The Basic Forces and Messenger Particles — Question 7

Question 7

A supplied running coupling decreases at shorter distances

Use the one-loop-shaped model $\alpha(Q)=1/[b\ln(Q^2/\Lambda^2)]$, with b=.60 and Lambda=.20 GeV. Apply only for Q much larger than Lambda.

1. Calculate alpha at Q=2 and 20 GeV.

2. Find the coupling ratio and the ratio of rates proportional to alpha squared with other factors held fixed.

Question 7 - Solution

A supplied running coupling decreases at shorter distances

(a) Method and calculation

$$\boxed{\alpha(2)=1/[.60\ln100]=.361912}.$$$$\boxed{\alpha(20)=1/[.60\ln10000]=.180956}.$$

(b) Interpretation and further calculation

$$\boxed{\alpha(20)/\alpha(2)=1/2}.$$$$\boxed{R_{20}/R_2=(1/2)^2=1/4}.$$

Physical check

This comparison isolates coupling dependence; actual cross sections also depend on energy, phase space and parton distributions. The model fails near its pole.