I have a question regarding the usage of the centre-of-mass definition (which I thought required the total momentum, $p_T=0$) in the solution to the question given below:
A particular centre-of-mass energy is needed to create a new particle. We will do the calculation in a so-called fixed-target configuration.
A particle of mass $m_1$ and total energy $E_1$ in the lab frame hits a stationary particle of mass $m_2$. Show that the required particle energy for a given $s$ is:
$$E_1=\frac{s-m_1^2c^4-m_2^2c^4}{2m_2c^2}$$ where $s$ is the square of the centre-of-mass energy. This is often called a ‘fixed target’ configuration as experiments were historically often done by colliding a beam of particles with a stationary target material.
The solution says:
In the fixed target experiment, the total energy $E_T=E_1+m_2c^2$ and the total momentum magnitude is $p_Tc=\sqrt{E_1^2-m_1^2c^4}$. The square of the centre-of-mass energy is, therefore, $$s=m_T^2c^4=E_T^2-p_T^2c^2=E_1^2+2E_1m_2c^2+m_2^2c^4-E_1^2+m_1^2c^4$$ $$=2E_1m_2c^2+m_1^2c^4+m_2^2c^4$$ Rearranging, gives the required particle energy: $$E_1=\frac{s-m_1^2c^4-m_2^2c^4}{2m_2c^2}$$
But, in the centre-of-mass frame $p_T=0$, as shown in this image* below:
So therefore,
$$p_T=\sqrt{E_1^2-m_1^2c^4}=0$$ and as a result, $$E_1=m_1c^2$$
My question is, why is the author using the centre-of-mass definition (which I thought required zero total momentum) when the total momentum is actually non-zero (only one particle is stationary)?
*Image in body is from ICL dept. of Physics
