The unsettled Sigma (and Lambda) single-particle potentials in dense matter: constraining sign, isospin dependence, and momentum dependence from chiral EFT and hyperon directed flow

The Lambda and Sigma single-particle (optical) potentials in dense nuclear matter are a central missing input for the strangeness sector of the neutron-star equation of state: they set the density at which hyperons appear and whether repulsive hyperon-nucleon-nucleon forces can support the observed two-solar-mass pulsars (the hyperon puzzle). While the Lambda potential at saturation density is well fixed at about -30 MeV by hypernuclear spectroscopy, the Sigma potential is poorly determined - no bound Sigma hypernuclei exist apart from 4-Sigma-He - and its sign, momentum dependence, isospin splitting, and behavior above saturation remain open. A 2025 chiral-EFT transport analysis (Jinno, Murase, Nara, Haidenbauer) showed that Lambda and Sigma-zero directed flow in Au+Au collisions at sqrt(sNN)=4.5 GeV is sensitive to the Sigma potential, but identified four unresolved systematics: three-body-force low-energy-constant choices, uncontrolled momentum dependence beyond the chiral cutoff, uncalculated hyperon-resonance potentials and feed-down, and missing experimental optical-potential information. The problem remains open: recent J-PARC E40 and BESIII Sigma-N data have moved chiral predictions of the Sigma potential at saturation density from weakly repulsive (about +10 MeV) to weakly attractive (about -10 MeV), sharpening rather than settling the question; no Sigma directed-flow measurement exists; and proposed new constraint channels (neutrino-induced hyperon final-state interactions, dedicated hyperon factories) remain forecasts.

Background

Hyperons are baryons containing a strange quark; the lightest are the isoscalar $\Lambda(uds)$ and the isotriplet $\Sigma^{+},\Sigma^{0},\Sigma^{-}$. Their interaction with ordinary nuclear matter is encoded in the single-particle (optical) potential $U_{Y}(\rho,k)$ - the energy shift of a hyperon $Y$ with momentum $k$ embedded in matter of baryon density $\rho$ - which controls hyperon propagation in nuclei and in heavy-ion collisions, and the density at which hyperons appear in neutron stars. In microscopic treatments $U_Y$ is obtained from a $G$-matrix (Brueckner-Hartree-Fock) resummation of the two-body hyperon-nucleon ($YN$) interaction, with in-medium corrections from three-body $YNN$ forces. Modern interactions are derived in $SU(3)$ chiral effective field theory (an effective low-energy theory of baryons and Goldstone mesons organized as an expansion in powers of momentum), whose short-distance couplings - low-energy constants, LECs - must be fixed from data; the $YNN$ LECs are commonly estimated by saturating intermediate states with the lightest decuplet baryons (the 'decuplet saturation' approximation), an assumption with limited empirical control.

Experimentally, the two hyperon isospin channels are strikingly asymmetric. The $\Lambda$ potential at nuclear saturation density $\rho_{0}\approx0.16~\mathrm{fm}^{-3}$ is pinned to $U_{\Lambda}(\rho_{0},k=0)\approx-30~\mathrm{MeV}$ by decades of $\Lambda$-hypernuclear spectroscopy (nuclei with a $\Lambda$ bound in place of a neutron). No bound $\Sigma$ hypernuclei are known except the quasi-bound $_{\Sigma}^{4}\mathrm{He}$ (a helium-4 nucleus with one nucleon replaced by a $\Sigma$ hyperon), and $\Sigma^{-}$-atom level shifts (small shifts of the atomic orbitals of a $\Sigma^{-}$ captured by a nucleus, probing the $\Sigma$-nucleus interaction near the nuclear surface) have long been interpreted as evidence for a repulsive, strongly isospin-dependent $\Sigma$ potential; the commonly quoted quasi-empirical isoscalar value, $U_{\Sigma}(\rho_{0})\approx+10$ to $30~\mathrm{MeV}$, carries a large uncertainty. The $\Sigma$ potential is thus the least determined piece of the hyperon-nuclear interaction exactly where it matters most.

The physical stakes are set by the hyperon puzzle: equations of state that allow hyperons in neutron-star cores are generically too soft to support the observed $\approx2\,M_{\odot}$ pulsars, and repulsive $YNN$ three-body forces that turn the $\Lambda$ potential repulsive above $\rho_{0}$ are a leading proposed resolution. Since the $\Sigma^{-}$ is typically the first hyperon expected in charge-neutral $\beta$-equilibrated matter, $U_{\Sigma}$ directly fixes the strangeness-onset density. A promising terrestrial probe is directed flow, $v_{1}=\langle p_{x}/\sqrt{p_{x}^{2}+p_{y}^{2}}\rangle$, the sidewards preference of emitted particles in non-central heavy-ion collisions, measured for $\Lambda$ hyperons by the STAR collaboration (the Solenoidal Tracker experiment at RHIC, the Relativistic Heavy-Ion Collider at Brookhaven) in fixed-target Au+Au collisions at $\sqrt{s_{NN}}=3$-$4.5~\mathrm{GeV}$ (the center-of-mass collision energy per nucleon pair), where the created matter reaches a few times $\rho_{0}$. The reconstructed $\Lambda$ sample, however, contains an irreducible $\Sigma^{0}\to\gamma\Lambda$ feed-down fraction, so its interpretation requires the very $\Sigma$ potential one wants to determine.

Recent scattering data sharpen the issue. J-PARC E40 (experiment E40 at the Japan Proton Accelerator Research Complex) measurements of $\Sigma^{\pm}p$ differential cross sections, incorporated into a next-to-next-to-leading-order ($\mathrm{N^{2}LO}$) chiral $YN$ interaction, shift the predicted $U_{\Sigma}(\rho_{0})$ from weakly repulsive ($\approx+10~\mathrm{MeV}$ in earlier NLO-based $G$-matrix calculations) to weakly attractive ($\approx-10~\mathrm{MeV}$), while the $YNN$ sector - which would push the potential repulsively - remains essentially unconstrained. The unresolved target is a quantitatively constrained determination of these potentials at $\rho_{0}$ and above.

Problem Statement

Determine, with controlled and quantified uncertainties, the Lambda and Sigma single-particle potentials $U_{Y}(\rho,k)$ in dense baryonic matter at and above nuclear saturation density - in particular the sign, magnitude, density dependence, isospin dependence (the Sigma neutron-proton splitting), and momentum dependence of the $\Sigma$ potential - from $SU(3)$ chiral effective field theory $YN$ and $YNN$ forces, to the standard that hyperon directed flow in heavy-ion collisions at $\sqrt{s_{NN}}\approx3$-$4.5~\mathrm{GeV}$ becomes a quantitative constraint on these potentials for neutron-star physics. Concretely, a complete answer must: (a) resolve or bound the present conflict on $U_{\Sigma}(\rho_{0})$ - weakly repulsive ($\approx+10~\mathrm{MeV}$) in NLO-based $G$-matrix calculations versus weakly attractive ($\approx-10~\mathrm{MeV}$) in the E40-constrained $\mathrm{N^{2}LO}$ two-body calculation versus the repulsive quasi-empirical range - by quantifying the sensitivity to $YNN$ three-body-force LEC choices (including the decuplet-saturation assumption and the 'maximal $\Sigma$ repulsion' set), to the regulator and cutoff treatment, and to the chiral truncation error; (b) control the momentum dependence of the chiral $G$-matrix beyond the interaction cutoff well enough that predictions of $\Lambda$ and $\Sigma^{0}$ directed flow are not dominated by that extrapolation; (c) compute the single-particle potentials of the hyperon resonances (for example within a parity-doublet framework, a scheme in which each baryon is paired with a heavier partner of opposite parity) and their feed-down effect on reconstructed ground-state hyperon flow observables, replacing the current untested assumption that resonances feel the same potentials as the ground-state hyperons; (d) demonstrate, against the measured hyperon $v_{1}$ data and the independent anchors (hypernuclear binding, $\Sigma^{-}$-atom constraints, $\Sigma N$ scattering from E40 and from BESIII (an experiment at the Beijing electron-positron collider)), that the resulting potentials and their uncertainty bands are simultaneously consistent across these observables. The completion boundary is a single internally consistent determination of $U_{\Lambda}(\rho,k)$ and $U_{\Sigma}(\rho,k)$ with error bands, valid from $\rho_{0}$ up to a few times $\rho_{0}$, that no published work currently provides.

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