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research30 min studyLesson 671 in Physics

Wick contractions and diagram symmetry factors

Derive scalar diagram coefficients from explicitly labeled contractions and distinguish combinatorics from coupling powers.

Before you begin
On this page

The question for today

What you’ll understand

  • Count contractions with fixed external labels.
  • Derive the one-vertex tadpole coefficient in quartic scalar theory.
  • Explain vacuum-bubble cancellation in normalized correlators.

A Feynman graph abbreviates many terms in a perturbative expansion. Its coefficient follows from counting those terms under a declared interaction normalization, rather than guessing from how symmetric the drawing looks. We use one real scalar field to make the counting visible. This module takes natural units, Minkowski signature (+,−,−,−), and the Feynman prescription throughout Minkowski expressions. Euclidean integrals will be identified explicitly. These conventions matter because combinatorial fractions, factors of i, and momentum integration measures perform different jobs.

State the interaction before counting

Choose Lint=−λϕ4/4!\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4! in four dimensions. The Dyson expansion provides one factor −iλ/4!-i\lambda/4! for each integrated interaction insertion and an additional inverse factorial for the number of insertions. Wick’s theorem replaces a free-vacuum time-ordered product by sums over pairings. Every scalar contraction is the Feynman two-point function. External insertions retain their labels; exchanging two distinct external points is not automatically an internal symmetry of a labeled graph.

For an even number of identical free scalar fields, the number of unrestricted pairings is a double factorial. That count changes when a required graph topology or connectedness condition is imposed. Normal ordering an interaction removes its internal same-vertex contractions and therefore changes which tadpoles are present; it must not be introduced silently after counting. Fermionic contractions additionally have permutation signs, whereas the examples here involve only a commuting real scalar field.

ΔF(x−y)=∫d4p(2π)4i e−ip⋅(x−y)p2−m2+i0,Npair(2n)=(2n−1)!!\Delta_F(x-y)=\int\frac{d^4p}{(2\pi)^4}\frac{i\,e^{-ip\cdot(x-y)}}{p^2-m^2+i0},\qquad N_{\mathrm{pair}}(2n)=(2n-1)!!
Mass and momentum have energy units in natural units. The contraction is a distribution, and coincident-point products require regularization. The pairing count applies before restricting topology or external connectivity.↔ Scroll wide equations horizontally.

A tadpole has a surviving factor one half

For a connected two-point function with one quartic insertion at z, choose one of its four fields to contract with the external point x. Choose one of the remaining three to contract with y. The last two vertex fields contract with one another. There are twelve contractions of this topology, and dividing by the interaction’s twenty-four gives one half. No extra permutation of the already labeled external points should be multiplied in a second time.

The surviving fraction is often described through a symmetry factor S equal to two, with the diagram weighted by its reciprocal. Some texts use the phrase symmetry factor for the reciprocal instead, so quote the actual coefficient when comparing conventions. Internal-line reversals and indistinguishable parallel lines can generate such factors, but a general rule must respect graph labels and vertex types. Counting directly in a small example is a reliable way to calibrate any later diagrammatic shortcut.

Gconn,1(2)(x,y)=−iλ2∫d4z ΔF(x−z)ΔF(0)ΔF(z−y)G^{(2)}_{\mathrm{conn},1}(x,y)=\frac{-i\lambda}{2}\int d^4z\,\Delta_F(x-z)\Delta_F(0)\Delta_F(z-y)
This is the formal first-order connected tadpole contribution for the stated non-normal-ordered interaction. The external propagators remain attached; amputation and ultraviolet regularization are additional operations.↔ Scroll wide equations horizontally.

Normalize away disconnected vacuum factors

A vacuum bubble can multiply a graph carrying external fields without changing its connections. In a normalized vacuum correlation function, the denominator contains the same vacuum diagrams. Their factor cancels order by order, leaving diagrams connected to the external insertions, including products of separate externally connected components when the requested correlator permits them. Taking the logarithm of a generating functional selects connected diagrams more generally. Neither step licenses discarding a disconnected contribution from an unnormalized object without explanation.

An internal loop introduces an undetermined momentum integral after vertex momentum conservation is used. It does not alone determine a graph’s coupling power or its symmetry coefficient. Coupling order is set by vertices, loop number by independent cycles, and symmetry by contraction multiplicity. The attached schematic scattering weight can illustrate a coupling power, but it has no Wick-contraction engine and cannot infer symmetry factors. Its lines and numerical weight must therefore be treated as a separate algebraic toy, rather than evidence for a physical process.

⟨Tϕ(x1)⋯ϕ(xn)⟩=⟨0∣TϕI(x1)⋯ϕI(xn)ei∫d4z Lint∣0⟩⟨0∣Tei∫d4z Lint∣0⟩\langle T\phi(x_1)\cdots\phi(x_n)\rangle=\frac{\langle0|T\phi_I(x_1)\cdots\phi_I(x_n)e^{i\int d^4z\,\mathcal L_{\mathrm{int}}}|0\rangle}{\langle0|Te^{i\int d^4z\,\mathcal L_{\mathrm{int}}}|0\rangle}
The free-vacuum perturbative expression assumes the usual adiabatic vacuum prescription. Numerator and denominator must be expanded consistently to the same order before canceling vacuum factors.↔ Scroll wide equations horizontally.

Work it through

Count the labeled tadpole

At first order in −λϕ4/4!-\lambda\phi^4/4!, count the connected two-point contractions with one same-vertex internal pair. Find their total coefficient relative to −iλ-i\lambda.

  1. Keep the two external fields labeled x and y. There are four choices for the field at z attached to x, followed by three choices for the one attached to y.

  2. The two remaining vertex fields form exactly one pair. Thus the requested topology has 4×3=124\times3=12 Wick contractions, each with the same scalar propagator product.

  3. The first-order Dyson term has no additional vertex-number factorial beyond one. Divide the contraction count by 4!=244!=24 to obtain 12/24=1/212/24=1/2, retaining the common factor −iλ-i\lambda.

The result

The connected tadpole carries coefficient −iλ/2, with the external propagators and one regularized coincident contraction still attached.

The living notebook

Explore the model

Interactive lab

Interactive lab · scattering

Coupling power is separate from combinatorics

Reset and set coupling g=0.5g=0.5, angle 90∘90^\circ; predict illustrative weight 1/161/16. Repeat from Reset with both controls g=1g=1, 90∘90^\circ; predict weight one. The lab uses only g4(1+cos⁡2θ)g^4(1+\cos^2\theta), a dimensionless schematic weight. It does not compute a cross section, Wick pairings, or a diagram symmetry coefficient.

Coupling power is separate from combinatoricsA schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Current readings: Coupling factor g⁴: 0.24; Angular factor 1 + cos²θ: 1.329; Illustrative weight: 0.319. All parameters are adjustable with the labeled controls below.Incoming termsOutgoing termsθSchematic interaction term · no particle trajectories
Incoming termsOutgoing termsAngle θ
Coupling factor g⁴
0.24
Angular factor 1 + cos²θ
1.329
Illustrative weight
0.319
t = 0 s

A schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA sixteenfold toy weight change tests a coupling power; it supplies no missing contraction count.

Pause & reason

Check your understanding

For the labeled one-vertex quartic tadpole, which coefficient multiplies the common factor −iλ?

Your turn

Try it yourself

At one quartic insertion with no external fields, count pairings of its four fields and find the coefficient relative to −iλ-i\lambda. Explain its role in a normalized correlator.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Choose a partner for one selected field; the remaining pair is then fixed. Keep the interaction factorial.

Compare with the solution

One chosen field has three possible partners, producing three complete pairings. The first-order vacuum graph therefore has coefficient 3/4!=1/83/4!=1/8 relative to −iλ-i\lambda, multiplying the vertex integral and two coincident propagators. When this disconnected vacuum factor multiplies an externally connected graph, the consistently expanded vacuum denominator cancels it. Its coefficient is meaningful before cancellation and must not be replaced by the tadpole’s one half.

Keep the idea

Take it with you

  • Diagram coefficients depend on interaction normalization and labeled contraction counting.
  • A reciprocal symmetry factor should be quoted as an actual numerical coefficient.
  • Vacuum factors cancel through normalization; loop count, coupling order, and combinatorics remain distinct.

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