# Feynman rules for derivative self-interactions

1. Nov 17, 2016

### spaghetti3451

1. The problem statement, all variables and given/known data

Consider a real scalar field with a derivative interaction

$$\mathcal{L} = \frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)+\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi.$$

What are the momentum-space Feynman rules for this theory?

2. Relevant equations

3. The attempt at a solution

To figure out the Feynman rules for this theory, it is helpful to compute the $n$-point correlation functions of the theory.

Let's start with the $2$-point correlation function.

$\displaystyle{\langle\Omega|T\{\phi(x_{1})\phi(x_{2})\}|\Omega\rangle}$

$\displaystyle{=\int\mathcal{D}\phi\ \phi(x_{1})\phi(x_{2})\exp\left[i\int d^{4}x\left(\frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)+\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi\right)\right]}$

$\displaystyle{=\int\mathcal{D}\phi\ \phi(x_{1})\phi(x_{2})\exp\left[i\int d^{4}x \left(\frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)\right)\right] \exp\left[i\int d^{4}x \left(\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi\right)\right]}$

$\displaystyle{=\int\mathcal{D}\phi\ \phi(x_{1})\phi(x_{2})\exp\left[i\int d^{4}x \left(\frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)\right)\right] \left(1+i\int d^{4}x \left(\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi\right) +\mathcal{O}\left(g^{2}\right)\right)}$

$\displaystyle{=\int\mathcal{D}\phi\ \phi(x_{1})\phi(x_{2})\exp\left[i\int d^{4}x \left(\frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)\right)\right] + \int\mathcal{D}\phi\ \phi(x_{1})\phi(x_{2})\exp\left[i\int d^{4}x \left(\frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)\right)\right] \left(i\int d^{4}x \left(\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi\right)\right) + \mathcal{O}\left(g^{2}\right)}$

$\displaystyle{=\langle0|T\{\phi(x_{1})\phi(x_{2})\}|0\rangle + \langle0|T\{\phi(x_{1})\phi(x_{2})\left(i\int d^{4}x \left(\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi\right)\right)\}|0\rangle + \mathcal{O}\left(g^{2}\right)}$

$\displaystyle{=\langle0|T\{\phi(x_{1})\phi(x_{2})\}|0\rangle + \frac{ig}{2} \int d^{4}x \langle0|T\{\phi(x_{1})\phi(x_{2}) \phi(x)\partial^{\mu}\phi(x)\partial_{\mu}\phi(x)\}|0\rangle + \mathcal{O}\left(g^{2}\right)}$.

I am a little confused by the derivatives in the second term. How can you apply Wick's theorem when there are derivative factors?

2. Nov 22, 2016

### Staff: Admin

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Feynman rules derivative Date
How to deduce the Feynman rules? Dec 26, 2017
How to find Lagrangian density from a vertex factor Mar 31, 2016
W boson decay Feb 1, 2016
Feynman rules for Lagrangian with derivative Interaction Oct 30, 2015
Vector QED Feynman rule Oct 16, 2015