  The spectra and role in the spin dynamical properties of bound states of
elementary magnetic excitations named Bethe strings that occur in some
integrable spin and electronic one-dimensional models have recently been
identified and realized in several materials by experiments. Corresponding
theoretical studies have usually relied on the one-dimensional spin-1/2
Heisenberg antiferromagnet in a magnetic field. At the isotropic point, it
describes the large onsite repulsion U limit of the spin degrees of freedom of
the one-dimensional fermionic Hubbard model with one electron per site in a
magnetic field h. In this paper we consider the thermodynamic limit and study
the effects of lowering the latter quantum problem ratio u=U/4t, where t is the
first-neighbor transfer integral, on the line-shape singularities in regions at
and just above the lower thresholds of the transverse and longitudinal spin
dynamical structure factors. The most significant spectral weight contribution
from Bethe strings leads to a gapped continuum in the spectrum of the spin
dynamical structure factor +-. Our study focuses on the line shape
singularities at and just above the gapped lower threshold of that continuum,
which have been identified in experiments. Our results are consistent with the
contribution of Bethe strings to the spin dynamical structure factor zz being
small at low spin densities and becoming negligible upon increasing that
density. Our results provide physically important information about how
electron itinerancy affects the spin dynamics.
