Resonance sum rules: an application to the square well potential

Abstract: We propose an extension of the Quantum Chromodynamics (QCD) sum rules, termed the Resonance sum rules (RSR), to access resonance poles in the complex energy plane. By strategically introducing a contour in the complex plane and conformal mapping, the method intends to reach resonance poles on the second Riemann sheet. To validate this approach, we apply RSR to the square-well potential model, for which the pole locations are known. The analysis demonstrates a successful extraction of the pole positions and residues for both the $S$-wave and $P$-wave resonances. The results are in good agreement with the analytic solutions, with discrepancies within $5\%$ for the pole positions and $20\%$ for the residues.This framework provides a basis for future applications to realistic hadronic resonances, promising new insights into spectral properties of QCD.