I want to know how to calculate the lowest-order-in-the-coupling-constant correction to $$M(x, y,k,p)=\langle k|\phi(x)\phi(y)|p\rangle$$ in $\phi^4$ scalar field theory in a relatively general renormalization scheme. Here $\langle k|$ is a one-particle state with 4-momentum k.

In free field theory, we can write $|p\rangle = a^\dagger(p)|0\rangle$ and expand the fields $\phi$ in terms of such creation and annhilation operators. Doing as such, and using the canonical commutation relations snags us, I believe,

$$M^{free}(x, y,k,p)=e^{ipx-iky}+e^{ipy-ikx}+\delta^3(\vec{k}-\vec{p})k^0 \int d^3q \frac{1}{q^0}e^{iq(x-y)}$$

Now, the interacting lagrangian is $$\mathcal L= -\frac{1}{2}Z_{\phi}\partial^{\mu}\phi\partial_{\mu}\phi -\frac{1}{2}Z_mm^2\phi^2-\frac{1}{24}Z_{\lambda}\lambda\phi^4.$$

I believe that in a general renormalization scheme under dimensional regularization, we have to one-loop that

$$Z_{\phi} = 1+O(\lambda^2)$$ $$Z_{m} = 1+\frac{\lambda}{16 \pi^2}\Big(\frac{1}{\varepsilon}+\kappa_m\Big)+O(\lambda^2)$$ $$Z_{\lambda} = 1+\frac{3\lambda}{16 \pi^2}\Big(\frac{1}{\varepsilon}+\kappa_{\lambda}\Big)+O(\lambda^2)$$

where $\kappa_m$ and $\kappa_{\lambda}$ are 0 in modified minimal subtraction but could have different finite values in some other renormalization scheme, like an on-shell one. I would like to keep them as non-zero for now.

I don't know the usual way that one finds corrections to excited state correlation functions. I would be tempted to try out Rayleigh-Schrodinger perturbation theory to find the wavefunction corrections to $|k \rangle$ to lowest order in $\lambda$, specifically corrections including factors of one-particle states with different momenta or three-particle states, as I could see those contributing to my $M$. If it seems fruitful I will add my results to this question.

My background knowledge is I know how to calculate corrections to the Z-factors via diagrams and how to find vacuum correlation functions to one-loop. I have also seen how to use the LSZ formula to translate between vacuum correlation functions and scattering amplitudes.

Please let me know if there is a usual perturbative technique for handling the changes in excited state correlation functions in quantum field theory.


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