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| | <center><math>\underline{\textbf{Navigation}}</math></center> |
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| | [[Lorentz_Transformation_to_Lab_Frame|<math>\vartriangleleft </math>]] |
| | [[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]] |
| | [[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]] |
| | </center> |
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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> | | <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_1\vec{p}_2- E_2\vec {p}_1|</math></center>
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| As shown earlier
| | <center><math>F_{cms}=4 \vec p_{1}^{*}\sqrt {s}</math></center> |
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| The s quantity is known as the square of the center of mass energy (invariant mass)
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| <center><math>s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2</math></center>
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| | <center><math>d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ</math></center> |
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| <center><math>s \equiv \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2} \equiv \mathbf P_1^{'*2}+2 \mathbf P_1^{'* }\mathbf P_2^{'*}+ \mathbf P_2^{'*2}</math></center> | | <center><math>d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'} d\Omega</math></center> |
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| | <center><math>(E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2}</math></center> |
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| As shown earlier, the square of a 4-momentum is
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| | <center><math>E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'}</math></center> |
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| <center><math>\mathbf P^{2} \equiv m^2</math></center>
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| This gives,
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| <center><math>s \equiv m_1^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2} \equiv m_1^{'2}+2 \mathbf P_1^{'*} \mathbf P_2^{'*}+ m_2^{'2}</math></center>
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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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| For the case <math>m_1=m_2=m</math>
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| | <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>s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^* \equiv 2m^{2}+2 \mathbf P_1^{'*} \mathbf P_2^{'*}</math></center>
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| Using the relationship
| | <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> |
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| <center><math>\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_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>s \equiv 2m^2+2(E_1^*E_2^*-\vec p \ _1^* \vec p \ _2^*)</math></center>
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| In the center of mass frame of reference,
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| <center><math>E_1^*=E_2^* \quad and \quad \vec p \ _1^*=-\vec p \ _2^*</math></center>
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| <center><math>s_{CM} \equiv 2m^2+2E_1^{*2}+2\vec p_1 \ ^{*2} </math></center>
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| Using the relativistic energy equation
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| <center><math>E^2 \equiv \vec p_1 \ ^2+m^2</math></center>
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| <center><math>s_{CM} \equiv 2m^2+2m^2+2\vec p_1 \ ^{*2}+\vec p_1 \ ^{*2})</math></center>
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| | <center><math>dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'}</math></center> |
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| <center><math>s_{CM}=4(m^2+\vec p_1 \ ^{*2})=(2E_1^*)^{2}</math></center>
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| The invariant form of F is
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| <center><math>F=4\sqrt{(\vec {p}_1 \cdot \vec {p}_2)^2-(m_1m_2)^2}</math></center>
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| <center><math>F_{cms}=4 \vec p_i\sqrt {s}</math></center>
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| <center><math>d\sigma=\frac{1}{4 \vec p_i\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>
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| <center><math>d^3 \vec p_1^'=\vec p^{'3}_1 d \vec p^' d\Omega</math></center>
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| <center><math>(E_1^')^2=(\vec p_1^')^2+(m_1)^2</math></center>
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| <center><math>E_1^' d E_1^'= \vec p_1^' d \vec p_1^'</math></center>
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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>
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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>
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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>
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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> |
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| <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> |
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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>\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><math>dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_f}{\sqrt s}d\Omega</math></center>
| | ---- |
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| | <center><math>\underline{\textbf{Navigation}}</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> |
| | [[Lorentz_Transformation_to_Lab_Frame|<math>\vartriangleleft </math>]] |
| | [[VanWasshenova_Thesis#Weighted_Isotropic_Distribution_in_Lab_Frame|<math>\triangle </math>]] |
| | [[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]] |
| | </center> |
[math]\displaystyle{ \underline{\textbf{Navigation}} }[/math]
[math]\displaystyle{ \vartriangleleft }[/math]
[math]\displaystyle{ \triangle }[/math]
[math]\displaystyle{ \vartriangleright }[/math]
Relativistic Differential Cross-section
[math]\displaystyle{ d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ }[/math]
dQ is the invariant Lorentz phase space factor
[math]\displaystyle{ 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]\displaystyle{ F_{cms}=4 \vec p_{1}^{*}\sqrt {s} }[/math]
[math]\displaystyle{ d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ }[/math]
[math]\displaystyle{ d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'} d\Omega }[/math]
[math]\displaystyle{ (E_{1}^{'})^2=(\vec p_{1}^{'})^2+(m_{1})^{2} }[/math]
[math]\displaystyle{ E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'} }[/math]
[math]\displaystyle{ 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]\displaystyle{ W_{i} \equiv E_{1}+E_{2} \qquad \qquad W_f \equiv E_{1}^{'}+E_{2}^{'} }[/math]
[math]\displaystyle{ 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]\displaystyle{ |\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]\displaystyle{ dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'} }[/math]
[math]\displaystyle{ dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_{f} dW_{f}}{W_{f}}d\Omega }[/math]
[math]\displaystyle{ dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_{f}}{\sqrt {s}}d\Omega }[/math]
[math]\displaystyle{ \frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2 }[/math]
[math]\displaystyle{ \underline{\textbf{Navigation}} }[/math]
[math]\displaystyle{ \vartriangleleft }[/math]
[math]\displaystyle{ \triangle }[/math]
[math]\displaystyle{ \vartriangleright }[/math]