Kleiss, R. H. P., Malamos, I., Papadopoulos, C. G., & Verheyen, R. (2012). Counting to one: reducibility of one- and two-loop amplitudes at the integrand level. J. High Energy Phys., 12(12), 038–24pp.
Abstract: Calculation of amplitudes in perturbative quantum field theory involve large loop integrals. The complexity of those integrals, in combination with the large number of Feynman diagrams, make the calculations very difficult. Reduction methods proved to be very helpful, lowering the number of integrals that need to be actually calculated. Especially reduction at the integrand level improves the speed and set-up of these calculations. In this article we demonstrate, by counting the numbers of tensor structures and independent coefficients, how to write such relations at the integrand level for one-and two-loop amplitudes. We clarify their connection to the so-called spurious terms at one loop and discuss their structure in the two-loop case. This method is also applicable to higher loops, and the results obtained apply to both planar and non-planar diagrams.
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Otten, S., Caron, S., de Swart, W., van Beekveld, M., Hendriks, L., van Leeuwen, C., et al. (2021). Event generation and statistical sampling for physics with deep generative models and a density information buffer. Nat. Commun., 12(1), 2985–16pp.
Abstract: Simulating nature and in particular processes in particle physics require expensive computations and sometimes would take much longer than scientists can afford. Here, we explore ways to a solution for this problem by investigating recent advances in generative modeling and present a study for the generation of events from a physical process with deep generative models. The simulation of physical processes requires not only the production of physical events, but to also ensure that these events occur with the correct frequencies. We investigate the feasibility of learning the event generation and the frequency of occurrence with several generative machine learning models to produce events like Monte Carlo generators. We study three processes: a simple two-body decay, the processes e(+)e(-)-> Z -> l(+)l(-) and pp -> tt<mml:mo><overbar></mml:mover> including the decay of the top quarks and a simulation of the detector response. By buffering density information of encoded Monte Carlo events given the encoder of a Variational Autoencoder we are able to construct a prior for the sampling of new events from the decoder that yields distributions that are in very good agreement with real Monte Carlo events and are generated several orders of magnitude faster. Applications of this work include generic density estimation and sampling, targeted event generation via a principal component analysis of encoded ground truth data, anomaly detection and more efficient importance sampling, e.g., for the phase space integration of matrix elements in quantum field theories. Here, the authors report buffered-density variational autoencoders for the generation of physical events. This method is computationally less expensive over other traditional methods and beyond accelerating the data generation process, it can help to steer the generation and to detect anomalies.
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Builtjes, L., Caron, S., Moskvitina, P., Nellist, C., Ruiz de Austri, R., Verheyen, R., et al. (2025). Attention to the strengths of physical interactions: Transformer and graph-based event classification for particle physics experiments. SciPost Phys., 19(1), 028–33pp.
Abstract: A major task in particle physics is the measurement of rare signal processes. Even modest improvements in background rejection, at a fixed signal efficiency, can significantly enhance the measurement sensitivity. Building on prior research by others that incorporated physical symmetries into neural networks, this work extends those ideas to include additional physics-motivated features. Specifically, we introduce energy-dependent particle interaction strengths, derived from leading-order SM predictions, into modern deep learning architectures, including Transformer Architectures (Particle Transformer), and Graph Neural Networks (Particle Net). These interaction strengths, represented as the SM interaction matrix, are incorporated into the attention matrix (transformers) and edges (graphs). Our results in event classification show that the integration of all physics-motivated features improves background rejection by 10% -40% over baseline models, with an additional gain of up to 9% due to the SM interaction matrix. This study also provides one of the broadest comparisons of event classifiers to date, demonstrating how various architectures perform across this task. A simplified statistical analysis demonstrates that these enhanced architectures yield significant improvements in signal significance compared to a graph network baseline.
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