Calculating Traces with FeynCalc I

To learn how to use FeynCalc to evaluate traces of Dirac matrices and Lorentz contractions we will give a few examples.

• Bhabha Scattering ( $e^- e^+ \rightarrow e^- e^+$) in QED

  
  

  The relevant matrix element can be written as

  $M= M_1 + M_2$

  where

  $$M_1=\frac{e^2}{t} \overline{u}(p_3) \gamma^{\mu} u(p_1) \overline... ...s} \overline{v}(p_2) \gamma^{\mu} u(p_1) \overline{u}(p_3) \gamma_{\mu} v(p_4)$$

  with

  $s=(p_1+p_2)^2, \qquad t=(p_1-p_3)^2$.

  Once summed and averaged over spins we get
  

  $${\frac{1}{4}\sum_{\rm spins} \vert M_1 +M_2\vert^2 =\, e^4\, \lef... ...}\right] Tr\left[p\!\!\! /_3 \gamma_{\mu}p\!\!\! /_4\gamma_{\nu}\right]\right.}$$

  $${\left. \hskip 20mm -\frac{2}{s t} Tr\left[p\!\!\! /_3 \gamma^{\m... ...\gamma^{\nu} p\!\!\! /_2 \gamma_{\mu}p\!\!\! /_4\gamma_{\nu}\right]\, \right\}}$$

  The mathematica program that evaluates the traces is Bhabha.m.

• $e^- \mu^- \rightarrow e^- \mu^-$ in QED

  
  

  Hence the spin averaged squared amplitude is

  $$\frac{1}{4}\sum_{\rm spins} \vert M \vert^2 = \frac{e^4}{4 t^2} ... ...ft[(q\!\!\! /'+m_{\mu} ) \gamma_{\mu}(q\!\!\! /+m_{\mu} ) \gamma_{\nu}\right]$$

  with

  $t=(p-p')^2$

  The program that calculates the traces is muemue.m.

• $e^- e^+ \rightarrow \mu^- \mu^+$ in the Standard Model

  
  

  Neglecting the electron mass, and considering a generic fermion f of charge $Q_f$, the spin averaged squared amplitude is (see, for example Introdução à Teoria do Campo by Jorge C. Romão, pag 128)

  
  

  $${ \frac{1}{4}\sum_{\rm spins} \vert M_1 +M_2\vert^2 =\frac{e^4}{s... ...ft[(q\!\!\! /_1 +m_f) \gamma_{\mu}(q\!\!\! /_2-m_f) \gamma_{\nu}\right]\right.}$$

  $$\hskip 10mm {+ \vert F(s)\vert^2 Tr\left[p\!\!\! /_2 \gamma^{\mu}... ...^f -g_A^f\gamma_5) (q\!\!\! /_2-m_f) \gamma_{\nu}(g_V^f -g_A^f\gamma_5)\right]}$$

  $$\hskip 10mm{\left. +2\, \hbox{Re\, }\left[ F(s) Tr\left[p\!\!\! /... ..._5)(q\!\!\! /_2-m_f) \gamma_{\nu}\right]\right] \right\} \phantom{\hskip 50mm}}$$

  where

  $$F(s)=\frac{1}{\sin^2\theta_W\cos^2\theta_W} \frac{s}{s-M_Z^2+i M_Z \Gamma_Z}$$

  The program that calculates the traces is eEmM.m.

• Compton Scattering ( $e^- \gamma \rightarrow e^- \gamma$) in QED

  The program that calculates the traces is eGeG.m.

• $e^- e^+ \rightarrow \gamma \gamma$ in QED

  The program that calculates the traces is eEGG.m.