Estimates and power counting in uniform nuclear matter with softened interactions

Modern softened nucleon-nucleon interactions are well suited for perturbative many-body calculations, but a many-body power counting scheme is lacking. Estimates of diagrammatic contributions at finite density are important ingredients in such a scheme. Here we show how to make quantitative estimates of the particle-particle and hole-hole channel in uniform nuclear matter for soft interactions. We also use estimates to assess the role of normal-ordered three-body forces for a pure contact interaction.

Figure 1

Second- and third-order Goldstone diagrams, for which we consider their relative size. Particles are upward-going arrows, holes are downward-going arrows, and dashed lines are two-body potential insertions.

Figure 2

(a) The absolute value of the second-order energy per particle in nuclear matter is plotted as a function of density $\rho$ for the ${}^{1}S_0$ partial wave using the AV18 potential. Both exact (solid) and estimates (dashed) are shown for four different SRG $\lambda$ scales. (b) Same as (a) but for third-order in the pp channel. (c) Same as (a) but for fourth order in the pp channel.

Figure 3

(a) The absolute value of the second-order energy per particle in nuclear matter is plotted as a function of density $\rho$ for the ${}^{3}S_1-{}^{3}D_1$ partial wave using the AV18 potential. Both exact (solid) and estimates (dashed) are shown for four different SRG $\lambda$ scales. (b) Same as (a) but for third order in the pp channel. (c) Same as (a) but for fourth order in the pp channel.

Figure 4

(a) The expansion parameter $\eta$ in Eq. (15) is plotted as a function of density $\rho$ for the ${}^{1}S_0$ partial wave using the AV18 potential. Four different SRG $\lambda$ scales are shown. (b) The same as (a) but for the ${}^{3}S_1-{}^{3}D_1$ partial wave.

Figure 5

Diagram of two Fermi spheres illustrating the hole and particle phase space available for two-body interactions. $\lambda$ here serves as a scale for which the phase space above is effectively decoupled from the phase space below. This is in contrast to other methods (e.g., $V_{\mathrm{low}k}$) where the $\mathrm{\Lambda }$ is an actual cutoff in the model space.

Figure 6

Goldstone diagrams of the second-order energy for $V_E$ and ${\overline{V}}_E$.

Figure 7

(a) The second-order energy per particle in nuclear matter is plotted as a function of density for the density-dependent (DD) and residual (RE) diagrams of the $3N$ contact $V_E$ at two different cutoffs ${\mathrm{\Lambda }}_{3N}$. Both exact (solid) and estimates (dashed) are shown for each diagram and cutoff. (b) The dimensionless ratio in Eq. (42) is plotted as a function of the Fermi momentum over the $3N$ cutoff.

Figure 8

(a) The absolute value of the third-order energy per particle in nuclear matter for the hh channel is plotted as a function of density $\rho$ for the ${}^{1}S_0$ partial wave using the AV18 potential. Both exact (solid) and estimates (dashed) are shown for four different SRG $\lambda$ scales. Estimates are done with $p_{\text{max}}=10$. (b) Same as (a) but for fourth order in the hh channel.

Figure 9

(a) The absolute value of the third-order energy per particle in nuclear matter for the hh channel is plotted as a function of density $\rho$ for the ${}^{3}S_1-{}^{3}D_1$ partial wave using the AV18 potential. Both exact (solid) and estimates (dashed) are shown for four different SRG $\lambda$ scales. Estimates are done with $p_{\text{max}}=10$. (b) Same as (a) but for fourth order in the hh channel.

Figure 10

(a) The absolute value of the ratio of the third-order pp to hh ladder term is plotted as a function of density $\rho$ for the ${}^{1}S_0$ partial wave using the AV18 potential. Both exact (solid) and estimate (dashed) calculations are shown for four different SRG $\lambda$ scales. (b) The same as (a) but for fourth-order.

Figure 11

(a) The absolute value of the ratio of the third-order pp to hh ladder term is plotted as a function of density $\rho$ for the ${}^{3}S_1-{}^{3}D_1$ partial wave using the AV18 potential. Both exact (solid) and estimate (dashed) calculations are shown for four different SRG $\lambda$ scales. (b) The same as (a) but for fourth order.

Figure 12

(a) The expansion parameter $\eta$ in Eq. (C7) is plotted as a function of density $\rho$ for the ${}^{1}S_0$ partial wave using the AV18 potential. Four different SRG $\lambda$ scales are shown. (b) The same as (a) but for the ${}^{3}S_1-{}^{3}D_1$ partial wave.