Insights in the T-matrix formalism

In many-body perturbation theory the self-energy $\Sigma=iGW\Gamma$ plays a key role since it contains all the many body effects of the system. The exact self-energy is not known and approximations are needed. As first approximation one can neglect the vertex $\Gamma$, and obtain the GW approximation. In some cases this is not sufficient, and one needs to go beyond this approximation. In this work we elucidate the concept of T-matrix [1] and its relation with Hedin's equations [2]: we look for a unified framework including GW, T-matrix, and GW$\Gamma$. We discuss this in relation to two main shortcomings of the GW approximation: the self-screening error and the incorrect atomic limit [3]. \\ \\ $[1]$ L.\ P.\ Kadano and G.\ Baym, Quantum Statistical Mechanics, W.\ A.\ Benjamin, Inc.\, New York, (1962).\\ $[2]$ L.\ Hedin, Phys.\ Rev.\ \textbf{139}, A796 (1965).\\ $[3]$ P. Romaniello, S. Guyont, and L. Reining, J.\ Chem.\ Phys.\textbf{131}, 154111 (2009).