  Starting from the Bethe ansatz solution we derive a set of coupled non-linear
integral equations for the fully packed double loop model (FPL^2) on the square
lattice. As an application we find exact expressions for the central charge and
for the scaling dimension corresponding to the simplest charge excitation. We
study numerically the low-lying excitations corresponding to more general
perturbations of the ground state and discover that the corresponding scaling
dimensions are well described by the Cartan matrix of sl_4.
