The experimental valence band photoemission spectrum of semiconductors exhibits multiple satellites that cannot be described by the GW approximation for the self-energy in the framework of many-body perturbation theory. Taking silicon as a prototypical example, we compare experimental high energy photoemission spectra with GW calculations and analyze the origin of the GW failure. We then propose an approximation to the functional differential equation that determines the exact one-body Green's function, whose solution has an exponential form. This yields a calculated spectrum, including cross sections, secondary electrons, and an estimate for extrinsic and interference effects, in excellent agreement with experiment. Our result can be recast as a dynamical vertex correction beyond GW, giving hints for further developments.Photoemission is a prominent tool to access information about electronic structure and excitations in materials. Modern synchrotron sources can provide detailed insight, thanks to their high intensity and broad photon energy range. But the interpretation of the experimental data is far from obvious, and theory is an essential complementary tool. However, ab initio calculations typically focus on bulk bandstructure [1, 2]; thus surface effects are ignored, and satellites are not included. The latter are a pure many-body effect due to coupling to excitations of the material. Such many-body effects are contained in approaches developed for correlated materials [3,4] however, these are usually based on models with short-range interactions, whereas satellites such as plasmons involve long-range effects. Plasmon satellites have been extensively studied in core-level experiments [5]. There they can be described by a theoretical model where a single dispersionless fermion couples to bosons. The resulting exact Green's function has an exponential form given by the so-called cumulant expansion (CE). A Taylor expansion of the exponential leads to a well defined quasi-particle (QP) peak followed by a decaying series of plasmon satellites at energy differences given by the plasmon energy, consistent with experimental observations [6][7][8][9][10]. In the valence region, plasmon satellites are much less studied, though ab initio approaches can provide a good starting point. At high photoelectron energies the photoemission spectrum is approximately proportional to the intrinsic spectral function A(ω) = −(1/π)Im G(ω), where G is the one-particle Green's function. The latter is typically calculated using the widely used GW approximation (GWA) [7,11,12]. In principle, the GWA contains correlations effects beyond the quasiparticle approximation. However, these additional features are rarely calculated due to computational complexity and, more importantly, the serious discrepancies between GWA and experiment (see e.g. [13][14][15][16]). The CE has also been used for the homogeneous electron gas [17] and simple metals [14,15], yielding an improved description of satellites over GW. Silicon [16] and graphite [18] ...

International audienceIn this joint experimental and theoretical work, we investigate collective electronic excitations (plasmons) in free-standing, single-layer graphene. The energy- and momentum-dependent electron energy-loss function was measured up to 50eV along two independent in-plane symmetry directions (ΓM and ΓK) over the first Brillouin zone by momentum-resolved electron energy-loss spectroscopy in a transmission electron microscope. We compare our experimental results with corresponding time-dependent density-functional theory calculations. For finite momentum transfers, good agreement with experiments is found if crystal local-field effects are taken into account. In the limit of small and vanishing momentum transfers, we discuss differences between calculations and the experimentally obtained electron energy-loss functions of graphene due to a finite momentum resolution and out-of-plane excitations