Finite-Width Dissolution of Radial Spectroscopy in Single-Top Mesonic Correlations

Abstract

Within the heavy-mass expansion, the pole width of a system containing one unstable heavy constituent inherits the constituent width up to $\mathcal{O}({\Lambda }_{\mathrm{kin}}^2/m_Q^2)$ corrections, while radial splittings remain $\mathcal{O}({\Lambda }_{\mathrm{rad}})$. The top quark is an extreme realization of this hierarchy. We implement the complex top pole mass in an instantaneous Bethe--Salpeter framework, where a biorthogonal Hellmann--Feynman relation realizes width inheritance at the operator level and an artificial heavy-mass scan confirms the predicted $m_Q^{-2}$ suppression. The low-pole source-projected response has a single broad maximum at the physical top width in the $t\bar{b}$, $t\bar{c}$, and $t\bar{u}$ channels. Full width-dependent non-Hermitian re-diagonalization and a direct full-matrix resolvent evaluation confirm the progressive dissolution of the small-width radial maxima. Thus stable-top eigenvalues survive as reference poles but not as a resolvable multi-peak spectrum; they may instead leave qualitative, process-dependent $Wb\bar{q}$ signatures, such as a broad threshold enhancement or modified color flow.