Consistent determination of chiral three-nucleon-force contact strengths: complete N3LO dynamics, simultaneous continuum and bound-state constraints, and consequences for nuclear structure

Beyond next-to-next-to-leading order (N2LO), the chiral three-nucleon force (3NF) acquires thirteen independent short-range contact terms whose strengths are unknown. A 2022 study fitted all of them, together with the N2LO constants $c_D$ and $c_E$, to elastic nucleon-deuteron (Nd) scattering data using the semilocal momentum-space-regularized (SMS) N4LO+ two-nucleon potential. The fit improved the continuum description, but the entire N3LO 3NF had to be omitted (making the fitted strengths preliminary and likely biased), the continuum-only fit failed to reproduce the triton binding energy and the doublet Nd scattering length, the impact on heavier nuclei was left unquantified, and the gap in the total Nd cross section above $\approx 200$ MeV persisted. The open problem is to determine these contact strengths within a Hamiltonian that includes the complete, symmetry-preserving N3LO 3NF, by a fitting strategy that simultaneously constrains continuum and bound-state observables, and to validate the resulting Hamiltonian in nuclear matter and heavier nuclei, clarifying at the same time the origin of the high-energy total-cross-section gap. As of August 2026 the problem is open: the required regularization formalism exists and its numerical implementation by the LENPIC collaboration is in progress, exploratory fits indicate that at most 9 of the 13 subleading strengths are determinable from elastic Nd data alone, and nuclear-matter tests show that the continuum-fitted strengths are not transferable to nuclear systems.

Background

Chiral effective field theory (chiral EFT) is the standard framework for nuclear forces: nucleons and pions interact through an expansion in $Q/\Lambda_b$ (typical external momentum over the breakdown scale), organized in chiral orders NLO, N2LO, N3LO, N4LO, and so on. Two-nucleon (NN) potentials in this framework, such as the semilocal momentum-space-regularized (SMS) N4LO+ potential of Reinert, Krebs, and Epelbaum (Eur. Phys. J. A 54, 86 (2018)), reproduce NN scattering data with near-perfect precision. Three-nucleon forces (3NF), needed for quantitatively correct predictions in systems with mass number $A\ge3$, first appear at N2LO: a parameter-free two-pion-exchange term plus two short-range contact terms whose unknown numerical strengths (low-energy constants, LECs) $c_D$ and $c_E$ are conventionally fixed from the triton ($^3$H) binding energy combined with Nd scattering data or the triton beta decay. At subleading order the purely short-range part of the 3NF contains thirteen independent contact operators (Girlanda, Kievsky, and Viviani, Phys. Rev. C 84, 014001 (2011)), with LECs commonly denoted $E_1,\dots,E_{13}$; the N3LO corrections to the 3NF were derived long ago using dimensional regularization (Ishikawa and Robilotta, Phys. Rev. C 76, 014006 (2007); Bernard, Epelbaum, Krebs, and Meißner, Phys. Rev. C 77, 064004 (2008)), but combining such pieces with cutoff regularization was later shown to violate chiral symmetry, so they cannot simply be added to the SMS Hamiltonian. Witała, Golak, and Skibiński (Phys. Rev. C 105, 054004 (2022)) asked whether all contact strengths can be determined from elastic nucleon-deuteron (Nd) scattering itself. They solved the Faddeev equations (the standard exact three-body scattering equations) with the SMS N4LO+ NN potential plus the N2LO 3NF and all thirteen N4LO contact terms, and fitted the strengths to cross sections and analyzing powers (polarization observables) at incoming-nucleon energies of 10, 70, and 135 MeV; since two pairs among the N4LO contact terms are fully equivalent for elastic Nd observables ($E_{11}$ with $E_9$ and $E_{12}$ with $E_{10}$), the 3N Hamiltonian depends altogether on thirteen independent parameters. The fit markedly improved the description of the data, including at the unfitted energies 190 and 250 MeV, supporting the hope that subleading contact terms cure notorious 3N puzzles: the long-standing underprediction of the elastic Nd differential cross section at its angular minimum, and of the low-energy nucleon analyzing power $A_y$ (the spin-dependent left-right asymmetry of the scattered nucleon). However, the authors themselves identified four unresolved issues. First, the N3LO 3NF was entirely omitted; pseudo-data tests showed that fitting with incomplete dynamics shifts the extracted strengths dramatically (for example $c_D$ from 1.00 to $\approx 3.98$ with $\chi^2/N$ rising from $\approx 0$ to $\approx 80$), and the final fit gave $\chi^2/N\approx 35$ with unusually large, strongly correlated strengths ($E_1\approx 6.40$, $E_2\approx 7.80$, $E_3\approx 6.97$, $E_7\approx -7.40$), so the authors call their own fit preliminary and ask that it be repeated once a consistently regularized N3LO 3NF becomes available. Second, the continuum-only fit fails to reproduce the triton binding energy $E^{\rm exp}_{^3{\rm H}}=-8.4820(1)$ MeV and the doublet Nd scattering length $^{2}a^{\rm exp}_{nd}=0.645\pm0.03$ fm (the scattering length of the nd channel with total spin $1/2$) — two observables tied to the continuum by the empirical Phillips line (the correlation between $E_{^3{\rm H}}$ and $^{2}a_{nd}$ along which predictions of realistic NN potentials, with or without standard 3NFs, cluster); hence the traditional strategy of fixing $c_D$ and $c_E$ along the triton correlation line must be revised when subleading contacts are present. Third, the fitted contact terms produce sizable potential-energy contributions, suggesting large — but unquantified — effects on the spectra of heavier nuclei. Fourth, the gap between theory and data in the total Nd cross section above $\approx 200$ MeV survives the contact terms; the authors conclude it very likely signals the opening of real pion production, which a purely nucleonic framework cannot describe.

Problem Statement

Determine the short-range part of the chiral three-nucleon force — the N2LO strengths $c_D$ and $c_E$ together with the thirteen N4LO contact strengths $E_1,\dots,E_{13}$ — within a three-nucleon Hamiltonian built on the SMS N4LO+ NN potential that also includes the complete, consistently (symmetry-preserving) regularized N3LO 3NF, such that: (1) the strengths, with uncertainties, are extracted by a fitting strategy that simultaneously describes elastic nucleon-deuteron scattering observables below the pion-production threshold and the three-nucleon bound-state benchmarks — the triton binding energy and the doublet Nd scattering length $^{2}a_{nd}$ — replacing the N2LO-era correlation-line practice; (2) the impact of the so-determined N4LO contact terms on nuclear bound systems is established, namely their contribution to binding energies and spectra of heavier nuclei and to nuclear and neutron matter, testing whether the so-determined strengths are compatible with nuclear structure; and (3) as the validity boundary of the purely nucleonic framework, the origin of the gap between theory and data in the total Nd cross section above $\approx 200$ MeV is identified — in particular whether explicit pion-production (inelastic) channels, relativistic dynamics, or another mechanism accounts for it. Each deliverable is complete only when its outcome is reported: (1) when a single fit simultaneously describes the sub-threshold elastic Nd observables and the 3N bound-state benchmarks — the triton binding energy and the doublet Nd scattering length $^{2}a_{nd}$ — and reports the strengths with uncertainties for the independently determinable N4LO combinations, stating explicitly which combinations cannot be separately fixed (for elastic Nd observables $E_{11}$ and $E_{12}$ are equivalent to $E_9$ and $E_{10}$, and elastic Nd data alone cannot fix all thirteen subleading strengths); (2) when the contribution of those contact terms to the binding energies and spectra of a representative set of heavier nuclei ($A\ge4$, e.g. a light $p$-shell nucleus) and to the energy per particle of symmetric nuclear and neutron matter is quantified, together with an explicit verdict on compatibility with nuclear structure, where 'compatible' means the induced shifts remain consistent with the data within the fit's propagated uncertainties rather than growing to dominate the observables; and (3) when the origin of the total Nd cross-section gap above $\approx 200$ MeV is decided from quantitative evidence — at least one named mechanism (explicit pion-production (inelastic) channels, relativistic dynamics, or another mechanism) is evaluated and its contribution compared with the measured gap, and it is stated whether these named mechanisms suffice or a further mechanism is required — rather than assumed from the onset of pion production. These three deliverables are jointly required and share one scientific objective: a complete and internally consistent chiral three-nucleon Hamiltonian at subleading contact order, validated across the 3N continuum, the 3N bound states, and nuclear matter/heavier nuclei.

References

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