Free-boson worldsheet propagator
= Free-boson worldsheet propagator
For the normalization $S=(2\pi\alpha')^{-1}\int d^2z\,\partial X\bar\partial X$, the free-boson worldsheet propagator has singular part
$$
X^\mu(z,\bar z)X^\nu(w,\bar w)\sim-\frac{\alpha'}2\eta^{\mu\nu}\log|z-w|^2.
$$
Differentiation gives the singular operator-product expansions of the embedding currents.