Shallow Bound, Deeply Wrong? Relativistic Effects Reshape Hadronic Molecule Predictions

For two decades, particle physicists have been cataloguing a zoo of exotic hadrons that refuse to fit into the tidy picture of the conventional quark model. States such as the X(3872), the hidden-charm pentaquarks, and the doubly charmed T_cc^+ sit tantalizingly close to the thresholds where two ordinary hadrons can pair up, fueling a popular interpretation: they may be hadronic molecules, loosely bound pairs of mesons or baryons held together by the same kind of force that binds protons and neutrons into nuclei. But a new theoretical study warns that the mathematical machinery most often used to calculate how tightly such molecules are bound may be far less reliable than the field has assumed, producing binding energies that differ by large factors depending on which equation is solved.

The study, carried out by Lin-Qing Song and Hai-Qing Zhou of Southeast University in Nanjing and published in The European Physical Journal C, presents a systematic comparison of three levels of approximation applied to exactly the same underlying interaction. The authors solved the full four-dimensional Bethe–Salpeter equation, the gold-standard relativistic bound-state equation of quantum field theory, alongside two of its commonly used three-dimensional reductions: the Salpeter equation, which freezes the interaction in time, and the nonrelativistic Schrödinger equation, which most phenomenological analyses of hadronic molecules employ. Crucially, the couplings and masses fed into each equation were kept identical, with no refitting of parameters to make the results agree. The differences that emerged therefore isolate a single question: how much does the dimensional reduction itself distort the answer?

The formalism begins with a one-boson-exchange model, in which two particles of mass m interact by swapping a boson of mass m_ex, the hadronic analogue of two electrons exchanging a photon. In the full Bethe–Salpeter treatment, the relative momentum between the particles carries both a spatial and a temporal component, and the interaction kernel depends on all four components. Solving this equation is notoriously difficult, so the authors employed a Wick rotation, continuously deforming the integration contour from real time into Euclidean space. They proved that for a massive exchanged boson, none of the poles of the constituent or exchange propagators crosses the rotating contour, so the continuation is clean and generates no extra residue terms. The resulting Euclidean equation was then discretized on an 800 by 800 grid, producing 640,000 coupled linear equations that were solved with the PETSc computational package, with bound-state energies located by a bisection search.

The benchmark calculations used a scalar Wick–Cutkosky-type model with constituent masses of 1 GeV and exchanged-boson masses ranging from 0.01 GeV, a long-range theoretical control, up to 0.5 GeV, representing genuinely short-ranged forces. The results were striking. When the Schrödinger equation predicted a binding energy of roughly 20 MeV, the full Bethe–Salpeter equation gave only about 60 percent of that value for the lightest exchanged boson, but a mere 5 percent for the heaviest one. In other words, for short-ranged, strongly coupled interactions, the nonrelativistic reduction can overstate the binding by more than an order of magnitude, even when the nominal binding energy is small compared with the constituent masses.

This finding cuts against a widespread intuition. It is often assumed that if a bound state is shallow, meaning its binding energy is tiny compared with the masses involved, then relativistic corrections must be negligible and the Schrödinger equation is safe. The new results show that shallow binding alone guarantees nothing of the sort. The reason lies in the structure of the reduction itself. When the Bethe–Salpeter kernel is reduced to a static potential, an exact diagrammatic series relates the original kernel to the reduced one, and the Schrödinger equation keeps only the leading term of that series. In quantum electrodynamics, the tiny fine-structure constant suppresses the neglected higher-order terms, which is why the Schrödinger equation describes hydrogen so beautifully. Hadronic molecules, by contrast, require couplings of order one or larger to bind at all, and under those conditions the discarded terms are anything but small.

The numerical analysis reinforced this picture in an unexpected way. The authors varied the momentum cutoff of their integration domain from 1 GeV up to 1000 GeV and watched how the binding energies responded. Raising the cutoff from 1 to 10 GeV changed the Schrödinger result dramatically, tripling its magnitude, while the Bethe–Salpeter answer was the least sensitive of the three. This demonstrates that momenta well outside the strict nonrelativistic regime contribute substantially even within the reduced equations, another sign that the nonrelativistic approximation is straining against its own assumptions. The authors were careful to note that this cutoff is purely a numerical device for testing convergence, not a physical regulator, and that the results stabilize completely beyond 10 GeV.

To connect the abstract scalar model to real hadrons, the authors repeated the comparison for the isoscalar D-Dbar system, a pair of charmed and anticharmed mesons interacting through the exchange of rho, omega, and sigma mesons, with hadronic form factors softening the short-distance behavior. Using the same parameter set that a previous phenomenological analysis had employed in a Schrödinger framework, they found that across the region where the Schrödinger equation predicts binding energies between roughly 1 and 50 MeV, the full Bethe–Salpeter equation yields only 10 to 30 percent of those values at the same couplings. The discrepancy grows steadily as the vector coupling strength increases, with all exchanged-meson masses and the cutoff held fixed. Their Schrödinger results reproduced the earlier work exactly, confirming that the gap is not a matter of input choices but of the equation itself.

Importantly, the authors emphasize that their conclusion does not overturn the well-established universality of shallow bound states. Effective field theory teaches that when a bound state is much larger than the range of its interaction, its low-energy properties depend only on a few parameters, such as the scattering length, once the short-distance dynamics have been properly matched. The deuteron, with its inverse binding momentum of about 4.3 femtometers, is described superbly by chiral nuclear effective theory, and a D-Dbar state bound by 1 MeV has a similar size of about 4.6 femtometers. But universality applies after parameters have been fitted or matched to the relevant dynamics. The new results address a different and more uncomfortable question: whether a fixed set of one-boson-exchange parameters can be carried unchanged from a Schrödinger calculation into a relativistic Bethe–Salpeter calculation. The answer, emphatically, is no.

The practical implications ripple across the entire exotic-hadron community. Most molecular-state analyses in the literature rely on Schrödinger or reduced Bethe–Salpeter equations with couplings and cutoffs fitted to data or borrowed from related calculations. The new study does not claim that such effective approaches are worthless; a good fitted potential can still reproduce the observables it was tuned to. Rather, the warning is about blind parameter transfer. If a coupling set calibrated in a nonrelativistic framework is plugged into a four-dimensional ladder equation, or vice versa, the resulting binding energies can differ by factors of three to ten or more. Any claim that a candidate molecule is bound by, say, a few hundred kiloelectronvolts rather than a few MeV, or that it is unbound altogether, may hinge entirely on which equation was solved.

The authors also set honest boundaries on their own work. The ladder Bethe–Salpeter equation serves here as a four-dimensional reference for testing the reductions, not as a complete physical prediction. Corrections beyond the ladder approximation, such as crossed-ladder diagrams, can themselves be large; in the scalar model with a heavy exchanged boson, earlier calculations show that including the crossed-ladder contribution can more than double the magnitude of the binding energy at strong coupling. Coupled-channel effects, where a molecular state mixes with nearby scattering channels, are likewise outside the comparison and may shift the numbers further. What remains is a clear and quantified message: the three-dimensional reduction of a relativistic bound-state equation is not a harmless technical step, and for the strongly coupled, short-ranged forces that govern hadronic molecules, it can be the dominant source of uncertainty in predicting how deeply nature binds its most puzzling particles.

Subject of Research: Relativistic corrections to the binding energies of two-body hadronic molecular states calculated via Bethe–Salpeter, Salpeter, and Schrödinger equations

Article Title: Relativistic correction to the binding energies of two-body hadronic molecular states

Article References: Song, L.-Q., & Zhou, H.-Q. (2026). Relativistic correction to the binding energies of two-body hadronic molecular states. The European Physical Journal C, 86(9), Article 1120. https://doi.org/10.1140/epjc/s10052-026-16420-6

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16420-6

Keywords: hadronic molecules, Bethe–Salpeter equation, Schrödinger equation, relativistic corrections, binding energy, exotic hadrons, one-boson exchange, D-Dbar system, quantum field theory, Wick rotation, effective field theory, XYZ states