In 'QCD and collider Physics' book by Ellis, Stirling and Webber, chapter 4.2 'The parton model from field theory' there is this equation for the amplitude of the process $e^- (k) P \rightarrow e^- (k^\prime) X$ (where $P$ stands for the proton and $X$ for an undetermined outgoing jet)
$$ A = e\bar{u}(k^\prime)\gamma^\alpha u(k)\frac{1}{q^2}\langle X| j_\alpha(0) |P\rangle \tag{1} $$
With $j_\alpha$ the electromagnetic current, but the book doesn't specify more. I've tried to get this formula via the scattering matrix $S$ in this way:
$$ S = \cal{T}\left( \exp\left[{i\int_{\mathbb{R}^4}d^4 x\ \cal{L}_{int}}\right] \right) \approx 1 + i\int_{\mathbb{R}^4}d^4 x\ \cal{L}_{int} - \frac{1}{2} \int_{\mathbb{R}^4}d^4 x d^4 y\ {\cal T}\left[{\cal L}_{int}(x){\cal L}_{int}(y)\right] \tag{2} $$ And
$$ {\cal L}_{int} = \sum_q\bar{q}e_q\gamma_\mu A^\mu q + \bar{\psi}_e e \gamma_\mu A^\mu \psi_e $$
Where $q$ is the quark, $\psi_e$ the electron and $A^\mu$ the photon fields; and $e_q$ the electric charge of quark $q$. Now the matrix element between intial and final state, considering $S$ till 1st order, is
$$ A = \langle e^- X| S | e^- P\rangle = i\sum_qe_q \langle e^- X| \int d^4 x \bar{q}\gamma_\mu A^\mu q | e^- P\rangle - ie\langle e^- X| \int d^4 x \bar{\psi}_e\gamma_\mu A^\mu \psi_e | e^- P\rangle \tag{3} $$
Since there is no photon in the final or initial states, $\langle 0_\gamma | A^\mu |0_\gamma \rangle = 0$. Therefore, you have to go to 2nd order in the $S$ expansion (last term in Eq. (2)).
This term is more complicated. First, you have to realise that due to the anticommutation relations of $q, \psi_e$ fields, you can write
$$ {\cal T}\left[ \bar{q}_x \gamma_\mu A^\mu_x q_x \bar{\psi}_{e, y}\gamma_\nu A^\nu_y \psi_{e, y}\right] = {\cal T}\left[ A^\mu_x A^\nu_y \right]\bar{q}_x \gamma_\mu q_x \bar{\psi}_{e, y}\gamma_\nu \psi_{e, y} $$
Where $q(x) = q_x, \psi_{e, y} = \psi_e(y)$.
Using this and after some calculations you end up with
$$ A = -ie\bar{u}_{k^\prime s^\prime}\gamma^\mu u_{ks}\frac{1}{(k - k^\prime)^2}\langle X| \sum_q e_q \int d^4 x\ e^{-i(k - k^\prime)x}\bar{q}_x \gamma_\mu q_x |P\rangle \tag{4}$$
This Eq. (4) is almost Eq. (1), but I don't see how to go on in order to get rid of the $-i$ factor and get an electromagnetic current evaluated at zero (I guess) momentum, $j_\mu (0)$. Any insights?
