Scalar susceptibility in QCD and the multiflavor Schwinger model

Abstract

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.