Difference between revisions of "Variables Used in Elastic Scattering"

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We can simiplify the expressions
 
We can simiplify the expressions
  
<center><math>\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+  m_1^'\right)= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_a}\right)^2=s</math></center>
+
<center><math>\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+  m_1^{'2}\right)= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_a}\right)^2=s</math></center>
  
  
<center><math>\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+  m_2^'\right)= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_b}\right)^2=s</math></center>
+
<center><math>\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+  m_2^{'2}\right)= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_b}\right)^2=s</math></center>
  
  
<center><math>\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+  m_1^'}\right)= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_c}\right)^2=s</math></center>
+
<center><math>\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+  m_1^{'2}\right)= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_c}\right)^2=s</math></center>
  
  
<center><math>\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+  m_2^'\right)= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_d}\right)^2=s</math></center>
+
<center><math>\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+  m_2^{'2}\right)= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_d}\right)^2=s</math></center>
  
 
=Mandelstam Representation=
 
=Mandelstam Representation=
  
 
[[File:Mandelstam.png | 400 px]]
 
[[File:Mandelstam.png | 400 px]]

Revision as of 21:11, 31 January 2016

Lorentz Invariant Quantities

Total 4-Momentums

As was shown earlier the scalar product of a 4-Momentum vector with itself ,

[math]{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s[/math]

,

and the length of a 4-Momentum vector composed of 4-Momentum vectors,

[math]{\mathbf P^2}=({\mathbf P_1}+{\mathbf P_2})^2=(E_1+E_2)^2-(\vec p_1 +\vec p_2 )^2=(m_1+m_2)^2=s[/math]

,

are invariant quantities.

It was further shown that

[math]{\mathbf P^*}^2={\mathbf P}^2[/math]


where [math]{\mathbf P^*}=({\mathbf P_1^*}+{\mathbf P_2^*})^2[/math] represents the 4-Momentum Vector in the CM frame


and [math]{\mathbf P}=({\mathbf P_1}+{\mathbf P_2})^2[/math] represents the 4-Momentum Vector in the initial Lab frame

which can be expanded to

[math]{\mathbf P^*}^2={\mathbf P}^2={\mathbf P^'}^2[/math]


where [math]{\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2[/math] represents the 4-Momentum Vector in the final Lab frame

New 4-Momentum Quantities

Working in just the Lab frame, we can form new 4-Momentum Vectors comprised of 4-Momenta in this frame, with

[math]{\mathbf P_1}- {\mathbf P_1^'}= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_a}[/math]


[math]{\mathbf P_1}- {\mathbf P_2^'}= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_b}[/math]


[math]{\mathbf P_2}- {\mathbf P_1^'}= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)={\mathbf P_c}[/math]


[math]{\mathbf P_2}- {\mathbf P_2^'}= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)={\mathbf P_d}[/math]

Using the algebraic fact

[math]\left({\mathbf a}- {\mathbf b}\right)^2=\left({\mathbf b}- {\mathbf a}\right)^2[/math]


and the fact that the length of these 4-Momentum Vectors are invariant,

[math]\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left({\mathbf P_1}^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+ {\mathbf P_1^'}\right)= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_a}\right)^2=s[/math]


[math]\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left({\mathbf P_1}^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+ {\mathbf P_2^'}\right)= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_b}\right)^2=s[/math]


[math]\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left({\mathbf P_2}^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+ {\mathbf P_1^'}\right)= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_c}\right)^2=s[/math]


[math]\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left({\mathbf P_2}^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+ {\mathbf P_2^'}\right)= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_d}\right)^2=s[/math]

Using the fact that the scalar product of a 4-momenta with itself is invariant,


[math]{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s[/math]


We can simiplify the expressions

[math]\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+ m_1^{'2}\right)= \left( \begin{matrix}E_1-E_1'\\ p_{1(x)}-p_{1(x)}^' \\ p_{1(y)}-p_{1(y)}^' \\ p_{1(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_a}\right)^2=s[/math]


[math]\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+ m_2^{'2}\right)= \left( \begin{matrix}E_1-E_2'\\ p_{1(x)}-p_{2(x)}^' \\ p_{1(y)}-p_{2(y)}^' \\ p_{1(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_b}\right)^2=s[/math]


[math]\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+ m_1^{'2}\right)= \left( \begin{matrix}E_2-E_1'\\ p_{2(x)}-p_{1(x)}^' \\ p_{2(y)}-p_{1(y)}^' \\ p_{2(z)}-p_{1(z)}^'\end{matrix} \right)^2=\left({\mathbf P_c}\right)^2=s[/math]


[math]\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+ m_2^{'2}\right)= \left( \begin{matrix}E_2-E_2'\\ p_{2(x)}-p_{2(x)}^' \\ p_{2(y)}-p_{2(y)}^' \\ p_{2(z)}-p_{2(z)}^'\end{matrix} \right)^2=\left({\mathbf P_d}\right)^2=s[/math]

Mandelstam Representation

Mandelstam.png