Chiral Perturbation to One Loop Including the $rho$ Meson

We formulate the chiral perturbation theory at the one loop level in the effective lagrangian including the $rho$ meson as a dynamical gauge boson of a hidden local symmetry(HLS). The size of radiative correction to the phenomenological parameter $a$ of HLS is estimated to be about $10$%. The complete list of ${cal O}(E^4)$ terms is given and the one loop counter terms are determined explicitly in the $N$ flavor model. We also obtain matching conditions to the conventional chiral perturbation of Gasser and Leutwyler in the chiral limit in a renormalization scale independent manner. We find that Gasser--Leutwylers estimates for $L_{9,10}$ are saturated by $rho$ and its one loop contributions without introducing non-minimal couplings of $pi$-$rho$ system, suggesting the absence of the tree level $a_1$ meson contributions.