Wick's theorem contributions to nucleon scattering

Question

I am following David Tong's notes on QFT. On page 58, he applies Wick's theorem to $\psi\psi\rightarrow\psi\psi$ scattering for a scalar field with a Yukwara interaction term. The claim is, that by Wick's theorem, the only contribution to $$\int d^4x_1d^4x_2 T[\psi^\dagger(x_1)\psi(x_1)\phi(x_1)\psi^\dagger(x_2)\psi(x_2)\phi(x_2)]\tag{3.46}$$ is $$:\psi^\dagger(x_1)\psi(x_1)\psi^\dagger(x_2)\psi(x_2):\Delta_F(x_1,x_2).\tag{3.47}$$ That is, contracting on the two $\phi$ terms. However, I do not see why we don't get a contribution by contracting on a $\psi^\dagger$ and $\psi$, i.e, a contribution from

$$:\psi^\dagger(x_1)\psi(x_2):\Delta_F(x_1, x_2)\Delta_F(x_1,x_2).$$

when the one of incoming and outgoing momenta is the same. That is to say, if the initial state is $|pq>$ and the final state is $<pq|$, why do we not get a further second order contribution.

Indeed, if I try it, I get

$$<pq|:\psi^\dagger(x_1)\psi(x_2):|pq>$$ $$<0|2\sqrt{E_pE_q}b_{p}b_{q}^\dagger e^{i(-q.x_1+p.x_2)}|0>,$$ so

$$<pq|:\psi^\dagger(x_1)\psi(x_2)\Delta_F(x_1, x_2)\Delta_F(x_1,x_2):|pq>$$ $$=<p|\int \frac{d^4k}{2\pi^2} \frac{\delta(p-q)}{(k^2-m^2)((p-k)^2-m^2)}|q>,$$

which certainly gives a non-zero contribution when $p=q$.

Answer 1

Remmeber that if you contract two scalar fields you get a scalar propagator $\Delta_S(k)$. If you contract two fermionic fields, you get the fermion propagator $\Delta_F(k)=(\gamma^\mu k_\mu-m)\Delta_S(k)$. Upload your notation! On the ohter hand, the interaction process $\psi\psi\rightarrow \psi\psi$ requires the anihilation of two fermions and the creation of two fermions, so you need four fermionic fields uncontracted in terms that contribute to this process.

The term you mention is different since it has a loop on it, it is similar to the Fermion-Self energy diagram of QED, but instead of a photon line you have a scalar one. So the term you mention does not contribute to the process $\psi\psi\rightarrow \psi\psi$, instead they are corrections to the fermion propagator.