A Pause for Reflection — Question 2

Question 2

A two-body decay can reconstruct its parent without knowing the boost

Two massless daughters have laboratory energies 4 and 9 GeV with opening angle 60 degrees. Use c=1.

1. Calculate their parent invariant mass and laboratory Lorentz factor.

2. Find their common energy in the parent rest frame and check why laboratory energies can differ.

Question 2 - Solution

A two-body decay can reconstruct its parent without knowing the boost

(a) Method and calculation

$$M^2=2E_1E_2(1-\cos\theta)=2(4)(9)(1-.5)=36.$$$$\boxed{M=6\ {\mathrm{GeV}},\quad\gamma=(4+9)/6=13/6}.$$

(b) Interpretation and further calculation

$$\boxed{E_1^*=E_2^*=M/2=3\ {\mathrm{GeV}}}.$$$$E_{1,2}=\gamma E^*(1\pm\beta\cos\theta^*),\quad\boxed{E_1+E_2=\gamma M}.$$

Physical check

Invariant mass is independent of the parent motion, whereas individual daughter energies are not.