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Confronting electron-positron annihilation into hadrons with QCD: an operator product expansion analysis

Experimental data on the total cross section of $e^+ e^-$ annihilation into hadrons are confronted with QCD and the operator product expansion using finite energy sum rules. Specifically, the power corrections in the operator product expansion, i.e. the vacuum condensates, of dimension $d = 2$, 4 and 6 are determined using recent isospin $I=0+1$ data sets. Reasonably stable results are obtained which are compatible within errors with values from $τ$-decay. However, the rather large data uncertainties, together with the current value of the strong coupling constant, lead to very large errors in the condensates. It also appears that the separation into isovector and isoscalar pieces introduces additional uncertainties and errors. In contrast, the high precision $τ$-decay data of the ALEPH collaboration in the vector channel allows for a more precise determination of the condensates. This is in spite of QCD asymptotics not quite been reached at the end of the $τ$ spectrum. We point out that isospin violation is negligible in the integrated cross sections, unlike the case of individual channels.