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ep-Finite Basis of Master Integrals for the Integration-By-Parts Method

K. G. Chetyrkin, M. Faisst, C. Sturm, M. Tentyukov

TL;DR

The paper introduces an $\epsilon$-finite basis of master integrals for Integration-By-Parts reductions in dimensional regularization, ensuring all coefficient functions remain finite as $\epsilon \to 0$. It presents a constructive algorithm to replace divergent masters with finite combinations, and applies the method to the set of all QED-like four-loop massive tadpoles, using a Padé-based semi-numerical approach to obtain analytical pole parts and numerical finite parts. The authors verify their results against recent Schröder–Vuorinen findings and demonstrate substantial analytic control over divergent structures, including new relations among master integrals. The work promises improved numerical stability and analytic tractability for high-loop IBP problems and suggests a potential universal basis for a broad class of four-loop, QED-like tadpole computations.

Abstract

It is shown that for every problem within dimensional regularization, using the Integration-By-Parts method, one is able to construct a set of master integrals such that each corresponding coefficient function is finite in the limit of dimension equal to four. We argue that the use of such a basis simplifies and stabilizes the numerical evaluation of the master integrals. As an example we explicitly construct the ep-finite basis for the set of all QED-like four-loop massive tadpoles. Using a semi-numerical approach based on Pade approximations we evaluate analytically the divergent and numerically the finite part of this set of master integrals. The calculations confirm the recent results of Schröder and Vuorinen. All the contributions found there by fitting the high precision numerical results have been confirmed by direct analytical calculation without using any numerical input.

ep-Finite Basis of Master Integrals for the Integration-By-Parts Method

TL;DR

The paper introduces an -finite basis of master integrals for Integration-By-Parts reductions in dimensional regularization, ensuring all coefficient functions remain finite as . It presents a constructive algorithm to replace divergent masters with finite combinations, and applies the method to the set of all QED-like four-loop massive tadpoles, using a Padé-based semi-numerical approach to obtain analytical pole parts and numerical finite parts. The authors verify their results against recent Schröder–Vuorinen findings and demonstrate substantial analytic control over divergent structures, including new relations among master integrals. The work promises improved numerical stability and analytic tractability for high-loop IBP problems and suggests a potential universal basis for a broad class of four-loop, QED-like tadpole computations.

Abstract

It is shown that for every problem within dimensional regularization, using the Integration-By-Parts method, one is able to construct a set of master integrals such that each corresponding coefficient function is finite in the limit of dimension equal to four. We argue that the use of such a basis simplifies and stabilizes the numerical evaluation of the master integrals. As an example we explicitly construct the ep-finite basis for the set of all QED-like four-loop massive tadpoles. Using a semi-numerical approach based on Pade approximations we evaluate analytically the divergent and numerically the finite part of this set of master integrals. The calculations confirm the recent results of Schröder and Vuorinen. All the contributions found there by fitting the high precision numerical results have been confirmed by direct analytical calculation without using any numerical input.

Paper Structure

This paper contains 12 sections, 56 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generalities and Notation
  3. Construction of the $\boldsymbol{\epsilon}$-finite set
  4. Applications to four-loop tadpoles
  5. Construction of the $\boldsymbol{\epsilon}$-finite set
  6. Calculation of the $\boldsymbol{\epsilon}$-finite basis
  7. Treatment of the basis integral $\boldsymbol{{T_{52}^f}}$
  8. Results for the $\boldsymbol{\epsilon}$-finite basis
  9. Analytical results for the master integrals
  10. General Comments
  11. Old masters in terms of $\boldsymbol{\epsilon}$-finite basis
  12. Proceeding in case of a "divergent" cut

Figures (3)

  • Figure 1: Subset of master integrals fully analytically known and therefore not considered in the construction of the $\epsilon$-finite basis.
  • Figure 2: Subset of master integrals for which the $\epsilon$-finite basis is constructed. The two numbers in brackets $(n_1,n_2)$ are decoded as follows: $n_1$ is the maximal power of a spurious pole $(\frac{1}{\epsilon})^{n_1}$ which could appear in front of the diagrams pictured above; ${n_2}$ is the maximal analytically known power $\epsilon^{n_2}$ of the $\epsilon$-expansion of the same integral as determined in Broadhurst:1992fiBroadhurst:1996az (for $T_{52}$) and in Schroder:2005vaSchroder:2005db for all the rest.
  • Figure 3: Diagrams of the $\epsilon$-finite basis.