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| − | Using the Feynman rules, each vertex contribute a factor <math>ie(p_{initial} + 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>p_{final}-p_{initial}</math> | + | Using the Feynman rules, each vertex contribute a factor |
| | + | <center><math>ie(p_{initial} + p_{final})^{\mu}</math></center> |
| | + | and the propagator gives |
| | + | <center><math> \frac{−ig_{\mu \nu}}{q^2}</math></center> |
| | + | where q is the momentum of the photon |
| | + | <center><math>p_{final}-p_{initial}</math></center> |
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Revision as of 16:52, 25 June 2017
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
pfinal−pinitial
-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
−iMe−e−=−ie2((p∗1+p′∗1)μ(p∗2+p′∗2)μ(p′∗2−p∗2)2−(p∗1+p′∗2)μ(p∗2+p′∗1)μ(p′∗1−p∗2)2)
Me−e−=e2(P∗1P∗2+P′∗1P′∗2+P′∗1P∗2+P∗1P′∗2(P′∗2−P∗2)2−P∗1P∗2+P′∗2P′∗1+P′∗2P∗2+P∗1P∗1(P′∗1−P∗2)2)
Using the fact that P′∗1P′∗2=P∗1P∗2P′∗1P∗1=P′∗2P∗2P∗1P′∗2=P∗2P′∗1
Me−e−=e2(2P∗1P∗2+2P′∗1P∗2(P′∗22−2P′∗2P∗2+P∗22)−2P∗1P∗2+2P∗1P′∗1(P′∗21−2P′∗1P′∗2+P′∗22))
Me−e−=e2(2P∗1P∗2+2P′∗1P∗2(P∗22−2P∗2P′∗2+P′∗22)−2P∗1P∗2+2P∗1P′∗1(P′∗22−2P′∗2P′∗1+P′∗21))
Me−e−=e2(2P∗1P∗2+2P′∗1P∗2(P∗2−P′∗2)2−2P∗1P∗2+2P∗1P′∗1(P′∗2−P′∗1)2)
Me−e−=e2((P∗21−2P∗1P′∗2+P′∗22)−(P∗21+2P∗1P∗2+P∗22)t−(P∗21−2P∗1P′∗1+P′∗21)−(P∗21+2P∗1P∗2+P∗22)u)
Me−e−=e2(u−st+t−su)
