$ΞNN$ three-baryon force from SU(3) chiral effective field theory: A femtoscopic study

Background: The development of SU(3) chiral effective field theory has opened the way to a systematic exploration of three-baryon forces (3BFs), a key ingredient in hypernuclear and dense matter physics. However, $ΞNN$ 3BF based on SU(3) chiral EFT has not been studied until now. Purpose: We apply SU(3) chiral EFT to derive $ΞNN$ potentials in momentum space. Then, we investigate how the $ΞNN$ 3BF affects the correlation function of deuteron–$Ξ^-$ pair created through heavy-ion collisions. Methods: To reduce the number of low-energy constants involved in the $ΞNN$ potentials, we employ the decuplet saturation approximation, by which only two of them remain unconstrained. The deuteron–$Ξ^-$ scattering is treated as an effective two-body problem with the $ΞNN$ 3BF incorporated into the potential between the deuteron and $Ξ^-$. Results: We found that the effect of the $ΞNN$ 3BF on the deuteron–$Ξ^-$ correlation function is at most about 4%. This small effect is not primarily due to the loosely-bound nature of the deuteron. Instead, this is because the deuteron and $Ξ^-$ interact with each other mainly at low momentum, corresponding to peripheral scattering, where the influence of the $ΞNN$ 3BF is limited. Conclusions: Since the correlation function shows limited sensitivity to the short-range 3BF, complementary approaches may be necessary.

💡 Research Summary

The paper presents the first systematic derivation of the ΞNN three‑baryon force (3BF) within SU(3) chiral effective field theory (χEFT) and investigates its impact on the femtoscopic deuteron–Ξ⁻ correlation function measured in heavy‑ion collisions. The authors start from the N²LO SU(3) χEFT Lagrangian, where three‑body forces appear for the first time, and decompose them into three topologies: two‑pion exchange (TPE), one‑pion exchange plus a contact vertex (OPE), and a pure contact term. Momentum‑space potentials are first obtained in the particle basis, then transformed to the isospin basis using appropriate Clebsch‑Gordan coefficients.

Because the full set of low‑energy constants (LECs) accompanying these diagrams is large (14 independent parameters), the authors employ the decuplet‑saturation approximation (DSA). In DSA the intermediate decuplet baryons (Δ, Σ*, etc.) are assumed to dominate, allowing the LECs to be expressed in terms of only two independent combinations. This dramatically reduces the parameter space while preserving the essential SU(3) symmetry structure. Explicit expressions for the SU(3) coefficients (e.g., N_{BBϕ}, N_{fϕ₁ϕ₂}) are given in the appendices.

To assess the phenomenological relevance of the derived 3BF, the authors embed it into an effective two‑body problem describing the scattering of a deuteron (treated as a bound proton–neutron pair) off a Ξ⁻. They introduce Jacobi coordinates: the relative p–n coordinate r and the deuteron‑Ξ⁻ separation R. The deuteron wave function φ(r) is taken from the Argonne V4′ nucleon‑nucleon potential and is restricted to the S‑wave component. The three‑body force is folded over φ(r) to produce a central, spin‑independent effective potential U_{3BF}(R) acting between the deuteron and the Ξ⁻. The folding integral involves a Fourier transform of the momentum‑space ΞNN potential, with the momentum‑conserving delta function ensuring only two independent momenta appear.

The total d–Ξ⁻ interaction consists of the conventional two‑body ΞN and NN forces plus the newly constructed U_{3BF}(R). Solving the Schrödinger equation for this system yields the scattering amplitude, which is then used to compute the correlation function C(k) as a function of the relative momentum k. The authors find that inclusion of the ΞNN 3BF modifies C(k) by at most about 4 % across the momentum range relevant to current femtoscopic measurements.

Two physical reasons are identified for this modest effect. First, the deuteron is weakly bound (binding energy ≈ 2.2 MeV), so its wave function is spatially extended; the short‑range three‑body interaction therefore samples only a small probability density. Second, the d–Ξ⁻ scattering in heavy‑ion collisions is dominated by low relative momenta (k ≲ 50 MeV/c), corresponding to peripheral collisions where the interaction occurs at relatively large distances. In this regime the contribution of a short‑range three‑body force is naturally suppressed.

The authors conclude that the deuteron–Ξ⁻ correlation function alone provides limited sensitivity to the short‑range ΞNN 3BF. They suggest that complementary observables—such as other hyperon‑hyperon correlation functions, bound Ξ‑nuclear states, or the equation‑of‑state of dense matter—should be combined with femtoscopic data to constrain the three‑body sector more robustly. Moreover, they recommend future work to test the decuplet‑saturation hypothesis against lattice QCD results and to extend the χEFT expansion to higher orders, thereby incorporating additional LECs and reducing theoretical uncertainties. This multi‑pronged strategy is essential for a reliable description of hypernuclear systems and for understanding the role of three‑baryon forces in the physics of neutron stars and dense baryonic matter.