The parity transformation property of a complex scalar field ϕ(x)\phi(x) is given by: Pϕ(t,x)P−1=ηPϕ(t,−x)P\phi(t,\textbf{x}) P^{-1}=\eta_P\phi(t,-\textbf{x}) where ηP=±1\eta_P=\pm 1 . The charge conjugation property of a complex scalar field ϕ(x)\phi(x) is given by: Cϕ(t,x)C−1=ηCϕ†(t,x)C\phi(t,\textbf{x}) C^{-1}=\eta_C\phi^\dagger(t,\textbf{x}) where ηC=±1\eta_C=\pm 1 . Therefore, the CP transformation property of ϕ(x)\phi(x) can be worked out to be (CP)ϕ(x)(CP)−1=C(Pϕ(t,x)P−1)C−1=ηPCϕ(t,−x)C−1=ηPηCϕ†(t,−x)(CP)\phi(x)(CP)^{-1}=C(P\phi(t,\textbf{x}) P^{-1})C^{-1}=\eta_PC\phi (t,-\textbf{x})C^{-1}=\eta_P\eta_C\phi^\dagger(t,-\textbf{x}) \Rightarrow (CP)\phi(x)(CP)^{-1}=\eta_{CP}\phi^\dagger(t,-\textbf{x})\tag{1} where ηCP=ηPηC=±1\eta_{CP}=\eta_P\eta_C=\pm 1 . How will the CPCP -transformation property change if H(x)H(x) is a SU(2) doublet, such as the Higgs field of the standard model H(x)=(ϕ1(x)ϕ2(x))TH(x)=\begin{pmatrix}\phi_1(x)& \phi_{2}(x)\end{pmatrix}^T ? Can I directly use (1) for the doublet H(x)H(x) itself? If yes, how do we work out the action of CP on the doublet H(x)H(x) ? Is it like (CP)H(CP)^{-1}=\begin{pmatrix}(CP)\phi_1(x)(CP)^{-1}\\ (CP)\phi_{2}(x)(CP)^{-1}\end{pmatrix}=\pm \begin{pmatrix}\phi^{\dagger}_1(t,-\textbf{x})\\ \phi_{2}^\dagger(t,-\textbf{x})\end{pmatrix}=\pm H^{\dagger}(t,-\textbf{x})?\tag{2} To be concrete, I want to check the CP-transformation property of the Higgs potential of the Standard model given by V(H)=μ2(H†H)+λ(H†H)2.V(H)=\mu^2(H^\dagger H)+\lambda(H^\dagger H)^2.