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ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2021/section-b/4/e/solution

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Past exam of the particle and nuclear physics course of the University of Cambridge / 2021 / Section B / 4 / e / Solution by

Codex  0 2026-10-18

Ignoring the requested final-mass difference and initially taking equal hadronic matrix elements, the common

W^{-} \to d \overset{u}{ˉ}

factor cancels. A decay width is proportional to the squared decay amplitude, so

\frac{B ( B _{d}^{0} \to D ^{+} π ^{-} )}{B ( B _{d}^{0} \to π ^{+} π ^{-} )} ≃ \frac{V _{c b}}{V _{u b}}^{2} ≃ \frac{s i n ^{4} θ _{C}}{s i n ^{6} θ _{C}} = \frac{1}{s i n ^{2} θ _{C}} .

(1)

With the supplied Cabibbo angle,

R_{B}^{rough} = \frac{1}{0.22 5 ^{2}} = 19.75 ≃ 20.

(2)

The quoted measurement, about

500 \pm 10

, is larger by a factor

500/19.75 ≃ 25.3

. If that whole difference were attributed to a ratio of other amplitudes, it would require a factor

25.3 ≃ 5.03

This does not invalidate quark mixing. The supplied Cabibbo-Kobayashi-Maskawa matrix is only a schematic hierarchy in powers of

sin θ_{C}

, particularly for

V_{u b}

; its omitted coefficients matter greatly after squaring. Equal phase space also does not imply equal hadronic decay amplitudes: the two transitions have different hadronic form factors and strong-interaction dynamics, and loop contributions can interfere with the charmless tree amplitude. Thus the calculation correctly predicts enhancement of the charmed channel, but the supplied CKM hierarchy alone does not quantitatively predict the measured ratio. It would be unjustified to treat

19.75

as a precision prediction with only the quoted experimental error.

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