A Deducing decay processes and Feynman diagrams using Lagrangian and conservation laws

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1. Feb 18, 2017

spaghetti3451

The decay processes of the $W$ bosons are completely governed by the charged current interaction terms of the Standard model:

$$\mathcal{L}_{cc} = ie_{W}\big[W_{\mu}^{+}(\bar{\nu}_{m}\gamma^{\mu}(1-\gamma_{5})e_{m} + V_{mn}\bar{u}_{m}\gamma^{\mu}(1-\gamma_{5})d_{n})\\ + W_{\mu}^{-}(\bar{e}_{m}\gamma^{\mu}(1-\gamma_{5})\nu_{m} + (V^{\dagger})_{mn}\bar{d}_{m}\gamma^{\mu}(1-\gamma_{5})u_{n})\big].$$

I would like to find the possible decay processes of the $W^{-}$ boson.

I would like to find the Feynman diagrams for each of these processes.

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I find that I have to use electric charge conservation and lepton charge conservation to deduce that the possible decay processes are

$$W^{-} \to e^{-} \bar{\nu}_{e}$$
$$W^{-} \to \mu^{-}\bar{\nu}_{\mu}$$
$$W^{-} \to \tau^{-}\bar{\nu}_{\tau}$$
$$W^{-} \to d\bar{u}$$
$$W^{-} \to s\bar{c}$$

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As the fermion is outgoing, the arrow on its external line in the Feynman diagram must point outwards.

As the anti-fermion is outgoing, the arrow on its external line in the Feynman diagram must point inwards.

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1. Is there a way to obtain these possible decay processes without using conservation laws and by simply inspecting the Lagrangian?

2. Is there a way to determine if the arrows on the external fermion and anti-fermion lines must point inwards and outwards without knowing using their particle/anti-particle property and by simply inspecting the Lagrangian?

2. Feb 23, 2017