Divergences in the quark number susceptibility : The origin and a cure

Quark number susceptibility on the lattice, obtained by merely adding a $\mu N$ term with $\mu$ as the chemical potential and $N$ as the conserved quark number, has a quadratic divergence in the cut-off $a$. We show that such a divergence already exist for free fermions with a cut-off regulator. While one can eliminate it in the free lattice theory by suitably modifying the action, as is popularly done, it can simply be subtracted off as well. Computations of higher order susceptibilities, needed for estimating the location of the QCD critical point, then need a lot fewer number of quark propagators at any order. We show that this method of divergence removal works in the interacting theory.