# A Bar on a fermion field, arrows on fermion lines and particle-antiparticle nature of a fermion

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1. Feb 18, 2017

### spaghetti3451

This question is about the use of bar on a fermionic field in a Lagrangian, the use of arrows on external fermion lines and the particle-antiparticle nature of a fermion.

For illustration of my question, I will use the following the charged-current interaction of the Standard model:

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

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1. What is the physical consequence of having a bar on one fermionic field in a Lagrangian? For example, the first term in the Lagrangian has a bar over the neutrino field $\nu$ while the third term has a bar over the electron field $e$. Is the role of the bar simply to ensure Lorentz invariance or is there some physical consequence to, say, the $W^{+}$-boson-to-lepton-decay due to having a bar on the neutrino field $\nu$ and not on the electron field $e$ in the first term of the Lagrangian?

2. How does the particle-antiparticle nature of a fermion show itself in the Lagrangian? Is it through the bar on a fermion field, or through the charge-conjugation operator?

2. Feb 23, 2017