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 $t-t^{prime}-t^{primeprime}-U$ 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 ($t^{prime}/tsim-0.16$) 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.
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