Someone argues: "The weak interaction is weak because

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Interacting fields, Feynman diagrams and QED · Electroweak theory and the Standard Model

Particle Physics, Ch Exchange, range and coupling: diagrams as bookkeeping

Problem

g_{\rm W}

, the weak coupling, is a tiny number, much smaller than the electric charge. In

\pi^+ \to \mu^+\nu_\mu

the internal line in the diagram is a genuine

W^+

boson that exists in flight between the two vertices; since

M_Wc^2 = 80.37\,\mathrm{GeV}

is far above the

139.6\,\mathrm{MeV}

available in pion decay, this real particle should be forbidden by energy conservation, and the fact that the decay happens anyway shows energy conservation is being violated at the level of individual vertices, compensated for by the weakness of

g_{\rm W}

." ((a)) Compare

g_{\rm W} = 0.653

with the dimensionless electric charge

\sqrt{4\pi\alpha} = 0.30

(with

\alpha = 1/137.0

) and say whether the first sentence is correct. ((b)) Explain, using the energy-time relation, why the internal line is not a real particle in flight and why no vertex-by-vertex energy conservation is violated. ((c)) State, in one sentence, the actual reason

\pi^+ \to \mu^+\nu_\mu

and other weak processes at low momentum transfer are rare.

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