A Pause for Reflection — Question 1

Question 1

Missing four-momentum tests an invisible-particle hypothesis

In a collider center-of-mass frame use c=1. Initial four-momentum is $(10,\mathbf0)$ GeV. The entire visible system has energy 6 GeV and momentum $(0,0,3)$ GeV.

1. Find the missing four-momentum and invariant mass.

2. Determine whether the missing system could be exactly one massless neutrino; state an allowed alternative.

Question 1 - Solution

Missing four-momentum tests an invisible-particle hypothesis

(a) Method and calculation

$$\boxed{P_{\mathrm{miss}}=(4,0,0,-3)\ {\mathrm{GeV}}}.$$$$\boxed{m_{\mathrm{miss}}=\sqrt{4^2-3^2}=\sqrt7=2.646\ {\mathrm{GeV}}}.$$

(b) Interpretation and further calculation

$$\text{One massless particle requires }E_{\mathrm{miss}}^2-|\mathbf p_{\mathrm{miss}}|^2=0.$$$$\boxed{m_{\mathrm{miss}}^2=7>0:\ \text{one massless neutrino is excluded}}.$$

Physical check

Multiple noncollinear massless particles can have positive combined invariant mass; so can one massive invisible particle. Detector losses must also be considered.