Callan–Symanzik equation

In physics, the Callan–Symanzik equation is a differential equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined and involves the beta-function of the theory and the anomalous dimensions. This equation has the following structure

$\left[M{\frac {\partial }{\partial M}}+\beta (g){\frac {\partial }{\partial g}}+n\gamma \right]G^{{(n)}}(x_{1},x_{2},\ldots ,x_{n};M,g)=0$

$\beta (g)$ being the beta function and $\gamma$ the scaling of the fields.

In quantum electrodynamics this equation takes the form

$\left[M{\frac {\partial }{\partial M}}+\beta (e){\frac {\partial }{\partial e}}+n\gamma _{2}+m\gamma _{3}\right]G^{{(n,m)}}(x_{1},x_{2},\ldots ,x_{n};M,e)=0$

n and m being the number of electrons and photons respectively.

It was discovered independently by Curtis Callan^[1] and Kurt Symanzik^[2][3] in 1970. Later it was used to understand asymptotic freedom.

This equation arises in the framework of renormalization group. It is possible to treat the equation using perturbation theory.

See also

• Exact renormalization group equation
• Beta function

Notes

1. C. G. Callan, Jr., Broken Scale Invariance in Scalar Field Theory, Phys. Rev. D 2, 1541–1547 (1970). APS
2. K. Symanzik, Small Distance Behaviour in Field Theory and Power Counting, Commun. math. Phys. 18, 227 (1970). SpringerLink
3. K. Symanzik, Small-Distance-Behaviour Analysis and Wilson Expansions, Commun. math. Phys. 23, 49 (1971). SpringerLink

References

• Jean Zinn-Justin, Quantum Field Theory and Critical Phenomena , Oxford University Press 2003, ISBN 0-19-850923-5
• John Clements Collins, Renormalization, Cambridge University Press 1986, ISBN 0-521-31177-2