Constraints for the scalar-isoscalar channel

For the standard CE in the scalar-isoscalar channel, Eq. (27), the constraints that we found from Eq. (48) are exactly the same as those defining the gauge-invariant functional up to N$^{3}$LO [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen]. This constitutes an explicit verification of the general result presented in Sec. 2.2.1. As in Ref. [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen], we obtained two sets of independent constraints, one for the isoscalar coupling constants $C_{a,\alpha}^{\beta,0}$ and one for the isovector coupling constants $C_{a,\alpha}^{\beta,1}$. Within each isospin channel, the linear conditions for the scalar coupling constants, $C_{a,n_\alpha L_\alpha 0 J_\alpha}^{n_\beta L_\beta 0 J_\beta,t}$, are disconnected from the linear conditions for the vector coupling constants, $C_{a,n_\alpha L_\alpha 1 J_\alpha}^{n_\beta L_\beta 1 J_\beta,t}$.