confusion about what Wikipedia says about Renormalization

Question

On the wikipedia page, on renormalization, it says the following: "Renormalization replaces the initially postulated mass and charge with new numbers such that the observed mass and charge matches those postulated initially."

I am confused as to what this means. It just said the initially postulated mass and charge replaced. Isn't it contradicting itself? The whole sentence sounds very confusing. Arent you supposed to measure some values first and then plug them into your calculation and get some values that get close to what the bare mass and charge should be ( although they are impossible to measure)

I remember being told the values that get measured have to do with the scales on which experiments are being conducted (electron mass and charge for example) and one can never measure bare mass and charge because of virtual particle cloud surrounding ans interacting with them. I would really appreciate it if someone could translate the wikipedia statement in plain english.

Answer 1

What they mean is that renormalization changes the original parameters appearing in the Lagrangian (such as the original mass parameter) in such a way that the OBSERVED parameters after taking the 'quantum' corrections into account is the same as the originally posited parameters.

Take the following example of $\phi^4$ theory. The Lagrangian for this theory is the following $$L = -\frac{1}2\partial_u\phi\partial^u\phi\space+\frac{1}2m_o^2\phi^2\space+\frac{1}{4!}\lambda_0\phi^4$$

Let's say we do an experiment, and we determine that the mass of our scalar particle is $m_r$. At tree level it would suffice to set $m_0$ to $m_r$. Problems arise however when we look at the one loop correction to the scalar propagator. If we use a cutoff regulator (we cutoff the momentum integral in the one loop diagram at a finite momentum $\Lambda$), we find that the new mass is going to be some function $F(\Lambda,m_0^2)$.

The key here conceptually is to find a bare mass $m_0(\Lambda)$ such that your new mass after incorporating the quantum corrections is equal to the experimentally determined mass $m_r$ for all values of the cutoff $\Lambda$.