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.

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As an example, for a quantum field theory with one massless scalar field and one self-coupling term, denote the bare field strength by ${\displaystyle \phi _{0}}$ and the bare coupling constant by ${\displaystyle g_{0}}$. In the process of renormalisation, a mass scale M must be chosen. Depending on M, the field strength is rescaled by a constant: ${\displaystyle \phi =Z\phi _{0}}$, and as a result the bare coupling constant ${\displaystyle g_{0}}$ 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

${\displaystyle G^{(n)}(x_{1},x_{2},\ldots ,x_{n};M,g)=\langle \phi (x_{1})\phi (x_{2})\cdots \phi (x_{n})\rangle }$

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 ${\displaystyle \phi }$. If the choice of ${\displaystyle M}$ is slightly altered by ${\displaystyle \delta M}$, then the following shifts will occur:

${\displaystyle M\to M+\delta M}$

${\displaystyle g\to g+\delta g}$

${\displaystyle \phi =Z\phi _{0}\to Z'\phi _{0}=(1+\delta \eta )\phi }$

${\displaystyle G^{(n)}\to (1+n\,\delta \eta )G^{(n)}}$

The Callan–Symanzik equation relates these shifts:

${\displaystyle n\,\delta \eta G^{(n)}={\frac {\partial G^{(n)}}{\partial M}}\delta M+{\frac {\partial G^{(n)}}{\partial g}}\delta g}$

After the following definitions

${\displaystyle \beta ={\frac {M}{\delta M}}\delta g}$

${\displaystyle \gamma =-{\frac {M}{\delta M}}\delta \eta }$

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

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

${\displaystyle \beta (g)}$ being the beta function.

In quantum electrodynamics this equation takes the form

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

where n and m are the numbers of electron and photon fields, respectively, for which the correlation function ${\displaystyle G^{(n,m)}}$ 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, ${\displaystyle \gamma _{2}}$ and ${\displaystyle \gamma _{3}}$, 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.