To understand the origin of unconventional charge-density-wave (CDW) states in cuprate superconductors, we establish the self-consistent CDW equation, and analyze the CDW instabilities based on the realistic Hubbard model, without assuming any q-dependence and the form factor. Many higher-order many-body processes, which are called the vertex corrections, are systematically generated by solving the CDW equation. When the spin fluctuations are strong, the uniform q = 0 nematic CDW with d-form factor shows the leading instability. The axial nematic CDW instability at q = Q_a = (δ,0) (δ ≈ π/2) is the second strongest, and its strength increases under the static uniform CDW order. The present theory predicts that uniform CDW transition emerges at a high temperature, and it stabilize the axial q = Q_a CDW at T = T_CDW. It is confirmed that the higher-order Aslamazov–Larkin processes cause the CDW orders at both q = 0 and Q_a.

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