Running of the heavy quark production current and 1/k potential in QCD

TL;DR

The paper develops and applies vNRQCD with a velocity RG to threshold heavy-quark production, addressing the need to sum logs from soft and ultrasoft scales. It computes the two-loop anomalous dimension for the $V^{(-1)}$ ($1/|{f k}|$) potential and derives the full NLL running for the $1/|{f k}|$ potentials, including explicit soft and ultrasoft contributions. It then determines the NLL running of the heavy-quark production current coefficient $c_1(\nu)$, showing that Kniehl-Penin $ olinebreak \alpha_s^3 \ln^2(\alpha_s)$ logs are reproduced within this framework. The results improve the perturbative stability of near-threshold $t\bar t$ predictions and confirm the consistency of velocity RG resummation with known non-RG logarithms, offering a path toward more precise threshold phenomenology.

Abstract

The 1/k contribution to the heavy quark potential is first generated at one loop order in QCD. We compute the two loop anomalous dimension for this potential, and find that the renormalization group running is significant. The next-to-leading-log coefficient for the heavy quark production current near threshold is determined. The velocity renormalization group result includes the alpha_s^3 ln^2(alpha_s) ``non-renormalization group logarithms'' of Kniehl and Penin.

Figures (7)

• Figure 1: QCD diagrams for tree level matching.
• Figure 2: Matching for the operator attaching an ultrasoft gluon to a potential interaction.
• Figure 3: Order $\alpha_s^3/v$ graphs containing soft gluons, ghosts, or massless quarks which are all denoted by a zigzag line. In (a) the dot denotes the Coulomb potential, while in (b) the dot denotes the order $v^0$ potential. The boxes denote soft vertices with insertions of the functions $U^{(\sigma)}$, $W^{(\sigma)}$, $Y^{(\sigma)}$, or $Z^{(\sigma)}$. In (a) the indices $\sigma+\sigma'=2$. In (c), (d), and (e) the $\otimes$ vertex is obtained from the one loop matching in Fig. \ref{['fig_sm']}.
• Figure 4: Contributions to the soft Lagrangian from one loop matching. In the full theory diagrams on the left the thick lines are massive quarks, while the thin lines are massless quarks. The matching in a) gives $\alpha_s$ corrections to the functions $U^{(1)}$, $W^{(1)}$, $Y^{(1)}$, and $Z^{(1)}$ in Eq. (\ref{['Lsoft']}). The matching in b) induces new operators that involve the scattering of soft gluons, ghosts, or light quarks off of a four fermion potential.
• Figure 5: Order $\alpha_s^3/v$ two loop graphs with an ultrasoft gluon and two potential insertions. The topologies shown give several different classes of diagrams depending on the vertices used as explained in the text. The diagrams obtained by flipping the graphs left-to-right and up-to-down are to be understood.