= Solution
Variation with respect to the <Dirac adjoint>, the Dirac field, and the real scalar field respectively gives
$$
\boxed{(i\gamma^\mu\partial_\mu-m-g\phi)\psi=0,\qquad i(\partial_\mu\bar\psi)\gamma^\mu+(m+g\phi)\bar\psi=0,\qquad(\Box+m^2)\phi=-g\bar\psi\psi.}
$$
The derivative in the adjoint equation acts to the left; its sign follows by integrating $i\bar\psi\gamma^\mu\partial_\mu\delta\psi$ by parts. The <Yukawa interaction> supplies a spacetime-dependent effective fermion mass and a scalar source.
Use $\not p=\gamma^\mu p_\mu$ and the convention $S_{fi}^{\rm conn}=i(2\pi)^4\delta^4(p_f-p_i)\mathcal M$. The momentum-space <Feynman rules> are: a scalar internal line contributes $i/(p^2-m^2+i0)$; an oriented fermion line contributes $i(\not p+m)/(p^2-m^2+i0)$; and each scalar-fermion vertex contributes $-ig$ times the identity in spinor space. Impose four-momentum conservation at each vertex. Incoming and outgoing fermions supply $u$ and $\bar u$, while incoming and outgoing antifermions supply $\bar v$ and $v$; scalar external legs supply one. Loop momenta are integrated with $d^4p/(2\pi)^4$, a closed fermion loop contributes a minus sign, and graph symmetry factors are included. Relative signs between distinct contractions of identical external fermions follow from their anticommutation relations. These specify the <Feynman rules> also beyond the tree approximation.
Label incoming momenta $p_1,p_2$ and outgoing momenta $p_3,p_4$, with all external particles on shell. Write the <Mandelstam variables> as $s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$, $u=(p_1-p_4)^2$, and abbreviate $D_r=r-m^2+i0$. Spin indices on $u_i,v_i$ are implicit. The six required <tree-level Feynman diagrams> are:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-41-yukawa-trees.png]
{title=The six Yukawa tree diagrams for fermion, antifermion and scalar scattering, with momentum labels and fermion-number arrows}
{height=960}
For two incoming fermions, the two diagrams exchange a scalar in the $t$ and $u$ channels. With external state ordering $b_1^\dagger b_2^\dagger|0\rangle$ and $b_3^\dagger b_4^\dagger|0\rangle$, the result is
$$
\boxed{\mathcal M_{\psi\psi}=-g^2\left[\frac{(\bar u_3u_1)(\bar u_4u_2)}{D_t}-\frac{(\bar u_4u_1)(\bar u_3u_2)}{D_u}\right].}
$$
Each scalar-exchange contraction has two factors $-ig$ and one scalar <Feynman propagator>. Exchanging the final fermions reverses the sign, as required by their identical-particle statistics.
For a fermion and antifermion, there is $t$-channel scalar exchange and $s$-channel annihilation. Take both initial and final states ordered as fermion creator followed by antifermion creator. Then
$$
\boxed{\mathcal M_{\psi\bar\psi}=g^2\left[\frac{(\bar u_3u_1)(\bar v_2v_4)}{D_t}-\frac{(\bar v_2u_1)(\bar u_3v_4)}{D_s}\right].}
$$
The relative minus is not optional. One way to track it is the <antifermion sign of a normal-ordered bilinear>: the antifermion scattering part of $:\bar\psi\psi:$ is $-d^\dagger d\,\bar vv$, whereas its annihilation part is $db\,\bar vu$. Thus the exchange contraction has the opposite fermionic sign to the annihilation contraction before multiplying by $(-ig)^2i/D_r$. A different overall phase convention for external states changes the common sign of this amplitude, but cannot change the relative sign.
For fermion-scalar scattering, the fermion can absorb the incoming scalar before emitting the outgoing one, or emit first and absorb afterwards. The internal momenta are $p_1+p_2$ and $p_1-p_4$ respectively, giving
$$
\boxed{\mathcal M_{\psi\phi}=-g^2\bar u_3\left[\frac{\not p_1+\not p_2+m}{D_s}+\frac{\not p_1-\not p_4+m}{D_u}\right]u_1.}
$$
Both orderings have the same sign: there is only one open fermion line and no exchange of identical external fermions. There is no scalar-exchange $t$ diagram because this <Yukawa interaction> has no three-scalar vertex. These amplitudes describe <tree scattering in Yukawa theory>.
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