arXiv subjects
Publications and source records attributed to Toni Teschke.
In this talk, we use several examples to elaborate on how a recently proposed algorithm can turn non-trivial Feynman integrals into an $\varepsilon $-factorised manner, regardless of their hidden geometric essence. In particular, some extra details about three-loop banana integrals with unequal-mass configuration are provided.
In this talk, we discuss how ideas from geometry help to improve Feynman integral reduction and the construction of $\varepsilon$-factorised differential equations. In particular, we outline a systematic procedure to obtain an $\varepsilon$-factorised differential equation for any Feynman integral.
In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric order relation in the integration-by-parts reduction to obtain a basis of master integrals, whose differential equations on the maximal cut are of a Laurent polynomial form in the regularisation parameter $\varepsilon$ and compatible with a filtration. This step works entirely with rational functions. In a second step, we provide a method to $\varepsilon$-factorise the aforementioned Laurent differential equations. The second step may introduce algebraic and transcendental functions. We illustrate the versatility of the algorithm by applying it to different examples with a wide range of complexity.
We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an $\varepsilon$-factorised differential equation, for which we rely on the algorithm presented in a recent publication. Equipping the space of differential forms in Baikov representation by a set of filtrations inspired by Hodge theory, we first obtain a differential equation with entries as Laurent polynomials in $\varepsilon$. Via a sequence of basis rotations we then remove any non-$\varepsilon$-factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.
We report on three improvements in the context of Feynman integral reduction and $\varepsilon$-factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the $\varepsilon$-dependence of the integration-by-parts identities. Secondly, we observe that with a specific choice of order relation in the Laporta algorithm, we directly obtain a basis of master integrals, whose differential equation on the maximal cut is in Laurent polynomial form with respect to $\varepsilon$ and compatible with a particular filtration. Thirdly, we prove that such a differential equation can always be transformed to an $\varepsilon$-factorised form. This provides a systematic algorithm to obtain an $\varepsilon$-factorised differential equation for any Feynman integral. Furthermore, the choices for the prefactors and the order relation significantly improve the efficiency of the reduction algorithm.
This paper combines the post-Minkowskian expansion of general relativity with the language of intersection theory. Because of the nature of the soft limit inherent to the post-Minkowskian expansion, the intersection-based approach is of enhanced utility in that theory compared to a generic quantum field theory. In the language of intersection theory, Feynman integrals are rephrased in terms of twisted cocycles. The intersection number is a pairing between two such cocycles and its existence allows for the direct projection onto a basis of master integrals. In this paper we use this approach to compute the second post-Minkowskian contribution to the scattering of two compact astronomical objects, getting results in agreement with previous findings.
The study investigates the gravitational scattering amplitude between two Schwarzschild black holes in a two to two interaction, focusing on the Second Post-Minkowskian correction (2 PM). Analyzing contributions from box and cross-box diagrams, the research interprets Feynman integrals as pairings between twisted co-cycles and cycles. The concept of twisted (co)-homology groups is introduced, leading to a master integral decomposition formula. The study successfully applies intersection theory to compute coefficients of the master integral basis, marking the first application of intersection theory in the quantum field theoretic description of gravity. The results align with existing literature on the 2PM correction.