We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-calledintersection numbersand describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.

Working within the post-Newtonian (PN) approximation to general relativity, we use the effective field theory (EFT) framework to study the conservative dynamics of the two-body motion at fourth PN order, at fifth order in the Newton constant. This is one of the missing pieces preventing the computation of the full Lagrangian at fourth PN order using EFT methods. We exploit the analogy between diagrams in the EFT gravitational theory and two-point functions in massless gauge theory, to address the calculation of four-loop amplitudes by means of standard multiloop diagrammatic techniques. For those terms which can be directly compared, our result confirms the findings of previous studies, performed using different methods.

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ₂ F ₁ hypergeometric function, and the Appell F ₁ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to n-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

We provide an overview of the status of Monte-Carlo event generators for high-energy particle physics. Guided by the experimental needs and requirements, we highlight areas of active development, and opportunities for future improvements. Particular emphasis is given to physics models and algorithms that are employed across a variety of experiments. These common themes in event generator development lead to a more comprehensive understanding of physics at the highest energies and intensities, and allow models to be tested against a wealth of data that have been accumulated over the past decades. A cohesive approach to event generator development will allow these models to be further improved and systematic uncertainties to be reduced, directly contributing to future experimental success. Event generators are part of a much larger ecosystem of computational tools. They typically involve a number of unknown model parameters that must be tuned to experimental data, while maintaining the integrity of the underlying physics models. Making both these data, and the analyses with which they have been obtained accessible to future users is an essential aspect of open science and data preservation. It ensures the consistency of physics models across a variety of experiments.