Abstract We present an investigation of the structural properties of heavy-light (D, Dₛ, B, Bₛ D, D s, B, B s) and heavy-heavy (Bc B c) mesons by incorporating the V -scheme strong coupling constant V α V up to the third-loop order (N³ N 3 LO). The inclusion of three-loop radiative corrections is critical to achieving perturbative convergence and effectively canceling renormalon-induced instabilities. A central feature of this work is the rigorous calibration of the string tension b using the universal scaling ratio R₍₅ = ₌ₒ^ (nf) /b R n f = Λ MS ¯ (n f) / b, matched to the non-perturbative Sommer scale r₀ r 0 from FLAG 2024 averages. By integrating these flavor-dependent N³ N 3 LO refinements (b 0. 1951 0. 0369 b ≈ 0. 1951 ± 0. 0369 GeV ² 2) into a variational Cornell potential framework, we demonstrate a systematic“stiffening”of the internal structure and a physical spatial contraction across successive loop orders. Our N³ N 3 LO results for root-mean-square (RMS) matter radii and charge radii exhibit remarkable agreement with established benchmarks: specifically, the Bc B c charge radius (0. 191 fm) and matter radius (0. 2967 fm) align closely with light-front predictions and relativistic quark model studies. Evaluation of the Isgur–Wise function reveals that while higher-order corrections primarily reshape the light-quark spectator cloud, the heavy core localization remains dominated by the constituent mass scale.
Choudhury et al. (Sun,) studied this question.