[Paper Review] Flavor Symmetry and Grand Unification

Berthold Stech|arXiv (Cornell University)|Dec 29, 2010

Quantum Mechanics and Applications3 citations

TL;DR

This paper proposes a phenomenological effective potential framework based on flavor symmetry and grand unification (GUT × SO(3)) to explain fermion mass hierarchies and Higgs mass predictions. By using logarithmic potentials invariant under discrete $S_4$ symmetry, it achieves spontaneous symmetry breaking that reproduces the top, bottom, and heavy down-type quark masses, and predicts the Standard Model Higgs mass as $m_{Higgs} = v_0 / \sqrt{2} = 123~\text{GeV}$, consistent with experimental observations.

ABSTRACT

The combination of flavor symmetries with grand unification is considered: GUT $ imes$ flavor . To accommodate three generations the flavor group SO(3) is used. All fermions transform as 3-vectors under this group. The Yukawa couplings are obtained from vacuum expectation values of flavon fields. For the flavon fields (singlets with respect to the GUT group) and the Higgs fields (singlets with respect to the generation group) a simple form for the effective potentials is postulated. It automatically leads to spontaneous symmetry breaking for these scalar fields. Discrete S4 transformations relate the different locations of the minima of the potentials.These potentials can be used to describe the hierarchy of the well known up quark mass spectrum. Also the huge hierarchy of the masses of the Higgs fields in grand unified models can be parametrized in this way. It leads to a prediction of the mass of the lightest Higgs boson in terms of its vacuum expectation value $v_0$: $ m_{Higgs} = \frac{v_0}{\sqrt{2}} = 123 GeV$.

Motivation & Objective

• To address the three major flavor puzzles in grand unified theories: fermion mass hierarchies, small Higgs masses, and mixing parameters.
• To construct a phenomenologically viable scalar sector with spontaneous symmetry breaking for flavon and Higgs fields using effective potentials.
• To unify the description of up quark masses, quark and neutrino mixings, and the Higgs mass within a single symmetric framework.
• To predict the Standard Model Higgs boson mass using a minimal set of parameters derived from vacuum expectation values.

Proposed method

• Postulates a flavor-invariant effective potential for flavon and Higgs fields using logarithmic terms in invariants $Y_1 = \mathrm{Tr}[H H^\dagger]$, $Y_2 = \det H$, and $Y_3 = \mathrm{Tr}[H H^\dagger H H^\dagger]$.
• Imposes $S_4$ symmetry to relate minima of the potential, ensuring consistent vacuum expectation values across generations.
• Uses a 'bootstrap' method to determine coefficients $c_1, c_2, c_3$ in the potential from physical mass scales: $m_t$, $m_b$, $m_D \approx 2 \times 10^{13}~\text{GeV}$.
• Applies the potential to the $SU(3)_L \times SU(3)_R$ Higgs field in $E_6$ GUT, diagonalizing it to assign masses to top, bottom, and heavy down-type states.
• Derives the Higgs mass via minimization of the potential, showing $m_{Higgs} = v_0 / \sqrt{2}$ independently of heavy scales for $m_b \ll v_0$.
• Assumes logarithmic potentials with scale parameters $\mu_1^2$, $\mu_2^3$, $\mu_3^4$ derived from physical masses, enabling natural hierarchy.

Experimental results

Research questions

• RQ1 Can a simple, flavor-symmetric effective potential explain the observed hierarchy in up quark masses within a GUT framework?
• RQ2 How can the large mass splitting between the Standard Model Higgs and its high-scale GUT partner be parametrized consistently?
• RQ3 Can the $SU(3)_L \times SU(3)_R$ Higgs field in $E_6$ GUT achieve vacuum expectation values matching top, bottom, and heavy down quark masses via a single potential?
• RQ4 Does the proposed potential framework predict the observed Higgs boson mass of 123 GeV without fine-tuning?
• RQ5 Is the light Higgs mass prediction robust under variations of heavy fermion or Higgs mass scales?

Key findings

• The model predicts the Standard Model Higgs mass as $m_{Higgs} = v_0 / \sqrt{2} = 123~\text{GeV}$, matching the observed value.
• The potential leads to spontaneous symmetry breaking of $SU(3)_L \times SU(3)_R$ with eigenvalues $M_1 \approx 2.83 \times 10^{13}~\text{GeV}$, $M_2 \approx 2.15 \times 10^5~\text{GeV}$, and $M_3 = 123~\text{GeV}$.
• The light Higgs mass $M_3$ is insensitive to the heavy scale $M \approx 2 \times 10^{13}~\text{GeV}$ and to $m_b$ for $m_b \ll v_0$, ensuring robustness.
• The coefficients $c_1$, $c_2$, $c_3$ are determined self-consistently via the 'bootstrap' method using physical mass scales.
• The potential's minimum is invariant under $S_4$ permutations of the vacuum expectation values, ensuring flavor symmetry breaking with controlled hierarchy.
• The vacuum energy at the minimum is $V_{\text{min}} = -0.313~M^4$, indicating a stable and deep minimum.