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

## TL;DR

This paper explores a supersymmetric $SU(5) imes U(1)_ ext{chi}$ GUT model consistent with the weak gravity conjecture, analyzing unification, proton decay, cosmic strings, and dark matter implications.

## Contribution

It introduces a novel supersymmetric GUT framework with unification at a high scale, incorporating the weak gravity conjecture and examining its phenomenological consequences.

## Key findings

- Proton decay into $e^+ ext{pi}^0$ may be observable in HyperKamiokande.
- Intermediate scale cosmic strings can survive inflation.
- Dimension five operators affect unification and proton lifetime estimates.

## 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.