# Finding the amplitude for the pair production $\gamma(p_1) + \gamma (p_2) \to e^- (p_3) + e^+ (p_4)$

I'm trying to find the amplitude for:

$$\gamma(p_1) + \gamma (p_2) \to e^- (p_3) + e^+ (p_4)\tag{1}$$

(My questions are stated in the end) My take on it: and so

$$\tag{2} \require{cancel} \mathcal{M}= e^2\{ \epsilon_1 \gamma \bar{u}_3 \frac{(\cancel{p_3}-\cancel{p_1})}{t} \nu_3 \gamma\epsilon_2 - \bar{u}_3 \gamma \epsilon_2 \frac{\cancel{p_2}-\cancel{p_3}}{u}\epsilon_1 \gamma\nu_4 \}$$

I know it is (v) but I don't' understand the following (as my answer is not present in the same form as the ones presented):

Where does the factor of 2 in $$(iv)$$ and $$(v)$$ come from?

• Also, note that in general you shouldn't post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines, and using MathJax for formulae. I've left the screenshot as is due to the challenge of correctly displaying slash marks over MathJax, but keep it in mind for future threads. – Emilio Pisanty Jun 9 at 11:24
• I've deleted a number of obsolete comments and/or responses to them. Let me repost the meta question discussing this question for reference. – David Z Jun 14 at 22:22

Where does the factor of 2 in $$(iv)$$ and $$(v)$$ come from?

It comes from the anticommutation relation for the Dirac gamma matrices,

$$\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2\eta_{\mu\nu}I_4\tag{a},$$

where $$\eta_{\mu\nu}$$ is the “mostly minus” Minkowski metric and $$I_4$$ (which is often omitted but implied) is the $$4\times 4$$ identity matrix.

In terms of "slashed" matrices ($$\not a\equiv a^\mu\gamma_\mu$$), this is

$$\not a\,\not b+\not b\,\not a=2(a \cdot b)I_4\tag{b}.$$

Your expression (2) has several errors. Once you correct them, you can use (b) to get the slashed matrices in the order that the answer wants.