  A $SU(N)_L\times SU(N)_R$ gauge theory for a scalar multiplet $Y$
transforming in the bi-fundamental representation $(N,\bar N)$ preserves, for
$N>4$, an accidental $U(1)$ symmetry firstly broken at operator dimension $N$.
Two configurations are possible for the vacuum expectation value of $Y$, which
correspond to the (maximal) little groups $\mathcal{H}_s=SU(N)_{L+R}$ and
$\mathcal{H}_h=SU(N-1)_L\times SU(N-1)_R\times U(1)_{L+R}$. In the first case
the accidental $U(1)$ gets also broken, yielding a pseudo Nambu-Goldstone boson
with mass suppression controlled by $N$, while in the second case a global
$U(1)$ remains unbroken. The strong CP problem is solved by coupling $Y$ to new
fermions carrying color. The first case allows for a Peccei-Quinn solution with
$U(1)_{PQ}$ protected up to order $N$ by the gauge symmetry. In the second case
$U(1)$ can get broken by condensates of the new strong dynamics, resulting in a
composite axion. By coupling $Y$ to fermions carrying only weak isospin, models
for axion-like particles can be constructed.
