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

TL;DR

This paper shows that even-integer zeta-function values cancel in Euclidean QCD quantities when using the $C$-scheme coupling, and predicts their disappearance at higher orders in this scheme.

Contribution

It demonstrates the cancellation of even-integer zeta values in Euclidean QCD quantities within the $C$-scheme and conjectures their absence at higher orders.

Findings

Cancellation of even-integer zeta values in the $C$-scheme

Prediction of $eta$-function terms involving zeta values at higher orders

Conjecture that zeta terms vanish in the $C$-scheme at all orders

Abstract

At order $\alpha_s^4$ in perturbative quantum chromodynamics, even-integer $\zeta$-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 $\hat\alpha_s$ which has recently been introduced in Ref. [1]. It is furthermore conjectured that a $\zeta_4$ term should arise in the Adler function at order $\alpha_s^5$ in the $\overline{\rm MS}$-scheme, and that this term is expected to disappear in the $C$-scheme as well.