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 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.  
  
 
<center>[[File:Feynman1stLevel.png | 600 px]]</center>
 
<center>[[File:Feynman1stLevel.png | 600 px]]</center>

Revision as of 15:22, 26 June 2017

Scattering Amplitude

In the Møller scattering (\mathbf P_1 + \mathbf P_2 \rightarrow \mathbf P_1^' + \mathbf P_2^') 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 (\mathbf P_1^' \leftrightarrow \mathbf P_2^' or P1P2). As a consequence there will be two Feynman diagrams to describe this scattering.

Feynman1stLevel.png


M=M1+M2


Using the Feynman rules, each vertex contribute a factor

ie(pinitial+pfinal)μ

and the propagator gives

igμνq2

where q is the momentum of the photon

qpfinalpinitial


-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}


-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}


-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 )


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


iMee=ie2((p1+p1)μ(p2+p2)μ(p2p2)2(p1+p2)μ(p2+p1)μ(p1p2)2)


Mee=e2(P1P2+P1P2+P1P2+P1P2(P2P2)2P1P2+P2P1+P2P2+P1P1(P1P2)2)



Using the fact that P1P2=P1P2P1P1=P2P2P1P2=P2P1


Mee=e2(2P1P2+2P1P2(P222P2P2+P22)2P1P2+2P1P1(P212P1P2+P22))


Mee=e2(2P1P2+2P1P2(P222P2P2+P22)2P1P2+2P1P1(P222P2P1+P21))


Mee=e2(2P1P2+2P1P2(P2P2)22P1P2+2P1P1(P2P1)2)



Mee=e2((P212P1P2+P22)(P21+2P1P2+P22)t(P212P1P1+P21)(P21+2P1P2+P22)u)


Mee=e2(ust+tsu)