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Pure QCD bounds and estimates for light quark masses

We consider bounds on light quark masses that follow from positivity of the pseudoscalar correlator spectral function plus the assumption that perturbative QCD is valid for the correlator and its derivatives up to order $N$ for momenta $t\geq\hat{t}$. We find that the bounds vary a lot depending on the assumed value of $\hat{t}$ and even, if it is too small ($\hat{t}\simeq 1.5, 1.6, 2.1 Gev^2$ for respectively $N=0, 1, 2$), that there is incompatibility between the asumption of validity of perturbative QCD and positivity. This allows us to establish a criterion for the values of $\hat{t}$ admissible, and to get upper and lower bounds for $m_s$ and upper bounds for $m_d+m_u, m_d-m_u$. The upper bounds are not particularly interesting, but the lower ones are very tight; specifically we find $$240 Mev\leq m_s; 16 Mev\leq m_d+m_u$$ if we assume perturbative QCD to give a valid description of the correlator for $t\geq 2.2 Gev^2$; or, if it only holds for $t\geq 4.5 Gev^2$ then $$150 Mev\leq m_s; 10 Mev\leq m_d+m_u.$$ Here the masses are running masses defined at $1 Gev$. We also show reasonable models where the bounds are saturated. The results suggest that some of the current estimates of the light quark masses are less precise than ordinarily claimed.