Supersymmetric $SU(5)\times U(1{)}_{\chi }$ and the Weak Gravity Conjecture

Abstract

The gauge symmetry $SU(5)\times U(1{)}_{\chi }$ 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 $U(1{)}_{\chi }$ 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 $U(1{)}_{\chi }$ at an ultraviolet cutoff $\Lambda \sim {\alpha }_{\Lambda }^{1/2}{M}_P\approx 5\times 10^{17}$ GeV, where ${\alpha }_{\Lambda }$ denotes the $SU(5)$ gauge coupling at $\Lambda$ and $M_P\approx 2.4\times 10^{18}$ GeV is the reduced Planck Scale. The impact of dimension five operators suppressed by $\Lambda$ on gauge coupling unification, proton lifetime estimates and $b-\tau$ Yukawa unification is discussed. In particular, the gauge boson mediated proton decay into $e^{+}{\pi }^0$ can lie within the $2-\sigma$ 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.

Comments: 6 pages, 6 figures