[Paper Review] Aspects of symmetry breaking in Grand Unified Theories

Luca Di Luzio|arXiv (Cornell University)|Oct 14, 2011

Particle physics theoretical and experimental studies209 references21 citations

TL;DR

This thesis re-evaluates spontaneous symmetry breaking in non-supersymmetric $SO(10)$ Grand Unified Theories, demonstrating that quantum corrections—previously overlooked at tree level—enable viable breaking patterns consistent with gauge coupling unification. The key result is that a minimal Higgs sector with only the 45- and 16-representations can achieve realistic symmetry breaking when quantum effects are included, overturning a longstanding tree-level 'no-go' theorem.

ABSTRACT

We reconsider the issue of spontaneous symmetry breaking in SO(10) grand unified theories. The emphasis is put on the quest for the minimal Higgs sector leading to a phenomenologically viable breaking to the standard model gauge group. Longstanding results claimed that nonsupersymmetric SO(10) models with just the adjoint representation triggering the first stage of the breaking cannot provide a successful gauge unification. The main result of this thesis is the observation that this no-go is an artifact of the tree level potential and that quantum corrections opens in a natural way the vacuum patterns favoured by gauge coupling unification. An analogous no-go, preventing the breaking of SO(10) at the renormalizable level with representations up to the adjoint, holds in the supersymmetric case as well. In this respect we show that a possible way-out is provided by considering the flipped SO(10) embedding of the hypercharge. Finally, the case is made for the hunting of the minimal SO(10) theory.

Motivation & Objective

• To resolve the longstanding issue that non-supersymmetric $SO(10)$ GUTs with only adjoint Higgses cannot achieve gauge unification at tree level.
• To investigate whether quantum corrections can lift the tree-level no-go theorems for minimal Higgs sectors in $SO(10)$ GUTs.
• To explore the viability of minimal $SO(10)$ models with only the 45 and 16 representations as the Higgs sector, focusing on vacuum stability and phenomenological consistency.
• To examine the role of flipped $SO(10)$ hypercharge embedding as a way-out in the supersymmetric case, where similar no-go theorems apply.
• To establish a framework for hunting the minimal viable $SO(10)$ GUT, both supersymmetric and non-supersymmetric, by analyzing vacuum structure and quantum corrections.

Proposed method

• Analyzes the one-loop effective potential in $SO(10)$ GUTs with a minimal Higgs sector composed of the 45 and 16 representations.
• Computes two-loop renormalization group equations (RGEs) for gauge couplings, including $U(1)$ mixing effects, to assess unification conditions.
• Derives one-loop stationary equations and scalar mass spectra, including pseudo-Goldstone bosons (PGBs), to study vacuum stability and symmetry breaking patterns.
• Applies the extended survival hypothesis to constrain intermediate-scale breaking chains and analyze vacuum manifolds.
• Compares the standard $SO(10)$ hypercharge embedding with the flipped $SO(10)$ embedding to evaluate differences in gauge group breaking and residual symmetries.
• Uses the $E_6$ embedding formalism to analyze vacuum manifolds and identify preserved generators, particularly in the context of $78 o 27 o ar{27}$ Higgs VEVs.

Experimental results

Research questions

• RQ1 Can quantum corrections in $SO(10)$ GUTs with only the 45 and 16 Higgs representations lift the tree-level no-go theorems preventing gauge coupling unification?
• RQ2 What is the structure of the one-loop vacuum in minimal $SO(10)$ models, and how does it differ from the classical tree-level vacuum?
• RQ3 How does the choice of hypercharge embedding—standard vs. flipped—impact the viability of minimal $SO(10)$ GUTs in the supersymmetric case?
• RQ4 Can a minimal Higgs sector (45 and 16 only) lead to a phenomenologically viable breaking pattern to the SM gauge group when quantum corrections are included?
• RQ5 What constraints do vacuum manifolds and preserved generators (e.g., $SU(5)$) impose on the structure of the Higgs potential and symmetry breaking in $E_6$-embedded models?

Key findings

• Quantum corrections to the one-loop effective potential open new vacuum directions that are forbidden at tree level, allowing viable symmetry breaking in minimal $SO(10)$ GUTs.
• The one-loop vacuum structure supports a stable breaking pattern to $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$, even when the tree-level potential forbids it.
• In the minimal $SO(10)$ model with only 45 and 16 Higgs representations, the one-loop scalar mass spectrum includes light pseudo-Goldstone bosons, consistent with the extended survival hypothesis.
• The flipped $SO(10}$ embedding of hypercharge provides a viable alternative in the supersymmetric case, where the standard embedding leads to a no-go theorem for minimal Higgs sectors.
• The vacuum manifold in the $E_6$-embedded model preserves an $SU(5)$ subgroup, and the preserved generators include $(\overline{3},2,\pm\frac{5}{6})$ leptoquark states, consistent with $SO(10)$ unification.
• When $\alpha_1 = \alpha_2 = 0$, the vacuum preserves an $SO(10) \otimes U(1)$ gauge group, with the extra $U(1)$ generated by $T_L^{(8)} - T_R^{(8)}$, showing that the minimal model can support a consistent unification scale.