Vertex factor for W^- -> e + anti neutrino_e

[Mithra]

Hi, I'm wondering if anyone can give me some advice on working out the vertex factor from a lagrangian. I think I know what I should be doing however it isn't quite getting the right answer so if anyone could guide me that would be great.

$\mathcal{L}_{W^-e^-\nu_e} = \frac{g_2}{\sqrt{2}}[\overline{\nu}_{eL}W^+_{\mu}\gamma^{\mu}e_L + \overline{e}_L W^-_{\mu}\gamma^{\mu}\nu_{eL}]$

Initially I just ignored the fields and so got a factor
$g_2 \sqrt{2} \gamma^\mu$
however I know this isn't right. From my notes I can see that there should be a 1-gamma^5 included, along with the factor being 1/(2*sqrt(2)) so I thought maybe I needed to convert the e_L s into just e using the helicity conversion

$e_L = \frac{1-\gamma^5}{2}$

but the factors still do not seem to be coming out correctly. I'm thinking maybe I should convert the W^(+/-) into W^1/W^2 but that doesn't look like its going to be hugely successful. I haven't yet changed the neutrino fields from left-handed as I assume neutrino fields are generically left handed anyway?

Any advice would be great, thanks!

 

[andrien]

multiply lagrangian by i,put plane wave form for those operators.Also write the amplitude in two spinors form by using those chiral operator.