Computing the pole mass from a given $\overline{MS}$ mass? Given a Yukawa coupling as a function of scale $\mu$ and a vev, therefore $m_R(μ)=Y(μ)⟨ϕ⟩$, how can I compute the corresponding pole mass $m_p$? Relations I was able to find are (page 39) 
$$m_p=m_R−Σ(m_P)$$ 
or specifically for the electron (page 17)
$$ m_P =m_R - \frac{e_R^2 }{16 \pi^2}\left[ 2 (m_P-m_R) + \int dx (4m_R-2m_Px)\log( \frac{\mu^2 }{(1-x)(m_R^2-xm_P^2)}) \right] .$$
Now, what I don't understand is how these equations can be used, in practice, to compute $m_P$, if $m_R(\mu)$ is given. Any tip or reference to an example computations would be much appreciated! As the pole mass should be indendent of $\mu$, I'm confused which $\mu$ I should use in the formulas above in order to compute $m_P$.
 A: You can't solve exactly, but you can solve order by order in perturbation theory. I'm essentially following chapter 27 of Srednicki. The relation $m_P = m_R - \Sigma(m_P)$ is the one you want, but $m_P$ can only be determined implicitly. Since $\Sigma$ begins at order $\alpha$, the difference between $m_R$ and $m_P$ is $\mathcal{O}(\alpha)$, so up to $\mathcal{O}(\alpha^2)$ you can replace $m_R$ by $m_P$ inside it. So we get:
$$m_P = m_R - \Sigma(m_R) + \mathcal{O}(\alpha^2)$$
If you want to go to a higher order, suppose you compute two-loop corrections to $\Sigma$ and get $\Sigma = \alpha \Sigma_1 + \alpha^2 \Sigma_2 + \mathcal{O}(\alpha^3)$. We have the relation
$$m_P = m_R - \alpha \Sigma_1(m_P) - \alpha^2 \Sigma_2(m_P) + \mathcal{O}(\alpha^3)$$
Now inside $\Sigma_1$ we can replace $m_P$ by its $\mathcal{O}(\alpha)$ expression in terms of $m_R$, while inside $\Sigma_2$ we just set $m_P = m_R$ like we did earlier, since it's already multiplied by $\alpha^2$:
$$\begin{align}
m_P &= m_R - \alpha \Sigma_1(m_R - \alpha \Sigma_1(m_R)) - \alpha^2 \Sigma_2(m_R) + \mathcal{O}(\alpha^3) \\
&= m_R - \alpha  [\Sigma_1(m_R) - \Sigma_1'(m_R)\alpha\Sigma_1(m_R)] - \alpha^2 \Sigma_2(m_R) + \mathcal{O}(\alpha^3)\\
&= m_R - \alpha \Sigma_1(m_R) + \alpha^2 [\Sigma_1(m_R)\Sigma_1'(m_R) - \Sigma_2(m_R)] + \mathcal{O}(\alpha^3)
\end{align}$$
And so on.
