Global Structure, Non-Invertible PQ Symmetry, and the DFSZ Domain Wall Problem

Abstract

In recent years it has become increasingly clear that the previously overlooked ``global structure'' of symmetry groups can encode significant theoretical structure and, more importantly, have substantial phenomenological implications. With this in mind we re-examine the DFSZ axion, which suffers from a domain wall problem due to the Standard Model generation structure. We show that global structure $(G_{\text{EW}} \times U(1)_{\text{PQ}})/\mathbb{Z}_2$ acting between the Peccei-Quinn symmetry and the electroweak gauge group plays a crucial role in determining the precise nature of the domain wall problem, which has important implications in both cubic and quartic DFSZ. We then demonstrate that the stability of the domain walls is enforced by a non-invertible chiral symmetry in quark flavor $Z'$ models which have additional global structure $(SU(3)_C \times G_F)/\mathbb{Z}_3$ acting between the color and the gauged quark flavor groups. The strategy of Non-invertible Naturalness then leads us to UV theories that resolve the domain wall problem through small-instanton-induced breaking of non-invertible symmetries. Finally, we sketch potential gravitational wave signatures arising from the annihilation of axion domain walls. Our work illustrates the importance of considerations of global structure in realistic models of particle physics.

Figures (5)

• Figure 1: The pure PQ string consists of winding along only the PQ direction (red). On the other hand, non-trivial global structure between PQ and hypercharge ($Y$) allows more minimal axion string which winds around both the PQ direction and the $Y$ direction by $\pi$ (blue). The quotient $(U(1)_Y \times U(1)_{\rm PQ})/\mathbb{Z}_2$ exactly sets the purple points to be equivalent.
• Figure 2: Axion domain wall-string configuration in a theory with $U(1)_{\rm PQ} \times G_{\rm EW}$ (left) vs in a theory with $[ U(1)_{\rm PQ} \times G_{\rm EW}] / \mathbb{Z}_2$ (right). Red color indicates PQ winding and blue color indicates winding along the gauge direction, e.g. $U(1)_Y$.
• Figure 3: One diagram in each $SU(9)$ UV completion generating $\mathbb{Z}_3$ non-invertible PQ symmetry violation. In each case the diagram with $\lambda$ removed also exists.
• Figure 4: Axion potential in the unit of $\Lambda_{\rm QCD}^{4}$ for $-\pi\leq\theta_{a}\equiv 2a/f_{a}\leq\pi$. The purple line shows the net axion potential contributed both by QCD and the constrained instanton of $SU(9)$ (orange dashed line). The induced bias is $V_{\rm bias}=V_{\rm tot}(\theta_{a}=\pm2\pi/3)$.
• Figure 5: Gravitational wave spectrum from the domain wall collapse for each specified $f_{a}$ and $T_{\rm ann}=10{\rm MeV}$. The purple dashed (gray dotdashed) line corresponds to $\Omega_{\rm GW,0}h^{2}$ for $f_{a}=10^{10}{\rm GeV}$ ($10^{9}{\rm GeV}$). The orange and red shaded regions are the projected sensitivities of the future detectors, SKA Carilli:2004nxJanssen:2014dkaWeltman:2018zrl and THEIA thetheiacollaboration2017theiafaintobjectsmotion.