Casimir-type effects in QCD as a source of Dark Energy

I discuss a Casimir-like behaviour in the $\theta$-dependent part of the energy in a ``deformed'' QCD. Defining the system on a manifold of size $L$, the energy takes the form $E = A\left[1+\frac{B}{L} + {\cal O}(L^{- 2})\right]$, despite the presence of a mass gap. In contrast, one would naively expect the form $E = A[1+Be^{-mL}]$ originating from any physical massive degrees of freedom. I explain how this form comes instead from a non-dispersive ``contact'' term which does not originate from any propagating degrees of freedom, so that the naive argument is not applicable. I then present some explicit results in a ``deformed'' QCD, which while weakly coupled and under full theoretical control still exhibits interesting properties of true QCD such as confinement, a mass gap, and non-trivial $\theta$-dependence. If the Dark Energy is defined as a mismatch between the energies of the system defined in a bounded system and in the Minkowski vacuum, then the discussed effect gives a Dark Energy estimated at $\Delta E \sim H\Lambda_{\mathrm{QCD}}^3 \sim (10^{-3}eV)^4$, which is astonishingly close to the observed value.