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.

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.

The fermion self-energy of an Electron in quantum electrodynamics. After renormalization, its contribution in a bound chemical atom is one source of the Lamb shift; other radiative corrections and nuclear structure also affect the observed energy levels.

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.