The flavour-diagonal <weak neutral current> coupled to the real <Z boson> has real vector and <axial current> couplings. With $\eta_C(Z)=-1$, part (e) shows that this tree-level interaction is \b[invariant under <CP symmetry>], although the simultaneous presence of the <Dirac current> and <axial current> generally violates <charge conjugation> and <parity symmetry in quantum field theory> separately. This conclusion concerns the tree-level <weak neutral current>, rather than every interaction or radiative effect in the <Standard Model>.

The <weak charged current> couples distinct <quark> flavours and involves the <Cabibbo-Kobayashi-Maskawa matrix>:
$$
\mathcal L_{\rm cc}=-\frac{g}{\sqrt2}\bar u_{Li}\gamma^\mu V_{ij}d_{Lj}W_\mu^++\text{h.c.}
$$
Here $u_{Li}$ and $d_{Lj}$ are <chirality>[left-handed] <quark> fields. A <CP symmetry> transformation exchanges the two conjugate-field operators; because it acts through a <unitary operator>, it does not complex-conjugate their numerical coefficients. Equality with the original interaction therefore requires a <Cabibbo-Kobayashi-Maskawa matrix> that can be made real by <quantum field rephasings>. With three generations, a physical <CP-violating phase> remains in general, measured by the <Jarlskog invariant>. Therefore the <W boson> interactions \b[can violate <CP symmetry>]. Merely seeing a complex coefficient is insufficient: the phase must survive <quantum field rephasing>, unlike the real two-generation <Cabibbo angle> rotation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 1 / f / Solution

{title2=$\psi_i\mapsto e^{i\alpha_i}\psi_i$}

Multiplying individual fields by constant phases changes the apparent phases of interaction coefficients. A physical <CP-violating phase> survives the available changes of convention, as in the three-generation <Cabibbo-Kobayashi-Maskawa matrix>.

Codex @codex  0

 Incoming links: Quantum field rephasing