Let's say that after dimensional regularization we have a one-loop effective action of the form $$\Gamma^{\text{div}}_1=\int\frac{\mu^\epsilon}{\epsilon}\Big[ c_1\phi^2(\partial_\mu A^\mu) +c_2 \phi A_\mu A^\mu+..\Big] $$ where $c_i$ are some real numbers, and no particular information of the fields is needed.
Is it possible, IN GENERAL, that higher order loops numerically contribute, or even cancel, such divergences in the broader effective action $\Gamma^{\text{div}}_1+\Gamma^{\text{div}}_2+$more loops? (Assuming the same tensorial structures appear.)
Because the question is posed so generally, my guess would be yes, as higher loops can contribute to both $\frac{1}{\epsilon}$ and $\frac{1}{\epsilon^2}$ terms.
My doubt is mainly about this last statement, as I am not sure that higher loops can be proportional to $\frac{1}{\epsilon}$, which would be a requirement for such cancellations.
