We present an analysis of four sum rules, each based on chiral symmetry and containing the difference ${\rho }_V\mathrm{}\left(S\right)-{\rho }_A\mathrm{}\left(S\right)\mathrm{}$ of isovector vector and axial-vector spectral functions. Experimental data from $\tau$ lepton decay and electron-positron scattering identify the spectral functions over a limited kinematic domain. We summarize the status of the existing database. However, a successful determination of the sum rules requires additional content, in the form of theoretical input. We show how chiral symmetry and the operator product expansion can be used to constrain the spectral functions, respectively, in the low energy and the high energy limits and proceed to perform a phenomenological test of the sum rules.

• Received 15 July 1993

DOI:https://doi.org/10.1103/PhysRevD.49.1513