  The collective modes observed in the loop-current ordered state in
under-doped cuprates by polarized neutron scattering require that the ground
state is a linear combination in each unit-cell of the four basis states which
are the possible classical magnetic moment configurations in each unit-cell.
The direction of such moments is in the c-axis of the crystals. The basis
states are connected by both time-reversal as well as spatial rotations about
the center of the unit-cells. Several new features arise in the theory of
polarized neutron scattering cross-section in this situation which appear not
to have been encountered before. An important consequence of these is that a
finite component transverse to the classical magnetic moment directions is
detected in the experiments. We show that this transverse component is of
purely quantum-mechanical origin and that its direction in the plane normal to
the c-axis is not detectable, even in principle, in experiments, at least in
the quantum-mechanical model we have adopted. We estimate the direction of the
"tilt" in the moment, i.e. the ratio of the transverse component to the c-axis
component, using parameters of the ground state obtained by fitting to the
observed dispersion of the collective modes in the ordered state. We can obtain
reasonable agreement with experiments but only by introducing a parameter for
which only an approximate magnitude can be estimated. Approximate calculations
of the form-factors are also provided.
