Sum rules and missing spectral weight in magnetic neutron scattering in the cuprates

We present estimates in the Hubbard and Heisenberg models for the spectral weight in magnetic neutron-scattering experiments on the cuprates. With the aid of spin-wave theory and the time-dependent Gutzwiller approximation, we discuss how the spectral weight is distributed among the different channels and between high and low energies. In addition to the well known total moment sum rule, we discuss sum rules for each component of the dynamical structure factor tensor which are peculiar for spin-1/2 systems. The various factors that reduce the spectral weight at the relevant energies are singled out and analyzed, such as shielding factors, weight at electronic energies, multimagnon process, etc. Although about 10-15 % of the naively expected weight is detected in experiments, after consideration of these factors the missing weight is within the experimental uncertainties. A large fraction of the spectral weight is hard to detect with present experimental conditions.