Life of \Pi

TL;DR

This paper evaluates the accuracy of advanced approximations for the heavy-quark contribution to the polarisation function at three- and four-loop orders, providing new precise results and testing Padé estimates in quantum chromodynamics.

Contribution

It presents the first high-precision three-loop calculation for general kinematics and tests Padé approximation predictions at four loops for the heavy-quark polarisation function.

Findings

First arbitrary precision three-loop result for general kinematics.

Validation of Padé approximation accuracy at four loops.

Enhanced understanding of heavy-quark polarisation function behavior.

Abstract

The heavy-quark contribution to the polarisation function $\Pi$ at higher perturbative orders is presently only known approximately. We scrutinise the accuracy of state-of-the-art approximations at three- and four-loop order. At three loops, we present for the first time a result with arbitrary numerical precision for general kinematics and compare to the best Pad\'e estimate. At four loops, we calculate the fourth (inverse) moment of the non-singlet heavy-quark vacuum polarisation in order to test the prediction for this moment based on Pad\'e approximation.