Virtual particles and screening of charges

[PeterPumpkin]

I'm reading The Lightness of Being by Frank Wilczek.

In a footnote talking about screening of a (real) positive charge by virtual particles (p47), he says "Thus the force falls off faster than 1 over the distance squared, as you'd have without screening" (by virtual particles).

How then, when we do experiments in the lab, do we find an inverse square law for (real) charges?

 

[vanhees71]

Frank Wilczek refers to QED here, and QED is under most circumstances involving a few elementary particles perturbative. The relevant coupling is the fine-structure constant $$\alpha \approx 1/137$$ and thus very small. Thus, quantum effects are very small. Here, Wilczek talks about the vacuum fluctuations of the electromagnetic field. The leading-order result is an electron-positron loop of the photon selfenergy. This means that a virtual electron-positron pair is excited and reabsorped by the same photon.

In empty space, you cannot observe this tiny effect, but if you bring in a real charge (say an electron), you find tiny changes to Coulomb's Law, i.e., the electrostatic potential doesn't go like $$1/r$$ but, in case of the Abelian gauge theory QED, is screened. With a grain of salt you can interpret this as a cloud of virtual electron-positron pairs, which shield the bare electric charge of the electron.

The main effect is that the electromagnetic coupling, i.e., the fine-structure constant becomes dependent on the momentum scale with which a charge is probed. E.g. from electron-positron scattering at high energies (around the Z-boson mass of about 90 GeV), you measure a fine-structure constant of about 1/128. The reason is that at higher energies you probe the charges of electron and positron at smaller distances, and thus their charge appears less "screened" by the virtual charge cloud than at very low energies, where von usually gives the value of 1/137, relevant for atomic physics.

Now, the interesting thing is that non-abelian gauge theories can show the opposite behavior: In QCD, which is a non-abelian gauge theory based on the color SU(3) gauge-symmetry group. There the same calculation (one loop) of the gluon self-energy leads to the conclusion that due to the self-interaction of gluons (i.e., gluons are color-charged contrary to the photons in QED which are uncharged) the strong fine-structure constant becomes smaller at higher energy-momentum scales. This is known as asymptotic freedom and has been discovered by Wilczek and Gross and independently by Politzer. For this very important discovery all three received the Nobel Prize in Physics 2004.