# What's the physical meaning of the length dependance of the running coupling constants?

I often hear this explanation and there's something that bugs me every time. What does it mean? Does the constant's value change when we observe it at different length scales? Is it something that changes with the observer then? Is it related to the energy scale of the particle or to the energy scale of the device necessary to measure it?

• Related answer here. – knzhou Sep 23 '16 at 1:06

For instance, consider Coulomb's law. In the mass scale, in the limit where $\vec{p}^{2} \gg m^{2}$, this law holds for an energy-dependant charge $$e_{\mu^{\prime}}^{2} = \frac{e_{\mu}^{2}}{1 - b e_{\mu}^{2} \ln \left(\frac{\mu^{\prime 2}}{\mu^{2}}\right)},$$ where $b$ is a constant (unimportant for the sake of this discussion), $\mu^{\prime}$ is a cutoff and $\mu$ is the energy scale. As the energy changes, so does the charge.
However, in the limit where $\vec{p}^{2} \ll m^{2}$, $e_{\mu^{\prime}}^{2}$ acquires a fixed value, $$\frac{e_{\mu^{\prime}}^{2}}{4\pi} = \frac{m}{\sqrt{c}},$$ and thus the constant stops running, so Coulomb's law holds for this constant value of the charge.