We examine the contribution of small instantons to the axion mass in various UV completions of QCD. We show that the reason behind the potential dominance of such contributions is the non-trivial embedding of QCD into the UV theory. The effects from instantons in the partially broken gauge group appear as fractional instanton corrections in the effective theory. These will exhibit unusual dependences on the various scales in the problem whenever the index of embedding is non-trivial. We present a full one-instanton calculation of the axion mass in the simplest product group models, carefully keeping track of numerical prefactors. Rather than using a t Hooft operator approximation we directly evaluate the contributions to the vacuum bubble, automatically capturing the effects of closing up external fermion lines with Higgs loops. This approach is manifestly finite and removes the uncertainty associated with introducing a cutoff scale for the Higgs loops. We verify that the small instantons may dominate over the QCD contribution for very high breaking scales and at least three group factors.
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