Extraction of light quark masses from sum rule analyses of axial vector and vector current Ward identities

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In light of recent lattice results for the light quark masses ${m}_{s}$ and ${m}_{u}{+m}_{d},$ we reexamine the use of sum rules in the extraction of these quantities, and discuss a number of potential problems with existing analyses. The most important issue is that of the overall normalization of the hadronic spectral functions relevant to the sum rule analyses. We explain why previous treatments, which fix this normalization by assuming complete resonance dominance of the continuum threshold region, can potentially overestimate the resonance contributions to spectral integrals by factors as large as $\ensuremath{\sim}5.$ We propose an alternate method of normalization based on an understanding of the role of resonances in chiral perturbation theory which avoids this problem. The second important uncertainty we consider relates to the physical content of the assumed location ${s}_{0}$ of the onset of duality with perturbative QCD. We find that the extracted quark masses depend very sensitively on this parameter. We show that the assumption of duality imposes very severe constraints on the shape of the relevant spectral function in the dual region and present rigorous lower bounds for ${m}_{u}{+m}_{d}$ as a function of ${s}_{0}$ based on a combination of these constraints and the requirement of positivity of ${\ensuremath{\rho}}_{5}(s).$ In the extractions of ${m}_{s},$ we find that the conventional choice of the value of ${s}_{0}$ is not physical. For a more reasonable choice of ${s}_{0},$ we are not able to find a solution that is stable with respect to variations of the Borel transform parameter. This problem can, unfortunately, be overcome only if the hadronic spectral function is determined up to significantly larger values of $s$ than is currently possible. Finally, we also estimate the error associated with the convergence of perturbative QCD expressions used in the sum rule analyses. Our conclusion is that, taking all of these issues into account, the resulting sum rule estimates for both ${m}_{u}{+m}_{d}$ and ${m}_{s}$ could easily have uncertainties as large as a factor of 2, which would make them compatible with the low estimates obtained from lattice QCD.