Validating the LO pionless-EFT $\Lambda$N+$\Lambda$NN hypernuclear Hamiltonian beyond the $s$ shell: $p$-shell binding, the dense-matter $\Lambda$ potential, and EFT systematics

At leading order (LO) in pionless effective field theory (πEFT), a $\Lambda$-nuclear Hamiltonian with five strangeness-sector parameters — spin-singlet and spin-triplet $\Lambda$N contact terms fixed by $\Lambda$N scattering lengths plus three $\Lambda$NN contact terms fitted to the $A\le4$ $s$-shell $\Lambda$ separation energies — was shown by Contessi, Barnea, and Gal to resolve the long-standing 1–3 MeV overbinding of $_\Lambda^5$He. Whether this minimal, purely $s$-shell-calibrated Hamiltonian is viable beyond the $s$ shell remains open: (i) can it reproduce $p$-shell ($6\le A\le16$) $\Lambda$ separation energies $B_\Lambda$ ab initio, including the ~1.8 MeV difference between $B_\Lambda({}_\Lambda^9\mathrm{Li})$ and $B_\Lambda({}_\Lambda^9\mathrm{Be})$ that phenomenology attributes to strongly repulsive $\Lambda$NN terms; and (ii) does it yield, in a rigorous many-body (e.g., quantum Monte Carlo) calculation rather than a Fermi-gas estimate, a $\Lambda$ single-particle potential in symmetric nuclear matter and pure neutron matter compatible with the empirical ~30 MeV well depth and stiff enough at neutron-star-core densities to bear on the hyperon puzzle. The most advanced follow-up (variational Monte Carlo with neural-network quantum states) reaches $A\le13$ but leaves $_\Lambda^9$Li uncomputed and $_\Lambda^9$Be compromised by its resonant $^8$Be core, so the splitting test and the dense-matter evaluation with this Hamiltonian are still undone.

Background

A $\Lambda$ hypernucleus is a nucleus in which one neutron is replaced by a $\Lambda$ baryon (a neutral hyperon carrying strangeness $S=-1$, with mass 1116 MeV and isospin 0), bound to the nuclear core by the hyperon–nucleon ($\Lambda$N) strong interaction. Its basic observable is the $\Lambda$ separation energy $B_\Lambda$, the energy required to remove the $\Lambda$; experimentally these are known from emulsion and spectroscopy measurements across the periodic table (reviewed by Gal, Hungerford, and Millener, Rev. Mod. Phys. 88, 035004 (2016)). The empirical ~30 MeV depth of the $\Lambda$ single-particle potential in symmetric nuclear matter (SNM, equal neutron and proton density) is inferred from these data, and the behavior of $\Lambda$ hyperons in pure neutron matter (PNM) at the several-times-saturation densities of neutron-star cores decides the 'hyperon puzzle': hyperons appearing in neutron-star cores soften the equation of state (EoS) and tend to reduce the maximum neutron-star mass below the observed ~2 solar masses, unless sufficiently repulsive hyperon–nucleon interactions — notably three-body $\Lambda$NN forces — counteract this softening. For light ($s$-shell, $A\le5$) hypernuclei, precise few-body calculations are possible, and most $\Lambda$N interaction models that reproduce $B_\Lambda$ for $A=3,4$ overbind $_\Lambda^5$He by 1–3 MeV — the 'overbinding problem'. Pionless effective field theory (πEFT) treats all baryon–baryon interactions as momentum-independent contacts smeared by a Gaussian regulator of width $\lambda^{-1}$, valid below the pion mass scale; its coupling constants are low-energy constants (LECs) fixed from data, with residual observables acquiring $O(Q/\lambda)$ cutoff dependence. Contessi, Barnea, and Gal (Phys. Rev. Lett. 121, 102502 (2018)) built an LO πEFT $\Lambda$-nuclear Hamiltonian with five LECs — two $\Lambda$N LECs tied to the spin-singlet and spin-triplet $\Lambda$N scattering lengths and three $\Lambda$NN LECs fitted to the three measured $A=3,4$ separation energies — and solved the few-body Schrödinger equation with the stochastic variational method (SVM), an essentially exact correlated-Gaussian basis technique. The resulting $B_\Lambda({}_\Lambda^5\mathrm{He})$ approaches the experimental $3.12\pm0.02$ MeV within a fraction of a MeV as $\lambda\to\infty$, largely resolving the overbinding problem, and the required $\Lambda$NN terms are repulsive, of qualitative strength similar to those used in quantum Monte Carlo (QMC) studies of the hyperon puzzle. In their outlook the authors pose two open directions: apply the derived $\Lambda$NN terms to $p$-shell hypernuclei ($6\le A\le16$), where shell-model studies using phenomenological inputs reproduce ground-state $B_\Lambda$ values except for the relatively large ~1.8 MeV difference between $B_\Lambda({}_\Lambda^9\mathrm{Li})$ and $B_\Lambda({}_\Lambda^9\mathrm{Be})$ — a splitting noted long ago to be settleable by strongly repulsive $\Lambda$NN terms — and apply the $\Lambda$N+$\Lambda$NN terms in state-of-the-art neutron-star matter calculations, since the paper's own evaluation of $\Lambda$ binding in SNM, a simple Fermi-gas estimate giving $B_\Lambda(\mathrm{SNM})\le27$ MeV at saturation density $\rho_0=0.16$ fm$^{-3}$, is only a lower bound demanding confirmation by rigorous many-body (Monte Carlo) techniques. Because the whole construction is truncated at LO, its robustness also requires NLO corrections (effective-range terms in $\Lambda$N, subleading three-body forces) and control of the $\lambda\to\infty$ extrapolation.

Problem Statement

Determine whether the LO pionless-EFT $\Lambda$N+$\Lambda$NN hypernuclear Hamiltonian of Contessi, Barnea, and Gal — two $\Lambda$N contact LECs fixed by the spin-singlet and spin-triplet $\Lambda$N scattering lengths and three $\Lambda$NN contact LECs fitted only to the $A=3,4$ $\Lambda$ separation energies — is a valid microscopic description of $\Lambda$ binding beyond the $s$ shell. This single objective is discharged by two linked deliverables. (1) $p$-shell test: compute $\Lambda$ separation energies for $p$-shell hypernuclei ($6\le A\le16$) with these $\Lambda$NN terms, for which both members of the $A=9$ mirror pair, $B_\Lambda({}_\Lambda^9\mathrm{Li})$ and $B_\Lambda({}_\Lambda^9\mathrm{Be})$, are required and together sufficient to resolve this deliverable (the remaining $p$-shell nuclei through $A=16$ are desirable but optional), and determine whether the observed $B_\Lambda$ systematics, in particular the ~1.8 MeV $B_\Lambda({}_\Lambda^9\mathrm{Li})$–$B_\Lambda({}_\Lambda^9\mathrm{Be})$ difference, are reproduced without refitting to $p$-shell data. (2) Dense-matter test: evaluate, by a rigorous many-body method (e.g., quantum Monte Carlo) rather than the source's Fermi-gas estimate, the $\Lambda$ single-particle potential or, equivalently, the $\Lambda$ binding/chemical potential in symmetric nuclear matter and in pure neutron matter with this Hamiltonian from saturation density up to neutron-star-core densities, with finite-size, statistical, and cutoff systematics, and conclude whether the repulsive $\Lambda$NN terms are compatible with the empirical ~30 MeV SNM well depth and sufficient to delay or suppress hyperon formation in neutron-star cores.

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