Absence of even-integer $ζ$-function values in Euclidean physical quantities in QCD

Abstract

At order $α_s^4$ in perturbative quantum chromodynamics, even-integer $ζ$-function values are present in Euclidean physical correlation functions like the scalar quark correlation function or the scalar gluonium correlator. We demonstrate that these contributions cancel when the perturbative expansion is expressed in terms of the so-called $C$-scheme coupling $\hatα_s$ which has recently been introduced in Ref. [1]. It is furthermore conjectured that a $ζ_4$ term should arise in the Adler function at order $α_s^5$ in the $\overline{\rm MS}$-scheme, and that this term is expected to disappear in the $C$-scheme as well.