Integral manipulation and Lagrangian simplification

Question

I'm trying to work out some expressions from [this paper](https://arxiv.org/abs/2008.02770), namely expression (4) from (11) and (10).

Consider the Lagrangian

\begin{equation}
  \mathcal{L}(M,W,\lambda) = g^2\left(1+\pi W(m^2)\right)+\int_{4m^2}^{\infty}ds\, \Im{\left[W(s)M(s)\right]}\tag{1}
\end{equation}

Using the fact that 

\begin{equation}
 W(s) = \frac{1}{\pi}\int_{4m^2}^{\infty}dz \,\frac{\omega(z)}{s-z+i0}-\frac{\omega(z)}{4m^2-s-z+i0}\tag{2}
\end{equation}

and the Sokhotski–Plemelj theorem,

\begin{equation}
\frac{1}{s-z\pm i0}= \mp i\pi \delta(s-z)+\mathcal{P}\frac{1}{(s-z)}\tag{3}
\end{equation}

I am trying to prove that (1) is equivalent to

\begin{equation}
\mathcal{L}(M,\omega,\lambda) = g^2+\int_{m^2}^{\infty}ds\,\omega(s)\mathcal{A}(s)\tag{4}
\end{equation}

where

\begin{equation}
\mathcal{A}(s) = M(s)-\left(M_{\infty}-\frac{g^2}{s-m^2}+\int_{4m^2}^{\infty}\frac{dz}{\pi}\, \frac{\Im{[M(z)]}}{s-z+i0}+\frac{\Im{[M(z)]}}{4-s-z+i0}\right)\tag{5}
\end{equation}

**My attempt**

First, using the second to last equation in the Appendix B.1 of the paper,

\begin{equation}
W(z) = \frac{1}{\pi}\int_{4m^2}^{\infty}ds\, \Im{ [W(s)]}\left(\frac{1}{s-z}-\frac{1}{s-t(z)}\right)\tag{6}
\end{equation}

where $t(z) = 4-z$ and $\Im{[W(s)]} = \omega(s)$ for $s > 4m^2$:

\begin{split}
g^2\pi W(m^2) &= g^2\pi \times\frac{1}{\pi}\int_{4m^2}^{\infty}ds\,\omega(s)\left(\frac{1}{s-m^2}+\frac{1}{4-s-m^2}\right)\\
&= -\frac{1}{\pi}\int_{4m^2}^{\infty}ds\,\omega(s)\left(-\frac{g^2}{s-m^2}-\frac{g^2}{4-s-m^2}\right)
\end{split}

For the integral, using $\Im{[M(s)\times W(s)]} = \Im{[M(s)]}\Re{[W(s)]}+\Re{[M(s)]}\Im{[W(s)]}$

\begin{equation}
\int_{4m^2}^{\infty}ds\, \Im{\left[W(s)M(s)\right]} = \int_{4m^2}^{\infty}ds\,\Im{[M(s)]}\left(W(s)-2i\Im{[W(s)]}\right)+\,\int_{4m^2}^{\infty}ds \Im{[W(s)]}M(s)
\end{equation}

The second integral can be identified with 

\begin{equation}
\int_{4m^2}^{\infty}ds \Im{[W(s)]}M(s) = \int_{4m^2}^{\infty}ds \, \omega(s)M(s)
\end{equation}

On the other hand, 

\begin{equation}
\int_{4m^2}^{\infty}ds\,\Im{[M(s)]}W(s) = \int_{4m^2}^{\infty}\omega(s)\int_{4m^2}^{\infty}\frac{dz}{\pi}\, \frac{\Im{[M(z)]}}{s-z+i0}-\frac{\Im{[M(z)]}}{4-s-z+i0}
\end{equation}

where eq. (2) and a Dirac-delta were used.

**Questions**

 - I tried using the aforementioned theorem to simplify the remaining of the expression, but to no avail. A Cauchy principal value appears which I am not being able to deal with;
 - How is it that eqs. (2) and (6) match? I tried to simplify one of them, but I could not show that they were the same expression.