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The macroscopic properties of compact stars in modified gravity theories canbe significantly different from the general relativistic (GR) predictions.Within the gravitational context of scalar-tensor theories, with a scalar field$\phi$ and coupling function $\Phi(\phi)= \exp[2\phi/\sqrt{3}]$, we investigatethe hydrostatic equilibrium structure of neutron stars for the simple potential$V(\phi)= \omega\phi^2/2$ defined in the Einstein frame (EF). From the scalarfield in the EF, we also interpret such theories as $f(R)$ gravity in thecorresponding Jordan frame (JF). The mass-radius relations, proper mass, andbinding energy are obtained for a polytropic equation of state (EoS) in the JF.Our results reveal that the maximum-mass values increase substantially as$\omega$ gets smaller, while the radius and mass decrease in thelow-central-density region as we move further away from the pure GR scenario.Furthermore, a cusp is formed when the binding energy is plotted as a functionof the proper mass, which indicates the appearance of instability.Specifically, we find that the central-density value where the binding energyis a minimum corresponds precisely to $dM/d\rho_c^J = 0$ on the$M(\rho_c^J)$-curve.

Compact stars in scalar-tensor theories with a single-well potential and the corresponding $f(R)$ theory