# Correlation functions in an interacting theory

Given the theory

$$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi+\mathcal{L}_{\text{int}},\qquad \mathcal{L}_{\text{int}}=-g\phi\chi^{*}\chi,$$

the time-correlation function ##\langle \Omega | \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|\Omega\rangle## is given by

$$\langle \Omega | \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|\Omega\rangle = \langle 0| \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|0\rangle -ig \int d^{4}x\ \langle 0 | \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})\phi(x)\chi^{*}(x)\chi(x)|0\rangle + \mathcal{O}(g^{2})$$

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Is ##\langle 0| \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|0\rangle = 0##?

## Answers and Replies

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.

nrqed
Homework Helper
Gold Member
Given the theory

$$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi+\mathcal{L}_{\text{int}},\qquad \mathcal{L}_{\text{int}}=-g\phi\chi^{*}\chi,$$

the time-correlation function ##\langle \Omega | \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|\Omega\rangle## is given by

$$\langle \Omega | \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|\Omega\rangle = \langle 0| \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|0\rangle -ig \int d^{4}x\ \langle 0 | \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})\phi(x)\chi^{*}(x)\chi(x)|0\rangle + \mathcal{O}(g^{2})$$

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Is ##\langle 0| \phi(x_{1})\chi^{*}(x_{2})\chi(x_{3})|0\rangle = 0##?
Yes since there is only one field $\phi$ so we have either a single creation operator or a single annihilation operator.