Abstract
We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of Feynman integrals, we present some selected new results for their expansions around 4 — 2ε dimensions.
Highlights
In high-energy physics experiments performed at current colliders such as the LHC, the flood of precision data requires matching theoretical efforts, in order extract the underlying event’s structure
We will showcase a few techniques and results related to investigations of the structure of higher-loop Feynman integrals which provide one of the basic building blocks of high-precision perturbative calculations within elementary particle physics
We have studied the class of fully massive vacuum diagrams, which are essential building blocks for a number of phenomenological applications such as QCD thermodynamics, anomalous dimensions, or moments
Summary
In high-energy physics experiments performed at current colliders such as the LHC, the flood of precision data requires matching theoretical efforts, in order extract the underlying event’s structure. Having all analytic ingredients for their solution in terms of infinite sums at hand, to obtain actual results for the master integrals it is most useful to resort to a numeric treatment This involves truncating the sum in Eq (3) at some smax, evaluating R consecutive values of I(x) around some large x (where the factorial series converges well), and using the difference equation Eq (2) to push their argument down to the required integrals, such as I(1). We have found that, following the program proposed by Stefano Laporta [4], one is faced with some problems and limitations when treating complex problems: the method is of limited use for multi-scale integrals; the complexity of coefficients in high-order equations can grow enormously; one typically obtains recurrence relations of high orders; in numerical evaluation, one often faces instabilities of the factorial series solutions To alleviate these problems and make feasible the computation of our set of 5-loop massive tadpoles, we had to considerably change the traditional setup. To name a few key ingredients [5]: we use coupled IBP equations, in order to tame the growth of complexity; we reduce (the orders of) recurrence relations, essentially by re-using the linear solver developed for the system of difference equations; our code predicts instability factors, in order to assign sufficient numerical precision
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