Quantization of free real scalar massless field in 2d

Is there a reference to literature where one explicitly constructs quantization of the free real scalar massless field in the 2-dimensional space-time? In particular, how the propagator looks like?

The 4d case is treated in many standard textbooks in QFT, but the 2d case seems to be different due to extra divergences which do not exist in higher dimensions.

For example, using the equation $Box_x langle Tphi(x)phi(y)rangle=delta^{(2)}(x-y)$ and the analogy with the 4d case (the Feynman rule) one might expect that the propagator should be proportional to $$frac{1}{p^2+ivarepsilon}=frac{1}{(p_0)^2-(p_1)^2+ivarepsilon}.$$
However in 2d this generalized function is not well defined, i.e. it diverges as $varepsilon to +0$ (even its imaginary part diverges).

ADD: Let’s prove the above claim that the imaginary part diverges. We have
$$Imleft(frac{1}{p^2+ivarepsilon}right)=frac{1}{2i}left(frac{1}{p^2+ivarepsilon}-frac{1}{p^2-ivarepsilon}right)=-frac{varepsilon}{(p^2)^2+varepsilon^2}.$$
Let $phi(p)$ be a smooth non-negative function which equals to 1 in the unit ball and vanishes outside some larger ball. Then
begin{eqnarray*}
-int dp^2Imleft(frac{1}{p^2+ivarepsilon}right)cdot phi(p)=int d^2p frac{varepsilon}{(p^2)^2+varepsilon^2}cdot phi(p)=int d^2p frac{1}{(p^2)^2+1}cdot phi(sqrt{varepsilon}p),
end{eqnarray*}
where the last equality is obtained by the change of variables $pmapsto sqrt{varepsilon}p$. As $varepsilonto +0$, the last integral becomes at least

begin{eqnarray}
int d^2p frac{1}{(p^2)^2+1}.
end{eqnarray}
Let us show that this is infinite. Let’s make change of variables $x=p_0-p^1,, y=p_0+p_1$. Then the last integral is
$$frac{1}{2}int dxdyfrac{1}{(xy)^2+1}=frac{1}{2} int dyint frac{dx}{(xy)^2+1}=frac{1}{2} int dy frac{1}{|y|}left(int frac{dz}{z^2+1}right)=infty,$$
where the second equality is obtained by the change of variables x=z/|y|$.
The result is proven.