Constraints for the vector-isoscalar channel

Validity of the CE for the vector-isoscalar density, Eq. (30) for $v=1$ and $t=0$, imposes through Eq. (48) at second order the following constraints on the coupling constants of the functional,

$$C_{00,1101}^{1101,t} = -\frac{1}{\sqrt{3}}C_{00,2011}^{0011,t},$$ (59)

$$C_{00,1110}^{1110,t} = -\frac{1}{\sqrt{3}}C_{00,2000}^{0000,t},$$ (60)

$$C_{00,1111}^{1111,t} = -C_{00,2000}^{0000,t},$$ (61)

$$C_{00,1112}^{1112,t} = -\sqrt{\frac{5}{3}}C_{00,2000}^{0000,t},$$ (62)

$$C_{20,0011}^{0011,t} = C_{22,0011}^{0011,t} = 0,$$ (63)

$$C_{11,1111}^{0000,t} = C_{11,0011}^{1101,t} = 0 ,$$ (64)

$$C_{00,2211}^{0011,t} = 0 ,$$ (65)

whereas the two coupling constants $C_{20,0000}^{0000,t}$ are left unrestricted. We note here that the constraints now connect scalar and vector coupling constants. Altogether, at second order, for the vector-isoscalar channel of the CE we have 6 free and 8 dependent coupling constants. Apart from that, 10 second-order coupling constants must vanish, which includes the surface ones in Eq. (63), spin-orbit ones of the Eq. (64), and tensor ones in Eq. (65).

For the fourth and sixth orders, analogous constraints are presented in Appendix B.