Eur. Phys. J. C (2026) 86: 589
https://doi.org/10.1140/epjc/s10052-026-15862-2

Regular Article - Theoretical Physics

Variationally improved structure of heavy-light mesons at $N^3$LO: stiffening effects and universal form factors in the V-scheme

Department of Physics, Tezpur University, Napaam, 784028, Tezpur, Assam, India

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Received: 13 April 2026
Accepted: 17 May 2026
Published online: 31 May 2026

Abstract

We present an investigation of the structural properties of heavy-light ($D,D_s,B,B_s$) and heavy-heavy ($B_c$) mesons by incorporating the V-scheme strong coupling constant ${\alpha }_V$ up to the third-loop order ($N^3$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 $R_{n_f}={\mathrm{\Lambda }}_{\overline{\text{MS}}}^{(n_f)}/\sqrt{b}$, matched to the non-perturbative Sommer scale $r_0$ from FLAG 2024 averages. By integrating these flavor-dependent $N^3$LO refinements ($b\approx 0.1951\pm 0.0369$ GeV$_{}^2$) 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 $N^3$LO results for root-mean-square (RMS) matter radii and charge radii exhibit remarkable agreement with established benchmarks: specifically, the $B_c$ 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.