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Feynman rules for derivative self-interactions

  1. Nov 17, 2016 #1
    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. jcsd
  3. Nov 22, 2016 #2
    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.
     
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