Coherent $\rho^0$ photoproduction in bulk matter at high energies

The momentum transfer $\Delta k$ required for a photon to scatter from a target and emerge as a $\rho^0$ decreases as the photon energy $k$ rises. For $k>3\times10^{14}$ eV, $\Delta k$ is small enough that the interaction cannot be localized to a single nucleus. At still higher energies, photons may coherently scatter elastically from bulk matter and emerge as a $\rho^0$, in a manner akin to kaon regeneration. Constructive interference from the different nuclei coherently raises the cross section and the interaction probability rises linearly with energy. At energies above $10^{23}$ eV, coherent conversion is the dominant process; photons interact predominantly as $\rho^0$. We compute the coherent scattering probabilities in slabs of lead, water and rock, and discuss the implications of the increased hadronic interaction probabilities for photons on ultra-high energy shower development.