The Mixed Vector Current Correlator <0|T(V^3_μV^8_ν)|0> To Two Loops in Chiral Perturbation Theory

Abstract

The isospin-breaking correlator of the product of flavor octet vector currents, $V^3_μ$ and $V^8_ν$, $Π^{38}_{μν}(q^2)$ is computed to next-to-next- to-leading (two-loop) order in Chiral Perturbation Theory. Large corrections to both the magnitude and $q^2$-dependence of the one-loop result are found, and the reasons for the slow convergence of the chiral series for the correlator given. The two-loop expression involves a single ${\cal O}(q^6)$ counterterm, present also in the two-loop expressions for $Π^{33}_{μν}(q^2)$ and $Π^{88}_{μν}(q^2)$, which counterterm contributes a constant to the scalar correlator $Π^{38}(q^2)$. The feasibility of extracting the value of this counterterm from other sources is discussed. Analysis of the slope of the correlator with respect to $q^2$ using QCD sum rules is shown to suggest that, even to two-loop order, the chiral series for the correlator may not yet be well-converged.