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I am puzzled by the $W^{\pm}$ vertex factor in weak interactions. In Griffiths' textbook "Introduction to Elementary Particles", the $W^{\pm}$ vertex factor is given by (10.92) on page 324: \begin{equation} \frac{-ig_{w}}{2\sqrt{2}}\gamma^{\mu}(1 - \gamma^{5}) \tag{10.92} \end{equation} However, In Srednicki's textbook "Quantum Field Theory", Problem 88.6 (on page 538) asks us to compute rates for the decay processes $W^{+}\rightarrow e^{+} \nu_{e}$, ... etc. The answer is given in the solutions manual. On page 146 of the solutions manual, it is stated

Consider a massive vector field $Z^{\mu}$ and a Dirac fermion field $\Psi$ with $\mathcal{L}_{int} = Z^{\mu}\overline{\Psi}(g_{v} - g_{A}\gamma^{5})\Psi$; then the amplitude for $Z\rightarrow e^{+}e^{-}$ is $\mathcal{T} = \varepsilon^{*\mu}\overline{v}_{2}\gamma_{\mu}(g_{v} - g_{A}\gamma^{5})u_{1}$. ... ... The amplitude is the same if $\overline{\Psi}$ is a different Dirac field that is unrelated to $\Psi$, so it also holds for a process like $W^{+}\rightarrow e^{+} \overline{\nu}$.

My question is: Why is there no $\varepsilon^{*\mu}$ in (10.92), whereas there is an $\varepsilon^{*\mu}$ (which seems to account for $W^{+}$) in the amplitude $\mathcal{T} = \varepsilon^{*\mu}\overline{v}_{2}\gamma_{\mu}(g_{v} - g_{A}\gamma^{5})u_{1}$ for the decay process $W^{+}\rightarrow e^{+} \overline{\nu}$?

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Those two things are related, but different.

Your equation $(10.92)$ indicates the value of a vertex, while $\mathcal T$ in Srednicki's book represents an amplitude.

Basically, the vertex is one of the two building blocks of the Feynmann diagrams. A diagram is a multiplication of vertices and propagators, and becomes a complex amplitude for the process when you multiply that amplitude with the external particle factors, such as $\epsilon^\mu$.

An example: Feynmann's QED vertex is given by $-ie\gamma^\mu$ (the sign depends on conventions, I'll follow Michele Maggiore's textbook). Now, let's take the typical first order contribution to the process $e^-\gamma\to e^-$: the relevant graph is

enter image description here

(here future and past are messed up, that graph here is just for reference). Now, the graph is composed by a vertex and three external legs: the vertex has value $-ie\gamma^\mu$, and the amplitude can be written as $$ \mathcal T=\epsilon_\mu(k)\bar u(p_1)(-i e\gamma^\mu)u(p_2), $$ where $k$ is the momentum of the photon, $p_1$ the momentum of the incoming electron and $p_2$ the momentum of the outgoing electron. The modulus of the amplitude squared, $|\mathcal T|^2$, is proportional to decay lengths and cross sections (more in general, in the $S$-matrix), and is used to understand "how much" a process happens.

P.s.: as a nice addendum, note that, if you consider the process $\gamma\to e^+e^-$, you can use the same rotated graph, so you have to change some external legs factor, obtaining $$ \mathcal T=\epsilon_\mu(k)\bar {u}(p_1)(-ie\gamma^\mu)v(p_2). $$ If you calculate the square of this amplitude, you obtain a value that is different from zero. But, from elementary considerations about 4-impulse conservation, you know that this process can't happen, as there is no way to sum the timelike momenta of the matter particles to obtain a lightlike momentum. So the amplitude can be different from zero even if a process is not observed: in this case, the $\delta$ that expresses momentum conservation in the $S$ matrix takes care of that, and the process $\gamma\to e^+e^-$ cannot happen, even if the amplitude is non zero.

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