Different methods for including retardation in hadronic interactions

TL;DR

The study addresses retardation in hadronic $q\bar{q}$ interactions by formulating a coordinate-space, Liénard-Wiechert retarded potential and constructing a Hermitian quantum operator for bound-state calculations. It derives the retarded distance and invariant $V_{\text{ret}}(\mathbf{r},\mathbf{p})$, expands the operator into a nonlocal series, and tests matrix elements with a harmonic-oscillator basis, providing numerical insight into retardation effects. A key finding is that, after appropriate substitutions, the LW retarded amplitude yields results consistent with the tree-level covariant Feynman amplitude $\hat{W}_{\text{Fey}}$, up to a Jacobian factor, supporting physical equivalence between the two descriptions. The work clarifies the role of retardation in coordinate-space quark models and informs momentum-space approaches for heavy-quarkonia, offering a foundation for more refined relativistic bound-state calculations.

Abstract

A study of the retardation contributions to the hadronic quark interaction is performed in the coordinate space elaborating a classical electrodynamics procedure. The possibility of constructing a corresponding quantum operator is critically analyzed also performing some numerical matrix element calculations. A comparison of the model with the Feynman diagram calculation at tree level is studied, showing a substantial physical correspondence of the two models.