Abstract
We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with 6th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K 3,4 at one edge.
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Communicated by A. Connes
CNRS.
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Brown, F. The Massless Higher-Loop Two-Point Function. Commun. Math. Phys. 287, 925–958 (2009). https://doi.org/10.1007/s00220-009-0740-5
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DOI: https://doi.org/10.1007/s00220-009-0740-5