Direct extraction of the chiral quark condensate and bounds on the light quark masses

We select sum rules from which one can extract directly reliable limits on the size of the chiral symmetry breaking light quark condensate $<\bar{\psi }\psi >$. Combined results from the nucleon and $B^{\ast }-B$ mass-splitting sum rules give a result compatible with the standard value: $<\bar\psi\psi> (1 GeV) \simeq (-229 MeV)^3$, through the determinations of the quark-gluon mixed condensate. The vector form factor of the $D\text{rar}{K}^{\ast }l\nu$ semi-leptonic decay leads to the range $0.6\le <\bar{\psi }\psi >(-229MeV{)}^3\le 1.5$. The upper limit combined with the Gell-Mann-Oakes-Renner (GMOR) relation implies the interesting lower bound on the sum of light quark masses:$(m_u+m_d) (1 GeV) \geq$ 9.4 MeV, which combined with the ratio of light quark masses from chiral perturbation theory leads to $m_s(1GeV)\ge (121\pm 12)$ MeV. The lower limit combined with the positivity of the $m_q^2$ contribution to the GMOR relation leads to the upper bound : $(m_u+m_d)(1GeV)\le$ 15.7 MeV, which is independent on the nature of chiral symmetry breaking.