New results for algebraic tensor reduction of Feynman integrals
We report on some recent developments in algebraic tensor reduction ofone-loop Feynman integrals. For 5-point functions, an efficient tensorreduction was worked out recently and is now available as numerical C++package, PJFry, covering tensor ranks until five. It is free of inverse 5-pointGram determinants, and inverse small 4-point Gram determinants are treated byexpansions in higher-dimensional 3-point functions. By exploiting sums oversigned minors, weighted with scalar products of chords (or, equivalently,external momenta), extremely efficient expressions for tensor integralscontracted with external momenta were derived. The evaluation of 7-pointfunctions is discussed. In the present approach one needs for the reductions a$(d+2)$-dimensional scalar 5-point function in addition to the usual scalarbasis of 1- to 4-point functions in the generic dimension $d=4-2 \epsilon$.When exploiting the four-dimensionality of the kinematics, this basis issufficient. We indicate how the $(d+2)$-dimensional 5-point function can beevaluated.