[Paper Review] Hypergeometric functions, their epsilon expansions and Feynman diagrams

M.Yu. Kalmykov, Bernd A. Kniehl+2 more|ArXiv.org|Oct 20, 2008

Advanced Mathematical Identities19 references9 citations

TL;DR

This paper presents a hypergeometric function approach to one-loop Feynman diagrams, enabling analytical computation of their $\varepsilon$ expansions via generalized hypergeometric functions. It derives exact representations for master integrals using ${}_2F_1$, ${}_3F_2$, and ${}_4F_3$ functions, with arguments tied to Gram and Cayley determinants, and provides all-order $\varepsilon$ expansions in terms of multiple polylogarithms for key diagrams.

ABSTRACT

We review the hypergeometric function approach to Feynman diagrams. Special consideration is given to the construction of the Laurent expansion. As an illustration, we describe a collection of physically important one-loop vertex diagrams for which this approach is useful.

Motivation & Objective

• To develop a systematic method for computing the Laurent series expansion (in $\varepsilon$) of one-loop Feynman diagrams using hypergeometric functions.
• To establish a direct correspondence between kinematic invariants and hypergeometric function parameters via Gram and Cayley determinants.
• To provide exact analytical expressions for master integrals of vertex-type diagrams in terms of generalized hypergeometric functions.
• To extend the applicability of hypergeometric techniques to diagrams with multiple mass scales and non-vanishing Cayley determinants.
• To demonstrate that all-order $\varepsilon$ expansions of certain diagrams can be expressed in terms of multiple polylogarithms of square roots of unity.

Proposed method

• Utilizes the hypergeometric function representation of one-loop $N$-point diagrams derived via Mellin-Barnes techniques and difference equations in space-time dimension.
• Expresses master integrals as generalized hypergeometric functions ${}_pF_q$ with arguments proportional to kinematic invariants $Q^2/m^2$ or $Q_i^2/m^2$.
• Relies on the one-to-one correspondence between hypergeometric function arguments and geometric invariants—Gram and Cayley determinants—characterizing the diagram.
• Applies the Davydychev-Tarasov algorithm to reduce arbitrary one-loop diagrams to a set of master integrals with shifted propagator powers.
• Employs the $\varepsilon$-expansion technique to compute Laurent series coefficients around $d = 4 - 2\varepsilon$, enabling numerical evaluation of UV- and IR-divergent diagrams.
• Uses contiguous relations and differential reduction techniques to systematically generate $\varepsilon$-expansions of hypergeometric functions appearing in the diagrams.

Experimental results

Research questions

• RQ1 How can one-loop Feynman diagrams be systematically represented using generalized hypergeometric functions?
• RQ2 What is the precise relationship between the kinematic invariants of a diagram and the parameters of the hypergeometric functions in its representation?
• RQ3 Can the $\varepsilon$-expansion of master integrals be computed analytically using hypergeometric function identities?
• RQ4 How do the vanishing or non-vanishing values of Gram and Cayley determinants affect the structure of the hypergeometric representation?
• RQ5 To what extent can all-order $\varepsilon$ expansions of master integrals be expressed in terms of multiple polylogarithms?

Key findings

• The one-loop vertex diagram $C_7$ with arbitrary propagator powers is expressed as a ${}_3F_2$ function with arguments proportional to $Q^2/m^2$, with coefficients involving gamma functions.
• Diagram $C_8$ is represented by a ${}_3F_2$ hypergeometric function with parameters dependent on $n/2 - 1$, $3 - n/2$, and $n/2$, and its argument is $Q^2/(4m^2)$.
• For $C_9$, the master integral is given as a difference of two ${}_3F_2$ functions with arguments $Q_1^2/m^2$ and $Q_2^2/m^2$, respectively, and vanishes when $Q_1^2 = Q_2^2$.
• Diagram $C_{10}$, where the Cayley determinant vanishes, reduces to a linear combination of propagator-type diagrams and is expressed via two ${}_2F_1$ functions with arguments involving squared momentum invariants.
• The master integral $C_{11}$ has an all-order $\varepsilon$-expansion expressible in terms of multiple polylogarithms of a square root of unity.
• Diagram $C_{12}$ is represented as a linear combination of two ${}_3F_2$ functions, with a structure similar to $C_8$, involving a difference of terms weighted by $Q_i^2$ and a denominator $Q_1^2 - Q_2^2$.