Feynman diagram

A Feynman diagram records a term in the perturbative expansion: edges represent propagators, vertices represent interactions, and loops represent unconstrained momentum integrals.

A Feynman integral is the momentum-space integral associated with a Feynman diagram: each independent loop momentum is integrated with measure $d^{d}k/(2\pi {)}^d$, and propagators, vertex numerators and momentum conservation determine the integrand. Its definition also requires a pole prescription and, when divergent, a regularization in quantum field theory. Feynman parameters combine propagator denominators and permit shifts of loop momenta. A massless two-propagator example is $\int d^{d}k/[(2\pi {)}^d{k}^2(p-k{)}^2]$, with its causal prescriptions understood. Subtracting local counterterms turns a regulated diagram into a finite contribution to a renormalized amplitude.

The arrow on a charged-fermion line in a Feynman diagram tracks fermion-number flow and fixes the order of the gamma matrices between external Dirac spinors. For an external particle it agrees with the particle direction; for an external antiparticle it is opposite to the antiparticle direction. It therefore need not point along the assigned four-momentum of every external leg.

A box diagram has a one-loop cycle of four propagator edges and four vertices connecting external legs. Its internal particle types depend on the theory. In a charged weak box contribution to kaon mixing, two charged weak-boson propagators connect two up-type quark lines, with flavor-changing charged-current vertices.

A closed chain of Dirac propagators is a fermion loop. Its closed fermion loop sign distinguishes it from a bosonic loop. Tracing spinor indices produces gamma matrix traces, and the high-momentum powers of the propagators determine possible ultraviolet divergences.

Three scalar Yukawa interactions on a closed Dirac field loop give two cyclic orientations. Odd gamma matrix traces vanish, while $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$ gives the displayed numerator. For external invariant mass squares $M^2,{\mu }^2,{\mu }^2$, write $D_a=a^2-m^2$ and similarly for the other two lines. The numerator becomes $4m[D_a+D_b+D_c+4m^2-{\mu }^2-M^2/2]$, reducing the fermion loop to three scalar bubbles and a scalar triangle Feynman integral. The closed fermion loop sign must be included. A massive fermion produces a logarithmic local divergence.

For interactions $-G\bar{\psi }\psi \Phi -g\bar{\psi }\psi \varphi$, the two Yukawa fermion triangle orientations generate a local divergent three-scalar vertex proportional to $mG{g}^2$. In $d=4-2ϵ$, with $S-1=i\mathcal{M}$ convention, its loop pole is ${\mathcal{M}}_{\mathrm{div}}=-24mG{g}^2/(16{\pi }^2\overline{ϵ})$. A $\Phi {\varphi }^2$ counterterm cancels it. The finite renormalized coefficient is additional data, fixed by a renormalization condition. Declaring this cubic absent from the displayed bare interaction is not by itself a prescription for its loop-generated finite value.

Four scalar Yukawa interactions on a closed fermion loop generate a scalar four-point one-particle-irreducible vertex. At high momentum four dirac propagators contribute $k^{-4}$, leaving a logarithmic four-dimensional ultraviolet divergence. Its local scalar-fourth-power term requires a quartic scalar coupling; scalar mass and field counterterms cannot cancel it.

A closed fermion loop carries an additional minus sign from the odd Grassmann parity of the fields in the Wick theorem contraction expansion. Equivalently, integrating out Grassmann fields gives a determinant, whose logarithmic interaction expansion has the fermionic sign relative to a bosonic inverse determinant. This sign must be included in addition to propagator and vertex phases.

A Feynman diagram with more than one connected component. Products of component contributions exponentiate; taking the logarithm of a Gaussian perturbation expansion removes disconnected products and keeps connected Feynman diagrams.

A triangle Feynman diagram has a loop of three propagators joining three vertices. In six-dimensional cubic scalar field theory it gives a one-loop three-point proper vertex. Attaching a cubic tree vertex makes a four-point connected diagram, but the connecting propagator is a bridge, so the latter is a one-particle-reducible Feynman diagram.

In mostly-minus signature, take $D_j=(\ell +r_j{)}^2-m^2+i0$. Feynman parameters and a momentum shift giveFor massive internal lines this is ultraviolet finite in four dimensions. Physical external kinematics can still create threshold cuts, encoded by the Feynman i-epsilon prescription.

A supergraph is a Feynman diagram formulated with superfield propagators and vertices in superspace. Performing its spinor-derivative algebra organizes perturbative corrections into superspace integration terms and provides a proof of the local superpotential non-renormalization theorem.

A one-loop Feynman diagram with four vertices joined into a box. Two W bosons and two internal quarks give a short-distance contribution to neutral D-meson mixing.

An interaction vertex joins the field insertions from one interaction term in a Feynman diagram. Its number of incident lines is the number of fields in that term, and its momentum-space Feynman rule includes the interaction coefficient and momentum conservation.

A squared-amplitude diagram consists of an amplitude graph and its complex-conjugate graph, with matching external states. Summing an external fermion spin sews the associated spinor indices using a fermion spin sum. The resulting closed spinor chains become gamma-matrix traces. This notation represents probabilities and must be distinguished from adding another physical interaction loop to the amplitude.

Feynman rules translate each diagram into momentum-space factors for propagators, vertices, external states, momentum conservation and loop integration.

With interaction density $\mu {\varphi }^3/3!-\lambda {\varphi }^4/4!$, the Dyson series contains $i{\mathcal{L}}_{\mathrm{int}}$ and hence cubic vertex $i\mu$ and quartic vertex $-i\lambda$. Wick contractions into the labelled slots cancel each factorial. Position integration gives the four-momentum delta function, and equivalent unlabelled contractions leave the Feynman-diagram symmetry factor. The free line carries $i/(p^2-m^2+i0)$ in the mostly-minus convention.

An interaction ${\mathcal{L}}_{\mathrm{int}}=+\lambda {\varphi }^4/4!$ produces a four-leg momentum-space vertex $i\lambda$. The $4!$ assignments of external legs cancel the coefficient factorial. With a negative interaction sign the vertex is $-i\lambda$. Graph Feynman-diagram symmetry factors are a separate combinatorial issue.

The symmetry factor of a Feynman diagram divides out permutations of indistinguishable internal lines and vertices that leave the labeled diagram unchanged. It is the order of the corresponding automorphism group after external labels are fixed.

A Feynman parameter combines propagator denominators; for example,After a momentum shift, this often converts a loop integral into a rotationally symmetric one.

For $\mathrm{Re}A>0$ and $\mathrm{Re}v>0$, rescaling the defining Gamma function integral proves this representation, with compatible complex-power branches. It turns denominator powers into exponential factors, making Gaussian integrals possible. Combining two parameters by their sum and ratio gives the usual Feynman parameter identity. Outside its convergence domain a further continuation or pole prescription is needed.

A vacuum Feynman diagram has no external lines and contributes a source-independent factor to the partition function. Taking the logarithm retains only connected vacuum diagrams, while normalizing $Z[0]=1$ removes all of them.

A perturbative correlator factors into an exponential of connected Vacuum Feynman diagrams times a sum of Feynman diagrams attached to external insertions. Dividing its path integral by the vacuum generating functional at zero source cancels that exponential, including all disconnected products of vacuum components. The corresponding operator formalism uses the same denominator in the vacuum expectation of the Dyson series. This leaves externally disconnected correlators intact; the connected generating functional removes their products.

The vacuum energy density is the bulk limit ${\rho }_{\mathrm{vac}}={\lim }_{\mathcal{V}\to \infty }{E}_0(\mathcal{V})/\mathcal{V}$, with boundary effects removed and a fixed convention for the Hamiltonian. In an effective field theory at a homogeneous vacuum it is the value of the renormalized scalar potential. A global supersymmetry algebra expresses the regulated vacuum energy as a sum of supercharge norms, giving nonnegative density; the scalar potential of supergravity instead includes a negative gravitational term.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.

The linked-cluster theorem states that the logarithm of a vacuum generating functional is the sum of connected vacuum diagrams. Exponentiating that sum reconstructs arbitrary products of disconnected connected components with their correct factorials.

A tadpole diagram contains an internal line that begins and ends at the same interaction vertex. In a scalar theory it commonly gives a momentum-independent correction to a mass or lower-order coupling.

Use $-g{\varphi }^3/3!$ and $D=6-2ϵ$ with $m>0$. In an unshifted vacuum, the one-loop mean field has pole $v_{\mathrm{div}}=-g{m}^2/[4(4\pi {)}^3ϵ]$. Attaching it to a cubic vertex gives the connected but one-particle-reducible two-point insertion $(-ig)v_{\mathrm{div}}=i{g}^2{m}^2/[4(4\pi {)}^3ϵ]$. It supplements the proper bubble unless a tadpole subtraction sets the renormalized mean field to zero. A disconnected product of two one-point functions is a separate two-loop term and is not a propagator insertion.

// Target: relativistic-quantum-field.bigb

A linear counterterm can be chosen so that the renormalized one-point function, or equivalently the chosen vacuum expectation value, is zero. This cancels attached tadpole diagrams in the perturbative expansion about that vacuum. In a massive six-dimensional cubic scalar field theory, the one-loop proper tadpole is $t=g{I}_1/2$; without subtraction the stationary background shifts by $v=-t/m^2+O(g^3)$ at one-loop order. One must specify this convention when comparing reducible exchange diagrams or mass corrections.

A bubble diagram has two interaction vertices joined by two internal propagators to form one loop. In scalar quartic theory it gives the one-loop correction to the four-point coupling.

The loop momentum integral with two massive quantum field theory propagators isIts momentum dependence gives a one-loop self-energy in cubic scalar field theory and a quartic effective vertex in quartic scalar field theory.

For one real phi-fourth theory field, each of the three labeled one-loop four-point channels has symmetry factor $1/2$. In $d=4-2ϵ$, the displayed expression is its amplitude through the finite part. The three poles require $\delta \lambda =3{\lambda }^2/(32{\pi }^2ϵ)$, and differentiating ${\lambda }_B={\mu }^{2ϵ}(\lambda +\delta \lambda )$ at fixed bare coupling gives $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$ in four dimensions. The minimal subtraction scheme retains $\log (4\pi )-{\gamma }_E$; modified minimal subtraction absorbs that constant. The Feynman-parameter logarithm is left unevaluated.

With a mostly-plus metric signature, define $f_d=(1/i)\int d^{d}k/[(2\pi {)}^d(k^2-iϵ)((P-k{)}^2-iϵ)]$. Wick rotation and a Feynman parameter give $f_d=\Gamma (2-d/2)(4\pi {)}^{-d/2}{\int }_0^1[x(1-x)P^2{]}^{d/2-2}dx$. At nonzero Euclidean $P^2$, the Gamma function supplies the pole. Exceptional massless momentum can introduce infrared ambiguity.

A sunset diagram is a two-point diagram with two interaction vertices joined by three internal propagators. Its dependence on the external momentum produces wave-function renormalization in scalar quartic theory.

A tree-level diagram has no loops and gives the leading classical contribution allowed by the interaction vertices.

For three real scalar fields with field interaction term ${\mathcal{L}}_I=-\frac{1}{2}(g_1{\varphi }^2\Phi +g_2{\psi }^2\Phi )$ and mediator mass $M$, the connected tree-level Feynman diagram for $\varphi \psi \to \varphi \psi$ has one exchanged $\Phi$ and Mandelstam invariant $t=(p-p^{\prime }{)}^2$. Each vertex has a factor $-i{g}_j$: its two identical-field Wick contractions cancel the factor $1/2$. Thus $i\mathcal{M}=(-i{g}_1)(-i{g}_2)i/(t-M^2+i0)$. There is no $s$- or $u$-channel at this order because the interaction has no mixed $\varphi \psi \Phi$ vertex. Distinguishable external particles also require no exchange symmetrization.

The loop order of a connected Feynman diagram is the number $L=I-V+1$ of independent momentum cycles, where $I$ is the number of internal lines and $V$ the number of vertices. Each loop introduces an unconstrained momentum integral.

For a connected Feynman diagram with external valence $n$ and cubic, quartic and sextic vertices, counting half-edges gives $2I+n=3V_3+4V_4+6V_6$. Using $L=I-(V_3+V_4+V_6)+1$ yields $L-1+n/2=V_3/2+V_4+2V_6$. Since each summand lies between half and twice its vertex count, the displayed bounds follow. More generally vertices of valence between $r_{\min }$ and $r_{\max }$ give $(r_{\min }/2-1)V\le L-1+n/2\le (r_{\max }/2-1)V$.

In a scalar field path integral with weight $\exp [i(S+\int J\varphi )/\hslash ]$, each internal propagator contributes $\hslash$, each interaction vertex contributes ${\hslash }^{-1}$, and each source insertion contributes ${\hslash }^{-1}$. A connected graph with $I$ internal lines and $V$ interaction vertices has loop order $L=I-V+1$. Its amputated amplitude, or its coefficient in $\log Z[J]$ with the source insertions included, is proportional to ${\hslash }^{I-V}={\hslash }^{L-1}$. An ordinary unamputated connected $n$-field correlation function instead has the additional $n$ external propagator factors and scales as ${\hslash }^{L+n-1}$. Multiplication by $-i\hslash$ to form the connected generating functional shifts the source-kernel counting to ${\hslash }^L$. These are compatible normalization conventions.

A loop momentum is an independent momentum integrated over in a Feynman diagram. Momentum conservation leaves one independent loop momentum for each cycle counted by the loop order.

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.

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.

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.