Constraints for the scalar-isovector channel

Validity of the CE for the scalar-isovector density, Eq. (30) for

and

, imposes through Eq. (48) certain specific constraints on the coupling constants of the functional. At second order we obtain,

$\displaystyle C_{00,2000}^{0000,t}$	$\textstyle =$	$\displaystyle -\frac{3^t}{\sqrt{3}}C_{00,1101}^{1101,1-t},$	(53)
$\displaystyle C_{00,1110}^{1110,t}$	$\textstyle =$	$\displaystyle -\frac{3^t}{3\sqrt{3}}C_{00,2011}^{0011,1-t}-\frac{3^t}{3}\sqrt{\frac{5}{3}}C_{00,2211}^{0011,1-t},$	(54)
$\displaystyle C_{00,1111}^{1111,t}$	$\textstyle =$	$\displaystyle -\frac{3^t}{3}C_{00,2011}^{0011,1-t}+\frac{3^t}{6}\sqrt{5}C_{00,2211}^{0011,1-t},$	(55)
$\displaystyle C_{00,1112}^{1112,t}$	$\textstyle =$	$\displaystyle -\frac{3^t}{3}\sqrt{\frac{5}{3}}C_{00,2011}^{0011,1-t}-\frac{3^t}{6\sqrt{3}}C_{00,2211}^{0011,1-t},$	(56)
$\displaystyle C_{11,1111}^{0000,t}$	$\textstyle =$	$\displaystyle C_{11,0011}^{1101,t} = 0 ,$	(57)
$\displaystyle C_{20,0000}^{0000,1}$	$\textstyle =$	$\displaystyle C_{20,0011}^{0011,1} = C_{22,0011}^{0011,1} = 0 .$	(58)

Constraints in Eqs. (53)-(56) connect the isoscalar and isovector coupling constants. The numerical coefficients of the corresponding linear combinations are the same as those for the scalar-isoscalar CE, see Eqs. (C1)-(C4) of Ref. [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen], apart from factors of $\sqrt{3}$ explained before Eq. (36). However, conditions for the scalar-isoscalar CE keep the coupling constants in the two isospin channels disconnected. Moreover, in the scalar-isovector channel the spin-orbit coupling constants must vanish, Eq. (57), along with the isovector surface coupling constants, Eq. (58). On the other hand, the corresponding isoscalar surface coupling constants, $C_{20,0000}^{0000,0}$ , $C_{20,0011}^{0011,0}$ , and $C_{22,0011}^{0011,0}$ , are left unrestricted.

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