# Zero interference between disconnected and connected diagrams in $2\to 4$ scattering in $\phi^6$ theory

In Schwartz's book Quantum Field Theory and the Standard Model Exercise 7.2 is to show that in $$2\to 4$$ scattering there is no interference between 6-point vertex diagrams and the disconnected diagrams with two three point vertex.

When I solve for amplitudes I get a delta function of the type $$\delta^4(p_1+p_2-p_3-p_4-p_5-p_6)$$ for the 6 point vertex and the following for the two 3 point vertex diagrams: $$\delta^4(p_1-p_3-p_4) \delta^4(p_2-p_5-p_6).$$ How to show that these lead to zero interference?

To paraphrase, the idea is that with a total $$S$$-matrix of $$S=-i\lambda \ \delta^4(\sum_{\text{total}} p) + -ig\delta^4(\sum_{\text{subset}} p)\delta^4(\sum_{\text{subset}} p),$$ the second term is only non-zero on a subset of the phase space with measure zero. More precisely the first term is non-zero on a 9-dimensional subset of the phase space while the latter only on a 6-dimensional subset (for 2->2 scattering). Thus for almost all points in the final phase space only the first term contributes.
In the (measure zero) regions where the second term is non-zero however, the double delta function translates to a factor of $$(VT)^2$$ whereas the first term is only proportional to $$VT$$. Thus whenever the second term contributes it is infinitely larger than the first term which can be ignored.
What this all means is that in the limit of large $$T$$ and $$V$$, only one term in $$S$$ ever matters and so there is never any interference. In practice what this says is that we only need to consider the fully connected diagram except in regions of the phase space where a disconnected process is allowed, in which case only the disconnected process contributes.