Cross sections are often expressed in millibarns, where 1 $\mathrm{mb}=10^{-3} \mathrm{~b}=10^{-27} \mathrm{~cm}^2$. Using $\mathrm{GeV}$ units, show that
$$
1 \mathrm{GeV}^{-2}=0.389 \mathrm{mb} \text {. }
$$
So far, we have not considered the elementary charge $e$, which measures how strongly electrons, say, interact electromagnetically with each other. To obtain a dimensionless measure of the strength of this interaction, we compare the electrostatic energy of repulsion between two electrons one natural unit of length apart with the rest mass energy of an electron:
$$
\alpha=\frac{1}{4 \pi} \frac{e^2}{(\hbar / m c)} / m c^2=\frac{e^2}{4 \pi \hbar c} \approx \frac{1}{137} .
$$
Conventional Mass, Length, Time Units, and Positron Charge in Terms of $h=c=1$ Energy Units
$$
\begin{array}{lcc}
\hline \text { Conversion Factor } & \begin{array}{l}
\hbar=c=1 \\
\text { Units }
\end{array} & \begin{array}{c}
\text { Actual } \\
\text { Dimension }
\end{array} \\
\hline 1 \mathrm{~kg}=5.61 \times 10^{26} \mathrm{GeV} & \mathrm{GeV} & \frac{\mathrm{GeV}}{c^2} \\
1 \mathrm{~m}=5.07 \times 10^{15} \mathrm{GeV}^{-1} & \mathrm{GeV}^{-1} & \frac{\hbar c}{\mathrm{GeV}} \\
1 \mathrm{sec}=1.52 \times 10^{24} \mathrm{GeV}^{-1} & \mathrm{GeV}^{-1} & \frac{\hbar}{\mathrm{GeV}} \\
e=\sqrt{4 \pi \alpha} & - & (\hbar c)^{1 / 2} \\
\hline
\end{array}
$$
TABLE $1.2 \mathrm{~b}$
Some Useful Conversion Factors
In (1.3), we have adopted the rationalized Heaviside-Lorentz system of electromagnetic units. That is, the $4 \pi$ factors appear in the force equations rather than in the Maxwell equations, and $\varepsilon_0$ is set equal to unity. This choice, which is conventional in particle physics, reduces Maxwell's equations to their simplest possible form. The value of $\alpha$ is, of course, the same in all systems of units, but the numerical value of $e$ is different.
For historical reasons, $\alpha$ is known as the fine structure constant. Unfortunately, this name conveys a false impression. We have seen that the charge of an electron is not strictly constant but varies with distance because of quantum effects; hence $\alpha$ must be regarded as a variable, too. The value $\frac{1}{137}$ is the asymptotic value of $\alpha$ shown in Fig. 1.5a.