Dark Matter as Ultralight Axion-Like particle in $E_6 \times U(1)_X$ GUT
  with QCD Axion
  Axion-like fields are naturally generated by a mechanism of anomaly
cancellation of one or more anomalous gauge abelian symmetries at the Planck
scale, emerging as duals of a two-form from the massless bosonic sector of
string theory. This suggests an analogy of the Green-Schwarz mechanism of
anomaly cancellation, at field theory level, which results in one or more
Stueckelberg pseudoscalars. In the case of a single Stueckelberg pseudoscalar
$b$, vacuum misalignments at phase transitions in the early Universe at the GUT
scale provide a small mass - due to instanton suppression of the periodic
potential - for a component of $b$, denoted as $\chi$ and termed the
"axi-Higgs", which is a physical axion-like particle. The coupling of the
axi-Higgs to the gauge sector via Wess-Zumino terms is suppressed by the Planck
mass, which guarantees its decoupling, while its angle of misalignment is
related to $M_{GUT}$. We build a gauged $E_6\times U(1)$ model with anomalous
$U(1)$. It contains both an automatic invisible QCD axion and an ultra-light
axi-Higgs. The invisible axion present in the model solves the strong CP
problem and has mass in the conventional range while the axi-Higgs, which can
act as dark matter, is sufficiently light ($10^{-22} \textrm{ eV} < m_{\chi} <
10^{-20} \textrm{ eV}$) to solve short-distance problems which confront other
cold dark matter candidates.
