The Basic Forces and Messenger Particles — Question 4

Question 4

A heavy propagator reduces to a contact interaction at low momentum

In a toy spacelike exchange amplitude use $A(Q)=g^2/(M^2+Q^2)$ with natural units. The contact approximation is $A_0=g^2/M^2$. Let M=80 GeV.

1. Find the exact-to-contact rate ratio at Q=8 GeV.

2. Find the maximum Q for which the exact amplitude is at least 95 percent of the contact amplitude.

Question 4 - Solution

A heavy propagator reduces to a contact interaction at low momentum

(a) Method and calculation

$$\frac{|A(Q)|^2}{|A_0|^2}=\left[1+(Q/M)^2\right]^{-2}.$$$$\boxed{R(8)=1.01^{-2}=.980296}.$$

(b) Interpretation and further calculation

$$[1+(Q/M)^2]^{-1}\ge.95\Rightarrow Q/M\le\sqrt{1/.95-1}.$$$$\boxed{Q\le18.3533\ {\mathrm{GeV}}}.$$

Physical check

The amplitude and rate errors differ. This scalar toy denominator isolates the scale dependence rather than a full electroweak cross section.