  This paper investigates the impact of small instanton effects on the axion
mass in composite axion models. In particular, we focus on the Composite
Accidental Axion (CAA) models, which are designed to address the axion quality
problem, and where the Peccei-Quinn (PQ) symmetry emerges accidentally. In the
CAA models, the QCD gauge symmetry is embedded in a larger gauge group at high
energy. These models contain small instantons not included in low-energy QCD,
which could enhance the axion mass significantly. However, in the CAA models,
our analysis reveals that these effects on the axion mass are non-vanishing but
are negligible compared to the QCD effects. The suppression of the small
instanton effects originates from the global chiral U(1) symmetries which are
not broken spontaneously and play a crucial role in eliminating $\theta$-terms
in the hidden sectors through anomalies. We find these U(1) symmetries restrict
the impact of small instantons in hidden sectors on the axion mass. Our study
provides crucial insights into the dynamics within the CAA models and suggests
broader implications for understanding small instanton effects in other
composite axion models.
