Confining two-body systems provide a useful setting for examining the internal consistency of perturbative treatments of bound-state wavefunctions. In this work, we investigate the applicability of Dalgarno–Lewis perturbation theory beyond first order within a nonrelativistic Cornell-type potential, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ V(r)=br-\tfrac{4\alpha _s}{3r}+c $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>b</mi> <mi>r</mi> <mo>-</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>α</mi> <mi>s</mi> </msub> </mrow> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </mfrac> </mstyle> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>, treating the linear confinement term as the parent Hamiltonian and incorporating the Coulombic interaction perturbatively so as to retain analytic control over the solutions. Going beyond standard first-order implementations, we carry out a systematic second-order Dalgarno–Lewis correction to the Airy-function ground state and analyze its impact on wavefunction moments. We show that while lower spatial moments are only mildly affected at first order, observables involving higher spatial moments are not fully stabilized at first order at the sub-percent level, indicating that first-order Dalgarno–Lewis theory does not by itself guarantee quantitative control of such quantities in confining two-body systems. Inclusion of second-order contributions is therefore necessary to ensure controlled perturbative accuracy and restore quantitative stability. As an explicit application, we evaluate effective overlap functions, their slopes and curvatures, as well as geometric and electromagnetic radii for representative heavy–light mesons (D, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$D_s$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>, B, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_s$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>). While the analysis is model dependent and is carried out within a nonrelativistic potential-model framework and does not rely on strict heavy-quark universality, the resulting magnitudes and systematic trends are compatible with those obtained in relativistic quark models, QCD sum-rule approaches, and continuum Dyson–Schwinger/Bethe–Salpeter studies. The present results provide semi-analytic benchmarks that clarify the role of higher-order wavefunction corrections in few-body bound-state problems.

Consistency of Dalgarno–Lewis perturbation theory for confining two-body systems: second-order corrections in the Cornell potential

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发表日期：

2026年4月5日

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Krishna Kingkar Pathak

摘要

Confining two-body systems provide a useful setting for examining the internal consistency of perturbative treatments of bound-state wavefunctions. In this work, we investigate the applicability of Dalgarno–Lewis perturbation theory beyond first order within a nonrelativistic Cornell-type potential, $ V(r)=br-\tfrac{4\alpha _s}{3r}+c $ $V (r) = b r - \frac{4 α_{s}}{3 r} + c$ , treating the linear confinement term as the parent Hamiltonian and incorporating the Coulombic interaction perturbatively so as to retain analytic control over the solutions. Going beyond standard first-order implementations, we carry out a systematic second-order Dalgarno–Lewis correction to the Airy-function ground state and analyze its impact on wavefunction moments. We show that while lower spatial moments are only mildly affected at first order, observables involving higher spatial moments are not fully stabilized at first order at the sub-percent level, indicating that first-order Dalgarno–Lewis theory does not by itself guarantee quantitative control of such quantities in confining two-body systems. Inclusion of second-order contributions is therefore necessary to ensure controlled perturbative accuracy and restore quantitative stability. As an explicit application, we evaluate effective overlap functions, their slopes and curvatures, as well as geometric and electromagnetic radii for representative heavy–light mesons (D, $D_s$ $D_{s}$ , B, and $B_s$ $B_{s}$ ). While the analysis is model dependent and is carried out within a nonrelativistic potential-model framework and does not rely on strict heavy-quark universality, the resulting magnitudes and systematic trends are compatible with those obtained in relativistic quark models, QCD sum-rule approaches, and continuum Dyson–Schwinger/Bethe–Salpeter studies. The present results provide semi-analytic benchmarks that clarify the role of higher-order wavefunction corrections in few-body bound-state problems.