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 ≥ t̂. We find that the bounds vary a lot depending on the assumed value of t̂ and even, if it is too small (t̂ ≃ 1.5, 1.6, 2.1 GeV 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 t̂ admissible, and to get upper and lower bounds for ms and upper bounds for md +mu, md −mu. The upper bounds are not particularly interesting, but the lower ones are very tight; specifically we find 240 MeV ≤ ms; 16 MeV ≤ md +mu if we assume perturbative QCD to give a valid description of the correlator for t ≥ 2.2 GeV; or, if it only holds for t ≥ 4.5 GeV then 150 MeV ≤ ms; 10 MeV ≤ md +mu. 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. * Permanent address. e-mail: [email protected] Typeset with PHysMaTEX