  In compact QED$_{2+1}$ quantum monopole fluctuations induce confinement by
expelling electric flux in a dual Meissner effect. Guided by Landau-Ginzburg
theory, one might guess that the inverse London penetration depth
$\lambda^{-1}$---the only physical mass scale---equals the photon propagator
mass pole $M_\gamma$. I show this is not true. Indeed, in the Villain
approximation the monopole part of the partition function factorizes from the
photon part, whose dynamical variables are Dirac strings. Since Dirac strings
are gauge-variant structures, I conclude that $M_\gamma$ is physically
irrelevant: it is not a blood relative of $\lambda$ or any other quantity in
the gauge-invariant sector. This result is confirmed by numerical simulations
in the full theory, where $M_\gamma$ is not sensitive to monopole prohibition
but essentially vanishes if Dirac strings are prohibited.
