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On the analytic computation of massless propagators in dimensional regularization

Erik Panzer

TL;DR

This work develops and applies a unified parametric, hyperlogarithmic framework to compute massless propagators in dimensional regularization, focusing on the ε-expansion of multi-loop p-integrals. By leveraging polynomial-reduction techniques and the notion of linear reducibility, the authors prove that all three-loop propagators are reducible and show that planar graphs yield expansions in rational combinations of MZVs (with non-planar cases allowing alternating sums), while extending these methods to certain four- and higher-loop graphs. They implement a Maple-based routine, validate results against known cases, and provide extensive analytic data (and ancillary files) for broad use, including counterterm handling for subdivergences. The results offer strong, general constraints on the kinds of periods that can appear and highlight the practical potential and current limits of hyperlogarithmic approaches for high-loop perturbative QFT calculations.

Abstract

We comment on the algorithm to compute periods using hyperlogarithms, applied to massless Feynman integrals in the parametric representation. Explicitly, we give results for all three-loop propagators with arbitrary insertions including order $\varepsilon^4$ and show examples at four and more loops. Further we prove that all coefficients of the $\varepsilon$-expansion of these integrals are rational linear combinations of multiple zeta values and in some cases possibly also alternating Euler sums.

On the analytic computation of massless propagators in dimensional regularization

TL;DR

This work develops and applies a unified parametric, hyperlogarithmic framework to compute massless propagators in dimensional regularization, focusing on the ε-expansion of multi-loop p-integrals. By leveraging polynomial-reduction techniques and the notion of linear reducibility, the authors prove that all three-loop propagators are reducible and show that planar graphs yield expansions in rational combinations of MZVs (with non-planar cases allowing alternating sums), while extending these methods to certain four- and higher-loop graphs. They implement a Maple-based routine, validate results against known cases, and provide extensive analytic data (and ancillary files) for broad use, including counterterm handling for subdivergences. The results offer strong, general constraints on the kinds of periods that can appear and highlight the practical potential and current limits of hyperlogarithmic approaches for high-loop perturbative QFT calculations.

Abstract

We comment on the algorithm to compute periods using hyperlogarithms, applied to massless Feynman integrals in the parametric representation. Explicitly, we give results for all three-loop propagators with arbitrary insertions including order and show examples at four and more loops. Further we prove that all coefficients of the -expansion of these integrals are rational linear combinations of multiple zeta values and in some cases possibly also alternating Euler sums.

Paper Structure

This paper contains 28 sections, 2 theorems, 84 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction and results
  2. All-order constraints on periods:
  3. Practical considerations and explicit calculations:
  4. Acknowledgements
  5. Parametric integration
  6. Polynomial reduction
  7. Propagators with three loops
  8. The finite planar graphs $L$, $Q$ and $V$
  9. The infrared divergent planar propagator $M$
  10. The non-planar convergent graph $N$
  11. Checks and symmetries
  12. Propagators with four loops
  13. Propagators without subdivergences
  14. The subdivergent propagators $M_{3,5}$ and $M_{5,1}$
  15. Symmetries
  16. ...and 13 more sections

Key Result

Theorem 1.1

All three-loop graphs of figure fig:3loops are linearly reducible. Every coefficient of their $\varepsilon$-expansions $G_0^{-3} \Phi$ is a rational linear combination of $\mathrm{MZV}$ for the planar graphs $L$, $M$, $Q$ and $V$. For the non-planar graph $N$ also alternating Euler sums may appear.

Figures (11)

  • Figure 1: The five different three-loop $p$-integrals were classified in ChetyrkinTkachov:IBP.
  • Figure 2: Glueing the external edges of $L$ and $M$ gives the triangular prism $Y_3$, while $Q$ and $V$ yield the wheel with four spokes $W_4$ and $N$ results in the complete bipartite graph $K_{3,3}$.
  • Figure 3: All five-loop graphs without one-scale subgraphs, divided into planar ($P$) and non-planar ($N$) ones. The Zig-zag graph ${_5 P_3}$ and ${_5 N_1}$ ($K_{3,3}$ with an additional edge) were considered in Brown:TwoPoint. Cutting any edge produces a propagator graph with four loops. The master integrals of BaikovChetyrkin:FourLoopPropagatorsAlgebraicLeeSmirnov:FourLoopPropagatorsWeightTwelve, some shown in figure \ref{['fig:4loops']}, give $\widetilde{M_{3,5}} = \widetilde{M_{3,6}} = {_5 P_1}$ (the complete graph $K_5$ minus one edge), $\widetilde{M_{4,4}} = {_5 P_3}$, $\widetilde{M_{4,5}} = {_5 N_1}$, $\widetilde{M_{5,1}} = {_5 N_2}$, $\widetilde{M_{6,1}} = {_5 P_7}$ (the cube) and $\widetilde{M_{6,2}} = \widetilde{M_{6,3}} = {_5 N_4}$.
  • Figure 4: Graphs with one-scale subdivergences like these two factorize into smaller graphs as \ref{["eq:insertions-F'"]} and \ref{['eq:insertions-M42']}, wherefore they do not necessitate a separate integration for evaluation. Other such reducible graphs are shown in figure \ref{['fig:4loops-reducibles']}.
  • Figure 5: These two auxiliary graphs are used in \ref{['eq:M-renormalized']} as counterterms to rewrite the divergent $\Phi_M$ in terms of $\Gamma$-functions and convergent parametric integrals.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Example 3.1
  • Remark 3.2
  • Remark 3.3