divergent integrals of the form (for illustrative purposes, we work with Euclidean integrals) $$I_Q = \int \frac{d^4 \ell}{(\ell^2 + \Delta)} \; ,$$ which, on dimensional regularization gives $I_Q = \Delta \left( \frac{\mu^2}{\Delta} \right)^{\delta/2} F(\delta)$, where, $\delta = 4-d$, $\mu$ is the fake renormalization scale (which inevitably gets introduced while performing dimensional regularization) and $F(\delta)$ is a dimensionless function of $\delta$ which contains poles of $\delta$. If $F$ has a simple pole and its Laurent series expansion of is $F = F_{-1} \delta^{-1} + F_0 + F \delta + \cdots $, then,

$$\label{eq:usual_qft_quad}
 I_Q = \Delta \left( \frac{F_{-1}}{\epsilon} + \frac{F_{-1}}{2} \log \left( \frac{\mu^2}{\Delta} \right) + F_0 + \cdots \right) \; .$$

Typically, $\Delta$ is a polynomial in the external momentum $k^2$($=-E^2+{\bf p}^2$) or the masses (and often, the Feynman parameters). The presence of $\Delta$ in the above expression causes the dimensionally regulated UV divergent integral to be a sum of two parts; the first part is polynomial in the external momentum with divergent coefficients and the second one is a finite function of the external momenta, the masses and the fake renormalization scale $\mu$. E.g. while evaluating the two-point function in $\phi^4$ theory in $d=4$, the corresponding $\Delta$ turns out to 