Integral of the classical part of a compact boson

Question

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 :

1. how that final integral is computed
2. 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)