S-Channel

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s Channel

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

$s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2$

$s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2$

$s \equiv \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2}$

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

$\mathbf P^{2} \equiv m^2$

$s \equiv m^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2}$

This gives

$s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^*$

Similarly, the scalar product of two 4-momentums

$s \equiv 2m^2+2(E_1^*E_2^*-\vec p_1^* \vec p_2^*)$

In the center of mass frame of reference,

$E_1^*=E_2^* \quad and \quad \vec p_1^*=-\vec p_2^*$

$s \equiv 2m^2+2(E_1^{*2}+\vec p_1^{*2} )$

Using the relativistic energy equation

$E^2 \equiv p^2+m^2$

$s \equiv 2m^2+2((m^2+p_1^{*2})+p_1^{*2})$

$s=4(m_{CM}^2+p_{CM}^2)$

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