Higher-order soft-gluon corrections for $t\bar{t}Z$ cross sections

We calculate higher-order soft-gluon corrections for the associated production of a top-antitop quark pair and a $Z$ boson ($t\bar{t}Z$ production) in proton-proton collisions. We find that these corrections are numerically dominant and large. We present approximate NNLO (aNNLO) and approximate N${}^3$LO (aN${}^3$LO) cross sections that add, respectively, second-order and third-order soft-gluon corrections to the exact NLO QCD result. We also add electroweak (EW) corrections through NLO. We estimate uncertainties that arise from scale dependence, which are smaller at higher orders, and from parton distribution functions. We compare our aN${}^3$LO QCD + NLO EW theoretical results to measurements of $t\bar{t}Z$ cross sections from the LHC and find good agreement. We also calculate single-particle-inclusive differential distributions in top-quark transverse momentum and rapidity.