Difference between revisions of "Limits based on Mandelstam Variables"

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In the center of mass frame of reference,  
 
In the center of mass frame of reference,  
  
<center><math>E_1^*=E_2^* \quad \vec p_1^*=-\vec p_2^*</math></center>
+
<center><math>E_1^*=E_2^* \quad and \quad \vec p_1^*=-\vec p_2^*</math></center>
  
  
<center><math>s \equiv 2m^2+2(E_1^*E_2^*-\vec p_1^* \vec p_2^*)</math></center>
+
<center><math>s \equiv 2m^2+2(E_1^{*2}+\vec p_1^{*2} )</math></center>
  
  

Revision as of 17:31, 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, the square of a 4-momentum is


[math]\mathbf P^{2} \equiv m^2[/math]


[math]s \equiv m^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2}[/math]


This gives

[math]s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^*[/math]


Similarly, the scalar product of two 4-momentums

[math]s \equiv 2m^2+2(E_1^*E_2^*-\vec p_1^* \vec p_2^*)[/math]


In the center of mass frame of reference,

[math]E_1^*=E_2^* \quad and \quad \vec p_1^*=-\vec p_2^*[/math]


[math]s \equiv 2m^2+2(E_1^{*2}+\vec p_1^{*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]
400px-CMcopy.png


[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