The problem of interpretability of machine learning architecture in particle physics has no agreed-upon definition, much less any proposed solution. We present a first modest step toward these goals by proposing a definition and corresponding practical method for isolation and identification of relevant physical energy scales exploited by the machine. This is accomplished by smearing or averaging over all input events that lie within a prescribed metric energy distance of one another and correspondingly renders any quantity measured on a finite, discrete dataset continuous over the dataspace. Within this approach, we are able to explicitly demonstrate that (approximate) scaling laws are a consequence of extreme value theory applied to analysis of the distribution of the irreducible minimal distance over which a machine must extrapolate given a finite dataset. As an example, we study quark versus gluon jet identification, construct the smeared likelihood, and show that discrimination power steadily increases as resolution decreases, indicating that the true likelihood for the problem is sensitive to emissions at all scales.

United States

Andrew Larkoski

A step toward interpretability: smearing the likelihood

Top quark polarization provides an important tool for studying its production mechanisms, spin correlations, top quark properties, and new physics searches. Unlike lighter quarks, the top quark’s polarization remains intact until its decay, enabling precise spin measurements. While the down-type fermions from boson decay are known to be effective spin analyzers, charged leptons have typically been the main target for most analyses. In this paper, we investigate the relevance of global jet dynamics – considering kinematics, jet charges, and particle multiplicity – for hadronic top quark polarimetry. The formalism used allows for analytical derivations obtained throughout the manuscript, offering deeper insights into the corresponding phenomenology.

Zhongtian Dong

Dorival Gonçalves

Kyoungchul Kong

Alberto Navarro

Analytical insights on hadronic top quark polarimetry

The Energy Mover’s Distance (EMD) has seen use in collider physics as a metric between events and as a geometric method of defining infrared and collinear safe observables. Recently, the Spectral Energy Mover’s Distance (SEMD) has been proposed as a more analytically tractable alternative to the EMD. In this work, we obtain a closed-form expression for the Riemannian-like $\textit{p}$ = 2 SEMD metric between events, eliminating the need to numerically solve an optimal transport problem. Additionally, we show how the SEMD can be used to define event and jet shape observables by minimizing the distance between events and parameterized energy flows (similar to the EMD), and we obtain closed-form expressions for several of these observables. We also present the Specter framework, an efficient and highly parallelized implementation of the SEMD metric and SEMD-derived shape observables as an analogue of the previously-introduced Shaper for EMD-based computations. We demonstrate that computing the SEMD with Specter can be up to a thousand times faster than computing the EMD with standard optimal transport libraries.

Rikab Gambhir

Jesse Thaler

SPECTER: efficient evaluation of the spectral EMD

Recently, a factorization theorem was proposed for partonic flavor evolution as defined by the net flavor of the Winner-Take-All axis of a jet. We validate the factorization theorem through explicit calculation at two-loop order, and in the process extract all anomalous dimensions and renormalization factors for any ultraviolet-to-infrared flavor transition at this order. These results can then be used to extract the renormalized hard function for flavored jet production at next-to-next-to-leading order for any process of interest.

Flavor factorization at two-loops

These lectures were presented at the 2024 QCD Masterclass in Saint-Jacut-de-la-Mer, France. They introduce and review fundamental theorems and principles of machine learning within the context of collider particle physics, focused on application to jet identification and discrimination. Numerous examples of binary discrimination in jet physics are studied in detail, including  $$H\rightarrow b{\bar{b}}$$  <math> <mrow> <mi>H</mi> <mo>→</mo> <mi>b</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   identification in fixed-order perturbation theory, generic one- versus two-prong discrimination with parametric power counting techniques, and up versus down quark jet classification by assuming the central limit theorem, isospin conservation, and a convergent moment expansion of the single particle energy distribution. Quark versus gluon jet discrimination is considered in multiple contexts, from using additive, infrared and collinear safe observables, to using hadronic multiplicity, and to including measurements of the jet charge. While many of the results presented here are well known, some novel results are presented, the most prominent being a parametrized expression for the likelihood ratio of quark versus gluon discrimination for jets on which hadronic multiplicity and jet charge are simultaneously measured. End-of-lecture exercises are also provided.

QCD masterclass lectures on jet physics and machine learning

How can one fully harness the power of physics encoded in relativistic N-body phase space? Topologically, phase space is isomorphic to the product space of a simplex and a hypersphere and can be equipped with explicit coordinates and a Riemannian metric. This natural structure that scaffolds the space on which all collider physics events live opens up new directions for machine learning applications and implementation. Here we present a detailed construction of the phase space manifold and its differential line element, identifying particle ordering prescriptions that ensure that the metric satisfies necessary properties. We apply the phase space metric to several binary classification tasks, including discrimination of high-multiplicity resonance decays or boosted hadronic decays of electroweak bosons from QCD processes, and demonstrate powerful performance on simulated data. Our work demonstrates the many benefits of promoting phase space from merely a background on which calculations take place to being geometrically entwined with a theory’s dynamics.

Tianji Cai

Junyi Cheng

Nathaniel Craig

Giacomo Koszegi

The phase space distance between collider events

Machine learning-based jet classifiers are able to achieve impressive tagging performance in a variety of applications in high-energy and nuclear physics. However, it remains unclear in many cases which aspects of jets give rise to this discriminating power, and whether jet observables that are tractable in perturbative QCD such as those obeying infrared-collinear (IRC) safety serve as sufficient inputs. In this article, we introduce a new classifier, Jet Flow Networks (JFNs), in an effort to address the question of whether IRC unsafe information provides additional discriminating power in jet classification. JFNs are permutation-invariant neural networks (deep sets) that take as input the kinematic information of reconstructed subjets. The subjet radius and a cut on the subjet’s transverse momenta serve as tunable hyperparameters enabling a controllable sensitivity to soft emissions and nonperturbative effects. We demonstrate the performance of JFNs for quark vs. gluon and Z vs. QCD jet tagging. For small subjet radii and transverse momentum cuts, the performance of JFNs is equivalent to the IRC-unsafe Particle Flow Networks (PFNs), demonstrating that infrared-collinear unsafe information is not necessary to achieve strong discrimination for both cases. As the subjet radius is increased, the performance of the JFNs remains essentially unchanged until physical thresholds that we identify are crossed. For relatively large subjet radii, we show that the JFNs may offer an increased model independence with a modest tradeoff in performance compared to classifiers that use the full particle information of the jet. These results shed new light on how machines learn patterns in high-energy physics data.

Dimitrios Athanasakos

James Mulligan

Mateusz Płoskoń

Felix Ringer

Is infrared-collinear safe information all you need for jet classification?

Binary discrimination between well-defined signal and background datasets is a problem of fundamental importance in particle physics. With detailed event simulation and the advent of extensive deep learning tools, identification of the likelihood ratio has typically been reserved as a computational problem. However, this approach can obscure overtraining or excessive sensitivity to tuned features of the simulation that may not be well-defined theoretically. Here, we present the first analysis of binary discrimination for signal and background distributions for which their likelihood ratio is infrared and collinear safe, and can therefore be calculated order-by-order in perturbation theory. We present explicit, general formulas for receiver operator characteristic curves and the area under it through next-to-leading order. These results can then establish absolute upper bounds on discrimination performance because any realistic implementation will have measurement errors, undetected particles, or restrictions on fiducial phase space. As a demonstration of this formalism, we apply it to discrimination of highly-boosted Higgs decays from gluon splitting to bottom quarks. Effects at next-to-leading order are first sensitive to the flow of color in the jet and significantly modify discrimination performance at leading-order. In the limit of infinite boost, these events can be perfectly discriminated because only the gluon will radiate at finite angles from the bottom quarks, and we find that large effects persist at energies accessible at the Large Hadron Collider. Next-to-leading order is therefore required to qualitatively understand results using machine-learning methods.

Binary discrimination through next-to-leading order

A definition of partonic jet flavor that is both theoretically well-defined and experimentally robust would have profound implications for measurements and predictions especially for heavy flavor applications. Recently, a definition of jet flavor was introduced as the net flavor flowing along the direction of the Winner-Take-All axis of a jet which is soft safe to all orders, but not collinear safe. Here, we exploit the lack of collinear safety and propose a factorization theorem of perturbative flavor fragmentation functions that resum collinear divergences and describe the evolution of flavor from the short distance of jet production to the long distance at which hadronization occurs. Collinear flavor evolution is governed by a small modification of the DGLAP equations. We present a detailed all-orders analysis and identify exact relations that must hold amongst the various anomalous dimensions by probability conservation and the existence of fixed points of the renormalization group flow. We explicitly validate the factorization theorem at one-loop order, and demonstrate its consistency at two loops in particular flavor channels. Starting at two-loops, constraints on phase space imposed by flavor measurements potentially allow for non-trivial soft contributions, but we demonstrate that they are scaleless and so explicitly vanish, ensuring that soft particles are summed inclusively and all divergences are exclusively collinear in nature. This factorization theorem opens the door to precision calculations with identified flavor in the infrared.

Duff Neill

Flavor fragmentation function factorization

By quantifying the distance between two collider events, one can triangulate a metric space and reframe collider data analysis as computational geometry. One popular geometric approach is to first represent events as an energy flow on an idealized celestial sphere and then define the metric in terms of optimal transport in two dimensions. In this paper, we advocate for representing events in terms of a spectral function that encodes pairwise particle angles and products of particle energies, which enables a metric distance defined in terms of one-dimensional optimal transport. This approach has the advantage of automatically incorporating obvious isometries of the data, like rotations about the colliding beam axis. It also facilitates first-principles calculations, since there are simple closed-form expressions for optimal transport in one dimension. Up to isometries and event sets of measure zero, the spectral representation is unique, so the metric on the space of spectral functions is a metric on the space of events. At lowest order in perturbation theory in electron-positron collisions, our metric is simply the summed squared invariant masses of the two event hemispheres. Going to higher orders, we present predictions for the distribution of metric distances between jets in fixed-order and resummed perturbation theory as well as in parton-shower generators. Finally, we speculate on whether the spectral approach could furnish a useful metric on the space of quantum field theories.