oneparticleirreducible

One-particle-irreducible, abbreviated 1PI, refers to a class of Feynman diagrams used in quantum field theory and related areas. A diagram is 1PI if it is connected and cannot be separated into two disconnected pieces by cutting a single internal line. If a diagram can be split into two parts by removing just one internal line, it is called one-particle-reducible (1PR).

1PI diagrams serve as the primitive building blocks for the one-particle-irreducible generating functional, also known as the effective action, denoted Γ[φ]. The functional derivatives of Γ with respect to the background field φ yield the 1PI n-point vertex functions Γ^(n). The sum of all 1PI diagrams with fixed external legs corresponds to the corresponding 1PI vertex function; these diagrams encode irreducible quantum corrections that cannot be decomposed into simpler scattering processes connected by a single propagator.

In practice, the two-point 1PI function is the proper self-energy, Σ(p). The full propagator G is related to the bare propagator G_0 via G^{-1} = G_0^{-1} − Σ, or equivalently G = [G_0^{-1} − Σ]^{-1}. Higher-point 1PI functions Γ^(n) are the proper vertices used in Dyson-Schwinger equations and in functional renormalization group analyses to organize perturbative and nonperturbative contributions.

1PI diagrams are important for renormalization because their divergences are primitive, guiding the definition of counterterms. The concept applies across scalar, fermionic, and gauge theories and also appears in condensed-matter and lattice contexts, where it facilitates systematic resummations and the construction of effective descriptions.