Variables Used in Elastic Scattering

Lorentz Invariant Quantities

Total 4-Momentums

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

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

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

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

where [math]\displaystyle{ {\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]\displaystyle{ {\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]\displaystyle{ {\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]\displaystyle{ {\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]\displaystyle{ {\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]\displaystyle{ \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]\displaystyle{ \left({\mathbf P_1}- {\mathbf P_1^'}\right)^2= \left(\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)\right)^2=\left({\mathbf P_a}\right)^2 }[/math]

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

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

[math]\displaystyle{ \left({\mathbf P_2}- {\mathbf P_2^'}\right)^2= \left(\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)\right)^2=\left({\mathbf P_d}\right)^2 }[/math]

Mandelstam Representation