TransWikia.com

How to verify my calculated amplitude under gauge invariance structure?

Physics Asked on December 21, 2021

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

One Answer

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 $epsilon_1 to p_1$.

Answered by Prahar on December 21, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP