# Peskin & Schroeder, section 12.2, page 409, analytic continuing from $d=2$ to $d=4$?

In Peskin & Schroeder section 12.2，page 409, consider the one-loop correction to the propagator in Yukawa theory, it has the form

The authors first spot the pole at $$d=2$$, and find out that it could be completely absorbed by counterterm $$\delta_m$$, then they analytically continuing this equation from $$d=2$$ to $$d=4$$, and claim that we don't need to worry about the $$\delta_m$$ anymore, since in the limit $$d\to4$$, this equation has the form Eq(12.33)

My question is, why couldn't we just stick with the situation $$d=4$$ and claims that in the massless limit of scalar field, there will be no contribution to the mass shift, why should we bother to consider the pole at $$d=2$$?

Additionally, even if we absorb the mass shift at $$d=2$$ with $$\delta_m$$, can we justify the analytic continuation? I think $$d=2$$ and $$d=4$$ are two completely different and independent situation, how could we just analytically continue one to the other?

Probably there are some important details that I do not fully understand. But on a high-level, we want to do this because we don't want $$\delta_m$$ to depend on $$M^2$$.
If we did it directly in $$d=4$$, (12.32) would have a pole depending on $$p^2$$, and therefore we would have to make $$\delta_m$$ depend on $$M^2$$. This would mess up all the following derivations.