Amplitude with Feynman diagrams and gluon propagators

[Aleolomorfo]

Homework Statement

Let's check that the term with $\frac{l_\alpha l_\beta}{k^2}$ in the following amplitude:
$$ \bar u(p) (-ig_s\gamma^\alpha t^a) \frac{i}{\displaystyle{\not}p+\not k} (-i\gamma^\nu e Q_f) \frac{i}{\displaystyle{\not}p'-\not k} (-ig_s \gamma^\beta t^a) v(p') \frac{i}{k^2} \biggl(-g_{\alpha\beta} + \frac{k_\alpha k_\beta}{k^2}\biggr) (\bar u(p) (-ieQ_f \gamma^\nu) v(p'))^* $$
is zero.
The amplitude comes from the interference of the NLO @ QCD of $e^+ e^- -> q \bar q$. I have attached the Feynman diagrams.
We consider everything massless.

Relevant Equations

$\displaystyle{\not}p u(p) = m u(p)$
If m = 0 ----> $\displaystyle{\not}p u(p) = 0$

The term which is relevant for the calculus is:
$$ \bar u(p) \gamma^\alpha \frac{1}{\displaystyle{\not}p+\not k} \gamma^\nu \frac{1}{\displaystyle{\not}p'-\not k} \gamma^\beta v(p') \frac{k_\alpha k_\beta}{k^2} $$
$$ \bar u(p) \displaystyle{\not}k \frac{1}{\displaystyle{\not}p+\not k} \gamma^\nu \frac{1}{\displaystyle{\not}p'-\not k} \displaystyle{\not}k \space v(p') $$
$$ \bar u(p) (\displaystyle{\not}k + \displaystyle{\not}p - \displaystyle{\not}p) \frac{1}{\displaystyle{\not}p+\not k} \gamma^\nu \frac{1}{\displaystyle{\not}p'-\not k} (\displaystyle{\not}k + \displaystyle{\not}p' - \displaystyle{\not}p') v(p') $$
$$ \bar u(p) \biggl(1 - \frac{\displaystyle{\not}p} {\displaystyle{\not}p+\not k}\biggr) \gamma^\nu \biggl(-1 + \frac{\displaystyle{\not}p'} {\displaystyle{\not}p'+\not k}\biggr) v(p') $$
The terms with $\displaystyle{\not}p'$ and $\displaystyle{\not}p$ vanish because of the relations mentioned above and I do not get zero. Where is my mistake?
Thanks in advance!

 

[member 553083]

The mistake is that you are not taking into account the fact that the spinor products $\bar{u}(p) v(p')$ and $\bar{u}(p') v(p)$ are both zero. This means that the terms with $\displaystyle{\not}p$ and $\displaystyle{\not}p'$ do not in fact vanish, but rather cancel each other out. Therefore, the final result is:$$ \bar u(p) \gamma^\nu v(p') $$