Triplet pairing in pure neutron matter

We study the zero-temperature BCS gaps for the triplet channel in pure neutron matter using similarity renormalization group (SRG) evolved interactions. We use the dependence of the results on the SRG resolution scale as a tool to analyze medium and many-body corrections. In particular, we study the effects of including the three-body interactions at leading order, which appear at next-to-next-to leading order (N2LO) in the chiral effective field theory (EFT), as well as that of the first-order self-energy corrections on the zero-temperature gap. In addition we also extract the transition temperature as a function of densities and verify the BCS scaling of the zero-temperature gaps to the transition temperature. We observe that the self-energy effects are very crucial in order to reduce the SRG resolution scale dependence of the results, while the three-body effects at the leading order do not change the two-body resolution scale dependence. On the other hand, the results depend strongly on the three-body cutoff, emphasizing the importance of the missing higher-order three-body effects. We also observe that self-energy effects reduce the overall gap as well as shift the gap closure to lower densities.

Figure 1

Zero-temperature gap for (a) the singlet and (b) the triplet obtained from chiral N3LO potential [29]. The ${}^{1}S_0$ gaps uses SRG evolved interaction at $\lambda =2.0{\text{fm}}^{-1}$, while the ${}^{3}P_2-{}^{3}F_2$ gap has the unevolved interaction as input. Obtaining the gaps for higher resolution scale for the ${}^{1}S_0$ channel using the stability method is complicated due to the presence of short-range components in the interaction, that is usually softened by the RG evolution [34].

Figure 2

Transition temperature as a function of $k_{\mathrm{F}}$ for (a) the singlet and (b) the triplet obtained from the chiral N3LO potential [29]. Again, $T_c$ for the ${}^{1}S_0$ channel uses the SRG-evolved interaction with $\lambda =2.0{\text{fm}}^{-1}$, while the triplet channel uses unevolved interaction as input.

Figure 3

SRG-evolved ${}^{3}P_2-{}^{3}F_2$ interactions as a function of $k^2$ and $k^{\prime {}^2}$ for (a) the N3LO EM 500 [29] interaction and (b) the ${\mathrm{AV}}_{18}$ [30] interaction. Note that the evolution drives the matrix elements towards the diagonal as function of the parameter $\lambda$.

Figure 4

Comparing the phase shift in the triplet channel for the N3LO EM 500 [29] and the ${\mathrm{AV}}_{18}$ [30] interaction. Note that the SRG evolution in each case preserves the phase shift as seen by the lack of dependence on the SRG resolution scale. For comparison, we have also included the partial wave analysis of the experimental phase shifts by Arndt et al. [41].

Figure 5

Zero-temperature gaps obtained from N3LO EM 500 and ${\mathrm{AV}}_{18}$ for different SRG resolution scales. The top panel (a) has the N3LO EM 500 bare and SRG-evolved interactions as inputs. The bottom panel (b) has the AV18 bare and SRG-evolved interactions for the same $\lambda$ values.

Figure 6

Transition temperature as a function of $k_{\mathrm{F}}$ for N3LO EM 500 and ${\mathrm{AV}}_{18}$ for different SRG resolution scales. The top panel (a) has the N3LO EM 500 bare and its SRG-evolved interactions as inputs. The bottom panel (b) has the bare AV18 and its SRG-evolved interactions as inputs.

Figure 7

Leading chiral $3N$ forces at N2LO [19].

Figure 8

Effective $2N$ generated by integrating the third particle over the states occupied in the Fermi sea.

Figure 9

Two-body SRG resolution-scale dependence for the N3LO EM 500, keeping ${\mathrm{\Lambda }}_{\text{3NF}}$ fixed when the two-body input is augmented by the effective $2N$ interaction. The left panel (a) has ${\mathrm{\Lambda }}_{\text{3NF}}=2.5{\text{fm}}^{-1}$, while the right panel (b) has ${\mathrm{\Lambda }}_{\text{3NF}}=2.0{\text{fm}}^{-1}$.

Figure 10

Three-body cutoff dependence for the N3LO EM 500, keeping the two-body SRG resolution scale $\lambda$ fixed, when the two-body input is augmented by the effective $2N$ interaction. The left panel (a) has the bare $2N$, while the right panel (b) has the SRG resolution scale fixed at $\lambda =2.0{\text{fm}}^{-1}$.

Figure 11

First-order self-energy.

Figure 12

Effective mass using the first-order self-energy term for (a) N3LO EM 500 and (b) ${\mathrm{AV}}_{18}$ interactions for bare and different SRG resolution scales.

Figure 13

First-order self-energy effects on the zero-temperature gap using (a) N3LO EM 500 and (b) AV18 respectively and the corresponding SRG-evolved interactions as inputs.

Figure 14

Relative error as a function of $k_{\mathrm{F}}$ for two different SRG resolution scales ($\lambda =2.0{\text{fm}}^{-1}$ and $\lambda =2.5{\text{fm}}^{-1}$) for the N3LO EM 500 and the AV18 SRG-evolved interactions.

Figure 15

Effect of first-order self-energy vs the free spectrum for (a) N3LO and (b) ${\mathrm{AV}}_{18}$. The log-scale details the effects of SRG running on the gaps as well as the closure.

Figure 16

Relative error between N3LO EM 500 and ${\mathrm{AV}}_{18}$ interactions for different cutoffs including the bare interaction with and without the self-energy corrections. The solid lines with filled symbols are the relative errors for the free spectrum, while the broken lines with the open symbols include self-energy corrections to first order.