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The Mixed-Isospin Vector Current Correlator in Chiral Perturbation Theory and QCD Sum Rules

The mixed-isospin vector current correlator, $\langle 0\vert T(V^ρ_μV^ω_ν)\vert 0\rangle$ is evaluated using both QCD sum rules and Chiral Perturbation Theory (ChPT) to one-loop order. The sum rule treatment is a modification of previous analyses necessitated by the observation that those analyses produce forms of the correlator that fail to be dominated, near $q^2=0$, by the most nearby singularities. Inclusion of contributions associated with the $ϕ$ meson rectify this problem. The resulting sum rule fit provides evidence for a significant direct $ω\rightarrowππ$ coupling contribution in $e^+e^-\rightarrowπ^+π^-$. It is also pointed out that results for the $q^2$-dependence of the correlator cannot be used to provide information about the (off-shell) $q^2$-dependence of the off-diagonal element of the vector meson propagator unless a very specific choice of interpolating fields for the vector mesons is made. The results for the value of the correlator near $q^2=0$ in ChPT are shown to be more than an order of magnitude smaller than those extracted from the sum rule analysis and the reasons why this suggests slow convergence of the chiral series for the correlator given.