"Zero overlap" of initial and final states in meson to nucleon + antinucleon scattering of scalar Yukawa

I’m currently studying QFT from David Tong’s lecture notes and video lectures. In meson to nucleon + antinucleon decay (section 3.2.1 in
this ) in scalar Yukawa theory to order $g$, without using Feynman diagrams, he talks about the term in the mode expansion with a meson creation operator $a_vec{p}^{dagger}$ not contributing to the scattering amplitude because the two-meson state has no overlap with the final state.

I think I get what this means on a physical level, obviously we don’t want any mesons in our final state, but I don’t see how to get the contribution to disappear mathematically. What does “zero overlap” mean? I think I’ve seen that it means $leftlangle a|b rightrangle = 0$ for whatever states you’re looking at, but it seems to me that at order $g$ this term will look like $leftlangle f|int d^4x; psi^{dagger}(x)psi (x) a^{dagger} a^{dagger}|0 rightrangle$ (with some factors that I think should be irrelevant suppressed) and I don’t see why this vanishes. Can anyone show me explicitly how this occurs?