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Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case

Vladimir V. Bytev, Mikhail Yu. Kalmykov, Bernd A. Kniehl

TL;DR

This work develops a differential-reduction framework to rewrite Feynman-diagram-derived hypergeometric functions ${}_{p+1}F_p$ with general parameters in terms of integer-shifted counterparts and their derivatives. By constructing step-up/step-down operators and their inverses, the authors express hypergeometric functions in a reduced basis, enabling all-order ${\e}$-expansions and connecting hypergeometric reducibility to IBP-based master-integral counts. The approach is demonstrated on a range of one-variable diagrams (one-loop to three-loop) via Mellin–Barnes representations, showing reductions to ${}_{2}F_{1}$, ${}_{3}F_{2}$, ${}_{4}F_{3}$, and higher, with explicit epsilon-series articulated in terms of polylogarithms and Remiddi–Vermaaseren functions. The results indicate a simple relation h = v + 1 between IBP master integrals and the maximal differential-reduction theta-order, validating the method as a unifying tool for analytic Feynman diagram evaluation and suggesting pathways for generalization to multivariable cases and automation (e.g., HYPERDIRE).

Abstract

The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.

Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case

TL;DR

This work develops a differential-reduction framework to rewrite Feynman-diagram-derived hypergeometric functions with general parameters in terms of integer-shifted counterparts and their derivatives. By constructing step-up/step-down operators and their inverses, the authors express hypergeometric functions in a reduced basis, enabling all-order -expansions and connecting hypergeometric reducibility to IBP-based master-integral counts. The approach is demonstrated on a range of one-variable diagrams (one-loop to three-loop) via Mellin–Barnes representations, showing reductions to , , , and higher, with explicit epsilon-series articulated in terms of polylogarithms and Remiddi–Vermaaseren functions. The results indicate a simple relation h = v + 1 between IBP master integrals and the maximal differential-reduction theta-order, validating the method as a unifying tool for analytic Feynman diagram evaluation and suggesting pathways for generalization to multivariable cases and automation (e.g., HYPERDIRE).

Abstract

The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.

Paper Structure

This paper contains 25 sections, 132 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Differential-reduction algorithm for the generalized hypergeometric function ${}_{p+1}F_{p}$
  3. Notation
  4. Non-exceptional values of parameters
  5. Differential-reduction algorithms for special values of parameters
  6. Criteria of reducibility of hypergeometric functions
  7. All-order $\varepsilon$ expansions of hypergeometric functions
  8. Reduction at $z=1$ and construction of $\varepsilon$ expansion
  9. Application to Feynman diagrams
  10. One-loop vertex: particular cases
  11. $E^{q}_{120}$
  12. $J^{q}_{22}$
  13. $E^{q}_{1220}$ and $B^{q}_{1220}$
  14. $V^{q}_{1220}$ and $J^{q}_{1220}$
  15. Two-loop vertex
  16. ...and 10 more sections

Figures (2)

  • Figure 1: One-loop vertex diagrams expressible in terms of generalized hypergeometric functions. Bold and thin lines correspond to massive and massless propagators, respectively.
  • Figure 2: Diagrams considered in the paper. Bold and thin lines correspond to massive and massless propagators, respectively.