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A quasi-finite basis for multi-loop Feynman integrals

Andreas von Manteuffel, Erik Panzer, Robert M. Schabinger

Abstract

We present a new method for the decomposition of multi-loop Euclidean Feynman integrals into quasi-finite Feynman integrals. These are defined in shifted dimensions with higher powers of the propagators, make explicit both infrared and ultraviolet divergences, and allow for an immediate and trivial expansion in the parameter of dimensional regularization. Our approach avoids the introduction of spurious structures and thereby leaves integrals particularly accessible to direct analytical integration techniques. Alternatively, the resulting convergent Feynman parameter integrals may be evaluated numerically. Our approach is guided by previous work by the second author but overcomes practical limitations of the original procedure by employing integration by parts reduction.

A quasi-finite basis for multi-loop Feynman integrals

Abstract

We present a new method for the decomposition of multi-loop Euclidean Feynman integrals into quasi-finite Feynman integrals. These are defined in shifted dimensions with higher powers of the propagators, make explicit both infrared and ultraviolet divergences, and allow for an immediate and trivial expansion in the parameter of dimensional regularization. Our approach avoids the introduction of spurious structures and thereby leaves integrals particularly accessible to direct analytical integration techniques. Alternatively, the resulting convergent Feynman parameter integrals may be evaluated numerically. Our approach is guided by previous work by the second author but overcomes practical limitations of the original procedure by employing integration by parts reduction.

Paper Structure

This paper contains 9 sections, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Method
  3. Regularizing dimension shifts
  4. Problems with the transformations
  5. Integration by parts reduction
  6. Existence of a quasi-finite basis
  7. Finding a minimal quasi-finite basis
  8. Illustrative Examples
  9. Outlook

Figures (3)

  • Figure 1: The two-loop tadpole diagram with a single massive propagator (thick line).
  • Figure 2: The two-loop non-planar form factor diagram. The two external momenta on the right are light-like, while the momentum entering from the left squares to $s$.
  • Figure 3: A minimal quasi-finite basis for the planar massless double box integral family.