Closer look at enhanced three-nucleon forces

TL;DR

This paper critically examines Cirigliano et al.'s claim that short-range two-pion–two-nucleon 3NFs, encoded by $D_2$ and $F_2$, are enhanced and should enter at lower orders. By dissecting RG scheme dependence and comparing with NN sector analogies, the authors show that the apparent enhancement of $D_2$ is not universal and depends on the renormalization conditions; when short-distance pieces in pion loops are removed via dispersive/finite-cutoff methods, the proposed 3NFs contribute only modestly to the nuclear EoS. The work demonstrates the importance of regulator choices in chiral EFT and provides updated estimates for the impact of these 3NFs on neutron and symmetric nuclear matter, indicating consistency with Weinberg power counting in a finite-cutoff framework. It also highlights that the large coefficients associated with subleading $2\pi$-exchange terms can produce apparent enhancements, but these are mitigated by proper regularization and, in some formulations, by moving certain contributions to lower orders. Overall, the findings support a conservative view: when properly renormalized and regulated, the newly proposed 3NFs have limited influence on nuclear matter properties, preserving the established EFT hierarchy.

Abstract

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.

Figures (7)

• Figure 1: Diagrams contributing to the 3NF in chiral EFT based on Weinberg's power counting with pions and nucleons as the only explicit degrees of freedom. Up to N$^4$LO, all possible 3NF topologies are shown in the leftmost column and include the two-pion exchange (a), two-pion-one-pion exchange (b), ring (c), one-pion-exchange-contact (d), two-pion-exchange-contact (e) and purely contact (f) diagrams. Dashed and solid lines denote pions and nucleons, respectively. Solid dots, filled circles, filled diamonds and filled squares denote the vertices from the effective chiral Lagrangian of dimension $\Delta_i = 0$, $1$, $2$ and $3$, respectively. The dominant N$^2$LO contributions are derived in Refs. vanKolck:1994yiEpelbaum:2002vt. The expressions for the N$^3$LO contributions, calculated using dimensional regularization, can be found in Refs. Ishikawa:2007zzBernard:2007spBernard:2011zr, while the N$^4$LO corrections of types (a), (b) and (c) have been worked out in Refs. Krebs:2012yvKrebs:2013kha. The subleading contact 3NF of type (f) is discussed in Ref. Girlanda:2011fh, while the N$^4$LO contributions of the type (d) and (e) have not yet been worked out. The diagram in the last column is considered by Cirigliano et al.Cirigliano:2024ocg and argued to be enhanced beyond NDA as explained in the text.
• Figure 2: The leading and subleading contributions to the S-wave NN scattering amplitude in pionless EFT. For notation see Fig. \ref{['fig1']}.
• Figure 3: Scaling of various diagrams for the KSW and Weinberg (W) choices of renormalization conditions specified in sections \ref{['sec:KSW']} and \ref{['sec:Weinberg']} are shown in the upper and lower rows, respectively. The scaling of diagrams involving two potential pions is shown for spin-triplet NN channels, while the corresponding contributions in spin-singlet channels are suppressed. Notice further that for the near-threshold kinematics, the contributions of the first two (subleading) diagrams in the Weinberg scheme get enhanced by the factor of $Q^{-2}$ after dressing them with the LO amplitude stemming from resummed diagrams on the right-hand side of the inequality. For remaining notation see Fig. \ref{['fig1']}.
• Figure 4: Examples of loop diagrams contributing to the $\pi N$ amplitude in covariant ChPT with a mismatch between the chiral order and the power of the ultraviolet divergence. Diagram (a) contributes at order $Q^4$ while diagram (b) with intermediate delta excitations shown by double lines appears at order $\epsilon^3$ in the small scale expansion. For remaining notation see Fig. \ref{['fig1']}.
• Figure 5: ChPT predictions for neutron-proton D-wave phase shifts. Dotted lines are LO results based on the $1\pi$-exchange potential, while dashed and solid lines emerge from taking into account the NLO and N$^2$LO $2\pi$-exchange in Eq. (\ref{['TPEPNN']}) calculated using dimensional regularization. Blue bands show the N$^2$LO predictions after removing the short-range components of the $2\pi$-exchange by imposing a sharp cutoff $\Lambda = 500-800$ MeV in the spectral integral. Filled circles are empirical phase shifts of the Nijmegen partial wave analysis Stoks:1993tb. Figure adapted from Refs. Epelbaum:2024gfgEpelbaum:2003gr.