I have a follow-up question to Dirac magnetic monopoles and quark fractional electric charge quantization, regarding whether the "unit of electric charge" in the Dirac quantization condition should be $e$ or $\frac{e}{3}$ because of the quarks' fractional charge. Ben Crowell's answer argues that you should use $e$, because quarks are confined so you could never take a single quark in a Wilson loop around the Dirac string. But above the Hagedorn temperature quarks deconfine, so couldn't you actually perform this experiment at a temperature above the Hagedorn temperature? Does this mean we should triple the minimum allowed magnetic monopole charge?


The answer can be found in section 6 of http://www.sciencedirect.com/science/article/pii/0550321381902376. Above the Hagedorn temperature, the strong interaction becomes long-ranged and falls of as $1/r$, just like the Coulomb interaction. As you braid a charge-$e/3$ quark around a Dirac string, it picks up an Aharonov-Bohm phase from the quark's electric charge circling the string's magnetic flux, but it also picks up an additional Aharonov-Bohm phase from the strong interaction's color tubes crossing the Dirac string, and it turns out that the strong interaction's Aharonov-Bohm phase is exactly twice that of the electromagnetic Aharonov-Bohm phase, thus making the total phase equivalent to what you would have gotten if you'd braided a color-neutral particle of charge $e$ (like an electron) instead of a quark. The strong interaction essentially makes up for the "missing" $(2/3) e$ of electric charge. Therefore the correct unit of electric charge to use in the Dirac quantization condition is $e$, not $e/3$.

In fact, you can even move a single quark around the monopole in the confined phase; you just need to move in a circle smaller than the confinement radius. Then the exact same effect as described above also occurs.

Also see pg. 468 of http://www.theory.caltech.edu/~preskill/pubs/preskill-1984-monopoles.pdf.

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