# QFT lagrangian

1. May 15, 2014

### beans73

I've been given the following lagrangian:

$$\mathcal{L}_{eff} = \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi - \frac{G}{4}(\bar{\psi}\psi)(\bar{\psi}\psi)$$

where I have been told that the coefficient G is real and has mass dimension -2.
I will eventually need to derive the feynman rules for this, but I just wanted to ask what exactly it is describing. the first part looks like the lagrangian for a dirac spinor, and i guess would contribute a propagator. what is the second term describing?

2. May 15, 2014

### dauto

The second term is an interaction between four fermions (two fermions and two anti fermions). It will allow the fermions to scatter off each other (two incoming (anti-)fermions scattering and producing two outgoing (anti-)fermions can be generated by a four-point interaction like the one in this Lagrangian). Without interaction terms in the Lagrangian, We would have fermions that do not interact with each other. Easy, but not very interesting. Ultimately that Lagrangian turns out to be non-renormalizable but that's for you to find out later in your studies, I suppose.