Reality of a momentum-space scalar correlator
= Reality of a momentum-space scalar correlator
For a real scalar field, $\phi(\mathbf k)^*=\phi(-\mathbf k)$. Its momentum-space $n$-point function therefore obeys
$$
\langle\phi(\mathbf k_1)\cdots\phi(\mathbf k_n)\rangle^*
=\langle\phi(-\mathbf k_1)\cdots\phi(-\mathbf k_n)\rangle.
$$
A parity-odd scalar correlator is consequently purely imaginary after removing its momentum-conserving delta function.