arXiv subjects

Kamel Benhaddou

Publications and source records attributed to Kamel Benhaddou.

2 recordsLinked to original sources

arXiv2017

A simple expression for the four-point scalar function from Gaussian integrals and Fourier transform

Recasting the $N$-point one loop scalar integral from Feynman to Schwinger parameters gives an integrand with a Gaussian form. By application of a Fourier transform, it is easy to derive explicit expressions for the two, three and four-point functions. The Fourier transformation disentangles singularities in the complex plane and extract their contribution as two-point functions in two dimensions. We explicitly derive a one dimensional expression for the (4D) four-point function whose integrand involves only square root and arcsine functions. This report is a condensed version of the approach developed in \cite{Benhaddou2016} which does not make use of probabilistic jargon.

hep-ph↗

arXiv2017

A Probabilistic Angle on One Loop Scalar Integrals

Recasting the $N$-point one loop scalar integral as a probabilistic problem, allows the derivation of integral recurrence relations as well as exact analytical expressions in the most common cases. $ε$ expansions are derived by writing a formula that relates an $N$-point function in decimal dimension to an $N$-point function in integer dimension. As an example, we give relations for the massive 5-point function in dimension $n=4-2ε$, $n=6-2ε$. The reduction of tensor integrals of rank 2 with $N=5$ is achieved showing the method's potential. Hypergeometric functions are not needed but only integration of arcsine function whose analytical continuation is well known.

math-ph↗