[Paper Review] The Missing Three-Nucleon Forces: Where Are They?

R. Machleidt|ArXiv.org|Sep 15, 2009

Nuclear physics research studies3 citations

TL;DR

This paper investigates the origin and magnitude of three-nucleon forces (3NF) in chiral effective field theory (ChEFT), arguing that next-to-next-to-next-to-leading order (N³LO) 3NF contributions in the Δ-less formulation are likely too weak to resolve outstanding nuclear physics puzzles. Instead, it identifies one-loop 3NF diagrams at N⁴LO (Δ-less) or N³LO (Δ-full) theory as the most promising candidates for the missing strong 3NF contributions needed to resolve issues like the Aₙ puzzle.

ABSTRACT

In recent years, there has been substantial progress in the derivation of nuclear forces from chiral effective field theory. Accurate two-nucleon forces (2NF) have been constructed up to next-to-next-to-next-to-leading order (N3LO) of chiral perturbation theory and applied in microscopic nuclear structure calculations with a good degree of success. However, chiral three-nucleon forces (3NF) have been used only at N2LO, improving some miscroscopic predictions, but leaving also several issues, like the "Ay puzzle'" of nucleon-deuteron scattering, unresolved. Thus, the 3NF at N3LO is needed for essentially two reasons: For consistency with the 2NF, and to (hopefully) improve some critical predictions of nuclear structure and reactions. However, there are indications that the 3NF at N3LO (in the so-called Delta-less version of the theory) is rather weak and may not solve any of the outstanding problems. If this suspicion is confirmed, we have to go beyond, which may be similar to opening Pandora's Box. In this talk, I will discuss the various possible scenarios and how to deal with them.

Motivation & Objective

• To identify the origin of missing three-nucleon forces (3NF) in chiral effective field theory (ChEFT) that could resolve persistent discrepancies in nuclear structure calculations.
• To assess whether next-to-next-to-next-to-leading order (N³LO) 3NF contributions in the Δ-less ChEFT framework are sufficiently large to address unresolved issues such as the Aₙ puzzle.
• To evaluate the role of Δ(1232) isobar degrees of freedom in improving the convergence and strength of 3NF contributions at higher orders.
• To determine whether higher-order loop diagrams—particularly those involving subleading LECs and Δ excitations—provide the necessary sizable 3NF effects.
• To guide future theoretical and experimental efforts toward identifying and measuring the missing 3NF components in few-nucleon systems.

Proposed method

• Uses chiral perturbation theory (ChPT) power counting to classify nuclear force contributions by their order ν, with ν = 2m − 4 for m-nucleon forces.
• Applies the power counting formula ν = 2L + ∑Δᵢ for irreducible diagrams, where Δᵢ = dᵢ + nᵢ/2 − 2 accounts for vertex dimensionality and nucleon legs.
• Analyzes one-loop 3NF diagrams at N⁴LO in the Δ-less theory and N³LO in the Δ-full theory, focusing on those with subleading LECs (cᵢ) and Δ-isobar vertices.
• Compares diagrams with and without explicit Δ-isobars, showing that Δ-inclusion shifts strength to lower orders and improves convergence.
• Evaluates the relative size of 3NF contributions using estimates from analogous 2NF calculations (e.g., 3PE diagrams with subleading vertices found to be ~10× larger than leading-order ones).
• Uses symbolic representations (e.g., Fig. 4a and 4b) to illustrate how loop diagrams with one cᵢ vertex or Δ excitation generate potentially large 3NF contributions.

Experimental results

Research questions

• RQ1 Why are next-to-next-to-next-to-leading order (N³LO) three-nucleon forces in the Δ-less chiral EFT framework unlikely to resolve the Aₙ puzzle?
• RQ2 What is the role of Δ(1232) isobar degrees of freedom in generating stronger, more convergent three-nucleon force contributions at higher orders?
• RQ3 Are one-loop 3NF diagrams involving subleading LECs (cᵢ) and Δ excitations the dominant missing contributions at N⁴LO (Δ-less) or N³LO (Δ-full) order?
• RQ4 How do the relative sizes of 3NF contributions compare across different orders and formulations (Δ-less vs. Δ-full), and what does this imply for predictive power?
• RQ5 Can the inclusion of Δ-isobars in ChEFT explain the apparent failure of N³LO 3NFs to resolve long-standing discrepancies in nucleon-deuteron scattering?

Key findings

• The N³LO three-nucleon force in the Δ-less chiral EFT framework is unlikely to produce sizable contributions due to unnaturally small values of the subleading LECs (cᵢ).
• One-loop 3NF diagrams at N⁴LO in the Δ-less theory or N³LO in the Δ-full theory, involving subleading LECs and Δ-isobar vertices, are expected to be the dominant missing contributions.
• These diagrams are predicted to be significantly larger than leading-order 3NF contributions, based on analogous 2NF calculations where subleading 3PE diagrams are ~10× larger than leading ones.
• The inclusion of Δ-isobars improves convergence by moving strength from large LECs (cᵢ) to lower orders, making the Δ-full theory more efficient and predictive.
• The failure of N³LO 3NFs to resolve the Aₙ puzzle suggests that the missing 3NFs may lie beyond N³LO, possibly requiring a reevaluation of the theoretical framework.
• The 3NF contributions from one-loop diagrams with one Δ excitation (N³LO in Δ-full theory) are identified as the most promising candidates to resolve outstanding issues in microscopic nuclear structure.