Why Do Electron, Muon, and Tau Masses Differ in Scale and Correction Effects?

[DuckAmuck]

This is a 3 part question.

1. I've come to understand that the mass values in the mass terms (pole mass) of the standard model don't represent what we actually measure. That there are loop corrections. (I understand the concept: there's screening either adding or subtracting from the true value). I get that this is also related to energy scale.
So now, my question is, why is it said that electrons, muons and tau masses are on different scales? Is it just because their masses are so different, or are their loop corrections different, or is it both?

2. Even though the lepton masses are on different scales, is it possible to have a relationship or some kind between them, sort of like the koide formula? Are the pole-masses on the same footing? Could something like the Koide formula have physical meaning?

3. How exactly are loop corrections added? All I know is how to solve for equations of motion from a lagrangian. Where do the loop corrections actually come in? Is there an easy-to-follow walk thru for this?

thank you

 

[RGevo]

Hello,

I would believe this statement of the scales refers just to the fact that they have different masses (which are orders of magnitudes apart).

In the SM, these are just numbers (as a result of yukawa interactions whose strength must be measured experimentally).

Do you know how to calculate a two point function for a fermion? I.e. I( Slashed(p) - m )

The inverse of this gives you the propagatir for a fermion, and the position of the pole gives you the pole mass. At leading order all mass definitions are equivalent/redundant.

Then, the next step is to calculate the one loop correction to the two point function. so all possible 1particle irreducible diagrams. (This is covered in all decent qft books).

The problem is then that these loop diagrams contain divergences. These are subtracted out, and in addition sometimes finite parts. Depending on what is subtracted gives you a different scheme.

The pole scheme corresponds to subtracting the divergent pieces and a residual scale dependence (a result of performing renormalisation in fixed order perturbation theory).

Some texts which discuss this well are the qft book of schwarz