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The parity transformation property of a complex scalar field $\phi(x)$ is given by: $$P\phi(t,\textbf{x}) P^{-1}=\eta_P\phi(t,-\textbf{x})$$ where $\eta_P=\pm 1$. The charge conjugation property of a complex scalar field $\phi(x)$ is given by: $$C\phi(t,\textbf{x}) C^{-1}=\eta_C\phi^\dagger(t,\textbf{x})$$ where $\eta_C=\pm 1$. Therefore, the CP transformation property of $\phi(x)$ can be worked out to be $$(CP)\phi(x)(CP)^{-1}=C(P\phi(t,\textbf{x}) P^{-1})C^{-1}=\eta_PC\phi (t,-\textbf{x})C^{-1}=\eta_P\eta_C\phi^\dagger(t,-\textbf{x})$$ $$\Rightarrow (CP)\phi(x)(CP)^{-1}=\eta_{CP}\phi^\dagger(t,-\textbf{x})\tag{1}$$ where $\eta_{CP}=\eta_P\eta_C=\pm 1$.

How will the CP-transformation property change if $H(x)$ is a SU(2) doublet, such as the Higgs field of the standard model $H(x)=\begin{pmatrix}\phi_1(x)& \phi_{2}(x)\end{pmatrix}^T$? Can I directly use (1) for the doublet $H(x)$ itself? If yes, how do we work out the action of CP on the doublet $H(x)$? 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 $$V(H)=\mu^2(H^\dagger H)+\lambda(H^\dagger H)^2.$$

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    $\begingroup$ By the Coleman-Mandula theorem, spacetime symmetries commute with the internal ones, so you can just use (1) for the components of the field, forgetting about the internal symmetry structure. $\endgroup$ – coconut Dec 15 '17 at 12:25
  • $\begingroup$ @coconut Parity is a spacetime symmetry. Right? Have I correctly implemented the CP symmetry in (2) on the doublet $H$? $\endgroup$ – SRS Dec 15 '17 at 13:05
  • $\begingroup$ @coconut You can put it as the answer if you wish. $\endgroup$ – SRS Dec 15 '17 at 13:58
  • $\begingroup$ Hmm, it seems that I might be wrong, as Coleman-Mandula is on the level of Lie algebras, so it doesn't take into account discrete symmetries such as C and P. Let's see if someone else can clarify this $\endgroup$ – coconut Dec 15 '17 at 14:27
  • $\begingroup$ These are certainly not the most general C and P transformations. In general, you can allow $\eta_P$ and $\eta_C$ to be arbitrary phase factors. And, in principle, you may even generalize the transformation of the Higgs by promoting the phase to a constant U(2) matrix. For the simple potential you are showing, this is not important, but once you go to more complicated models with complex coupling coefficients, these things start to become important. $\endgroup$ – user178876 Dec 16 '17 at 22:06
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Let me start by remarking that these are certainly not the most general C and P transformations. In general, you can allow $\eta_C$ and $\eta_P$ to be arbitrary phase factors. And, in principle, you may even generalize the transformation of the Higgs by promoting the phase to a constant U(2) matrix. For the simple potential you are showing, this is not important, but once you go to more complicated models with complex coupling coefficients, these things start to become important.

In any case, you can work in a field basis in which $\eta_{CP}=1$. CP is unbroken in this sector.

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