Difference between revisions of "Scattering Amplitude"

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=Scattering Amplitude=
 
=Scattering Amplitude=
  
In the Møller scattering <math>(\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^')</math> we have deal with identical particles in the initial and final states, which means that the amplitude has to be symmetric under interchange of particles <math>(\mathbf P_1^' \leftrightarrow \mathbf P_2^' </math> or <math> \mathbf P_1  \leftrightarrow \mathbf P_2)</math>. As a consequence there will be two Feynman diagrams to describe this scattering.  
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In the Møller scattering <math>(\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^')</math> we have identical particles in the initial and final states.  This that the amplitude to be symmetric under interchange of particles <math>(\mathbf P_1^' \leftrightarrow \mathbf P_2^' </math> or <math> \mathbf P_1  \leftrightarrow \mathbf P_2)</math>. Due to this symmetry we can determine two 1st level Feynman diagrams to describe this scattering.  
  
 
<center>[[File:Feynman1stLevel.png | 600 px]]</center>
 
<center>[[File:Feynman1stLevel.png | 600 px]]</center>
  
 
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The amplitudes of the individual Feynman diagrams add linearly to form the total amplitude
 
<center><math>\mathfrak{M}=\mathfrak{M}_1+\mathfrak{M}_2</math></center>
 
<center><math>\mathfrak{M}=\mathfrak{M}_1+\mathfrak{M}_2</math></center>
  

Revision as of 15:30, 27 June 2017

Scattering Amplitude

In the Møller scattering [math](\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^')[/math] we have identical particles in the initial and final states. This that the amplitude to be symmetric under interchange of particles [math](\mathbf P_1^' \leftrightarrow \mathbf P_2^' [/math] or [math] \mathbf P_1 \leftrightarrow \mathbf P_2)[/math]. Due to this symmetry we can determine two 1st level Feynman diagrams to describe this scattering.

Feynman1stLevel.png

The amplitudes of the individual Feynman diagrams add linearly to form the total amplitude

[math]\mathfrak{M}=\mathfrak{M}_1+\mathfrak{M}_2[/math]


Using the Feynman rules, each vertex contribute a factor

[math]ie(\mathbf p_{initial} + \mathbf p_{final})^{\mu}[/math]

and the propagator gives

[math] \frac{−ig_{\mu \nu}}{q^2}[/math]

where q is the momentum of the photon

[math]q \equiv \mathbf p_{final}-\mathbf p_{initial}[/math]

and [math]g_{\mu \nu}[/math] is the Mandelstam metric which allows the transformation from the contravariant to covariant form needed for tensor multiplication. Examining both Feynman diagrams seperately, we find for their individual amplitudes


[math]-i \mathfrak{M}_1=ie(\mathbf p_1+\mathbf p_1^')^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf p_2^')^{\nu} \qquad \qquad -i \mathfrak{M}_2=ie(\mathbf p_1+\mathbf p_2^')^{\mu} \left (\frac{-ig_{\mu \nu}}{q^2} \right ) ie ( \mathbf p_{2}+\mathbf p_1^')^{\nu}[/math]


[math]-i \mathfrak{M}_1=ie(\mathbf p_1+\mathbf p_1^')^{\mu} \left (\frac{-ig_{\mu \nu}}{(\mathbf p_2^'-\mathbf p_2)^2} \right ) ie( \mathbf p_{2}+\mathbf p_2^')^{\nu} \qquad \qquad -i \mathfrak{M}_2=ie(\mathbf p_1+\mathbf p_2^')^{\mu} \left (\frac{-ig_{\mu \nu}}{(\mathbf p_1^'-\mathbf p_2)^2} \right ) ie( \mathbf p_{2}+\mathbf p_1^')^{\nu}[/math]


[math]-i \mathfrak{M}_1=ie^2\left (\frac{(\mathbf p_1+\mathbf p_1^')_{\mu} (\mathbf p_{2}+\mathbf p_2^')^{\mu}}{(\mathbf p_2^'-\mathbf p_2)^2} \right ) \qquad \qquad -i \mathfrak{M}_2=ie^2\left (\frac{(\mathbf p_1+\mathbf p_2^')_{\mu} (\mathbf p_{2}+\mathbf p_1^')^{\mu}}{(\mathbf p_1^'-\mathbf p_2)^2} \right ) [/math]


Without loss of generality, we can extend this to the center of mass frame


[math]-i \mathfrak{M}_{e^-e^-}=-ie^2 \left ( \frac{(\mathbf p_1^*+\mathbf p_{1}^{'*})_{\mu}(\mathbf p_2^*+\mathbf p_{2}^{'*})^{\mu}}{(\mathbf p_{2}^{'*}-\mathbf p_2^*)^2}- \frac{(\mathbf p_1^*+\mathbf p_{2}^{'*})_{\mu}(\mathbf p_2^*+\mathbf p_{1}^{'*})^{\mu}}{(\mathbf p_{1}^{'*}-\mathbf p_2^*)^2} \right )[/math]


[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{\mathbf P_1^* \mathbf P_2^*+\mathbf P_{1}^{'*} \mathbf P_{2}^{'*}+\mathbf P_{1}^{'*} \mathbf P_2^*+\mathbf P_1^* \mathbf P_{2}^{'*}}{(\mathbf P_{2}^{'*}-\mathbf P_2^*)^2}- \frac{\mathbf P_1^* \mathbf P_2^*+\mathbf P_{2}^{'*} \mathbf P_{1}^{'*}+\mathbf P_{2}^{'*} \mathbf P_2^*+\mathbf P_1^* \mathbf P_{1}^*}{(\mathbf P_{1}^{'*}-\mathbf P_2^*)^2} \right )[/math]



Using the fact that [math]\mathbf P_1^{'*} \mathbf P_2^{'*}=\mathbf P_1^* \mathbf P_2^* \quad \quad \mathbf P_1^{'*} \mathbf P_1^{*}=\mathbf P_2^{'*} \mathbf P_2^* \quad \quad \mathbf P_1^{*} \mathbf P_2^{'*}=\mathbf P_2^* \mathbf P_1^{'*}[/math]


[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_{1}^{'*} \mathbf P_2^*}{(\mathbf P_{2}^{'*2}-2\mathbf P_{2}^{'*}\mathbf P_2^{*}+\mathbf P_2^{*2})}- \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_1^* \mathbf P_{1}^{'*}}{(\mathbf P_{1}^{'*2}-2\mathbf P_{1}^{'*}\mathbf P_2^{'*}+\mathbf P_2^{'*2})} \right )[/math]


[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_{1}^{'*} \mathbf P_2^*}{(\mathbf P_2^{*2}-2\mathbf P_2^{*}\mathbf P_{2}^{'*}+\mathbf P_{2}^{'*2})}- \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_1^* \mathbf P_{1}^{'*}}{(\mathbf P_2^{'*2}-2\mathbf P_2^{'*}\mathbf P_{1}^{'*}+\mathbf P_{1}^{'*2})} \right )[/math]


[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_{1}^{'*} \mathbf P_2^*}{(\mathbf P_2^*-\mathbf P_{2}^{'*})^2}- \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_1^* \mathbf P_{1}^{'*}}{(\mathbf P_2^{'*}-\mathbf P_{1}^{'*})^2} \right )[/math]



[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{ (\mathbf P_1^{*2}-2 \mathbf P_1^* \mathbf P_2^{'*}+ \mathbf P_2^{'*2})-(\mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2})}{t}- \frac{(\mathbf P_1^{*2}-2 \mathbf P_1^* \mathbf P_1^{'*}+ \mathbf P_1^{'*2})-( \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2})}{u} \right )[/math]


[math]\mathfrak{M}_{e^-e^-}=e^2 \left (\frac{u-s}{t}+\frac{t-s}{u} \right )[/math]