Searching for realistic 4d string models with a Pati-Salam symmetry -- Orbifold grand unified theories from heterotic string compactification on a Z6 orbifold

TL;DR

This work constructs three-family Pati-Salam models within a non-prime ${ m Z}_6$ heterotic orbifold using two discrete Wilson lines, producing a 5d-like limit on ${ m S^1}/{ m Z}_2$ with a distinctive 2+1 family split and a ${ m D}_{4}$ family symmetry for the first two families. The models connect to 5d orbifold GUT setups based on ${ m SO}_{10}$ or ${ m E}_{6}$, realize PS at low energies, and yield gauge-Yukawa unification for the third family in the ${ m E}_{6}$-bulk scenario, while predicting realistic but challenging Yukawa textures for the lighter families via higher-dimensional operators. A central tension arises from residual color-triplet states and proton stability, which compel detailed vacuum choices and may require non-perturbative string boundary conditions to reconcile gauge coupling unification with the string scale. The work highlights a promising direction for string-model-building of realistic GUT-like theories with controlled flavor structures, while outlining clear avenues for refinement, such as reducing exotics and ensuring proton stability across all orders.

Abstract

Motivated by orbifold grand unified theories, we construct a class of three-family Pati-Salam models in a Z6 abelian symmetric orbifold with two discrete Wilson lines. These models have marked differences from previously-constructed three-family models in prime-order orbifolds. In the limit where one of the six compactified dimensions (which lies in a Z2 sub-orbifold) is large compared to the string length scale, our models reproduce the supersymmetry and gauge symmetry breaking pattern of 5d orbifold grand unified theories on an S1/Z2 orbicircle. We find a horizontal 2+1 splitting in the chiral matter spectra -- 2 families of matter are localized on the Z2 orbifold fixed points, and 1 family propagates in the 5d bulk -- and identify them as the first-two and third families. Remarkably, the first two families enjoy a non-abelian dihedral D4 family symmetry, due to the geometric setup of the compactified space. In all our models there are always some color triplets, i.e. (6,1,1) representations of the Pati-Salam group, survive orbifold projections. They could be utilized to spontaneously break the Pati-Salam symmetry to that of the Standard Model. One model, with a 5d E6 symmetry, may give rise to interesting low energy phenomenology. We study gauge coupling unification, allowed Yukawa couplings and some of their phenomenological consequences. The E6 model has a renormalizable Yukawa coupling only for the third family. It predicts a gauge-Yukawa unification relation at the 5d compactification scale, and is capable of generating reasonable quark/lepton masses and mixings. Potential problems are also addressed, they may point to the direction for refining our models.

Figures (5)

• Figure 1: 5d ${\rm {E}_{6}}$ orbifold GUT model with bulk and brane states. The bulk gauge symmetry is broken to ${\rm {SO}_{10}}$ on the end of world brane at $x^5=0$ and to ${\rm {SU}_{6}}\times{\rm {SU}_{2R}}$ at $x^5=\pi R$. The massless sector of the 4d effective theory has a PS gauge symmetry. In addition, the bulk contains four hypermultiplets, and the ${\rm {SO}_{10}}$ brane contains two spinor representations, giving rise to the first two matter families.
• Figure 2: Fundamental region of the root lattice ${{\rm {G}_{2}}\oplus{\rm {SU}_{3}}\oplus{\rm {SO}_{4}}}$. The filled circles, crosses and squares represent fixed points in the $T_1$, $T_{2,4}$ and $T_3$ twisted sectors. See appendix \ref{['app:models']} for further details.
• Figure 3: ${{\rm {G}_{2}}\oplus{\rm {SU}_{3}}\oplus{\rm {SO}_{4}}}$ lattice with ${\mathbb Z}_3$ fixed points. The fields $V, \ \Sigma, \ $ and ${\bf 27} (\in U_1) + {\bf\overline{27}} (\in U_2)$ are bulk states from the untwisted sectors. On the other hand, $3\times({\bf 27} + {\bf\overline{27}})$ are "bulk" states located on the $T_{(0,1)}/T_{(0,2)}$ twisted sector (${\rm {G}_{2}}$, ${\rm {SU}_{3}}$) fixed points.
• Figure 4: ${{\rm {G}_{2}}\oplus{\rm {SU}_{3}}\oplus{\rm {SO}_{4}}}$ lattice with ${\mathbb Z}_6$ fixed points. The $T_{(1,1)}/ T_{(1,2)}$ twisted sector states sit at these fixed points.
• Figure 5: ${{\rm {G}_{2}}\oplus{\rm {SU}_{3}}\oplus{\rm {SO}_{4}}}$ lattice with ${\mathbb Z}_2$ fixed points. The $T_{(1,0)}$ twisted sector states sit at these fixed points.