Supersymmetric $SU\mathbf{(}5\mathbf{)}\times U\mathbf{(}1{\mathbf{)}}_{\chi }$ and the weak gravity conjecture

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 $\mathrm{\Lambda }\sim {\alpha }_{\mathrm{\Lambda }}^{1/2}{M}_P\approx 5\times {10}^{17}\mathrm{GeV}$, where ${\alpha }_{\mathrm{\Lambda }}$ denotes the $SU(5)$ gauge coupling at $\mathrm{\Lambda }$ and $M_P\approx 2.4\times {10}^{18}\mathrm{GeV}$ is the reduced Planck scale. The impact of dimension five operators suppressed by $\mathrm{\Lambda }$ on gauge coupling unification, proton lifetime estimates and $b-\tau$ Yukawa unification is discussed. In particular, the modified proton lifetime estimate for decay into $e^{+}{\pi }^0$ can be tested at Hyper-Kamiokande. 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.

Figure 1

Running of the gauge couplings in MSSM and $\chi SU(5)$. Unification of the $\chi SU(5)$ gauge couplings occurs at $\mathrm{\Lambda }\approx 5\times {10}^{17}\mathrm{GeV}$. ${\mu }_{\chi }={10}^{14}\mathrm{GeV}$ denotes the $U(1{)}_{\chi }$ symmetry breaking scale and $M_P=2.4\times {10}^{18}\mathrm{GeV}$ is the reduced Planck scale.

Figure 2

Proton lifetime vs. $\epsilon$ (blue line). The green line denotes the $2\sigma$ experimental bound on proton lifetime set by Super-K [19], and the red line is the expected $2\sigma$ sensitivity at Hyper-K [20].

Figure 3

$y_b/y_{\tau }$ versus $\mu$, the energy scale, for $\tan \beta =50$. $y_b-y_{\tau }$ at $M_{GUT}$ are 0.06 (top), 0 (middle) and $-0.04$ (bottom). ${\delta }_b^{\text{finite}}$ denote the size of the finite one loop corrections to $y_b$.