Quantum Field Theory – LSZ Reduction Formula in Peskin and Schroeder: Disconnected Diagrams and Pole Structure

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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.

Best Answer

  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.

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