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.

Amputation removes the external propagator factors from a momentum-space correlation function. For a connected four-point function, its diagrammatic external lines each carry a full two-point function; multiplying by the corresponding inverse two-point kernels leaves an interaction kernel.

This operation helps separate propagation to and from the interaction from the interaction itself. It is part of the connection between Green functions and scattering amplitudes, although physical external states also require on-shell limits and the appropriate pole-residue normalization. An amputated connected function need not be one-particle irreducible: it may still contain internal lines whose removal disconnects it. External-line removal and internal irreducibility are distinct operations.

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 proper two-point correction to the gluon propagator in QCD. At one loop it receives quark, gluon, Faddeev-Popov ghost field and four-gluon tadpole diagram contributions. The Slavnov-Taylor identity makes the total tensor transverse, $k_{\mu }{\Pi }^{\mu \nu ,ab}=0$, although the individual gluon and ghost loop contributions need not be transverse. The sign of the symbol must be tied to whether the amputated insertion is defined as $i\Pi$ or $-i\Pi$.

With insertion convention $i\Pi$, ghost propagator $i/p^2$, and oriented ghost-gluon Feynman rule $g{f}^{abc}{p}^{\mu }$ with $p$ pointing away along the ghost arrow, the ghost loop is $i{\Pi }_{\mathrm{gh}}=-g^2{C}_A{\delta }^{ab}\int d^{D}l(2\pi {)}^{-D}(l+k{)}^{\mu }{l}^{\nu }/[l^2(l+k{)}^2]$. In dimensional regularization with $D=4-ϵ$, a Feynman parameter and a shift give the displayed ultraviolet pole. Its contraction is $g^2{C}_A{\delta }^{ab}{k}^2{k}^{\nu }/(32{\pi }^2ϵ)$; it cancels the nontransverse part of the two-three-gluon-vertex contribution. A different insertion or ghost-vertex convention must be translated before comparing signs.

The two-point, one-particle-irreducible correction for a scalar field. With an insertion denoted $-i\Sigma (p^2)$, the full scalar propagator is $i/[p^2-m^2-\Sigma (p^2)+i0]$. A divergent coefficient of $p^2$ requires wave-function renormalization; a momentum-independent divergent term requires a mass counterterm.

The coefficient of a simple one-particle pole in a propagator is its pole residue. If $G(p)=i/[p^2-m_0^2-\Sigma (p^2)]$ and $m$ is the pole mass, then $Z_{\mathrm{pole}}=[1-{\Sigma }^{\prime }(m^2){]}^{-1}$. An on-shell renormalization scheme may impose unit residue by a finite or divergent wave-function renormalization; mere ultraviolet finiteness does not force the residue to be one.

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 Feynman gauge, with insertion $-i\Sigma (p)$ and $d=4-2ϵ$, the one-loop QED result is ${\Sigma }_{\mathrm{UV}}=e^2(-/p+4m)/(16{\pi }^2ϵ)$. The coefficient of $/p$ is the electron kinetic counterterm ${\delta }_2=-e^2/(16{\pi }^2ϵ)$ when its insertion is $i/p{\delta }_2$. The scalar part fixes a separate mass counterterm in the chosen normalization convention.

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.

For a stable particle represented by an isolated propagator pole, the pole mass $M$ is defined by the pole at invariant momentum squared $p^2=M^2$. It is a property of the complete propagator pole, distinct from an arbitrary running mass parameter defined in a renormalization scheme.

An unstable resonance generally has a complex pole, often parametrized as $s_p=(M-i\Gamma /2{)}^2$; its real mass and width conventions must then be specified. A peak position in a measured distribution need not coincide exactly with that pole parameter because of backgrounds, thresholds and resolution. Confined fields or infrared-singular charged states require further qualifications before the simple isolated stable-particle picture applies. The pole location and the chosen field's pole residue are different quantities.

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.