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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 3 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

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:

Figure 1.
Favoured and doubly Cabibbo-suppressed anti-D decays
. Details.

For

\overset{ˉ}{D}^{0} \to K^{+} π^{-}

,

\overset{c}{ˉ} \to \overset{s}{ˉ} W^{-}

and

W^{-} \to d \overset{u}{ˉ}

. The spectator quark combines with

\overset{s}{ˉ}

into the kaon

K^{+} = u \overset{s}{ˉ}

, while

d \overset{u}{ˉ}

forms the pion

π^{-}

. The Cabibbo-Kobayashi-Maskawa matrix factor is

V_{cs} V_{u d}^{*}

.

For

\overset{ˉ}{D}^{0} \to K^{-} π^{+}

,

\overset{c}{ˉ} \to \overset{ˉ}{d} W^{-}

and

W^{-} \to s \overset{u}{ˉ}

. The spectator quark combines with

\overset{ˉ}{d}

into the pion

π^{+} = u \overset{ˉ}{d}

, while

s \overset{u}{ˉ}

forms the kaon

K^{-}

. The Cabibbo-Kobayashi-Maskawa matrix factor is

V_{c d} V_{u s}^{*}

.

Neglecting neutral D-meson mixing and assuming comparable strong interaction matrix elements, the relative direct decay widths are

\frac{Γ ( D ˉ ^{0} \to K ^{-} π ^{+} )}{Γ ( D ˉ ^{0} \to K ^{+} π ^{-} )} \approx \frac{V _{c d} V _{u s}^{*}}{V _{cs} V _{u d}^{*}}^{2} \approx tan^{4} θ_{C} .

Here

θ_{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.

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