Two-Loop Master Integrals for $\gamma^* \to 3$ Jets: The planar topologies
By Thomas Gehrmann, Ettore Remiddi. 2000
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The calculation of the two-loop corrections to the three jet production rate and to event shapes in electron-positron annihilation requires the computation of a number of up to now unknown two-loop four-point master integrals with one off-shell and three on-shell legs. In this paper, we compute those master integrals which correspond to planar topologies by solving differential equations in the external invariants which are fulfilled by the master integrals. We obtain the master integrals as expansions in $\e=(4-d)/2$, where $d$ is the space-time dimension. The results are expressed in terms of newly introduced two-dimensional harmonic polylogarithms, whose properties are shortly discussed. For all two-dimensional harmonic polylogarithms appearing in the divergent parts of the integrals, expressions in terms of Nielsen's polylogarithms are given. The analytic continuation of our results to other kinematical situations is outlined.
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Citations in the corpus
- The full four-loop cusp anomalous dimension in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{do (2019, unknown): “ has already been observed and stressed in a number of calculations, 8Typically, one needs to expand to some order ǫn+k, with k > 0, in order to obtain all constraints at order ǫn. – 15 – see e.g. [53, 56, 65]. Finally, we fix the overall normalization by explicitly evaluating one trivial integral”
- Algorithm FIRE—Feynman Integral REduction (2008, methodology): “The most commonly used source consists of applying the Laporta algorithm (see [3, 4 ]) to derive proper expressions from IBP relations [2].”
- Event shapes and jet rates in electron-positron annihilation at NNLO (2009, background): “It should be mentioned that the results of this paper rely heavily on research carried out in the past years: Integration techniques for two-loop amplitudes [13–16], the calculation of the relevant tree-, one- and two-loop-amplitudes [17–29], routines for the numerical evaluation of polylogarithms [”
- Bootstrapping the three-loop hexagon (2011, background): “ degree zero pure functions are constants. The definition (8) includes logarithms and classical polylogarithms, as well as other iterated integrals, such as harmonic polylogarithms of one [58] or more [59, 60, 61, 62] variables. The symbol[37] S(f) of a pure function fis defined recursively with respe”
- From polygons and symbols to polylogarithmic functions (2011, unknown): “thereof [26, 27].”
- Bootstrapping an NMHV amplitude through three loops (2014, methodology): “This means that hexagon functions approach a subset of the 2dHPL function space introduced by Gehrmann and Remiddi [90] in order to solve for the master integrals for the process γ∗ → qgq̄ at two loops.”