Inverse relations

In Section 2 we noticed the fact that once either the Galilean or gauge invariance is imposed, the numbers of parameters of the pseudopotential are the same, at each order, as the numbers of coupling constants of the EDF for each isospin. This situation allows us to obtain the inverse relations, namely, expressions relating the coupling constants of the EDF to the parameters of the pseudopotential. For the case of gauge invariance, at second order they are given in Tables 12 and 13, at fourth order in Tables 14 and 15, and at sixth order in Tables 16 and 17. As sets of independent coupling constants of the gauge-invariant EDF we selected the ones used in Appendix C of Ref. [2]. Note that in each case, the parameters of the pseudopotential can be expressed either by the isoscalar or by the isovector coupling constants. For the case of Galilean invariance, analogous expressions are available in the supplemental material [24].

Table 12: Second-order parameters of the pseudopotential as functions of the coupling constants of the isoscalar EDF $(t=0)$ when the gauge invariance is imposed, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= A ( a C_{20,0000}^{00... ...101,0}+ e C_{11,0011}^{1101,0}+ f C_{00,2011}^{0011,0}+ g C_{00,2211}^{0011,0})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$	$g$
$C_{00,00}^{20}$	$-\frac{2}{3}$	10	2$\sqrt{3}$	0	5	0	$-$$\sqrt{3}$	0
$C_{00,20}^{20}$	$\frac{2}{3}$	$2\sqrt{3}$	6	0	$\sqrt{3}$	0	$-$3	0
$C_{00,22}^{22}$	$-4$	0	0	$-$2	0	0	0	$1$
$C_{11,00}^{11}$	$-\frac{2}{3}$	$-$6	$2\sqrt{3}$	0	3	0	$\sqrt{3}$	0
$C_{11,20}^{11}$	$-\frac{2}{3}$	$-2\sqrt{3}$	10	0	$\sqrt{3}$	0	5	0
$C_{11,11}^{11}$	$-\frac{4}{3}$	0	0	0	0	1	0	0
$C_{11,22}^{11}$	$-\frac{4}{3}$	0	0	2	0	0	0	1

Table 13: Same as in Table 12 but for the isovector EDF $(t=1)$, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= A ( a C_{20,0000}^{00... ...101,1}+ e C_{11,0011}^{1101,1}+ f C_{00,2011}^{0011,1}+ g C_{00,2211}^{0011,1})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$	$g$
$C_{00,00}^{20}$	$\frac{2}{3}$	$2\sqrt{3}$	6	0	$\sqrt{3}$	0	$-$3	0
$C_{00,20}^{20}$	$-\frac{2}{3}$	6	$-2\sqrt{3}$	0	3	0	$\sqrt{3}$	0
$C_{00,22}^{22}$	$\frac{4}{3}$	0	0	$-2\sqrt{3}$	0	0	0	$\sqrt{3}$
$C_{11,00}^{11}$	$-\frac{2}{3}$	$-$2$\sqrt{3}$	$-$6	0	$\sqrt{3}$	0	$-$3	0
$C_{11,20}^{11}$	$\frac{2}{\sqrt{3}}$	$-$2$\sqrt{3}$	2	0	$\sqrt{3}$	0	1	0
$C_{11,11}^{11}$	$-\frac{4}{\sqrt{3}}$	0	0	0	0	1	0	0
$C_{11,22}^{11}$	$-\frac{4}{\sqrt{3}}$	0	0	2	0	0	0	1

Table: Same as in Table 12 but for the fourth-order parameters of the pseudopotential, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= A ( a C_{00,2202}^{22... ...011,0}+ d C_{40,0000}^{0000,0}+ e C_{40,0011}^{0011,0}+ f C_{42,0011}^{0011,0})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{11,00}^{31}$	$\frac{2}{15}$	$18\sqrt{5}$	$-18\sqrt{5}$	$-7\sqrt{15}$	$-120$	$40\sqrt{3}$	$0$
$C_{11,20}^{31}$	$\frac{2}{15}$	$6\sqrt{15}$	$-30\sqrt{15}$	$-35\sqrt{5}$	$-40\sqrt{3}$	$200$	$0$
$C_{11,22}^{33}$	$\frac{8}{3}\sqrt{\frac{7}{15}}$	0	0	$-1$	0	0	$2$
$C_{22,00}^{22}$	$\frac{1}{9}$	$30$	18	$7\sqrt{3}$	$40\sqrt{5}$	$8\sqrt{15}$	$0$
$C_{22,20}^{22}$	$-\frac{1}{9\sqrt{5}}$	$6\sqrt{15}$	$18\sqrt{15}$	$21\sqrt{5}$	$40\sqrt{3}$	120	$0$
$C_{22,22}^{22}$	$-\frac{4}{3}\sqrt{7}$	0	0	$1$	0	0	$2$

Table: Same as in Table 13 but for the fourth-order parameters of the pseudopotential, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= A ( a C_{00,2202}^{22... ...011,1}+ d C_{40,0000}^{0000,1}+ e C_{40,0011}^{0011,1}+ f C_{42,0011}^{0011,1})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{11,00}^{31}$	$-\frac{2}{15}$	$-6\sqrt{15}$	$-18\sqrt{15}$	$-21\sqrt{5}$	$40\sqrt{3}$	$120$	$0$
$C_{11,20}^{31}$	$-\frac{2}{15}$	$18\sqrt{5}$	$-18\sqrt{5}$	$-7\sqrt{15}$	$-120$	$40\sqrt{3}$	$0$
$C_{11,22}^{33}$	$\frac{8}{3}\sqrt{\frac{7}{5}}$	0	0	$-1$	0	0	$2$
$C_{22,00}^{22}$	$\frac{1}{9}$	$-6\sqrt{3}$	$-18\sqrt{3}$	$-21$	$-8\sqrt{15}$	$-24\sqrt{5}$	$0$
$C_{22,20}^{22}$	$\frac{1}{9\sqrt{5}}$	$18\sqrt{5}$	$-18\sqrt{5}$	$-7\sqrt{15}$	120	$-40\sqrt{3}$	$0$
$C_{22,22}^{22}$	$\frac{4}{3}\sqrt{\frac{7}{3}}$	0	0	$1$	0	0	$2$

0.75pt

Table: Same as in Table 12 but for the sixth-order parameters of the pseudopotential, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= A ( a C_{00,4212}^{22... ...011,0}+ d C_{60,0000}^{0000,0}+ e C_{60,0011}^{0011,0}+ f C_{62,0011}^{0011,0})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{11,22}^{53}$	$-\frac{16}{3}\sqrt{\frac{7}{15}}$	0	0	$1$	0	0	$2$
$C_{22,00}^{42}$	$\frac{2}{21}$	$21$	$-15\sqrt{105}$	$42\sqrt{3}$	$-208\sqrt{5}$	$-56\sqrt{15}$	$0$
$C_{22,20}^{42}$	$-\frac{2}{3}\sqrt{\frac{1}{7}}$	$3{\sqrt{21}}$	$-9\sqrt{5}$	$18{\sqrt{7}}$	$-8{\sqrt{105}}$	$-24{\sqrt{35}}$	$0$
$C_{22,22}^{44}$	$-\frac{16}{\sqrt{5}}$	0	0	$1$	0	0	$-2$
$C_{33,00}^{33}$	$-\frac{2}{45}$	$\sqrt{105}$	$45$	$6\sqrt{35}$	$-40\sqrt{21}$	$40\sqrt{7}$	$0$
$C_{33,20}^{33}$	$\frac{2}{9}\sqrt{\frac{1}{15}}$	$-5\sqrt{21}$	$-9\sqrt{5}$	$-30\sqrt{7}$	$8\sqrt{105}$	$-40\sqrt{35}$	$0$

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Table: Same as in Table 13 but for the sixth-order parameters of the pseudopotential, according to the formula $C_{\tilde{n} \tilde{L},v_{12} S}^{\tilde{n}' \tilde{L}'}= A ( a C_{00,4212}^{22... ...011,1}+ d C_{60,0000}^{0000,1}+ e C_{60,0011}^{0011,1}+ f C_{62,0011}^{0011,1})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{11,22}^{53}$	$-\frac{16}{3}\sqrt{\frac{7}{5}}$	0	0	$1$	0	0	$2$
$C_{22,00}^{42}$	$\frac{2}{7\sqrt{3}}$	$-21$	$3\sqrt{105}$	$-42\sqrt{3}$	$56\sqrt{5}$	$56\sqrt{15}$	$0$
$C_{22,20}^{42}$	$-\frac{2}{21}$	$21$	$9\sqrt{105}$	$42\sqrt{3}$	$168\sqrt{5}$	$-56\sqrt{15}$	$0$
$C_{22,22}^{44}$	$\frac{16}{\sqrt{15}}$	0	0	$1$	0	0	$-2$
$C_{33,00}^{33}$	$-\frac{2}{45}$	$-3\sqrt{35}$	$15\sqrt{3}$	$-6\sqrt{105}$	$-40\sqrt{7}$	$-40\sqrt{21}$	$0$
$C_{33,20}^{33}$	$\frac{2}{9}\sqrt{\frac{1}{15}}$	$3\sqrt{7}$	$9\sqrt{15}$	$6\sqrt{21}$	$-24\sqrt{35}$	$8\sqrt{105}$	$0$