Unitarity potentials and neutron matter at the unitary limit

We study the equation of state of neutron matter using a family of unitarity potentials, all of which are constructed to have infinite ${}^1{S}_0$ scattering lengths $a_s$. For such a system, a quantity of much interest is the ratio $\xi =E_0/E_0^{\mathrm{free}}$, where $E_0$ is the true ground-state energy of the system and $E_0^{\mathrm{free}}$ is that for the noninteracting system. In the limit of $a_s\to \pm \infty$, often referred to as the unitary limit, this ratio is expected to approach a universal constant, namely $\xi ~0.44(1)$. In the present work we calculate this ratio $\xi$ using a family of hard-core square-well potentials whose $a_s$ can be exactly obtained, thus enabling us to have many potentials of different ranges and strengths, all with infinite $a_s$. We have also calculated $\xi$ using a unitarity CDBonn potential obtained by slightly scaling its meson parameters. The ratios $\xi$ given by these different unitarity potentials are all close to each other and also remarkably close to 0.44, suggesting that the above ratio $\xi$ is indifferent to the details of the underlying interactions as long as they have infinite scattering length. A sum-rule and scaling constraint for the renormalized low-momentum interaction in neutron matter at the unitary limit is studied. The importance of pairing in our all-order ring diagram and model-space HF calculations of neutron matter is discussed.

Figure 1

Diagrams included in the all-order pphh ring-diagram summation for the ground-state energy shift of nuclear matter. Each dashed line represents a $V_{\mathrm{low}-\mathrm{k}}$ vertex.

Figure 2

Ring-diagram unitary ratios $\xi$ given by different unitarity potentials.

Figure 3

Diagonal matrix elements $V_{\mathit{NN}}(k,k)$ of different unitarity potentials.

Figure 4

The ring-diagram unitary ratios $\xi$ near the unitary limit. $k_F=1.2$ fm${}^{-1}$ is used.

Figure 5

MSHF unitary ratios $\xi$ given by different unitarity potentials.

Figure 6

Diagonal matrix elements of $V_{\mathrm{low}-\mathrm{k}}^{k_F}$ from different unitarity potentials. $\Lambda =k_F=1.2$ fm${}^{-1}$ is used.

Figure 7

MSHF effective mass $m^{*}$ and well depth $\Delta$ at the unitary limit. Results calculated from the unitarity potentials are denoted by star, dot, and triangle respectively for $k_F$ in the ranges (0.8–0.9), (0.9–1.4), and (1.4–1.5) fm${}^{-1}$. The solid line represents the linear expression Eq. (18) with $\xi =0.44$. See text for other explanations.