LENPIC NN Interaction

• LENPIC NN interactions are high-precision chiral EFT potentials built up to advanced orders, systematically incorporating both two- and three-nucleon forces.
• They utilize semilocal configuration- and momentum-space regularization to achieve accurate fits to nucleon-nucleon data while controlling cutoff artifacts.
• Rigorous LEC determination and Bayesian uncertainty quantification underpin precise ab initio predictions in few-nucleon and light nuclei research.

The LENPIC nucleon-nucleon (NN) interaction refers to a class of high-precision chiral effective field theory (EFT) nuclear potentials developed under the aegis of the Low Energy Nuclear Physics International Collaboration (LENPIC). These interactions are constructed up to high orders in the chiral expansion and systematically incorporate both two- and three-nucleon forces (2NF and 3NF), employing rigorous regularization, data-constrained low-energy constants (LECs), and state-of-the-art uncertainty quantification. LENPIC potentials are available in both semilocal configuration-space (SCS) and semilocal momentum-space (SMS) regulated versions, with the latter enabling robust ab initio calculations for few-body and light-to-medium nuclei.

1. Chiral EFT Framework and Operator Structure

LENPIC interactions are formulated within Weinberg's chiral EFT, which encodes the symmetries and breaking patterns of QCD at low energies using nucleon and pion degrees of freedom. The chiral expansion is organized in powers of the soft scale $Q~m_\pi$ over the breakdown scale $\Lambda_b~650$ MeV. The nuclear Hamiltonian at chiral order $\nu$ consists of:

• Long-range contributions: one-pion exchange (OPE), two-pion exchange (TPE), and, at higher orders, three-pion exchange (3PE). The TPE and 3PE kernels involve LECs $c_1$, $c_3$, $c_4$ from $\pi N$ scattering and higher-order subleading $\bar d_i$, $\bar e_i$.
• Short-range contact terms:
  • LO: two S-wave operators ($C_S$, $C_T$)
  • NLO: seven derivative operators
  • N$^2$LO: no new contacts
  • N$^3$LO and higher: additional derivative operators (up to eleven local operators at N$^3$LO)
• Three-nucleon forces: Appear at N$^2$LO, with structure:
  • 2$\pi$-exchange (2$\pi$E) (proportional to $c_1$, $c_3$, $c_4$)
  • 1$\pi$-contact (proportional to $c_D$)
  • Pure contact (proportional to $c_E$)

Each operator is multiplied by a regulator function to provide a smooth cutoff at high momenta or small distances.

2. Semilocal Momentum-space Regularization and Regulator Choices

The LENPIC SMS regularization scheme applies the same functional regulator to both long- and short-range terms to maintain locality in the IR while controlling the UV behavior:

• Long-range pion exchange:

$$f_{\mathrm{LR}}(q) = \exp\left[-\frac{q^2 + M_\pi^2}{\Lambda^2}\right]$$

• Short-range contact terms:

$$f_{\mathrm{SR}}(p',p) = \exp\left[-\frac{p'^2 + p^2}{\Lambda^2}\right]$$

The cutoff parameter $\Lambda$ is set to values (e.g., 450 and 500 MeV in current LENPIC SMS implementations) large enough to preserve pion physics but sufficiently low to suppress inconsistent short-distance dynamics and regulator artifacts.

Compared to earlier SCS (coordinate-space) regulators, SMS regulation:

• Removes regulator oscillations in partial-wave decompositions,
• Allows direct matching to high-quality NN data fits,
• Reduces cutoff artifacts in three-nucleon observables without altering long-range physics for each cutoff,
• Yields very similar $\chi^2$ fits in the two-nucleon sector at N$^4$LO and beyond.

3. Low-energy Constants and Fitting Protocol

The LECs controlling long-range chiral interactions are determined from independent pion-nucleon scattering analyses (Roy–Steiner input for N$^2$LO: $c_1=-0.74$, $c_3=-3.61$, $c_4=+2.44$ GeV$^{-1}$), ensuring consistency across NN and 3NF sectors. NN contact LECs at LO and NLO are adopted from global fits (e.g., to the Granada $np+pp$ database below 300 MeV lab energy). 3NF LECs, $c_D$ and $c_E$, are fixed in two steps:

1. The triton ($^3$H) binding energy is used to determine a correlation line $c_E(c_D)$.
2. The neutron–deuteron elastic differential cross-section minimum at $E_N=70$ MeV fixes $c_D$.

Representative fitted values: | $\Lambda$ (MeV) | $c_D$ | $c_E$ | |-----------------|---------|---------| | 450 | +2.485 | –0.528 | | 500 | –1.626 | –0.063 |

This protocol ensures direct control of 3NF effects and stable reproduction of few-nucleon observables (Maris et al., 2020).

4. Three-Nucleon Force (3NF) at N$^2$LO in Momentum Space

The regularized N$^2$LO SMS 3NF reads: $$V^{3N}_\Lambda = \frac{g_A^2}{8 F_\pi^4}e^{-(q_1^2+M_\pi^2)/\Lambda^2}e^{-(q_3^2+M_\pi^2)/\Lambda^2} \frac{\vec{\sigma}_1\cdot\vec{q}_1\,\vec{\sigma}_3\cdot\vec{q}_3}{(q_1^2+M_\pi^2)(q_3^2+M_\pi^2)} \Big[2c_1M_\pi^2T_{13} + c_3T_{13}(\vec{q}_1\cdot\vec{q}_3) + \frac{c_4}{2}T_{132}\,\vec{\sigma}_2\cdot(\vec{q}_1\times\vec{q}_3)\Big]$$

$$- \frac{g_A c_D}{8F_\pi^4\Lambda_\chi}T_{13}e^{-(p_{12}^2+p_{12}'^2)/\Lambda^2} \frac{\vec{\sigma}_3\cdot\vec{q}_3}{q_3^2+M_\pi^2} + \frac{c_E}{2F_\pi^4\Lambda_\chi}T_{12}e^{-(p_{12}^2+p_{12}'^2)/\Lambda^2} + \text{(5 perms)}$$

with

• $q_i = p'_i - p_i$ (momentum transfers),
• $T_{ij} = \vec{\tau}_i\cdot\vec{\tau}_j$, $$T_{ijk} = \vec{\tau}_i\times\vec{\tau}_j\cdot\vec{\tau}_k$$,
• $g_A$, $F_\pi$, $M_\pi$ standard constants, $\Lambda_\chi = 700$ MeV.

A subtraction constant $C(\Lambda)$ is introduced so that the regulated long-range 2$\pi$E force vanishes at zero separation in coordinate space, ensuring proper short-distance behavior.

5. Implementation, Uncertainty Quantification, and Performance

• Partial-wave decompositions are performed numerically in momentum space, with regulator choices ensuring rapid convergence and avoiding spurious artifacts.
• Few-nucleon properties ($A=3,4$) are computed with Faddeev-Yakubovsky solvers using the full 3NF structure, and larger nuclei ($A\leq16$) are treated via No-Core Configuration Interaction (NCCI, MFDn), with similarity renormalization group (SRG) evolution to handle many-body operators.
• Uncertainty estimation uses a Bayesian truncation-error model, where observables $X$ are expanded as $$X = X^{(0)} + \Delta X^{(2)} + \Delta X^{(3)} + \ldots$$, with $Q = \max(p/\Lambda_b, M_\pi^\text{eff}/\Lambda_b)$. Gaussian priors for the EFT expansion coefficients and marginalization over the next $h$ omitted orders provide correlated truncation error estimates (Maris et al., 2020).
• Performance:
  • Nd scattering and induced-deuteron breakup reactions are accurately reproduced up to 135 MeV within combined experimental and theoretical errors,
  • Reproduction of $A=3,4$ binding energies and radii once 3NF parameters are fitted,
  • In light nuclei ($A\leq 16$), a modest overbinding trend for $A\gtrsim 10$ appears, within a few MeV, consistent with other chiral interactions,
  • SMS regulators yield efficient and artifact-suppressed results, setting the stage for higher-order ($N^3$LO and above) consistent implementations.

6. Extension to Higher Orders and Convergence Properties

Order-by-order studies reveal the dominant role of two-pion– and three-pion–exchange at $N^4$LO and $N^5$LO, with explicit formulas and phase-shift predictions available from Entem, Kaiser, and Machleidt (Entem et al., 2014, Entem et al., 2015). Key findings:

• At $N^4$LO, new TPE loops contribute significant mid-range repulsion, compensating excessive attraction at NNLO and $N^3$LO, thereby achieving near-perfect reproduction of peripheral partial waves,
• Genuine 3$\pi$-exchange is numerically small but its inclusion improves the overall convergence pattern, particularly for $F$ and $G$ waves (Entem et al., 2014, Kaiser, 2015).
• At $N^5$LO, new long-range contributions are predominantly repulsive and about $30-50\%$ smaller than the $N^4$LO corrections, confirming the rapid convergence of the chiral expansion (Entem et al., 2015). Peripheral phase shifts are predicted to within $0.05^\circ$ up to 200 MeV, with N$^5$LO corrections entirely consistent with minimal higher-order systematic uncertainties.

7. Impact, Applications, and Comparative Assessment

LENPIC NN interactions, in both SCS and SMS forms, enable precise ab initio studies of light-nuclei structure and reactions, neutron matter, and electroweak observables. SMS regularization facilitates high-precision NN fits, stable 3NF effects, and reliable uncertainty quantification within correlated Bayesian frameworks. In practical applications, LENPIC interactions:

• Match or exceed the predictive power of other modern chiral and phenomenological potentials,
• Offer a systematically improvable, controlled expansion for both the NN force and many-body nuclear observables,
• Demonstrate robust convergence, consistency between sectors, and reduced sensitivity to cutoff artifacts (Maris et al., 2020, Entem et al., 2014, Entem et al., 2015).

The rigorous protocols for LEC determination, regulator choice, and error estimation position the LENPIC NN interaction as a backbone for precision calculations and EFT-based uncertainty quantification in nuclear theory.