4-momentum of meson in nucleon scattering Consider the nucleon scattering in scalar Yukawa theory.
Suppose that we are NOT using Feynman diagram (rules) and instead use the tedious Dyson- Wick more formal method.
How do we establish or derive this relationship between 4-momentum of meson and 4-momentum of nucleon:
$$
k=p_{1}-p_{1}^\prime
\tag1
$$
In particular why not 
$$
k=p_{1}+p_{2}-p_{1}^\prime-p_{2}^\prime
\tag2
$$
 A: I don't know if this answer will satisfy you completely since I'm not familiar with the Dyson-Wick formalism.  But if I have scattering between two nucleons, with initial momenta $p_1, p_2$ and final momenta $p_1', p_2'$, and no additional particles in the final state, the conservation of momentum gives me
\begin{align}
\text{initial momentum } p &= p_1 + p_2
\\
\text{final momentum } p' &= p_1' + p_2'
\\
p-p' &= (p_1 + p_2) - (p_1' + p_2') \equiv 0.
\end{align}
So your proposed definition for $k$ vanishes identically.
Because momentum is conserved in elastic scattering, we can determine the momentum transferred between the two nucleons by looking at either of them in both the initial and the final state.
What characterizes the interaction, therefore, is the change in the momentum of either particle:
$$
k \equiv p_1'- p_1 = -(p_2'- p_2)
$$
The Feynman diagram suggests very strongly that we should assign this momentum to some virtual particle.  In a scattering-matrix approach, we declare our ignorance of what's happening at the scattering vertex and look for some operator that transforms our initial momenta $(p_1,p_2)$ to our final momenta $(p_1', p_2')$.
The only parameter that's available to characterize this matrix is the momentum transfer $k$.
If the matrix is associated with some scalar field, then $k$ must be the four-momentum associated with that field.
