Lists of terms of the pseudopotential $\hat{V}$ order by order

In Tables 2-5 are listed, respectively, all possible terms of the pseudopotential (1) in zero, second, fourth, and sixth order. In each order, the numbers of terms equal 2, 7, 15, and 26, giving the total number of 50 terms up to N$^3$LO. We see that these numbers of terms are exactly equal to those corresponding to the EDF in each isospin channel with the Galilean invariance imposed, cf. Table VI of Ref. [2]. One should note that each term of the pseudopotential (2) is Galilean-invariant by construction, because it is built with relative-momentum operators $K_ {\tilde{n }\tilde{L}}$; therefore, the pseudopotential is not changed by a transformation to a system moving with a constant velocity. When both isoscalar and isovector channels are considered in the EDF, the number of EDF terms becomes in each order twice larger than the number of terms of the pseudopotential.

This means that the EDF obtained by averaging the pseudopotential is constrained by as many conditions as there are terms in each isospin channel. One possible solution is than to find a one-to-one correspondence between the EDF and the pseudopotential by relating the isoscalar part of the EDF to its isovector part, in a way that will be showed explicitly in the following Sections of this work.

Table 2: Zero-order terms of the pseudopotential (2).

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$S$	gauge
1	0	0	0	0	0	0	Y
2	0	0	0	0	2	0	Y

Table 3: Same as in Table 2 but for the second order terms.

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

Table 4: Same as in Table 2 but for the fourth order terms.

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$S$	gauge
1	4	0	0	0	0	0	D
2	4	0	0	0	2	0	D
3	4	2	0	0	2	2	D
4	3	1	1	1	0	0	Y
5	3	1	1	1	2	0	Y
6	3	1	1	1	1	1	N
7	3	1	1	1	2	2	D
8	3	3	1	1	2	2	I
9	2	0	2	0	0	0	D
10	2	0	2	0	2	0	D
11	2	2	2	0	2	2	D
12	2	2	2	2	0	0	I
13	2	2	2	2	2	0	I
14	2	2	2	2	1	1	N
15	2	2	2	2	2	2	I

Table 5: Same as in Table 2 but for the sixth order terms.

No.	$\tilde{n}'$	$\tilde{L}'$	$\tilde{n}$	$\tilde{L}$	$v_{12}$	$S$	gauge
1	6	0	0	0	0	0	D
2	6	0	0	0	2	0	D
3	6	2	0	0	2	2	D
4	5	1	1	1	0	0	D
5	5	1	1	1	2	0	D
6	5	1	1	1	1	1	N
7	5	1	1	1	2	2	D
8	5	3	1	1	2	2	I
9	4	0	2	0	0	0	D
10	4	0	2	0	2	0	D
11	4	2	2	0	2	2	D
12	4	0	2	2	2	2	D
13	4	2	2	2	0	0	I
14	4	2	2	2	2	0	I
15	4	2	2	2	1	1	N
16	4	2	2	2	2	2	D
17	4	4	2	2	2	2	I
18	3	1	3	1	0	0	D
19	3	1	3	1	2	0	D
20	3	1	3	1	1	1	N
21	3	1	3	1	2	2	D
22	3	3	3	1	2	2	D
23	3	3	3	3	0	0	I
24	3	3	3	3	2	0	I
25	3	3	3	3	1	1	N
26	3	3	3	3	2	2	D

To make the connection between the pseudopotential and the standard form of the Skyrme interaction more transparent, we give here the relations of conversion between the parameters of the zero- and second-order pseudopotential and those of the Skyrme interaction, see Ref. [19] for the definitions used. They read,

$$t_0 = C_{00,00}^{00}+\frac{1}{\sqrt{3}}C_{00,20}^{00},$$ (9)

$$t_0x_0 = -\frac{2}{\sqrt{3}}C_{00,20}^{00},$$ (10)

$$t_1 = \frac{1}{\sqrt{3}}C_{00,00}^{20}+\frac{1}{3}C_{00,20}^{20},$$ (11)

$$t_1x_1 = -\frac{2}{3}C_{00,20}^{20},$$ (12)

$$t_2 = \frac{1}{\sqrt{3}}C_{11,00}^{11}+\frac{1}{3}C_{11,20}^{11},$$ (13)

$$t_2x_2 = -\frac{2}{3}C_{11,20}^{11},$$ (14)

$$W_0 = \frac{1}{\sqrt{6}}C_{11,11}^{11},$$ (15)

$$t_o = -\frac{1}{3\sqrt{5}}C_{11,22}^{11},$$ (16)

$$t_e = -\frac{1}{3\sqrt{5}}C_{00,22}^{22}.$$ (17)

In relations of Eqs. (2.2), parameters $t_3$ and $t_3x_3$ are missing: they are related to the terms of the Skyrme interaction depending on density, which have been introduced to mimic the effects of the three-body force in the phenomenological interaction and to get the saturation feature of the nuclear force. In the same way, the zero-order parameters $C_{00,00}^{00}$ and $C_{00,20}^{00}$ of the pseudopotential, see Eqs. (9) and (10), should become density-dependent.

In his effective nuclear potential, Skyrme also introduced [18] one additional term of the fourth order, which he justified through the presence of considerable D-waves in the nucleon-nucleon interaction energies around 100MeV. Also in this case, we give the relation between the corresponding parameter $t_{D}$ and the parameter of our full pseudopotential,

$$t_D =\frac{1}{2}C_{20,20}^{00} .$$ (18)