Perturbative $χ$EFT calculations of the deuteron and triton up to N$^2$LO

TL;DR

This work extends perturbative χEFT calculations for the deuteron and triton to next-to-next-to-leading order ($N^2LO$) within the Long–Yang power counting, treating subleading interactions perturbatively and iterating only selected LO components nonperturbatively. It analyzes exceptional cutoff values in the two-nucleon sector, including the $^3P_0$ and $^3S_1 ext{-}^3D_1$ channels, and shows how these affect NN observables and the three-nucleon system, notably the triton, using a no-core shell-model framework. A key finding is that shifting the LO wave function can mitigate exceptional cutoffs in the NN sector (notably the $^3P_0$ channel) and render some $N^2LO$ predictions finite, though the $^3S_1 ext{-}^3D_1$ channel presents a more stubborn divergence in the triton. The study documents convergence trends up to $oldsymbol{ extΛ \\lesssim 1200 ext{ MeV}}$, highlights regulator-related challenges, and points to regulator choices and possible inclusion of three-nucleon forces as important directions for achieving RG-invariant, high-precision few-nucleon predictions.

Abstract

We extend previous studies of the deuteron and triton ground-state energies to next-to-next-to-leading order (N$^2$LO) in chiral effective field theory, employing a power counting in which subleading interactions are treated perturbatively. Triton calculations are performed using the no-core shell model, and we demonstrate converged perturbative results for regulator cutoffs up to $Λ\approx 1200$ MeV. We analyze exceptional cutoffs in the $^3P_0$ and $^3S_1 \mathrm{-}^3D_1$ nucleon-nucleon channels and find a resulting cutoff dependence in the triton ground-state energy at N$^2$LO. The effect associated with the exceptional cutoff in the $^3P_0$ channel can be mitigated by redefining the leading-order wave function within the freedom allowed by the effective field theory. The same approach applied in the $^3S_1 \mathrm{-}^3D_1$ channel remedies the effect of this exceptional cutoff on the ground state energy of the deuteron, but not for the triton.

Figures (19)

• Figure 1: Values of the LECs at LO (top panel) and N$^2$LO (remaining panels) in the $^3P_0$ channel as a function of the momentum cutoff, $\Lambda$. The solid vertical line marks the location of the limit-cycle-like cutoff, while the dashed vertical line marks the exceptional cutoff in the given cutoff interval.
• Figure 2: Values of the LECs at LO (top panel) and N$^2$LO (remaining panels) in the $^3S_1 \mathrm{-}^3D_1$ channel as a function of the cutoff, $\Lambda$. The vertical solid line indicates the location of a limit-cycle-like cutoff, while the vertical dashed lines indicate the location of exceptional cutoffs.
• Figure 3: The ground-state energy of the deuteron, $E_d$, at LO and N$^2$LO as a function of the cutoff $\Lambda$. The vertical solid lines indicate the locations of the limit-cycle-like cutoffs, while the vertical dashed lines indicate the location of the exceptional cutoffs. The horizontal dashed line shows the experimental value for $E_d$.
• Figure 4: Phase shifts in the $^3S_1$ channel as a function of the laboratory scattering energy, $T_\mathrm{lab}$, for orders LO (green) and N$^2$LO (orange). The bands indicate the envelope of cutoff variation from $\Lambda=750$ MeV (dashed lines) to $\Lambda=1000$ MeV (solid lines). Note that the NLO contribution in $^3S_1 \mathrm{-}^3D_1$ is zero. The red star shows the phase shift used to calibrate the LO LEC, which is shifted $-3^\circ$, $0^\circ$ and $+3^\circ$ compared to the Nijmegen Stoks:1993tb phase shift in the top, middle, and bottom panel, respectively. The black dashed line shows the phase shift from the Nijmegen partial wave analysis Stoks:1993tb.
• Figure 5: Predicted deuteron ground-state energy at N$^2$LO as a function of $\Lambda$ for different values of the shift $\Delta$. The solid lines in each cutoff region correspond to shifts $\Delta=+3^\circ,-3^\circ$ and $0^\circ$, respectively, as defined in \ref{['eq:3S1_phase_shift_shift']}. The dashed-dotted line shows the result for $\Delta=0^\circ$, and the dashed line shows the experimental value for $E_d$.