Regularizing the quark-level $\sigma$ model
  We show that the finite difference, $-i\pi^2 m^2$, between quadratic and logarithmic divergent integrals $\int d^4p[m^2(p^2-m^2)^{-2}-(p^2-m^2)^{-1}]$, as encountered in the linear $\sigma$ model, is in fact regularization independent. 
R. Delbourgo, A.A. Rawlinson and M.D. Scadron