Collective modes and electromagnetic response of a chiral superconductor

Motivated by the recent controversy surrounding the Kerr effect measurements in strontium ruthenate [J. Xia et al., Phys. Rev. Lett. 97, 167002 (2006)], we examine the electromagnetic response of a clean chiral $p$-wave superconductor. When the contributions of the collective modes are accounted for, the Hall response in a clean chiral superconductor is smaller by several orders of magnitude than previous theoretical predictions and is too small to explain the experiment. We also uncover some unusual features of the collective modes of a chiral superconductor, namely, that they are not purely longitudinal and couple to external transverse fields.