Ultrasoft renormalization in nonrelativistic QCD

For nonrelativistic 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 $t\overline{t}$ cross section near threshold is very small, being at the $\lesssim 1\%$ level and essentially independent of the energy. We also discuss implications for analyzing situations where ${\mathrm{mv}}^2\sim {\Lambda }_{\mathrm{QCD}}.$