Constraints on the Energy Density Functional

The zero range of the pseudopotential is at the origin of the specific constraints induced upon the resulting coupling constants of the EDF. Indeed, elimination of the pseudopotential parameters from pairs of relationships defined by Tables 12-13, 14-15, and 16-17 leaves us with sets of linear equations that the EDF coupling constants must obey. At second order, that is, for the standard Skyrme interaction, this fact is well known and allows us to express the time-odd coupling constants through the time-even ones, see Ref. [4] for the complete set of expressions. We do not yet know if the analogous property may hold at higher orders, because this fact crucially depend on the arbitrary choice of the independent coupling constants that define the Galilean or gauge symmetries.

In the present paper, we derive the set of constraints on the EDF coupling constants that can be obtained by inverting the relations for the isovector coupling constants, given in Tables 13, 15, and 17. This allows us to express, at each order, the isovector coupling constants through the isoscalar ones. For the case of gauge invariance, at second, fourth, and sixth order, such relations are listed in Tables 18, 19, and 20, respectively. For the case of Galilean invariance, analogous expressions are available in the supplemental material [24].

Table: Constraints on the EDF that is derived by averaging the second-order gauge-invariant pseudopotential, expressed by the formula $C_{mI,\tilde{n} L \nu J}^{\tilde{n}' L' \nu' J', 1}= a C_{00,1101}^{1101,0}+ b ... ...1101,0}+ e C_{20,0000}^{0000,0}+ f C_{20,0011}^{0011,0}+ g C_{22,0011}^{0011,0}$.

	$a$	$b$	$c$	$d$	$e$	$f$	$g$
$C_{00,1101}^{1101,1}$	$-\frac{1}{\sqrt{3}}$	0	0	0	$-\frac{2}{\sqrt{3}}$	$-2$	0
$C_{00,2011}^{0011,1}$	$0$	$-\frac{1}{\sqrt{3}}$	0	0	$2$	$-\frac{2}{\sqrt{3}}$	0
$C_{00,2211}^{0011,1}$	0	0	$-\frac{1}{\sqrt{3}}$	0	0	0	$\frac{4}{\sqrt{3}}$
$C_{11,0011}^{1101,1}$	0	0	0	$\frac{1}{\sqrt{3}}$	0	0	0
$C_{20,0000}^{0000,1}$	$-\frac{1}{2\sqrt{3}}$	$\frac{1}{2}$	0	0	$-\frac{1}{\sqrt{3}}$	$0$	0
$C_{20,0011}^{0011,1}$	$-\frac{1}{2}$	$-\frac{1}{2\sqrt{3}}$	0	0	0	$-\frac{1}{\sqrt{3}}$	0
$C_{22,0011}^{0011,1}$	0	0	$\frac{1}{\sqrt{3}}$	0	0	0	$-\frac{1}{\sqrt{3}}$

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Table 19: Same as in Table 18 but for the fourth-order terms, according to the formula $C_{mI,n L v J}^{n' L' v' J', 1}= A( a C_{00,2202}^{2202,0}+ b C_{00,2212}^{2212... ...011,0}+ d C_{40,0000}^{0000,0}+ e C_{40,0011}^{0011,0}+ f C_{42,0011}^{0011,0})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{40,0000}^{0000,1}$	$\frac{1}{120}$	$-6\sqrt{15}$	$-18\sqrt{15}$	$-21\sqrt{5}$	$-40\sqrt{3}$	$0$	$0$
$C_{40,0011}^{0011,1}$	$\frac{1}{120}$	$-18\sqrt{5}$	$18\sqrt{5}$	$7\sqrt{15}$	$0$	$-40\sqrt{3}$	$0$
$C_{42,0011}^{0011,1}$	$-\frac{1}{\sqrt{3}}$	0	0	$1$	0	0	$1$
$C_{00,2202}^{2202,1}$	$\frac{1}{9}$	$-3\sqrt{3}$	$0$	$0$	$-4\sqrt{15}$	$-12\sqrt{5}$	$0$
$C_{00,4211}^{0011,1}$	$-\frac{1}{\sqrt{3}}$	0	0	$1$	0	0	$4$
$C_{00,2212}^{2212,1}$	$\frac{1}{9}$	$0$	$-3\sqrt{3}$	$0$	$-4\sqrt{15}$	$4\sqrt{5}$	$14$

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Table 20: Same as in Table 18 but for the sixth-order terms, according to the formula $C_{mI,n L v J}^{n' L' v' J', 1}=A( a C_{00,4212}^{2212,0}+ b C_{00,3303}^{3303,... ...011,0}+ d C_{60,0000}^{0000,0}+ e C_{60,0011}^{0011,0}+ f C_{62,0011}^{0011,0})$.

	$A$	$a$	$b$	$c$	$d$	$e$	$f$
$C_{60,0000}^{0000,1}$	$\frac{1}{840}$	$21\sqrt{15}$	$-45\sqrt{7}$	$126\sqrt{5}$	$-280\sqrt{3}$	$0$	$0$
$C_{60,0011}^{0011,1}$	$\frac{1}{840}$	$-21\sqrt{5}$	$-45\sqrt{21}$	$-42\sqrt{15}$	0	$-280\sqrt{3}$	$0$
$C_{62,0011}^{0011,1}$	$\frac{1}{\sqrt{3}}$	0	0	$1$	0	0	$-1$
$C_{00,3303}^{3303,1}$	$\frac{1}{9}$	$0$	$-3\sqrt{3}$	$0$	$-8\sqrt{7}$	$-8\sqrt{21}$	$0$
$C_{00,6211}^{0011,1}$	$-\frac{1}{\sqrt{3}}$	0	0	$1$	0	0	$-4$
$C_{00,4212}^{2212,1}$	$\frac{1}{3}$	$-\sqrt{3}$	$0$	$0$	$8\sqrt{15}$	$-8\sqrt{5}$	$-24$