Probing the three-body force in hadronic systems with specific charge parity

Three-body forces, a type of non-perturbative strong interaction, are widely studied in nuclear physics. However, whether their inclusion is necessary in nuclear systems remains a topic of intense debate. In this letter, we propose that the existence of three-body forces in certain three-body hadronic systems with definite $C$-parity is certain. Such systems consist of two components whose interactions are mediated by three-body forces–a mechanism not easily realized in conventional three-nucleon systems. We investigate two specific three-body hadronic systems, $\bar{D}_sDK$ and $\bar{D}^*Dη$, using contact-range potentials. The two-body hadron-hadron interactions are constrained by reproducing their scattering lengths, while the three-body couplings are constrained by charge symmetry. Our results indicate that three-body forces play a minor role in binding the $I(J^{PC})=0(0^{–})$ $\bar{D}_sDK$ system, but a crucial one in binding the $I(J^{PC})=0(1^{-+})$ $\bar{D}^*Dη$ system. In fact, three-body forces determine whether $\bar{D} ^*Dη$ forms a bound state, making this system a promising candidate for exploring three-body forces in hadronic physics.

💡 Research Summary

The paper investigates the role of three‑body forces in hadronic systems that possess a well‑defined charge‑parity (C‑parity). While three‑nucleon (3N) forces are known to contribute only a few percent to the binding of light nuclei, their direct experimental identification remains difficult. The authors propose that replacing nucleons with mesons in a three‑body system that carries a definite C‑parity creates a natural setting where a three‑body interaction emerges as a transition between charge‑conjugated components of the wave function. Two specific systems are studied: the I(J^PC)=0(0^−−) (\bar D_s D K) system and the I(J^PC)=0(1^−+) (\bar D^* D \eta) system.

The theoretical framework combines pionless effective field theory (EFT) with the Gaussian Expansion Method (GEM). Two‑body interactions are modeled by Gaussian contact potentials (V(r)=C_a \exp(-r^2/b^2)). The strengths (C_a) are fixed by reproducing scattering lengths and, where available, effective ranges obtained from lattice QCD or phenomenology: D K scattering length a₀=−1.49 fm (r₀=0.20 fm), (\bar D_s D) (a₀≈0.525 fm), (\bar D^* D) (a₀≈−1.7 fm), D η (a₀≈0.29 fm), etc. Heavy‑quark spin symmetry and SU(3) relations are used to infer the remaining couplings. The cutoff b for the two‑body potentials is varied between 0.5 fm and 1.0 fm to assess regulator dependence.

A three‑body contact term is introduced to encode the C‑parity‑dependent transition: (V_3(\mathbf r,\mathbf R)=C_3 \exp(-r^2/b_3^2)\exp(-R^2/b_3^2)), with a shorter range cutoff (b_3=0.3)–0.5 fm. The sign of (C_3) is dictated by the C‑parity: the (\bar D_s D K) system (0^−−) receives a repulsive three‑body force (positive (C_3)), whereas the (\bar D^* D \eta) system (1^−+) receives an attractive one (negative (C_3)). (C_3) itself is treated as a free parameter.

The three‑body Schrödinger equation is solved in the GEM basis, expanding the total wave function as a sum over the three Jacobi partitions. Only S‑wave components are retained. A bound state is claimed only if the lowest eigenvalue is negative and the root‑mean‑square (rms) radii of both Jacobi coordinates remain finite and stable under basis variations.

Results for (\bar D_s D K): Varying (C_3) from 0 up to ≈1000 MeV (the regime where the three‑body term becomes strongly repulsive) produces only modest changes in the binding energy, which stays close to the two‑body (D K) binding energy. The rms radius of the (D K) subsystem grows from ~1 fm to ~1.5 fm, while the distance between (\bar D_s) and the (D K) pair expands toward the computational box limit. The three‑body contribution to the total potential energy never exceeds about 4 % of the sum of all two‑body contributions. Only for unrealistically large repulsion ((C_3>1000) MeV) does the bound state dissolve. Hence, the three‑body force plays a minor, almost negligible role in this system, confirming earlier expectations that three‑body effects in such configurations are small.

Results for (\bar D^* D \eta): With an attractive three‑body term, the situation is dramatically different. For (C_3) near zero the system is unbound. As (C_3) becomes more negative (≈−200 MeV or stronger), a bound state appears, with binding energies increasing rapidly and rms radii shrinking to a few femtometers, indicating a compact three‑body molecule. The three‑body interaction thus determines whether the state exists at all. This sensitivity makes the (\bar D^* D \eta) system a promising laboratory for probing three‑body forces in hadronic physics.

The authors discuss the practical implications: the (\bar D^* D \eta) state carries exotic quantum numbers (J^PC=1^−+) that cannot be formed by a simple (c\bar c) pair, reducing contamination from conventional charmonium and enhancing its suitability as a molecular candidate. Experimental searches at facilities such as BESIII, LHCb, and Belle II could look for signals in the relevant invariant‑mass distributions. Observation (or non‑observation) of a bound (\bar D^* D \eta) state would provide direct evidence for the existence and magnitude of three‑body forces in hadronic systems, a topic that has been largely inaccessible in nuclear physics.

In summary, the paper demonstrates that three‑body forces, while negligible in the (\bar D_s D K) system, are essential for binding the (\bar D^* D \eta) system. By exploiting charge‑parity selection rules, the authors propose a novel avenue to isolate and study three‑body interactions in the hadron sector, offering clear predictions that can be tested in upcoming experiments.