  A resonant inelastic x-ray scattering (RIXS) study of overdamped
spin-excitations in slightly underdoped La$_{2-x}$Sr$_{x}$CuO$_4$ (LSCO) with
$x=0.12$ and $0.145$ is presented. Three high-symmetry directions have been
investigated: (1) the antinodal $(0,0)\rightarrow (1/2,0)$, (2) the nodal
$(0,0)\rightarrow (1/4,1/4)$ and (3) the zone boundary direction
$(1/2,0)\rightarrow (1/4,1/4)$ connecting these two. The overdamped excitations
exhibit strong dispersions along (1) and (3), whereas a much more modest
dispersion is found along (2). This is in strong contrast to the undoped
compound La$_{2}$CuO$_4$ (LCO) for which the strongest dispersions are found
along (1) and (2). The $t-t^{\prime}-t^{\prime\prime}-U$ Hubbard model used to
explain the excitation spectrum of LCO predicts $-$ for constant $U/t$ $-$ that
the dispersion along (3) scales with $(t^{\prime}/t)^2$. However, the diagonal
hopping $t^{\prime}$ extracted on LSCO using single-band models is low
($t^{\prime}/t\sim-0.16$) and decreasing with doping. We therefore invoked a
two-orbital ($d_{x^2-y^2}$ and $d_{z^2}$) model which implies that $t^{\prime}$
is enhanced. This effect acts to enhance the zone-boundary dispersion within
the Hubbard model. We thus conclude that hybridization of $d_{x^2-y^2}$ and
$d_{z^2}$ states has a significant impact on the zone-boundary dispersion in
LSCO.
