Testing the Heterotic String with the Axion-Photon Coupling

TL;DR

The work assesses axion-like particles in heterotic string theories with non-standard hypercharge embedding, showing that two ALPs, $\theta_2$ and $\varphi$, can couple to photons without coupling to QCD, while a QCD axion persists as $\theta_1$. It derives tree-level and loop-corrected upper bounds on the ratio $g_{a\gamma}/m_a$ that depend on the SUSY-breaking scale and $\alpha_{\rm GUT}$, and shows that threshold corrections and UV/worldsheet instantons typically reduce these bounds within perturbative control. The analysis demonstrates that ALPs above the QCD line are only viable if perturbativity is lost or if substantial UV instanton effects occur, and discusses the cosmological and phenomenological implications of heavy ALPs and the potential for using ALP searches to test string compactifications. Overall, the paper provides a framework linking axion couplings to the detailed gauge-background data of heterotic embeddings, constraining the landscape of viable string vacua via axion phenomenology and guiding experimental searches.

Abstract

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 $g_{aγ}/m_a$ 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 $\sin^2θ_w$ 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 $g_{aγ}/m_a$ 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.

Figures (3)

• Figure 1: Summary of results. Axion parameter space for the heterotic string axiverse with non-standard embedding of hypercharge. Experimental contraints adapted from AxionLimits. The black dashed line corresponds to the standard GUT QCD axion prediction with $E/N=8/3$. The black solid line corresponds to the QCD axion prediction with $E/N=49/24$, which exhibits an accidental cancellation with the pion mixing contribution that leads to $g_{a\gamma}$ a factor $\sim 6$ smaller than the standard GUT prediction. Different dashed lines correspond to different $g_{a\gamma}/m_a$ predictions for $\theta_2$ and $\varphi$. (As we are neglecting the effect of threshold corrections in the ALP mass, we have a comparable ratio for both ALPs, $g_{\theta_2\gamma}/m_{\theta_2}\sim g_{\varphi\gamma}/m_{\varphi}$.) The purple and dark green dashed lines correspond to unbroken $SU(2)$ and $SU(3)$ groups in the second $E_8$. The light blue dashed line corresponds to the second $E_8$ fully broken to $U(1)$s and $\alpha_{\rm UV}=1/24$ and $M_{\rm susy}=10^4$ GeV. The light green dashed line corresponds to $\alpha_{\rm UV}=1/30$ and $M_{\rm susy}=10^9$ GeV. The gold dashed line corresponds to a situation where there is no low-scale supersymmetry and the gauge couplings are given by extrapolating the measured values at the weak scale assuming SM matter content, $\alpha_{\rm UV}=1/37$. In any of the three cases, the charged matter spectrum is given by 3-generation MSSM-like models. The shaded regions to the right of each of the $g_{a\gamma}/m_a$ lines can be populated, for example via axion mixing, in each situation, e.g. the shaded light green region is allowed for the cases $\alpha_{\rm UV}=1/30$ and $M_{\rm susy}=10^9$ GeV, and for $\alpha_{\rm UV}=1/37$ but not for $\alpha_{\rm UV}=1/24$ and $M_{\rm susy}=10^4$ GeV. The shaded red region is incompatible with the heterotic string even with non-standard $U(1)_Y$ embedding. Orange corresponds to the prediction for a model-dependent axion receiving its mass from a worldsheet instanton with action $S_{ws}\sim 2\pi \text{Vol}(C)$ and 2-cycle volume $\text{Vol}(C)=10$. As shown later in figures \ref{['fig:ParameterSpace_with_small_k1']} and \ref{['fig:ParameterSpace_with_threshold_correction']} any mechanism that allows to recover the measured gauge couplings (e.g. small level of hypercharge embedding or threshold corrections) tend to decrease the leading order estimates for $g_{a\gamma}/m_a$ in this figure.
• Figure 2: Same as Figure \ref{['fig:ParameterSpace']} but with small level of $U(1)_Y$ embedding into the second $E_8$, $k_1^{(2)}=1/9$. The smaller level of embedding leads to $E/N=41/16$ for $\theta_1$ and shifts the solid black line closer to the standard GUT prediction. The instanton action and axion masses are unchanged but the $\theta_2$ coupling to photons is reduced as $g_{\theta_2\gamma}\propto (k_1^{(2)})^2$. The coupling-to-mass ratio for $\varphi$, $g_{\varphi\gamma}/m_{\varphi}$ also vanishes in the limit of small $k_1^{(2)}$ (see text for details). For simplicity other lines with confining sectors are not shown.
• Figure 3: Same as Figure \ref{['fig:ParameterSpace']} but including negative threshold corrections, $\widehat{\Delta}_0<0$. As an example, we have taken $5\widehat{\Delta}_0 = \frac{3}{4}\frac{k_1^{(2)}}{\alpha_{\rm GUT}}$, that is 75% of the tree-level contribution and fixed the $E_8^{(2)}$ and $U(1)_1$ instanton actions accordingly, see Eq. \ref{['eq:inst_actions_negative_threshold']}. The $U(1)_1$ instanton has the smallest action implying that the $\varphi$ has $g_{\varphi\gamma}/m_\varphi\ll g_{\theta_2\gamma}/m_{\theta_2}$ (see text for details). For this reason, we focus on the prediction of the coupling-to-mass ratio for $\theta_2$. Situations with $\alpha_{\rm GUT}^{-1}=24$ (blue) and $\alpha_{\rm GUT}^{-1}=30$ (light green) lead to $g_{\theta_2\gamma}/m_{\theta_2}$ smaller than the QCD axion prediction (solid black line). For the benchmark $\alpha_{\rm GUT}^{-1}=37$, set by the gold shaded region, the ALP $\theta_2$ can lie above the QCD band albeit by a small margin. Smaller threshold corrections will lead to even smaller coupling-to-mass ratio for both ALPs. For simplicity, lines corresponding to heavy ALPs from confining groups are not included. The worldsheet instanton line (orange dashed) is kept for reference. See text for the case with positive threshold corrections.