Tensor-like form of the pseudopotential

In this Section, we present the tensor-like form of the pseudopotential, which is, in fact, a different form of coupling of the relative-momentum operators with the spin operators, just like in the tensor term of the standard Skyrme interaction. In this form, the pseudopotential of Eq. (1) is a sum of the following terms,

$$\hat{V}=\sum_{\tilde{n}' \tilde{L}', \tilde{n} \tilde{L},v_... ...e{V}}_{\tilde{n} \tilde{L},v_{12} J}^{\tilde{n}' \tilde{L}'} ,$$ (22)

where

$$\hat{\tilde{V}}_{\tilde{n} \tilde{L}, v_{12}J}^{\tilde{n}' \tilde{L}'} = \frac{1}{2} i^{v_{12}} \left(1-{\textstyle{\frac{1}{2}}}\delta_{v_1,v_2}\right) \times$$

$$\left( \left[ \left[K'_{\tilde{n}'\tilde{L}'} \sigma^{(1)}_{v_1} ... ... \left[K_{\tilde{n}\tilde{L}} \sigma^{(2)}_{v_2} \right]_{J}\right]_{0} \right.$$

$$\left. + \left[ \left[K'_{\tilde{n}'\tilde{L}'} \sigma^{(2)}_{v_1... ... \left[K_{\tilde{n}\tilde{L}} \sigma^{(1)}_{v_2} \right]_{J}\right]_{0} \right.$$

$$\left. + \left[ \left[K'_{\tilde{n}\tilde{L}} \sigma^{(1)}_{v_1} ... ...left[K_{\tilde{n}'\tilde{L}'} \sigma^{(2)}_{v_2} \right]_{J}\right]_{0} \right.$$

$$\left. + \left[ \left[K'_{\tilde{n}\tilde{L}} \sigma^{(2)}_{v_1} ... ..._{\tilde{n}'\tilde{L}'} \sigma^{(1)}_{v_2} \right]_{J}\right]_{0} \right)\times$$

$$\left(1-\hat{P}^{M}\hat{P}^{\sigma}\hat{P}^{\tau}\right) \hat{\delta}_{12}(\bm{r}'_1\bm{r}'_2;\bm{r}_1\bm{r}_2) .$$ (23)

The lists of the zero-, second-, fourth-, and sixth-order terms $\hat{\tilde{V}}_{\tilde{n} \tilde{L},v_{12} J}^{\tilde{n}' \tilde{L}'}$ of the pseudopotential are given, respectively, in Tables 6-9, which are the analogues of Tables 2-5 given in Section 2.1.

Table: Zero-order terms of the recoupled pseudopotential ($\ref{eq:9}$).

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$J$
1	0	0	0	0	0	0
2	0	0	0	0	2	0

Table 7: Same as in Table 6 but for the second-order terms.

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$J$
1	1	1	1	1	0	1
2	1	1	1	1	1	1
3	1	1	1	1	2	0
4	1	1	1	1	2	1
5	1	1	1	1	2	2
6	2	0	0	0	0	0
7	2	0	0	0	2	1
8	2	2	0	0	2	1

Table 8: Same as in Table 6 but for the fourth-order terms.

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$J$
1	2	0	2	0	0	0
2	2	0	2	0	2	1
3	2	2	2	2	0	2
4	2	2	2	2	1	2
5	2	2	2	0	2	1
6	2	2	2	2	2	1
7	2	2	2	2	2	2
8	2	2	2	2	2	3
9	3	1	1	1	0	1
10	3	1	1	1	1	1
11	3	1	1	1	2	0
12	3	1	1	1	2	1
13	3	1	1	1	2	2
14	3	3	1	1	2	2
15	4	0	0	0	0	0
16	4	0	0	0	2	1
17	4	2	0	0	2	1

Table 9: Same as in Table 6 but for the sixth-order terms.

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$J$
1	3	1	3	1	0	1
2	3	1	3	1	1	1
3	3	1	3	1	2	0
4	3	1	3	1	2	1
5	3	1	3	1	2	2
6	3	3	3	3	0	3
7	3	3	3	3	1	3
8	3	3	3	1	2	2
9	3	3	3	3	2	2
10	3	3	3	3	2	3
11	3	3	3	3	2	4
12	4	0	2	0	0	0
13	4	0	2	0	2	1
14	4	0	2	2	2	1
15	4	2	2	2	0	2
16	4	2	2	2	1	2
17	4	2	2	0	2	1
18	4	2	2	2	2	1
19	4	2	2	2	2	2
20	4	2	2	2	2	3
21	4	4	2	2	2	3
22	5	1	1	1	0	1
23	5	1	1	1	1	1
24	5	1	1	1	2	0
25	5	1	1	1	2	1
26	5	1	1	1	2	2
27	5	3	1	1	2	2
28	6	0	0	0	0	0
29	6	0	0	0	2	1
30	6	2	0	0	2	1

By means of the recoupling technique, it is possible to determine relations between the two different coupling schemes of the pseudopotential. This derivation, along with the relationships between the corresponding parameters $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}$ and $\tilde{C}_{\tilde{n} \tilde{L},v_{12} J}^{\tilde{n}' \tilde{L}'}$, is presented in Appendix C.

The reader might have noticed that the two forms of the pseudopotential do not have the same numbers of terms: the tensor-like form of the pseudopotential (Tables 7, 8, and 9) has more terms than the central-like form (Tables 3, 4, and 5). This means that not all of the terms of the tensor-like form are linearly independent from one another, even though they are all allowed by the symmetries, and thus some terms can be expressed as linear combinations of others, or, equivalently, some linear combinations of terms are identically equal to zero. This fact, can be expressed in the form of the following explicit dependencies between the parameters of the tensor-like pseudopotential.

For the second-order terms we have,

$$\tilde{C}_{11,21}^{11}=-\frac{2}{\sqrt{3}}\tilde{C}_{11,20}^{11}+\sqrt{\frac{5}{3}}\tilde{C}_{11,22}^{11} ,$$ (24)

whereas the fourth-order dependencies read,

$$\tilde{C}_{22,21}^{22} = -\frac{\sqrt{15}}{9}\tilde{C}_{22,22}^{22} + \frac{2}{9}\sqrt{21}\tilde{C}_{22,23}^{22} ,$$ (25)

$$\tilde{C}_{11,21}^{31} = -\frac{2}{\sqrt{3}}\tilde{C}_{11,20}^{31}+\sqrt{\frac{5}{3}}\tilde{C}_{11,22}^{31} ,$$ (26)

and finally at sixth order we have,

$$\tilde{C}_{31,21}^{31} = -\frac{2}{\sqrt{3}}\tilde{C}_{31,20}^{31} + \sqrt{\frac{5}{3}}\tilde{C}_{31,22}^{31} ,$$ (27)

$$\tilde{C}_{33,23}^{33} = -4 \sqrt{\frac{5}{7}}\tilde{C}_{33,22}^{33}+\frac{9}{\sqrt{7}}\tilde{C}_{33,24}^{33} ,$$ (28)

$$\tilde{C}_{22,21}^{42} = -\frac{\sqrt{15}}{9}\tilde{C}_{22,22}^{42}+\frac{2}{9}\sqrt{21}\tilde{C}_{22,23}^{42} ,$$ (29)

$$\tilde{C}_{11,21}^{51} = -\frac{2}{\sqrt{3}}\tilde{C}_{11,20}^{51}+\sqrt{\frac{5}{3}}\tilde{C}_{11,22}^{51} .$$ (30)