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 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 $^{∗}$) / ( $_{i}$√$_{2) is a potential DM candidate.}$ 5 O$_{1}$ = Tr(Δ † $_{L}$Δ$_{L}$ + Δ † $_{R}$Δ$_{R}$ ) , O$_{2}$ = Tr( φ $^{†}$φ ) , O$_{3}$ = Tr( φ † ˜ φ + ˜ φ $^{†}$φ ) O$_{4}$ = Tr( χ $^{†}$χ ) , O$_{5}$ = Tr( χ $^{†}$˜ χ + ˜ χ $^{†}$χ ) . (7) − V 0 = 1 √$_{2}$ ˜ µ 3 $_{0}$( S + S $^{∗}$) − ˜ µ 2 $_{S}$SS ∗ − 1 4 ˜ µ 2 $_{σ}$( S + S $^{∗}$) 2 $_{+}$√$_{2˜}$ µ$_{σS}$ ( S + S $^{∗}$) SS ∗ + 1 $_{6}$√$_{2}$ ˜ µ$_{3}$$_{σ}$ ( S + S $^{∗}$) 3 + ˜ λ$_{S}$ ( SS $^{∗}$) 2 − 1 4 ˜ λ$_{σS}$ ( S + S $^{∗}$) $^{2}$SS ∗ − 1 16 ˜ λ$_{σ}$ ( S + S $^{∗}$) 4 + 5 ∑ i =1 [ − 1 √$_{2}$ ˜ µ$_{i,σ}$ ( S + S $^{∗}$) + ˜ λ$_{i,S}$SS ∗ − 1 4 ˜ λ$_{i,σ}$ ( S + S $^{∗}$) 2 ] O$_{i}$ , (6) V = V 0 + V $_{1}$. (5) (4) P CP φ φ † φ ∗ χ χ † χ ∗ Δ$_{L}$$_{(}$$_{R}$$_{)}$ Δ$_{R}$$_{(}$$_{L}$$_{)}$ Δ ∗ L ( R ) S S S ∗