# 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