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
L(M,W,λ)=g2(1+πW(m2))+∫∞4m2dsℑ[W(s)M(s)]
Using the fact that
W(s)=1π∫∞4m2dzω(z)s−z+i0−ω(z)4m2−s−z+i0
and the Sokhotski–Plemelj theorem,
1s−z±i0=∓iπδ(s−z)+P1(s−z)
I am trying to prove that (1) is equivalent to
L(M,ω,λ)=g2+∫∞m2dsω(s)A(s)
where
A(s)=M(s)−(M∞−g2s−m2+∫∞4m2dzπℑ[M(z)]s−z+i0+ℑ[M(z)]4−s−z+i0)
**My attempt**
First, using the second to last equation in the Appendix B.1 of the paper,
W(z)=1π∫∞4m2dsℑ[W(s)](1s−z−1s−t(z))
where t(z)=4−z and ℑ[W(s)]=ω(s) for s>4m2:
g2πW(m2)=g2π×1π∫∞4m2dsω(s)(1s−m2+14−s−m2)=−1π∫∞4m2dsω(s)(−g2s−m2−g24−s−m2)
For the integral, using ℑ[M(s)×W(s)]=ℑ[M(s)]ℜ[W(s)]+ℜ[M(s)]ℑ[W(s)]
∫∞4m2dsℑ[W(s)M(s)]=∫∞4m2dsℑ[M(s)](W(s)−2iℑ[W(s)])+∫∞4m2dsℑ[W(s)]M(s)
The second integral can be identified with
∫∞4m2dsℑ[W(s)]M(s)=∫∞4m2dsω(s)M(s)
On the other hand,
∫∞4m2dsℑ[M(s)]W(s)=∫∞4m2ω(s)∫∞4m2dzπℑ[M(z)]s−z+i0−ℑ[M(z)]4−s−z+i0
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.