Accurate Test of Chiral Dynamics in the \boldmath$\vec{\gamma} p \rightarrow \pi^0p$ Reaction

A precision measurement of the differential cross sections $d\sigma/d\Omega$ and the linearly polarized photon asymmetry $\Sigma \equiv (d\sigma_\perp - d\sigma_\parallel) \slash (d\sigma_\perp + d\sigma_\parallel)$ for the $\vec{\gamma} p \rightarrow \pi^0p$ reaction in the near-threshold region has been performed with a tagged photon beam and almost $4\pi$ detector at the Mainz Microtron. The Glasgow-Mainz photon tagging facility along with the Crystal Ball/TAPS multi-photon detector system and a cryogenic liquid hydrogen target were used. These data allowed for a precise determination of the energy dependence of the real parts of the $S$- and all three $P$-wave amplitudes for the first t…