# 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.

As an example, for a quantum field theory with one massless scalar field and one self-coupling term, denote the bare field strength by and the bare coupling constant by . In the process of renormalisation, a mass scale *M* must be chosen. Depending on *M*, the field strength is rescaled by a constant: , and as a result the bare coupling constant is correspondingly shifted to the renormalised coupling constant *g*.

Of physical importance are the renormalised *n*-point functions, computed from connected Feynman diagrams, schematically of the form

For a given choice of renormalisation scheme, the computation of this quantity depends on the choice of *M*, which affects the shift in *g* and the rescaling of . If the choice of is slightly altered by , then the following shifts will occur:

The Callan-Symanzik equation relates these shifts:

After the following definitions

the Callan-Symanzik equation can be put in the conventional form:

being the beta function.

In quantum electrodynamics this equation takes the form

where *n* and *m* are the numbers of electron and photon fields, respectively, for which the correlation function is defined. The renormalised coupling constant is now the renormalised elementary charge *e*. The electron field and the photon field rescale differently under renormalisation, and thus lead to two separate functions, and , respectively.

The Callan-Symanzik equation 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[edit]

## Notes[edit]

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

## References[edit]

- 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 - Michael E. Peskin and Daniel V. Schroeder,
*An Introduction to Quantum Field Theory*, Addison-Wesley, Reading, 1995.*2nd edition, pbk*. Westview Press. 2015.^{[1]}

**^**Berg, Michael (10 February 2016). "Review of

*An introduction to quantum field theory*by Peskin & Schroeder".

*MAA Reviews, maa.org*.