We study the pseudoscalar ($J^{ PC}=0^{ -+}$) and vector ($1^{ --}$) top--antitop (toponium) systems within the rainbow--ladder truncation of the Dyson--Schwinger and Bethe--Salpeter equations, employing the Qin--Chang effective interaction. After validating the framework in the charmonium and bottomonium sectors, we extend it consistently to the top sector, incorporating renormalisation-group running of the current quark mass and a careful treatment of the number of active flavours. We compute masses and leptonic decay constants for $N_f=5$ and $6$, then analyse their dependence on the renormalisation scale in the range $\mu=400-800\,\text{ GeV}$. The resulting toponium masses lie near $344-346\,\text{ GeV}$ with hyperfine splittings below $0.14-0.17\,\text{ GeV}$, while the decay constants are large, $6-7\,\text{ GeV}$, and exhibit the expected heavy-quark scaling behaviour. We find only mild sensitivity to the renormalisation point and a systematic reduction of binding when increasing $N_f$. Although the physical top quark decays weakly before hadronisation, our results demonstrate that, within a Poincar\'e-covariant nonperturbative framework, quantum chromodynamics (QCD) generates tightly correlated pseudoscalar and vector toponium systems in that extreme heavy-quark limit.
Categories: hep-ph hep-ex hep-lat nucl-ex nucl-th