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Solving recurrence relations for multiloop integrals in the limit of large values of the dimensional regularization parameter

P. A. Baikov

TL;DR

The paper introduces a method to compute $1/d$ expansion coefficients for solutions of integration by parts relations in multiloop Feynman integrals by reformulating the relations as formal $1/d$-series and applying linear substitutions to render them explicitly recursive. By deriving a structured substitution framework and solving bilinear equations for the substitution parameters $\lambda$, the authors obtain recursive relations that connect integrals in a controlled, finite manner, demonstrated for vacuum integrals up to $7$ loops (including cases with one massive line). The approach seeks to overcome the computational blow-up typical of $1/d$ expansions and to provide a scalable path toward high-loop calculations, including potential extensions to external kinematics. This work offers new algebraic tools for IBP reduction, enabling both reconstruction and cross-checking of master-integral coefficients and potentially reducing the resource burden of high-precision multiloop computations.

Abstract

A method for calculating the $1/d$ expansion coefficients for solutions of integration by parts relations for Feynman integrals is presented. The idea is to use linear substitutions to transform these relations to an explicitly recursive form. A possible type of such substitutions is proposed for the case of vacuum integrals. Its applicability is shown for several families of massless (with one massive line) vacuum integrals up to the 7-loop level.

Solving recurrence relations for multiloop integrals in the limit of large values of the dimensional regularization parameter

TL;DR

The paper introduces a method to compute expansion coefficients for solutions of integration by parts relations in multiloop Feynman integrals by reformulating the relations as formal -series and applying linear substitutions to render them explicitly recursive. By deriving a structured substitution framework and solving bilinear equations for the substitution parameters , the authors obtain recursive relations that connect integrals in a controlled, finite manner, demonstrated for vacuum integrals up to loops (including cases with one massive line). The approach seeks to overcome the computational blow-up typical of expansions and to provide a scalable path toward high-loop calculations, including potential extensions to external kinematics. This work offers new algebraic tools for IBP reduction, enabling both reconstruction and cross-checking of master-integral coefficients and potentially reducing the resource burden of high-precision multiloop computations.

Abstract

A method for calculating the expansion coefficients for solutions of integration by parts relations for Feynman integrals is presented. The idea is to use linear substitutions to transform these relations to an explicitly recursive form. A possible type of such substitutions is proposed for the case of vacuum integrals. Its applicability is shown for several families of massless (with one massive line) vacuum integrals up to the 7-loop level.
Paper Structure (6 sections, 55 equations, 1 figure)

This paper contains 6 sections, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integration by parts relations for Feynman integrals
  3. Linear substitutions in recurrence relations: one-parameter case
  4. Linear substitutions in recurrence relations: Feynman integrals
  5. Examples
  6. Conclusion

Figures (1)

  • Figure 1: $\overline{\lambda}_a$ (roots of eqs. (\ref{['condla']})) for some vacuum diagrams. Ordinary lines are massless, double line is massive with $\mu=1$; $a_L=L$, $b_L=-\frac{L-2}{2}e_L$, $c_L=\frac{L(L-2)}{2}e_L-L(L-1)$; for planar diagrams (first row) $e_5= \frac{20}{9}$, $e_6= \frac{15}{8}$, $e_7=(\pm 28\,(61)^{1/2} + 1232)/625$; for nonplanar (second row) $e_6=(\pm 3\,(41)^{1/2} + 57)/16$, $e_7=(\pm 168\,(26)^{1/2} + 2268)/625$.