Connected Feynman diagram

A connected Feynman diagram has a path between every pair of its vertices. Normalized correlation functions discard disconnected vacuum bubbles, while full correlators can still factor into disconnected components carrying external insertions.

For a real phi cubed theory with labelled external points, the connected three-point function at third order in the coupling constant has seven labelled one-loop Feynman diagrams: one triangle, three external-branch bubble insertions, and three tadpole-ended chains. The symmetry factors are respectively $1,2,2$, with graph weight $1/S$. To prove completeness, use $3V=2I+3$ and $L=I-V+1$: $V=3$ gives $I=3$ and $L=1$. The unique internal cycle is then a triangle, a pair of parallel edges, or a self-loop; the degree-three condition fixes the external attachments. Only the triangle remains one-particle irreducible after amputation of external propagators. A tadpole-free prescription removes the three chains but must be stated separately.

A connected Feynman diagram is one-particle reducible if cutting one internal propagator disconnects it. These diagrams occur in connected correlation functions but do not contribute to a quantum effective action defined by a Legendre transform. For a strict Wilsonian effective action, a bridge also cannot be a high-momentum propagator when the sum of external momenta on one side lies below the integrated shell.

In the six-dimensional cubic scalar field theory, use full external-propagator amputation and tadpole subtraction. The exchange part of the zero-momentum four-point amplitude isThe triangle corrects either end of each of three exchanges, and the symmetry factor of an internal bubble is $1/2$. Equivalently insert ${\Gamma }^{(3)}=g+g^3{I}_3$ and ${\Gamma }^{(2)}=m^2-g^2{I}_2/2$ into $3[{\Gamma }^{(3)}{]}^2/{\Gamma }^{(2)}$. Without tadpole subtraction, the shift $v=-g{I}_1/(2m^2)$ adds $3g^4{I}_1/(2m^6)$. None of these is an additional local 1PI quartic coupling; zero-momentum bridges are also excluded from a strict high-mode Wilsonian shell. If bare rather than full external propagators are amputated, external-line decorations additionally contribute $6g^4{I}_2/m^4$, and $6g^4{I}_1/m^6$ without tadpole subtraction. Every displayed term has mass dimension minus two. In a renormalized expansion, the exchange diagrams also receive the counterterm insertions $6g\delta g/m^2-3g^2\delta m^2/m^4$. Without enforcing a zero one-point function, a linear counterterm $\delta h\varphi$ contributes $3g^3\delta h/m^6$ through the background shift; choosing $\delta h=-g{I}_1/2$ cancels the attached tadpole. A kinetic counterterm is proportional to the exchanged momentum squared, so its insertion vanishes at this zero-momentum point.

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.

For a connected real-scalar Feynman diagram, degree counting and $L=I-V+1$ give $V_3+2V_4=E+2L-2$. A one-loop one-particle-irreducible Feynman diagram is a cycle: every vertex has two internal half-edges. For three labelled external legs, the possibilities are one cubic triangle with symmetry factor one and three mixed cubic-quartic bubbles with symmetry factor two. For four labelled external legs, there are three quartic bubbles with symmetry factor two, six mixed triangles with symmetry factor one, and three cubic boxes with symmetry factor one. Cyclic ordering modulo reversal explains the three boxes.

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.

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.

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.