The Basic Forces and Messenger Particles — Question 1

Question 1

A mediator mass sets the exponential range of a Yukawa potential

Use $V(r)=A e^{-r/\lambda}/r$ and $\lambda=\hbar c/(Mc^2)$, with $\hbar c=.1973$ GeV fm. Compare mediators of rest energies 80 GeV and .140 GeV.

1. Calculate both characteristic ranges and their ratio.

2. At three characteristic ranges, find the potential relative to an unscreened A/r potential at the same separation.

Question 1 - Solution

A mediator mass sets the exponential range of a Yukawa potential

(a) Method and calculation

$$\boxed{\lambda_{80}=.1973/80=.00246625\ {\mathrm{fm}}}.$$$$\boxed{\lambda_{.140}=1.40929\ {\mathrm{fm}},\quad\lambda_{.140}/\lambda_{80}=571.429}.$$

(b) Interpretation and further calculation

$$V(r)/(A/r)=e^{-r/\lambda}.$$$$\boxed{V(3\lambda)/(A/3\lambda)=e^{-3}=.0497871}.$$

Physical check

The .140 GeV scale models residual nuclear exchange, not the mass of a gluon. The exponential factor and geometric 1/r factor play different roles.