Four loop massless propagators: An algebraic evaluation of all master integrals

The old glue-and-cut symmetry of massless propagators, first established in Ref. [1] (Chetyrkin and Tkachov, 1981), leads after reduction to master integrals is performed to a host of non-trivial relations between the latter. The relations constrain the master integrals so tightly that they all can be analytically expressed in terms of only few, essentially trivial, watermelon-like integrals. As a consequence we arrive at explicit analytical results for all master integrals appearing in the process of reduction of massless propagators at three and four loops. The transcendental structure of the results suggests a clean explanation of the well-known mystery of the absence of even zetas (zeta(2n)) in the Adler function and other similar functions essentially reducible to massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicitly performing the corresponding five-loop master integrals. (C) 2010 Elsevier B.V. All rights reserved.