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

At fourth order we found the following constraints,

$$C_{00,4000}^{0000,t} = (\sqrt{3})^{t}\frac{C_{00,4011}^{0011,0}}{\sqrt{3}},$$ (144)

$$C_{00,2000}^{2000,t} = (\sqrt{3})^{t}\frac{5 C_{00,4011}^{0011,0}}{3 \sqrt{3}},$$ (145)

$$C_{00,3101}^{1101,t} = -(\sqrt{3})^{t}\frac{4 C_{00,4011}^{0011,0}}{\sqrt{3}},$$ (146)

$$C_{00,2202}^{2202,t} = (\sqrt{3})^{t}\frac{2}{3} \sqrt{\frac{5}{3}} C_{00,4011}^{0011,0},$$ (147)

$$C_{00,3110}^{1110,t} = -(\sqrt{3})^{t}\frac{4}{3} C_{00,4011}^{0011,0},$$ (148)

$$C_{00,3111}^{1111,t} = -(\sqrt{3})^{t}\frac{4 C_{00,4011}^{0011,0}}{\sqrt{3}},$$ (149)

$$C_{00,3112}^{1112,t} = -(\sqrt{3})^{t}\frac{4}{3} \sqrt{5} C_{00,4011}^{0011,0},$$ (150)

$$C_{00,4011}^{0011,1} = \sqrt{3} C_{00,4011}^{0011,0},$$ (151)

$$C_{00,2011}^{2011,t} = (\sqrt{3})^{t}\frac{5}{3} C_{00,4011}^{0011,0},$$ (152)

$$C_{00,2211}^{2211,t} = (\sqrt{3})^{t}\frac{2}{3} C_{00,4011}^{0011,0},$$ (153)

$$C_{00,2212}^{2212,t} = (\sqrt{3})^{t}\frac{2}{3} \sqrt{\frac{5}{3}} C_{00,4011}^{0011,0},$$ (154)

$$C_{00,2213}^{2213,t} = (\sqrt{3})^{t}\frac{2}{3} \sqrt{\frac{7}{3}} C_{00,4011}^{0011,0},$$ (155)

$$C_{00,2211}^{2011,t} = C_{00,3312}^{1112,t}=C_{00,4211}^{0011,t}=0,$$ (156)

and only coupling constant $C_{40,0000}^{0000,0}$ 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,

$$C_{00,6000}^{0000,t} = -(\sqrt{3})^{t}\frac{C_{00,5112}^{1112,0}}{2 \sqrt{15}},$$ (157)

$$C_{00,4000}^{2000,t} = -(\sqrt{3})^{t}\frac{7 C_{00,5112}^{1112,0}}{2 \sqrt{15}},$$ (158)

$$C_{00,4202}^{2202,t} = -(\sqrt{3})^{t}\frac{2 C_{00,5112}^{1112,0}}{\sqrt{3}},$$ (159)

$$C_{00,5101}^{1101,t} = (\sqrt{3})^{t}\sqrt{\frac{3}{5}} C_{00,5112}^{1112,0},$$ (160)

$$C_{00,3101}^{3101,t} = (\sqrt{3})^{t}\frac{7}{5} \sqrt{\frac{3}{5}} C_{00,5112}^{1112,0},$$ (161)

$$C_{00,3303}^{3303,t} = (\sqrt{3})^{t}\frac{2}{9} \sqrt{\frac{7}{5}} C_{00,5112}^{1112,0},$$ (162)

$$C_{00,5110}^{1110,t} = (\sqrt{3})^{t}\frac{C_{00,5112}^{1112,0}}{\sqrt{5}},$$ (163)

$$C_{00,5111}^{1111,t} = (\sqrt{3})^{t}\sqrt{\frac{3}{5}} C_{00,5112}^{1112,0},$$ (164)

$$C_{00,5112}^{1112,1} = \sqrt{3} C_{00,5112}^{1112,0},$$ (165)

$$C_{00,3110}^{3110,t} = (\sqrt{3})^{t}\frac{7 C_{00,5112}^{1112,0}}{5 \sqrt{5}},$$ (166)

$$C_{00,3111}^{3111,t} = (\sqrt{3})^{t}\frac{7}{5} \sqrt{\frac{3}{5}} C_{00,5112}^{1112,0},$$ (167)

$$C_{00,3112}^{3112,t} = (\sqrt{3})^{t}\frac{7}{5} C_{00,5112}^{1112,0},$$ (168)

$$C_{00,3312}^{3312,t} = (\sqrt{3})^{t}\frac{2}{9} C_{00,5112}^{1112,0},$$ (169)

$$C_{00,3313}^{3313,t} = (\sqrt{3})^{t}\frac{2}{9} \sqrt{\frac{7}{5}} C_{00,5112}^{1112,0},$$ (170)

$$C_{00,3314}^{3314,t} = (\sqrt{3})^{t}\frac{2 C_{00,5112}^{1112,0}}{3 \sqrt{5}},$$ (171)

$$C_{00,6011}^{0011,t} = -(\sqrt{3})^{t}\frac{C_{00,5112}^{1112,0}}{2 \sqrt{5}},$$ (172)

$$C_{00,4011}^{2011,t} = -(\sqrt{3})^{t}\frac{7 C_{00,5112}^{1112,0}}{2 \sqrt{5}},$$ (173)

$$C_{00,4211}^{2211,t} = -(\sqrt{3})^{t}\frac{2 C_{00,5112}^{1112,0}}{\sqrt{5}},$$ (174)

$$C_{00,4213}^{2213,t} = -(\sqrt{3})^{t}2 \sqrt{\frac{7}{15}} C_{00,5112}^{1112,0},$$ (175)

$$C_{00,4212}^{2212,t} = -(\sqrt{3})^{t}\frac{2 C_{00,5112}^{1112,0}}{\sqrt{3}},$$ (176)

$$C_{00,4011}^{2211,t} = C_{00,4413}^{2213,t}=C_{00,3312}^{3112,t}= 0,$$ (177)

$$C_{00,4211}^{2011,t} = C_{00,6211}^{0011,t}=C_{00,5312}^{1112,t}=0 ,$$ (178)

and only coupling constant $C_{60,0000}^{0000,0}$ 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 $v=1$ and $t=1$ 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.