Connected, Disconnected, and Vacuum Diagrams

Vacuum diagrams, connected correlators, and disconnected external processes are different combinatorial sectors. Source-independent vacuum bubbles exponentiate and cancel from normalized correlators. The logarithm of the normalized generating functional generates diagrams connected to all of their external insertions. Full correlators are then reconstructed as sums over set partitions into connected correlators. Disconnected scattering terms are not always “nothing”: they include the identity contribution and independent spectator or multiprocess components, which must be separated according to the observable.

Required background. Diagrammatics and Symmetry Factors supplies connected components and the factorials for repeated identical components.

Helpful background. Connected Correlators and Cumulants gives the primary moment–cumulant definition independently of perturbative graphs.

Vacuum bubbles cancel after normalization

Let

$$Z[J]=\int\mathcal D\phi\, \exp\!\left(iS[\phi]+i\int\mathrm d^d x\,J\phi\right).$$

A vacuum diagram has no source attachment and no external insertion. Every diagram contributing to $Z[J]$ decomposes uniquely into connected components, some source dependent and some vacuum only. If a connected vacuum graph $C_\alpha$ occurs $n_\alpha$ times, exchanging its identical copies contributes $n_\alpha!$ to the symmetry factor. Summing over every multiplicity therefore produces

$$\prod_\alpha \sum_{n_\alpha=0}^{\infty} \frac{C_\alpha^{n_\alpha}}{n_\alpha!} =\exp\!\left(\sum_\alpha C_\alpha\right).$$

Thus all vacuum diagrams exponentiate. Since they multiply every source-dependent diagram by the same factor, the normalized functional

$$\mathcal Z[J]\equiv\frac{Z[J]}{Z[0]}, \qquad \mathcal Z[0]=1,$$

contains no source-independent vacuum component. The cancellation of disconnected vacuum fluctuation subgraphs in the vacuum-amplitude ratio is stated explicitly in Weinberg 1995, § 9.3, p. 389, while the repeated-component factorial and exponentiation are derived in Srednicki 2007, § 9, p. 76.

This cancellation does not say that the vacuum persistence amplitude is intrinsically trivial. Its phase contributes to vacuum energy, and an imaginary part can diagnose instability in an appropriate background. Normalization removes that common factor from the correlator being computed.

The logarithm selects connected external diagrams

Define

$$W[J]=-i\log\mathcal Z[J], \qquad \mathcal Z[J]=e^{iW[J]}.$$

Connected time-ordered correlators are

$$G_c^{(n)}(x_1,\ldots,x_n) =\left. \frac{1}{i^{n-1}} \frac{\delta^n W[J]} {\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0}.$$

The logarithm removes products of independent source-dependent components just as the ordinary cumulant-generating function removes products of moments. Diagrammatically, every graph contributing to $G_c^{(n)}$ has one connected component containing all $n$ labeled external insertions.

For a centered field, the four-point relation is

$$\begin{aligned} G^{(4)}(1,2,3,4) &=G_c^{(4)}(1,2,3,4)\\ &\quad+G_c^{(2)}(1,2)G_c^{(2)}(3,4)\\ &\quad+G_c^{(2)}(1,3)G_c^{(2)}(2,4)\\ &\quad+G_c^{(2)}(1,4)G_c^{(2)}(2,3). \end{aligned}$$

The three products are disconnected with respect to the four external labels, yet none is a vacuum bubble: every component carries external insertions. They remain in the full four-point function. In a free Gaussian vacuum $G_c^{(4)}=0$; in an interacting theory it is generally nonzero.

The general partition formula

Let $\Pi_n$ be the set of partitions of $\{1,\ldots,n\}$. Then

$$\boxed{ G^{(n)}(x_1,\ldots,x_n) =\sum_{\pi\in\Pi_n} \prod_{B\in\pi}G_c^{(|B|)}(x_B) }.$$

Each block $B$ identifies the external labels belonging to one connected component. This is a structural identity between moments and cumulants, not a weak-coupling approximation. Perturbation theory supplies a graph expansion for each connected block.

The combinatorics can be checked before drawing graphs. If $W[J]$ has no one-point term, the fourth derivative of $e^{iW[J]}$ receives either one fourth derivative acting on the same exponent or two second derivatives acting on two copies. The latter assignments are the three unordered partitions $12|34$, $13|24$, and $14|23$; there is no extra $1/2$ because the exponential coefficient and the interchange of its two identical factors cancel.

If the one-point function is nonzero, singleton blocks contribute $G_c^{(1)}$. Shifting to a background with vanishing tadpole can simplify the partition formula, but that shift is a dynamical and renormalization choice rather than part of the definition of connectedness.

Connectedness in scattering

Write

$$S=1+iT.$$

The identity term describes no scattering. Matrix elements of $S$ between multiparticle states can also contain disconnected pieces in which some particles are spectators or two separated subgroups scatter independently. Their momentum delta distributions factor accordingly. A connected scattering amplitude is the coefficient with one overall momentum-conserving delta distribution after external poles are amputated.

This distinction is important in three settings:

• single hard scattering: one normally reports the connected amputated amplitude;
• spectator processes: disconnected delta factors encode unchanged external particles and must be treated consistently with state normalization; and
• multiple independent scatterings: products of connected components may contribute to an inclusive experimental event model, but their probabilistic combination is not supplied by deleting diagrams at amplitude level.

Ordinary LSZ acts on the full correlator. It converts each connected block into its corresponding connected scattering component when the external particles have stable isolated poles; it does not turn a disconnected partition into one connected interaction.

Three cancellation tests

Candidate component	Depends on sources or external insertions?	Survives $Z[J]/Z[0]$?	Appears in $W=-i\log\mathcal Z$?
isolated vacuum bubble	no	no	no
connected graph containing every external insertion	yes	yes	yes
product of two externally connected components	yes	yes	no, but reconstructed by exponentiating $W$
identity or spectator factor in an $S$-matrix element	through external states	yes	not one connected amplitude

These columns diagnose a common overcorrection: dividing by $Z[0]$ removes only source-independent vacuum factors; taking a logarithm then selects the single externally connected component.

Limits of the linked-cluster argument

The combinatorial exponential assumes that the regulated perturbative expansion and its products are defined. It does not establish convergence, the existence of the infinite-volume vacuum, or cluster decomposition at arbitrary separation. Nor does it remove infrared divergences: soft and collinear sectors can prevent ordinary charged-particle amplitudes from being finite even after vacuum normalization.

In an in–in or thermal contour, the normalization and connected generator are defined on that contour. Copying the in–out formula without its branch labels can cancel the wrong object. Similarly, backgrounds with vacuum decay require keeping track of the persistence amplitude before choosing normalized observables.

Common pitfalls

Calling every disconnected graph a vacuum bubble. A vacuum bubble has no external or source attachment. A product of two externally connected components is disconnected but survives normalized correlators.

Equating “connected” with “one-particle irreducible.” A connected graph can be separated by cutting one internal line. One-particle irreducibility is a stronger property used by the effective action and self-energy organization.

Dropping the identity contribution without stating the observable. $S=1+iT$ separates no-scattering from interaction. Cross sections, interference, and spectator normalization determine which pieces are relevant.

Check your understanding

Expand $e^{iW[J]}$ through fourth order for a centered field and recover the connected/disconnected decomposition without a diagram.

Solution

Write the source expansion schematically as

$$iW[J]=\frac{i^2}{2!}G_c^{(2)}J^2 +\frac{i^4}{4!}G_c^{(4)}J^4+O(J^6),$$

with spacetime integrations and labels suppressed. In $e^{iW}$, the $J^4$ coefficient comes from the connected fourth-order term and from $\tfrac12[(i^2/2!)G_c^{(2)}J^2]^2$. Four labeled derivatives assign the labels to the two quadratic factors in six ways; division by $2!$ for exchanging the factors leaves the three partitions. Hence $G^{(4)}=G_c^{(4)}+G_{c,12}^{(2)}G_{c,34}^{(2)}+G_{c,13}^{(2)}G_{c,24}^{(2)}+G_{c,14}^{(2)}G_{c,23}^{(2)}$. Since $\mathcal Z[0]=1$, no source-independent term appears.

Where to continue

• Generate the same decomposition by source differentiation: Functional Derivation of Perturbation Theory shows why the logarithm selects connected graphs.
• Amputate connected external poles: Asymptotic States, LSZ, and Scattering Observables separates identity, connected amplitudes, and state normalization.
• Study cluster decomposition as a structural statement: Foundations of Quantum Field Theory supplies the nonperturbative assumptions beyond this graph-combinatorial argument.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge: Cambridge University Press, 1995. First edition; 2005 paperback, 2012 printing consulted. DOI.