The research, carried out by Pranami Baishya and Bhaskar Jyoti Hazarika of the Center for Theoretical Studies at Pandu College in Guwahati, India, focuses on tetraquarks, hypothetical particles built from two quarks and two antiquarks. Rather than treating all four quarks as loose constituents, the authors adopt the diquark-antidiquark picture: a tightly bound quark pair acts as a single effective object, the diquark, which then binds with its antimatter counterpart. This two-body simplification transforms an intractable four-body problem into something a desktop computer can handle, while still capturing what many theorists believe is the essential binding mechanism of these exotic states.

The mathematical engine of the study is the spinless Salpeter equation, a relativistic upgrade of the familiar Schrödinger equation from introductory quantum mechanics. Where the Schrödinger equation assumes particles move slowly compared to light, the Salpeter equation replaces the kinetic energy with the square-root form demanded by special relativity: the square root of the momentum squared plus the mass squared. This matters enormously for tetraquarks, because the constituent masses and binding energies involved often sit in a regime where non-relativistic approximations begin to break down. The equation retains relativistic accuracy while remaining far more tractable than fully relativistic frameworks like the Bethe-Salpeter equation.

To describe the force between the diquark and antidiquark, the team employed the celebrated Cornell potential, a phenomenological formula that has anchored heavy-quark physics since the late 1970s. The potential combines two essential behaviors of the strong nuclear force: a Coulomb-like term that dominates at short distances, proportional to the strong coupling constant divided by the separation, and a linear term that grows with distance, reflecting the string-like confinement that prevents quarks from ever being pulled apart. The string tension was fixed at 0.1 GeV squared, with a phenomenological constant of minus 0.8 GeV, while the strong coupling constant was scanned across a physically motivated range from 0.20 to 0.64, chosen from a parameter space established by earlier rigorous constraints on the Cornell model.

The technical heart of the paper lies in a variational calculation. The authors assumed a simple exponential trial wave function for the ground state, characterized by a single adjustable parameter beta that controls the spatial size of the bound state. By computing the expectation value of the full Hamiltonian and minimizing it with respect to beta, they derived an analytical mass formula for tetraquark ground states. Relativistic corrections were then handled through first-order perturbation theory: the leading correction to the kinetic energy, suppressed by factors of velocity squared over the speed of light squared, was treated as a small perturbation. Remarkably, the team also derived the first-order correction to the wave function itself, showing explicitly how relativistic effects mix the first excited state into the ground state and thereby reshape the internal structure of the tetraquark.

The model was put to the test against four landmark tetraquark candidates spanning the heavy-quark spectrum. The X(3872), discovered by the Belle Collaboration in 2003, is the most famous exotic hadron and served as the benchmark hidden-charm state. The X(4700) probes heavier hidden-charm configurations. The X(6900), reported by the LHCb Collaboration in 2020 as a narrow structure in the J/psi-pair mass spectrum, is widely interpreted as a fully charmed tetraquark containing two charm quarks and two charm antiquarks. Finally, the fully bottom tetraquark Tbbbb, composed of four bottom quarks, remains experimentally unobserved but theoretically compelling as a potentially deeply bound multiquark system.

The results reveal a striking pattern of successes and shortfalls. For the X(3872), the calculated mass deviated from the experimental value of 3.871 GeV by a mere 0.013 GeV, an agreement the authors describe as excellent and which validates the variational approach for certain heavy-light configurations. Even more impressive was the fully bottom sector: the predicted mass of 18.853 GeV for Tbbbb landed just 0.027 GeV from the theoretical benchmark of 18.826 GeV reported by Karliner and colleagues, suggesting that the spinless Salpeter framework with the Cornell potential is particularly well suited to fully heavy systems, where spin-dependent interactions are comparatively suppressed by the sheer mass of the quarks involved.

The charm sector proved more stubborn. The calculated mass of the X(6900) candidate fell short of the LHCb measurement of 6.905 GeV by roughly 0.785 GeV, and the X(4700) showed a similarly large deviation. The authors suggest these gaps may signal that additional physics is at play in charm systems: higher-order relativistic corrections, coupled-channel dynamics in which the tetraquark mixes with nearby meson-meson states, or spin-dependent interactions that the spinless framework deliberately omits. The discrepancy is itself scientifically valuable, pointing theorists toward which dynamical ingredients a more complete description must include.

Perhaps the most intriguing finding concerns the behavior of the strong coupling constant across excitation modes. For ground states, light tetraquarks favored a coupling of 0.20 while heavy tetraquarks preferred 0.34, and an optimal global description of all states emerged around 0.36. But for excited 2S and 1P states, the pattern inverted: light tetraquarks demanded a much larger coupling of 0.64, while heavy tetraquarks converged to 0.20. The authors interpret this as evidence of distinct effective interaction regimes. Enhanced coupling in excited light tetraquarks may indicate a transition toward more spatially extended, molecular-like configurations probing the long-distance confinement regime, whereas the persistently low coupling in excited heavy tetraquarks implies the heavy-quark core stays compact even under radial or orbital excitation. This differential response could ultimately help distinguish between compact diquark-antidiquark and loose molecular interpretations of tetraquark states, one of the liveliest debates in hadron physics.

The study also found that calculated masses decrease mildly as the coupling constant grows, a direct consequence of the attractive Coulomb term deepening the potential well, and that 2S excited states sit systematically above their 1P counterparts within the model. The variational parameter beta grew steadily with coupling strength for every configuration, confirming that stronger interactions squeeze the bound state into a more compact spatial footprint, with the fully bottom system exhibiting by far the largest beta values. Looking ahead, the authors identify clear paths for refinement: incorporating spin-dependent forces, extending the perturbative treatment to higher orders in velocity squared, and investigating decay properties that would offer direct experimental handles on internal structure. As accelerator experiments continue to churn out new four-quark candidates, simple yet disciplined frameworks like this one provide the quantitative scaffolding on which our understanding of the nonperturbative strong force, the last great uncharted territory of the Standard Model, will continue to be built.

Subject of Research: Theoretical calculation of heavy tetraquark mass spectra using the spinless Salpeter equation and Cornell potential in a diquark-antidiquark model

Article Title: Mass spectra of tetraquark states from the spinless Salpeter equation

Article References: Baishya, P., & Hazarika, B. J. (2026). Mass spectra of tetraquark states from the spinless Salpeter equation. The European Physical Journal C, 86(10), Article 1155. https://doi.org/10.1140/epjc/s10052-026-16342-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16342-3

Keywords: tetraquarks, spinless Salpeter equation, Cornell potential, diquark-antidiquark model, quantum chromodynamics, exotic hadrons, X(3872), X(6900), fully heavy tetraquarks, variational method, relativistic corrections, strong coupling constant

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