# Callan-Szymanzick equation and renormalization

#### eljose79

When i coursed the quantum field theory at university our teacher told us about this equation..i have searched information aobut it in many books and have the form...it seems is a partial-differential equation but i have doubts..

a)what is have to do with renormalization?...(in fact how to use it in renormalization theory
b)If theory is not renormalizable then..what would be the c-s equation then?.exist if not why?

hope someone can help....

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#### vanesch

Staff Emeritus
Gold Member
C-Z equation

In QFT, the objects from which everything follows (well...) are the green functions, (the vacuum expectations of products of field operators) but usually those greenfunctions blow up. So we use a cutoff scale to make those green functions finite (but dependent on the cutoff). As such, we can calculate what are the masses, charges, field renormalizations etc... as a function of the cutoff. When we say, for example, what is the charge of an electron, we actually talk about the outcome of an experiment to be performed, for example: put an electron in such and such an E-field and look at its acceleration. That's a non-trivial operation in QFT, and it turns out that this outcome, "physical charge" is a function of the bare charge e0 and the cutoff in a complicated way. So we can solve for e0 as a function of the physical charge e and the cutoff. But the "physical charge" is defined at a certain energy scale: for example, we consider the electron at rest (energy is the rest mass), or in motion or whatever. So this procedure depends on what is called the "renormalisation scale" M. Renormalization is nothing else but substituting everywhere e0 by the function of the cutoff, the physical charge e and the renormalization scale M. In a renormalizable theory, we then get finite answers, even in the limit of the cutoff going to infinity. In that limit, there's still a functional dependence of every physical quantity on the physical charge e and the renormalization scale. The C-Z equation is a partial differential equation that sets a condition on the dependence on e and on M.

cheers,
Patrick.

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