move a first step into the direction of applying expert-knowledge in particle physics to calculate the optimal decision function and test whether it is achieved by standard training, thus making the aforementioned black-box more transparent. In particular, we consider the binary classification problem of discriminating quark-initiated jets from gluon-initiated ones. We construct a new version of the widely used N-subjettiness, which features a simpler theoretical behaviour than the original one, while maintaining, if not exceeding, the discrimination power. We input these new observables to the simplest possible neural network, i.e. the one made by a single neuron, or perceptron, and we analytically study the network behaviour at leading logarithmic accuracy. We are able to determine under which circumstances the perceptron achieves optimal performance. We also compare our analytic findings to an actual implementation of a perceptron and to a more realistic neural network and find very good agreement., Comment: 32 pages, 9 figures; v2 brings extra clarifications, as accepted by JHEP

We study the fragmentation function of jets propagating through a dense quark-gluon plasma within perturbative QCD. Our results for its nuclear modification factor are in qualitative agreement with the experimental data in Pb+Pb collisions at the LHC. In particular, we reproduce the enhancements seen in the data at both relatively soft and relatively large transverse momenta, with clear physical interpretations. The perturbative predictions however are quite sensitive to the value of the infrared cutoff mimicking the confinement scale, due to the fact that the fragmentation function is not an infrared safe quantity. To remedy this, we propose a new observable -- the (primary) subjet fragmentation function -- which is infrared safe and has features similar to the fragmentation function. We provide predictions for this observable in the vacuum and in heavy-ion collisions that could be tested against the experimental data., 4 pages, 4 figures; 10th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions

We study a puzzle raised recently regarding the running coupling prescription used in the calculation of forward particle production in proton-nucleus collisions at next-to-leading order: using a coordinate space prescription which is consistent with the one used in the high energy evolution of the target leads to results which can be two orders of magnitude larger than the ones obtained with a momentum space prescription. We show that this is an artefact of the Fourier transform involved when passing between coordinate and momentum space and propose a new coordinate space prescription which avoids this problem., Comment: 4 pages, 1 figure. Quark Matter 2018 proceedings

Using a pQCD picture for jet evolution in a dense QCD medium, in which medium-induced parton branchings are factorized from vacuum-like emissions, we study two jet substructure observables: the $z_g$ and the Soft Drop multiplicity distributions. We compute the respective nuclear modification factors using a Monte-Carlo implementation of the parton showers. Our results are in qualitative agreement with LHC data for Pb+Pb collisions. We identify the physical mechanisms explaining our results: incoherent jet energy loss, semi-hard medium-induced emissions and the bias introduced by the steeply falling jet spectrum., Comment: Presented at the XXVIIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2019), Wuhan, China

When computed to next-to-leading order in perturbative QCD, the non-linear Balitsky-Kovchegov (BK) equation for the high-energy evolution of the dipole-hadron scattering appears to be unstable. We show that this instability can be avoided by using the rapidity of the dense hadronic target (instead of that of the dilute dipole projectile) as the evolution time. Using this variable, we construct a collinearly-improved version of the BK equation, where the dominant radiative corrections to the kernel -- those enhanced by double collinear logarithms -- are resummed to all orders., Comment: Presented at the XXVIIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2019), Wuhan, China

Parton showers are among the most widely used tools in collider physics. Despite their key importance, none so far has been able to demonstrate accuracy beyond a basic level known as leading logarithmic (LL) order, with ensuing limitations across a broad spectrum of physics applications. In this letter, we propose criteria for showers to be considered next-to-leading logarithmic (NLL) accurate. We then introduce new classes of shower, for final-state radiation, that satisfy the main elements of these criteria in the widely used large-$N_C$ limit. As a proof of concept, we demonstrate these showers' agreement with all-order analytical NLL calculations for a range of observables, something never so far achieved for any parton shower., 6 pages, 2 figures, plus supplemental material; v2 brings extra references and clarifications, as accepted by Physical Review Letters

Within perturbative QCD, we develop a new picture for the parton shower generated by a jet propagating through a dense quark-gluon plasma. This picture combines in a simple, factorised, way multiple medium-induced parton branchings and standard vacuum-like emissions, with the phase-space for the latter constrained by the presence of the medium. We implement this picture as a Monte Carlo generator that we use to study two phenomenologically important observables: the jet nuclear modification factor $R_{AA}$ and the $z_g$ distribution reflecting the jet substructure. In both cases, the outcome of our Monte Carlo simulations is in good agreement with the LHC measurements. We provide basic analytic calculations that help explaining the main features observed in the data. We find that the energy loss by the jet is increasing with the jet transverse momentum, due to a rise in the number of partonic sources via vacuum-like emissions. This is a key element in our description of both $R_{AA}$ and the $z_g$ distribution. For the latter, we identify two main nuclear effects: incoherent jet energy loss and hard medium-induced emissions. As the jet transverse momentum increases, we predict a qualitative change in the ratio between the $z_g$ distributions in PbPb and pp collisions: from increasing at small $z_g$, this ratio becomes essentially flat, or even slightly decreasing., 49 pages, 23 figures

To maximise the potential for new measurements and discoveries at the LHC, the machine delivers as high as possible collision rates. As a consequence, multiple proton-proton collisions occur whenever two bunches cross. Interesting high-energy (hard) collisions are therefore contaminated by several soft, zero-bias, ones. This effect known as pileup pollutes the final state of the collision. It complicates the reconstruction of the objects in this final state, resulting in increased experimental uncertainties. To reduce these uncertainties, and improve the quality and precision of LHC measurements, techniques are devised to correct for the effects of pileup. This document provides a theoretical review of the main methods used during Run I and II of the LHC to mitigate pileup effects. I start with an in-depth presentation of the area--median used for the majority of applications, including several refinements of the original idea, their practical implementation and an assessment of their efficiency and robustness. I then focus on several theoretical calculations that can provide both quantitative and qualitative information on the area--median approach. In the case of boosted jets, a field that has seen a wide interest recently, a set of methods, known as grooming techniques has also been used. I describe these techniques, address their performance and briefly show that they are amenable to a theoretical, analytic, understanding. The last part of this review focuses on ideas oriented towards future pileup mitigation techniques. This includes new methods that have recently been proposed as well as a large series of alternative ideas. The latter are yet unpublished and have not received the same amount of investigation than the former but they have the potential to bring new developments and further improvement over existing techniques in a future where pileup mitigation will be crucial., 259 pages, 101 figures. Version to appear in Phys. Rept

The high-energy evolution in perturbative QCD suffers from a severe lack-of-convergence problem, due to higher order corrections enhanced by double and single transverse logarithms. We resum double logarithms to all orders within the non-linear Balitsky-Kovchegov equation, by taking into account successive soft gluon emissions strongly ordered in lifetime. We further resum single logarithms generated by the first non-singular part of the splitting functions and by the one-loop running of the coupling. The resummed BK equation admits stable solutions, which are used to successfully fit the HERA data at small $x$ for physically acceptable initial conditions and reasonable values of the fit parameters., Comment: 4 pages, based on a talk presented at Quark Matter 2015, 27 Sep-3 Oct 2015, Kobe, Japan

A main difficulty in understanding the dynamics of jets produced in the high-density environment of ultrarelativistic heavy ion collision, is to provide a unified description for the two sources of radiation that are a priori expected: the "vacuum-like" emissions responsible for the parton shower from large virtualities down to the hadronisation scale and the "medium-induced" emissions responsible for the energy loss by the jet. In the recent paper arXiv:1801.09703, we demonstrated that these two mechanisms can be factorized from each other within a controlled, "double-logarithmic" approximation in perturbative QCD. In this proceeding we recall the main features of the jet evolution in a dense QCD medium. We emphasize that the in-medium parton showers differ from those in the vacuum in two crucial aspects: their phase-space is reduced and the first emission outside the medium can violate angular ordering. Based on this factorized picture, which is Markovian, we have recently developed a Monte Carlo event generator which includes both vacuum-like and medium-induced emissions and goes beyond the double logarithmic approximation. We here present our first results for the fragmentation function and the energy loss by the jet., 8 pages, 6 figures, for the International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions, 30 September - 5 October 2018 in Aix-Les-Bains, Savoie, France

Since many facets and applications of jet substructure have been covered in this book, it is useless to try and summarise them all individually. Instead, in this concluding chapter, we will briefly summarise the main lessons we have learned from about a decade of jet substructure studies and from the aspects covered in this book.

In this chapter, we give a brief overview of existing experimental performance studies, as well as measurements and searches performed by the ATLAS and CMS collaborations using jet substructure techniques. Our goal is not to provide a comprehensive discussion of all searches and measurements performed by LHC experiments, but rather to select and showcase results with close connection to the topics discussed before.

The next-to-leading order (NLO) Balitsky-Kovchegov (BK) equation describing the high-energy evolution of the scattering between a dilute projectile and a dense target suffers from instabilities unless it is supplemented by a proper resummation of the radiative corrections enhanced by (anti-)collinear logarithms. Earlier studies have shown that if one expresses the evolution in terms of the rapidity of the dilute projectile, the dominant anti-collinear contributions can be resummed to all orders. However, in applications to physics, the results must be re-expressed in terms of the rapidity of the dense target. We show that although they lead to stable evolution equations, resummations expressed in the rapidity of the dilute projectile show a strong, unwanted, scheme dependence when their results are translated in terms of the target rapidity. Instead, in this paper, we work directly in the rapidity of the dense target where anti-collinear contributions are absent but where new, collinear, instabilities arise. These are milder since disfavoured by the typical BK evolution. We propose several prescriptions for resumming these new double logarithms and find only little scheme dependence. The resummed equations are non-local in rapidity and can be extended to full NLO accuracy., Comment: 74 pages, 12 figures

There is no jet substructure without jets. Jets are ubiquitous objects in collider physics, which have been used for 40 years. This Chapter discusses the concept jets as collimated sprays of particles (or flows of energy). It also introduces the algorithms which are used for their practical definitions, basic experimental aspects and the software implementation.

The study of the internal structure of hadronic jets has become in recent years a very active area of research in particle physics. Jet substructure techniques are increasingly used in experimental analyses by the LHC collaborations, both in the context of searching for new physics and for Standard Model measurements. On the theory side, the quest for a deeper understanding of jet substructure algorithms has contributed to a renewed interest in all-order calculations in QCD. This has resulted in new ideas about how to design better observables and how to provide a solid theoretical description for them. In the last years, jet substructure has seen its scope extended, for example, with an increasing impact in the study of heavy-ion collisions, or with the exploration of deep-learning techniques. Furthermore, jet physics is an area in which experimental and theoretical approaches meet together, where cross-pollination and collaboration between the two communities often bear the fruits of innovative techniques. Despite the wealth of literature on this topic, we feel that a comprehensive and, at the same time, pedagogical introduction to jet substructure is still missing. This makes the endeavour of approaching the field particularly hard, as newcomers have to digest an increasing number of substructure algorithms and techniques, too often characterised by opaque terminology and jargon. Furthermore, while first-principle calculations in QCD have successfully been applied in order to understand and characterise the substructure of jets, they often make use of calculational techniques, such as resummation, which are not the usual textbook material. Thus, the idea of combining our experience in different aspects of jet substructure phenomenology to put together this set of lecture notes, which we hope could help and guide someone who moves their first steps in the physics of jet substructure.

Two-prong taggers aim at discriminating massive objects that decay into two hard QCD partons (usually quarks), from the background of QCD jets. This signal is often an electroweak boson (H/W/Z) but it can also be a new particle (see Chap. 10 for examples). Our goal in this chapter is two-folded and it closely follows what was done in the previous chapter for quark-gluon tagging. First, we want to give a brief insight into analytic properties of two-prong taggers, mainly selecting a few representative substructure tools and comparing their behaviour in Monte Carlo simulations with analytic results. Then, we will perform a comparative Monte Carlo study of the taggers discriminating properties, assessing both their performance and their resilience against non-perturbative effects.

Since both jets and their substructure are deeply rooted in Quantum Chromodynamics (QCD), the fundamental theory of strong interactions, we give a brief introduction to QCD in the context of collider experiments. Since QCD is a vast field, we focus here on the introductory aspects which are directly relevant for the purpose of this book.

In this chapter we will revisit the calculations performed in Chap. 4 and extend them in order to describe jet mass distributions with grooming algorithms. In what follows, we are not going to present state-of-the art theoretical calculations, but instead we aim to keep the our discussion as simple as possible. Therefore, the theoretical accuracy of the calculations that we will present will be the minimum one which is required to capture the essential feature of the distributions. We will mostly concentrate of QCD jets, which present the most interesting and intricate features, while a discussion about jets originated to a boosted heavy particles will be presented in Sect. 6.4.

One of our main goal in this book is to connect the phenomenology of jet substructure to analytic calculations in QCD. This Chapter studies an important example: the jet mass distribution. This simple and typical property of a jet allows us to discuss the need for all-order calculations in the boosted regime and to introduce resummation techniques which will be used repeatedly for all the more advanced jet substructure tools we study in later chapters.

In this Chapter we go back to the SoftDrop algorithm and we study two variables deeply connected to it: the angular separation of the two subjets that pass SoftDrop and their momentum sharing. We find that while the former can be described using the all-order techniques described so far, the latter one cannot be. It instead exhibits peculiar features. This observable is not collinear safe for all values of the angular exponent β but it belongs to the wider class of Sudakov-safe observables.

Within the framework of the dipole factorisation, we use a recent collinearly-improved version of the Balitsky-Kovchegov equation to fit the HERA data for inclusive deep inelastic scattering at small Bjorken $x$. The equation includes an all-order resummation of double and single transverse logarithms and running coupling corrections. Compared to similar equations previously proposed in the literature, this work makes a direct use of Bjorken $x$ as the rapidity scale for the evolution variable. We obtain excellent fits for reasonable values for the four fit parameters. We find that the fit quality improves when including resummation effects and a physically-motivated initial condition. In particular, the resummation of the DGLAP-like single transverse logarithms has a sizeable impact and allows one to extend the fit up to relatively large photon virtuality $Q^2$., Comment: 16 pages, 4 figures, 2 tables

We investigate some well-known problematic aspects of the single-jet inclusive cross-section, specifically its non-unitarity and the possibly related issue of apparent perturbative instability at low orders. We study and clarify their origin by introducing possible alternative weighted definitions of the observable which restore unitarity. We show that the perturbative instability of the standard definition is an accidental artefact of the smallness of the NLO $K$ factor which only manifests itself for values of the jet radius in the range $R\sim 0.3-0.6$, and that its non-unitarity is necessary in order to ensure cancellation of logs of the momentum cutoff used in the jet definition. We also show that alternative unitary definitions do not have better perturbative properties compared to the conventional non-unitary definition, while suffering from lack of cancellation of large logs., Comment: 19 pages, 6 figures, accepted for publication on PRD

This chapter is dedicated to the specific applications of substructure techniques to discriminate quark-initiated jets from gluon-initiated jets. We begin with a brief discussion on how to tentatively define quark- and gluon-initiated jets. We then concentrate on two broad classes of observables: (i) shape observables like angularities or energy-correlation functions, and (ii) multiplicity-based observables like the iterated soft-drop multiplicity. In each case, we provide a basic analytic understanding of the tools based on the resummation techniques introduced in Chap. 4. We then discuss the key properties of the tools and assess their performance on Monte-Carlo simulations, highlighting the features which are reproduced by the simple analytic calculations.

In this chapter we genuinely start to discuss jet substructure per se. We begin with a generic description of the physics concepts which are at the foundation of jet substructure techniques. We then discuss ideas on how to assess the performance of these tools, including their sheer discriminating power as well as their resilience against non-perturbative effects. The remaining part of the chapter is dedicated to a description of jet substructure techniques themselves, covering an extensive list of tools which we have organised, as much as possible, according to the basic concepts discussed at the beginning of the chapter.