Variables Used in Elastic Scattering - Revision history

Vanwdani: Blanked the page

2017-06-01T19:10:31Z

Blanked the page

Vanwdani: /* Mandelstam Representation */

2016-02-01T15:17:14Z

Mandelstam Representation

Vanwdani: /* Mandelstam Representation */

2016-02-01T15:14:00Z

Mandelstam Representation

Vanwdani: /* Mandelstam Representation */

2016-02-01T00:36:00Z

Mandelstam Representation

Vanwdani: /* New 4-Momentum Quantities */

2016-02-01T00:35:22Z

New 4-Momentum Quantities

Vanwdani: /* New 4-Momentum Quantities */

2016-02-01T00:33:21Z

New 4-Momentum Quantities

Vanwdani: /* New 4-Momentum Quantities */

2016-02-01T00:32:36Z

New 4-Momentum Quantities

Vanwdani: /* New 4-Momentum Quantities */

2016-02-01T00:28:05Z

New 4-Momentum Quantities

Vanwdani: /* New 4-Momentum Quantities */

2016-02-01T00:24:53Z

New 4-Momentum Quantities

Vanwdani: /* New 4-Momentum Quantities */

2016-02-01T00:02:21Z

New 4-Momentum Quantities

← Older revision		Revision as of 19:10, 1 June 2017
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−	~~=Lorentz Invariant Quantities=~~
−	~~==Total 4-Momentums==~~
−	~~As was [[DV_Calculations_of_4-momentum_components#4-Momentum_Invariants \| shown earlier]] the scalar product of a 4-Momentum vector with itself ,~~
−	~~<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2=s</math></center> ,~~

−	~~and the length of a 4-Momentum vector composed of 4-Momentum vectors,~~
−
−	~~<center><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></center>,~~
−
−	~~are invariant quantities.~~
−
−	~~It was [[DV_Calculations_of_4-momentum_components#Equal_masses \| further shown ]] that~~
−
−	~~<center><math>{\mathbf P^*}^2={\mathbf P}^2</math></center>~~
−
−
−	~~<center>''where'' <math>{\mathbf P^}=({\mathbf P_1^}+{\mathbf P_2^*})^2</math> ''represents the 4-Momentum Vector in the CM frame''</center>~~
−
−
−	~~<center> ''and'' <math>{\mathbf P}=({\mathbf P_1}+{\mathbf P_2})^2</math> ''represents the 4-Momentum Vector in the initial Lab frame''</center>~~
−
−	~~which can be expanded to~~
−
−	~~<center><math>{\mathbf P^}^2={\mathbf P^{'}}^2={\mathbf P}^2={\mathbf P^'}^2</math></center>~~
−
−
−	~~<center>''where'' <math>{\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2</math> ''represents the 4-Momentum Vector in the final Lab frame''</center>~~
−
−
−	~~<center>''and'' <math>{\mathbf P^{'}}=({\mathbf P_1^{'}}+{\mathbf P_2^{'*}})^2</math> ''represents the 4-Momentum Vector in the final CM frame''</center>~~
−
−	~~==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~~
−
−	<center><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></center>
−
−
−	<center><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></center>
−
−
−	<center><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></center>
−
−
−	<center><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></center>
−
−	~~Using the algebraic fact~~
−
−	~~<center><math>\left({\mathbf a}- {\mathbf b}\right)^2=\left({\mathbf b}- {\mathbf a}\right)^2</math></center>~~
−
−
−	~~and the fact that the length of these 4-Momentum Vectors are invariant,~~
−
−	<center><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</math></center>
−
−
−	<center><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</math></center>
−
−
−	<center><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</math></center>
−
−
−	<center><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</math></center>
−
−	~~Using the fact that the scalar product of a 4-momenta with itself is invariant,~~
−
−
−
−	~~<center><math>{\mathbf P_1}\cdot {\mathbf P^1}=E_1E_1-\vec p_1\cdot \vec p_1 =m_{1}^2</math></center>~~
−
−
−	~~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^{'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{\mathbf P_1^}\cdot {\mathbf 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{\mathbf P_2^}\cdot {\mathbf 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{\mathbf P_2^}\cdot {\mathbf P_2^{'}}+ m_2^{'2}\right)=\left({\mathbf P_d^*}\right)^2</math></center>~~
−
−	~~Finding the cross terms,~~
−
−	<center><math>{\mathbf P_1^}\cdot {\mathbf P^{'}}=\left(\begin{matrix} E_1^\\ p_{1(x)}^ \\ p_{1(y)}^* \\ p_{1(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_{1(x)}^{'} & p_{1(y)}^{'} & p_{1(z)}^{'} \end{matrix} \right)=E_1^E_1^{'}-\vec p_1^\cdot \vec p_1^{'} </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^{'}-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^{'}-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^{'}-2\vec p_2^\cdot \vec p_2^{'}}+ m_2^{'2}\right)=\left({\mathbf P_d^*}\right)^2</math></center>~~
−
−
−	~~Using the fact that in the CM frame,~~
−
−	~~<center>[[File:CM.png \|400 px]]</center>~~
−
−
−	~~<center><math>\vec p_1^=-\vec p_2^</math></center>~~
−
−
−	~~<center><math>\vec p_1^{'}=-\vec p_2^{'}</math></center>~~
−
−
−	~~Since this is an ellastic collision between identical particles, Energy is conserved,~~
−
−	~~<center><math>E_1^=E_1^{'}</math></center>~~
−
−
−	~~<center><math>E_2^=E_2^{'}</math></center>~~
−
−
−	~~Lastly as [[DV_Calculations_of_4-momentum_components#Scattered_and_Moller_Electron_energies_in_CM \|shown earlier]], <math>E_1^=E_2^</math>~~
−
−
−	~~We can further simplify~~
−
−	~~<center><math>\left({\mathbf P_1^}- {\mathbf P_1^{'}}\right)^2=\left( m_1^{2}-2{E_2^E_2^{'}-2\vec p_2^\cdot \vec p_2^{'}}+ m_1^{'2}\right)=\left({\mathbf P_a^*}\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>~~
−
−
−
−	~~<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_1^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_2^{'}}\right)^2</math></center>~~
−
−
−
−
−	~~<center><math>\left({\mathbf P_1^}- {\mathbf P_2^{'}}\right)^2=\left( m_1^{2}-2{E_2^E_1^{'}-2\vec p_2^\cdot \vec p_1^{'}}+ 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^{'}-2\vec p_2^\cdot \vec p_1^{'}}+ m_1^{'2}\right)=\left({\mathbf P_c^*}\right)^2</math></center>~~
−
−
−
−	~~<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_2^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_1^{'}}\right)^2</math></center>~~
−
−	~~=Mandelstam Representation=~~
−
−	~~<center>[[File:Mandelstam.png \| 400 px]]</center>~~
−
−
−	~~<center><math>\Longrightarrow \left({\mathbf P_1^}+ {\mathbf P_2^{}}\right)^2=\left({\mathbf P_1^{'}}+ {\mathbf P_2^{'}}\right)^2\equiv s</math></center>~~
−
−
−
−	~~<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_1^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_2^{'}}\right)^2\equiv t</math></center>~~
−
−
−
−	~~<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_2^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_1^{'}}\right)^2\equiv s</math></center>~~

← Older revision		Revision as of 15:17, 1 February 2016
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−	<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_1^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_2^{'}}\right)^2</math></center>	+	<center><math>\Longrightarrow \left({\mathbf P_1^}+ {\mathbf P_2^{}}\right)^2=\left({\mathbf P_1^{'}}+ {\mathbf P_2^{'}}\right)^2\equiv s</math></center>
		+
		+
		+
		+	<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_1^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_2^{'}}\right)^2\equiv t</math></center>
		+
		+
		+
		+	<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_2^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_1^{'}}\right)^2\equiv s</math></center>

← Older revision		Revision as of 15:14, 1 February 2016
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	<center>[[File:Mandelstam.png \| 400 px]]</center>		<center>[[File:Mandelstam.png \| 400 px]]</center>
		+
		+
		+	<center><math>\Longrightarrow \left({\mathbf P_1^}- {\mathbf P_1^{'}}\right)^2=\left({\mathbf P_2^}- {\mathbf P_2^{'}}\right)^2</math></center>

← Older revision		Revision as of 00:36, 1 February 2016
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	=Mandelstam Representation=		=Mandelstam Representation=

−	[[File:Mandelstam.png \| 400 px]]	+	<center>[[File:Mandelstam.png \| 400 px]]</center>

@@ Line 127: / Line 127: @@
 <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 133: / Line 139: @@
 <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>
 =Mandelstam Representation=
 [[File:Mandelstam.png | 400 px]]

@@ Line 110: / Line 110: @@
-Since this is an ellastic collision, Energy is conserved,
+Since this is an ellastic collision between identical particles, Energy is conserved,
 <center><math>E_1^*=E_1^{'*}</math></center>
@@ Line 118: / Line 118: @@
-Lastly
+Lastly as [[DV_Calculations_of_4-momentum_components#Scattered_and_Moller_Electron_energies_in_CM |shown earlier]], <math>E_1^*=E_2^*</math>
 We can further simplify
-<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_2^{'*}}+  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_2^*E_2^{'*}-2\vec p_2^*\cdot \vec p_2^{'*}}+  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^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+  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_2^*E_1^{'*}-2\vec p_2^*\cdot \vec p_1^{'*}}+  m_2^{'*2}\right)=\left({\mathbf P_b^*}\right)^2</math></center>

@@ Line 105: / Line 105: @@
 <center><math>\vec p_1^*=-\vec p_2^*</math></center>
 <center><math>\vec p_1^{'*}=-\vec p_2^{'*}</math></center>
 We can further simplify

@@ 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
-<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=
 [[File:Mandelstam.png | 400 px]]