Family Unification in $SO(16)$ Grand Unification

Abstract: We propose a unified model for the three Standard Model (SM) gauge symmetries and $SU(3)$ family symmetry based on $SO(16)$ grand unified gauge symmetry on six-dimensional (6D) spacetime. In this model, three chiral generations of quarks and leptons are unified into a 6D Weyl fermion in the spinor representation of $SO(16)$. The 6D $SO(16)$ gauge anomaly is canceled by the vectorlike nature of the model, and the 4D gauge anomalies are canceled by introducing suitable 4D localized fermions at the fixed point. The 4D gauge coupling constant of $SO(16)$ has the property of becoming weaker at high energies.

• The paper introduces a 6D SO(16) framework that unifies three chiral generations within a single Weyl fermion.
• It employs orbifold compactification and scalar VEVs to break the symmetry into a SM-like gauge group while effectively canceling anomalies.
• Renormalization group analysis reveals asymptotic freedom of the SO(16) gauge coupling due to negative contributions from KK modes.

Family Unification in $SO(16)$ Grand Unification

Introduction

The paper explores the theoretical framework of unifying the three chiral generations of quarks and leptons in the Standard Model (SM) using a novel approach based on $SO(16)$ grand unified gauge symmetry. The framework is set in a six-dimensional spacetime to address long-standing issues in particle physics regarding the chiral nature and mass hierarchy of fermions. The authors propose a model where three generations are consolidated into a single 6D Weyl fermion. This is facilitated by the spinor representation of $SO(16)$, a novel strategy that aims to bypass the limitations of earlier Grand Unified Theories (GUTs) which often generated unwanted additional generations.

$SO(16)$ GUT Framework

In this paper, the construction utilizes a richer gauge structure afforded by $SO(16)$ in a 6D spacetime characterized by orbifold compactification. This allows for the breaking of the $SO(16)$ gauge symmetry into $SO(10)\times SU(3)\times U(1)$, mimicking areas of the SM while incorporating additional family symmetries. The choice of orbifold spacetime $M^4 \times D^2/\mathbb{Z}_N$ facilitates this breakdown via the properties of fixed points, naturally leading to anomaly cancellation mechanisms inherent in the model design.

Anomaly Cancellation

An essential aspect of constructing a consistent GUT lies in addressing gauge and gravitational anomalies. The model uses both positive and negative Weyl fermions to maintain vectorlike balance in six dimensions, which assures cancellation of 6D $SO(16)$ gauge anomalies. In four dimensions, the anomalies are managed by strategically incorporating 4D localized fermions at fixed points, rendering the theoretical buildup free from the inconsistencies that often plague unified models.

Symmetry Breaking and VEVs

To fully realize the model's reduction to the SM gauge group, scalar fields are introduced, whose vacuum expectation values (VEVs) perform successive symmetry breaking stages: from $SO(16)$ to $SO(10)\times SU(3)\times U(1)$ and subsequently to the SM gauge group. This layered breakdown entails specific scalar content, crucially relying on nonsingular VEVs at grand unification scale magnitudes to terminate with the low-energy interactions visible today, minus unwanted exotics.

Renormalization Group Analysis

The renormalization group equation (RGE) for the gauge coupling constants above the compactification scale ($1/R$) is critically examined. The model supports asymptotic freedom of the $SO(16)$ gauge coupling. This arises from negative contributions to the $\beta$-function by the KK modes in the 6D field content, cementing the robustness of coupling behavior at high energies, which is a staple requirement for any realistic GUT model.

Conclusion

By engaging a 6D framework with rich $SO(16)$ symmetry, the paper proposes a comprehensive avenue for family unification, absent of exotic fermions while retaining gauge anomaly cancellation. This model holds promising implications for new physics scenarios, offering a novel direction for familial symmetry in unification plans. The asymptotic freedom of the $SO(16)$ gauge coupling and computed RGEs suggest noteworthy extensions into investigation realms, such as proton decay and dark matter insights, highlighting future theoretical advancements stimulated by this groundwork.