Few-Nucleon Systems within Finite-Cutoff Pionless EFT

TL;DR

This work demonstrates that a finite-cutoff pionless EFT, equipped with Gaussian-regulated two- and three-body forces and solved via stochastic variational methods in a weak harmonic-oscillator trap, can accurately describe light-nucleus binding energies up to A=5 and a broad set of low-energy S-wave scattering observables. By fitting NN scattering lengths and effective ranges and calibrating a three-body force to reproduce ${}^3 ext{H}$, ${}^3 ext{He}$, and ${}^4 ext{He}$ binding energies, the approach yields deuteron binding in excellent agreement with experiment and reproduces NLO πEFT results across multiple channels, including proton–proton and various N–N–N scattering channels in both neutral and Coulomb-modified contexts. The results confirm the predictive power of finite-cutoff πEFT for few-body nuclear physics, reveal where three-body forces are essential (notably in the Nd doublet channel), and provide a solid benchmark suite for A=2–5 systems, with potential extensions to larger nuclei and reaction processes. The methodology offers computational efficiency and a clear path toward systematic improvements by higher-order corrections, making it a practical framework for describing low-energy nuclear dynamics.

Abstract

We investigate pionless effective field theory (\nopieft) with finite-cutoff regularization as a framework for describing few-nucleon systems. This formulation incorporates effective-range effects already at leading order (LO), thereby reaching next-to-leading-order (NLO) accuracy while maintaining computational efficiency. Using correlated-Gaussian stochastic variational methods in a weak harmonic-oscillator trap, together with neutral and Coulomb-modified quantization conditions, we calculate binding energies and low-energy $S$-wave scattering parameters for systems with up to five nucleons. At an optimal cutoff, the computed binding energies of the deuteron, triton, helion, and alpha particle reproduce experimental values at the percent level once a three-body force is included. Scattering parameters for proton--proton, nucleon--deuteron, nucleon--triton, proton--helion, deuteron--deuteron, and nucleon--alpha channels are obtained and found to be consistent with both experimental data and existing NLO \nopieft\ calculations. These results demonstrate that finite-cutoff \nopieft\ offers a robust and predictive framework for few-body nuclear physics.

Figures (5)

• Figure 1: Deuteron ground-state energy in $\slashed{\pi}$EFT as a function of the cutoff $\Lambda$. The LO (blue squares) and NLO (red dots) results coincide with the experimental value (green dashed line) at $\Lambda \approx 1.25 \,\text{fm}^{-1}$, where the empirical effective range is already reproduced at LO.
• Figure 2: Effective-range expansion for nucleon--deuteron scattering in the $S=3/2$ channel. Results for neutral $n$--$d$ scattering (cyan squares, Eq. \ref{['eq:ere']}) and charged $p$--$d$ scattering (blue dots, Eq. \ref{['eq:coulomb_ere']}) are shown together with second-order polynomial fits (solid lines). Available experimental data are also included: Phillips et al. PhiBar69 for $n$--$d$, and Arvieux Arv74, Huttel et al. HutArnBau83, Kievski et al. KieRosTor96, and Wood et al. WooBruFis02 for $p$--$d$.
• Figure 3: Modified effective-range expansion for nucleon--deuteron scattering in the $S=1/2$ channel. Results for $n$--$d$ (cyan squares) and $p$--$d$ (blue dots) are fitted using Eq. \ref{['eq:mere']} (solid lines). Experimental data are included for comparison: Van Oers et al. OerBro67 ($n$--$d$), and Arvieux Arv74, Huttel et al. HutArnBau83, Kievski et al. KieRosTor96, and Wood et al. WooBruFis02 ($p$--$d$).
• Figure 4: Spectrum of four nucleons in a harmonic trap with $S=0$, $I=0$, and $L=0$. The bound $^4$He state is shown together with the $p$--$t$, $n$--$h$, $d$--$d$, and $d$--$n$--$p$ thresholds (dashed lines), as well as the corresponding trapped energies (solid lines).
• Figure 5: Spectrum of four nucleons in a harmonic trap with $S=0$, $I=0$, and $L=0$. The deuteron--deuteron scattering states are shown by solid lines and the proton--triton states by dashed lines. An avoided crossing is visible when $d$--$d$ energies approach those of the $p$--$t$ channel.