# Why does this amplitude not factorise into subamplitudes?

Consider the process $$Xq\rightarrow Yq$$ at tree level via exchange of a photon. It is depicted in the following Feynman diagram. In various literature it is said that for a nearly on-shell photon this amplitude factorises into the product of two amplitudes of the subprocesses $$X\gamma\rightarrow Y$$, $$q\rightarrow q\gamma$$ and the denominator squared of the photon propagator. But there is one subtlety that I do not understand about this.

The tensorial structure admits to writing the matrix element as the product $$M_\mu\Delta^{\mu\nu}N_\nu=M_\mu(-g^{\mu\nu})N_\nu\frac{i}{k^2}$$. The polarisation sum of massless vector bosons is $$\sum_i\epsilon^\mu_i\epsilon^{\nu\ast}_i=-g^{\mu\nu}+\epsilon^\mu_+\epsilon^{\nu\ast}_-+\epsilon^\mu_-\epsilon^{\nu\ast}_+$$, which allows us to substitute the metric for this sum. If the photon is nearly on-shell, the terms with $$\epsilon_\pm$$ nearly vanish, leading to the expression $$\sum_iM_\mu\epsilon^\mu_i\epsilon^{\nu\ast}_iN_\nu\frac{i}{k^2}$$. Now this almost looks like the product of two distinct matrix elements, except there is the sum over $$i$$, which couples both factors. If we calculate the amplitude and define $$M_{\mu\nu}:=M^\dagger_\mu M_\nu$$, $$N_{\alpha\beta}:=N^\dagger_\alpha N_\beta$$, we find that $$|M|^2_{Xq\rightarrow Yq}=\frac{1}{k^4}\sum_{i,j}M_{\mu\nu}\epsilon^{\mu\ast}_i\epsilon^\nu_j\times N_{\alpha\beta}\epsilon^\alpha_i\epsilon^{\beta\ast}_j.$$ But this is not necessarily equal to the product $$\frac{1}{k^4}|M|^2_{X\gamma\rightarrow Y}\times|M|^2_{q\rightarrow q\gamma}=\frac{1}{k^4}\sum_iM_{\mu\nu}\epsilon^{\mu\ast}_i\epsilon^\nu_i\times\sum_jN_{\alpha\beta}\epsilon^\alpha_j\epsilon^{\beta\ast}_j$$ that you find in literature (e.g. Peskin & Schroeder pp.578).

I suppose there must be a profound error in my thinking somewhere but I just cannot see where. I hope somebody can explain.

$$\lim_{P^2\rightarrow0} M(1,\cdots,n) = \sum_{ i\in \text{helicities}} M(1,\cdots,k,P^i)\frac{i}{P^2} M(P^{-i}, k+1,\cdots,n)\,,$$