Question about interacting fields

[creepypasta13]

I had some questions about the equations that David Tong derives in his lecture notes here:
http://www.damtp.cam.ac.uk/user/tong/qft/three.pdf

He gets defines the time evolution operator according to equations 3.20 and 3.23 as
U(t, t_{0}) = T*exp(-i* \int[H_{I}(t') * dt']) = 1 - i*\int[dt' * H_{I}(t')] + ...

According to eq 3.26, in the limit as t approaches +/- infinity, U is the same as the S-matrix:

lim <f|U(t_{+}, t_{-})|i> = <f|S|i>

but according to eq3.25, the Hamiltonian for the Yukawa theory is:
H_{I} = g * \int[d^{3}x * \psi^{dagger} \psi \varphi]

But according to the series expansion formula for U above, and plugging in the Hamiltonian into it, the series expansion for U should be:

U(t, t_{O}) = 1 - i*\int[d^{4}x * g*\psi^{dagger} \psi \varphi] + ...

I see that the leading term in g just has the \psi^{dagger} \psi \varphi in it. But in eq 3.46, he calculates <f| S-1| i>, which is SECOND order in g^2. Now it contains two terms each of \psi^{dagger} \psi \varphi

My question is, when he says 'S-1', does he mean "S without the 1st order term" ? Or do he mean "S without the number '1' "? If the former, that would make sense. But the latter makes no sense at all

 

[creepypasta13]

I figured it out. We have the -1 because it corresponds to when |i> = |f>, which gives <i|f>=1, which we want to exclude