Closer look at enhanced three-nucleon forces

In a recent publication, Cirigliano {\it et al.} [Phys. Rev. Lett. 135, 022501 (2025)] argue that three-nucleon forces (3NFs) involving short-range operators that couple two pions with two nucleons are enhanced beyond what is expected in chiral effective field theory based on naive dimensional analysis. Here, we scrutinize the arguments and conclusions of that paper by taking into account renormalization scheme dependence of the corresponding low-energy constants. We gain further insights into the expected impact of these 3NFs by comparing them with contributions of similar type, induced by pion-exchange diagrams at lower orders in the chiral expansion. We also estimate the impact of these 3NFs on properties of nuclear matter. After removal of scheme-dependent short-distance components in pion loops, the 3NFs considered by Cirigliano {\it et al.} are shown to yield reasonably small contributions to the equation of state of neutron and symmetric nuclear matter in agreement with expectations based on Weinberg’s power counting.

💡 Research Summary

In this paper the authors critically examine the claim made by Cirigliano et al. (Phys. Rev. Lett. 135, 022501 2025) that three‑nucleon forces (3NFs) involving short‑range two‑pion–two‑nucleon operators are enhanced by two inverse powers of the chiral expansion parameter Q, thereby promoting a nominal Q⁶ (N⁵LO) contribution to the level of Q⁴ (N³LO). The analysis proceeds in several steps.

First, the renormalization‑group (RG) behavior of the low‑energy constant D₂, which multiplies a quark‑mass‑dependent NN contact interaction, is revisited. The authors emphasize that the scaling of D₂ is scheme‑dependent. In the Kaplan‑Savage‑Wise (KSW) scheme, where the subtraction point μ is chosen of order the soft scale (μ∼p≪Λ_b), the RG flow forces D₂ to scale as Q⁻², exactly as assumed by Cirigliano et al. In contrast, Weinberg’s power‑counting scheme adopts a hard‑scale subtraction (μ∼Λ_b) and treats all LECs according to naive dimensional analysis (NDA). In this scheme D₂ and the related constant F₂ are naturally of order unity (in appropriate units) and do not acquire the large enhancement. By extracting D₂ and F₂ from state‑of‑the‑art chiral NN potentials (e.g., EMN, SMS) the authors find values |D₂|≈0.5–1 fm⁴ and |F₂|≈0.2–0.5 fm⁴, fully consistent with NDA.

Second, the authors discuss the necessity of removing scheme‑dependent short‑range pieces that arise in dimensional‑regularization (DR) loop calculations. The two‑pion‑exchange loops contain polynomial terms that depend on the regularization scheme; these must be subtracted (e.g., via t‑channel dispersion relations) to isolate the genuine long‑range physics. After this subtraction the (e)‑type 3NF proportional to D₂ (and the analogous F₂ term) remains a genuine Q⁶ contribution. Its magnitude is then compared with the parameter‑free N³LO and N⁴LO 3NFs of type (e) generated by three‑pion‑exchange diagrams of type (b). The comparison shows that the D₂‑induced 3NF is smaller by roughly two orders of magnitude relative to the leading long‑range three‑pion contributions.

Third, the paper investigates the convergence pattern of the chiral expansion in the NN sector. By applying the same subtraction of short‑distance pieces to the 2π‑exchange NN potential, the authors demonstrate that the hierarchy of LO, NLO, N²LO, etc., becomes manifest: higher‑order corrections are naturally suppressed, confirming the internal consistency of Weinberg’s counting.

Fourth, the impact of the D₂‑ and F₂‑driven 3NFs on the equation of state (EoS) of pure neutron matter and symmetric nuclear matter is evaluated. Using regularized three‑body forces together with the subtracted NN potentials, the authors compute the energy per particle and pressure as functions of density. The D₂‑type 3NF contributes at most a 1 % correction to the EoS in the density range relevant for nuclear structure and neutron‑star physics, whereas the established N³LO/N⁴LO three‑pion 3NFs provide 5–10 % effects, in line with empirical constraints.

Finally, the authors conclude that the “enhancement” reported by Cirigliano et al. is an artifact of adopting the KSW renormalization scheme and of retaining scheme‑dependent short‑range loop pieces. When the Weinberg scheme is employed and the short‑distance contributions are removed, the D₂‑type 3NF behaves exactly as predicted by naive dimensional analysis and Weinberg power counting: it is a higher‑order, numerically small correction. Consequently, the existing hierarchy of nuclear forces used in modern ab‑initio calculations remains valid, and no revision of the standard chiral EFT power counting is required.