Pseudoscalar and vector toponia in a Dyson--Schwinger--Bethe--Salpeter framework

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 Poincaré-covariant nonperturbative framework, quantum chromodynamics (QCD) generates tightly correlated pseudoscalar and vector toponium systems in that extreme heavy-quark limit.