A Pause for Reflection — Question 7

Question 7

A tiny reconstructed mass is sensitive to the opening-angle calibration

Two photons of equal energy E=10 GeV have small opening angle theta=.020 rad, with uncertainty .002 rad. Energy uncertainties are negligible.

1. Use the exact invariant relation to derive the small-angle mass formula and estimate M.

2. Propagate the angle uncertainty and compare its fractional size with a 1 percent common energy-scale error.

Question 7 - Solution

A tiny reconstructed mass is sensitive to the opening-angle calibration

(a) Method and calculation

$$M^2=2E^2(1-\cos\theta)\simeq E^2\theta^2.$$$$\boxed{M\simeq E\theta=.200\ {\mathrm{GeV}}/c^2}\quad(c=1\text{ in the derivation}).$$

(b) Interpretation and further calculation

$$\boxed{\sigma_M\simeq E\sigma_\theta=.020\ {\mathrm{GeV}}/c^2,\quad\sigma_M/M=.10}.$$$$\boxed{\delta M/M=\delta E/E=.01}\quad\text{for a common energy-scale shift}.$$

Physical check

The angle uncertainty dominates this example. Nearly collinear photons make a small invariant mass despite a large total laboratory energy.