We investigate the spectra of heavy quarkonia–charmonium (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$c\bar{c}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mover accent="true"> <mrow> <mi>c</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>) and bottomonium (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b\bar{b}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>) by solving the Klein–Gordon equation in D dimensions with a potential combining equal scalar–vector quadratic confinement and a modified screened Yukawa core. Using the Nikiforov–Uvarov method, we derive closed-form expressions for bound-state energies and wave functions, applicable in both relativistic and nonrelativistic regimes. We test the model against PDG-2024 data in two complementary ways: first, by fitting each sector independently (‘non-joint’) and second, by enforcing a common parameter set across both families (‘joint’). Analyses are performed for individual resonances and for spin-averaged center-of-gravity (COG) masses, which minimize hyperfine effects and reveal the underlying level structure. Non-joint fits reproduce each spectrum with typical absolute deviations of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}(0.1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mn>0.1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> GeV. The joint COG fit, based on 15 data points (8 for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$c\bar{c}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mover accent="true"> <mrow> <mi>c</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>, 7 for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$b\bar{b}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>), achieves <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\chi ^2_{\textrm{tot}}=1.1584$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>χ</mi> <mtext>tot</mtext> <mn>2</mn> </msubsup> <mo>=</mo> <mn>1.1584</mn> </mrow> </math></EquationSource> </InlineEquation> for 6 degree of freedom, corresponding to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\chi ^2/\textrm{pt}=0.077$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mtext>pt</mtext> <mo>=</mo> <mn>0.077</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\chi ^2/\textrm{dof}=0.193$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mtext>dof</mtext> <mo>=</mo> <mn>0.193</mn> </mrow> </math></EquationSource> </InlineEquation>. With this strategy (common parameters and COG inputs with sector-specific theory uncertainties), most residuals lie in the 0.02−0.23 GeV range. The largest occur in higher radial excitations: the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$b\bar{b}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation> 3S and 4S levels are overestimated (underbound) by <InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\sim }0.23~\textrm{GeV}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <mn>0.23</mn> <mspace width="3.33333pt" /> <mtext>GeV</mtext> </mrow> </math></EquationSource> </InlineEquation> and the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$c\bar{c}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mover accent="true"> <mrow> <mi>c</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation> 4S lies <InlineEquation ID="IEq12"> <EquationSource Format="TEX">${\sim }0.18$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <mn>0.18</mn> </mrow> </math></EquationSource> </InlineEquation> GeV high. The <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$b\bar{b}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation> 1P centroid is underestimated by <InlineEquation ID="IEq14"> <EquationSource Format="TEX">${\sim }0.15$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <mn>0.15</mn> </mrow> </math></EquationSource> </InlineEquation> GeV. These patterns suggest that physics beyond a common-parameter static potential, e.g., mild flavor dependence in screening/curvature or threshold-induced coupled-channel effects, becomes relevant at intermediate radii, especially for excited S waves. The model further predicts unobserved levels: <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$c\bar{c}(1F)\approx 4.124$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mover accent="true"> <mrow> <mi>c</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mn>1</mn> <mi>F</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>4.124</mn> </mrow> </math></EquationSource> </InlineEquation> GeV and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$b\bar{b}(1D,2D,1F)\approx (10.132,10.545,10.450)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mn>1</mn> <mi>D</mi> <mo>,</mo> <mn>2</mn> <mi>D</mi> <mo>,</mo> <mn>1</mn> <mi>F</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mo stretchy="false">(</mo> <mn>10.132</mn> <mo>,</mo> <mn>10.545</mn> <mo>,</mo> <mn>10.450</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> GeV, with estimated theoretical uncertainties of 0.10−0.20 GeV. Overall, compact static potentials capture the gross spectral structure across quark flavors while highlighting where dynamical refinements, such as flavor-sensitive screening and coupled-channel effects, will be most impactful for future theory and experiment.

Heavy meson spectra in quadratic plus modified Yukawa potential

期刊论文
发表日期：

2026年4月2日

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Sounia Abualkishik,
Ahmed Al-Jamel

摘要

We investigate the spectra of heavy quarkonia–charmonium ( $c\bar{c}$ $c \bar{c}$ ) and bottomonium ( $b\bar{b}$ $b \bar{b}$ ) by solving the Klein–Gordon equation in D dimensions with a potential combining equal scalar–vector quadratic confinement and a modified screened Yukawa core. Using the Nikiforov–Uvarov method, we derive closed-form expressions for bound-state energies and wave functions, applicable in both relativistic and nonrelativistic regimes. We test the model against PDG-2024 data in two complementary ways: first, by fitting each sector independently (‘non-joint’) and second, by enforcing a common parameter set across both families (‘joint’). Analyses are performed for individual resonances and for spin-averaged center-of-gravity (COG) masses, which minimize hyperfine effects and reveal the underlying level structure. Non-joint fits reproduce each spectrum with typical absolute deviations of $\mathcal {O}(0.1)$ $O (0.1)$ GeV. The joint COG fit, based on 15 data points (8 for $c\bar{c}$ $c \bar{c}$ , 7 for $b\bar{b}$ $b \bar{b}$ ), achieves $\chi ^2_{\textrm{tot}}=1.1584$ $χ_{tot}^{2} = 1.1584$ for 6 degree of freedom, corresponding to $\chi ^2/\textrm{pt}=0.077$ $χ^{2} / pt = 0.077$ and $\chi ^2/\textrm{dof}=0.193$ $χ^{2} / dof = 0.193$ . With this strategy (common parameters and COG inputs with sector-specific theory uncertainties), most residuals lie in the 0.02−0.23 GeV range. The largest occur in higher radial excitations: the $b\bar{b}$ $b \bar{b}$ 3S and 4S levels are overestimated (underbound) by ${\sim }0.23~\textrm{GeV}$ $\sim 0.23 GeV$ and the $c\bar{c}$ $c \bar{c}$ 4S lies ${\sim }0.18$ $\sim 0.18$ GeV high. The $b\bar{b}$ $b \bar{b}$ 1P centroid is underestimated by ${\sim }0.15$ $\sim 0.15$ GeV. These patterns suggest that physics beyond a common-parameter static potential, e.g., mild flavor dependence in screening/curvature or threshold-induced coupled-channel effects, becomes relevant at intermediate radii, especially for excited S waves. The model further predicts unobserved levels: $c\bar{c}(1F)\approx 4.124$ $c \bar{c} (1 F) \approx 4.124$ GeV and $b\bar{b}(1D,2D,1F)\approx (10.132,10.545,10.450)$ $b \bar{b} (1 D, 2 D, 1 F) \approx (10.132, 10.545, 10.450)$ GeV, with estimated theoretical uncertainties of 0.10−0.20 GeV. Overall, compact static potentials capture the gross spectral structure across quark flavors while highlighting where dynamical refinements, such as flavor-sensitive screening and coupled-channel effects, will be most impactful for future theory and experiment.