The Charmed Meson Spectrum Using One-Loop Corrections to the One-Gluon Exchange Potential

Abstract

We investigate the charmed meson spectrum using a constituent quark model (CQM) with one-loop corrections applied to the one-gluon exchange (OGE) potential. The study aims to understand if the modified version of our CQM sufficiently account for the charmed meson spectrum observed experimentally, without invoking exotic quark and gluon configurations such as hybrid mesons or tetraquarks. Within this model, charmed mesons’ masses are computed, comparing theoretical predictions to experimental data. The results, within uncertainties, suggest that our theoretical framework generally reproduces mass splittings and level ordering observed for charmed mesons. Particularly, large discrepancies between theory and experiment found in P-wave states are, at least, significantly ameliorated by incorporating higher-order interaction terms. Therefore, the findings emphasize that while the traditional quark model is limited in fully describing charmed mesons, enhanced potential terms may bridge the gap with experimental observations. The study contributes a framework for predicting excited charmed meson states for future experimental validation.

1. Introduction

A simple analysis of the properties of mesons containing a single heavy quark, $Q=c$ or b, can be carried out in the limit of $m_Q\to \infty$. In such a regime, the heavy quark acts as a static color source for the rest of the heavy-light meson, i.e., its spin $s_Q$ is decoupled from the total angular momentum of the light antiquark, $j_q$, and they are separately conserved. As a result, heavy-light mesons are grouped into doublets, each associated with a specific value of $j_q$ and parity. The members of each doublet differ from the orientation of $s_Q$ with respect to $j_q$ and they are degenerate in the heavy quark symmetry (HQS) limit [1], whose mass degeneracy is broken at order $1/m_Q$.

For $Q\overline{q}$ states, and following HQS, one can write ${\overrightarrow{j}}_q={\overrightarrow{s}}_q+\overrightarrow{\mathsf{\ell }}$, where $s_q$ is the light antiquark spin and ℓ is its orbital angular momentum relative to the static heavy quark. Therefore, the lowest-lying $Q\overline{q}$ mesons correspond to $\mathsf{\ell }=0$ with $j_q^P={{\displaystyle \frac{1}{2}}}^{-}$. This doublet comprises two S-wave states with spin-parity $J^P=(0^{-},1^{-})$, where $\overrightarrow{J}={\overrightarrow{j}}_q+{\overrightarrow{s}}_Q$. For $\mathsf{\ell }=1$, it could be either $j_q^P={{\displaystyle \frac{1}{2}}}^{+}$ or $j_q^P={{\displaystyle \frac{3}{2}}}^{+}$, and thus the two corresponding doublets are $J^P=(0^{+},1^{+})$ and $J^P=(1^{+},2^{+})$, respectively. The mesons with $\mathsf{\ell }=2$ are collected either in the $j_q^P={{\displaystyle \frac{3}{2}}}^{-}$ doublet, consisting of states with $J^P=(1^{-},2^{-})$, or in the $j_q^P={{\displaystyle \frac{5}{2}}}^{-}$ with $J^P=(2^{-},3^{-})$; and so forth and so on.

If we now focus on the spectrum of charmed mesons, $\left(c\overline{n}\right)$-states with $n=u$ or d quark, it contains a number of long known and well-established states collected in the Review of Particle Physics (RPP) of Particle Data Group (PDG) [2]. We find the lowest-lying S-wave states with quantum numbers $J^P=0^{-}$ and $1^{-}$, denoted as D and $D^{\mathsf{\ast }}$ mesons. The P-wave ground states with spin-parity quantum numbers $0^{+}$ ($D_0^{\mathsf{\ast }}\left(2300\right)$), $1^{+}$ ($D_1\left(2420\right)$ and $D_1\left(2430\right)$) and $2^{+}$ ($D_2^{\mathsf{\ast }}\left(2460\right)$) are also given in Ref. [2]. In addition, the PDG lists as a well-established state a highly-excited charmed meson, with spin-parity $J^P=3^{-}$, denoted as $D_3^{\mathsf{\ast }}\left(2750\right)$. It was observed as a resonant substructure in the $B^0\to {\overline{D}}^0{\pi }^{+}{\pi }^{-}$ and $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ decays analyzed with the Dalitz plot technique [3,4].

Over the past 15 years or so, several new signals in the charmed meson sector have been observed. The signals now named $D_0\left(2550\right)$, $D_1^{\mathsf{\ast }}\left(2600\right)$, $D_2\left(2740\right)$, and $D_3^{\mathsf{\ast }}\left(2750\right)$ were observed for the first time by the BaBar collaboration in 2010 [5], and were confirmed by the LHCb experiment with slightly different masses in 2013 [6]. Furthermore, the LHCb collaboration reported in Ref. [6] two new higher D-meson excitations, $D_J^{\mathsf{\ast }}\left(3000\right)$ and $D_J\left(3000\right)$, with natural and unnatural parities, respectively, collectively named by the PDG as $D{\left(3000\right)}^0$ (Natural parity means that the bosonic meson field behaves under reflection as $+1$ for even spin and $-1$ for odd spin; note then that, for heavy-light mesons, the superindex “∗” is used for those having natural parity). In 2015, a new state $D_1^{\mathsf{\ast }}\left(2760\right)$, with spin-parity quantum numbers $J^P=1^{-}$, was observed by the LHCb collaboration in the $D^{+}{\pi }^{-}$ channel by analyzing the $B^{-}\to D^{+}{K}^{-}{\pi }^{-}$ decay [7]. Finally, there have been observed two more states which are not collected in the RPP of PDG. The first one is the named $D^{\mathsf{\ast }}{\left(2640\right)}^{\pm }$ seen in Z decays by Abreu et al. [8] but missing in the searches performed in Refs. [9,10], thus requiring confirmation. The second was observed in 2016 by the LHCb collaboration in the $D^{+}{\pi }^{-}$ channel when analyzing the $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ decay [4]; they assigned to this signal the name $D_2^{\mathsf{\ast }}\left(3000\right)$ with spin-parity $J^P=2^{+}$ because its resonance parameters were inconsistent with the previously observed $D{\left(3000\right)}^0$ [6].

Theoretical predictions of the spectrum of charmed mesons dates from the early days of phenomenological quark models [11,12,13,14]. In the last years, many studies have been carried out within different theoretical approaches such as lattice-regularized QCD [15,16,17,18,19,20], unitarized coupled-channels T-matrix analyses [21,22,23,24], heavy meson and chiral effective theories [25,26,27,28,29], Regge-based phenomenology [30,31], and phenomenological quark models [32,33,34,35,36,37,38,39,40,41,42,43]. This is mainly because of two reasons. The first one is the recent experimental measurements in the subject which provide sustained progress in the field as well as the breadth and depth necessary for a vibrant theoretical research environment. The second is related mostly to the fact that $D_0^{\mathsf{\ast }}\left(2300\right)$ and $D_1\left(2420\right)$ charmed mesons, which belong to the doublet $j_q^P={{\displaystyle \frac{1}{2}}}^{+}$ predicted by HQS, have surprisingly light masses, compared with naive quark model expectations, and are located below $D\pi$ and $D^{\mathsf{\ast }}\pi$ thresholds, respectively. This implies that these states are narrow. These facts have stimulated a fruitful line of research, suggesting that their structure is much richer than what one might guess assuming the $Q\overline{q}$ picture [44,45,46].

The quark model has been notably successful in describing the heavy quark-antiquark system since the early days of charmonium studies (see, for example, Refs. [47,48,49,50,51,52,53,54]). Moreover, predictions from this framework on the properties of heavy quarkonia, including those related to decays and interactions, have proven highly valuable for guiding experimental searches. Additionally, the quark model’s adaptability makes it well-suited for exploratory research on exotic matter. In this work, we present a study on the role of one-loop corrections to the one-gluon exchange potential in the description of the spectrum of open charmed mesons within a constituent quark model (CQM). The topic is relevant as there are still significant discrepancies between theoretical predictions and experimental results regarding the masses of multiple charmed mesons. This approach demonstrates that refinements to the conventional quark model can explain much of the observed spectrum without invoking exotic states.

The theoretical results presented here are based on a constituent quark model (CQM), initially proposed in Ref. [55], and recently applied to conventional mesons containing heavy quarks, capturing a broad range of physical observables related to spectra [56,57], strong decays [58,59], hadronic transitions [60,61], and both electromagnetic and weak reactions [62,63]. To improve the accuracy of mass splittings, we adopt the approach in Ref. [64] and incorporate one-loop corrections to the One-Gluon Exchange (OGE) potential as derived by Gupta and Radford [65]. These corrections include, for the first time in the perturbative series, a spin-dependent term that affects only mesons composed of quarks of different flavors. Our main objective is to assess whether the experimentally observed charmed meson spectrum can be fully described within the quark-antiquark model, without invoking exotic configurations.

The manuscript is organized as follows. After this introduction, the theoretical framework is briefly presented in Section 2. Section 3 is mainly devoted to the analysis and discussion of our results. Finally, we summarize and draw some conclusions in Section 4.

2. Theoretical Formalism

The dynamical braking of chiral symmetry in Quantum Chromodynamics (QCD) is responsible, among other phenomena, for generating constituent quark masses and Goldstone-boson exchanges between light quarks. This together with the one-gluon exchange interaction and the color confining force consist on the main pieces of our constituent quark model [55,66].

Under chiral transformations, the following Lagrangian

$$\mathcal{L}=\overline{\psi }(/\partial -M\left(q^2\right)U^{{\gamma }_5})\psi ,$$

(1)

is invariant [67]. In Equation (1), $M\left(q^2\right)$ is the dynamical momentum-dependent constituent quark mass and $U^{{\gamma }_5}=e^{i{\lambda }_a{\varphi }^a{\gamma }_5/f_{\pi }}$, with $\varphi =\{\overrightarrow{\pi },K,{\eta }_8\}$, is the matrix of Goldstone-boson fields that can be expanded as

$$U^{{\gamma }_5}=1+{\displaystyle \frac{i}{f_{\pi }}}{\gamma }^5{\lambda }^a{\varphi }^a-{\displaystyle \frac{1}{2f_{\pi }^2}}{\varphi }^a{\varphi }^a+\dots$$

(2)

One can guess that the first term of the expansion provides the constituent quark mass, the second gives rise to one-boson exchange interactions between light quarks, and the main contribution of the third term comes from the two-pion exchange which is simulated in our case by means of a scalar-meson exchange interaction. In the presence of heavy quarks, chiral symmetry is explicitly broken and Goldstone-boson exchanges do not appear. However, it constrains the model parameters through the light-meson phenomenology [68].

At energy scales higher than that of dynamical breaking of chiral symmetry, the CQM incorporates QCD perturbative effects by taking into account one-gluon fluctuations around the instanton vacuum through the vertex Lagrangian

$${\mathcal{L}}_{qqg}=i\sqrt{4\pi {\alpha }_s}\overline{\psi }{\gamma }_{\mu }{G}_c^{\mu }{\lambda }^c\psi ,$$

(3)

with ${\lambda }^c$ being the $SU\left(3\right)$ color matrices and $G_c^{\mu }$ the gluon field. The ${\alpha }_s$ is a scale-dependent effective strong coupling constant that allows a comprehensive description of light, strange and heavy meson spectra [55,66]:

$${\alpha }_s\left({\mu }_{ij}\right)={\displaystyle \frac{{\alpha }_0}{ln\left(\frac{{\mu }_{ij}^2+{\mu }_0^2}{{\Lambda }_0^2}\right)}},$$

(4)

in which ${\mu }_{ij}$ is the reduced mass of the meson’s constituent $q\overline{q}$ pair and ${\alpha }_0$, ${\mu }_0$ and ${\Lambda }_0$ are parameters of the quark model.

The potential derived from Equation (3) contains central, tensor, and spin-orbit contributions given by

$$\begin{array}{cc}\hfill V_{\mathrm{OGE}}^{\mathrm{C}}\left({\overrightarrow{r}}_{ij}\right)=& {\displaystyle \frac{1}{4}}{\alpha }_s({\overrightarrow{\lambda }}_i^c·{\overrightarrow{\lambda }}_j^c)\left[{\displaystyle \frac{1}{r_{ij}}}-{\displaystyle \frac{1}{6m_i{m}_j}}({\overrightarrow{\sigma }}_i·{\overrightarrow{\sigma }}_j){\displaystyle \frac{e^{-r_{ij}/r_0\left({\mu }_{ij}\right)}}{r_{ij}{r}_0^2\left({\mu }_{ij}\right)}}\right],\hfill \\ \hfill V_{\mathrm{OGE}}^{\mathrm{T}}\left({\overrightarrow{r}}_{ij}\right)=& -{\displaystyle \frac{1}{16}}{\displaystyle \frac{{\alpha }_s}{m_i{m}_j}}({\overrightarrow{\lambda }}_i^c·{\overrightarrow{\lambda }}_j^c)\left[{\displaystyle \frac{1}{r_{ij}^3}}-{\displaystyle \frac{e^{-r_{ij}/r_g\left({\mu }_{ij}\right)}}{r_{ij}}}\left({\displaystyle \frac{1}{r_{ij}^2}}+{\displaystyle \frac{1}{3r_g^2\left({\mu }_{ij}\right)}}+{\displaystyle \frac{1}{r_{ij}{r}_g\left({\mu }_{ij}\right)}}\right)\right]S_{ij},\hfill \\ \hfill V_{\mathrm{OGE}}^{\mathrm{SO}}\left({\overrightarrow{r}}_{ij}\right)=& -{\displaystyle \frac{1}{16}}{\displaystyle \frac{{\alpha }_s}{m_i^2{m}_j^2}}({\overrightarrow{\lambda }}_i^c·{\overrightarrow{\lambda }}_j^c)\left[{\displaystyle \frac{1}{r_{ij}^3}}-{\displaystyle \frac{e^{-r_{ij}/r_g\left({\mu }_{ij}\right)}}{r_{ij}^3}}\left(1+{\displaystyle \frac{r_{ij}}{r_g\left({\mu }_{ij}\right)}}\right)\right]\times \hfill \\ \hfill & \times \left[({(m_i+m_j)}^2+2m_i{m}_j)({\overrightarrow{S}}_{+}·\overrightarrow{L})+(m_j^2-m_i^2)({\overrightarrow{S}}_{-}·\overrightarrow{L})\right],\hfill \end{array}$$

(5)

where $S_{ij}=3({\overrightarrow{\sigma }}_i·{\widehat{r}}_{ij})({\overrightarrow{\sigma }}_j·{\widehat{r}}_{ij})-{\overrightarrow{\sigma }}_i·{\overrightarrow{\sigma }}_j$ is the quark tensor operator and ${\overrightarrow{S}}_{\pm }={\displaystyle \frac{1}{2}}({\overrightarrow{\sigma }}_i\pm {\overrightarrow{\sigma }}_j)$ are the symmetric and antisymmetric spin-orbit operators, respectively. Besides, $r_0\left({\mu }_{ij}\right)={\widehat{r}}_0{\displaystyle \frac{{\mu }_{nn}}{{\mu }_{ij}}}$ and $r_g\left({\mu }_{ij}\right)={\widehat{r}}_g{\displaystyle \frac{{\mu }_{nn}}{{\mu }_{ij}}}$ are regulators that depend on ${\mu }_{ij}$, which is again the reduced mass of the meson’s constituent $q\overline{q}$ pair. The contact term of the central potential has been regularized as $\delta \left({\overrightarrow{r}}_{ij}\right)\approx (1/4\pi r_0^2)·e^{-r_{ij}/r_0}/r_{ij}$.

To improve the description of charmed mesons, we follow the proposal of Ref. [64] and include one-loop corrections to the OGE potential as derived by Gupta et al. [65]. As in the case of $V_{\mathrm{OGE}}$, $V_{\mathrm{OGE}}^{1-\mathrm{loop}}$ contains central, tensor and spin-orbit contributions, given by

$$\begin{array}{c}{V}_{\mathrm{OGE}}^{1-\mathrm{loop},\mathrm{C}}\left({\overrightarrow{r}}_{ij}\right)=0,\hfill \\ \begin{array}{cc}\hfill V_{\mathrm{OGE}}^{1-\mathrm{loop},\mathrm{T}}\left({\overrightarrow{r}}_{ij}\right)={\displaystyle \frac{C_F}{4\pi }}{\displaystyle \frac{{\alpha }_s^2}{m_i{m}_j}}{\displaystyle \frac{1}{r^3}}{S}_{ij}& \left[{\displaystyle \frac{b_0}{2}}\left(ln\left(\mu r_{ij}\right)+{\gamma }_E-{\displaystyle \frac{4}{3}}\right)+{\displaystyle \frac{5}{12}}{b}_0-{\displaystyle \frac{2}{3}}{C}_A\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{2}}\left(C_A+2C_F-2C_A\left(ln\left(\sqrt{m_i{m}_j}{r}_{ij}\right)+{\gamma }_E-{\displaystyle \frac{4}{3}}\right)\right)\right],\hfill \end{array}\hfill \\ \begin{array}{c}{V}_{\mathrm{OGE}}^{1-\mathrm{loop},\mathrm{SO}}\left({\overrightarrow{r}}_{ij}\right)={\displaystyle \frac{C_F}{4\pi }}{\displaystyle \frac{{\alpha }_s^2}{m_i^2{m}_j^2}}{\displaystyle \frac{1}{r^3}}\times \hfill \\ \begin{array}{cc}\hfill \times \left\{\right({\overrightarrow{S}}_{+}·\overrightarrow{L})& [\left({(m_i+m_j)}^2+2m_i{m}_j\right)\left(C_F+C_A-C_A\left(ln\left(\sqrt{m_i{m}_j}{r}_{ij}\right)+{\gamma }_E\right)\right)\hfill \\ \hfill & +4m_i{m}_j\left({\displaystyle \frac{b_0}{2}}\left(ln\left(\mu r_{ij}\right)+{\gamma }_E\right)-{\displaystyle \frac{1}{12}}{b}_0-{\displaystyle \frac{1}{2}}{C}_F-{\displaystyle \frac{7}{6}}{C}_A+{\displaystyle \frac{C_A}{2}}\left(ln\left(\sqrt{m_i{m}_j}{r}_{ij}\right)+{\gamma }_E\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}(m_j^2-m_i^2)C_{A}ln\left({\displaystyle \frac{m_j}{m_i}}\right)]\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill +({\overrightarrow{S}}_{-}·\overrightarrow{L})& [(m_j^2-m_i^2)\left(C_F+C_A-C_A\left(ln\left(\sqrt{m_i{m}_j}{r}_{ij}\right)+{\gamma }_E\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{(m_i+m_j)}^2{C}_{A}ln\left({\displaystyle \frac{m_j}{m_i}}\right)\left]\right\},\hfill \end{array}\hfill \end{array}\hfill \end{array}$$

(6)

where $C_F=4/3$, $C_A=3$, $b_0=9$, ${\gamma }_E=0.5772$, and the scale $\mu \approx 1\mathrm{GeV}$.

Finally, an important non-perturbative term of our CQM is the color confining interaction between quarks and antiquarks to ensure colorless hadrons. The potential used here is linearly-rising for short interquark distances, but acquires a plateau at large distances to mimic the effect of sea quarks, which induces the breakdown of the color binding string [69]. Its explicit expression is

$$\begin{array}{cc}\hfill V_{\mathrm{CON}}^{\mathrm{C}}\left({\overrightarrow{r}}_{ij}\right)=& \left[-a_c(1-e^{-{\mu }_c{r}_{ij}})+\Delta \right]({\overrightarrow{\lambda }}_i^c·{\overrightarrow{\lambda }}_j^c),\hfill \\ \hfill V_{\mathrm{CON}}^{\mathrm{SO}}\left({\overrightarrow{r}}_{ij}\right)=& -\left({\overrightarrow{\lambda }}_i^c·{\overrightarrow{\lambda }}_j^c\right){\displaystyle \frac{a_c{\mu }_c{e}^{-{\mu }_c{r}_{ij}}}{4m_i^2{m}_j^2{r}_{ij}}}\left[((m_i^2+m_j^2)(1-2a_s)+4m_i{m}_j(1-a_s))({\overrightarrow{S}}_{+}·\overrightarrow{L})\right.\hfill \\ \hfill & \left.+(m_j^2-m_i^2)(1-2a_s)({\overrightarrow{S}}_{-}·\overrightarrow{L})\right],\hfill \end{array}$$

(7)

where the model parameters are $a_c$, $\Delta$, ${\mu }_c$ and $a_s$, the last one being the mixture between scalar and vector Lorentz structures of the confinement.

Among the different methods to solve the Schrödinger equation in order to find the quark-antiquark bound states, we use the Gaussian Expansion Method (GEM) [70], which is a variational approach widely used in hadronic and nuclear. The central idea is to expand the radial part of the wave function in terms of Gaussian basis functions with ranges chosen in a geometric progression. This choice ensures that both the short- and long-distance behavior of the system are efficiently captured with a relatively small number of basis functions. Compared with other basis sets, Gaussians are computationally convenient: matrix elements of the kinetic energy, confinement potentials, and one-gluon exchange terms can be evaluated analytically, which greatly reduces numerical cost. The geometric progression of ranges avoids redundancy in the basis and provides flexibility to describe different spatial scales, from tightly bound states to more diffuse excitations. In practice, GEM has been successfully applied to meson and baryon spectroscopy, multiquark systems, and light nuclei. Its accuracy and efficiency make it a standard tool for constituent quark model studies, especially when exploring spectra and structural properties of systems where exact solutions are not tractable.

Therefore, as explained above, the radial wave function solution of the Schrödinger equation is written as an expansion in terms of Gaussian basis functions

$$R_{\alpha }\left(r\right)=\sum _{n=1}^{n_{max}}{c}_n^{\alpha }{\varphi }_{nl}^G\left(r\right),$$

(8)

where $\alpha$ refers to the channel quantum numbers and the coefficients $c_n^{\alpha }$, and eigenvalue E, are determined from

$$\sum _{n=1}^{n_{max}}\left[\left(T_{n^{\prime }n}^{\alpha }-E{N}_{n^{\prime }n}^{\alpha }\right)c_n^{\alpha }+\sum _{{\alpha }^{\prime }}{V}_{n^{\prime }n}^{\alpha {\alpha }^{\prime }}{c}_n^{{\alpha }^{\prime }}=0\right],$$

(9)

where $T_{n^{\prime }n}^{\alpha }$, $N_{n^{\prime }n}^{\alpha }$ and $V_{n^{\prime }n}^{\alpha {\alpha }^{\prime }}$ are the matrix elements of the non-relativistic kinetic energy, the normalization, and the potential, respectively. The matrices $T_{n^{\prime }n}^{\alpha }$ and $N_{n^{\prime }n}^{\alpha }$ are diagonal whereas the mixing between different channels is given by $V_{n^{\prime }n}^{\alpha {\alpha }^{\prime }}$.

Estimates of the average momentum, $\langle p\rangle$, of a light constituent quark with mass M within a meson generally indicate that $\langle p\rangle \sim M$. This observation implies that theoretical analyses of bound states involving light quarks should, at some level, incorporate relativistic dynamics (see, for instance, Ref. [71] where it is found that relativistic effects and the proper choice of the Lorentz structure of the quark-antiquark interaction in a meson are crucial for bringing theoretical predictions into accord with experimental data). Therefore, the possible shortcomings of the non-relativistic quark model in this regard have been discussed for many years [72,73,74]. Collectively, a consistent conclusion may have been reached: despite its intrinsic limitations, the non-relativistic quark model has proven to be highly effective, providing a coherent framework that successfully accounts for a wide variety of observables. This observation motivates the use of a non-relativistic framework in the present work. The argument is pragmatic: the non-relativistic quark model is parameterized, with its parameters fixed by data, ensuring numerical accuracy within the fitted domain. Incorporating relativistic effects simply introduces additional parameters, likewise constrained by experiment, so the resulting predictions do not differ markedly from those of the original formulation. Although the fitted values change, the potential is not an observable; hence, no physical content is altered.

3. Results

Model parameters relevant for this analysis are shown in

Table 1. Constituent quark model parameters.

		Original Set	Fine-Tuned Set
Quark masses	$m_n$ (MeV)	313	313
	$m_c$ (MeV)	1763	1763
OGE	${\alpha }_0$	$2.118$	$2.118$
	${\Lambda }_0$$\left({\mathrm{fm}}^{-1}\right)$	$0.113$	$0.113$
	${\mu }_0$ (MeV)	$36.976$	$36.976$
	${\widehat{r}}_0$ (fm)	$0.181$	$0.181$
	${\widehat{r}}_g$ (fm)	$0.259$	$0.259$
Confinement	$a_c$ (MeV)	$507.4$	$478.0$
	${\mu }_c$$\left({\mathrm{fm}}^{-1}\right)$	$0.576$	$0.551$
	$\Delta$ (MeV)	$184.432$	$178.019$
	$a_s$	$0.81$	$0.81$

. As stated in the Introduction, our main objective is to assess whether the full spectrum of experimentally observed charmed mesons can be roughly explained within the quark-antiquark model, without requiring more exotic configurations. All model parameters were constrained based on prior investigations of hadron phenomenology (see, for instance, Refs. [55,66,75]). However, for the lightest heavy-light meson sector, we must acknowledge that these are not the most suitable, and we have therefore taken the liberty of slightly modifying those that influence the slope of the linear confinement potential. This is why we show a column of finely tuned parameters in Table 1.

Two primary sources of theoretical uncertainty affect our results: numerical and parametric. The numerical uncertainty, arising from the computational algorithm, is negligible, as convergence tests confirm that discretization errors, iteration tolerances, and numerical precision play no significant role within the accuracy required for our analysis. The dominant contribution stems from the determination of model parameters, which are tuned to reproduce a selected set of hadron observables within a defined level of agreement with experiment. Such parameters typically include effective couplings, cutoff scales, and quark masses, all of which encode physics not explicitly resolved by the model. Their extraction is subject to both statistical limitations of the data sets used and systematic ambiguities in the fitting strategy, such as the choice of observables, the weight assigned to each in the fit, and the interplay between correlated quantities. As a result, assigning explicit uncertainties to these parameters, and thus to derived quantities, is not straightforward, since different but equally reasonable parameter sets may yield comparably good fits while producing variations in predictions outside the fitted region. This issue is particularly pronounced in phenomenological hadronic models, where limited experimental constraints and the effective nature of the theory prevent a direct, model-independent determination of parameters. For this reason, throughout this work we adopt a conservative estimate of theoretical uncertainty of approximately 10–20% for the binding mass of a given meson, reflecting both the spread observed when alternative parameter choices are considered and the typical level of agreement achieved between effective models and experimental spectroscopy.

In

Table 2. Charmed meson masses, in MeV, from constituent quark model (CQM) and experiment [2,6,8]. We show, for CQM’s energy levels, the quark-antiquark value taking into account the one-gluon exchange potential $\mathcal{O}\left({\alpha }_s\right)$ and including its one-loop correction $\mathcal{O}\left({\alpha }_s^2\right)$. For experiment, we distinguish between well-established states ([2]) and those levels that still need confirmation and so have been omitted from the summary table ([2]*). Bottom part of the table indicates two additional states distinguished experimentally whose $J^P$ quantum numbers are not known and thus $J^P={?}^{?}$.

Meson	${\mathit{J}}^{\mathit{P}}$	n	The. $\mathcal{O}\left({\mathit{\alpha }}_{\mathit{s}}\right)$	The. $\mathcal{O}\left({\mathit{\alpha }}_{\mathit{s}}^2\right)$	Exp.	Ref.
D	$0^{-}$	1	1868	1868	$1867.95\pm 0.27$	[2]
		2	2619	2619	$2549\pm 19$	[2]*
		3	3053	3053
$D_0^{\mathsf{\ast }}$	$0^{+}$	1	2445	2281	$2343\pm 10$	[2]
		2	2934	2820
		3	3252	3172
$D^{\mathsf{\ast }}$	$1^{-}$	1	1982	1977	$2009.12\pm 0.04$	[2]
		2	2677	2675	$2627\pm 10$	[2]*
		3	2841	2810	$2781\pm 18\pm 13$	[2]*
$D_1$	$1^{+}$	1	2410	2438	$2422.1\pm 0.8$	[2]
		2	2519	2461	$2412\pm 9$	[2]
		3	2918	2940
$D_2$	$2^{-}$	1	2736	2744	$2747\pm 6$	[2]*
		2	2876	2865
		3	3125	3132
$D_2^{\mathsf{\ast }}$	$2^{+}$	1	2452	2483	$2461.1\pm 0.7$	[2]
		2	2944	2966
		3	3108	3095
$D_3^{\mathsf{\ast }}$	$3^{-}$	1	2767	2783	$2763.1\pm 3.2$	[2]
		2	3145	3157
		3	3309	3303
$D_3$	$3^{+}$	1	2995	2998
		2	3130	3127
		3	3296	3299
$D^{\mathsf{\ast }}\left(2640\right)$	${?}^{?}$				$2637\pm 2\pm 6$	[8]
$D\left(3000\right)$	${?}^{?}$				$3214\pm 29\pm 49$	[6]

, we show the charmed meson masses, in MeV, from constituent quark model (CQM) and experiment [2,6,8]. We show, for CQM’s energy levels, the quark-antiquark value taking into account the one-gluon exchange potential $\mathcal{O}\left({\alpha }_s\right)$ and including its one-loop correction $\mathcal{O}\left({\alpha }_s^2\right)$. For experiment, we distinguish between well-established states ([2]) and those levels which still need confirmation and so have been omitted from the summary table ([2]*).

Two charmed mesons with quantum numbers $J^P=0^{-}$ have been experimentally observed, D and $D_0\left(2550\right)$. The first one is the ground level of charmed mesons and it is well-established in the RPP of PDG [2]. The second is still omitted from the summary table because even though two experiments observed this state, its mass is different. Our theoretical prediction is slightly higher than the average mass reported by the RPP of PDG [2]; note that the experimental masses measured until now go from 2518 to 2580 MeV for this state. Another important feature to highlight is that $\mathcal{O}\left({\alpha }_s^2\right)$ OGE corrections are zero in this $J^P$-channel and thus our naïve quark model must predict correctly these two states from the global fit of hadron phenomenology.

The partner of the D-meson, which belongs to the $j_q^P={{\displaystyle \frac{1}{2}}}^{-}$ doublet in heavy quark spin symmetry, is the $D^{\mathsf{\ast }}$ meson. As one can see in Table 2, there are three candidates: $D^{\mathsf{\ast }}$, $D_1^{\mathsf{\ast }}\left(2600\right)$, and $D_1^{\mathsf{\ast }}\left(2760\right)$; the first one is well-established in PDG the other two are omitted from the summary table since they need confirmation. The 1-loop OGE corrections are small to moderate in this channel, producing mass shifts from 2 to 30 MeV. One may state that our results for $J^P=1^{-}$ channel agrees reasonably well with the experimental masses reported until now.

There are four P-wave states measured experimentally and denoted in the RPP of PDG as $D_0^{\mathsf{\ast }}\left(2300\right)$, $D_1\left(2420\right)$, $D_1\left(2430\right)$, $D_2^{\mathsf{\ast }}\left(2460\right)$. As one can see in Table 2, our theoretical results reproduce correctly the level ordering and they are also in global agreement with the experimental reported masses once the one-loop OGE corrections are incorporated. The addition of the $\mathcal{O}\left({\alpha }_s^2\right)$ OGE corrections was proposed by Lakhina et al. in [64] motivated by the fact that in the one-loop computation there is a spin-dependent term which affects only to mesons with different flavor quarks and it is not negligible for P-wave states where theory and experiment find their most significant differences. We demonstrate herein that naïve quark models cannot reproduce the P-wave charmed meson spectrum but, instead of resorting first to more complicated solutions such as exotic hadron structures, one should investigate the possibility of having missed potential terms that may be relevant for this sector.

The RPP of PDG reports two more charmed mesons with well-established spin-parity quantum numbers, the $D_2\left(2740\right)$ and $D_3^{\mathsf{\ast }}\left(2750\right)$ mesons. The first one is omitted from the summary table whereas the second is a well-established charmed meson. Theoretically, both are dominant D-wave states whose masses are close to the experimental measurements; therefore, we may confirm that the experimental assignment is plausible. When incorporating the $\mathcal{O}\left({\alpha }_s^2\right)$ OGE corrections, the theoretical masses of these states grow moderately but the change is not dramatic.

We focus now on the two states whose quantum numbers have not been assigned (see the bottom part of Table 2). The $D^{\mathsf{\ast }}{\left(2640\right)}^{\pm }$ seems to have a mass similar to the expected one for the first excitation of the $D^{\mathsf{\ast }}$ meson. In fact, there is no other possible case attending to the mass only. The $D{\left(3000\right)}^0$, whose mass is actually $(3214\pm 29\pm 49)\mathrm{MeV}$, could be fitted as the first excitation of either $D_3^{\mathsf{\ast }}$ or $D_3$, but could be also assigned as the second excitation of either $D_2$ or $D_2^{\mathsf{\ast }}$ mesons.

The naïve constituent quark model is able to globally reproduce the spectrum of charmed mesons. In particular, there are higher-order terms of the gluon exchange potential that seem to be very significant in those channels of charmed mesons where there is a larger discrepancy between theory and experiment. As can be guessed from the discussion so far, and seen in Figure 1, when the next-to-leading (NLO) order term of the OGE interaction is included in the model, the spectrum of charmed mesons is described reasonably well. The numerical value of the scale-dependent effective strong coupling constant, ${\alpha }_s$, is $0.43$ in the D meson sector. This implies that $\mathcal{O}\left({\alpha }_s^2\right)$ one-gluon-exchange (OGE) potentials provide a mass correction of approximately $20\%$. The next-to-next-to-leading order (NNLO) corrections may introduce an additional $8\%$ variation in mass, which remains well within the theoretical uncertainties of the quark model and thus going beyond the scope of this manuscript.

The discussion in the paragraph above does not mean that higher Fock-state components or more complex structures such as tetraquark or meson-meson degrees of freedom in the meson’s wave function cannot play a role but, before resorting to them, one should explore simpler refinements. Some exotic hadrons, such as $T_{cc}$, $T_{cs\left(\overline{s}\right)}$, and $Z_c$, possess quantum numbers that unequivocally rule out a simple quark-antiquark interpretation, necessitating alternative descriptions such as multiquark configurations. These states do not couple to conventional mesons, implying that their dynamics must be understood beyond the naïve quark model, possibly as tetraquarks whose structure—whether molecular or compact—follows the same fundamental quark interactions [76,77,78].

Note that most of the original model parameters were constrained in the charmonium and bottomonium sectors and, as shown in Equation (6), there are one-loop OGE terms that impact mesons made by an equally-flavor quark-antiquark pair. However, as one can see in Equation (6), the next-to-leading order corrections to the OGE potential are only tensor and spin-orbit terms, which are $1/\left(m_i{m}_j\right)$ suppressed contributions, meaning that the 1-loop corrections must be a factor 6 smaller in the charmonium sector and a factor 50 in the bottomonium one. Moreover, as one can see in Table 2 and Figure 1, the one-loop corrections to the original OGE potential produces mass shifts which are relatively small for most D-meson states. The largest effect is experienced by the $J^P=0^{+}$, $1^{+}$, and $2^{+}$D mesons, being notable for scalar states and just moderate for pseudo-vector and tensor states. Besides, in the charmonium sector, the operators $S_{ij}$ and ${\overrightarrow{S}}_{+}·\overrightarrow{L}$ are active whereas the ${\overrightarrow{S}}_{-}·\overrightarrow{L}$ operator is not, eliminating this additional effect. Therefore, the impact of the 1-loop OGE corrections in the charmonium sector is quite small, within the theoretical uncertainty of the CQM.

Table 3. Charmed meson masses, in MeV, computed within our modified constituent quark model framework, compared with those predicted by Lattice-QCD [16,19], unitary coupled-channels model [22], quark-model based coupled channels calculation, and experiment [2,6,8]. For experiment, we distinguish between well-established states ([2]) and those levels which still need confirmation and so have been omitted from the summary table ([2]*).

Meson	${\mathit{J}}^{\mathit{P}}$	n	Herein	Lattice-QCD	Unitary-CCs	QM-CCs	Exp.	Ref.
D	$0^{-}$	1	1868	$1868\left(11\right)$	-	1865	$1867.95\pm 0.27$	[2]
		2	2619	$2624\left(20\right)$	-	2551	$2549\pm 19$	[2]*
		3	3053	$3055\left(28\right)$	-	-
$D_0^{\mathsf{\ast }}$	$0^{+}$	1	2281	$2256\left(18\right)$	$\left[\begin{array}{c}{2105}_{-8}^{+6}\\ {2451}_{-26}^{+35}\end{array}\right]$	2261	$2343\pm 10$	[2]
		2	2820	$2980\left(27\right)$	-	2816
		3	3172	$3360\left(28\right)$	-	-
$D^{\mathsf{\ast }}$	$1^{-}$	1	1977	$1995\left(15\right)$	-	2007	$2009.12\pm 0.04$	[2]
		2	2675	$2719\left(24\right)$	-	2658	$2627\pm 10$	[2]*
		3	2810	$2801\left(23\right)$	-	2759	$2781\pm 18\pm 13$	[2]*
$D_1$	$1^{+}$	1	2438	$2367\left(20\right)$	$\left[\begin{array}{c}{2247}_{-6}^{+5}\\ {2555}_{-30}^{+47}\end{array}\right]$	2378	$2422.1\pm 0.8$	[2]
		2	2461	$2470\left(17\right)$	-	2454	$2412\pm 9$	[2]
		3	2940	$3045\left(29\right)$	-	3131
$D_2$	$2^{-}$	1	2744	$2838\left(22\right)$	-	2732	$2747\pm 6$	[2]*
		2	2865	$2915\left(22\right)$	-	2803
		3	3132	$3398\left(35\right)$	-	-
$D_2^{\mathsf{\ast }}$	$2^{+}$	1	2483	$2506\left(19\right)$	-	2468	$2461.1\pm 0.7$	[2]
		2	2966	$3109\left(28\right)$	-	3230
		3	3095	$3151\left(31\right)$	-	-
$D_3^{\mathsf{\ast }}$	$3^{-}$	1	2783	$2927\left(22\right)$	-	2756	$2763.1\pm 3.2$	[2]
		2	3157	$3518\left(28\right)$	-	-
		3	3303	-	-	-
$D_3$	$3^{+}$	1	2998	$3210\left(23\right)$	-	-
		2	3127	$3229\left(23\right)$	-	-
		3	3299	-	-	-

presents a comparison between our theoretical results and those obtained from Lattice QCD [16,19], unitarized coupled-channel chiral Lagrangian approaches [22,24] and quark-model based coupled-channels calculation [79]. Lattice QCD yields results that are broadly consistent with our refined CQM for low-energy bound states, in particular the ground and first excited states with $J^P=0^{-}$ and $1^{-}$. For quantum numbers $0^{+}$, $1^{+}$, and $2^{\pm }$, the agreement remains satisfactory for the lowest bound states but gradually deteriorates for higher radial excitations. The most pronounced discrepancies are observed in the case of the highest spin states. This behavior may be attributed to the confining interaction in our model, which saturates at high energies rather than increasing linearly without bound. By contrast, Lattice QCD systematically predicts higher mass values than those reported by the PDG. Unitary coupled-channel chiral Lagrangian approaches typically focus on the spectrum of charmed mesons with spin-parity $J^P=0^{+}$ and $1^{+}$, as in Refs. [22,24], since these channels exhibit the largest discrepancies between experiment and quark-model predictions. As shown in Table 3, however, such analyses often provide only a partial picture, as results for other quantum numbers are generally not reported; Ref. [24] only provides the mass of the $D_0^{\mathsf{\ast }}$ meson to be $2318\left(29\right)\mathrm{MeV}$. Moreover, even in some cases, the agreement with experiment is not always as satisfactory as one might expect. Finally, a recent study has presented an analysis of the charmed-meson spectrum within a closely related naive quark model, incorporating one-loop corrections to the OGE potential together with coupled-channel effects [79]. The resulting spectrum appears to be broadly consistent with ours (see Table 3), suggesting that, once the model parameters are fitted to experimental data, the inclusion of meson–meson degrees of freedom may have only a limited influence on those states that are particularly sensitive to the one-loop corrections of the OGE potential.

4. Summary

We have evaluated the effectiveness of one-loop corrections to the one-gluon exchange potential in describing the spectrum of charmed mesons within a well-established constituent quark model. By incorporating these corrections, particularly spin-dependent terms that mainly affect P-wave states of mesons with different flavor quarks, the model aims to bridge gaps between theoretical predictions and experimental measurements across the charmed meson spectrum. The study investigates both well-established and recently observed states listed in the RPP of PDG.

The model successfully reproduces masses of many S- and D-wave states. Notably, P-wave states initially posed significant discrepancies with naïve quark model predictions. However, incorporating one-loop corrections to the OGE potential mitigate these differences, bringing theoretical predictions closer to the observed values, though the model remains unable to reproduce the exact mass spectrum with full quantitative accuracy. This adjustment suggests that refinements of the naïve constituent quark model can effectively reproduce the charmed meson spectrum without resorting first to exotic configurations, such as quark-gluon hybrids, compact tetraquarks, or meson-meson molecules.

Overall, this enhanced CQM provides a refined framework for describing the heavy-light meson spectra, particularly offering insight into the nature of charmed mesons and the dynamics governing their mass structure. The results set a foundation for predicting higher-excited charmed states, potentially guiding future experimental searches and broadening the understanding of charmed meson interactions.