Mass relations in heavy hadrons from Jensen-like inequalities

Understanding the nonperturbative regime of strong interactions remains a central challenge in quantum chromodynamics (QCD). At short distances, the interaction exhibits Coulomb-like behavior with asymptotic freedom, which can lead to concave binding-energy patterns through inverse-mass dependence. In contrast, at long distances, confinement manifests as a linear potential, giving rise to string-like binding that may break at critical scales. This behavior introduces strong condensation and unquenched effects, complicating mass predictions for heavy hadronic states. This issue is particularly timely in light of the recent LHCb discovery of the doubly charmed baryon $ \Xi_{cc}^+ $ [1] and the surge in mass predictions for triply heavy baryons [2].

Empirical mass inequalities among hadrons containing heavy or strange quarks provide valuable insights into these dynamics. Mixed-flavor mesons and baryons systematically exceed the mean value of their same-flavor counterparts, as in

with $ x \neq y $ and at least one heavy flavor [3−11]. Here, the symbols x and y denote quark-flavor labels (e.g., c for charm and b for bottom), not numerical variables; the relations are applied to the mass function evaluated at these discrete labels. These relations, valid for pseudoscalar and vector mesons as well as ground-state baryons, indicate concavity of the mass function with respect to the energy scale, arising from flavor-dependent interactions in the quark model. Similar mass decompositions into constituent quark masses, two-body binding energies, and hyperfine (chromomagnetic) terms have been widely used for heavy hadrons and tetraquarks [10, 12−22].

In this work, we interpret these inequalities phenomenologically in terms of the concavity of chromoelectric binding energies $ B_{i\bar{j}} $, which are extracted empirically from spin-averaged hadron masses to ensure model independence and reliance on experimental data. By fitting $ B(\mu) $ and $ B(1/\mu) $, interpreted as the mean values of the Cornell potential, we explore their concave dependence on the reduced mass $ \mu_{ij} $, yielding Jensen-like relations under flavor permutations. This fit further reveals a critical confinement scale for positive binding energies, in agreement with predictions for string-breaking distances. We validate the approach with $ \Delta M_{ \rm{EXP}} \approx \Delta B + \Delta C $ and obtain approximate mass estimates for unobserved doubly and triply heavy baryons; we also discuss potential decay and scattering processes. This phenomenological study focuses on empirical relations for interaction energies extracted from experimental meson masses, rather than on a fully rigorous three-body derivation.

The mass inequalities (1) can be interpreted through a Jensen-like relation for a concave function f,

with weights $ \alpha_{i} $ summing to 1. For $ n=2 $, $ \alpha_{i} = 1/2 $, and flavor labels $ x_1 = x $, $ x_2 = y $, one obtains the meson and baryon relations in Eq. (1). For $ n=3 $, weights $ \alpha_{i} = 1/3 $ yield $ m_{xyz} \gt \dfrac{1}{3}(m_{xxx} + m_{yyy} + m_{zzz}) $. These relations hold for mesons with at least one heavy quark and for baryons in which the xy pair exceeds the ud diquark mass [11]. Such patterns arise from the concavity of flavor-dependent interactions [7−9], particularly under the exchange of heavy and light flavors, indicating flavor symmetry breaking.

The concavity of the mass function can be motivated by the underlying dynamics of the quark model. To illustrate this, consider the hadron mass in second-order perturbation theory:

where $ H_0 $ is the leading-order Hamiltonian, V denotes the perturbative interactions, and the variational parameter λ is associated with the inverse reduced mass $ 1/\mu $. The concavity can be examined by analyzing the second derivative of the mass with respect to λ, providing a phenomenological basis for the observed Jensen-like behavior. In the quark model, the mass includes both leading-order and perturbative interactions:

where

Here, $ E_{ \rm{BD}} $ captures confinement and short-range chromoelectric binding, while $ E_{{\rm{CMI}}} $ represents the perturbative chromomagnetic interaction. The binding energies $ B_{ij(\bar{j})} $ are regarded as the expectation values of the Cornell potential for $ {\bf{1}}_c $ ($ i\bar{j} $) and $ {\bf{\bar{3}}}_c $ (ij). The meson coupling parameters $ C_{i\bar{j}}=A_{i\bar{j}}/m_{i}m_{j} $ are taken from Ref. [23], with $ A_{i\bar{j}} $ fitted to a convex function of the reduced mass μ, while the baryon couplings are set to $ C_{ij}=0.85\,C_{i\bar{j}} $ (see Sec. III). The constituent masses used for μ are extracted from the Regge-like framework [23, 24]. The bindings $ B_{ij(\bar{j})} $ and couplings $ C_{i\bar{j}} $ themselves are fixed from experimental meson masses and hyperfine splittings and do not depend on this mass choice. Further investigation of the concavity of interaction energies follows from condition (3) when λ is associated with $ 1/\mu $, leading to the fitting of the binding energies $ B(1/\mu) $.

From another perspective, the mass inequalities (1) can be related to Eqs. (4)–(6) by decomposing the differences:

This decomposition allows us to examine the mass inequalities and thus obtain approximate mass relations by evaluating $ \Delta B $ and $ \Delta C $.

The binding energy is mainly generated by short-range chromoelectric interactions in heavy-quark or heavy-strange systems, as considered in various works [12−14, 23, 25−27]. Ref. [12] employed five binding energies, $ B_{cs} $, $ B_{bs} $, $ B_{cc} $, $ B_{bc} $, and $ B_{bb} $, to study doubly heavy systems. In the QCD string picture [23, 26, 27], which incorporates perturbative gluon interactions and confinement, we extract the binding energies from the spin-averaged meson masses:

Here $ \bar{M} $ denotes the spin-averaged meson mass, and the light non-strange flavor is labeled by n (u or d). The mass of the pseudoscalar meson $ \eta_s $, 686.9 MeV, is obtained from [6]

supported by lattice QCD simulations [28−31]. The mass of 6332 MeV for the vector meson $ B_c^{\ast} $ is adopted from [32], and is also supported by lattice QCD simulations [33−35]. The binding energies are listed collectively in Table 1, including a potential binding energy of −40.6 MeV for $ s\bar{s} $, where s and $ \bar{s} $ are both heavier than non-strange light flavors. For baryon systems, the binding energies $ B_{ij} $ under the $ {\bf{\bar{3}}}_c $ representation can be obtained by scaling with color factors, $ B_{ij} = B_{i\bar{j}}/2 $, following the principle of chromoelectric interactions.

Using the effective (non-vanishing) binding energies in Table 1, we obtain fitted relations for meson systems as functions of the reduced mass $ \mu_{ij} = m_i m_j/(m_i + m_j) $:

where $ \Lambda=200\, $MeV is the QCD scale, the masses $ m_i $ and $ m_j $ of the flavors are given below Eq. (6), and the basis functions are chosen as in Ref. [23]. A least-squares fit of this three-parameter form to the six effective binding energies yields an rms residual of $ \approx 15.8\, $MeV and $ \chi^{2}/{\rm{dof}}\approx 497 $ with unit MeV weights. The large unweighted $ \chi^{2} $ reflects the simplicity of the parametrization and the scatter in the light sector; this form is used to demonstrate the concave trend in $ B(1/\mu) $, not as a precision mass formula. Applying the condition (3) with λ associated with $ 1/\mu $, the concavity of the binding energies can be examined through second derivatives, as well as in Fig. 1, providing a phenomenological motivation for the observed mass patterns. Another contribution comes from chromomagnetic interactions, where $ A_{i\bar{j}} $ exhibits concavity with respect to $ 1/\mu $ and convexity with respect to μ in Ref. [23].

Figure 1.  (color online) Plot of fitted binding energies for meson systems as a function of the reduced mass μ of the quark-antiquark pair. The gray curve denotes Eq. (10), with discrete data from Table 1 shown as filled circles. The red curve represents Eq. (11) as a function of $ 1/\mu $.

As shown in Fig. 1, the binding energy should be dominated by the Coulombic potential at very short distances, with contributions from the long-range confinement potential, consistent with theoretical expectations and the fitted data. At a critical reduced mass of 147.4 MeV, the binding energy becomes positive and is no longer effective. Using $ \hbar c\approx 197.3\, $MeV fm gives a length $ 197.3/147.4\approx 1.34\, $fm, which is identified with the string-breaking distance. Lattice QCD and other studies predict this distance to be 1.2–1.4 fm [36−43], with the upper limit from Ref. [43] close to our result. This scale implies strong condensation and unquenched effects, leading to the instability of pseudo-Goldstone mesons such as $ s\bar{n} $ and $ n\bar{n} $.

Following the mass relations (7), one can evaluate $ \Delta B $ and $ \Delta C $ to confirm that the mass patterns are consistent with binding energies and chromomagnetic interactions. The values of $ C_{i\bar{j}} $ are extracted from meson data, and the ratio $ C_{ij} = 0.85C_{i\bar{j}} $ is adopted to minimize the standard error σ. The traditional ratio 2/3 used in early quark-model analyses [17, 44, 45] is a model hypothesis for meson/baryon hyperfine scaling rather than a requirement of the present decomposition; here $ r=C_{ij}/C_{i\bar{j}} $ is fixed by fitting $ \Delta M_{ \rm{EXP}}\approx\Delta B+\Delta C $. Scanning r shows that $ r=0.85 $ yields the minimum $ \sigma=2.07\, $MeV, while $ r=2/3 $ and $ r=1 $ give $ \sigma\approx4.5 $ and 4.4 MeV, respectively. Varying r within [0.75, 0.95] shifts the masses by only a few MeV for doubly heavy states and by $ \lesssim 15\, $MeV for $ \Omega_{ccc} $—well within the typical theoretical uncertainties of quark-model and lattice estimates. The comparisons are shown explicitly in Table 2, where $ \Delta B + \Delta C $ closely tracks $ \Delta M_{ \rm{EXP}} $ for baryons, with slight deviations for mesons due to isospin splittings. The small standard error $ \sigma = 2.07\, $MeV confirms that the observed patterns are consistent with flavor-dependent interactions.

The $ K^{\ast} $ entry in Table 2 is the only case with $ \Delta M_{ \rm{EXP}}<0 $. Light mesons are governed mainly by long-range confinement, where the potential grows approximately linearly and no clear concavity emerges. The Jensen-like inequality is therefore not expected to hold in the same way as for systems with at least one heavy quark, which probe short-distance Coulombic binding. Both the experimental value (−3.8 MeV) and the decomposition (−1.8 MeV) are expected to be close to zero, due to the lack of binding contrast. The small negative sign is thus naturally attributed to experimental and theoretical uncertainties, rather than to a robust inverted inequality. All other entries involve heavy flavors, for which the concave binding pattern dominates and the inequalities are satisfied. The $ K^{\ast} $ row is retained as a light-sector boundary check and does not affect the heavy-hadron predictions.

The four baryon rows in Table 2 that exchange s and n ($ \Xi^{\ast} $, $ \Xi_c^{\ast} $, $ \Xi_c^{\prime} $, $ \Xi_b^{\prime} $) all give $ \Delta B+\Delta C=0.8\, $MeV, while experiment yields 3.5–4.7 MeV. With $ B_{nn}=B_{sn}=0 $, the s–n exchange carries no binding contrast in those channels, so the few-MeV underestimation is a light-sector effect. It can arise from (i) the schematic vanishing of the bindings, (ii) a small residual in s–n systems from the strange-quark mass and infrared confinement not captured when $ B_{sn}=0 $, and (iii) hyperfine uncertainties in $ C_{ij} $. It does not indicate a failure of the heavy-quark concavity mechanism. The same s–n pattern appears in $ \Xi_b^{\ast}>(\Omega_b^{\ast}+\Sigma_b^{\ast})/2 $ used for $ M(\Omega_b^{\ast}) $.

Using the binding energies $ B_{ij(\bar{j})} $ and coupling parameters $ C_{ij(\bar{j})} $ given in Table 1, we evaluate the decomposition $ \Delta B + \Delta C \approx \Delta M $ to obtain approximate numerical mass relations. This phenomenological approach allows us to estimate the masses of unobserved heavy baryons and to identify potential decay and scattering channels. Table 3 lists these estimates for doubly and triply heavy baryons, including extensions to relations involving averages over three baryons. These relations are constructed for the same spin configurations on both sides to avoid interference from spin multiplets.

The values of $ \Delta B+\Delta C $ in Table 3 span a wide range, from 0.8 to 201.9 MeV. This spread is expected from the concavity of $ B(\mu) $. Large differences arise when a heavy quark is exchanged for a much lighter flavor (e.g., $ \Xi_b^{\prime}>(\Xi+\Xi_{bb})/2 $ or the $ n=3 $ routes involving $ \Omega_{bbb} $), so that the concave binding is sampled over a wide interval in $ 1/\mu $ and the binding contrast dominates. By contrast, the smallest entries—such as $ \Xi_b^{\ast}>(\Omega_b^{\ast}+\Sigma_b^{\ast})/2 $, giving 0.8 MeV, and $ \Omega_{bc}>(\Omega_{bcc}+\Omega_b)/2 $, giving 1.0 MeV—correspond to s–n exchanges. With $ B_{nn}=B_{sn}=0 $, those channels carry essentially no binding contrast, so $ \Delta B+\Delta C $ is expected to lie close to zero. The residual few MeV arise mainly from hyperfine contributions and the light-sector systematics already discussed after Table 2. The $ \sim 0.8\, $ MeV cases remain physically meaningful, aligning with the few-MeV s–n underestimation noted above.

For the relation $ \Xi_c^{\prime}>(\Xi+\Xi_{cc})/2 $ in Table 2, one can explicitly estimate the mass of $ \Xi_{cc} $ as 3623.6 MeV, consistent with the reported value of 3621.6 MeV [46, 47]. Similarly, more estimates can be made if other hadrons are discovered experimentally, such as

Comparisons are listed in Table 4, together with several notable references. Several relations can be used to predict the mass of $ M(\Omega_{ccc}) $, but only Eq. (12) is considered here. Chained $ n=2 $ routes for $ M(\Omega_{ccc}) $ via $ M(\Omega_{cc}^{\ast}) $ or $ M(\Xi_{cc}^{\ast}) $ differ from the direct $ n=3 $ value by up to $ \sim 45\, $MeV, since residual errors are amplified by the 1/2 and 1/3 inversion factors and accumulate under chaining. We therefore quote only the single-step $ n=3 $ estimates from the measured $ \Xi_{c}^{\ast} $ and $ \Xi_{b}^{\ast} $ in Eq. (12).

The remaining mass relations are expected to be confirmed in future experiments, both in mass spectra and scattering channels. For example, the inequality $ D> (\eta_c+\pi)/2 $ implies a scattering channel $ D\bar{D} \to \eta_c + \pi $, with a mass gap of 307.8 MeV, indicating a favored process. Baryon scatterings could also be discovered in channels from Table 3, such as $ \Xi_c^{\ast}\Xi_c^{\ast}\to\Xi^{\ast}+\Xi_{cc}^{\ast} $ (exchanging c–n), $ \Xi_b^{\ast}\Xi_b^{\ast}\to\Xi^{\ast}+\Xi_{bb}^{\ast} $ (exchanging b–n), and $ \Xi_{cc}^{\ast}\Xi_{cc}^{\ast}\to\Omega_{ccc}+\Sigma_{c}^{\ast} $ (exchanging c–s), that is, by exchanging a heavy quark with a light or strange quark. This phenomenon is consistent with the quark-exchange model for hadron scatterings [50−52], as the strong bindings (5) favor the exchange of heavy quarks or heavy-strange systems.

In this work, we investigate a set of mass inequalities for hadrons containing one or more heavy quarks and interpret them in terms of the concavity of the binding energies $ B_{ij(\bar{j})} $ as mean values of the Cornell potential. By employing the quark model and decomposing the hadron mass into leading-order binding contributions (5) from confinement and chromomagnetic interactions (6), we establish a mass relation $ \Delta M \approx \Delta B + \Delta C $ under flavor permutations. This phenomenological approach yields Jensen-like relations for distinct flavors, such as $ m_{xyz} \gt \dfrac{1}{3}(m_{xxx} + m_{yyy} + m_{zzz}) $ for averages among three baryons, attributing the concavity to flavor-dependent interactions with respect to $ 1/\mu $ in second-order perturbation theory.

We extract effective two-body binding energies $ B_{i\bar{j}} $ and hyperfine couplings $ C_{i\bar{j}} $ from meson masses, including an additional binding of −40.6 MeV for the $ s\bar{s} $ system. These bindings are parametrized as functions of the reduced mass $ \mu_{ij} $ with logarithmic scaling, yielding a concave form $ B(1/\mu) $ that satisfies the second-derivative condition (3). This fit reveals a critical confinement scale of 1.34 fm, beyond which the binding becomes positive, consistent with predictions for the string-breaking distance and indicating the onset of strong unquenched effects.

For baryonic systems, we adopt the scaling relations $ B_{ij} = B_{i\bar{j}}/2 $ from color factors and $ C_{ij} = 0.85 C_{i\bar{j}} $ to minimize the standard error to $ \sigma = 2.07 $ MeV across $ \Delta M_{ \rm{EXP}} \approx \Delta B + \Delta C $. This close agreement confirms that the observed mass patterns are consistent with the concavity of the binding energies and chromomagnetic interactions. Using this decomposition, we obtain approximate mass estimates for unobserved doubly and triply heavy baryons, as well as potential scattering channels such as $ D\bar{D} \to \eta_c + \pi $ and $ \Xi_c^* \Xi_c^* \to \Xi^* + \Xi_{cc}^* $. Relations with large differences support the physical picture of the quark-exchange model, due to strong binding and positive differences. These estimates are expected to be validated in future experiments.

No new experimental data were created in this study. The numerical codes and scripts used to generate the results are available at https://github.com/SerialCore/MassInequality.