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Due to constraints from low energy phenomenology, the mixing between W$_{L}$ and W$_{R}$ is rather small. Thus the SM gauge boson W ± is mostly W ± L . 

In this work, we shall focus on the scalar sector which is relevant to the DM particle. Under the P and CP transformations, the scalar fields transform as follows 

The whole scalar potential contains two parts 

It is required that the dominant part V 0 is both P - and CP -invariant while a small part V 1 contains the soft C -violating interactions. The most general form for V 0 is given by [15] 

where 

<img src='content_image/375435.jpg'>

<img src='content_image/375438.jpg'>

<img src='content_image/375436.jpg'>

(4) 

<img src='content_image/375434.jpg'>

Due to the C and CP symmetries, V 0 only involves combinations of ( S + S $^{∗}$) and SS $^{∗}$. The interactions containing odd powers of ( S − S $^{∗}$) are forbidden as they are P -even but C -odd. Furthermore, ( S − S $^{∗}$) cannot mix with the Higgs multiplets in O$_{i}$ because the five independent gauge-invariant combinations O$_{i}$ ( i = 1 , . . . , 5) in Eq. (7) are both P - and C -even. Other possible Higgs multiplet combinations such as Tr( φ † ˜ φ − ˜ φ $^{†}$φ ) and Tr(Δ † $_{L}$Δ$_{L}$ − Δ † $_{R}$Δ$_{R}$ ) are P -odd, thus cannot couple to ( S − S $^{∗}$). The terms proportional to even powers of ( S − S $^{∗}$) can be rewritten in terms of ( S + S $^{∗}$) 2 and SS $^{∗}$. The absence of odd power term of ( S − S $^{∗}$) means that S$_{D}$ = ( S − S $^{∗}$) / ( $_{i}$√$_{2) is a potential DM candidate.}$ 

