Top-quark pair production close to threshold: Top-quark mass, width, and momentum distribution

The complete next-to-next-to-leading order (NNLO) QCD corrections to the total cross section $\sigma {(e}^{+}{e}^{-}\to Z^{*},{\gamma }^{*}\to t\overline{t}),$ in the kinematic region close to the top-quark–top-antiquark threshold, are calculated by solving the corresponding Schrödinger equations exactly in momentum space in a consistent momentum cutoff regularization scheme. The corrections coming from the same NNLO QCD effects to the top-quark three-momentum distribution $d\sigma /d|k_t|$ are determined. We discuss the origin of the large NNLO corrections to the peak position and the normalization of the total cross section observed in previous works and propose a new top-quark mass definition, the $1S$ mass $M_{1S},$ which stabilizes the peak in the total cross section. If the influence of beamstrahlung and initial state radiation on the mass determination is small, a theoretical uncertainty on the $1S$ top-quark mass measurement of 200 MeV from the total cross section at the linear collider seems possible. We discuss how well the $1S$ mass can be related to the $\overline{\mathrm{MS}}$ mass. We propose a consistent way to implement the top-quark width at NNLO by including electroweak effects into the NRQCD matching coefficients, which can then become complex.