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

Abstract We present an investigation of the structural properties of heavy-light ( $$D, D_s, B, B_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> </mml:math> ) and heavy-heavy ( $$B_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> ) mesons by incorporating the V -scheme strong coupling constant $$\alpha _V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>V</mml:mi> </mml:msub> </mml:math> up to the third-loop order ( $$N^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> 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 $$R_{n_f} = \Lambda _{\overline{\text {MS}}}^{(n_f)}/\sqrt{b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mover> <mml:mtext>MS</mml:mtext> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>b</mml:mi> </mml:msqrt> </mml:mrow> </mml:math> , matched to the non-perturbative Sommer scale $$r_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> from FLAG 2024 averages. By integrating these flavor-dependent $$N^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO refinements ( $$b \approx 0.1951 \pm 0.0369$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>0.1951</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.0369</mml:mn> </mml:mrow> </mml:math> GeV $$^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mn>2</mml:mn> </mml:mmultiscripts> </mml:math> ) 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 $$N^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO results for root-mean-square (RMS) matter radii and charge radii exhibit remarkable agreement with established benchmarks: specifically, the $$B_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> 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.