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 ,

${\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s$

,

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

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

,

are invariant quantities.

It was further shown that

${\mathbf P^*}^2={\mathbf P}^2$

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

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

which can be expanded to

${\mathbf P^*}^2={\mathbf P^{'*}}^2={\mathbf P}^2={\mathbf P^'}^2$

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

and ${\mathbf P^{'*}}=({\mathbf P_1^{'*}}+{\mathbf P_2^{'*}})^2$ represents the 4-Momentum Vector in the final CM frame

New 4-Momentum Quantities

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

${\mathbf P_1^*}- {\mathbf P_1^{'*}}= \left( \begin{matrix}E_1^*-E_1^{'*}\\ p_{1(x)}^*-p_{1(x)}^{'*} \\ p_{1(y)}^*-p_{1(y)}^{'*} \\ p_{1(z)}^*-p_{1(z)}^{'*}\end{matrix} \right)={\mathbf P_a^*}$

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

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

${\mathbf P_2^*}- {\mathbf P_2^{'*}}= \left( \begin{matrix}E_2^*-E_2^{'*}\\ p_{2(x)}^*-p_{2(x)}^{'*} \\ p_{2(y)}^*-p_{2(y)}^{'*} \\ p_{2(z)}^*-p_{2(z)}^{'*}\end{matrix} \right)={\mathbf P_d^*}$

Using the algebraic fact

$\left({\mathbf a}- {\mathbf b}\right)^2=\left({\mathbf b}- {\mathbf a}\right)^2$

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

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

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

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

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

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

${\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s$

We can simiplify the expressions

$\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_1^'}+ m_1^{'2}\right)=\left({\mathbf P_a}\right)^2=s$

$\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2{\mathbf P_1}\cdot {\mathbf P_2^'}+ m_2^{'2}\right)=\left({\mathbf P_b}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_1^'}+ m_1^{'2}\right)=\left({\mathbf P_c}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2{\mathbf P_2}\cdot {\mathbf P_2^'}+ m_2^{'2}\right)=\left({\mathbf P_d}\right)^2=s$

Finding the cross terms,

${\mathbf P_1}\cdot {\mathbf P^'}=\left(\begin{matrix} E\\ p_x \\ p_y \\ p_z \end{matrix} \right)\cdot \left( \begin{matrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 &0 & 0 &-1\end{matrix} \right)\cdot \left(\begin{matrix} E' & p_x^' & p_y^' & p_z^' \end{matrix} \right)=E_1E_1^'-\vec p_1\cdot \vec p_1^'$

This gives

$\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2\left(E_1E_1^'-\vec p_1\cdot \vec p_1^'\right)+ m_1^{'2}\right)=\left({\mathbf P_a}\right)^2=s$

$\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2\left(E_1E_2^'-\vec p_1\cdot \vec p_2^'\right)+ m_2^{'2}\right)=\left({\mathbf P_b}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2\left(E_2E_1^'-\vec p_2\cdot \vec p_1^'\right)+ m_1^{'2}\right)=\left({\mathbf P_c}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2\left(E_2E_2^'-\vec p_2\cdot \vec p_2^'\right)+ m_2^{'2}\right)=\left({\mathbf P_d}\right)^2=s$

Using the fact that energy is conserved in this elastic collision

$\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2\left((E-E_2)(E-E_2^')-\vec p_1\cdot \vec p_1^'\right)+ m_1^{'2}\right)=\left({\mathbf P_a}\right)^2=s$

$\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2\left((E-E_2)(E-E_1^')-\vec p_1\cdot \vec p_2^'\right)+ m_2^{'2}\right)=\left({\mathbf P_b}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2\left((E-E_1)(E-E_2^')-\vec p_2\cdot \vec p_1^'\right)+ m_1^{'2}\right)=\left({\mathbf P_c}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2\left((E-E_1)(E-E_1^')-\vec p_2\cdot \vec p_2^'\right)+ m_2^{'2}\right)=\left({\mathbf P_d}\right)^2=s$

$\left({\mathbf P_1}- {\mathbf P_1^'}\right)^2=\left( m_1^2-2\left((E^2-EE_2-EE_2^'+E_2E_2^')-\vec p_1\cdot \vec p_1^'\right)+ m_1^{'2}\right)=\left({\mathbf P_a}\right)^2=s$

$\left({\mathbf P_1}- {\mathbf P_2^'}\right)^2=\left( m_1^2-2\left((E^2-EE_2-EE_1^'+E_2E_1^')-\vec p_1\cdot \vec p_2^'\right)+ m_2^{'2}\right)=\left({\mathbf P_b}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_1^'}\right)^2=\left( m_2^2-2\left((E^2-EE_1-EE_2^'+E_1E_2^')-\vec p_2\cdot \vec p_1^'\right)+ m_1^{'2}\right)=\left({\mathbf P_c}\right)^2=s$

$\left({\mathbf P_2}- {\mathbf P_2^'}\right)^2=\left( m_2^2-2\left((E^2-EE_1-EE_1^'+E_1E_1^')-\vec p_2\cdot \vec p_2^'\right)+ m_2^{'2}\right)=\left({\mathbf P_d}\right)^2=s$

Mandelstam Representation