The Basic Forces and Messenger Particles — Question 5

Question 5

Weak effective-interaction scaling cannot be extrapolated indefinitely

In a low-energy model $\sigma=C G^2E^2$, where G has dimensions energy to the power -2 and C is dimensionless in natural units. The approximation requires E much smaller than a mediator mass M=80 GeV.

1. Find the cross-section ratio at energies 2 and 5 GeV and check dimensions.

2. If a reference unitarity scale is $\sigma_{\max}=4\pi/E^2$, solve symbolically where the extrapolated formulas meet.

Question 5 - Solution

Weak effective-interaction scaling cannot be extrapolated indefinitely

(a) Method and calculation

$$\boxed{\sigma(5)/\sigma(2)=(5/2)^2=6.25}.$$$$[G^2E^2]=E^{-4}E^2=E^{-2}\quad\text{as required for area in natural units}.$$

(b) Interpretation and further calculation

$$CG^2E^2=4\pi/E^2\Rightarrow E^4=4\pi/(CG^2).$$$$\boxed{E_{\mathrm{meet}}=(4\pi/C)^{1/4}G^{-1/2}}.$$

Physical check

The meeting scale is an estimate within the supplied toy comparison. A complete high-energy theory changes the low-energy growth.