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Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams

M. Yu. Kalmykov

TL;DR

<3-5 sentence high-level summary> Kalmykov develops a reduction algorithm that expresses any Gauss hypergeometric function $_2F_1$ with arbitrary parameters as a linear combination of a small master set of fixed-parameter $_2F_1$ functions, enabling systematic ε-expansions in dimensional regularization. He classifies the integer/half-integer parameter cases into six basis types (A–F), showing A–D reduce to A-type and deriving integral relations between expansion coefficients; A, B, C, D, and E have ε-expansions up to weight 4 in terms of Nielsen polylogarithms, while type F requires new elliptic-like generalizations. The paper provides explicit expansions for the basis types (with detailed expressions for type A, and compact forms for B–E) and discusses the role of F in introducing elliptic structures at higher weight. Finally, the approach is demonstrated on q-loop off-shell propagator diagrams and q-loop bubbles, yielding master-integral counts (two for sunset-type beyond one loop, four for bubbles beyond two loops) and illustrating practical utility for multi-loop Feynman diagram computations, with code available online.</p>

Abstract

The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and epsilon-expansion of q-loop off-shell propagator diagrams with one massive line and q massless lines and q-loop bubble with two-massive lines and q-1 massless lines are considered. The code (Mathematica/FORM) is available via the www at this URL http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams

TL;DR

<3-5 sentence high-level summary> Kalmykov develops a reduction algorithm that expresses any Gauss hypergeometric function with arbitrary parameters as a linear combination of a small master set of fixed-parameter functions, enabling systematic ε-expansions in dimensional regularization. He classifies the integer/half-integer parameter cases into six basis types (A–F), showing A–D reduce to A-type and deriving integral relations between expansion coefficients; A, B, C, D, and E have ε-expansions up to weight 4 in terms of Nielsen polylogarithms, while type F requires new elliptic-like generalizations. The paper provides explicit expansions for the basis types (with detailed expressions for type A, and compact forms for B–E) and discusses the role of F in introducing elliptic structures at higher weight. Finally, the approach is demonstrated on q-loop off-shell propagator diagrams and q-loop bubbles, yielding master-integral counts (two for sunset-type beyond one loop, four for bubbles beyond two loops) and illustrating practical utility for multi-loop Feynman diagram computations, with code available online.</p>

Abstract

The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and epsilon-expansion of q-loop off-shell propagator diagrams with one massive line and q massless lines and q-loop bubble with two-massive lines and q-1 massless lines are considered. The code (Mathematica/FORM) is available via the www at this URL http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

Paper Structure

This paper contains 14 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reduction of Gauss hypergeometric functions
  3. Relations between basis hypergeometric functions with integer or half-integer values of parameters
  4. Laurent expansion of basis functions with integer or half-integer values of parameters
  5. type A
  6. type B
  7. type C
  8. type D
  9. type E
  10. type F
  11. Application to Feynman diagrams
  12. $q$--loop propagator with $q$ massless line
  13. $q$--loop bubble with $q-1$ massless lines
  14. Conclusion

Figures (1)

  • Figure 1: Diagrams considered in the paper. Bold and thin lines correspond to massive and massless propagators, respectively.