Testing the Heterotic String with the Axion-Photon Coupling

The discovery of an axion-like particle above the QCD line would rule out Grand Unified Theories, including the perturbative heterotic string with the Standard Model embedded in a single $E_8$ factor or $SO(32)$. In this work we study a possible loophole to this observation, given by compactifications of the $E_8\times E_8$ heterotic string with a non-standard embedding of the Standard Model into the 10-dimensional gauge group. If electromagnetism is embedded into both $E_8$ factors, axions can couple to photons via the anomaly without coupling to QCD. We obtain upper bounds to the coupling-to-mass ratio $g_{a\gamma}/m_a$ for these axion-like particles as a function of the supersymmetry breaking scale and the unified gauge coupling. To be compatible with the measured gauge couplings and the weak mixing angle $\sin^2\theta_w$ at low-energies, phenomenologically viable models with non-standard $U(1)_Y$ embedding require sizeable one-loop threshold corrections from string states and/or charged matter at intermediate energy scales. We study how these effects modify the tree-level upper bounds to $g_{a\gamma}/m_a$ and show that, in the perturbative regime, they reduce the leading order estimates. Axion-like particles far above the QCD line are only possible in certain models where perturbation theory is lost. The main conclusion is that the discovery of an axion violating the bounds found in this work would be incompatible with large classes of otherwise phenomenologically viable string models, including the perturbative heterotic $SO(32)$ and $E_8\times E_8$ string, the type-I string, and certain heterotic M-theories. The role of small gauge instantons and worldsheet instantons in making some of the axion-like particles heavy and cosmologically relevant is briefly discussed.