Abstract
We show that axions interacting with Abelian gauge fields obtain a potential from loops of magnetic monopoles. This is a consequence of the Witten effect: the axion field causes the monopoles to acquire an electric charge and alters their energy spectrum. The axion potential can also be understood as a type of instanton effect due to a Euclidean monopole worldline winding around its dyon collective coordinate. We calculate this effect, which has features in common with both non-Abelian instantons and Euclidean brane instantons. To provide consistency checks, we argue that this axion potential vanishes in the presence of a massless charged fermion and that it is robust against the presence of higher-derivative corrections in the effective Lagrangian. Finally, as a first step toward connecting with particle phenomenology and cosmology, we discuss the regime in which this potential is important in determining the dark matter relic abundance in a hidden sector containing an Abelian gauge group, monopoles, and axions.
Highlights
Introduction.—Axions are naturally light scalar bosons that are of great interest in solving the strong CP problem [1,2,3,4], as dark matter candidates [5,6,7], and for many other applications
It is well known that instanton effects can generate a potential for an axion θ [3,4] when it is coupled to a non-Abelian gauge field via the topological coupling θtrðF ∧ FÞ
We argue that axions coupled to Abelian gauge fields through a θF ∧ F coupling acquire a potential through an instanton effect whenever there are monopoles magnetically charged under F, due to the Witten effect [13]
Summary
We show that axions interacting with Abelian gauge fields obtain a potential from loops of magnetic monopoles. The completeness hypothesis postulates that any UVcomplete theory of an interacting Uð1Þ gauge field (which has quantized charge) contains magnetic monopoles [17], which break a would-be 1-form global symmetry [18]. A consistent description of this effect requires that the effective theory on the magnetic monopole world volume contains, in addition to the usual translational zero modes xμ, a collective coordinate interacting with the field θ. This takes the form of a compact scalar boson σ ≅ σ þ 2π, with an action that (expanding around a monopole worldline extended in time) contains [27]
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