  We propose a $d$-density wave induced by the spin-orbit coupling in the CuO
plane. The spectral function of high-temperature superconductors in the under
doped and lightly doped regions is calculated in order to explain the Fermi arc
spectra observed recently by angle-resolved photoemission spectroscopy. We take
into account the tilting of CuO octahedra as well as the on-site
Coulombrepulsive interaction; the tilted octahedra induce the staggered
transfer integral between $p_{x,y}$ orbitals and Cu $t_{2g}$ orbitals, and
bring about nontrivial effects of spin-orbit coupling for the $d$ electrons in
the CuO plane. The spectral weight shows a peak at around ($\pi/2$,$\pi/2$) for
light doping and extends around this point forming an arc as the carrier
density increases, where the spectra for light doping grow continuously to be
the spectra in the optimally doped region. This behavior significantly agrees
with that of the angle-resolved photoemissionspectroscopy spectra. Furthermore,
the spin-orbit term and staggered transfer effectively induce a flux state, a
pseudo-gap with time-reversal symmetry breaking. We have a nodal metallic state
in the light-doping case since the pseudogap has a $d_{x^2-y^2}$ symmetry.
