How Many Counterterms Are Needed for Logarithmic Divergences?

[zetafunction]

If i have ONLY logarithmic divergences as $$\lambda \rightarrow \infty$$ of the form

$$log(a+\lambda ^{n})$$ or $$log (\lambda )$$ or $$log^{k}(\lambda)$$ for some real numbers a,n and k HOW many counterterms should i put into de Lagrangian in order to make it FINITE ?? , the idea is let us suppose we use DIMENSIONAL REGULARIZATION so we only had logarithmic divergent integrals (and assuming that power law divergences can be reduced by dimensional regularization or other method to only logarithmic divergences), how many counterterms should i add to the original lagrangian to obtain finite results ??

 

[shiekh]

One way around is to use operator-regularization (a generalization of the zeta-function) that will take care of everything, and preclude the need to add extra terms to the original Lagrangian (should they be needed).