Consider a theory with only a single massless scalar field $\phi(x)$ and a current $J^\mu(x)$ which can be polynomially expanded as fields and their derivatives and spacetime \begin{align} J^\mu(x) = f_1(x)\mathcal{O}_1\phi(x) + f_2(x) [\mathcal{O}_2\phi(x)] [\mathcal{O}_3\phi(x)] + \ldots \end{align} where $\mathcal{O}_i$ space time derivatives operators and $f_i(x)$ are functions of space time, for instance $f(x)\mathcal{O} = x_\mu \partial^\mu$, the explicit form does not matter at this point. Lorentz index for the current is not important for the analysis as well. I am trying to calculate the following LSZ reduction \begin{align} \lim_{p^0\rightarrow E_{p}}\prod_{a=1}^n\lim_{p_a^0 \rightarrow E_a}(p_a^0-E_a +i\epsilon)(p_a^0 + E_a)\int_{x}e^{ip\cdot x}\int_{x_a}e^{i p_a\cdot x_a}\langle\Omega|T\{J^\mu(x)\phi(x_1)\ldots\phi(x_n)\} |\Omega\rangle \end{align} where $E_p = |\vec{p}|, E_a = |\vec{p}_a|$, $\langle\Omega|$ and $|\Omega\rangle$ are the interaction vacuum and $T$ is the usual time ordering. For simplicity I've put all particles in the out state. Herein we only consider contribution from the far future integral i.e. $t>T^+$.
For $\mathcal{O}_1\phi(x) \subset J^\mu$, everything works out as desired, we insert complete basis and use the normalised matrix element \begin{align} \mathcal{O}_1\langle\Omega|\phi(x)|\vec{p}\rangle = \tilde{\mathcal{O}}_1(E_p,\vec{p}) e^{-i( E_p t - \vec{p}\cdot\vec{x})} \end{align} where $|\vec{p}\rangle$ is the one particle state, $\tilde{\mathcal{O}}_1(E_p,\vec{p})$ is the momentum dependent function we get when $\mathcal{O}_1$ acts on the matrix element. Following step by step in Peskin&Schroeder QFT text book section 7 calculation, we arrive at \begin{align} \lim_{p^0\rightarrow E_{p}}\tilde{f}_1(\partial_{p^0},\partial_{p_i})\frac{\tilde{\mathcal{O}}_1(E_p,\vec{p}) e^{i(p^0 - E_p +i\epsilon)T^+}\langle \vec{p},\vec{p}_1,\ldots, \vec{p}_n|\Omega\rangle}{E_p(p^0-E_p +i\epsilon)} \end{align} where $\tilde{f}(\partial_{p^0},\partial_{\vec{p}})$ is the Fourier transform of $f(x)$ and $\langle \vec{p},\vec{p}_1,\ldots, \vec{p}_n|\Omega\rangle$ is the $S$ matrix element. We have a well handle on this term.
The problem arises when I try to compute the second term in current. I'm skipping a few steps, but they amount to computing the following identity \begin{align} \lim_{p^0\rightarrow E_p}\tilde{f}_2(\partial_{p^0},\partial_{\vec{p}})\int_{t>T^+}\int_{\vec{x}} e^{ip\cdot x} \int_{p'} \frac{d^3 \vec{p}'}{2 E_p'}\;\langle\Omega|\mathcal{O}_2\phi(x)\mathcal{O}_3 \phi(x)|\vec{p}\rangle \;\; \langle p_1,\ldots,p_n,p|\Omega\rangle \end{align} The problem here is that we don't have non-perturbative control to the matrix element \begin{align} \langle\Omega|\mathcal{O}_2\phi(x)\mathcal{O}_3 \phi(x)|\vec{p}\rangle \end{align} Literatures like Tree level scattering amplitude in nlsm Inflationary adler zero suggest that we could do the following factorisation at tree level, \begin{align} &\langle\Omega|\mathcal{O}_2\phi(x)\mathcal{O}_3 \phi(x)|\vec{p}\rangle \;\; \langle p_1,\ldots,p_n,p|\Omega\rangle = \sum_{L,R}\left[\mathcal{O}_2\langle p_L|\phi(x)|\Omega\rangle \mathcal{O}_3 \langle\Omega|\phi(x)|\vec{p}\rangle + 2\leftrightarrow 3\right]\langle p_R|\Omega\rangle \\ \end{align} where $\langle p_1,\ldots,p_n,p|$ is split into two disjoint sets $\langle p_L|,\langle p_R|$ and sum over all partitions. The split grant us better control to the matrix element.
However, the matrix element $\langle\Omega|\mathcal{O}_2\phi(x)\mathcal{O}_3 \phi(x)|\vec{p}\rangle$ could clearly be computed peturbatively by expanding $\langle\Omega|= \langle 0| e^{i \int dt H_{int}}$ and $|\vec{p}\rangle=e^{-i \int dt H_{int}}|\vec{p}\rangle_I$ where $|\vec{p}\rangle_I$ is the state in interaction picture. These two treatment are clearly different, one does not mingle other particle insertion $p_1, p_2, \ldots p_n$ with $\phi(x)$ and one does. I wonder what is the correct prescription for computation.
Any help is welcome. Thanks in advance.
