I would like to know what is callan-symanzik equation used for in renormalization theory , if this can give you the renormalizated quantities and why can not be used when the theory is non-renormalizable.
The Callan-Symanzik equation is not a part of renormalization per se. It is a key part of Renormalization Group theory, which is someting different erected on the same base. In Kaku's Quantum Field Theory textbook he discusses the derivation of the Callan-Symanzik relation on p. 486, in the chapter QCD and the renormalization group.
"We begin with the obvious identity that the derivative of a propagator with respect to the unroneomalized mass=squared, simply squares the propagator.,,,
"Now assume that [the propagator] occurs in some vertex function .. of arbitrary order. From a field theory point of view, the squaring of the propagator (with the same momentum) is equivalent to the insertion of an operator φ^{2}(x) in the diagram with zero momentum.
"This means that the derivative of an arbitrary vertex function with respect to [unrenomalized mass squared] yields another vertex function where φ^{2}(x) has been inserted.
"We now make the transition from the unrenormalized vertices to the renormalized ones. This means the introduction of yet another renormalized constant Z_{φ2} to renormalize the insertion of the composite field operator."
He then does the math, simple algebra and calculus. The result is a partial differential equation stating that the linear combination of the derivatives of a vertex function with respect to normalized mass and to the coupling stength is equal to the renormalized mass squared times the same (renormalized) vertex function with φ squared inserted.
I won't reproduce the equations here, but you can see how they were derived by working back and forth between the unrenomalized Feynmann diagrams and the renormalized ones.
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