Abstract
The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.
Highlights
Based on the assumption that a rational transformation exists that transforms a differential equation into canonical form, section 3 first explores some general features of such transformations, which are useful for devising the algorithm
Assuming the existence of a rational transformation that transforms a differential equation of master integrals to an ǫ-form, the algorithm presented here can be used to compute such a transformation
It is applicable to differential equations involving multiple scales and allows for a rational dependence of the differential equation on the dimensional regulator and extends previous approaches
Summary
Higher order corrections in quantum field theory involve integrations over the unconstrained loop momenta, in general in the form of tensor integrals. The master integrals can be considered as functions of a set {xj} of M dimensionless kinematic invariants and the dimensional regulator ǫ defined through d = 4 − 2ǫ, where d denotes number of spacetime dimensions. The fact that the matrices ai(ǫ, {xj}) are rational functions of the kinematic invariants and ǫ follows from the structure of the integration-by-parts relations. Where it has been used that the master integrals are linearly independent over the field of rational functions in the invariants This condition can serve as a consistency check of the differential equation. L=1 with Al being constant m × m matrices In this form, which is called canonical form or ǫ-form, it is easy to solve the differential equation in terms of iterated integrals [71, 72]
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