Difference between revisions of "Variables Used in Elastic Scattering"
Jump to navigation
Jump to search
where
represents the 4-Momentum Vector in the CM frame and
represents the 4-Momentum Vector in the initial Lab frame
where
represents the 4-Momentum Vector in the final Lab frameand
represents the 4-Momentum Vector in the final CM frame
![CM.png](/./images/thumb/2/2c/CM.png/400px-CM.png)
Line 87: | Line 87: | ||
− | <center><math>\left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_1^{'*}-\vec p_1^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_1^{'*}-2\vec p_1^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</math></center> |
− | <center><math>\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_2^{'*}-\vec p_1^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_2^{'*}-2\vec p_1^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center> |
− | <center><math>\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_1^{'*}-\vec p_2^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_1^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</math></center> |
− | <center><math>\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_2^{'*}-\vec p_2^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_2^{'*}-2\vec p_2^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center> |
Line 110: | Line 110: | ||
We can further simplify | We can further simplify | ||
− | <center><math>\left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_1^{'*}+\vec p_2^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_1^{'*}+2\vec p_2^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_a^*}\right)^2</math></center> |
− | <center><math>\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_2^{'*}+\vec p_2^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_1^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_1^{*2}-2{E_1^*E_2^{'*}+2\vec p_2^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center> |
− | <center><math>\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_1^{'*}-\vec p_2^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_2^*}- {\mathbf P_1^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_1^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+ m_1^{'*2}\right)=\left({\mathbf P_c^*}\right)^2</math></center> |
− | <center><math>\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_2^{'*}-\vec p_2^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center> | + | <center><math>\left({\mathbf P_2^*}- {\mathbf P_2^{'*}}\right)^2=\left( m_2^{*2}-2{E_2^*E_2^{'*}-2\vec p_2^*\cdot \vec p_2^{'*}}+ m_2^{'*2}\right)=\left({\mathbf P_d^*}\right)^2</math></center> |
=Mandelstam Representation= | =Mandelstam Representation= | ||
[[File:Mandelstam.png | 400 px]] | [[File:Mandelstam.png | 400 px]] |
Revision as of 00:02, 1 February 2016
Lorentz Invariant Quantities
Total 4-Momentums
As was shown earlier the scalar product of a 4-Momentum vector with itself ,
,
and the length of a 4-Momentum vector composed of 4-Momentum vectors,
,
are invariant quantities.
It was further shown that
which can be expanded to
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
Using the algebraic fact
and the fact that the length of these 4-Momentum Vectors are invariant,
Using the fact that the scalar product of a 4-momenta with itself is invariant,
We can simiplify the expressions
Finding the cross terms,
Using the fact that in the CM frame,
![CM.png](/./images/thumb/2/2c/CM.png/400px-CM.png)
We can further simplify