  Deviations from General Relativity can alter the quasi-normal mode (QNM)
ringdown of perturbed black holes. It is known that a shift-symmetric (hence
massless) scalar can only introduce black hole hair if it couples to the
Gauss-Bonnet invariant, in which case the scalar charge is fixed with respect
to the black hole mass and controlled by the strength of that coupling. The
charge per unit mass decreases with the mass and can, therefore, be used as a
perturbative parameter for black holes that are sufficiently large with respect
to the scale suppressing the deviation from General Relativity or the Standard
model. We construct an effective field theory scheme for QNMs using this
perturbative parameter to capture deviations from Kerr for both the background
and the perturbations. We demonstrate that up to second order in the charge per
unit mass, QNMs can be calculated by solving standard linearised perturbation
equations for the Kerr metric with sources depending on solutions of the same
equations up to first order. It follows that corrections to the QNM frequencies
are heavily suppressed for sufficiently massive black holes, meaning that LISA
is very unlikely to detect any evidence of scalar hair in ringdown signals.
