Callan-Symanzik equation and renormalization

[eljose79]

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.

 

[selfAdjoint]

The Callan-Symanzik equation is not a part of renormalization per se. It is a key part of Renormalization Group theory, which is something 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 φ²(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 φ²(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 Feynman diagrams and the renormalized ones.

 

[R0man]

The Callan-Symanzik equation is a fundamental equation in the field of renormalization theory. It is used to study the behavior of physical quantities in quantum field theories, and specifically how they change under a change in the energy scale at which they are measured. This equation is important in the process of renormalization, which is a mathematical technique used to remove infinities that arise in quantum field theories.

In renormalization theory, the Callan-Symanzik equation is used to calculate the beta function, which describes how a physical quantity changes as the energy scale changes. This allows us to determine the renormalized value of a quantity at a particular energy scale, given its value at a different energy scale. Essentially, it allows us to connect the physical quantities we measure in experiments to the underlying theory.

However, the Callan-Symanzik equation is only applicable in theories that are renormalizable. This means that the theory has a finite number of parameters that need to be renormalized in order to remove infinities. If a theory is non-renormalizable, it means that there are an infinite number of parameters that would need to be renormalized, making the process impossible. In this case, the Callan-Symanzik equation cannot be used to determine the renormalized values of physical quantities, and alternative techniques must be used.