# Singularities of matrix element of composite local operator in QFT

Consider a state $|\psi\rangle$ in a quantum field theory and a local operator $\mathcal{O}(x)$. It's known that the $n$-point function

$\langle \psi | \mathcal{O}(x_1) \cdots \mathcal{O}(x_n) | \psi\rangle$

is singular for certain $x_i$ (for example, when two of the $x_i$ are coincident or null separated). My question is: given two states $|\psi\rangle$ and $|\phi\rangle$ (which for simplicity we can take to be orthogonal), can anything be said about the divergences of the matrix element

$\langle \psi | \mathcal{O}(x_1) \cdots \mathcal{O}(x_n) | \phi\rangle$?

For instance, does the above object have any singularities? If it does, are they related in any known way to singularities of the corresponding correlator in the states $|\psi\rangle$ and $|\phi\rangle$? (Also note that I'm only interested in Hadamard states.)