LSZ reduction formula in Peskin and Schroeder: How do we see that disconnected diagrams have incorrect pole structure?

Question

In derivation of the LSZ reduction formula in Peskin and Schroeder, on page 227, the book says

Let us analyze the relation between the diagrammatic expansion of the scalar field four-point function and the $S$-matrix element for the 2-particle $\rightarrow$ 2-particle scattering. We will consider explicitly the fully connected Feynman diagrams contributing to the correlator. By a similar analysis, it is easy to confirm that disconnected diagrams should be disregarded because they do not have the singularity structure, with a product of four poles, indicated on the right-hand side of (7.42).

$$\tag{7.42}\prod_i^n \int d^4x_i e^{ip_i\cdot x_i}\prod_1^m\int d^4y_j e^{-ik_j\cdot y_j} \langle \Omega|T\{\phi(x_1)...\phi(x_n)\phi(y_1)...\phi(y_m)\}|\Omega\rangle\thicksim \bigg(\prod_{i=1}^n\frac{\sqrt{Z}i}{p_i^2-m^2+i\epsilon}\bigg)\bigg(\prod_{j=1}^n\frac{\sqrt{Z}i}{k_j^2-m^2+i\epsilon}\bigg)\langle\boldsymbol{p}_1...\boldsymbol{p}_n|S|\boldsymbol{k}_1...\boldsymbol{k}_m\rangle.$$

My question is: How do we see that disconnected diagrams have incorrect pole structure? If a diagram is disconnected, its value would be the product of its disconnected pieces, which I think should give the correct pole structure.

Answer 1

1. Closely related is the fact that in 4D momentum space, a Feynman diagram/correlation function with $r$ connected components contains $r$ 4D momentum Dirac delta distributions, due to spacetime translation symmetry.

   Hence in the space $\mathbb{R}^{4(m+n)}$ of $m$ incoming and $n$ outgoing off-shell 4D momenta, the support of a Feynman diagram with $r$ connected components is $4(m+n-r)$-dimensional.

   Therefore for external momenta that satisfy total momentum conservation, disconnected diagrams contribute almost nowhere relatively speaking as compared to connected diagrams.

2. Moreover, if a connected component has only 2 legs, i.e. is a connected propagator, its singularity structure is subdominant compared to the RHS of eq. (7.42). See also e.g. this related Phys.SE post.

   In particular for $2\to 2$ scattering, all the disconnected diagrams consist of 2 connected propagators, and hence have a subdominant singularity structure.