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| =In the CM Frame= | | =In the CM Frame= |
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| | <center><math>{\mathbf P_1^{*}}=-{\mathbf P_2^{*}}</math></center> |
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| | <center><math>{\mathbf P_1^{'*}}=-{\mathbf P_2^{'*}}</math></center> |
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| | <center><math>E_1^{*}=E_1^{'*}=E_2^{*}=E_2^{'*}</math></center> |
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| | <center><math>\left | \vec p_1^* \right |=\left | \vec p_1^{'*} \right |=\left | \vec p_2^* \right |=\left | \vec p_2^{'*} \right |</math></center> |
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| | <center><math>u =\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_1^{'*}}\right)^2</math></center> |
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| | <center><math>u={\mathbf P_1^{*2}}+ {\mathbf P_2^{'*2}}-2 {\mathbf P_1^*} {\mathbf P_2^{'*}}={\mathbf P_2^{*2}}+ {\mathbf P_1^{'*2}}-2 {\mathbf P_2^*} {\mathbf P_1^{'*}}</math></center> |
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| =In the Lab Frame= | | =In the Lab Frame= |
Revision as of 16:33, 15 March 2018
The quantity is known as the
[math]\displaystyle{ u \equiv \left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_1^{'*}}\right)^2 }[/math]
In the CM Frame
[math]\displaystyle{ {\mathbf P_1^{*}}=-{\mathbf P_2^{*}} }[/math]
[math]\displaystyle{ {\mathbf P_1^{'*}}=-{\mathbf P_2^{'*}} }[/math]
[math]\displaystyle{ E_1^{*}=E_1^{'*}=E_2^{*}=E_2^{'*} }[/math]
[math]\displaystyle{ \left | \vec p_1^* \right |=\left | \vec p_1^{'*} \right |=\left | \vec p_2^* \right |=\left | \vec p_2^{'*} \right | }[/math]
[math]\displaystyle{ u =\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_1^{'*}}\right)^2 }[/math]
[math]\displaystyle{ u={\mathbf P_1^{*2}}+ {\mathbf P_2^{'*2}}-2 {\mathbf P_1^*} {\mathbf P_2^{'*}}={\mathbf P_2^{*2}}+ {\mathbf P_1^{'*2}}-2 {\mathbf P_2^*} {\mathbf P_1^{'*}} }[/math]
In the Lab Frame
[math]\displaystyle{ u={\mathbf P_1^{2}}+ {\mathbf P_2^{'2}}-2 {\mathbf P_1} {\mathbf P_2^{'}}={\mathbf P_2^{2}}+ {\mathbf P_1^{'2}}-2 {\mathbf P_2} {\mathbf P_1^{'}} }[/math]
[math]\displaystyle{ u=2m^2-2E_1E_2^{'}+2 \vec p_1 \vec p_2^{'}=2m^2-2E_2E_1^{'}+2 p_2 p_1^{'} }[/math]
with [math]\displaystyle{ p_2=0 }[/math]
and [math]\displaystyle{ E_2=m }[/math]
[math]\displaystyle{ u=2m^2-2E_1E_2^{'}+2 \vec p_1 \vec p_2^{'}=2m^2-2mE_1^{'} }[/math]
[math]\displaystyle{ u=2m^2-2E_1E_2^{'}+2 \left | p_1 \right | \left | p_2^{'} \right | \cos \theta_{1\ 2^{'}}=2m^2-2mE_1^{'} }[/math]