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SO(10) Unified Theories in Six Dimensions

Lawrence Hall, Yasunori Nomura, Takemichi Okui, David Smith

TL;DR

This work demonstrates that breaking $SO(10)$ to the Standard Model gauge group with an extra $U(1)_X$ via orbifold compactification in six dimensions can yield natural doublet-triplet splitting and proton stability, while addressing right-handed neutrino masses.It develops three explicit 6D models on different orbifolds: $T^2/Z_2$, $T^2/Z_6$, and $T^2/(Z_2\times Z_2')$, analyzing bulk versus brane matter, anomaly cancellation (including Green–Schwarz in the $T^2/Z_2$ case), and the implications for fermion mass relations and SUSY breaking.A key finding is that, unlike 5D $SU(5)$ constructions, $SO(10)$ requires at least two extra dimensions to avoid unwanted zero modes, with fixed-point or fixed-line gauge structures shaping Yukawas and mass relations; however, achieving precise gauge coupling unification and communicating $U(1)_X$ breaking to generate realistic neutrino masses remains challenging across the models.Overall, the orbifold GUT approach in six dimensions offers a promising framework to address longstanding GUT issues, while highlighting tensions between anomaly cancellation, neutrino mass generation, and exact predictions for the weak mixing angle.

Abstract

We construct supersymmetric models of SO(10) unification in which the gauge symmetry is broken by orbifold compactification. We find that using boundary conditions to break the gauge symmetry down to $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_X$ without leaving unwanted massless states requires at least two extra dimensions, motivating us to work with 6D orbifolds. SO(10) is broken by two operations, each of which induces gauge-breaking to either the Georgi-Glashow, Pati-Salam, or flipped $SU(5) \otimes U(1)$ subgroups; assigning different unbroken subgroups to the two operations leaves only the standard model gauge group and $U(1)_X$ unbroken. The models we build employ extra-dimensional mechanisms for naturally realizing doublet-triplet splitting, suppressing proton decay, and avoiding unwanted grand-unified fermion mass relations. We find some tension between being free of anomalies of the 6D bulk, accommodating a simple mechanism for generating right-handed neutrino masses, and preserving the precise prediction of the weak mixing angle.

SO(10) Unified Theories in Six Dimensions

TL;DR

This work demonstrates that breaking $SO(10)$ to the Standard Model gauge group with an extra $U(1)_X$ via orbifold compactification in six dimensions can yield natural doublet-triplet splitting and proton stability, while addressing right-handed neutrino masses.It develops three explicit 6D models on different orbifolds: $T^2/Z_2$, $T^2/Z_6$, and $T^2/(Z_2\times Z_2')$, analyzing bulk versus brane matter, anomaly cancellation (including Green–Schwarz in the $T^2/Z_2$ case), and the implications for fermion mass relations and SUSY breaking.A key finding is that, unlike 5D $SU(5)$ constructions, $SO(10)$ requires at least two extra dimensions to avoid unwanted zero modes, with fixed-point or fixed-line gauge structures shaping Yukawas and mass relations; however, achieving precise gauge coupling unification and communicating $U(1)_X$ breaking to generate realistic neutrino masses remains challenging across the models.Overall, the orbifold GUT approach in six dimensions offers a promising framework to address longstanding GUT issues, while highlighting tensions between anomaly cancellation, neutrino mass generation, and exact predictions for the weak mixing angle.

Abstract

We construct supersymmetric models of SO(10) unification in which the gauge symmetry is broken by orbifold compactification. We find that using boundary conditions to break the gauge symmetry down to without leaving unwanted massless states requires at least two extra dimensions, motivating us to work with 6D orbifolds. SO(10) is broken by two operations, each of which induces gauge-breaking to either the Georgi-Glashow, Pati-Salam, or flipped subgroups; assigning different unbroken subgroups to the two operations leaves only the standard model gauge group and unbroken. The models we build employ extra-dimensional mechanisms for naturally realizing doublet-triplet splitting, suppressing proton decay, and avoiding unwanted grand-unified fermion mass relations. We find some tension between being free of anomalies of the 6D bulk, accommodating a simple mechanism for generating right-handed neutrino masses, and preserving the precise prediction of the weak mixing angle.

Paper Structure

This paper contains 20 sections, 29 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. SO(10) Gauge Symmetry Breaking on a Torus
  3. A Model on $T^2/Z_2$
  4. Gauge fields in the bulk
  5. Gauge symmetries at the orbifold fixed points
  6. Doublet-triplet splitting
  7. Bulk anomalies
  8. Quarks and leptons in the bulk
  9. Minimal matter content from anomaly cancellation
  10. Supersymmetry breaking, $U(1)_X$ breaking, and neutrino masses
  11. Quarks and leptons on branes
  12. A Model on $T^2/Z_6$
  13. Orbifold structure
  14. Matter configuration
  15. $U(1)_X$ breaking and fermion masses
  16. ...and 5 more sections

Figures (2)

  • Figure 1: A convenient grouping of the $SO(10)$ generators. The parities of the corresponding gauge bosons under torus translations are given in Eq. (\ref{['eq:gaugemodes']}).
  • Figure 2: The $T^2/Z_2$ orbifold in the $z=x_5+ix_6$ plane. Each orbifold fixed point is denoted by a cross and labelled by the non-trivial gauge transformations acting on it: 10 for $SO(10)$, 5-1 for $SU(5) \otimes U(1)_X$, $5'$-$1'$ for $SU(5)' \otimes U(1)'_X$ and 4-2-2 for $SU(4)_C \otimes SU(2)_L \otimes SU(2)_R$. The physical space may be taken as the two-sided rectangle formed by folding the shaded region along the dotted line and then gluing together the touching edges.