Is the "center-of-mass energy" in particle physics related to the center-of-mass frame or not? For collision of particles the invariant quantity "center-of-mass energy" is defined, as done here.
Since it is an invariant quantity it does not depend on the frame of reference chosen. So the name "center-of-mass" here is not to be related to the center-of-mass frame? Does this have anything to do with the center-of-mass frame or not?
 A: The center-of-mass energy is the energy as measured in the frame where:
$$0=\sum_i \vec p_i\tag{1}$$
i.e. the sum of all 3-momenta is zero.

In a general inertial frame $r$ where the 4-momenta is given by $P_i^r$ we have the quantity:
$$\Lambda^2=\left(\sum_i P_i^r\right)^\mu \left(\sum_j P_j^r\right)_\mu\tag{2}$$
here all Lorentz indices are contracted (using the $\eta=(+---)$ metric) and such $\Lambda$ is Lorentz invariant. 

Now let us calculate $\Lambda$ in the CofM frame (I actually prefer the term 'zero momentum frame') here we have that (due to (1)):
$$\sum_i P_i^r=\begin{pmatrix} E_{CM} \\0 \\0 \\0\end{pmatrix}$$
where $E_{cm}=\sum_i E_i$ hence subbing this into (2) we get:
$$\Lambda^2=E_{cm}^2$$
so $\Lambda=E_{cm}$ (spare a sign which we define to be $+$). 
Thus to summarize:


*

*$\Lambda$ is Lorentz invariant.

*When calculated in the cofm frame it is equal to the total energy $E_{cm}$.

*Thus in any frame it is equal to $E_{cm}$ hence we call $\Lambda$ the center of mass energy.

