Two-Loop Analysis of Vector Current Propagators in Chiral Perturbation Theory

We perform a calculation of the isospin and hypercharge vector current propagators (${\mathrm{\Delta }}_{V33}^{\mu \nu }(q^2)$ and ${\mathrm{\Delta }}_{V88}^{\mu \nu }(q^2)$) to two loops in chiral perturbation theory. The analysis is carried out with straightforward Feynman diagram methods by making appropriate use of external vector sources. Counterterms from the $\mathcal{O}(q^6)$ chiral lagrangian, required to absorb divergences and scale dependence encountered at the two-loop level, are constructed. Our final results are finite, covariant, and scale-independent. Several applications are described, including a comparison of the two-loop isospin vector spectral function with data and the construction of new chiral sum rules.