10.1016/j.nuclphysb.2003.09.004
Fleischer, Jochem
Jochem
Fleischer
Jegerlehner, F
F
Jegerlehner
Tarasov, OV
OV
Tarasov
A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions
ELSEVIER SCIENCE BV
2003
2010-04-28T12:58:41Z
2018-07-24T12:59:55Z
journal_article
https://pub.uni-bielefeld.de/record/1609642
https://pub.uni-bielefeld.de/record/1609642.json
A difference equation w.r.t. space-time dimension $d$ for $n$-point one-loopintegrals with arbitrary momenta and masses is introduced and a solutionpresented. The result can in general be written as multiple hypergeometricseries with ratios of different Gram determinants as expansion variables.Detailed considerations for $2-,3-$ and $4-$point functions are given. For the$2-$ point function we reproduce a known result in terms of the Gausshypergeometric function $_2F_1$. For the $3-$point function an expression interms of $_2F_1$ and the Appell hypergeometric function $F_1$ is given. For the$4-$point function a new representation in terms of $_2F_1$, $F_1$ and theLauricella-Saran functions $F_S$ is obtained. For arbitrary $d=4-2\epsilon$,momenta and masses the $2-,3-$ and $4-$point functions admit a simple one-foldintegral representation. This representation will be useful for the calculationof contributions from the $\epsilon-$ expansion needed in higher orders ofperturbation theory. Physically interesting examples of $3-$ and $4-$pointfunctions occurring in Bhabha scattering are investigated.