Solution
= Solution
The Green function for the normalization in the question is
$$
\langle X(z,\bar z)X(w,\bar w)\rangle=-\frac12\log|z-w|^2+ ext{constant}.
$$
Differentiating at distinct points gives the holomorphic two-point function
$$
\boxed{\langle\partial_zX(z)\partial_wX(w)\rangle
=-\frac1{2(z-w)^2}}.
$$