Isospin-breaking effects in the extraction of isoscalar and isovector spectral functions from e+e−→ hadrons

We investigate the problem of the extraction of the isovector and isoscalar spectral functions from data on ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}$ hadrons, in the presence of non-zero isospin breaking. It is shown that the conventional approach to extracting the isovector spectral function in the $\ensuremath{\rho}$ resonance region, in which only the isoscalar contribution associated with $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\pi}\ensuremath{\pi}$ is subtracted, fails to fully remove the effects of the isoscalar component of the electromagnetic current. The additional subtractions required to extract the pure isovector and isoscalar spectral functions are estimated using results from QCD sum rules. It is shown that the corrections are small $(\ensuremath{\sim}2%)$ in the isovector case (though relevant to precision tests of the CVC hypothesis), but very large $(\ensuremath{\sim}20%)$ in the case of the $\ensuremath{\omega}$ contribution to the isoscalar spectral function. The reason such a large effect is natural in the isoscalar channel is explained, and implications for other applications, such as the extraction of the sixth order chiral low-energy constant, $Q$, are discussed.