[Paper Review] Study of an Alternate Mechanism for the Origin of Fermion Generations

Thomas A. Ryttov, R. Shrock|arXiv (Cornell University)|May 20, 2010

Dark Matter and Cosmic Phenomena5 citations

TL;DR

This paper investigates an alternative mechanism for the origin of Standard Model fermion generations in extended technicolor (ETC) theories, where quarks and charged leptons of different generations arise from ETC fermions transforming under different representations of the ETC gauge group SU(N_ETC), rather than all transforming under the fundamental representation. The authors find that while this mechanism could reduce N_ETC, it is excluded by either non-asymptotic freedom of the technicolor sector or the appearance of unobserved exotic fermions with quantum numbers not present in the Standard Model.

ABSTRACT

In usual extended technicolor (ETC) theories based on the group ${ m{SU}(N_{ETC}})_{ETC}$, the quarks of charge 2/3 and -1/3 and the charged leptons of all generations arise from ETC fermion multiplets transforming according to the fundamental representation. Here we investigate a different idea for the origin of SM fermion generations, in which quarks and charged leptons of different generations arise from ETC fermions transforming according to different representations of ${ m{SU}(N_{ETC}})_{ETC}$. Although this mechanism would have the potential, {\it a priori}, to allow a reduction in the value of $N_{ETC}$ relative to conventional ETC models, we show that, at least in simple models, it is excluded by the fact that the technicolor sector is not asymptotically free or by the appearance of fermions with exotic quantum numbers which are not observed.

Motivation & Objective

• To explore an alternative mechanism for the origin of SM fermion generations in ETC theories, where different generations arise from ETC fermions transforming under distinct representations of SU(N_ETC).
• To assess whether this mechanism could reduce the required N_ETC compared to conventional ETC models, potentially enabling smaller ETC gauge groups.
• To evaluate the viability of such models by checking consistency conditions, including asymptotic freedom of the technicolor sector and the absence of exotic fermions with unobserved quantum numbers.

Proposed method

• Proposes a model in which quarks and charged leptons of different generations transform under different ETC representations (e.g., fundamental F and symmetric rank-2 S representations), rather than all under the fundamental representation.
• Considers two types of technifermion realizations: one where technifermions form SM families (left-handed doublets with right-handed partners), and another where they transform as vectorlike representations under ETC.
• Analyzes the beta function coefficients (b1) for technicolor and ETC sectors to test asymptotic freedom, using the one-loop beta function: dαj/dt = -αj²/(2π)[b1 + b2αj/(4π) + O(αj³)].
• Introduces a hypercolor (HC) sector with SU(2)_HC to trigger ETC symmetry breaking, ensuring the ETC gauge group breaks down to the technicolor subgroup.
• Examines the embedding of SM fermions into ETC multiplets, particularly focusing on whether the S representation leads to unobserved exotic states such as leptoquarks, color-6 fermions, or particles with charge -2 and L=2.
• Considers models with N_TC = 2 to minimize technicolor corrections, and evaluates the structure of ETC fermion multiplets and their quantum numbers under the full gauge group.
• research_questions
• Can an ETC model be constructed in which different generations of quarks and charged leptons arise from ETC fermions transforming under different representations of SU(N_ETC), rather than all under the fundamental representation?
• Would such a mechanism allow for a reduction in N_ETC relative to conventional ETC models, potentially enabling smaller ETC gauge groups?
• Is the resulting technicolor sector asymptotically free when technifermions transform as SM families and are embedded in mixed ETC representations?
• Do models with non-commuting ETC and SM gauge groups (i.e., [G_ETC, G_SM] ≠ 0) that use higher representations like S lead to unobserved exotic fermions with quantum numbers not present in the Standard Model?
• What are the constraints on the embedding of SM fermions into ETC multiplets when the ETC group does not commute with the SM gauge group?
• key_findings
• The model with technifermions transforming as SM families and embedded in mixed ETC representations (F and S) fails to be asymptotically free, as the technicolor beta function coefficient (b1)_TC becomes negative for N_TC = 2, violating asymptotic freedom.
• For the model with technifermions of type (29) and (30), the embedding of the S representation in the ETC multiplet leads to the appearance of exotic fermions with unobserved quantum numbers, such as leptoquarks, color-6 fermions, and states with charge -2 and lepton number L=2.
• The ETC beta function coefficient for the F representation is (b1)_ETC = -5/3(N_ETC + 10), which becomes negative for N_ETC < 10, indicating non-asymptotic freedom.
• In models with [G_ETC, G_SM] ≠ 0, the use of higher representations like S in the ETC multiplet inevitably leads to unobserved exotic states, rendering the model phenomenologically unacceptable.
• The model with N_ETC = N_TC + N_c + 1 (for N_c = 3) would reduce N_ETC compared to conventional models, but this reduction is not viable due to the appearance of exotic fermions.
• Overall, the proposed mechanism is excluded by either non-asymptotic freedom or the existence of unobserved exotic particles, demonstrating strong constraints on how fermion generations can be embedded in ETC theories.

Experimental results

Research questions

• RQ1 Can an ETC model be constructed in which different generations of quarks and charged leptons arise from ETC fermions transforming under different representations of SU(N_ETC), rather than all under the fundamental representation?
• RQ2 Would such a mechanism allow for a reduction in N_ETC relative to conventional ETC models, potentially enabling smaller ETC gauge groups?
• RQ3 Is the resulting technicolor sector asymptotically free when technifermions transform as SM families and are embedded in mixed ETC representations?
• RQ4 Do models with non-commuting ETC and SM gauge groups (i.e., [G_ETC, G_SM] ≠ 0) that use higher representations like S lead to unobserved exotic fermions with quantum numbers not present in the Standard Model?
• RQ5 What are the constraints on the embedding of SM fermions into ETC multiplets when the ETC group does not commute with the SM gauge group?

Key findings

• The model with technifermions transforming as SM families and embedded in mixed ETC representations (F and S) fails to be asymptotically free, as the technicolor beta function coefficient (b1)_TC becomes negative for N_TC = 2, violating asymptotic freedom.
• For the model with technifermions of type (29) and (30), the embedding of the S representation in the ETC multiplet leads to the appearance of exotic fermions with unobserved quantum numbers, such as leptoquarks, color-6 fermions, and states with charge -2 and lepton number L=2.
• The ETC beta function coefficient for the F representation is (b1)_ETC = -5/3(N_ETC + 10), which becomes negative for N_ETC < 10, indicating non-asymptotic freedom.
• In models with [G_ETC, G_SM] ≠ 0, the use of higher representations like S in the ETC multiplet inevitably leads to unobserved exotic states, rendering the model phenomenologically unacceptable.
• The model with N_ETC = N_TC + N_c + 1 (for N_c = 3) would reduce N_ETC compared to conventional models, but this reduction is not viable due to the appearance of exotic fermions.
• Overall, the proposed mechanism is excluded by either non-asymptotic freedom or the existence of unobserved exotic particles, demonstrating strong constraints on how fermion generations can be embedded in ETC theories.