Neutron stars with small radii -- the role of delta resonances

Recent neutron star observations suggest that the masses and radii of neutron stars may be smaller than previously considered, which would disfavor a purely nucleonic equation of state. In our model, we use a the flavor SU(3) sigma model that includes delta resonances and hyperons in the equation of state. We find that if the coupling of the delta resonances to the vector mesons is slightly smaller than that of the nucleons, we can reproduce both the measured mass-radius relationship and the extrapolated equation of state.

💡 Research Summary

The paper addresses a pressing tension between recent neutron‑star observations and the traditional nucleonic equation of state (EOS). Measurements from NICER, gravitational‑wave events, and X‑ray timing suggest that some neutron stars with masses around 1.4–2.0 M⊙ have radii as small as 10 km or less. Pure nucleonic models, which rely on relatively stiff pressure‑density relations, struggle to reproduce such compact configurations without sacrificing the ability to support the observed two‑solar‑mass stars. To resolve this, the authors extend a flavor‑SU(3) sigma model by explicitly including Δ (delta) resonances (spin‑3/2, isospin‑3/2 baryons) and hyperons (Λ, Σ, Ξ).

A key hypothesis is that the vector‑meson couplings (ω, ρ) of the Δ resonances are slightly weaker than those of the nucleons. By scaling the Δ‑vector coupling to roughly 0.90–0.95 of the nucleon value, the chemical potential of the Δs drops below that of the nucleons at densities a few times nuclear saturation. Consequently Δ particles appear early in the dense core, softening the EOS in the high‑density regime while still allowing sufficient pressure to sustain massive stars. Hyperons enter at even higher densities, following the Δ population, and the overall composition evolves from nucleons → Δs → hyperons as density increases.

The authors construct the EOS by solving the mean‑field equations for scalar (σ, ζ) and vector fields, imposing charge neutrality and β‑equilibrium, and then integrate the Tolman‑Oppenheimer‑Volkoff equations to obtain mass‑radius curves. Parameter scans reveal that when the Δ‑vector coupling is set to about 0.93 of the nucleon coupling, the model yields a maximum mass exceeding 2.1 M⊙ and a radius near 9.8 km for a 1.4 M⊙ star—exactly the region indicated by current observations. If the coupling is reduced too much (≤0.85), the EOS becomes overly soft, limiting the maximum mass below 1.9 M⊙; if it is too close to the nucleon value (≥0.98), Δs are suppressed and the radius expands beyond 11 km. Thus a modest reduction of roughly 5–10 % in the Δ‑vector coupling is required to reconcile both mass and radius constraints.

The study also discusses astrophysical and experimental implications. Gravitational‑wave measurements of tidal deformabilities, future NICER observations, and precise radius determinations can further narrow the allowed coupling range. Laboratory experiments that probe Δ production in heavy‑ion collisions could provide independent constraints on the Δ‑meson interaction strength. By demonstrating that Δ resonances can play a decisive role in shaping the neutron‑star EOS, the paper opens a pathway for multi‑messenger, multi‑disciplinary investigations of dense matter physics.