[Paper Review] A Minimal Supersymmetric $\mathrm{E}_6$ Unified Theory

K. S. Babu, Borut Bajc+1 more|arXiv (Cornell University)|Apr 3, 2015

Particle physics theoretical and experimental studies19 references3 citations

TL;DR

This paper proposes a minimal supersymmetric E₆ Grand Unified Theory with a Higgs sector comprising 27 + 27̄ + 351′ + 351̄′ + 78 representations, achieving realistic electroweak symmetry breaking and successful doublet-triplet splitting via the 78 representation. The model uses only two symmetric Yukawa matrices to generate all fermion masses and mixings, yielding a good numerical fit to quark and lepton data, making it the most predictive renormalizable supersymmetric E₆ GUT to date.

ABSTRACT

We show explicitly that supersymmetric $E_6$ Grand Unified Theory with a Higgs sector consisting of $\{27+\bar{27}+351'+\bar{351'}+78\}$ fields provides a realistic scenario for symmetry breaking and fermion mass generation. While gauge symmetry breaking can be achieved without the $78$ field, its presence is critical for a successful doublet-triplet mass splitting. The Yukawa sector of the model consists of only two symmetric matrices describing all of quark, lepton and neutrino masses and mixings. The fermion mass matrices are computed at low energy and a fit to the second and third generation masses and mixings is performed. We find a good numerical fit to the low-energy data. Thus, this model, having $11$ superpotential parameters, alongside the two symmetric Yukawa matrices, seems to be the best realistic candidate for a minimal renormalizable supersymmetric $E_6$ unified theory.

Motivation & Objective

• To construct a minimal, renormalizable supersymmetric E₆ Grand Unified Theory that successfully breaks gauge symmetry to the Standard Model (SM) group.
• To resolve the doublet-triplet splitting problem in E₆ GUTs, which previously failed in models with only 27+27̄+351′+351̄′ Higgs fields.
• To reduce the number of Yukawa matrices to just two, enhancing predictivity compared to prior models with three or more matrices.
• To achieve a realistic fit to fermion masses and mixing angles using only 11 superpotential parameters and two symmetric Yukawa matrices.

Proposed method

• Introduce the adjoint 78 representation to the minimal Higgs sector (27+27̄+351′+351̄′), which is essential for doublet-triplet splitting without coupling to matter fields.
• Use equations of motion and an ansatz for vacuum expectation values (VEVs) to solve the symmetry breaking pattern, ensuring the SM gauge group is realized.
• Construct the Yukawa sector with two symmetric matrices coupling the 3 generations of matter fields (27_F) to the 351′ and 27 Higgs fields.
• Project the 27 representation into SM quantum numbers, identifying the 3 light generations and 6 vector-like states (3 down-type quarks/charged leptons, 3 neutrino doublet-antidoublet pairs, 6 singlets).
• Compute fermion mass matrices at low energy and perform a numerical fit to second and third generation quark and lepton masses and mixing angles.
• Suppress proton decay via a split supersymmetry scenario, ensuring compatibility with experimental bounds without altering the model's core predictions.

Experimental results

Research questions

• RQ1 Can the doublet-triplet splitting problem in E₆ GUTs be solved using only two Yukawa matrices and a minimal Higgs sector?
• RQ2 Does the inclusion of the 78 representation enable a realistic vacuum solution that breaks E₆ to the SM gauge group while preserving the MSSM Higgs doublet structure?
• RQ3 Can a numerical fit to fermion masses and mixing angles be achieved with only two symmetric Yukawa matrices in a renormalizable E₆ GUT?
• RQ4 Is the neutrino mass scale naturally adjustable in this model, avoiding the tension seen in minimal SO(10) GUTs?
• RQ5 Can proton decay be suppressed without altering the model's predictive power or symmetry breaking structure?

Key findings

• The 78 representation is essential for doublet-triplet splitting; without it, the model fails to generate a light Higgs doublet due to identical mass matrix structures for doublets and triplets.
• The model achieves a realistic vacuum solution with the correct SM gauge group, confirmed via explicit solution of the equations of motion and F-term constraints.
• A good numerical fit to the second and third generation quark and lepton masses and mixing angles is obtained using only two symmetric Yukawa matrices and 11 superpotential parameters.
• The neutrino mass scale is naturally adjustable via the VEV of a Pati-Salam (1,3,10) component of the 351′, avoiding the tension seen in minimal SO(10) GUTs.
• Proton decay amplitudes are suppressed in a split supersymmetry scenario, ensuring compatibility with experimental limits without modifying the model's low-energy phenomenology.
• The model is the first minimal, renormalizable, supersymmetric E₆ GUT with only two Yukawa matrices that successfully unifies symmetry breaking, fermion masses, and gauge unification.