Analyzing the Fierz rearrangement freedom for local chiral two-nucleon potentials

Chiral effective field theory is a framework to derive systematic nuclear interactions. It is based on the symmetries of quantum chromodynamics and includes long-range pion physics explicitly, while shorter-range physics is expanded in a general operator basis. The number of low-energy couplings at a particular order in the expansion can be reduced by exploiting the fact that nucleons are fermions and therefore obey the Pauli exclusion principle. The antisymmetry permits the selection of a subset of the allowed contact operators at a given order. When local regulators are used for these short-range interactions, however, this “Fierz rearrangement freedom” is violated. In this paper, we investigate the impact of this violation at leading order (LO) in the chiral expansion. We construct LO and next-to-leading order (NLO) potentials for all possible LO-operator pairs and study their reproduction of phase shifts, the ${}^4\mathrm{He}$ ground-state energy, and the neutron-matter energy at different densities. We demonstrate that the Fierz rearrangement freedom is partially restored at NLO where subleading contact interactions enter. We also discuss implications for local chiral three-nucleon interactions.

Figure 1

Phase shifts for the ${}^{1}S_0, {}^{3}S_1$, and ${}^{3}D_1$ channels and the $J=1$ mixing angle ${\epsilon }_1$ for $R_0=1.0\mathrm{fm}$ at LO and NLO. The $S$-wave phase shifts are fit and independent of the choice of the LO short-range operators. The lines for different operator choices overlap except in the case of the mixing angle ${\epsilon }_1$ and the ${}^{3}D_1$ partial wave at NLO.

Figure 2

Phase shifts for the ${}^{1}P_1$ and triplet ${}^{3}P_J$ waves for $R_0=1.0\mathrm{fm}$. The legend is as in Fig. 1.

Figure 3

Phase shifts for the ${}^{1}D_2, {}^{3}D_2, {}^{1}F_3$, and ${}^{3}F_3$ partial waves for $R_0=1.0\mathrm{fm}$. The legend is as in Fig. 1.

Figure 4

Phase shift as a function of laboratory energy in the ${}^{1}P_1$ channel for the two interactions that give the most extreme results: the ${\sigma }_{12},{\tau }_{12}$ and ${\tau }_{12},{\sigma }_{12}{\tau }_{12}$ interactions. We show the results for $R_0=1.0\mathrm{fm}$ with EKM uncertainty estimates (bands, see text) and the results for $R_0=1.2\mathrm{fm}$ as solid lines. The color scheme is as in Fig. 1.

Figure 5

Many-body observables at LO and NLO for different LO operator combinations. Here we drop the operator subscripts for legibility; i.e. ${\sigma }_{12}\to \sigma$, etc. The shaded regions show the spread for the various operator choices, while the dashed line denotes the centroid. When calculating the band and the centroid, we exclude the complete potentials in all cases and the $\mathbb{1},\tau$ operator combination in the case of neutron matter. Note that for neutron matter at LO, we only show the results for five interactions, because we find bound neutron matter for the ${\text{LO}}_P$ and the $\mathbb{1},{\tau }_{12}$ interactions. Left panel: Ground-state energy of ${}^4\mathrm{He}$. Middle panel: Neutron matter energy per particle at half saturation density. Right panel: Neutron matter energy per particle at saturation density.