[Paper Review] Further thoughts on supersymmetric $E_8$ as a family and grand unification theory

Stephen L. Adler|ArXiv.org|Jan 26, 2004

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper proposes a viable supersymmetric $E_8$ grand unification model where the supersymmetric partners of standard model fermions are vectors, not scalars, with chiral superfields limited to symmetry-breaking Higgs fields. It shows that gravitational couplings stabilize the vacuum via an $\mathcal{F}$-type superpotential, generating a gluino mass of $\sim10^{-3}$ eV and enabling dynamical $SO(10)\times SU(3)\times U(1)$ breaking, offering a new paradigm for low-energy model building.

ABSTRACT

We continue the analysis of the possibility of supersymmetric $E_8$ as a family unification and grand unification theory, this time under the assumption that there is a vacuum gluino condensate, but that this condensate is {\it not} accompanied by dynamical generation of a mass gap in the pure $E_8$ gauge theory. Arguments supporting these assumptions are given. When the $E_8$ theory is coupled to supergravity, assuming vanishing of the cosmological constant and a supersymmetry breaking scale of around a TeV, we show that the gluino mass induced by gravitational coupling to the condensate is of order $10^{-3}$ eV or smaller, compatible with the fermion (and particularly the neutrino) mass spectrum. We suggest that composite scalar Higgs superfields can arise from a chiral glueball in the attractive 3875 channel (and possibly other channels), permitting the breaking of the original $E_8$ gauge group to a SO(10) grand unification group, times a SU(3) family symmetry group and an extra U(1) factor. A general analysis of the Higgs superpotential shows that in the absence of gravitational couplings, there is always a supersymmetric vacuum in the (unphysical) limit of infinite Higgs superfields. However, when gravitational couplings are included, dimensional analysis of the superpotential shows that the vacuum can be stabilized for finite Higgs superfields, with the occurrence of dynamical ${\cal F}$-type supersymmetry breaking. We conclude that an $E_8$ unification may be theoretically viable, providing an alternative paradigm for low energy model building, in which the supersymmetric partners of the standard model fermions are vectors, and in which the only chiral superfields are symmetry breaking Higgs fields.

Motivation & Objective

• To explore the viability of $E_8$ as a grand unification and family unification group in a supersymmetric framework, despite the failure of the condensate-free vacuum proposal.
• To address the challenge of dynamical supersymmetry breaking and gauge symmetry breaking in $E_8$-based models without relying on perturbative Higgs fields.
• To demonstrate that gravitational couplings can stabilize the vacuum and generate a natural hierarchy, enabling phenomenologically acceptable low-energy models.
• To propose a new paradigm in which only Higgs superfields are chiral, and supersymmetric partners of fermions are vector-like, differing from the MSSM.

Proposed method

• Assumes a gluino condensate in $E_8$ supersymmetric Yang–Mills theory without a dynamically generated mass gap, motivated by group-theoretic and non-perturbative considerations.
• Analyzes the coupling of the $E_8$ theory to linearized supergravity, showing that gravitational couplings allow the formation of effective superpotentials with dimensionless parameters $\mathcal{M}/M_{\text{Planck}}$.
• Constructs an $\mathcal{F}$-type superpotential involving composite Higgs superfields, using dimensionless functions that vanish at infinity, to stabilize the vacuum at finite field values.
• Demonstrates that the superpotential leads to spontaneous supersymmetry breaking via non-vanishing $F$-terms, with the breaking scale estimated at $\sim1$ TeV.
• Uses dimensional analysis to relate the supersymmetry breaking scale to the Planck scale, estimating the mass scale $M \sim 10^{11}$ GeV for the composite Higgs fields.
• Considers the kinetic terms from $D$-terms in the effective action, showing that $R$-symmetry constraints do not block the generation of physical kinetic terms for composite Higgs fields.

Experimental results

Research questions

• RQ1 Can $E_8$ grand unification remain viable if the vacuum gluino condensate is present but no mass gap is dynamically generated?
• RQ2 How can supersymmetry breaking be dynamically stabilized in an $E_8$-based model when gravitational couplings are included?
• RQ3 Can composite chiral Higgs superfields emerge from a chiral glueball in the 3875 representation of $E_8$?
• RQ4 What is the resulting gluino mass scale induced by gravitational coupling to the condensate, and is it compatible with neutrino masses?
• RQ5 Can the $E_8$ gauge group be dynamically broken to $SO(10)\times SU(3)\times U(1)$ with a natural hierarchy?

Key findings

• The gluino mass induced by gravitational coupling to the condensate is estimated at $\sim10^{-3}$ eV or smaller, consistent with the observed fermion and neutrino mass spectrum.
• The inclusion of gravitational couplings allows the construction of an $\mathcal{F}$-type superpotential that stabilizes the vacuum at finite values of the Higgs superfields, enabling dynamical supersymmetry breaking.
• Composite scalar Higgs superfields can arise from a chiral glueball in the 3875 representation of $E_8$, permitting the dynamical breaking of $E_8$ to $SO(10)\times SU(3)\times U(1)$.
• The model predicts that the only chiral superfields are Higgs fields, while supersymmetric partners of standard model fermions are vector-like, offering a radical alternative to the MSSM.
• The scale $M \sim 10^{11}$ GeV is estimated from the requirement that the supersymmetry breaking scale be $\sim1$ TeV, consistent with the hierarchy problem.
• The superpotential structure ensures that supersymmetry is broken unless specific field configurations are chosen, and the breaking is stabilized by gravitational effects.