A Clockwork Axion

Abstract: We present a renormalizable theory of scalars in which the low energy effective theory contains a pseudo-Goldstone Boson with a compact field space of 2{\pi} F and an approximate discrete shift symmetry Z_Q with Q>>1, yet the number of fields in the theory goes as log(Q). Such a model can serve as a UV completion to models of relaxions and is a new source of exponential scale separation in field theory. While the model is local in `theory space', it appears to not have a continuum generalization (i.e., it cannot be a deconstructed extra dimension). Our framework shows that super-Planckian field excursions can be mimicked while sticking to renormalizable four-dimensional quantum field theory. We show that a supersymmetric extension is straightforwardly obtained and we illustrate possible UV completions based on a compact extra-dimension, where all global symmetries arise accidentally as a consequence of gauge invariance and 5D locality.

• The paper demonstrates a novel discrete scalar model achieving super-Planckian field excursions via a pseudo-Goldstone boson.
• The paper employs a network of N+1 complex scalar fields with nearest-neighbor interactions to create an exponential scale separation in the effective decay constant.
• The paper outlines a clear pathway to supersymmetric extensions, suggesting promising links to extra-dimensional and string theory frameworks.

Analysis of "A Clockwork Axion" by Kaplan and Rattazzi

This paper introduces a renormalizable theory of scalars that results in a pseudo-Goldstone Boson with a substantial effective field range. This framework extends the ultraviolet (UV) completion of certain field models capable of mimicking super-Planckian field excursions while remaining consistent with renormalizable four-dimensional quantum field theory. The authors leverage the concept of a discretized field space, driven by a compact field space of $2\pi F$ with a discrete shift symmetry $Z_Q$ where $Q \gg 1$.

Key Contributions

1. Scale Separation: The authors develop a model featuring exponential scale separation in field theory which impedes direct continuum analogs, a condition pivotal for UV completions of certain field theories such as relaxions. Here, the field excursion is dramatically larger than the Planck scale, indicating large effective decay constants and showcasing a novel source for such separation.
2. Discrete Field Model: The model comprises $N+1$ complex scalar fields, manifesting nearest-neighbor type interactions. This discrete arrangement, lacking a meaningful continuum limit, serves to construct a pseudo-Goldstone Boson with a significant field range, specifically achieved with a low number of fields due to the $F \geq 3f$ relationship, a substantial scale of compact field space.
3. Supersymmetry and UV Completion: A straightforward pathway to a supersymmetric extension suggests potential for how these constructions could be embedded in higher dimensional or string theory settings. The paper proposes an influential understanding of global symmetries as accidentals within gauge-invariant contexts, grounded in extra-dimensional locality.

Numerical Results and Theoretical Implications

The proposed eigenvalue problem analysis reveals a systematic access to zero and massive modes via a matrix form. Notably, the exponential suppression of overlap of the zero-mode with UV operators produces effective decay constants exponentially surpassing the theoretical scale $f$. Such significant theoretical insights suggest profound implications for the formulation and constraints of field theories managing the electroweak hierarchy problem, entailing large scalar field excursions.

Future Directions

The exploration of accidental global symmetries discovered by the authors may inform broader investigations into multi-axion models and enrich the dialogue on imposing global symmetries within the constraints of quantum gravity. Additionally, the study recommends consideration of embedding these constructs within string theory frameworks.

The findings regarding discrete field configurations provide a rich terrain for constructing UV-complete models of cosmological inflation and potentials akin to those of relaxions, with further investigations merited into string-theoretical and gauge-invariance manifestations.

Conclusion

In sum, Kaplan and Rattazzi's work demonstrates a decisive method for achieving drastically large field excursions via a modest number of fields. It contributes substantially to the discussion of UV completions and their relationship with gauge interactions and quantum field theory, while also offering a template for further exploration into new dimensions and categories of physics models.