Resonance determination by stabilized analytic continuation of theoretical data, and comparison with the moments method

The conventional methods used so far with QCD sum rules stem from the observation that $s_1$-s ($s_1$ being the position of the first resonance) can be written as the limit of the ratio of the nth- and (n+1)th-order moments, or are based on similar techniques such as those of Borel operators. Unfortunately, apart from the fact that practical instabilities appear because one uses high derivatives of very smooth functions, these methods do not have internal controls which would enable them to adapt to the varying precision of the theoretical information for different values of the energy in the spacelike region, to finite resonance widths, threshold behaviors, etc. There are alternative procedures, however, based on extremal problems and leading to simple (Fredholm) integral equations, which are flexible enough to accommodate these various practical requirements. Computer tests have been carried out, using these procedures, on a number of completely soluble quantum-mechanical examples, and the comparison with the conventional moments techniques is described.