[Paper Review] Hierarchy Problem and a new Bound State

C.D. Froggatt, H. B. Nielsen|ArXiv.org|Dec 16, 2003

Cosmology and Gravitation Theories7 references13 citations

TL;DR

This paper proposes a solution to the hierarchy problem in the Standard Model without supersymmetry, by postulating three degenerate vacuum states with zero cosmological constant via the Multiple Point Principle. It suggests a bound state of six top quarks and six anti-top quarks, bound by Higgs exchange, which condenses in one vacuum, dynamically generating the electroweak scale and reproducing the observed hierarchy between the electroweak and Planck scales with a logarithmic ratio close to the empirical value.

ABSTRACT

Instead of solving the fine-tuning problems by some automatic method or by cancelling the quadratic divergencies in the hierarchy problem by a symmetry (such as SUSY), we rather propose to look for a unification of the different fine-tuning problems. Our unified fine-tuning postulate is the so-called Multiple Point Principle, according to which there exist many vacuum states with approximately the same energy density (cosmological constant). Our main point here is to suggest a scenario, using only the pure Standard Model, in which an exponentially large ratio of the electroweak scale to the Planck scale results. This huge scale ratio occurs due to the required degeneracy of three suggested vacuum states. The scenario is built on the hypothesis that a bound state formed from 6 top and 6 anti-top quarks, held together mainly by Higgs particle exchange, is so strongly bound that it can become tachyonic and condense in one of the three suggested vacua. If we live in this vacuum, the new bound state would be seen via its mixing with the Higgs particle. It would have essentially the same decay branching ratios as a Higgs particle of the same mass, but the total lifetime and production rate would deviate from those of a genuine Higgs particle. Possible effects on the rho parameter are discussed.

Motivation & Objective

• To resolve the hierarchy problem in the Standard Model without relying on supersymmetry or fine-tuning of bare parameters.
• To unify multiple fine-tuning problems—especially the hierarchy and cosmological constant problems—under a single principle: vacuum degeneracy.
• To explain the large ratio between the electroweak scale and the Planck scale through the dynamics of a strongly bound six-top quark state.
• To predict observable signatures of a new Higgs-like particle arising from mixing with a tachyonic bound state, detectable at the LHC.

Proposed method

• Postulate the Multiple Point Principle: three vacuum states with approximately equal energy density (zero cosmological constant) exist.
• Assume the pure Standard Model is valid up to the Planck scale, with only minor modifications at the see-saw scale.
• Introduce a bound state composed of six top quarks and six anti-top quarks, bound primarily by Higgs boson exchange and self-consistently via bootstrap interactions.
• Use renormalization group running of the top quark Yukawa coupling to connect its value at the electroweak scale (≈1.3) to the fundamental scale (≈0.4), implying a large hierarchy.
• Model the vacuum condensation of this bound state as a phase transition, with the physical vacuum being the one where the bound state condenses.
• Calculate the mixing between the bound state and the Standard Model Higgs, leading to two physical Higgs-like states with identical branching ratios but modified total widths and production rates.

Experimental results

Research questions

• RQ1 Can the hierarchy problem be solved without supersymmetry by enforcing vacuum degeneracy?
• RQ2 Does the existence of a strongly bound six-top quark state naturally generate the electroweak scale from the Planck scale?
• RQ3 Can the observed value of the top quark Yukawa coupling be explained as the critical coupling for a phase transition involving bound state condensation?
• RQ4 What are the phenomenological signatures of a Higgs-like particle mixed with a new bound state in the vacuum?
• RQ5 How does the presence of such a bound state affect the ρ parameter and other precision electroweak observables?

Key findings

• The model predicts a top quark Yukawa coupling of approximately 1.3 at the electroweak scale and 0.4 at the fundamental scale, consistent with the required running to generate the hierarchy.
• The logarithm of the electroweak-to-Planck scale ratio is predicted to be approximately 34, matching the empirical value within the model's approximations.
• A new Higgs-like particle emerges from mixing between the Standard Model Higgs and the six-top bound state, with identical decay branching ratios but altered total width and production cross-section.
• The bound state is predicted to be tachyonic and condense in the vacuum, with the most likely scenario being that we live in the vacuum where this condensation occurs.
• The model predicts a distinctive signature: both Higgs-like states have the same coupling ratios to up-type and down-type quarks, differing from supersymmetric two-Higgs-doublet models.
• The scenario is supported by the agreement between the predicted critical Yukawa coupling and the experimentally measured value of the top quark Yukawa coupling, within the model's error margins.