Scalar susceptibility in QCD and the multiflavor Schwinger model

We evaluate the leading infrared behavior of the scalar susceptibility in QCD and in the multiflavor Schwinger model for a small nonzero quark mass $m$ and/or small nonzero temperature as well as the scalar susceptibility for the finite-volume QCD partition function. In QCD, it is determined by one-loop chiral perturbation theory, with the result that the leading infrared singularity behaves as $\sim \ln m$ at zero temperature and as $\sim \frac{T}{\sqrt{m}}$ at finite temperature. In the Schwinger model with several flavors we use exact results for the scalar correlation function. We find that the Schwinger model has a phase transition at $T=0\mathrm{}$ with critical exponents that satisfy the standard scaling relations. The singular behavior of this model depends on the number of flavors with a scalar susceptibility that behaves as $\sim m^{-\frac{2}{(N_f+1)}}$. At finite volumes $V$ we show that the scalar susceptibility is proportional to $\frac{1}{m^{2}V}$. Recent lattice calculations of this quantity by Karsch and Laermann are discussed.