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

At fourth order we found that the isoscalar and isovector coupling constants are connected or not depending on the parity of quantum numbers $n$. Moreover, similarly as for the scalar-isoscalar channel of the CE, the scalar and vector coupling constants are kept apart. The constraints among the scalar coupling constants read,

$$C_{00,4000}^{0000,t} = \frac{3 C_{00,2202}^{2202,t}}{2 \sqrt{5}},$$ (76)

$$C_{00,2000}^{2000,t} = \frac{1}{2} \sqrt{5} C_{00,2202}^{2202,t},$$ (77)

$$C_{00,3101}^{1101,t} = -2\sqrt{\frac{3}{5}}3^t C_{00,2202}^{2202,1-t},$$ (78)

and those among the vector coupling constants read,

$$C_{00,3110}^{1110,t} = -\frac{2}{\sqrt{5}} 3^t C_{00,2212}^{2212,1-t}-\frac{7}{\sqrt{15}} 3^t C_{00,4211}^{0011,1-t},$$ (79)

$$C_{00,3111}^{1111,t} = -2 (3^t)\sqrt{\frac{3}{5}} C_{00,2212}^{2212,1-t},$$ (80)

$$C_{00,3112}^{1112,t} = -2 (3^t)C_{00,2212}^{2212,1-t}-\frac{14}{5}\frac{3^t}{\sqrt{3}} C_{00,4211}^{0011,1-t},$$ (81)

$$C_{00,3312}^{1112,t} = -\frac{2}{3^{1-t}} \sqrt{\frac{7}{5}} C_{00,4211}^{0011,1-t},$$ (82)

$$C_{00,4011}^{0011,t} = \frac{3}{2} \sqrt{\frac{3}{5}} C_{00,2212}^{2212,t}+\frac{7 C_{00,4211}^{0011,t}}{4 \sqrt{5}},$$ (83)

$$C_{00,2011}^{2011,t} = \frac{1}{2} \sqrt{15} C_{00,2212}^{2212,t}+\frac{7}{12} \sqrt{5} C_{00,4211}^{0011,t},$$ (84)

$$C_{00,2211}^{2011,t} = \frac{7}{3} C_{00,4211}^{0011,t},$$ (85)

$$C_{00,2211}^{2211,t} = \sqrt{\frac{3}{5}} C_{00,2212}^{2212,t}+\frac{7 C_{00,4211}^{0011,t}}{3 \sqrt{5}},$$ (86)

$$C_{00,2213}^{2213,t} = \sqrt{\frac{7}{5}} C_{00,2212}^{2212,t}+\frac{1}{2} \sqrt{\frac{21}{5}} C_{00,4211}^{0011,t} .$$ (87)

We also found that the fourth-order surface isovector coupling constants must vanish,

$$C_{40,0000}^{0000,1} = C_{40,0011}^{0011,1} = C_{42,0011}^{0011,1} = 0,$$ (88)

whereas the corresponding isoscalar coupling constants are unrestricted. Apart from the coupling constants discussed above, all the remaining 54 fourth-order coupling constants are forced to be equal to zero.

In the same way, at sixth order we found the following constraints for the scalar,

$$C_{00,6000}^{0000,t} = -3^t\frac{3 C_{00,3303}^{3303,1-t}}{4 \sqrt{7}},$$ (89)

$$C_{00,4000}^{2000,t} = -3^t\frac{3}{4} \sqrt{7} C_{00,3303}^{3303,1-t},$$ (90)

$$C_{00,4202}^{2202,t} = -3^t3 \sqrt{\frac{5}{7}} C_{00,3303}^{3303,1-t},$$ (91)

$$C_{00,5101}^{1101,t} = \frac{9}{2} \sqrt{\frac{3}{7}} C_{00,3303}^{3303,t},$$ (92)

$$C_{00,3101}^{3101,t} = \frac{9}{10} \sqrt{21} C_{00,3303}^{3303,t},$$ (93)

and vector coupling constants,

$$C_{00,5110}^{1110,t} = -3^t\frac{C_{00,4212}^{2212,1-t}}{2 \sqrt{5}}-3^t3 \sqrt{\frac{3}{5}} C_{00,6211}^{0011,1-t},$$ (94)

$$C_{00,5111}^{1111,t} = -3^t\frac{1}{2} \sqrt{\frac{3}{5}} C_{00,4212}^{2212,1-t},$$ (95)

$$C_{00,5112}^{1112,t} = -3^t\frac{1}{2} C_{00,4212}^{2212,1-t}-3^t\frac{6}{5} \sqrt{3} C_{00,6211}^{0011,1-t},$$ (96)

$$C_{00,5312}^{1112,t} = -\frac{4}{3^{1-t}} \sqrt{\frac{7}{5}} C_{00,6211}^{0011,1-t},$$ (97)

$$C_{00,3110}^{3110,t} = -3^t\frac{7 C_{00,4212}^{2212,1-t}}{10 \sqrt{5}}-3^t\frac{21}{5} \sqrt{\frac{3}{5}} C_{00,6211}^{0011,1-t},$$ (98)

$$C_{00,3111}^{3111,t} = -3^t\frac{7}{10} \sqrt{\frac{3}{5}} C_{00,4212}^{2212,1-t},$$ (99)

$$C_{00,3112}^{3112,t} = -3^t\frac{7}{10} C_{00,4212}^{2212,1-t}-3^t\frac{42}{25} \sqrt{3} C_{00,6211}^{0011,1-t},$$ (100)

$$C_{00,3312}^{3112,t} = -3^t\frac{12}{5} \sqrt{\frac{7}{5}} C_{00,6211}^{0011,1-t},$$ (101)

$$C_{00,3312}^{3312,t} = -3^t\frac{1}{9} C_{00,4212}^{2212,1-t}-3^t\frac{2}{5} \sqrt{3} C_{00,6211}^{0011,1-t},$$ (102)

$$C_{00,3313}^{3313,t} = -\frac{3^t}{9} \sqrt{\frac{7}{5}} C_{00,4212}^{2212,1-t},$$ (103)

$$C_{00,3314}^{3314,t} = -\frac{C_{00,4212}^{2212,1-t}}{3^{1-t} \sqrt{5}}-\frac{8 C_{00,6211}^{0011,1-t}}{3^{1-t} \sqrt{15}},$$ (104)

$$C_{00,6011}^{0011,t} = \frac{1}{4} \sqrt{\frac{3}{5}} C_{00,4212}^{2212,t}+\frac{3 C_{00,6211}^{0011,t}}{2 \sqrt{5}},$$ (105)

$$C_{00,4011}^{2011,t} = \frac{7}{4} \sqrt{\frac{3}{5}} C_{00,4212}^{2212,t}+\frac{21 C_{00,6211}^{0011,t}}{2 \sqrt{5}},$$ (106)

$$C_{00,4211}^{2011,t} = 6 C_{00,6211}^{0011,t},$$ (107)

$$C_{00,4011}^{2211,t} = \frac{21}{5} C_{00,6211}^{0011,t},$$ (108)

$$C_{00,4211}^{2211,t} = \sqrt{\frac{3}{5}} C_{00,4212}^{2212,t}+\frac{12 C_{00,6211}^{0011,t}}{\sqrt{5}},$$ (109)

$$C_{00,4213}^{2213,t} = \sqrt{\frac{7}{5}} C_{00,4212}^{2212,t}+18 \sqrt{\frac{3}{35}} C_{00,6211}^{0011,t},$$ (110)

$$C_{00,4413}^{2213,t} = \frac{4 C_{00,6211}^{0011,t}}{\sqrt{5}} .$$ (111)

We also found that the sixth-order surface isovector coupling constants must vanish,

$$C_{60,0000}^{0000,1} = C_{60,0011}^{0011,1} = C_{62,0011}^{0011,1} = 0,$$ (112)

whereas the corresponding isoscalar coupling constants are unrestricted. Apart from the coupling constants discussed above, all the remaining 200 sixth-order coupling constants are forced to be equal to zero.

We have seen that for the scalar-isovector channel, the coupling constants are diagonal or nondiagonal in the isospin quantum number $t$. We can understand this point considering the fact that in order to separate the scalar-isovector channel of the CE and obtain condition (48) for $t=1$, we have to multiply Eq. (41) by the isospin operator $\tau_{\tau'\tau}$. Then, the isospin index $t''$ tells us in which half of the isospin space the coupling constants is. Nondiagonal constraints mean, in fact, that the same pair of secondary densities in the final form of condition (48) can be produced by two terms of the functional that are isoscalar and isovector. This is possible, because the coupling to rank $t=1$ allows for pairs of densities nondiagonal in the isospin space, and this, in turn, boils down to constraints for coupling constants nondiagonal in the isospin space.