Difference between revisions of "Relativistic Differential Cross-section"

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<center><math>\underline{\textbf{Navigation}}</math></center>
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<center>
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[[Lorentz_Transformation_to_Lab_Frame|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]]
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[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]]
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=Relativistic Differential Cross-section=
 
=Relativistic Differential Cross-section=
  
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<center><math>dQ=(2\pi)^4\delta^4(\vec p_1 +\vec p_2 - \vec p_1^' -\vec p_2^')\frac{d^3 \vec p_1^'}{(2\pi)^3 2E_1^'}\frac{d^3 \vec p_2^'}{(2\pi)^3 2E_2^'}</math></center>
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<center><math>dQ=(2\pi)^4\delta^4 \left(\vec p_{1} +\vec p_{2} - \vec p_{1}^{'} -\vec p_{2}^{'} \right)\frac{d^3 \vec p_{1}^{'}}{(2\pi)^3 2E_{1}^{'}}\frac{d^3 \vec p_{2}^{'}}{(2\pi)^3 2E_{2}^{'}}</math></center>
  
  
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<center><math>F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|E_1E_2\vec v_{21}|</math></center>
 
  
  
where <math>v_{21}</math> is the relative velocity between the particles in the frame where particle 1 is at rest
 
  
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<center><math>F_{cms}=4 \vec p_{1}^{*}\sqrt {s}</math></center>
  
<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)= E_{1}E_{2}</math></center>
 
  
  
Using the relativistic definition of energy
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<center><math>d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>
  
<center><math>E^2 \equiv p^2+m^2=m^2</math></center>
 
  
  
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<center><math>d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'}  d\Omega</math></center>
  
<center><math>\rightarrow \mathbf P_1 \cdot \mathbf P_2 =mE_{2}</math></center>
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<center><math>(E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center>
  
  
Letting <math>E_{21}\equiv E_2</math> be the energy of particle 2 wiith respect to particle 1, the relativistic energy equation can be rewritten such that
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<center><math>E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'}</math></center>
  
  
<center><math>|p_{21}^2| =E_{21}^2-m^2=\frac{(\mathbf P_1 \cdot \mathbf P_2)^2}{m^2}-m^2=\frac{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}{m^2}</math></center>
 
  
where similarly <math>p_{21}</math> is defined as the momentum of particle 2 with respect to particle 1.
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<center><math>dQ=\frac{1}{(4\pi)^2}\delta (E_{1}+E_{2}-E_{1}^{'}-E_{2}^{'})\frac{\vec p_{1}^{'}dE_{1}^{'}}{E_{2}^{'}}d\Omega</math><\center>
  
  
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<center><math>W_{i} \equiv E_{1}+E_{2}  \qquad \qquad W_f \equiv E_{1}^{'}+E_{2}^{'}</math></center>
  
  
The relative velocity can be expressed as
 
  
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<center><math>dW_f=dE_{1}^{'}+dE_{2}^{'}=\frac{\vec p_{1}^{'} d \vec p_{1}^{'}}{E_{1}^{'}}+\frac{p_{2}^{'} dp_{2}^{'}}{E_{2}^{'}}</math></center>
  
<center><math> v_{21}=\frac{|\vec p_{21}|}{E_{21}}</math></center>
 
  
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In the center of mass frame
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<center><math>|\vec p_{1}^{'}|=|\vec p_{2}^{'}|=|\vec p_{f}^{'}| \rightarrow |\vec p_{1}^{'} d \vec p_{1}^{'}|=|\vec p_{2}^{'} d \vec p_{2}^{'}|=|\vec p_{f}^{'} d \vec p_{f}^{'}|</math></center>
  
  
<center><math>F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|mE_{21}\vec v_{12}|=4|mE_{21}\frac{|\vec p_{21}|}{E_{21}}|=4m|\vec p_{21}|</math></center>
 
 
 
 
The invariant form of F is
 
 
<center><math>F=4\sqrt{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}</math></center>
 
 
 
<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)</math></center>
 
 
 
 
where in the center of mass frame <math>E_1=E_2</math> and <math> \vec p_1^*=-\vec p_2^*</math>
 
 
 
<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_1^2+\vec p_1\,^{*2}</math></center>
 
 
 
<center><math>F=4\sqrt{(E_1^2+\vec p_1\,^{*2})^2-m^4}=4\sqrt{(\vec p_1 \,^{*2}+m^2+\vec p_1\,^{*2})^2-m^4}</math></center>
 
 
 
 
<center><math>F=4\sqrt{(2\vec p_1\,^{*2}+m^2)^2-m^4}=4\sqrt{4\vec p_1\,^{*4}+m^4+4\vec p_1\,^{*2}m-m^4}</math></center>
 
 
 
 
 
<center><math>F_{cms}=4 \sqrt {4m^2\vec p_1\,^{*2}+4\vec p_1 \,^{*4}}</math></center>
 
 
 
 
<center><math>F_{cms}=4 \sqrt {\vec p_1\,^{*2}4(m^2+\vec p_1 \,^{*2})}</math></center>
 
 
 
 
As shown earlier
 
<center><math>s_{CM}=4(m^2+\vec p_1 \,^{*2})</math></center>
 
 
 
 
 
<center><math>F_{cms}=4 \sqrt {\vec p_1^*\,^2s}</math></center>
 
 
 
 
<center><math>F_{cms}=4 \vec p_1^*\sqrt {s}</math></center>
 
 
 
 
<center><math>d\sigma=\frac{1}{4 \vec p_1\,^*\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>
 
 
 
 
<center><math>d^3 \vec p_1^'=\vec p^{'3}_1 d \vec p^'  d\Omega</math></center>
 
 
 
<center><math>(E_1^')^2=(\vec p_1^')^2+(m_1)^2</math></center>
 
 
 
<center><math>E_1^' d E_1^'= \vec p_1^' d \vec p_1^'</math></center>
 
 
 
 
<center><math>dQ=\frac{1}{(4\pi)^2}\delta (E_1+E_2-E_1^'-E_2^')\frac{\vec p_1^'dE_1^'}{E_2^'}d\Omega</math><\center>
 
 
 
<center><math>W_i \equiv E_1+E_2  \qquad \qquad W_f \equiv E_1^'+E_2^'</math></center>
 
  
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<center><math>dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'}</math></center>
  
 
<center><math>dW_f=dE_1^'+dE_2^'=\frac{\vec p_1^' d \vec p_1^'}{E_1^'}+\frac{p_2^' dp_2^'}{E_2^'}</math></center>
 
 
 
In the center of mass frame
 
<center><math>|\vec p_1^'|=|\vec p_2^'|=|\vec p_f^'| \rightarrow |\vec p_1^' d \vec p_1^'|=|\vec p_2^' d \vec p_2^'|=|\vec p_f^' d \vec p_f^'|</math></center>
 
  
  
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<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_{f} dW_{f}}{W_{f}}d\Omega</math></center>
  
<center><math>dW_f=\frac{W_f}{E_2^'}dE_1^'</math></center>
 
  
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<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_{f}}{\sqrt {s}}d\Omega</math></center>
  
  
<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_f dW_f}{W_f}d\Omega</math></center>
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<center><math>\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2</math></center>
  
  
<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_f}{\sqrt s}d\Omega</math></center>
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----
  
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<center><math>\underline{\textbf{Navigation}}</math></center>
  
<center><math>\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_f}{\mathbf p_i}|\mathcal {M}|^2</math></center>
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<center>
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[[Lorentz_Transformation_to_Lab_Frame|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]]
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[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]]
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</center>

Latest revision as of 20:51, 29 December 2018

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]

Relativistic Differential Cross-section

[math]d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ[/math]

dQ is the invariant Lorentz phase space factor


[math]dQ=(2\pi)^4\delta^4 \left(\vec p_{1} +\vec p_{2} - \vec p_{1}^{'} -\vec p_{2}^{'} \right)\frac{d^3 \vec p_{1}^{'}}{(2\pi)^3 2E_{1}^{'}}\frac{d^3 \vec p_{2}^{'}}{(2\pi)^3 2E_{2}^{'}}[/math]


and F is the flux of incoming particles



[math]F_{cms}=4 \vec p_{1}^{*}\sqrt {s}[/math]


[math]d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ[/math]


[math]d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'} d\Omega[/math]


[math](E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2}[/math]


[math]E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'}[/math]


[math]dQ=\frac{1}{(4\pi)^2}\delta (E_{1}+E_{2}-E_{1}^{'}-E_{2}^{'})\frac{\vec p_{1}^{'}dE_{1}^{'}}{E_{2}^{'}}d\Omega[/math]<\center>


[math]W_{i} \equiv E_{1}+E_{2} \qquad \qquad W_f \equiv E_{1}^{'}+E_{2}^{'}[/math]


[math]dW_f=dE_{1}^{'}+dE_{2}^{'}=\frac{\vec p_{1}^{'} d \vec p_{1}^{'}}{E_{1}^{'}}+\frac{p_{2}^{'} dp_{2}^{'}}{E_{2}^{'}}[/math]


In the center of mass frame

[math]|\vec p_{1}^{'}|=|\vec p_{2}^{'}|=|\vec p_{f}^{'}| \rightarrow |\vec p_{1}^{'} d \vec p_{1}^{'}|=|\vec p_{2}^{'} d \vec p_{2}^{'}|=|\vec p_{f}^{'} d \vec p_{f}^{'}|[/math]


[math]dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'}[/math]


[math]dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_{f} dW_{f}}{W_{f}}d\Omega[/math]


[math]dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_{f}}{\sqrt {s}}d\Omega[/math]


[math]\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2[/math]



[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]