# How to verify my calculated amplitude under gauge invariance structure?

I calculated the Compton amplitude of three diagrams but how can I verify it under gauge invariance structure? $${\cal A} = 2 e^2 \left[ \frac{ p_3 \cdot \epsilon_1 p_2 \cdot \epsilon_4^* }{ p_2 \cdot p_4 } - \frac{ p_2 \cdot \epsilon_1 p_3 \cdot \epsilon_4^*}{ p_2 \cdot p_1 } + \epsilon_1 \cdot \epsilon_4^* \right]$$

• On this site all math is expected to be in MathJax. Scans of handwriting are not considered acceptable. Here is a MathJax tutorial. Jul 24, 2020 at 0:05
• Thank you so much 😊 Jul 24, 2020 at 0:06
• Writing four 4-vectors next to each other doesn’t mean anything. Jul 24, 2020 at 0:07
• Oh no the p1 and p4 are the initial and final photon momenta and p2 and p4 is for electron and of course the epsilon is the polarization for photons. Jul 24, 2020 at 0:26
• I was referring to your omission of the scalar products, which Prahar added when he did the MathJax for you. You mean $p_2$ and $p_3$ for the electron. Jul 26, 2020 at 0:07

Gauge invariance requires that the amplitude vanishes if you replace the polarization with its corresponding momentum. If we replace $$\epsilon_4 \to p_4$$, we get \begin{align} {\cal A} &\to 2 e^2 \left[ \frac{ p_3 \cdot \epsilon_1 p_2 \cdot p_4 }{ p_2 \cdot p_4 } - \frac{ p_2 \cdot \epsilon_1 p_3 \cdot p_4}{ p_2 \cdot p_1 } + \epsilon_1 \cdot p_4 \right]\\ &= 2 e^2 ( p_3 + p_4 - p_2 ) \cdot \epsilon_1 \\ &= 2 e^2 p_1 \cdot \epsilon_1 \\ &= 0 \end{align} In the second line, I have used the fact that $$p_3 \cdot p_4 = p_1 \cdot p_2$$ which is implied by momentum conservation. In the third, we have again used momentum conservation, $$p_1 + p_2 = p_3 + p_4$$.
You can try to verify gauge invariance by also replacing $$\epsilon_1 \to p_1$$.