$USp(32)$ Special Grand Unification

Abstract: We discuss a grand unified theory (GUT) based on a $USp(32)$ GUT gauge group broken to its subgroups including a special subgroup. A GUT based on an $SO(32)$ GUT gauge group has been discussed on six-dimensional (6D) orbifold space $M^4\times T^2/\mathbb{Z}_2$. It is inspired by the $SO(32)$ string theory behind the $SU(16)$ GUT whose $SU(16)$ is broken to a special subgroup $SO(10)$. Alternative direction is to embed an $SU(16)$ gauge group into a $USp(32)$ GUT gauge group, which is inspired by a non-supersymmetric symplectic-type $USp(32)$ string theory. In a $USp(32)$ GUT, one generation of the SM fermions is embedded into a 6D bulk Weyl fermion in a $USp(32)$ defining representation. For a three generation model, all the 6D and 4D gauge anomalies in the bulk and on the fixed points are canceled out without exotic chiral fermions at low energies. The SM Higgs scalar is embedded into a 6D bulk scalar field in a $USp(32)$ adjoint representation.

• The paper introduces a novel USp(32) GUT framework in a 6D orbifold space, embedding one generation of SM fermions into its defining representation to ensure anomaly cancellation.
• It employs a three-stage symmetry breaking mechanism using orbifold boundary conditions and 5D brane scalars to reduce USp(32) down to the Standard Model gauge group.
• The model builds on non-supersymmetric string theories, providing a viable unification approach that avoids low-energy exotic fermions and aligns with observable phenomena.

$USp(32)$ Special Grand Unification

Introduction

The paper presents a grand unified theory (GUT) based on a $USp(32)$ gauge group, exploring an alternative unification strategy distinct from traditional gauge groups like $SU(5)$, $SO(10)$, and $E_6$, which are commonly involved in GUTs (2007.08067). The proposed model is motivated by both the $SO(32)$ string theory and a non-supersymmetric symplectic-type $USp(32)$ string theory, grounded in higher-dimensional orbifold space $M^4\times T^2/\mathbb{Z}_2$. This approach aims to provide a non-supersymmetric framework that embeds the Standard Model (SM) within a larger symmetry group, and attempts to address the unification of the gauge interactions.

The Theoretical Framework

The paper proposes a new GUT framework, "USp(32) Special Grand Unification" that operates in a six-dimensional (6D) orbifold space $M^4 \times T^2/\mathbb{Z}_2$ and is inspired by symplectic-type non-supersymmetric string theories. A notable feature of this theory is the embedding of one generation of Standard Model (SM) fermions into a 6D bulk Weyl fermion situated within the defining representation of the $USp(32)$ group. The model accommodates three SM generations in a manner that naturally cancels all gauge anomalies, vital for a consistent field theory, without introducing exotic chiral fermions at low energies.

Symmetry Breaking Mechanism

In the proposed $USp(32)$ special GUT, the large symmetry group is systematically broken down to $SU(3)_C \times SU(2)_L \times U(1)_Y$ through three stages of symmetry breaking:

1. Orbifold Boundary Conditions (BCs): The starting point is the 6D $USp(32)$ gauge group, which is initially broken to $SU(16)\times U(1)$ by choosing appropriate boundary conditions on the orbifold $M^4\times T^2/\mathbb{Z}_2$.
2. Spontaneous Symmetry Breaking with Brane Scalars: The introduction of 5D brane scalar fields on the UV brane at $y=0$ leads to the spontaneous symmetry breaking of $SU(16) \times U(1)$ to $SO(10)$, and further down to either $SU(5)$ or $G_{\rm PS}$, and finally to the Standard Model gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$. This breaking pattern involves 5D brane scalars $\Phi_{SU(16)}$, $\Phi_{SO(10)}$, and $\Phi_{SO(5,1)}$ with hierarchically different VEVs.
3. Sector Responsibilities:
   • The 6D bulk gauge field $A_M$ as part of an adjoint representation, contains the gauge bosons corresponding to the SM gauge fields.
   • An $SU(16)$ special GUT model utilizes a 6D $USp(32)$ defining representation to incorporate the SM fermions and Higgs boson.
   • One generation of the Standard Model fermions is embedded into a 6D bulk $\mathbb{W}_{2\pm}^{(a)}$ Weyl fermion in the $-0.125em(32)$ defining representation, ensuring anomaly cancellation without introducing exotic chiral fermions at low energies.

Anomaly Cancellation

The model addresses the anomaly cancellation, an essential requirement in building GUTs, to ensure that all the 6D and 4D gauge anomalies in both the bulk and at the fixed points are canceled. This is achieved by a careful selection and arrangement of the representations and corresponding boundary conditions for the fermionic and scalar fields.

Practical and Theoretical Implications

Practically, the $USp(32)$ GUT proposed provides a novel approach to embedding the SM within a larger gauge group without encountering problematic chiral fermion surplus at low energies. Theoretical implications of this model suggest a new avenue in special GUTs, linking non-supersymmetric string theories to observable low-energy phenomena.

Conclusion

The $USp(32)$ Special Grand Unification offers a compelling exploration of higher dimensional orbifold GUT models, showing potential for incorporation into the framework of both symplectic groups and string theory without generating exotic particles at low energies. Future efforts can focus on exploring the GHU extensions of the model to further align this large gauge group unorthodox approach with known particle physics and potential unknowns, such as dark matter. This development signifies a unique yet consistent path in the continuous exploration of theories beyond the SM, hinting at unexplored possibilities for future research in unification scenarios.