Four-boson first excited state near two-body unitarity

Near two -body unitarity, the three -boson system is characterized by an approximate discrete scale invariance manifest in a geometric tower of bound states (the Efimov effect). In the absence of a strong four -body force, the four -boson system has two states associated with each Efimov state, one very nearly unstable, the other several times deeper. We study correlations between the excited- and ground -state properties, such as binding energies and radii, up to next -to -leading order in an effective field theory for short-range forces. We obtain the parameters in these correlations from similar correlations arising from existing precise calculations based on short-range potentials. We also derive correlations among excited -state properties that emerge from the proximity of the state to the break-up threshold into a boson and a three -boson bound state, using an effective field theory for "halo" states.