In most of the books on QFT, the author talks about various methods of regularization but in the end chooses the dimensional regularization and MS-bar scheme when discussing the final renormalization, I have not seen any review, books or lecture notes where the author actually takes momentum cut-off as regularization and proceeds towards renormalizing the theory. I totally understand and appreciate the elegance and usefulness of dimensional method, but in certain situations we may need to take hard momentum cut-off ($ \Lambda $) route and do the renormalization, but in doing that we face the problem of exactly how to handle the polynomial divergence ($ \Lambda^n $ kind, if any ) and the logarythmic one !
Suppose I have a one-loop calculation of an amplitude, \begin{align} A = A_{\text{finite}} + a_n \left(\frac{\Lambda}{m}\right)^n + b \log\frac{\Lambda^2}{m^2}. \end{align}
Where $ m $ is any mass. How exactly should we proceed to derive various renormalized parameters of our theory? What subtraction schemes exist in this regularization?