Two-Loop Gluon-Condensate Contributions To Heavy-Quark Current Correlators: Exact Results And Approximations

TL;DR

This paper delivers complete analytic two-loop coefficient functions for the gluon condensate in heavy-quark current correlators across all kinematic regimes, using an on-shell scheme and advanced symbolic techniques. It verifies consistency with known limiting cases, fixes scheme-related issues, and provides precise expressions in all four channels (V, A, S, P) along with their limiting behaviors. Through rigorous comparison with numerical approximation methods, the work demonstrates that high-accuracy predictions are achievable with only a few moments and well-chosen threshold/asymptotic inputs, enabling reliable gluon-condensate extractions (≈0.021 GeV^4) and informing prospects for three-loop calculations. The study thus advances both analytic results and practical approximation techniques, with implications for QCD sum rules and multi-loop photon/QCD correlator computations.

Abstract

The coefficient functions of the gluon condensate $<G^2>$, in the correlators of heavy-quark vector, axial, scalar and pseudoscalar currents, are obtained analytically, to two loops, for all values of $z=q^2/4m^2$. In the limiting cases $z\to0$, $z\to1$, and $z\to-\infty$, comparisons are made with previous partial results. Approximation methods, based on these limiting cases, are critically assessed, with a view to three-loop work. High accuracy is achieved using a few moments as input. A {\em single} moment, combined with only the {\em leading} threshold and asymptotic behaviours, gives the two-loop corrections to better than 1% in the next 10 moments. A two-loop fit to vector data yields $<\frac{α_{\rm s}}πG^2>\approx0.021$ GeV$^4$.