On the amelioration of quadratic divergences

Once massless quadratically divergent tadpole diagrams are discarded, because they contain no intrinsic scale, it is possible to convert other divergences into logarithmic form, using partial fraction identities; this includes the case of quadratic divergences, as has been applied to the linear sigma model. However the procedure must be carried out with due care, paying great attention to correct numerator factors. 11.10.Gh, 12.90.+b Typeset using REVTEX 1 In QED one knows that the formally quadratically divergent photon vacuum polarization graph is reduced to a logarithmically divergent singularity by invoking gauge invariance. However, in a linear σ model field theory, involving spinless pion and σ mesons, no principle like gauge invariance can be invoked; yet the quadratic divergence can be tamed and converted into a logarithmic one. This fact was first noted for a quark-level SU(2) model, by using the 2l-dimensional regularization lemma (DRL) [1]: