Scale Setting in QCD and the Momentum Flow in Feynman Diagrams

TL;DR

The paper generalizes scale setting in QCD by introducing a distribution function $w(t)$ that represents the momentum flow in one-loop diagrams with a running coupling at the vertices, yielding an all-orders resummed quantity $S_{ m res}(M^2)$ that is renormalization-scale and -scheme invariant. This approach unifies the BLM idea with a full resummation of the leading $\beta_0^{n-1}\alpha_s^n$ terms, clarifies the appearance of infrared renormalons, and provides a natural separation of short- and long-distance contributions via a hard factorization scale $\lambda$, thereby connecting perturbative results with non-perturbative effects through the operator product expansion. The method is illustrated across heavy-quark systems and light-vector current correlators, revealing how the distribution function controls the effective momentum scales, renormalon ambiguities, and the relative size of perturbative and non-perturbative contributions. The work offers a principled framework for improving perturbative predictions and sets the stage for extending the resummation to cross sections and inclusive processes in a subsequent part, with implications for precision QCD phenomenology.

Abstract

We present a formalism to evaluate QCD diagrams with a single virtual gluon using a running coupling constant at the vertices. This method, which corresponds to an all-order resummation of certain terms in a perturbative series, provides a description of the momentum flow through the gluon propagator. It can be viewed as a generalization of the scale-setting prescription of Brodsky, Lepage and Mackenzie to all orders in perturbation theory. In particular, the approach can be used to investigate why in some cases the ``typical'' momenta in a loop diagram are different from the ``natural'' scale of the process. It offers an intuitive understanding of the appearance of infrared renormalons in perturbation theory and their connection to the rate of convergence of a perturbative series. Moreover, it allows one to separate short- and long-distance contributions by introducing a hard factorization scale. Several applications to one- and two-scale problems are discussed in detail.

Figures (7)

• Figure 1: Distribution function for the ratio $m_b/m_b^{\rm R}$ in the two renormalization schemes R1 (solid line) and R2 (dashed-dotted line). The dotted line shows the unsubtracted distribution function for the ratio of the pole mass and the bare mass. The long arrows indicate the position of the BLM scale in the schemes R1 and R2. The small arrow shows the factorization point, which separates short- and long-distance contributions.
• Figure 2: The function $\gamma(r)$ defined in (\ref{['gamdef']}).
• Figure 3: Distribution function for the ratio $f_B^{\rm stat}/f_{B^*}^{\rm stat}$. The right arrow indicates the BLM scale, the left one the factorization point.
• Figure 4: Distribution functions for $\eta_V$ (solid line) and $(\eta_V-\eta_A)$ (dashed-dotted line). The long arrows show the BLM scales, the short arrow indicates the factorization point.
• Figure 5: Short-distance contributions to $m_b(\lambda)/\overline{m}_b(m_b)$ (solid line) and $(f_B/f_{B^*})(\lambda)$ (dashed-dotted line) as a function of the factorization scale.