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Analytic Epsilon Expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight

R. N. Lee, V. A. Smirnov

TL;DR

The paper develops analytic $\epsilon$-expansions for massless three-loop master integrals relevant to quark/gluon form factors and the electron $g-2$, extending results to transcendentality weights characteristic of four-loop calculations. By combining the dimensional-recurrence (DRA) method with sector decomposition, Mellin--Barnes representations, and PSLQ, the authors obtain high-precision, analytic expressions for a large set of integrals, including the most complex cases, and establish homogeneous transcendentality bases. These results, featuring high-weight constants and multiple zeta values, pave the way for future four-loop determinations of form factors and $g-2$ contributions. The work demonstrates the strength of integrating DRA with complementary techniques to achieve analytic control over challenging multiloop integrals.

Abstract

We evaluate analytically higher terms of the epsilon-expansion of the three-loop master integrals corresponding to three-loop quark and gluon form factors and to the three-loop master integrals contributing to the electron g-2 in QED up to the transcendentality weight typical to four-loop calculations, i.e. eight and seven, respectively. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition implemented in FIESTA, the method of Mellin--Barnes representation, and the PSLQ algorithm.

Analytic Epsilon Expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight

TL;DR

The paper develops analytic -expansions for massless three-loop master integrals relevant to quark/gluon form factors and the electron , extending results to transcendentality weights characteristic of four-loop calculations. By combining the dimensional-recurrence (DRA) method with sector decomposition, Mellin--Barnes representations, and PSLQ, the authors obtain high-precision, analytic expressions for a large set of integrals, including the most complex cases, and establish homogeneous transcendentality bases. These results, featuring high-weight constants and multiple zeta values, pave the way for future four-loop determinations of form factors and contributions. The work demonstrates the strength of integrating DRA with complementary techniques to achieve analytic control over challenging multiloop integrals.

Abstract

We evaluate analytically higher terms of the epsilon-expansion of the three-loop master integrals corresponding to three-loop quark and gluon form factors and to the three-loop master integrals contributing to the electron g-2 in QED up to the transcendentality weight typical to four-loop calculations, i.e. eight and seven, respectively. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition implemented in FIESTA, the method of Mellin--Barnes representation, and the PSLQ algorithm.

Paper Structure

This paper contains 4 sections, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Master integrals for quark and gluon form factors
  3. Master integrals for the $g-2$ factor
  4. Conclusion

Figures (4)

  • Figure 1: Master integrals for the three-loop form factors. Internal lines denote massless propagators $1/(k^2+i 0)$.
  • Figure 2: Master integrals with a homogeneous transcendentality weight to replace the corresponding integrals without numerator.
  • Figure 3: Master integrals for the three-loop $g-2$ factor. Each line corresponds to a definite complexity level indicated to the left.
  • Figure 4: The integral $I_{11}$ (left) and the auxiliary diagram used for its reduction (right).