The Two-Loop Scale Dependence of the Static QCD Potential including Quark Masses

Abstract

The interaction potential V (Q 2 ) between static test charges can be used to define an effective charge α V (Q 2 ) and a physically-based renormalization scheme for quantum chromodynamics and other gauge theories. In this paper we use recent results for the finite-mass fermionic corrections to the heavyquark potential at two-loops to derive the next-to-leading order term for the Gell Mann-Low function of the V -scheme. The resulting effective number of flavors N F (Q 2 /m 2 ) in the α V scheme is determined as a gauge-independent and analytic function of the ratio of the momentum transfer to the quark pole mass. The results give automatic decoupling of heavy quarks and are independent of the renormalization procedure. Commensurate scale relations then provide the next-to-leading order connection between all perturbatively calculable observables to the analytic and gauge-invariant α V scheme without any scale ambiguity and a well defined number of active flavors. The inclusion of the finite quark mass effects in the running of the coupling is compared with the standard treatment of finite quark mass effects in the MS scheme.

Form factors of the gauge-invariant three-gluon vertex

“…The negative value of N F at 0 < ∼ Q < ∼ 4M is not entirely novel, as a similar behavior was also found in the context of two-loop quark mass corrections to Vscheme effective charge [40]. It is essentially due to the anti-screening of color charge, this case in the triangle interaction, and does not arise in the one-loop single scale effective charge, as seen in Fig.…”

An Analytic Approach to Perturbative QCD

“…Using the relation (42) we verify that the analytic expression (43) follows from the representation (40). Note that (43) does not possess a pole at Q 2 = 0 and it has a more strong singularity then the corresponding amplitude for the free field.…”

“…Generally, the running coupling is a gauge-and scheme-dependent quantity (it is nonphysical object in this sense [9]). Presumably, more useful quantities, in the infrared region, are the scheme-independent effective charge (the combination of a vertex and propagators) [41]) or the running couplings of physical schemes [15,16,42]. Most satisfactory approach can be based on the notion of scheme-and gauge-independent universal charge (the analogous of the QED effective charge) [6].…”