  In this work, we take all the color-antitriplet diquarks, such as the scalar,
pseudoscalar, vector, axialvector and tensor diquarks, as the basic
constituents to construct the local pseudoscalar four-quark currents without
importing the explicit P-waves to implement the negative-parity, and
investigate the mass spectroscopy of the pseudoscalar hidden-charm tetraquark
states without strange, with strange and with hidden-strange in the framework
of the QCD sum rules in a consistent and comprehensive way. We obtain the
lowest mass $4.56\pm0.08\,\rm{GeV}$ for the pseudoscalar tetraquark state with
the symbolic quark constituents $c\bar{c}u\bar{d}$, which is much larger than
the experimental value $4239\pm18{}^{+45}_{-10}\,\rm{MeV}$ extracted by the
LHCb collaboration.
