  The angle-resolved photoemission spectra of the superconductor
(Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ have been investigated both experimentally and
theoretically. Our results explain the previously obscured origins of all
salient features of the ARPES response of this paradigm pnictide compound and
reveal the origin of the Lifshitz transition. Comparison of calculated ARPES
spectra with the underlying DMFT band structure shows an important impact of
final state effects, which results for three-dimensional states in a deviation
of the ARPES spectra from the true spectral function. In particular, the
apparent effective mass enhancement seen in the ARPES response is not an
entirely intrinsic property of the quasiparticle valence bands but may have a
significant extrinsic contribution from the photoemission process and thus
differ from its true value. Because this effect is more pronounced for low
photoexcitation energies, soft-X-ray ARPES delivers more accurate values of the
mass enhancement due to a sharp definition of the 3D electron momentum.
