Troubleshooting Mandl & Shaw QFT: Deriving eqn (9.94) on pg 192 of 2nd edition

[Vic Sandler]

I tried to derive eqn (9.94) on page 192 of the second edition of Mandl and Shaw QFT and failed. Can someone help me see what I am doing wrong?

Ignoring factors that do not change from eqn (9.92) to eqn (9.94), noting that f(k) has been set to 1, and dropping terms linear in k and k squared as described in the text, I get:

$$\frac{\gamma^{\alpha}(\not{p'}+m)\gamma^{\mu}(\not{p}+m)\gamma_{\alpha}}{((p'-k)^2 - m^2)((p-k)^2 - m^2)}$$

$$= \gamma^{\mu}\frac{(-2p'p)}{(-2p'k)(-2pk)} + \frac{4m(p' + p)^{\mu}}{(-2p'k)(-2pk)} + \gamma^{\mu}\frac{(-2m^2)}{(-2p'k)(-2pk)}$$

The first term on the right hand side is the same as in the book, but divided by -2. Perhaps the other two terms combine in some way to fix it up, but I don't see it. I also don't see what terms are meant by the author when he says "the dots indicate terms which are finite in the limit $\lambda \rightarrow 0$" since none of the terms involve $\lambda$.

 

[andrien]

Let us see only the p/ . p'/ term.
γ^α/p'γ_μ/pγ_α=[-/p'γ^α+2p'^α]γ_μ[γ_α(-/p)+2p_α),now /p and /p' operating on free particles states will give m which will cancel with the m already written in (/p+m) and (/p'+m) terms.You are left with
2p'^αγ_μ2p_α,so you will get 4 in the numerator not 2.

 

[Vic Sandler]

Thanks Andrien, that helps a lot. But it still leaves the middle term:

$$\frac{4m(p' + p)^{\mu}}{(-2p'k)(-2pk)}$$

Do you have any ideas about it?

 

[andrien]

There are no such terms because as I have written earlier owing to -/p(-/p') acting on free particle spinor you get a factor of -m which will cancel with m in (/p -/k +m) and also dropping linear terms and quadratic terms in k,you are left only with the first term already written in the book.