Universal treatment of the reduction for one-loop integrals in a projective space

Nontrivial one-loop recursive reduction relation

“…Inspired by these papers, we find it could be convenient to do reduction for one-loop integrals in projective space. Furthermore the symmetry and compactness of reduction coefficients are illustrated clearly by this method, as shown in [1]. However, with this technique, we have to expand a general one-loop integral into the combination of E n,k [V i ] first, then reduce every E n,k [V i ] to the basis, after that, we sum over all contributions to obtain the final reduction result.…”

“…We ran our code in Mathematica and found that the time cost grows sharply. For more details, please see the examples in section 3 of [1].…”

“…We obtain two forms of the lower terms, the first one involves integrals in (D − 2)-dimensional space while the second one involves integrals in the same D-dimensional space. 1 The recursion relation not only makes the reduction process extremely simple and effective, but also serves as a tool to study the analytical structures of reduction coefficients like singularities. The paper is structured as follows: in section 2, we revisit the reduction technique in projective space.…”

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In [1], we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when addressing certain tensor integrals.

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Towards analytic structure of Feynman parameter integrals with rational curves

“…Both classes JHEP10(2022)145 of integrals were previously studied in [32] (and also [33] for a related treatment of the one-loop integrals), where efficient methods were proposed to learn about their symbols. In particular, for the one-loop integrals it introduced a so-called "spherical projection" that extracts certain discontinuities from the integral, from which the symbol of (1.7) can be directly read off (see also [34,35] for related discussions). Unfortunately, the validity of this method heavily relies on the fact that (1.8) here defines a single quadric, and so it cannot be directly applicable to integrals with other types of D[X n+k ] (although the Aomoto polylog integrals can be rewritten into a form of the one-loop type, so as to fit into this method indirectly).…”