Codex @codex  0

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 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:Here $u_{Li}$ and $d_{Lj}$ are 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 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.

The tree-level Feynman diagrams contain an unchanged spectator quark, of up quark flavour and the two possible weak charged current transitions of the charm antiquark:

For ${\bar{D}}^0\to K^{+}{\pi }^{-}$, $\bar{c}\to \bar{s}{W}^{-}$ and $W^{-}\to d\bar{u}$. The spectator quark combines with $\bar{s}$ into the kaon $K^{+}=u\bar{s}$, while $d\bar{u}$ forms the pion ${\pi }^{-}$. The Cabibbo-Kobayashi-Maskawa matrix factor is $V_{cs}{V}_{ud}^{\ast }$.

For ${\bar{D}}^0\to K^{-}{\pi }^{+}$, $\bar{c}\to \bar{d}{W}^{-}$ and $W^{-}\to s\bar{u}$. The spectator quark combines with $\bar{d}$ into the pion ${\pi }^{+}=u\bar{d}$, while $s\bar{u}$ forms the kaon $K^{-}$. The Cabibbo-Kobayashi-Maskawa matrix factor is $V_{cd}{V}_{us}^{\ast }$.

Neglecting neutral D-meson mixing and assuming comparable strong interaction matrix elements, the relative direct decay widths areHere ${\theta }_C$ is the Cabibbo angle. The second process is doubly Cabibbo suppressed: it contains two small Cabibbo suppression factors in its amplitude. The Cabibbo angle estimate assumes similar Quantum chromodynamics matrix elements; it is not an exact rate equality.