QCD Finite Energy Sum Rules and the Isoscalar Scalar Mesons

Abstract

We apply QCD Finite Energy Sum Rules to the scalar-isoscalar current to determine the lightest $u \bar{u} + d \bar{d}$ meson in this channel. We use `pinch-weights' to improve the reliability of the QCD predictions and reduce the sensitivity to the cut-off $s_0$. A decaying exponential is included in the weight function to allow us to focus on the contribution from low mass states to the phenomenological integral. On the theoretical side we include OPE contributions up to dimension six and a contribution due to instantons taken from the Instanton Liquid Model. Phenomenologically, we incorporate experimental data by using a coupling scheme for the scalar current which links the vacuum polarisation to the $ππ$ scattering amplitude via the scalar form factor. We find that the sum rules are well saturated for certain instanton parameters. We conclude that the $f_0(400-1200)$ definitely contains a large $u \bar{u} + d \bar{d}$ component, whereas the $f_0(980)$ most likely does not. We are able to estimate the average light quark mass and find $m_q(1 \GeV) = 5.2 \pm 0.6$ MeV.