Collider physics at the precision frontier

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorithm, which avoids algebraic extensions.

On the computation of intersection numbers for twisted cocycles

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.

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Decomposition of Feynman integrals by multivariate intersection numbers

/pdf/the-three-loop-form-factor-in-n-4-super-yang-mills-1totgx1kiw.pdf

The three-loop form factor in N=4 super Yang-Mills

: We review recent progress and future prospects for harnessing powerful tools from theoretical high-energy physics, such as scattering amplitudes and eﬀective ﬁeld theory, to develop a precise and systematically improvable framework for calculating gravitational-wave signals from binary systems composed of black holes and/or neutron stars. This eﬀort aims to provide state-of-the-art predictions that will enable high-precision measurements at future gravitational-wave detectors. In turn, applying the tools of quantum ﬁeld theory in this new arena will uncover theoretical structures that can transform our understanding of basic phenomena and lead to new tools that will further the cycle of innovation. While still in a nascent stage, this research direction has already derived new analytic results in general relativity, and promises to advance the development of highly accurate waveform models for ever more sensitive detectors.

Snowmass White Paper: Gravitational Waves and Scattering Amplitudes

Constructing canonical Feynman integrals with intersection theory


 In this paper, we present the universal structure of the alphabet of one-loop Feynman integrals. The letters in the alphabet are calculated using the Baikov representation with cuts. We consider both convergent and divergent cut integrals, and find that letters in the divergent cases can be easily obtained from the convergent cases by taking certain limits. The letters are written as simple expressions in terms of various Gram determinants. The knowledge of the alphabet makes it easy to construct the canonical differential equations of the $d\log$ form, and helps to bootstrap the symbols of the solutions. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

Alphabet of one-loop Feynman integrals

https://link.springer.com/content/pdf/10.1007/JHEP07(2022)066.pdf

Baikov representations, intersection theory, and canonical Feynman integrals

https://link.springer.com/content/pdf/10.1007/JHEP02(2023)178.pdf

Module intersection and uniform formula for iterative reduction of one-loop integrals

https://link.springer.com/content/pdf/10.1007/JHEP04(2022)025.pdf

Jiaqi Chen

Papers

Constructing canonical Feynman integrals with intersection theory

Alphabet of one-loop Feynman integrals

Baikov representations, intersection theory, and canonical Feynman integrals

Module intersection and uniform formula for iterative reduction of one-loop integrals

Two-loop infrared singularities in the production of a Higgs boson associated with a top-quark pair