I am trying to understand the difference between the 2-point vertex function and the self energy. In many presentations, they are described in ways that seem nearly equivalent, yet as I work through the details I run into seemingly nonsensical statements.
To start with definitions, we know that the 2-point vertex $\Gamma^{(2)}$ is characterized by $\Gamma^{(2)} = G_c^{-1}$, the inverse of the 2-point connected Green's function. The self-energy $\Sigma$ is the sum of all 1-particle irreducible (1PI) diagrams of the two-point function after amputating the external legs.
It is also said that $\Gamma^{(n)}$, the $n$-point vertex, is said to be the sum of all amputated 1PI diagrams with $n$ external legs. This statement is not a definition, but a derived consequence. However, this statement seems strange to me in the case $n=2$, in which case it would be equivalent to saying $\Gamma^{(2)} = \Sigma$.
We know that $G_c^{-1} = G_0^{-1} - \Sigma$, where $G_0$ is the free propagator. If $\Gamma^{(2)} = \Sigma$, then we would arrive at the nonsensical statement $2\Gamma^{(2)} = G_0^{-1}$. This seems to imply that $\Gamma^{(2)}$ should not be thought of as the sum of all the 1PI diagrams for the amputated 2-point correlator.
Am I missing something here, or are textbooks being imprecise in the statement that $\Gamma^{(n)}$ is the sum of all amputated 1PI diagrams with $n$ external legs?
