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On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

S. Groote, J. G. Körner, A. A. Pivovarov

TL;DR

The paper surveys configuration-space (x-space) methods for evaluating sunrise-type Feynman diagrams at arbitrary loop order, showing that UV poles can be extracted analytically and finite parts reduce to tractable one-dimensional integrals. It establishes a dispersion framework linking the correlator to a finite spectral density ρ(s) and clarifies how multi-particle phase space emerges from the discontinuity across the physical cut. The authors develop a practical toolbox, including pole extraction, ε-expansion via Bessel-function techniques, and threshold/large-momentum expansions, and they extend the approach to non-standard propagators and related topologies like spectacle diagrams. The work provides extensive explicit results up to four loops and five-loop vacuum bubbles, cross-validates with momentum-space results, and demonstrates broad applicability to HQET/NRQCD regimes, offering a robust analytic-numeric toolkit for multi-loop sunrise-type calculations.

Abstract

We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We present explicit formulae for their analytical evaluation for arbitrary mass configurations and arbitrary dimensions at any loop order. We discuss several limiting cases of their kinematical regimes which are e.g. relevant for applications in HQET and NRQCD.

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

TL;DR

The paper surveys configuration-space (x-space) methods for evaluating sunrise-type Feynman diagrams at arbitrary loop order, showing that UV poles can be extracted analytically and finite parts reduce to tractable one-dimensional integrals. It establishes a dispersion framework linking the correlator to a finite spectral density ρ(s) and clarifies how multi-particle phase space emerges from the discontinuity across the physical cut. The authors develop a practical toolbox, including pole extraction, ε-expansion via Bessel-function techniques, and threshold/large-momentum expansions, and they extend the approach to non-standard propagators and related topologies like spectacle diagrams. The work provides extensive explicit results up to four loops and five-loop vacuum bubbles, cross-validates with momentum-space results, and demonstrates broad applicability to HQET/NRQCD regimes, offering a robust analytic-numeric toolkit for multi-loop sunrise-type calculations.

Abstract

We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We present explicit formulae for their analytical evaluation for arbitrary mass configurations and arbitrary dimensions at any loop order. We discuss several limiting cases of their kinematical regimes which are e.g. relevant for applications in HQET and NRQCD.

Paper Structure

This paper contains 50 sections, 307 equations, 16 figures.

Table of Contents

  1. Introduction and motivation
  2. Basic notions and relations for sunrise-type diagrams
  3. Definitions and notation
  4. Momentum versus configuration space representation: Fourier transform in the evaluation of sunrise--type integrals
  5. Regularization and renormalization
  6. The spectrum as discontinuity across the physical cut
  7. "Tool box" for a practical analysis
  8. Extraction of poles in dimensional regularization
  9. Singular and finite parts for particular parameter values of the genuine sunrise: warm-up example demonstrating the technique
  10. Three-loop sunrise-type diagram for special momenta
  11. Infinite parts of sunrise-type diagrams with different masses at any number of loops: general techniques
  12. Finite part for the genuine sunrise with three different masses
  13. Calculation of the finite part: further examples
  14. Higher order expansion in $\varepsilon$: general techniques and the four-loop vacuum bubble
  15. $\varepsilon$-expansion of the five-loop vacuum bubble
  16. ...and 35 more sections

Figures (16)

  • Figure 1: Genuine sunrise (a) and general topology of the class of sunrise-type diagrams (b)
  • Figure 2: The spectral density for the four-line sunrise-type diagram with equal masses for $D=e=2.718\ldots$, $D=3$, and $D=\pi=3.14\ldots$ space-time dimensions.
  • Figure 3: Three-loop massive sunrise-type bubble. A dot on a line corresponds to one differentiation with respect to the propagator mass squared
  • Figure 4: Comparison of functions $f_n(z)$ for different values of $n$, plotted on a logarithmic scale
  • Figure 5: Integrands for numerical integration in case of the four-loop bubble at different orders $O(\varepsilon^n)$ in $\varepsilon$
  • ...and 11 more figures