One-particle-irreducible correlation function (${\Gamma }^{(n)}$)

An $n$-point one-particle-irreducible correlation function is obtained by differentiating the quantum effective action $n$ times. Its diagrammatic expansion contains the amputated one-particle-irreducible diagrams with $n$ external legs.

A self-energy is the loop contribution to a one-particle-irreducible two-point function. Repeated self-energy insertions form a geometric series that shifts the propagator pole and changes its residue.

The pole mass is the value of the invariant momentum at a pole of the full propagator. It is independent of the renormalization scheme, whereas a running mass parameter generally depends on the renormalization scale.

The four-point one-particle-irreducible correlation function is the fourth functional derivative of the quantum effective action. It defines a renormalized quartic coupling after evaluation at the kinematic point specified by a renormalization condition.