$1/m_t^2$ and cannot contribute to the cross section matrix elements at NNLL level without even having accounted for additional powers of the coupling constants. It remains to discuss the space-like ultrasoft ${\mathbf A}$ gluons which couple to the quarks with the ${\mathbf p}.{\mathbf A}/m_t$ coupling. In pure QCD space-like ultrasoft gluon exchange contributes to the cross section matrix elements at NNLL order and also to the renormalization group running of the $t\bar t$ production operators at the NLL level [@LMR] through operator mixing. Accounting for the $g^2$-suppression from an additional electroweak loop correction to the interaction vertex then also leads to a contribution beyond the NNLL order.

At NNLL order let us first consider whether one needs to account for any operator in addition to those present already at the LL level. Ultrasoft gluon interactions have been discussed above. Soft gluon operators first contribute at the NLL level in pure QCD (for example as corrections to the Coulomb potential [@amis; @amis2]), so $g^2$ electroweak corrections to these operators are beyond NNLL order. Such corrections can also not contribute at NNLL order through mixing since the Coulomb potential does not cause UV divergences. It remains to discuss four-quark operators. Because an 