- #1

etotheipi

2019 Award

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## Main Question or Discussion Point

Wikipedia says this about the missing mass of a reaction:

Is this just then an arbitrary definition, which is not directly related to ##M^2 = E^2 - p^2##? Thank you .

I wondered where such an expression is coming from? The invariant mass of a system ##M##, in natural units, satisfies $$M^2 = \left(\sum E \right)^2 - ||\sum \mathbf{p}||^2$$ If anything, then the "missing" mass (which they also, confusingly, term as invariant mass) should go as $$\Delta [M^2] = \Delta \left[ \left(\sum E \right)^2 \right] - \Delta \left[ ||\sum \mathbf{p}||^2 \right]$$ I'm not sure why the equation they give is valid, since surely it is incorrect to bring the ##\Delta## inside the squared terms?The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units): $$W^2 = \left(\sum E_{in} - \sum E_{out} \right)^2 - ||\sum \mathbf{p}_{in} - \sum \mathbf{p}_{out}||^2$$

Is this just then an arbitrary definition, which is not directly related to ##M^2 = E^2 - p^2##? Thank you .