Feynman integral in $\mathbb R^1\oplus\mathbb R^m$ and complex expansion of $_2F_1$

TL;DR

This paper derives closed-form expressions for a specific Feynman integral in mixed real spaces, expressing it through complex hypergeometric functions and establishing relations between their real and imaginary parts.

Contribution

It provides novel closed-form formulas for the Feynman integral in the case D=1, involving complex hypergeometric functions and their interrelations.

Findings

Expressions involving $_2F_1$, $_2F_2$, $_3F_2$, $H_4$, and $F_2$ functions.

Relations between real and imaginary parts of $_2F_1$ and $H_4$ functions.

Explicit formulas for the Feynman integral in the special case D=1.

Abstract

Closed form expressions are proposed for the Feynman integral $$ I_{D, m}(p,q) = \int\frac{d^my}{(2\pi)^m}\int\frac{d^Dx}{(2\pi)^D} \frac1{(x-p/2)^2+(y-q/2)^4} \frac1{(x+p/2)^2+(y+q/2)^4} $$ over $d=D+m$ dimensional space with $(x,y),\,(p,q)\in \mathbb R^D \oplus \mathbb R^m$, in the special case $D=1$. We show that $I_{1,m}(p,q)$ can be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function $_2F_1$, as well as generalized hypergeometric $_2F_2$ and $_3F_2$, Horn $H_4$ and Appell $F_2$ functions. Several interesting relations are derived between the real and imaginary parts of $_2F_1$ and the function $H_4$.