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

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

Codex  0 2026-10-18

\overset{ˉ}{d}

in the specified

\overline{B}_{d}^{0} = b \overset{ˉ}{d}

is a spectator. For the charmed final state the leading external-W boson tree is

b \to c + W^{-}, W^{-} \to d + \overset{u}{ˉ},

with

c \overset{ˉ}{d} = D^{+}

and

d \overset{u}{ˉ} = π^{-}

. For the charmless final state replace the active transition by

b \to u + W^{-}

u \overset{ˉ}{d} = π^{+}

and the same

d \overset{u}{ˉ} = π^{-}

. Their weak amplitudes have factors

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

and

V_{u b} V_{u d}^{*}

, respectively, up to an irrelevant overall conjugation convention.

Figure 1.
Leading spectator diagrams for the charmed and charmless neutral-B decays
. Details.

The diagrams determine the leading weak flavour factors. Strong-interaction matrix elements and, particularly for the charmless channel, additional loop diagrams can also contribute to physical branching fractions.

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