Abstract
We apply the differential equation technique to the calculation of the one-loop massless diagram with five onshell legs. Using the reduction to ϵ-form, we manage to obtain a simple one-fold integral representation exact in space-time dimensionality. The expansion of the obtained result in ϵ and the analytical continuation to physical regions are discussed.
Highlights
JHEP02(2016)021 obtained using dimensional recurrence relation [8, 9]
The integral was expressed in terms of the Appell function F3 and hypergeometric functions pFq
We apply the approach first introduced in ref. [10], based on the reduction of the differential equations for master integrals to the Fuchsian form with factorized dependence of the right-hand side on
Summary
We introduce the following notation rn = (−1)isn+isn+i+1 , ∆ = det (2pi · pj |i,j=1,...4) = riri+2 , S = 4s1s2s3s4s5/∆. Using techniques described in detail in the succeeding sections, we obtain the following exact in d representation for P (6−2 ) for real si (of arbitrary signs). By Re arctan√( x+i0/r) we understand the function x+i0 f (x, r). Any order of -expansion can be trivially written as a one-fold integral of elementary functions. In the appendix A we explain how to rewrite this integral in terms of the Goncharov’s polylogarithms. We demonstrate the cancellation of O( −1) terms. In order to crosscheck our result, we have performed comparison with the numerical results for pentagon obtained using Fiesta 3, ref. Some results of the comparison are presented in table 1
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