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| − | In the center of mass frame, the momentum of the particles interacting are equal and opposite, i.e. <math>p_1^*=-p_2^*</math>. However, the 4-momentum still retains an energy component, which as a scalar quantity, can not be countered by another particle's direction of motion.
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| | + | As shown earlier, for identical masses in the center of mass frame, and the mass is invariant between frames |
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| − | Similarly, by the relativistic definition of energy
| + | <center><math>P^{*2}=m^2</math></center> |
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| − | <center><math>E^2 \equiv p^2+m^2</math></center>
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| − | where both particles have the same mass, this implies
| + | <center><math>s \equiv m^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2}</math></center> |
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| | + | <center><math>s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^*</math></center> |
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| − | <center><math> \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=4E_{CM}^2=4(m_{CM}^2+p_{CM}^2)=s</math></center>
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Revision as of 17:21, 8 June 2017
Limits based on Mandelstam Variables
s Channel
[math]s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2[/math]
[math]s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2[/math]
[math]s \equiv \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2}[/math]
As shown earlier, for identical masses in the center of mass frame, and the mass is invariant between frames
[math]P^{*2}=m^2[/math]
[math]s \equiv m^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2}[/math]
[math]s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^*[/math]
[math]s=4(m_{CM}^2+p_{CM}^2)[/math]
t Channel
[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}+ {\mathbf P_2^{'*}}\right)^2[/math]
[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2[/math]
[math]t \equiv P_1^{*2}-2P_1^*P_1^{'*}+P_1^{'*2}[/math]
[math]t \equiv 2m_1^2-2E_1^*E_1^{'*}+2p_1^*p_1^{'*}[/math]
[math]t \equiv 2m_1^*-2E_1^{*2}+2p_1^{*2}cos\ \theta[/math]
[math]t \equiv -2p_1^{*2}(1-cos\ \theta)[/math]
u Channel
