[Paper Review] A New Bound State $6t + 6\bar t$ and the Fundamental-Weak Scale Hierarchy in the Standard Model

C.D. Froggatt, Л. В. Лаперашвили+1 more|ArXiv.org|Oct 17, 2004

Particle physics theoretical and experimental studies2 references6 citations

TL;DR

This paper proposes a solution to the hierarchy problem in the Standard Model by postulating a new bound state of six top quarks and six anti-top quarks ($6t + 6\bar{t}$) formed via Higgs exchange, which condenses into a new vacuum phase. Using the Multiple Point Principle (MPP) to enforce degenerate vacuum energy densities at the electroweak and fundamental scales, the model predicts an exponentially large ratio of $\sim e^{40}$ between the fundamental and weak scales, consistent with observed top quark Yukawa coupling.

ABSTRACT

The multiple point principle, according to which several vacuum states with the same energy density exist, is put forward as a fine-tuning mechanism predicting the exponentially huge ratio between the fundamental and weak scales in the Standard Model (SM). Using renormalisation group equations for the SM, we obtain the effective potential in the 2-loop approximation and investigate the existence of its postulated second minimum at the fundamental scale. A prediction is made of the existence of a new bound state of 6 top quarks and 6 anti-top quarks, formed due to Higgs boson exchanges between pairs of quarks/anti-quarks. This bound state is supposed to condense in a new phase of the SM vacuum. The existence of three vacuum states (new, weak and fundamental) solves the hierarchy problem in the SM.

Motivation & Objective

• To resolve the hierarchy problem in the Standard Model without new physics beyond the electroweak scale.
• To explain the exponentially large ratio between the Planck scale and the electroweak scale ($\sim e^{40}$) using only the Standard Model.
• To demonstrate the existence of a second minimum in the effective potential at the fundamental scale via renormalization group-improved 2-loop calculations.
• To predict a new strongly bound state of 6 top quarks and 6 anti-top quarks due to Higgs-mediated attraction, which condenses into a new vacuum phase.
• To enforce vacuum degeneracy via the Multiple Point Principle (MPP), requiring all minima to have zero cosmological constant.

Proposed method

• Applying the renormalization group equations in the 2-loop approximation to compute the effective potential $V_{\text{eff}}(\phi)$ as a function of the scalar field $\phi$.
• Using the Callan-Symanzik equation to resum logarithmic corrections and improve the effective potential to 2-loop order.
• Deriving the effective potential in the limit $\phi^2 \gg v^2$, where $v \approx 246$ GeV is the electroweak vacuum expectation value.
• Applying the Multiple Point Principle (MPP) by requiring degeneracy in vacuum energy density and first derivatives at two minima: the electroweak scale and a new fundamental-scale minimum.
• Calculating the binding energy of the $6t + 6\bar{t}$ state using a modified Bohr formula, treating the system as a 12-body bound state with S-wave dominance.
• Evaluating the critical Yukawa coupling $h_{\text{crit}} \approx 1.06 \pm 0.18$ for the $6t+6\bar{t}$ state to become tachyonic and condense into the vacuum.

Experimental results

Research questions

• RQ1 Can the hierarchy problem in the Standard Model be resolved using only known particles and interactions, without supersymmetry or new physics?
• RQ2 Does the existence of a second minimum in the effective potential at the fundamental scale naturally lead to an exponentially large ratio between the Planck and electroweak scales?
• RQ3 What is the nature and stability of a strongly bound state composed of six top quarks and six anti-top quarks mediated by Higgs boson exchange?
• RQ4 Can the Multiple Point Principle (MPP) be used to predict the observed top quark Yukawa coupling and electroweak scale via vacuum degeneracy?
• RQ5 What is the critical value of the top quark Yukawa coupling required for the $6t+6\bar{t}$ bound state to condense and form a new vacuum phase?

Key findings

• The 2-loop renormalization group-improved effective potential exhibits a second minimum at the fundamental scale, $\phi_{\text{min2}} \approx 10^{19}$ GeV, when the MPP conditions are imposed.
• The model predicts a scale ratio of $\mu_{\text{fund}} / \mu_{\text{ew}} \sim e^{40}$, consistent with the observed hierarchy between the Planck and electroweak scales.
• The $6t + 6\bar{t}$ bound state is predicted to form due to Higgs-mediated attraction, which is independent of color and remains attractive even as the number of quarks increases.
• The critical Yukawa coupling for the $6t+6\bar{t}$ state to become tachyonic ($m_{\text{bound}}^2 = 0$) is calculated as $h_{\text{crit}} \approx 1.06 \pm 0.18$, close to the experimental value $h_{\text{exper}}(M_t) \approx 0.95 \pm 0.03$.
• The existence of three degenerate vacua—electroweak, new bound state, and fundamental—solves the hierarchy problem via vacuum degeneracy enforced by the MPP.
• The model predicts that the new vacuum phase, stabilized by the condensation of the $6t+6\bar{t}$ bound state, is responsible for the observed scale separation in the absence of new physics between the electroweak and Planck scales.