# Parity Violation in Feynman diagrams

I'm trying to understand how parity violation in a lagrangian traduces to changes in Feynman's rules/diagrams. To illustrate, consider the following self-interaction case for fermions which contains P violation:

Te following interacting lagrangian which includes a very interesting self-interaction:

$$L_{int}=-\lambda(\bar{\psi}\psi)(\bar{\psi}\gamma^5\psi)$$

The addition of the second factor $$(\bar{\psi)\gamma^5\psi})$$ means the lagrangian not only contains a self-interaction, but also violates parity!

While it's obvious from the lagrangian that I fould have the same fermionic propagators and vertices with 4-legs, I don' know how the parity violation can be traduced to the Feynman rules or in the way I draw the diagrams.

For example, if I consider a scattering of two incoming fermions and I only consider vertices with 4-legs, the most simple case would be a final state with two outcoming fermions. However, this process is identical as if I considered a non-parity violating self-interacting lagrangian like $$L_{int}=-\lambda(\bar{\psi}\psi)(\bar{\psi}\psi)$$, so I'm sure I'm missing something but can't find what.

I've read Perskin, Greiner, and Weinberg books on QFT, but they all deal with basic QED and scalar interactions. Also, most online lectures online deal with CP violation with quarks, while none deal with potential P violation of fermions.

• Just as a note, quarks are fermions. – Triatticus Nov 21 '18 at 18:52
• How do you define parity violation? Is $\lambda(\bar{\psi}\gamma^5\psi)(\bar{\psi}\gamma^5\psi)$ parity violating? Or is $\lambda(\bar{\psi}\psi)(\bar{\psi}\psi) - \lambda(\bar{\psi}\gamma^5\psi)(\bar{\psi}\gamma^5\psi)$ parity violating? The second one is the Nambu-Jona-Lasinio interaction. – MadMax Nov 21 '18 at 19:48
• $\gamma_5$ treats $\psi_L$ very differently from $\psi_R$, leaving the latter alone but multiplying the former by -1. So two incoming Rs will go into two Ls, but two incoming Ls will go into - two Rs. This is not what the parity preserving vertex gives you. – Cosmas Zachos Nov 21 '18 at 20:25
• Triatticus, yeah, I meant I only wanted to deal with P violation of general fermions (not limited to CP violation of quarks). Madmax, I'm talking about parity violation of only $L_{int}=-\lambda(\bar{\psi}\psi)(\bar{\psi}\gamma^5\psi)$. It is parity violating since when I apply the parity operation, the term $(\bar{\psi}\gamma^5\psi)$ changes sign due to the $\gamma^5$. Actually, the full lagrangian in this theory is $L=i\bar{\psi}\gamma^\mu\partial_\mu\psi-m\bar{\psi}\psi\lambda(\bar{\psi}\psi)(\bar{\psi}\gamma^5\psi)$. – Charlie Nov 21 '18 at 21:13
• Cosmas Zachos, I see, then that's how the parity violation manifests. You're right, since I have the $\gamma^5$ in the Lagrangian then I should have different particles when considering the vertices for $\psi_L$ and $\psi_R$, which was what I was looking to understand. I'll try some diagrams to see if I understand how this comes from. Is it convenient I represent the $\psi_R$ and $\psi_L$ with two different styles of lines? – Charlie Nov 21 '18 at 21:16