- #1
spaghetti3451
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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?