Variationally improved structure of heavy-light mesons at N³LO: stiffening effects and universal form factors in the V-scheme

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 α 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 Rnf = Λ MS̄(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.