FIG. 4: Generic one-loop Feynman diagrams for the \( Z Z Z ^ { * } \) coupling. The complete set of diagrams is obtained by permuting \( Z _ { \alpha } ( p _ { 1 } ) \) and \( Z _ { \beta } ( p _ { 2 } ) \) and exchanging the charged scalars bosons.

\( Z Z Z ^ { * } \) vertex can be generated. Notice that diagrams (2) and (3) are obtained from diagram (1) after performing the permutations \( Z _ { \alpha } ( p _ { 1 } ) \leftrightarrow Z _ { \mu } ( q ) \), and \( Z _ { \beta } ( p _ { 2 } ) \leftrightarrow Z _ { \mu } ( q ) \), respectively. Additional diagrams are obtained from these diagrams following a similar procedure as that described in Fig. 3: for each one of the Feynman diagrams of Fig. 4 there are three more diagrams that are obtained similarly as diagrams (b)-(d) of Fig. 3, which are obtained from diagram (a) by permuting the Z gauge bosons and exchanging the charged scalars. Therefore, there are a total of twelve Feynman diagrams for the \( Z Z Z ^ { * } \) coupling. By using the appropriate simplifications, the amplitude of each diagram of Figure 4 reduces to those of the \( Z Z ^ { \ast } \gamma \) and \( Z Z \gamma ^ { * } \) couplings. The diagrams (2) and (3) of Fig. 4 not only are required by Bose symmetry but, once their amplitudes are added up, ultraviolet divergences cancel out. After the Passarino-Veltman method is applied, we obtain the following result

\[ \begin{array} { r l } { f _ { 4 } ^ { Z } ( q ^ { 2 } , m _ { i } ^ { 2 } , m _ { j } ^ { 2 } ) \! \! } & { = \, \frac { m _ { Z } ^ { 2 } g _ { i i } ^ { Z } \mathrm { I m } \left( g _ { i j } ^ { Z } g _ { j i } ^ { Z ^ { + } } \right) } { 1 2 \pi ^ { 2 } s _ { W } q ^ { 2 } ( q ^ { 2 } - m _ { Z } ^ { 2 } ) ( q ^ { 2 } - 4 m _ { Z } ^ { 2 } ) ^ { 2 } } } \\ { \times } & { \Big \{ q ^ { 2 } \left( q ^ { 2 } \left( m _ { i } ^ { 2 } + 3 m _ { j } ^ { 2 } + 7 m _ { Z } ^ { 2 } \right) + 6 \left( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } \right) ^ { 2 } - 1 6 m _ { i } ^ { 2 } m _ { Z } ^ { 2 } - 6 m _ { Z } ^ { 4 } - q ^ { 4 } \right) B _ { \mathrm { i i } } ( q ^ { 2 } ) } \\ { + \, \left( m _ { Z } ^ { 2 } q ^ { 2 } \left( 1 0 m _ { Z } ^ { 2 } - 1 3 m _ { i } ^ { 2 } - 3 m _ { j } ^ { 2 } \right) + 6 m _ { Z } ^ { 2 } ( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } ) \left( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } + 2 m _ { Z } ^ { 2 } \right) + q ^ { 4 } \left( 4 m _ { i } ^ { 2 } - m _ { Z } ^ { 2 } \right) \right) B _ { \mathrm { i i } } ( m _ { Z } ^ { 2 } ) } \\ { - \, \left( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } \right) \left( 2 q ^ { 2 } \left( 4 m _ { Z } ^ { 2 } - q ^ { 2 } \right) ( 1 - B _ { i j } ( 0 ) ) - 3 q ^ { 4 } B _ { i j } ( q ^ { 2 } ) - 3 m _ { Z } ^ { 2 } \left( 4 m _ { Z } ^ { 2 } - 7 q ^ { 2 } \right) B _ { i j } ( m _ { Z } ^ { 2 } ) \right) } \\ { - \, \left. 6 \left( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } + 2 m _ { Z } ^ { 2 } \right) \left( m _ { Z } ^ { 2 } q ^ { 2 } \left( m _ { Z } ^ { 2 } - 3 m _ { i } ^ { 2 } - m _ { j } ^ { 2 } \right) + m _ { Z } ^ { 2 } \left( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } \right) ^ { 2 } + m _ { i } ^ { 2 } q ^ { 4 } \right) C _ { i i j } ( q ^ { 2 } ) } \\ { - \, \left. 6 q ^ { 2 } \left( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } - m _ { Z } ^ { 2 } \right) \left( m _ { i } ^ \]

where q is now the four-momentum of the off-shell Z boson. We note that all the properties discussed above are also present in this form factor. In the next section we will evaluate the CP-violating TNGBCs for illustrative values of the charged scalar boson masses and the four-momentum of the virtual gauge boson.