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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.

Both the D meson and kaon are spin-zero pseudoscalars. The strong interaction states obey parity symmetry in quantum field theory. Consequently an axial current matrix element between them would have to be a pseudovector formed from only $p$ and $k$, which is impossible: an totally antisymmetric tensor would require more linearly independent vectors. ThusLorentz covariance then leaves two linearly independent vectors for the vector current matrix element, giving the pseudoscalar-to-pseudoscalar form factorsThe vanishing axial current here follows from strong interaction parity symmetry in quantum field theory, not parity symmetry in quantum field theory of the weak interaction.

To obtain the requested decay formula, use naive factorization of a nonleptonic meson decay: approximate the four-quark matrix element by the product of the ${\bar{D}}^0\to K^{+}$ current matrix element and the vacuum-to-pion matrix element. This is an additional hadronic approximation; tree-level weak vertices alone do not establish it, and nonfactorizable Quantum chromodynamics effects can change the result. The vacuum-to-pion vector current matrix element vanishes by parity symmetry in quantum field theory, while the specified pion decay constant normalization givesThe $\sqrt{2}$ cancels the $1/\sqrt{2}$ in the four-fermion interaction. Up to an irrelevant overall phase, the scattering amplitude is thereforeFor $m_{\pi }=0$, $q^2=0$, so only $f_{+}(0)$ survives. Integrating the two-body decay phase space in the D meson rest frame givesHenceThis coefficient uses exactly the stated pion decay constant convention and the stated massless-pion approximation.