Low-energy quasiparticle (QP) excitations in the heavy-fermion superconductor URu$_2$Si$_2$ were investigated by specific-heat $C(T, H, phi, theta)$ measurements of a high-quality single crystal. The occurrence of QP excitations due to the Doppler-shift effect was detected regardless of the field direction in $C(H)$ of the present clean sample, which is in sharp contrast to a previous report. Furthermore, the polar-angle-dependent $C(theta)$ measured under a rotating magnetic field within the ac plane exhibits a shoulder-like anomaly at $theta sim 45$ deg and a sharp dip at $theta = 90$ deg ($H parallel a$) in the moderate-field region. These features are supported by theoretical analyses based on microscopic calculations assuming the gap symmetry of $k_z(k_x+ik_y)$, whose gap structure is characterized by a combination of a horizontal line node at the equator and point nodes at the poles. The present results have settled the previous controversy over the gap structure of URu$_2$Si$_2$ and have authenticated its chiral $d$-wave superconductivity.
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