Griffiths Introduction to Elementary Particles · Problem 6.6

Quantum Electrodynamics — David J. Griffiths

Physics — QFT · decay-lifetime

Problem

As an application of Example 6.5, consider the decay of

\pi^0 \to \gamma + \gamma

. Of course, the

\pi^0

is a composite object (

u\bar{u}

and

d\bar{d}

), and so equation (6.23) does not really apply. But let's pretend that the

\pi^0

is a true elementary particle, and see how close we come. Unfortunately, we don't know the amplitude

\mathcal{M}

; however, it must have the dimensions of mass times velocity [eq. (6.44)], and there is only one mass and one velocity available. Moreover, the emission of each photon introduces a factor of

\sqrt{\alpha}

(the fine structure constant) into

\mathcal{M}

, as we shall see in Chapter 7, so the amplitude must be proportional to

\alpha

. On this basis, estimate the lifetime of the

\pi^0

. Compare the experimental value. [Evidently, the decay of the

\pi^0

is a much more complicated process than this crude model suggests. See C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, Reading, MA: Benjamin/Cummings, 1983, eq. (1.2.25).]

Related theory

Proper Lifetime From Decay Distance

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