Nonexistence of $d_{x^2-y^2}$ superconductivity in the Hubbard model

It is rigorously shown that the two-dimensional Hubbard model with narrow bands (including next-nearest-neighbor hopping, etc.) does not exhibit $d_{x^2-y^2}$-wave pairing long-range order at any nonzero temperature. This kind of pairing long-range order will also be excluded at zero temperature if an excited energy gap opens in the charge excitation spectrum of the system. These results hold true for both repulsive and attractive Coulomb interactions and for any electron fillings, and are consistent with quantum Monto Carlo calculations.