I don't understand how to evaluate the diagram for electron-photon vertex with help of Cutosky's rule:

enter image description here

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 enter image description here

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.

  • $\begingroup$ Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. $\endgroup$
    – Qmechanic
    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.


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