we use the dimensionally-regulated expression for the vacuum-polarization insertions in the gluon propagator.)

Because the renormalization parts of an operator matrix element are short-distance quantities, we can analyze them perturbatively.[^6] Furthermore, the renormalization parts are independent of the external states. Therefore, we are free to make a choice of the external states that is convenient for a perturbative analysis, namely, states consisting of on-shell elementary quanta (quarks, gluons, etc.).

We note that, for purposes of computing the ambiguities in the dimensionally-regulated matrix elements, any reference cutoff regulator will do. At the one-loop level, a particularly convenient choice of cutoff regulator is a simple cutoff on the magnitude of the three-momentum of the gluon in the loop. While such a cutoff is not, in general, consistent with gauge invariance, it is compatible with the QED-like nature of the gauge invariance at one loop. With this choice, we compute $$\langle {\cal K} \rangle_{\hbox{dim}}-\langle {\cal K} 
\rangle_{\hbox{cutoff}}
=\int_0^{\rm dim} d^4l\, I_{\cal K}-\int dl_0\, \int_0^{\lambda} 
d^3\mbox{\boldmath $l$} 
\, I_{\cal K}
=\int dl_0\, \int_{\lambda}^{\rm dim} d^3\mbox{\boldmath $l$} 
\, I_{\cal K},
\label{matrix-diff}$$ where ${\cal K}$ is an operator, $I_{\cal K}$ is the integrand corresponding to its matrix element between on-shell states in one-loop perturbation theory, and 