One-particle-irreducible Feynman diagram (1PI Feynman diagram)

A one-particle-irreducible Feynman diagram is a connected Feynman diagram that remains connected after any one internal line is cut. The vertices of the quantum effective action generate precisely these diagrams.

These Feynman diagrams generate the vertices of the quantum effective action.

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.

The second derivative of a Legendre effective action is the inverse connected two-point kernel. In a translation-invariant scalar theory it is ${\tilde{\Gamma }}^{(2)}(p)=\tilde{G}(p{)}^{-1}$. Its loop expansion contains the reference inverse propagator, any separately displayed counterterms, and the self-energy. A generic amputated connected higher-point function is not the same as a one-particle-irreducible vertex.

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.

Dyson resummation sums repeated self-energy insertions as a geometric series. If $G_0$ is the free propagator and $\Sigma$ is the proper two-point insertion, thenfor the convention in which an insertion contributes $+\Sigma$ inside the series.

The fermion self-energy is the amputated one-particle-irreducible two-point function $\Sigma (/p)$. It corrects both the coefficient of $/p$ and the fermion mass and thereby shifts the pole and residue of the exact propagator.

In quantum electrodynamics, the modified minimal subtraction scheme running mass differs from the perturbative pole mass by a finite correction. The displayed relation follows by evaluating the renormalized fermion self-energy at the mass shell and imposing a zero of the inverse Dirac propagator. The sign assigned to the self-energy insertion must be matched to its denominator convention; changing that naming convention does not change the conversion. A wave-function coefficient counterterm and multiplicative mass renormalization are also distinct: the coefficient of the mass term includes both at first order.

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.

The small-external-momentum expansion of a four-point one-particle-irreducible correlation function generates a local quartic term followed by four-field terms with derivatives, for example ${\varphi }^2(\partial \varphi {)}^2$ and higher-derivative operators, modulo integration by parts. This is an analytic expansion when there is an appropriate gap and regulator smoothness. A sharp projector on every internal momentum can instead generate nonanalytic boundary artifacts; then the local expansion is a smooth-regulator or fixed-loop-cutoff convention, not an automatic theorem about the entire sharp-cutoff vertex.