High-Quality Axion Models with the Anomalous $U(1)_X$ Gauge Symmetry

TL;DR

The paper tackles the axion quality problem by embedding the Peccei–Quinn mechanism in a framework with an anomalous $U(1)_X$ gauge symmetry whose anomalies cancel via the Green--Schwarz mechanism. It develops a generic construction with two vector-like fermion pairs and two Higgs fields $\Phi_1,\Phi_2$ that break $U(1)_X$, yielding a PQ axion with decay constant $f_a$ and forcing PQ-violating operators to have dimension $p+q \ge 11$, thereby suppressing harmful Planck-scale effects. Three concrete realizations (Models I–III) embed these ingredients in different GUT contexts (SU(5) and flipped SU(5)), achieving gauge coupling unification at high scales: $M_{\rm GUT} \approx 1.9\times 10^{16}$ GeV (Model I, with $\Delta\lesssim 1\%$), $4.5\times 10^{16}$ GeV (Model II, $\Delta \approx 0.1\%$), and $2.14\times 10^{16}$ GeV (Model III, $\Delta \approx 0.7\%$), with vector-like masses ranging from TeV to $10^{11}$ GeV. The results demonstrate that axion quality can be preserved with minimal extra content while achieving precise gauge coupling unification, and they also illustrate how TeV-scale or higher-scale vector-like states can influence unification patterns. This framework has implications for axion phenomenology and grand unification, offering a robust path to high-quality axions compatible with unification constraints.

Abstract

We propose the generic high-quality axion models with anomalous $U(1)_X$ gauge symmetry and vector-like particles. We briefly review the gauge anomaly cancellations via the Green-Schwarz mechanism, study the breaking of the $U(1)_X$ gauge symmetry, as well as derive the Nambu-Goldstone boson, Peccei-Quinn (PQ) axion, and axion decay constant in general. The high-dimensional operators, which break the $U(1)_{PQ}$ global symmetry, have dimension eleven or higher due to the anomalous $U(1)_X$ gauge symmetry, and thus the axion quality problem is solved. In particular, unlike the high-quality axion models with anomaly free $U(1)$ gauge symmetry, we only need to introduce two pairs of vector-like particles. To be concrete, we present three specific models with two pairs of vector-like particles. We show that gauge anomalies in all three models can be canceled via the Green-Schwarz mechanism. To achieve gauge coupling unification, we need to introduce additional vector-like particles only in Model I. We find that gauge coupling unification is achieved at the unification scale around $10^{16}$ GeV with a relative error of less than 1\%. Notably, gauge coupling unification in Model II is achieved naturally with the smallest relative error of 0.1\%.

Figures (3)

• Figure 1: The evolution of two-loop gauge couplings in the Model I with vector-like particles $(XF, \overline{XF})$ and $(XF', \overline{XF'})$. The masses of the vector-like particles are set as follows: $M_{XW}=M_{(XL,\overline{XL})}=800$ GeV for $XW$ and $(XL, \overline{XL})$, $M_{XB}=M_{XG}=5$ TeV for $XB$ and $XG$, and $M_{(XF,\overline{XF})}=M_{(XF',\overline{XF'})}=10^{11}$ GeV for $(XF,\overline{XF})$ and $(XF',\overline{XF'})$.
• Figure 2: The evolution of two-loop gauge couplings in the Model II with vector-like particles $(XT, \overline{XT})$ and $(XT', \overline{XT'})$. The masses of these particles are set as $M_{(XT, \overline{XT})}=M_{(XT', \overline{XT'})}=10^9$ GeV.
• Figure 3: The evolution of two-loop gauge couplings in the Model III with vector-like particles $(XQ, \overline{XQ})$ and $(XD', \overline{XD'})$. The masses of the vector-like particles are set as follows: $M_{(XQ, \overline{XQ})}=1.5$ TeV for $(XQ, \overline{XQ})$, and $M_{(XD', \overline{XD'})}=5.5$ TeV for $(XD', \overline{XD'})$.