Phase Space Matching and Finite Lifetime Effects for Top-Pair Production Close to Threshold

The top-pair $t\bar t$ production cross section close to threshold in $e^+e^-$ collisions is strongly affected by the small lifetime of the top quark. Since the cross section is defined through final states containing the top decay products, a consistent definition of the cross section depends on prescriptions how these final states are accounted for the cross section. Experimentally, these prescriptions are implemented for example through cuts on kinematic quantities such as the reconstructed top quark invariant masses. As long as these cuts do not reject final states that can arise from the decay of a top and an anti-top quark with a small off-shellness compatible with the nonrelativistic power-counting, they can be implemented through imaginary phase space matching conditions in NRQCD. The prescription-dependent cross section can then be determined from the optical theorem using the $e^+e^-$ forward scattering amplitude. We compute the phase space matching conditions associated to cuts on the top and anti-top invariant masses at next-to-next-to-leading logarithmic (NNLL) order and partially at next-to-next-to-next-to-leading logarithmic (N${}^3$LL) order in the nonrelativistic expansion and, together with finite lifetime and electroweak effects known from previous work, analyze their numerical impact on the $t\bar t$ cross section. We show that the phase space matching contributions are essential to make reliable NRQCD predictions, particularly for energies below the peak region, where the cross section is small. We find that irreducible background contributions associated to final states that do not come from top decays are strongly suppressed and can be neglected for the theoretical predictions.