Mass from interactions

Classical Mechanics Level 5

In high energy physics, fundamental particles acquire mass through their interactions with the Higgs boson, and the Higgs particles themselves can acquire mass through by interacting with other Higgs. The interaction of fundamental particles with the actual Higgs is a little too complicated for a problem, but we can ask a simpler problem that will illuminate how it's possible. Our problem is: how is it possible that an interaction can give otherwise massless particles a mass/rest energy?

To see this, consider two massless particles that move around a central point (evenly spaced from each particle), thereby making a bound state and traveling together as one particle. Go to the frame where this central point is not moving at all. Assume that the distance between the massless particles is unchanged with time, and the rest mass of this 2-particle bound state is defined as the lowest possible energy/\(c^2\).

Find the rest mass of two particle bound state in ng.

Details and assumptions

  • The potential energy between the particles is \(V(r)=\alpha e^{\beta r}r^{-1/2}\) (where \(r\) is the distance between these particles, \(\alpha= 1 ~\mbox{Nm}^{3/2}\) and \(\beta=10^{15}~\mbox{m}^{-1}~=1~\mbox{fm}^{-1}~\))

  • Since the particles are massless, their speed is always equal to the speed of light \(c=3 \times 10^{8} ~\mbox{m/s}\) and their energy is defined as \(E=|\vec{p}|c=pc\) where \(\vec{p}\) is their momentum.

  • From the Einstein's equation, we have the relation between the rest energy \(E_r\) and mass \(m\) is \(E_r=mc^2\)

  • \(1~\mbox{ng} = 10^{-9}~\mbox{g}=10^{-12} ~\mbox{kg}\)


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