  Recently two branches of weakly dispersive collective modes have been
discovered in under-doped cuprates by inelastic neutron scattering.
Polarization analysis reveals that the modes are magnetic excitations. They are
only visible for temperatures below the transition temperature to a broken
symmetry phase which was discovered earlier and their intensity increases as
temperature is further decreased. The broken symmetry phase itself has
symmetries consistent with ordering of orbital current loops within a unit-cell
without breaking translational symmetry. In order to calculate the collective
modes of such a state we add quantum terms to the Ashkin-Teller (AT) model with
which the classical loop current order has been described. We derive that the
mean field ground state of the quantum model is a product over all unit-cells
of linear combination of the four possible classical configurations of the loop
current order in each unit-cell. The collective modes are calculated by using a
generalized Holstein-Primakoff boson representation of orbital moment operators
and lead to three branches of gapped weakly dispersive collective modes. The
experimental results are consistent with the two lower energy branches; the
third mode is at a higher energy than looked for by present neutron scattering
experiments and might also be over-damped. Implications of the discovery of the
collective modes are discussed.
