Particles, Particles, Particles — Question 6

Question 6

Momentum resolution limits mass identification near the speed of light

Infer mass from $m=p\sqrt{1-\beta^2}/(\beta c)$. Momentum and speed errors are independent. At $\beta=.900$, let $\sigma_p/p=.0100$ and $\sigma_\beta=.00200$.

1. Derive the leading fractional standard uncertainty in mass.

2. Evaluate it and identify the dominant contribution.

Question 6 - Solution

Momentum resolution limits mass identification near the speed of light

(a) Method and calculation

$$d\ln m=d\ln p-\frac{d\beta}{\beta(1-\beta^2)}.$$$$\boxed{\left(\frac{\sigma_m}{m}\right)^2=\left(\frac{\sigma_p}{p}\right)^2+\left[\frac{\sigma_\beta}{\beta(1-\beta^2)}\right]^2}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_m/m=\sqrt{.0100^2+[.00200/(.900\cdot.190)]^2}=.0153881}.$$$$\boxed{\text{speed term}=.0116959>\text{momentum term}=.0100}.$$

Physical check

The speed sensitivity diverges as beta approaches one. This is linear uncertainty propagation and assumes errors small enough for local derivatives to apply.