In this paper, we embed the ${\mathbf{Z}}_{4R}$ parity as a discrete subgroup of a global symmetry ${\mathrm{U}(1)}_{\mathrm{R}}$ obtained from ${\mathbf{Z}}_{12-I}$ compactification of a heterotic string ${\mathrm{E}}_8\times {\mathrm{E}}_8^{\prime }$. A part of ${\mathrm{U}(1)}_{\mathrm{R}}$ transformation is the shift of the anticommuting variable $\vartheta$ to $e^{i\alpha }\vartheta$, which necessarily incorporates the transformation of the internal space coordinate. Out of six internal spaces, we identify three U(1)’s whose charges are denoted as $Q_{18}$, $Q_{20}$, and $Q_{22}$. The ${\mathrm{U}(1)}_{\mathrm{R}}$ is defined as ${\mathrm{U}(1)}_{\mathrm{EE}}\times {\mathrm{U}(1)}_{\mathrm{KK}}$, where ${\mathrm{U}(1)}_{\mathrm{EE}}$ is the part from the ${\mathrm{E}}_8\times {\mathrm{E}}_8^{\prime }$ and ${\mathrm{U}(1)}_{\mathrm{KK}}$ is the part generated by $Q_{18}$, $Q_{20}$, and $Q_{22}$. We propose a method to define a ${\mathrm{U}(1)}_{\mathrm{R}}$ direction. The needed vacuum expectation values for breaking gauge U(1)’s except for ${\mathrm{U}(1)}_Y$ of the standard model carry a ${\mathrm{U}(1)}_{\mathrm{R}}$ charge 4 modulo 4 such that ${\mathrm{U}(1)}_{\mathrm{R}}$ is broken down to ${\mathbf{Z}}_{4R}$ at the grand unification scale. ${\mathbf{Z}}_{4R}$ is broken to ${\mathbf{Z}}_{2R}$ between the intermediate ($\sim 10^{11}\mathrm{GeV}$) and the electroweak scales ($100\mathrm{GeV}\sim 1\mathrm{TeV}$). The conditions we impose are proton longevity, a large top quark mass, and acceptable magnitudes for the $\mu$ term and neutrino masses.

• Received 6 December 2018

DOI:https://doi.org/10.1103/PhysRevD.99.093004

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Published by the American Physical Society