  An ongoing challenge in the study of quantum materials, is to reveal and
explain collective quantum effects in spin systems where interactions between
different modes types are important. Here we approach this problem through a
combined experimental and theoretical study of interacting transverse and
longitudinal modes in an easy-plane quantum magnet near a continuous quantum
phase transition. Our inelastic neutron scattering measurements of
Ba$_{2}$FeSi$_{2}O$_{7}$ reveal the emergence, decay, and renormalization of a
longitudinal mode throughout the Brillouin zone. The decay of the longitudinal
mode is particularly pronounced at the zone center. To account for the
many-body effects of the interacting low-energy modes in anisotropic magnets,
we generalize the standard spin-wave theory. The measured mode decay and
renormalization is reproduced by including all one-loop corrections. The
theoretical framework developed here is broadly applicable to quantum magnets
with more than one type of low energy mode.
