# Logic for Eq.(9.68) in Griffiths' “Intro. to Elementary Particles” (2nd edition)

Treating charged pion decay $$\pi^{-} \rightarrow \ell^{-} + \bar{\nu}_{\ell}$$ (where $$\ell$$ designates a lepton of first- or second-generation) by the representation of the figure below,

Griffiths (Eq.~ 9.68) states that the amplitude "must have the general form

$$\mathcal{M} = \frac{g_w^2}{8(M_W c)^2} \; \left[\bar{u}(3) \gamma_{\mu} (1 - \gamma^5) v(2) \right] \, F^{\mu}$$ where $$F^{\mu}$$ is a 'form factor' describing the $$\pi \rightarrow W$$ blob."

What is the basis for this? I understand the appearance of one weak vertex factor

$$\frac{-i g_w}{2 \sqrt{2}} \gamma^{\nu}(1-\gamma^5)$$

(for the normal vertex in the diagram) and the presence of the (approximated) propagator $$i g_{\mu\nu}/(M_W c)^2$$. However the appearance of the factor $$[(g_w/(2\sqrt{2})]^2$$ suggests an implicit second vertex proportional to $$g_w$$, or rather, to $$i g_w/(2\sqrt{2})$$ (though clearly numbers like $$2\sqrt{2}$$ could as easily have been placed into the magnitude of the needed vector $$F^{\mu}$$).

• There are two weak vertices: $\bar{u} d$ coupling to the W and the lepton vertex, no? The form factor describes the $\pi$ converting to the quark-antiquark pair. – Cosmas Zachos Aug 1 at 18:55
• Yes, but I thought ostensibly the purpose of the "blob" representation was to back away from the (in some sense privileged) knowledge that the $\pi$ is composed of a quark and anti-quark. – JayDee.UU Aug 1 at 20:11
• Possibly; however, the complication is in the strong interactions, and not the weak vertex itself, spectacularly simple. So, even ignoring quarks, you want to separate the hadronic (pion) current $F^\mu$ from the W as much as you can, and keep complete control of the weak parts. Call it a convention. – Cosmas Zachos Aug 1 at 22:22
• Thanks for the clarification. – JayDee.UU Aug 1 at 22:39