Lifting asymptotic degeneracies with the mirror TBA

We describe a qualitative feature of the AdS5 x S5 string spectrum which is not captured by the asymptotic Bethe ansatz. This is reected by an enhanced discrete symmetry in the asymptotic limit, whereby extra energy degeneracy enters the spectrum. We discuss how finite size corrections should lift this degeneracy, through both perturbative (Lüscher) and non-perturbative approaches (the Mirror TBA), and illustrate this explicitly on two such asymptotically degenerate states