[Physics] Why are only logarithmic divergence relevant for the Callan-Symanzik equation? Intuitive understanding

quantum-field-theoryregularizationrenormalization

I may be wrong, but it seems that only logarithmic divergences need to be retained when using the Callan-Symanzik equation, finding running couplings, etc. Why is this the case? Is there some simple intuitive understanding for why the logarithmic divergences are most important for these applications?

Best Answer

Of course, you can't just neglect power divergent terms. However, we have a regulator (dimensional regularization, DR) that automatically eliminates power divergent terms. To the extent that all consistent regularization schemes are equivalent, we would expect that nothing new can be learned by including power divergent terms. There are, however, cases where DR, while not wrong, obscures the physics or leads to apparently poorly converging expansions. See http://arxiv.org/abs/nucl-th/9802075 for an example in which the authors modify DR to retain power divergences (``PDS''), and solve the associated RG equations.