I'm reading a book of conformal field theory (the one by Di Francesco, Pierre Mathieu and David Senechal), and I'm having trouble understanding the math in a section (p. 468)

The aim of the section is to compute the two-point function, on a plane, of a magnetic operator, for a compact boson.

The bosonic field is decomposed between a classical and a quantum part, the classical part having the following form :

\[\varphi^{cl} = m R \ln\left( \frac{z-z_1}{z-z_2} \right)\]

And the action (\(S[\Phi] = \frac{-1}{8\pi} \int (\nabla \Phi)^2\), with their conventions) gives :

\[S[ \tilde{\varphi} + \varphi^{\text{cl}}] = S[\tilde{\varphi}] + S[\varphi^{cl}] - \frac{1}{4\pi} \int \nabla \tilde{\varphi} \nabla \varphi^{cl}\]

The third term is evaluated to zero, as \(\Delta \varphi^{cl} = 0\). And the second one is given by :

\[\frac{1}{|z_1 - z_2|^{\frac{mR^2}{2}}}\]

The result is classic and I know it can be obtained differently, but I don't understand :

- how that final integral is computed
- why the third term gives zero

Thank you for your help,

(and this is my first post, so nice to meet you, and sorry for any mistake)