Abstract
We apply differential equations technique to the calculation of the one-loop massless diagram with one offshell legs. Using reduction to $\epsilon$-form, we managed to obtain a simple one-fold integral representation exact in space-time dimensionality. Expansion of the obtained result in $\epsilon$ and analytical continuation to physical region are discussed.
Highlights
In this paper we consider the one-loop integral with massless internal lines and one offshell leg exactly in the dimension of space-time, which we call below the pentagon integral
This type of integrals arises for example in a system of differential equations on one loop master integrals with massless internal lines and with more than five legs, for example on massless on-shell hexagon integral
In the previous paper [3] we considered the pentagon integral with all on-shell legs exactly in d dimensions
Summary
In this paper we consider the one-loop integral with massless internal lines and one offshell leg exactly in the dimension of space-time, which we call below the pentagon integral. This type of integrals arises for example in a system of differential equations on one loop master integrals with massless internal lines and with more than five legs, for example on massless on-shell hexagon integral. In the previous paper [3] we considered the pentagon integral with all on-shell legs exactly in d dimensions. We introduce the notation and present the result for the Pentagon integral. In Appendix B and C we present the calculation of the easy box and the hard box master integrals respectively
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