FIG. 3: One-loop level Feynman diagrams inducing the \( Z Z ^ { \ast } \gamma \)(\( V = \gamma \)) coupling.

which is necessary to obtain nonzero results and is consistent with the Lorentz structure of the Lagrangian. We also have introduced the shorthand notation \( B _ { a b } ( c ^ { 2 } ) \, = \, B _ { 0 } ( c ^ { 2 } , m _ { a } ^ { 2 } , m _ { b } ^ { 2 } ) \), \( C _ { a b c } ( p _ { 2 } ^ { 2 } ) \, = \, C _ { 0 } ( m _ { Z } ^ { 2 } , 0 , p _ { 2 } ^ { 2 } , m _ { a } ^ { 2 } , m _ { b } ^ { 2 } , m _ { c } ^ { 2 } ) \) and \( C _ { a b c } ( q ^ { 2 } ) \, = \, C _ { 0 } ( m _ { Z } ^ { 2 } , m _ { Z } ^ { 2 } , q ^ { 2 } , m _ { a } ^ { 2 } , m _ { b } ^ { 2 } , m _ { c } ^ { 2 } ) \), where \( B _ { 0 } \) and \( C _ { 0 } \) stand for Passarino-Veltman scalar functions. From the above expressions, it is evident that the form factors vanish when the masses of the charged scalar bosons are degenerate. It is also straightforward to show that ultraviolet divergences cancel out.

# B. \( Z Z \gamma ^ { * } \) coupling

The Feynman diagrams for the \( Z _ { \alpha } ( p _ { 1 } ) Z _ { \beta } ( p _ { 2 } ) \gamma _ { \mu } ^ { * } ( q ) \) couplings are similar to those inducing the \( Z Z ^ { * } \gamma \) coupling, but in this case the photon is off-shell. The corresponding form factor is given by:

\[ \begin{array} { r l } { f _ { 4 } ^ { \gamma } ( q ^ { 2 } , m _ { i } ^ { 2 } , m _ { j } ^ { 2 } ) \; = \; \frac { m _ { Z } ^ { 2 } \, \mathrm { I m } \, \left( g _ { i j } ^ { Z } g _ { j  i } ^ { Z ^ { + } } \right) } { 1 2 m _ { Z } ^ { 2 } \pi ^ { 2 } q ^ { 2 } ( 4 m _ { Z } ^ { 2 } - q ^ { 2 } ) ^ { 2 } } \Big \{ ( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } ) \, \big ( 1 2 m _ { Z } ^ { 2 } B _ { i j } ( m _ { Z } ^ { 2 } ) + ( 4 m _ { Z } ^ { 2 } - q ^ { 2 } ) ( 1 - B _ { i j } ( 0 ) ) \big ) } \\ { \; + \; \; \left( q ^ { 2 } \, ( m _ { i } ^ { 2 } + 3 m _ { j } ^ { 2 } + 7 m _ { Z } ^ { 2 } - q ^ { 2 } ) + 6 \, \big ( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } \big ) ^ { 2 } - 2 m _ { Z } ^ { 2 } ( 8 m _ { i } ^ { 2 } - 3 m _ { Z } ^ { 2 } ) \right) B _ { i i } ( q ^ { 2 } ) } \\ { \; - \; \; 6 \, \big ( m _ { i } ^ { 2 } - m _ { j } ^ { 2 } - m _ { Z } ^ { 2 } \big ) \, \Big ( m _ { i } ^ { 4 } - 2 m _ { i } ^ { 2 } \, \big ( m _ { j } ^ { 2 } + m _ { Z } ^ { 2 } \big ) + \big ( m _ { j } ^ { 2 } - m _ { Z } ^ { 2 } \big ) ^ { 2 } + m _ { j } ^ { 2 } q ^ { 2 } \Big ) \, C _ { i j i } ( q ^ { 2 } ) - \big ( i \leftrightarrow j \big ) \Big \} , \mathrm { \small ~ ( 1 0 ) } } \end{array} \]

where q is the photon four-momentum. As expected, this form factor is proportional to \( \mathrm { I m } ( g _ { i j } ^ { Z } g _ { j i } ^ { Z ^ { * } } ) \) and vanishes when \( m _ { i } = m _ { j } \) .

# C. \( Z Z Z ^ { * } \) coupling

Because of Bose symmetry, in the case of the \( Z Z Z ^ { * } \) coupling, there are several more diagrams than those inducing the \( Z Z ^ { \ast } \gamma \) vertex. We present in Figure 4 the generic Feynman diagrams from which all diagrams inducing the

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.