Tachyons and Misaligned Supersymmetry in Orientifold Vacua

We extend the analysis of Misaligned Supersymmetry to orientifold vacua. The lack of modular invariance in the Klein bottle, annulus and M\"obius strip amplitudes implies that the IR properties of the physical spectrum are related to the UV behaviour of closed strings freely propagating between D-branes and O-planes, and \emph{vice versa}. We thus build sector averaged sums $\langle d (n) \rangle$ associated to both representations of each Riemann surface. We find that the $\langle d (n) \rangle$'s associated to the loop channel control the coupling of closed-string tachyons to D-branes and/or O-planes, and therefore their vanishing is a necessary, but not sufficient, condition for stability in the closed sector. Similarly, the sector averaged sums associated to the tree-level amplitudes encode the presence of tachyons in the physical spectrum, and their vanishing is a sufficient, but not necessary, condition for classical stability. Using this technology, it is difficult, if not impossible, to formulate necessary and sufficient criteria for the absence of tachyons since, in orientifold vacua, they can be removed from the spectrum by the orientifold projection. Although in these cases one would naively expect a cancellation between $\langle d (n)\rangle$'s associated to different amplitudes, this does not occur in practice. We provide several examples in ten and lower dimensions supporting our results.