Feynman rules for derivative self-interactions

[spaghetti3451]

Homework Statement

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?

Homework Equations

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?

 

[mark726]

Also, the last term in the expansion, $\mathcal{O}(g^2)$, is not a Feynman diagram, so it should not be included in the Feynman rules. To find the Feynman rules, we need to expand the exponential term in the path integral and then use Wick's theorem to obtain the connected diagrams. In this case, the Feynman rules are as follows:

1. Propagator: $\langle0|T\{\phi(x_1)\phi(x_2)\}|0\rangle = \frac{i}{p^2-m^2}$, where $p$ is the momentum of the field.

2. Vertex factor: $-ig(p_1+p_2)^\mu$, where $p_1$ and $p_2$ are the momenta of the two fields at the vertex.

3. Momentum conservation: $\delta(p_1+p_2+p_3)$, where $p_3$ is the momentum of the external field.

Using these rules, we can construct the Feynman diagrams for any $n$-point correlation function and compute them using the standard techniques of Feynman diagram calculations.