Proton Stability: From the Standard Model to Beyond Grand Unification

TL;DR

This paper investigates conditions under which the Standard Model and certain beyond Standard Model theories can have a stable proton, focusing on discrete symmetries and anomaly considerations using cobordism theory.

Contribution

It systematically classifies anomalies in the Standard Model related to discrete baryon plus lepton symmetries, exploring implications for proton stability in various theoretical frameworks.

Findings

Discrete baryon plus lepton symmetry can protect proton stability.

Certain anomaly-free symmetry combinations allow a stable proton.

Cobordism theory helps classify potential anomalies in the Standard Model.

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)$_{\bf B - L}$ symmetry, there is also a compatible discrete baryon plus lepton $\mathbb{Z}_{2N_f, \bf B + L}$ symmetry. The $\mathbb{Z}_{2N_f, \bf B + L}$ is discrete due to the ABJ anomaly under the BPST SU(2) instanton. Although both U(1)$_{\bf B - L}$ and $\mathbb{Z}_{2N_f, \bf B + L}$ 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)$_{\bf B - L}$ with a discrete variant $\mathbb{Z}_{4,X}$ for $X \equiv 5({\bf B - L})-\frac{2}{3} {\tilde Y}$ 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 ${\bf B + L}$ symmetry protection, in particular (U(1)$_{\bf B - L} \times \mathbb{Z}_{2N_f,\bf B + L})/{\mathbb{Z}_2^{\rm F}}$ or $(\mathbb{Z}_{4,X} \times \mathbb{Z}_{2N_f, \bf B + L})/{\mathbb{Z}_2^{\rm F}}$ symmetry with the fermion parity $\mathbb{Z}_2^{\rm F}$.