Renormalization of mass of an electron inside the crystal

In Cheng and Li's book, Gauge theory of Elementary Particle Physics, he essentially says that renormalization has nothing to do with infinities. Even in a totally finite theory, we would still have to renormalize physical quantities. For example, the mass $m^*$ of an electron inside the crystal is renormalized from the mass $m$ it has outside the crystal (due to the interaction inside the crystal). However, unlike relativistic QFT, both $m$ and $m^*$ are measurable and finite. Therefore, the correction $\delta m=m-m^*$ should also be finite.

How does one calculate this correction $\delta m$? If one uses quantum field theory, he finds that the correction to electron mass is logarithmically divergent.

• In QFT the masses are not divergent if we take some cut-off. The divergence only shows up if you push our QFT to arbitrary small distance. In your case, the QFT is an effective one, so is wrong to push it to arbitrary small distance, you know that there is a free electron at a given (small distance/high energy), with a bare mass $m$. The corrections will not be divergent because the limits of our integrals in momentum space will not be pushed to infinity. The $\delta m$ will clearly be $\sim\Lambda\log \Lambda$ at first orders. With $\Lambda$ being the the cut-off. – Nogueira Feb 16 '18 at 1:02

There are also typical examples from the high-energy effective field theories, like chiral perturbation theory, which describes the pseudo-scalar mesons octet interactions below the scale of the QCD chiral symmetry spontaneous breaking. In this theory the natural cut-off is the proton mass. Without EM interactions the masses $\pi^{0}, \pi^{\pm}$ are equal. However, once the EM interactions are turned on, loop corrections to the charged mesons masses appear. With finite cut-off, these corrections are, of course, finite.