  Composite Higgs models feature new strong dynamics leading to the description
of the Higgs boson as a bound state arising from the breaking of a global
(flavour) symmetry. These models generally include light states generated by
the same dynamics, the detection of which may present the first observable
signs of compositeness. One such state is a pseudo-scalar boson resulting from
the breaking of a $U(1)$ symmetry common to most composite setups, and whose
hints are expected to be visible through low-mass resonance searches at present
and future hadron and lepton colliders. In this work we study the phenomenology
of this pseudo-scalar field. We show that, for a light state, bottom quark loop
effects dominantly impact the production cross section and considerably modify
the decay pattern. Moreover, we make a case for targeted low-mass analyses at
future lepton colliders, with an emphasis on high-luminosity machines aiming to
operate at low centre-of-mass energies. We present a simplified outline of a
search for this light pseudo-scalar at one such machine, considering
electron-positron collisions at the $Z$-pole. We focus on a signature arising
from the pseudo-scalar decay into a pair of hadronic taus and a production mode
association with a pair of leptons of opposite electric charges, and compare
cut and count methods with machine learning methods.
