Tale of two-$U(1)$ axion models

$U(1)$ global symmetry to solve the strong $CP$ problem could be a remnant of multi-$U(1)$ symmetries from QCD and hidden strong dynamics. Both Peccei-Quinn $U(1)$ and dynamical $U(1)$ are described uniformly, based on which we classify various mixed two-$U(1)$ models to solve both strong $CP$ and quality problems. We propose a moose diagram method with different fermion assignments to directly read relations between $CP$ phases, which illustrate how the strong $CP$ problem is solved in terms of cancellation between $CP$ phases. In two-axion models, we find that the lightest axion is still the same as a QCD axion in the infrared region, while a one-axion model with $Z_2$ symmetry enhances the axion mass spectrum. Our discussions can be extended to multiaxion cases.

Figure 1

(a) Solid line between $Q$ and $SU(3{)}_c$ showing that $Q$ is under representation 3 of QCD and the solid line between $Q^{\prime }$ and $SU(3{)}_h$ showing that $Q$ is under representation 3 of hidden QCD. The red dashed line between $a_1$ and $Q(Q^{\prime }$) shows that the charge of $Q(Q^{\prime }$) under $U(1{)}_{\mathrm{PQ}}$ is $m_1$ ($n_1$). Similarly, the blue dashed line shows that the charge of $Q(Q^{\prime })$ under $U(1{)}_{{\mathrm{PQ}}^{\prime }}$ is $m_2$ ($n_2$). $1/f_{a_1}$ and $1/f_{a_2}$ are scale factors for two $U(1)$’s. (b) Solid lines showing that $\psi$ is under representation 3 for both QCD and hidden QCD, and that $\chi$ is a singlet for QCD but under representation 3 for hidden QCD. The red dashed lines show that the ratio of $\psi$ and $\chi$ in the ${\eta }_h^{\prime }$ current is $1:1$, which is similar to the $U(1)$ charge in (a) in the infrared region. The only difference is one more normalization factor $\sqrt{2}/2$ in front of the scale factor $1/f_{{\pi }_h}$. Similarly, blue lines show that the ratio of $\psi$ and $\chi$ in the ${\pi }_h$ current is 1:$-3$. The normalization factor is $\sqrt{6}/6$ and the scale factor is $1/f_{{\pi }_h}$.

Figure 2

(a) Structure of $U(1{)}_{\mathrm{PQ}}$ models with mirror symmetry. (b) Structure of $U(1{)}_A$ dynamic solutions with mirror symmetry.

Figure 3

Structure of models with mixed $U(1)\mathrm{s}$. (a) Model A. The spontaneous breaking of $U(1{)}_{\mathrm{PQ}}$ is independent of the hidden QCD. (b) Model B. The dynamical symmetry breaking of hidden QCD induces $U(1{)}_{\mathrm{PQ}}$ breaking.