Feynman integral in ${\mathbb{R}}^1\oplus {\mathbb{R}}^m$ and complex expansion of ${}_2{F}_1$

Abstract

Closed form expressions are proposed for the Feynman integral $$I_{D,m}(p,q)=\int \frac{d^{m}y}{(2\pi {)}^m}\int \frac{d^{D}x}{(2\pi {)}^D}\frac{1}{(x-p/2{)}^2+(y-q/2{)}^4}\frac{1}{(x+p/2{)}^2+(y+q/2{)}^4}$$ 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 ${}_2{F}_1$, as well as generalized hypergeometric ${}_2{F}_2$ and ${}_3{F}_2$, Horn $H_4$ and Appell $F_2$ functions. Several interesting relations are derived between the real and imaginary parts of ${}_2{F}_1$ and the function $H_4$.

To appear in Integral Transforms and Special Functions after a major revision

References in corpus (7)

• Finding new relationships between hypergeometric functions by evaluating Feynman integrals
• New relationships between Feynman integrals
• A massive Feynman integral and some reduction relations for Appell functions
• The Clausenian hypergeometric function ${}_3{F}_2$ with unit argument and negative integral parameter differences
• Large-n expansion for m-axial Lifshitz points
• Lifshitz-point correlation length exponents from the large-n expansion
• On computing some special values of hypergeometric functions

Cited by in corpus (1)

• A Feynman integral in Lifshitz-point and Lorentz-violating theories in $R^D\oplus R^m$