10.3 Solutions 597
The two vertices 1 and 2 where the W boson couples pertain to weak
interactions and have strengths√
αw. Vertex 3 is electromagnetic and has
strength√
αem. The overall strength of the diagram is√
αw.√
αw.√
αemor
αw√
αem10.105 The decay ofD^0 →K−+π+is accomplished by the exchange ofW+
boson as illustrated by the Feynman diagram Fig. 10.23.
Fig. 10.23
The quark composition (D^0 =cu,K−=su,π+=ud) is also indicated.
The favoured route for the decay is viac→s. Hence the decay occurs via
lowest order charge current weak interaction. However, forD+→K^0 +
π+the c-quark is required to decay to d-quark via W emission and the
subsequent decay ofW+toπ+. This would mean that thedquark in the
D+decay to ansquark in theK^0 which is not possible as they both have
the same charge. Furthermore the transitionc→dis not favoured in the
Cabibbo scheme.10.106 Consider the Feynman diagrams for the decaysπ+→μ++νμandπ^0 →
γ+γ
Fig. 10.24
For charged pion there are two vertices of strength
√
αw, and a propagator
1
Q^2 +Mw^2 c^2∼=^1
Mw^2 c^2
In the limitQ<<Mwcmomentum transfer squaredQ^2 carried byW
boson is quite small. Therefore, the decay rate will be proportional to
(√
αw√
αw
Mw^2) 2
=
αw^2
Mw^4