# LSZ reduction formula in Peskin and Schroeder

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.

• Those disconnected diagrams cancel out with the denominator of the cross section when doing things with the path integral formulation. It's been a while though so it's hard for me to be more specific. But you can factor them out in the numerator in a clever way and it's actually quite remarkable. Oct 19, 2022 at 18:14

Closely related is the fact that in 4D momentum space, a Feynman diagram/correlation function with $$n$$ connected components contains $$n$$ 4D momentum Dirac delta functions, due to spacetime translation symmetry.
• $\uparrow$ Yes. Oct 23, 2022 at 2:02