Pseudo-Goldstones from Supersymmetric Wilson Lines on 5D Orbifolds

We consider a U(1) gauge theory on the five dimensional orbifold ${\mathcal{M}}_4\times S^1\mathrm{/}{Z}_2$, where $A_5$ has even $Z_2$ parity. This leads to a light pseudoscalar degree of freedom $W(x)$ in the effective 4D theory below the compactification scale arising from a gauge-invariant brane-to-brane Wilson line. As noted by Arkani-Hamed et al in the non-supersymmetric $S^1$ case the 5D bulk gauge-invariance of the underlying theory together with the non-local nature of the Wilson line field leads to the protection of the 4D theory of $W(x)$ from possible large global-symmetry violating quantum gravitational effects. We study the $S^1\mathrm{/}{Z}_2$ theory in detail, in particular developing the supersymmetric generalization of this construction, involving a pseudoscalar Goldstone field (the axion') and its scalar and fermion superpartners (saxion’ and `axino’). The global nature of $W(x)$ implies the absence of independent Kaluza-Klein excitations of its component fields. The non-derivative interactions of the (supersymmetric) Wilson line in the effective 4D theory arising from U(1) charged 5D fields $\mathrm{\Phi }$ propagating between the boundary branes are studied. We show that, similar to the non-supersymmetric $S^1$ case, these interactions are suppressed by $\exp(-\pi R m_{\Phi})$where$\pi R$ is the size of the extra dimension.