The 2025 relativistic mean-field study of ω\omega-ρ0\rho^0 meson mixing by Tanimura, Naito, Sagawa, and Cheoun (DD-ME-CSB) fitted the mixing strength Δmv2\Delta m_{\mathrm{v}}^2 to mirror-nucleus mass differences and mapped it onto Skyrme-type charge-symmetry-breaking (CSB) energy-density-functional coefficients, but left the mechanism quantitatively uncontrolled at four points. The open problem is to determine the effective CSB interaction generated by ω\omega-ρ0\rho^0 mixing in covariant density functional theory with controlled approximations: including Fock (exchange) terms so that pionic CSB enters explicitly, establishing whether the effective mixing strength is density dependent, testing the robustness of the gradient-expansion mapping to Skyrme-type CSB functionals, and deciding whether the same mixing mechanism accounts for the binding-energy differences of mirror Λ\Lambda hypernuclei.

Background

Isospin symmetry treats the proton and the neutron as two states of one nucleon; nuclear physics breaks it slightly through the up-down quark mass difference and through electromagnetism. The breaking is conventionally decomposed into charge-independence breaking (CIB, the class II interaction, which differs between like-nucleon and proton-neutron pairs) and charge symmetry breaking (CSB, the class III interaction, which makes the proton-proton and neutron-neutron interactions differ). CSB is observable in the binding-energy differences of mirror nuclei (nuclei with proton and neutron numbers interchanged, such as 17^{17}O-17^{17}F): after the Coulomb energy is removed, a residual difference remains, known since the 1960s as the Okamoto-Nolen-Schiffer anomaly. In the meson-exchange picture, CSB of the nuclear force is dominated by mixing between the isoscalar ω\omega meson and the neutral isovector ρ0\rho^0 meson (with smaller π0\pi^0-η\eta mixing): the physical mesons propagate as mixed states and generate a proton-neutron asymmetric potential. Covariant (relativistic) density functional theory describes nuclei from a Lagrangian in which nucleons couple to σ\sigma, ω\omega, and ρ\rho meson fields. In the widely used relativistic Hartree-Bogoliubov (RHB) scheme, only direct (Hartree) mean fields are kept and the Fock (exchange) terms are dropped; consequently the pion, which contributes to nuclei only through exchange, cannot be treated explicitly. In the source paper, Tanimura, Naito, Sagawa, and Cheoun started from the density-dependent DD-ME2 relativistic mean-field Lagrangian, introduced an ω\omega-ρ0\rho^0 mixing off-diagonal mass term of strength Δmv2\Delta m_{\mathrm{v}}^{2}, and refitted all parameters to the mass differences of T=1/2T=1/2 mirror nuclei together with ground-state properties of magic nuclei, including refined electromagnetic corrections (nucleon form factors, first-order vacuum polarization, perturbative exact Coulomb exchange and pairing). The resulting DD-ME-CSB parameter set reproduces mirror mass differences reasonably well up to T=2T=2 (TT being the total isospin quantum number; the fit used T=1/2T=1/2 mirror pairs and the predictions extend to T=1T=1 and T=2T=2), demonstrating the importance of ω\omega-ρ0\rho^0 mixing, and a gradient expansion of the energy density functional (EDF) maps the meson-mixing CSB onto Skyrme-type CSB coefficients, a0a_0 and a1=(1/mω2+1/mρ2)a0a_1=(1/m_\omega^2+1/m_\rho^2)a_0. The mapping neglected the spatial dependence of the density-dependent couplings gω(ρ)g_\omega(\rho) and gρ(ρ)g_\rho(\rho) (evaluating them at saturation density), kept only first-order ∇2\nabla^2 terms, and does not generate the ρτ\rho\tau-type EDF terms (with τ\tau the kinetic-energy density) that a first-order Skyrme interaction produces; the next-to-leading coefficient a1a_1 varies significantly across existing CSB functionals. For hypernuclei (nuclei containing a Λ\Lambda hyperon in place of a nucleon), the analogous CSB appears as binding-energy differences between mirror Λ\Lambda hypernuclei such as Λ4^{4}_{\Lambda}H-Λ4^{4}_{\Lambda}He, where Λ\Lambda-Σ0\Sigma^0 mixing, the electromagnetic mixing of the Λ\Lambda and Σ0\Sigma^0 hyperons, is a mechanism specific to this channel. The authors state the limitations of their construction explicitly: the Fock terms are neglected so pionic CSB cannot be included; the fitted mirror-mass residuals decrease for heavier nuclei, suggesting the effective mixing strength may be density dependent from many-body effects; the robustness of the Skyrme mapping is unquantified; and the extension of the ω\omega-ρ0\rho^0 mixing CSB scheme to the NN-Λ\Lambda interaction was proposed but not implemented.

Problem Statement

Determine the effective charge-symmetry-breaking interaction generated by ω\omega-ρ0\rho^0 mixing in covariant density functional theory at a quantitatively controlled level, with one common objective: fixing the mixing strength Δmv2\Delta m_{\mathrm{v}}^{2} (or its density-dependent generalization) and its energy-density-functional image, together with the domain of validity of the mechanism, for both ordinary nuclei and hypernuclei. Four linked questions, all bearing on that same object and jointly deciding whether the meson-mixing mechanism gives a quantitatively reliable account of charge symmetry breaking: (1) Exchange and pionic CSB — include the Fock (exchange) terms, moving from relativistic Hartree-Bogoliubov to relativistic Hartree(-Bogoliubov)-Fock so that the pion and related exchange contributions to isospin breaking enter explicitly, and quantify how the inferred Δmv2\Delta m_{\mathrm{v}}^{2} and the derived Skyrme-like CSB coefficients change. (2) Density dependence — establish whether the effective ω\omega-ρ0\rho^0 mixing strength is density dependent (as the decrease of mirror-mass residuals for heavier nuclei suggests), and if so determine its functional form, magnitude, and microscopic origin in many-body effects. (3) Robustness of the EDF mapping — assess quantitatively the impact of retaining the spatial (density) dependence of the couplings gω(ρ)g_\omega(\rho) and gρ(ρ)g_\rho(\rho), of higher-order gradient terms beyond the leading ∇2\nabla^2 order, and of the ρτ\rho\tau-type terms that appear in Skyrme formulations, in order to establish the robustness of the mapping onto Skyrme-type CSB energy density functionals and to reconcile the spread of the next-to-leading-order coefficient a1a_1 among models. (4) Hypernuclear extension — determine whether ω\omega-ρ0\rho^0 mixing suffices to account for the observed binding-energy differences of mirror Λ\Lambda hypernuclei, or whether other mechanisms such as Λ\Lambda-Σ0\Sigma^0 mixing must be included together with it, within the same covariant framework. The four parts, in order, concern: (1) a quantified change, with its uncertainty, in Δmv2\Delta m_{\mathrm{v}}^{2} and in the derived coefficients relative to the Hartree-level fit; (2) a determination of whether the effective mixing strength depends on density and, if so, of its functional form (parameters fitted to mirror-mass data or derived microscopically) and its microscopic origin; (3) a separate pricing of each of the three refinements — coupling density dependence, higher-order gradient terms, and ρτ\rho\tau-type terms — through their individual effect on a0a_0 and a1a_1 (with uncertainty), together with an account of the sign and magnitude of the a1a_1 spread among existing CSB functionals; (4) an explicit comparison of the ω\omega-ρ0\rho^0-only prediction with one that adds Λ\Lambda-Σ0\Sigma^0 mixing against the measured mirror-Λ\Lambda binding-energy differences.

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