S-Channel

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s Channel

The s quantity is known as the square of the center of mass energy (invariant mass)

[math]\displaystyle{ s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2 }[/math]

[math]\displaystyle{ 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]

As shown earlier, the square of a 4-momentum is

[math]\displaystyle{ \mathbf P^{2} \equiv m^2 }[/math]

This gives,

[math]\displaystyle{ 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]

For the case [math]\displaystyle{ m_1=m_2=m }[/math]

[math]\displaystyle{ s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^* \equiv 2m^{2}+2 \mathbf P_1^{'*} \mathbf P_2^{'*} }[/math]

Using the relationship

[math]\displaystyle{ \mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2) }[/math]

[math]\displaystyle{ s \equiv 2m^2+2(E_1^*E_2^*-\vec p \ _1^* \vec p \ _2^*) }[/math]

In the center of mass frame of reference,

[math]\displaystyle{ E_1^*=E_2^* \quad and \quad \vec p \ _1^*=-\vec p \ _2^* }[/math]

[math]\displaystyle{ s_{CM} \equiv 2m^2+2E_1^{*2}+2\vec p_1 \ ^{*2} }[/math]

Using the relativistic energy equation

[math]\displaystyle{ E^2 \equiv \vec p_1 \ ^2+m^2 }[/math]

[math]\displaystyle{ s_{CM} \equiv 2m^2+2m^2+2\vec p_1 \ ^{*2}+\vec p_1 \ ^{*2}) }[/math]

[math]\displaystyle{ s_{CM}=4(m^2+\vec p_1 \ ^{*2})=(2E_1^*)^{2} }[/math]

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