Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals

TL;DR

This paper leverages the glue-and-cut symmetry to analytically evaluate all master integrals for massless propagators up to four loops, revealing their structure and implications for higher-loop calculations.

Contribution

It introduces an algebraic method based on symmetry relations to explicitly compute all master integrals in massless propagator diagrams up to four loops.

Findings

All master integrals at three and four loops are expressed in terms of trivial watermelon integrals.

The results explain the absence of even zeta values in certain functions related to massless propagators.

The approach is extendable to five-loop integrals once reduction techniques are available.

Abstract

The old "glue--and--cut" symmetry of massless propagators, first established in [1], 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 the massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicit performing the corresponding five-loop master integrals.