$A=2,3,4$ nuclear contact coefficients in the generalized contact formalism

TL;DR

This work applies the generalized contact formalism (GCF) to light nuclei with $A = 2, 3, 4$ using both local and non-local chiral interactions, solving the bound-state problem with Hyperspherical Harmonics to obtain 2-body momentum distributions and density functions. By examining plateaus in $n_{N_1N_2,A}^{S}(k)/| ilde{eta}_{N_1N_2}^{S}(k)|^2$ and $ ho_{N_1N_2,A}^{S}(r)/|eta_{N_1N_2}^{S}(r)|^2$, the authors extract nuclear contact coefficients $C_{N_1N_2,A}^{S}$ and $ ilde{C}_{N_1N_2,A}^{S}$ for $S=0,1$ across several spin-isospin channels and nuclei, testing the GCF prediction $ ilde{C} = C$ and the model-independence of coefficient ratios. Their results show robust agreement in the dominant $np$ $S=1$ channel across potentials and nuclei, but reveal tensions in the $S=0$ channels when non-local chiral interactions are used, particularly for $^4$He. The study finds that ratios of contact coefficients across nuclei are largely model-independent (especially for local potentials), enabling reliable extrapolation to heavier systems, while indicating limitations of the GCF in non-local frameworks for certain channels. Overall, the work extends SRC analysis beyond local interactions and highlights the need for further exploration of non-local effects and higher-mass systems, with estimated theoretical uncertainties on ratios at the 10–20% level.

Abstract

We present a theoretical calculation for the $A = 2, 3$ and 4 nuclear contact coefficients within the generalized contact formalism, using both local and non-local chiral potentials. The Hyperspherical Harmonics method is employed to calculate the nuclear wave functions, from which we derive two-body momentum distributions and density functions to extract the contact coefficients. We have extracted the contact coefficients from two-body momentum distributions or from density functions, for a given nucleus and potential, and we have found that the generalized contact formalism predictions are verified in the triplet spin channel for local and non-local potentials. On the other hand, some significant tensions exist for the singlet channels, especially when studied with non-local potentials. We have also analyzed the model-independence of the ratios between the contact coefficients, and we have found to be quite satisfied. This study extends previous works based on local interaction models only.

Figures (25)

• Figure 1: 1BMDs $n_{p, A=3}(k)$ (dashed line) and $n_{p, A=3}^{\rm GCF} (k)$(solid line) for $^3\text{He}$ with local potentials. The upper panel shows the results in a logarithmic scale, while the lower panel displays the ratio $n_{p, A=3}(k)/n_{p, A=3}^{\rm GCF} (k)$. The gray band indicates the 20% deviation from unity.
• Figure 2: Same as Fig. \ref{['fig:1bmd_loc_he3']}, but for the $^4$He. Here the gray band indicates the 30% deviation from unity.
• Figure 3: Same as Fig. \ref{['fig:1bmd_loc_he3']}, but for the N2LO, N3LO and N4LO chiral potentials with fixed cutoff $\Lambda=550$ MeV.
• Figure 4: Same as Fig. \ref{['fig:1bmd_loc_he4']}, but for the N2LO, N3LO and N4LO chiral potentials with fixed cutoff $\Lambda=550$ MeV.
• Figure 5: Ratios between the 2BDFs and the universal functions, with (black lines) or without (red lines) the $L=0$ assumption for the $^3$He nucleus, calculated using the Norfolk potentials NV2+3/Ia* and NV2+3/Ib*.