An extended study of the prompt photon photoproduction at HERA with $k_T$-factorization

We reconsider the prompt photon photoproduction at HERA in the framework of the $k_T$-factorization QCD approach. The proposed method is based on the ${\cal O}(\alpha^2\alpha_s)$ amplitudes for $\gamma q\to\gamma gq$ and $\gamma g^*\to\gamma q\bar q$ partonic subprocesses. Additionally, we take into account the ${\cal O}(\alpha^2\alpha_s^2)$ box contributions $\gamma g\to\gamma g$ to the production cross sections. The unintegrated (or transverse momentum dependent) parton densities in the proton are determined using Kimber-Martin-Ryskin (KMR) prescription. Our consideration covers both inclusive and jet associated prompt photon photoproduction rates. We find that our numerical predictions agree well with the recent data taken by H1 and ZEUS collaborations at HERA. We demonstrate that the box contributions are sizeable and amount up to $\sim15$ of the calculated total cross section.