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 ultraunification (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 Lie group ${G}_{{\mathrm{SM}}_{q}}\ensuremath{\equiv}\frac{\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1{)}_{\stackrel{\texttildelow{}}{Y}}}{{\mathbb{Z}}_{q}}$ of $q=1$, 2, 3, 6 and ${N}_{f}$ families of 15 or 16 Weyl fermions, in addition to the continuous baryon minus lepton $\mathrm{U}(1{)}_{\mathbf{B}\ensuremath{-}\mathbf{L}}$ symmetry, there is also a compatible discrete baryon plus lepton ${\mathbb{Z}}_{2{N}_{f},\mathbf{B}+\mathbf{L}}$ symmetry. The ${\mathbb{Z}}_{2{N}_{f},\mathbf{B}+\mathbf{L}}$ is discrete due to the Adler-Bell-Jackiw anomaly under the BPST SU(2) instanton. Although both $\mathrm{U}(1{)}_{\mathbf{B}\ensuremath{-}\mathbf{L}}$ and ${\mathbb{Z}}_{2{N}_{f},\mathbf{B}+\mathbf{L}}$ symmetries are anomaly free under the dynamically gauged ${G}_{{\mathrm{SM}}_{q}}$, it is important to check whether they have mixed anomalies with the gravitational background field (spacetime diffeomorphism under Spin group rotation) and higher symmetries (whose charged objects are Wilson electric or 't Hooft magnetic line operators) of SM. We can also replace the $\mathrm{U}(1{)}_{\mathbf{B}\ensuremath{-}\mathbf{L}}$ with a discrete variant ${\mathbb{Z}}_{4,X}$ for $X\ensuremath{\equiv}5(\mathbf{B}\ensuremath{-}\mathbf{L})\ensuremath{-}\frac{2}{3}\stackrel{\texttildelow{}}{Y}$ of electroweak hypercharge $\stackrel{\texttildelow{}}{Y}$. 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 ultraunification in the presence of discrete $\mathbf{B}+\mathbf{L}$ symmetry protection, in particular $(\mathrm{U}(1{)}_{\mathbf{B}\ensuremath{-}\mathbf{L}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2{N}_{f},\mathbf{B}+\mathbf{L}})/{\mathbb{Z}}_{2}^{F}$ or $({\mathbb{Z}}_{4,X}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2{N}_{f},\mathbf{B}+\mathbf{L}})/{\mathbb{Z}}_{2}^{F}$ with the fermion parity ${\mathbb{Z}}_{2}^{F}$.