Proton Stability: From the Standard Model to Beyond Grand Unification

Abstract

A proton is known for its longevity, but what is its lifetime? While many Grand Unified Theories predict the proton decay with a finite lifetime, we show that the Standard Model (SM) and some versions of Ultra Unification (which replace sterile neutrinos with new exotic gapped/gapless sectors, e.g., topological or conformal field theory under global anomaly cancellation constraints) with a discrete baryon plus lepton symmetry permit a stable proton. For the 4d SM with $N_f$ families of 15 or 16 Weyl fermions, in addition to the continuous baryon minus lepton U(1)${}_{\mathbf{B}-\mathbf{L}}$ symmetry, there is also a compatible discrete baryon plus lepton ${\mathbb{Z}}_{2N_f,\mathbf{B}+\mathbf{L}}$ symmetry. The ${\mathbb{Z}}_{2N_f,\mathbf{B}+\mathbf{L}}$ is discrete due to the ABJ anomaly under the BPST SU(2) instanton. Although both U(1)${}_{\mathbf{B}-\mathbf{L}}$ and ${\mathbb{Z}}_{2N_f,\mathbf{B}+\mathbf{L}}$ symmetries are anomaly-free under the dynamical SM gauge field, it is important to check whether they have mixed anomalies with the gravitational background field and higher symmetries (whose charged objects are Wilson electric or 't Hooft magnetic line operators) of SM. We can also replace the U(1)${}_{\mathbf{B}-\mathbf{L}}$ with a discrete variant ${\mathbb{Z}}_{4,X}$ for $X\equiv 5(\mathbf{B}-\mathbf{L})-\frac{2}{3}\tilde{Y}$ of electroweak hypercharge $\tilde{Y}$. We explore a systematic classification of candidate perturbative local and nonperturbative global anomalies of the 4d SM, including all these gauge and gravitational backgrounds, via a cobordism theory, which controls the SM's deformation class. We discuss the proton stability of the SM and Ultra Unification in the presence of discrete $\mathbf{B}+\mathbf{L}$ symmetry protection, in particular (U(1)${}_{\mathbf{B}-\mathbf{L}}\times {\mathbb{Z}}_{2N_f,\mathbf{B}+\mathbf{L}})/{\mathbb{Z}}_2^{\mathrm{F}}$ or $({\mathbb{Z}}_{4,X}\times {\mathbb{Z}}_{2N_f,\mathbf{B}+\mathbf{L}})/{\mathbb{Z}}_2^{\mathrm{F}}$ symmetry with the fermion parity ${\mathbb{Z}}_2^{\mathrm{F}}$.

arXiv comment6 pages. Sequel to arXiv:2112.14765. v2: refinement. Related talks: https://www.youtube.com/results?search_query=cobordism+deformation+standard+model+crticiality. The original title "Proton Stability: From the Standard Model to Ultra Unification" is adjusted in PRD

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Cited by in corpus (9)

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• Cobordism and Deformation Class of the Standard Model
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Gauge Enhanced Quantum Criticality Beyond the Standard Model
• Anomalies of non-Abelian finite groups via cobordism
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• Non-Invertible Peccei-Quinn Symmetry, Natural 2HDM Alignment, and the Visible Axion
• A Note on Proton Stability in the Standard Model
• Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states