Solution
= Solution
In the usual real-vacuum phase convention the complex scalar <Higgs doublet> obeys
$$
\boxed{\phi(x)\longmapsto\phi^*(x_P).}
$$
<Charge conjugation> complex-conjugates the doublet, while <parity> changes its argument without a Lorentz-index factor because it has spin zero. A fixed gauge rotation or intrinsic scalar phase may be included in an equivalent convention; the displayed choice leaves the standard real neutral Higgs expectation value invariant.