Formula Identifies Domain Wall Structures and Reveals a Four-Group Symmetry in Axion Models

Scientists are increasingly focused on resolving the domain wall problem that plagues many axion models and challenges standard big-bang cosmology. Motoo Suzuki from SISSA International School for Advanced Studies, INFN, Sezione di Trieste, and IFPU, working with Ryo Yokokura from the Department of Physics & Research and Education Center for Natural Sciences, Keio University, present a novel field theory formulation of the Lazarides-Shafi mechanism to address this issue. Their research clarifies the generalized symmetry structure, including higher-form symmetries and higher-group aspects, of this mechanism and establishes a master formula for calculating the domain wall number. This work, a collaborative effort between SISSA, INFN, IFPU, and Keio University, is significant because it provides a model-independent way to identify degenerate vacua and demonstrates the existence of a symmetry-protected topological phase even when higher-form global symmetries are seemingly eliminated.

Researchers have developed a novel theoretical framework to address a longstanding problem in axion cosmology, the potential for stable domain walls to form during the universe’s evolution. Axions, hypothetical particles proposed to resolve the strong CP problem in particle physics, often predict the existence of these domain walls, which could disrupt the standard model of cosmology.

This work introduces a topological quantum field theory (TQFT) that isolates the essential mechanisms of the Lazarides-Shafi mechanism, a proposed solution involving degenerate vacua identified through continuous gauge symmetry. The resulting TQFT provides a master formula for calculating the domain wall number, offering a model-independent way to determine whether these problematic structures will actually form.

The TQFT is a four-dimensional generalisation of the Dijkgraaf-Witten theory, independent of the specific details of any particular axion model. The analysis reveals that the system possesses a rich symmetry structure, characterised by a four-group structure and residing in a symmetry-protected topological (SPT) phase, a state of matter characterised by robust, collective behaviour.

This discovery suggests a deeper connection between axion physics and the broader field of topological quantum matter. Clarifying the precise conditions required for complete vacuum identification reveals that certain higher-form symmetries must be broken to successfully resolve the domain wall issue. A four-dimensional TQFT underpins the work, designed to isolate the essential structure of the Lazarides-Shafi mechanism for resolving the domain wall problem in axion models.

This TQFT action, formulated in the low-energy limit, describes a system of N1 degenerate vacua and incorporates a zero-form field, φ, representing the axion, alongside one-form (A) and two-form (B) U gauge fields, and a three-form field (C) originating from the QCD topological term. The construction deliberately omits kinetic terms for dynamical fields, focusing solely on the symmetries to define the action and ensure it remains metric-independent, a characteristic of TQFTs.

Central to the methodology is the inclusion of a Stueckelberg coupling, achieved by modifying the differential of the axion field, dφ → dφ + lA, which facilitates vacuum identification. This approach allows for a systematic computation of the domain wall number, providing a master formula to assess the effectiveness of the Lazarides-Shafi mechanism in axion models.

Further refinement involves analysing the action’s invariance under specific gauge transformations affecting C, A, φ, and B, establishing the symmetry generators for global symmetries. By integrating out the two-form field, B, the researchers simplified the TQFT, revealing a gauge-invariant axion defined as a linear combination of φ and the resulting field, χ.

This reduction in symmetry, from Z N1 to Z N1/K, directly demonstrates the vacuum identification achieved through the implemented mechanism and provides a clear criterion for diagnosing the domain wall problem in axion models. Calculations reveal a master formula for N1, N2, and M. Specifically, N2 represents the centre symmetry of the gauge group G involved in vacuum identification, while N1 arises from the anomaly coefficient of the PQ symmetry with QCD, quantified as A(UPQ −SU2 c).

The parameter M is derived from the gauge-equivalent shift of θ under the centre symmetry transformation of G. In scenarios aiming for a domain-wall-number-one solution (NDW = 1), the condition N1/K = 1 must be satisfied, equivalent to gcd(N1, M) = 1 and gcd(N1, N2) = N1. Achieving this necessitates that the remaining global symmetries are Z N2/N1 × Z N2/N1.

Analysis of SU(N) gauge sectors, Higgsed by a field in the symmetric representation of SU(N), demonstrates that complete vacuum identification occurs only for odd N, reflecting the full breaking of the one-form centre symmetry when N is odd. Furthermore, the study details a four-group structure emerging in the M = 0 limit, where the theory separates into a BF sector and an axion, three-form sector.

Topological line and surface operators create sources for differential forms, exhibiting correlations analogous to the Aharonov, Bohm phase. Specifically, the correlation between a particle charged under dB and a string magnetically charged under A, as well as between a domain wall and an instanton, reveals a quantized phase. When M = 1 and N1/K = 1, the operator U1(V) can split into two W2 via pair-creating ei H B, demonstrating the interconnectedness of these topological objects.

Scientists grappling with the fundamental nature of dark matter and the universe’s earliest moments have long been stymied by theoretical inconsistencies. Models invoking axions, hypothetical particles proposed as dark matter candidates, often run into the ‘domain wall problem’, a cosmological issue arising from the way these particles can create defects in spacetime.

This recent work offers a significant step forward by providing a robust, mathematically grounded framework for understanding the conditions under which these troublesome domain walls can be avoided. Rather than patching up individual models, the researchers have identified a ‘master formula’ governing the domain wall number, effectively offering a blueprint for constructing viable axion models.

The implications extend beyond cosmology, revealing a surprising connection to symmetry-protected topological phases of matter, a field gaining prominence in condensed matter physics. However, this isn’t a complete resolution; while the framework clarifies how to avoid domain walls, it doesn’t dictate which specific axion model is correct. Furthermore, the intricate higher-group symmetries involved remain largely unexplored, presenting a challenge for future theoretical investigations. The next phase will likely involve applying this framework to specific particle physics models, testing its predictions against experimental data, and perhaps uncovering new, unexpected phenomena at the intersection of cosmology, particle physics, and materials science.