Supersymmetric $SU(5) \times U(1)_χ$ and the Weak Gravity Conjecture

Abstract

The gauge symmetry $SU(5) \times U(1)_χ$ is the unique maximal subgroup of SO(10) which retains manifest unification at $M_{GUT}$ of the Standard Model gauge couplings, especially if low scale supersymmetry is present. The spontaneous breaking of $U(1)_χ$ at some intermediate scale leaves unbroken a $Z_2$ symmetry which is precisely `matter' parity. This yields a stable supersymmetric dark matter particle as well as topologically stable cosmic strings. Motivated by the weak gravity conjecture we impose unification of $SU(5)$ and $U(1)_χ$ at an ultraviolet cutoff $Λ\sim α_Λ^{1/2} M_{P} \approx 5 \times 10^{17}$ GeV, where $α_Λ$ denotes the $SU(5)$ gauge coupling at $Λ$ and $M_P \approx 2.4 \times 10^{18}$ GeV is the reduced Planck Scale. The impact of dimension five operators suppressed by $Λ$ on gauge coupling unification, proton lifetime estimates and $b-τ$ Yukawa unification is discussed. In particular, the gauge boson mediated proton decay into $e^+π^0$ can lie within the $2-σ$ sensitivity of HyperKamiokande. We also discuss how the intermediate scale strings may survive inflation while the $SU(5)$ monopoles are inflated away. The unbroken $Z_2$ symmetry provides an intriguing link between dark matter, black holes carrying `quantum hair' and cosmic strings.