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:
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 chargee. The electron field and the photon field rescale differently under renormalisation, and thus lead to two separate functions, and , respectively.