Understanding Renormalization: Exploring Mass and Cutoff in Quantum Field Theory

[Student57]

I have a couple of questions regarding renormalization.

1. If it is possible to change mass as long as we do it simultaneously with changing ultraviolet cutoff, that would imply that the value we pick for mass is more or less arbitrary. If so, how come we have exact decimal value of mass of an electron, which doesn't look arbitrary at all?

2. When we write renormalized mass, m=m_0+dm, we have dm >>m_0. If so, why even include m_0 at all, why not throw it out as something negligible?

Thanks!

 

[The_Duck]

1. If it is possible to change mass as long as we do it simultaneously with changing ultraviolet cutoff, that would imply that the value we pick for mass is more or less arbitrary. If so, how come we have exact decimal value of mass of an electron, which doesn't look arbitrary at all?

The propagator has a pole at a definite energy; this is the mass of the physical particle and is not arbitrary. We distinguish this "pole mass" from the mass parameter that appears in the Langrangian, which depending on your renormalization scheme may have a different value from the pole mass.

2. When we write renormalized mass, m=m_0+dm, we have dm >>m_0. If so, why even include m_0 at all, why not throw it out as something negligible?

Just because dm is formally infinite doesn't mean the finite parts don't matter. We carefully arrange for the infinities to cancel in calculations of physical processes. Then changing the finite parts of the parameters changes the finite physical results.