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

[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:

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

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?

 

[eljose79]

1. The physical consequence of having a bar on one fermionic field in a Lagrangian is that it represents the antiparticle of that field. In the Standard Model, fermions are divided into two categories: particles and antiparticles. The bar on a fermionic field indicates that it represents the antiparticle of that particular fermion. For example, in the first term of the Lagrangian, the bar over the neutrino field $\nu$ indicates that it represents the antineutrino, while in the third term, the bar over the electron field $e$ indicates that it represents the positron. This is consistent with the fact that the charged-current interaction involves the exchange of $W^{\pm}$ bosons, which can only interact with the corresponding antiparticles.

The role of the bar is not just to ensure Lorentz invariance, but it also has physical consequences. For instance, in the first term of the Lagrangian, the presence of the bar over the neutrino field $\nu$ allows for the decay of a $W^{+}$ boson into an antineutrino and an electron, while in the third term, the absence of a bar over the electron field $e$ prevents the decay of a $W^{-}$ boson into a positron and a neutrino. In other words, the presence or absence of the bar affects the possible interactions between particles and antiparticles, which has important physical consequences.

2. The particle-antiparticle nature of a fermion is also reflected in the charge-conjugation operator, which is represented by the bar. The charge-conjugation operator is responsible for transforming a particle into its antiparticle, and vice versa. In the Lagrangian, the bar over a fermion field indicates that it is being acted upon by the charge-conjugation operator, thus transforming it into its antiparticle. This is consistent with the fact that in the Standard Model, the charge-conjugation operator is a fundamental symmetry, and the presence or absence of a bar on a fermion field reflects this symmetry.

In summary, the bar on a fermionic field in a Lagrangian represents the antiparticle of that field, and it has physical consequences in terms of the possible interactions between particles and antiparticles. Additionally, it also reflects the charge-conjugation symmetry in the Standard Model.