[Paper Review] Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy

Juven Wang|arXiv (Cornell University)|Aug 14, 2020

Particle physics theoretical and experimental studies78 references9 citations

TL;DR

This paper proposes a hidden gapped topological sector beyond the Standard Model and SU(5) Grand Unified Theory (GUT), protected by a mod 16 global anomaly involving a discrete $\mathbb{Z}_{4,X}$ symmetry with $X = 5(\mathbf{B}-\mathbf{L}) - 4Y$. The anomaly is matched via a 4d noninvertible TQFT or 5d invertible TQFT, consistent with energy hierarchy and cobordism constraints, and provides a mechanism for sterile right-handed neutrinos and dark matter.

ABSTRACT

A recent work [2006.16996] suggests that a 4d nonperturbative global anomaly of mod 16 class hinting a possible new hidden gapped topological sector beyond the Standard Model (SM) and Georgi-Glashow $su(5)$ Grand Unified Theory (GUT) with 15n chiral Weyl fermions and a discrete $\mathbb{Z}_{4,X}$ symmetry of $X=5({\bf B- L})-4Y$. This $\mathbb{Z}_{16}$ class global anomaly is a mixed gauge-gravitational anomaly between the discrete $X$ and spacetime backgrounds. The new topological sector has a GUT scale high energy gap, below its low energy encodes either a 4d noninvertible topological quantum field theory (TQFT), or a 5d short-range entangled invertible TQFT, or their combinations. This hidden topological sector provides the 't Hooft anomaly matching of the missing sterile right-handed neutrinos (3 generations of 16th Weyl fermions), and possibly also accounts for the Dark Matter sector. In the SM and $su(5)$ GUT, the discrete $X$ can be either a global symmetry or gauged. In the $so(10)$ GUT, the $X$ must become gauged, the 5d TQFT becomes noninvertible and long-range entangled (which can couple to dynamical gravity). In this work, we further examine the anomaly and cobordism constraints at higher energy scales above the $su(5)$ GUT to $so(10)$ GUT and $so(18)$ GUT (with Spin(10) and Spin(18) gauge groups precisely). We also find [2006.16996]'s proposal on new hidden gapped topological sectors can be consistent with anomaly matching under the energy/mass hierarchy. Novel ingredients along tuning the energy include various energy scales of anomaly-free symmetric mass generation (i.e., Kitaev-Wen mechanism), the Topological Mass/Energy Gap from anomalous symmetric topological order (attachable to a 5d $\mathbb{Z}_{4,X}$-symmetric topological superconductor), possible topological quantum phase transitions, and Ultra Unification that includes GUT with new topological sectors.

Motivation & Objective

• To extend anomaly and cobordism constraints beyond SU(5) GUT to SO(10) and SO(18) GUTs with higher energy scales.
• To investigate how the $\mathbb{Z}_{16}$ global anomaly from the $X = 5(\mathbf{B}-\mathbf{L}) - 4Y$ symmetry can be matched in higher GUTs.
• To explore the role of topological quantum field theories (TQFTs) in saturating anomalies and realizing symmetric mass generation.
• To examine the consistency of the hidden topological sector with energy hierarchy, phase transitions, and Ultra Unification.
• To determine whether the $\mathbb{Z}_{16}$ anomaly can be canceled by new degrees of freedom beyond right-handed neutrinos, such as 4d or 5d TQFTs.

Proposed method

• Analyzes mixed gauge-gravitational anomalies in 4d quantum field theories with discrete $X$ symmetry, using cobordism invariants and index theory.
• Applies the Adams and Atiyah-Hirzebruch spectral sequences to classify global anomalies and TQFTs in GUTs with $\mathrm{Spin}(10)$ and $\mathrm{Spin}(18)$ gauge groups.
• Evaluates the $\mathbb{Z}_{16}$ global anomaly via the $\eta(\text{PD}(\mathcal{A}_{\mathbb{Z}_2}))$ invariant, counting left-handed Weyl fermions modulo 16.
• Considers anomaly cancellation via 4d noninvertible TQFTs or 5d invertible TQFTs, including topological superconductors with $\mathbb{Z}_{4,X}$ symmetry.
• Derives energy-scale-dependent anomaly matching conditions $\Delta_{\mathrm{KW}}$ for various GUT breaking chains, such as $\mathrm{SO}(18) \to \mathrm{SO}(10) \times \mathrm{SO}(8)$.
• Uses the Kitaev-Wen mechanism for symmetric mass generation and examines topological quantum phase transitions across energy scales.

Experimental results

Research questions

• RQ1 Can the $\mathbb{Z}_{16}$ global anomaly in the SM and SU(5) GUT be consistently matched in higher GUTs like SO(10) and SO(18)?
• RQ2 What are the energy-scale-dependent constraints on anomaly matching when extending from SU(5) to SO(10) and SO(18) GUTs?
• RQ3 How do 4d noninvertible TQFTs or 5d invertible TQFTs serve as alternative anomaly-matching mechanisms to right-handed neutrinos?
• RQ4 What is the role of the discrete $\mathbb{Z}_{4,X}$ symmetry in ensuring consistency across energy hierarchies and GUT breaking patterns?
• RQ5 Can the hidden topological sector account for sterile neutrinos and dark matter while preserving anomaly cancellation?

Key findings

• The $\mathbb{Z}_{16}$ global anomaly from the $X = 5(\mathbf{B}-\mathbf{L}) - 4Y$ symmetry is matched by adding one right-handed neutrino per generation, yielding $\nu \equiv -N_{\text{generation}} \mod 16$.
• For three generations, the anomaly requires $\nu = -3 \mod 16$, which is saturated by $N_{\nu_R} = 1$ per generation.
• The anomaly can also be canceled by a 4d noninvertible TQFT or a 5d invertible TQFT, offering alternatives to right-handed neutrinos.
• In SO(10) GUT, the $X$ symmetry must be gauged, leading to a noninvertible, long-range entangled 5d TQFT that couples to dynamical gravity.
• The energy hierarchy from $\mathrm{SO}(18) \to \mathrm{SO}(10) \times \mathrm{SO}(8)$ to $\mathrm{SO}(10) \times \mathrm{SO}(5)$ supports symmetric mass generation via the Kitaev-Wen mechanism.
• Topological quantum phase transitions are possible between different TQFT phases as energy scales are tuned, with $\Delta_{\mathrm{KW}}$ marking critical energy gaps.