# A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions

Fleischer J, Jegerlehner F, Tarasov OV (2003)
NUCLEAR PHYSICS B 672(1-2): 303-328.

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Journal Article | Original Article | Published | English

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Abstract
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.
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Fleischer J, Jegerlehner F, Tarasov OV. A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions. NUCLEAR PHYSICS B. 2003;672(1-2):303-328.
Fleischer, J., Jegerlehner, F., & Tarasov, O. V. (2003). A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions. NUCLEAR PHYSICS B, 672(1-2), 303-328. doi:10.1016/j.nuclphysb.2003.09.004
Fleischer, J., Jegerlehner, F., and Tarasov, O. V. (2003). A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions. NUCLEAR PHYSICS B 672, 303-328.
Fleischer, J., Jegerlehner, F., & Tarasov, O.V., 2003. A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions. NUCLEAR PHYSICS B, 672(1-2), p 303-328.
J. Fleischer, F. Jegerlehner, and O.V. Tarasov, “A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions”, NUCLEAR PHYSICS B, vol. 672, 2003, pp. 303-328.
Fleischer, J., Jegerlehner, F., Tarasov, O.V.: A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions. NUCLEAR PHYSICS B. 672, 303-328 (2003).
Fleischer, Jochem, Jegerlehner, F, and Tarasov, OV. “A new hypergeometric representation of one-loop scalar integrals in $d$ dimensions”. NUCLEAR PHYSICS B 672.1-2 (2003): 303-328.
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