Difference between revisions of "Limit of Scattering Angle Theta in Lab Frame"

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<center><math>u=2m^2-2E_1^{*2}+2 \left | p_1^{*2}\right | \cos \theta_{1\ 2'}=2m^2-2E_2^{*2}+2 \left | p_2^{*2}\right | \cos \theta_{2\ 1'}</math></center>
+
<center><math>u=2m^2-2E_1^{*2}+2 \left | p_1^{*2}\right | \cos \theta_{1^*\ 2^{'*}}=2m^2-2E_2^{*2}+2 \left | p_2^{*2}\right | \cos \theta_{2^*\ 1^{'*}}</math></center>
  
  
where <math>\theta_{1\ 2'}</math> and <math>\theta_{2\ 1'}</math>is the angle between the before and after momentum in the CM frame
+
where <math>\theta_{1^*\ 2^{'*}}</math> and <math>\theta_{2^*\ 1^{'*}}</math>is the angle between the before and after momentum in the CM frame
  
 
=In the Lab Frame=
 
=In the Lab Frame=

Revision as of 16:40, 15 March 2018

The quantity is known as the

[math]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]{\mathbf P_1^{*}}=-{\mathbf P_2^{*}}[/math]


[math]{\mathbf P_1^{'*}}=-{\mathbf P_2^{'*}}[/math]


[math]E_1^{*}=E_1^{'*}=E_2^{*}=E_2^{'*}[/math]


[math]\left | \vec p_1^* \right |=\left | \vec p_1^{'*} \right |=\left | \vec p_2^* \right |=\left | \vec p_2^{'*} \right |[/math]


[math]u =\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_1^{'*}}\right)^2[/math]


[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]


[math]u=2m^2-2E_1^*E_2^{'*}+2 \vec p_1^* \vec p_2^{'*}=2m^2-2E_2^*E_1^{'*}+2 p_2^* p_1^{'*}[/math]


[math]u=2m^2-2E_1^{*2}+2 \left | p_1^{*2}\right | \cos \theta_{1^*\ 2^{'*}}=2m^2-2E_2^{*2}+2 \left | p_2^{*2}\right | \cos \theta_{2^*\ 1^{'*}}[/math]


where [math]\theta_{1^*\ 2^{'*}}[/math] and [math]\theta_{2^*\ 1^{'*}}[/math]is the angle between the before and after momentum in the CM frame

In the Lab Frame

[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]


[math]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]p_2=0[/math]

and [math]E_2=m[/math]

[math]u=2m^2-2E_1E_2^{'}+2 \vec p_1 \vec p_2^{'}=2m^2-2mE_1^{'}[/math]


[math]u=2m^2-2E_1E_2^{'}+2 \left | p_1 \right | \left | p_2^{'} \right | \cos \theta_{1\ 2^{'}}=2m^2-2mE_1^{'}[/math]