In this talk the norms of the Bethe states for the spin 1/2 Heisenberg chain in the critical regime are discussed. A set of conjectures will be formulated concerning the scaling behavior of the norms, which were obtained from a combination of analytic techniques based on the ODE/IQFT correspondence and numerical analysis. Also I will discuss the role of the different Hermitian structures associated with the integrable structures studied in the series of works of Bazhanov, Lukyanov and Zamolodchikov in the mid nineties.