Physical meaning of mass renormalization In the case of charge renormalization, one can present a neat and nice physical idea that brings a physical ground to it called "Vacuum Polarization". Which can be even extended to non-abelian gauge groups(as Coleman did)
I want to know if there's any mechanism that gives some physical sprit to mass renormalization, since I can't understand how mass can be screened with virtual electron-positron pairs.
 A: Mass renormalization (in QED) does not involve virtual electron-positron pairs.  It involves the emission and reabsorption of a virtual photon by an electron in motion.  This is a quantum-mechanical interaction between the electron and the electron’s own electromagnetic field.
The intuitive idea of mass renormalization actually can be explained without quantum mechanics, however.  A charged electron produces an electromagnetic field around itself.  That field makes contributions to the energy-momentum tensor, so it possesses energy and inertia.  If you want to accelerated the electron, you also have to accelerate the electromagnetic field attached to it, and this changes the effective, observable mass of the electron.  This is the essence of mass renormalization, with the electromagnetic field adding to mass of a charged particle (adding an infinite amount, if the calculation is not regularized).
A: It is very good and deep question. Let me try to answer and clarify some subtleties.
In QED we have:
$$
e^2_{phys}(\Lambda) = e_0^2\left(1-\frac{e_0^2}{6\pi^2}\ln \frac{\Lambda}{m_0}+\dots\right)
=
\frac{e^2_0}{1+\frac{e_0^2}{6\pi^2}\ln \frac{\Lambda}{m_0}}
$$
$$
m_{run}(\Lambda) = m_0\left(1+ \frac{3e_0^2}{8\pi^2}\ln \frac{\Lambda}{m_0}+\dots\right)
=
m_0 \left(\frac{e_0^2}{e^2_{phys}}\right)^{9/4}
$$

*

*Here $m_0$ is bare mass, $e_0$ is bare charge (physical charge on scale $\Lambda = m_0$). So running mass is directly related to value of physical charge. Physical interpretation: energy of electromagnetic field around electron give contribution to mass.


*Note, that on scale $\Lambda>m_0$:
$$
e^2_{phys} < e_0^2
$$
$$
m_{run} > m_0
$$


*$e^2_{phys}$ is real experimentally observable quantity, $e^2_{phys}$  change with scale, and has an interpretation as "vacuum polarization", as you said in the question.


*'Running mass' $m_{run}$ is a parameter in the Lagrangian. It depends on the regularisation scheme and can be reproduced from physical (pole) mass.


*There is also notion of pole mass -- pole in full propagator. Pole mass is the physical mass, scale-independent and independent of any renormalization scheme we use to subtract any infinite parts of the loop corrections. It is what we observe.
See also What is the difference between pole and running mass?
A: Let us start from QED.
In this theory, vacuum polarization in the lowest order is given by one-fermion loop correction to bare photon propagator (note that photon has zero mass). It is quite straightforward (or I can provide a derivation) that this correction modifies charge but not mass. For me, physical ground becomes clear after calculation.
If you consider electron propagator renormalization, one loop correction shifts the pole in propagator and in this sense electron mass changes. But there is nothing relating to electron-positron pairs and vacuum polarization.
