# Cutkosky's rule for vertex correction

I don't understand how to evaluate the diagram for electron-photon vertex with help of Cutosky's rule: According to the rule, one should cut diagram "in the middle". I denote final state as F and initial state as I, "middle" as N. Then, one should write down two expressions: $$i\mathcal{T}_{NI};\quad i\mathcal{T}_{NF},$$ but I don't understand how to write down these expressions.

I have read this paper (page 9) where vertex diagram is calculated but it is unclear for me how the author finds $$i\mathcal{T}_{NI}$$ and $$i\mathcal{T}_{NF}$$. Let me try it. I start from and write down $$i\mathcal{T}_{NI}=\bar{u}(k+q)(-ie\gamma^{\mu})v(-k),$$ which I just rewrite from the paper. But is it right or not?... It seems right. Then, for $$i\mathcal{T}_{NF}$$: $$i\mathcal{T}_{NF}=-\bar{u}(k+q)(-ie\gamma^{\mu})u(p')\bar{v}(-p)(-ie\gamma^{\mu})v(k)D_{\mu\nu}(p-k),$$ where $$D_{\mu\nu}(p-k)$$ is photon propagator.

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. Jan 16 '19 at 10:14

The Expression for $$iT_{NI}$$ is Right: You have a vertex (factor $$-ie \Gamma^\mu$$) and two Fermions that ending on the red line (These give factors $$\bar u (k+q)$$ and $$v(-k)$$).

In the Expression for $$iT_{NF}$$ you are also on the Right way. There are two Vertices and four external Fermions. Moreover, there is one Photon Propagator that is $$D_{\mu \nu}(p-k)$$. Energy-Momentum conservation implies:

$$p - k = k' - p' := r$$

Adding both expressions for the Photon Momentum $$r$$:

$$2r = (p-p') - (k-k') = (p-p') - q$$.

The relative 4-momentum for the Fermions is $$s = (p-p')/2$$ and thus

$$r =s-q/2$$.

Note that in the paper there is due to Energy-Momentum conservation(I simply relabel the $$p$$ in paper as the $$q$$ and the $$q$$ in paper as the $$r$$, etc.)

$$-q = k'-k = (p'-r)-(p+r) = p'-p + 2r$$

which is true.

So all in all you have correct results, only the 4-momenta have different labels than in the paper. Note that a massive Klein-Gordon Propagator is used in the paper for the photon.