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 small non-zero 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 $\sim \log m$ at zero temperature and as $\sim T/\sqrt m$ 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 $T=0$ 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 $\sim m^{-2/(N_f+1)}$. At finite volume $V$ we show that the scalar susceptibility is proportional to $1/m^2V$. Recent lattice calculations of this quantity by Karsch and Laermann are discussed.