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Massive Feynman diagrams and inverse binomial sums

A. I. Davydychev, M. Yu. Kalmykov

TL;DR

The paper develops and applies a framework connecting massive Feynman diagram epsilon-expansions to inverse binomial sums via hypergeometric-function derivatives. By leveraging generating-function techniques and analytic continuation, it derives explicit, high-weight sum expressions in terms of log-sine and polylogarithmic functions, and demonstrates its utility through concrete two- and three-loop master integrals. Key contributions include new epsilon-term results, a recursive method for sums, and analytic-continuation schemes that translate generalized log-sine results into Nielsen polylogarithms. The findings advance analytical capabilities for multi-loop calculations, providing compact representations of master integrals across off-shell momentum regions and two-mass configurations.

Abstract

When calculating higher terms of the epsilon-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the epsilon-expansion of the polarization function of the neutral gauge bosons are constructed.

Massive Feynman diagrams and inverse binomial sums

TL;DR

The paper develops and applies a framework connecting massive Feynman diagram epsilon-expansions to inverse binomial sums via hypergeometric-function derivatives. By leveraging generating-function techniques and analytic continuation, it derives explicit, high-weight sum expressions in terms of log-sine and polylogarithmic functions, and demonstrates its utility through concrete two- and three-loop master integrals. Key contributions include new epsilon-term results, a recursive method for sums, and analytic-continuation schemes that translate generalized log-sine results into Nielsen polylogarithms. The findings advance analytical capabilities for multi-loop calculations, providing compact representations of master integrals across off-shell momentum regions and two-mass configurations.

Abstract

When calculating higher terms of the epsilon-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the epsilon-expansion of the polarization function of the neutral gauge bosons are constructed.

Paper Structure

This paper contains 18 sections, 194 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Inverse binomial sums and hypergeometric functions
  3. The $\varepsilon$-expansion of the $_2F_1$ function
  4. Expansion of higher functions
  5. Further results for the sums
  6. Analytical continuation
  7. Application to Feynman diagrams
  8. One-loop vertex
  9. Two-loop self-energy integral $F_{10101}$
  10. Two-loop sunset-type diagram $J_{011}$
  11. Two-loop vertex diagrams
  12. Three-loop vacuum diagram $D_4$
  13. Conclusion
  14. Harmonic polylogarithms of complex arguments
  15. Additional identities between inverse binomial sums
  16. ...and 3 more sections

Figures (2)

  • Figure 1: One- two- and three-loop diagrams considered in the paper. Bold and thin lines correspond to massive and massless propagators, respectively.
  • Figure 2: Two-loop contributions to the off-shell polarization function of a neutral gauge boson. Bold and thin lines correspond to the massive quark propagator and the massless boson (gluon or photon) propagator, respectively.