I am trying to reproduce the equivalent photon approximation as discussed in chapter 17.5 in Peskin and Schroeder but cannot justify equation (17.93).
The process we are considering is the scattering of an electron off of an unpolarized target $e^-X \rightarrow e^-Y$ by exchanging a single photon. The matrix element for this process is \begin{equation} \label{eq:melement} \mathcal{M}=(-ie)\bar{u}(k)\gamma^\mu u(p) \frac{-ig_{\mu\nu}}{q^2}\tilde{\mathcal{M}^\nu}(q), \end{equation} where $\tilde{\mathcal{M}^\nu}(q)$ describes the scattering of $X$ into $Y$ and its interaction with the photon.
If $p$ and $k$ are collinear, then $q\approx 0$ and one can replace the metric in the photon propagator to leading order by a sum over the transverse polarizations $g^{\mu\nu}\rightarrow - (\epsilon^\mu_R\epsilon^\nu_L+\epsilon^\nu_R\epsilon^\mu_L)$.
P&S go on and compute first the contribution from the first interaction vertex between states of definite helicity that I will call $\mathcal{M}'$. One finds \begin{equation} \mathcal{M'}_{LL}=\bar{u}_L(k)(-ie \gamma^\mu)u_L(p)\epsilon^\mu_L=ie\frac{\sqrt{2(1-z)}}{z}p_{\perp}\qquad \mathcal{M'}_{LR}=\bar{u}_L(k)(-ie \gamma^\mu)u_L(p)\epsilon^\mu_R=ie\frac{\sqrt{2(1-z)}}{z(1-z)}p_{\perp}\, \end{equation} where $(1-z)$ is the ratio of energies between initial and final electron and $p_\perp$ is the deviation from collinearity of the electrons. Squaring and averaging over initial polarizations and using the fact that parity invariance implies that these expressions do not change when flipping the polarizations, one finds \begin{equation} \frac{1}{2}\sum_{pols}|\mathcal{M}'|^2=\frac{2e^2p^2_\perp}{z(1-z)}\left(\frac{1+(1-z)^2}{z}\right). \end{equation}
So far I agree with everything. Now, P&S go back to the original matrix element $\mathcal{M}$ and state that computing the square and averaging over initial polarizations yields equation (17.93) \begin{equation} |\mathcal{M}|^2=\left[\frac{1}{2}\sum_{pols}|\mathcal{M}'|^2\right]\left(\frac{1}{q^2}\right)^2 |\tilde{\mathcal{M}}|^2. \end{equation}
This is the step I do now understand. In computing $\frac{1}{2}\sum_{pols}|\mathcal{M}'|^2$ we first squared the individual matrix elements and then summed over polarizations. However, in computing $|\mathcal{M}|^2$ what we have is rather \begin{equation} \frac{1}{2}\sum_{pols}|\mathcal{M}|^2=e^2\sum_i(\bar{u}_i(k)\gamma_\mu(\epsilon^\mu_R+\epsilon^\mu_L) u_i(p))^*\,(\bar{u}_i(k)\gamma_\mu(\epsilon^\mu_R+\epsilon^\mu_L) u_i(p))|\,\frac{\tilde{\mathcal{M}}|^2}{q^4}. \end{equation}
In particular, there are now cross terms $\mathcal{M'}_{LL} \mathcal{M'}^*_{LR}$ that do not seem to be there in the result by P&S. Are they using an approximation here, or is there something else I'm missing?
