Ultrasoft Renormalization in Non-Relativistic QCD

TL;DR

The paper analyzes ultrasoft renormalization in non-relativistic QCD within the vNRQCD framework, revealing that ultrasoft gluons induce mixing of soft operators into new four-quark operators ${\cal O}_{2i}^{(2)}$ that influence the running of spin-independent ${1/m^2}$ potentials. It also develops a subtraction procedure for mixed ultrasoft-potential diagrams, showing that new operators ${\cal O}_{ki}$ and ${\cal O}_{ci}$ are necessary to properly renormalize such graphs. Numerically, these refinements lead to small but non-negligible adjustments to the NNLL top threshold cross section and to a revised understanding of running in the regime where mv^2 approaches $\Lambda_{\rm QCD}$, with important implications for comparisons to pNRQCD and potential nonperturbative effects. Overall, the work strengthens the perturbative control of NRQCD near threshold while highlighting the subtle interplay between ultrasoft, soft, and potential scales. The results support the continued use of scale-correlated vNRQCD in precision top and quarkonium studies, albeit with caution in nonperturbative regimes.

Abstract

For Non-Relativistic QCD the velocity renormalization group correlates the renormalization scales for ultrasoft, potential and soft degrees of freedom. Here we discuss the renormalization of operators by ultrasoft gluons. We show that renormalization of soft vertices can induce new operators, and also present a procedure for correctly subtracting divergences in mixed potential-ultrasoft graphs. Our results affect the running of the spin-independent potentials in QCD. The change for the NNLL t-tbar cross section near threshold is very small, being at the 1% level and essentially independent of the energy. We also discuss implications for analyzing situations where mv^2 ~ Lambda_QCD.

Figures (11)

• Figure 1: Examples of mixed soft-ultrasoft graphs with $A^0$ ultrasoft gluons which could renormalize ${\cal L}_s$ in Feynman gauge. The zigzag lines denote soft gluons, quarks, or ghosts.
• Figure 2: The zigzag lines denote soft gluons, quarks, or ghosts. Graphs (a) and (b) are examples of mixed soft-ultrasoft graphs, while (c) denotes an operator for soft Compton scattering off a potential.
• Figure 3: Example of the matching calculation for ${\cal O}_{2\varphi}^{(\sigma)}$. Here the zig-zag lines denote soft massless fermions. At the high scale ($\nu=1$) the graphs on the left exactly cancel so the coefficient of the operator on the right is zero.
• Figure 4: Example of renormalization of two-body potential graphs by an ultrasoft gluon. The ultrasoft couplings are the order $v^0$ term from Eq. (\ref{['Lp']}) and the order $v^1$ term ${\bf p}\cdot {\bf A}$ in Eq. (\ref{['Lus']}).
• Figure 5: a) ultrasoft gluon graphs with ${\bf p}\cdot {\bf A}$ vertices (with wavefunction renormalization on the other line understood), b) soft graph with $Z_0^{(\sigma=0)}$ and $Z_0^{(\sigma'=2)}$ vertices, c) soft graph involving ${\cal O}_{2\varphi}^{(2)}$. The zig-zag lines here denote massless soft fermions.