UV sensitivity of the axion mass from instantons in partially broken gauge groups

Abstract

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.

The axion mass from 5D small instantons

“…The contribution of small instantons from a compactified 5th dimension shares some features with the 4D moose model [24,25]. In moose models the enhanced contributions of small instantons arise due to a high index of embedding of a QCD instanton into the gauge group of the microscopic theory [29]. Indeed, in the simplest moose model the QCD gauge group, SU (3) c , arises as a diagonal subgroup of a larger product group, SU (3) 1 × SU (3) 2 at some UV scale, with a theta angle and axion at each site.…”

“…The QCD instanton of the two-site moose model corresponds to a multi-instanton configuration of the microscopic theory (specifically, the (1, 1) configuration). In other words, the small instantons in the broken gauge group factors correspond to "fractional" instantons of QCD, and their weights, Λ b 1 1 and Λ b 2 2 , are large compared to the weight of the QCD instanton contribution, Λ b QCD [29] (where b 1 , b 2 , and b are the β function coefficients of the corresponding gauge groups). As a result, the small instanton contributions to the axion masses in the broken factors are enhanced.…”

“…The detailed calculation in Ref. [29] shows that the small instanton enhancement is not sufficiently large in a two-site model 4 but could be significant in k-site models with k ≥ 3. Increasing the number of sites invokes the analogy between moose models and the deconstructed description of 5D theories.…”

“…ConclusionWe have shown that if QCD gluons propagate in a 5th dimension at high energies, then the effective action receives a power-law term R/ρ due to the positive frequency modes 4 This is a consequence of interactions between the Higgs field responsible for breaking the product gauge group, and the small instantons in the broken gauge group factors[29,30].…”

The QCD axion at finite density

“…This prediction seems however to be dependent on the UV dynamics of QCD, in particular on the size of the so-called small instanton contributions, see e.g [10][11][12]…”