Constraints for the vector-isovector channel (fourth and sixth orders)

At fourth order we found the following constraints,

and only coupling constant is unrestricted. Apart from these 2 free and 23 dependent coupling constants, the vector-isovector channel of the CE requires that all the remaining fourth-order coupling constants are forced to be equal to zero. In particular in the Eq. (156) we showed the vanishing coupling constants, which were found to be non-vanishing in the scalar-isoscalar channel.

At sixth order we have,

(157) | |||

(158) | |||

(159) | |||

(160) | |||

(161) | |||

(162) | |||

(163) | |||

(164) | |||

(165) | |||

(166) | |||

(167) | |||

(168) | |||

(169) | |||

(170) | |||

(171) | |||

(172) | |||

(173) | |||

(174) | |||

(175) | |||

(176) | |||

(177) | |||

(178) |

and only coupling constant is unrestricted. Apart from 2 free and 39 dependent coupling constants, the vector-isovector channel of the CE requires that all the remaining sixth-order coupling constants are forced to be equal to zero. In particular, in the Eqs. (177)-(178) we showed the vanishing coupling constants that were found to be non-vanishing in the scalar-isoscalar channel.

The results presented in this section show simultaneously both features we saw respectively in Appendices A and B. At all orders, one can express all coupling constants through only one independent coupling constant, in such a way that the constraints are nondiagonal in both spin and isospin space. Again, this fact is due to the rank of and in the pairs of densities in the final form of condition (48), which allows the coupling constants at different spins and isospins to enter into the same constraints.