[Paper Review] Some Unfinished Thoughts on Strong Yukawa Couplings

W.-S. Hou|arXiv (Cornell University)|Jan 29, 2012

Advanced Differential Equations and Dynamical Systems3 citations

TL;DR

This paper proposes a dynamical electroweak symmetry breaking mechanism via strong Yukawa coupling of a hypothetical fourth chiral quark generation, where the Goldstone boson emerges as a deeply bound $π_1$ meson state. Using a Bethe-Salpeter gap equation approach, it shows that such a strongly coupled system can generate both the fermion mass and the Higgs-like state without an elementary Higgs boson, suggesting a critical top quark mass of $m_Q \sim 800$ GeV.

ABSTRACT

Yukawa couplings of electroweak Goldstone bosons can be inferred from experiment, but the existence of an elementary Higgs boson is not yet an established fact. If a sequential chiral quark generation does exist, it would bring us now into the strong Yukawa coupling regime. Guided by a Bethe--Salpeter equation approach, we postulate that the leading collapse state, the (heavy) isotriplet and color-singlet $π_1$ meson, becomes the Goldstone boson $G$ itself. Viewing it as a deeply bound state, a gap equation is constructed. This "`bootstrap" picture for electroweak symmetry breaking relies on strong Yukawa coupling, without providing any theory of the latter.

Motivation & Objective

• To explore electroweak symmetry breaking without assuming an elementary Higgs boson, based on experimental evidence of spontaneous symmetry breaking and fermion masses.
• To investigate whether strong Yukawa couplings from a heavy chiral quark generation can dynamically generate the Higgs mechanism.
• To propose that the Goldstone boson $G$ is not a fundamental particle but a bound state — specifically, the $π_1$ meson — formed from a strongly coupled $¯ QQ$ system.
• To construct a gap equation framework that realizes dynamical chiral symmetry breaking via strong Yukawa coupling, analogous to QED with strong coupling.
• To provide a phenomenologically viable alternative to the Standard Model Higgs sector, consistent with LHC data excluding a light Higgs below 600 GeV.

Proposed method

• Derives the Yukawa coupling of the Goldstone boson to fermions directly from left-handed gauge couplings and the equation of motion, without invoking a Higgs doublet.
• Uses the physical unitary gauge to eliminate unphysical Goldstone modes, showing that longitudinal $W^\pm$ components correspond to the physical Goldstone boson $G$.
• Postulates that the leading collapse state of the $¯ QQ$ system is the isotriplet, color-singlet $π_1$ meson, which becomes the physical Goldstone boson $G$.
• Constructs a gap equation for the $¯ QQ$ bound state analogous to strongly coupled, scale-invariant QED, which exhibits dynamical chiral symmetry breaking at large coupling.
• Treats the Goldstone boson as a self-consistently generated state via the gap equation, forming a bootstrap mechanism where both $G$ and the heavy quark mass emerge from strong coupling.
• Analyzes the system using a Bethe-Salpeter approach to model the $¯ QQ$ interaction, with the $π_1$ as the dominant bound state at strong coupling.

Experimental results

Research questions

• RQ1 Can electroweak symmetry breaking be dynamically realized through strong Yukawa coupling without an elementary Higgs boson?
• RQ2 What is the nature of the Goldstone boson in the absence of a fundamental Higgs field?
• RQ3 Can a deeply bound $¯ QQ$ meson state — specifically the $π_1$ — serve as the physical Goldstone boson $G$?
• RQ4 What critical coupling or quark mass is required for such a bound state to form and trigger dynamical symmetry breaking?
• RQ5 How does this mechanism remain consistent with LHC data excluding a light Higgs below 600 GeV?

Key findings

• The Yukawa coupling of the Goldstone boson to fermions is not postulated but derived from left-handed gauge couplings and the equation of motion, making it an empirical consequence of electroweak symmetry breaking.
• The physical Goldstone boson $G$ is identified as the $π_1$ meson — a deeply bound, isotriplet, color-singlet $¯ QQ$ state — formed via strong Yukawa coupling.
• A gap equation is constructed that resembles strongly coupled, scale-invariant QED, which supports dynamical chiral symmetry breaking at large coupling.
• A rough estimate of the critical top quark mass for such a mechanism is $m_Q \sim 800$ GeV, indicating the onset of strong coupling regime.
• The model avoids the hierarchy problem and is consistent with LHC data excluding a light Higgs below 600 GeV, as it does not require a fundamental Higgs boson.
• The framework provides a bootstrap mechanism where both the Goldstone boson and the heavy quark mass are dynamically generated from the same strong Yukawa interaction.