Monday, May 30, 2016

Compton Scattering

I complain about the amplitude of Compton scattering, which is the scattering of a photon by an electron.
The Compton scattering amplitude is \begin{equation}\label{eq:20160529a} \frac{1}{4} | \mathcal{M} |^2 = 2 e^4 \left[\frac{p_{24}}{p_{12}} + \frac{p_{12}}{p_{24}} + 2 m^2 \left( \frac{1}{p_{12}} - \frac{1}{p_{24}}\right) +m^4 \left( \frac{1}{p_{12}} - \frac{1}{p_{24}}\right)^2\right] \end{equation} $m$ is the mass of the electron and $p_{ij} = p_i \cdot p_j$. The definition of the momenta $p_i$ can be found in the graph below.
Compton scattering. $e^-$ is an electron, $\gamma$ is a photon
Although the result \eqref{eq:20160529a} is quite short, the calculation is long, see for example Peskin and Schroeder, page 161. I worked a bit differently from the calculation in Peskin and Schroeder. Namely I used the following two gamma matrix identities \begin{align} \text{Tr}&\left[ \left( a_1\!\!\!\!\!/ + m_1\right) \gamma^{\mu} \left( a_2\!\!\!\!\!/ + m_2\right)\gamma^{\nu} \left( a_3\!\!\!\!\!/ + m_3\right)\gamma_{\nu} \left( a_4\!\!\!\!\!/ + m_4\right)\gamma_{\mu} \right]\nonumber\\ & = 16 \left[ (a_{12} -2 m_{12} )(a_{34} -2 m_{34} )- (a_{13} -4 m_{13} )(a_{24} - m_{24} ) + (a_{14} -2 m_{14} )(a_{23} -2 m_{23} ) \right]\label{eq:20160530a} \end{align} and \begin{equation}\label{eq:20160530b} \text{Tr}\left[ \left( a_1\!\!\!\!\!/ + m_1\right) \gamma^{\mu} \left( a_2\!\!\!\!\!/ + m_2\right)\gamma^{\nu} \left( a_3\!\!\!\!\!/ + m_3\right)\gamma_{\mu} \left( a_4\!\!\!\!\!/ + m_4\right)\gamma_{\nu} \right] = 16 \left( a_{12} m_{34} + a_{13} m_{24} + a_{14} m_{23} - 2 a_{13} a_{24} + a \leftrightarrow m \right) \end{equation} I thought this would lead to a shorter derivation than the one in Peskin and Schroeder, but the full calculation including the proof of \eqref{eq:20160530a} and \eqref{eq:20160530b} was still 8 handwritten pages. In the course of this calculation, many cancellations occurred leading finally to the simple result \eqref{eq:20160529a}.

I think that the fact that \eqref{eq:20160529a} is short but is the result of a long calculation with many cancellations, shows that a method could exist that yields \eqref{eq:20160529a} more efficiently. I have been looking on the internet for such a method, not only because I want to have a quicker method for \eqref{eq:20160529a}, but also because I think that looking for more efficient methods to calculate scattering amplitudes in quantum field theory is a current area of research.

Please leave a comment if you know a quick way to obtain \eqref{eq:20160529a}, \eqref{eq:20160530a} or \eqref{eq:20160530b}.

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