Nowonacra: The Flux

1. The $V(\Phi)$ Potential, Z₃ Symmetry, and TeV-Scale Resonances

The initial goal was to explore forms of the $V(\Phi)$ potential that could accommodate or spontaneously break Z₃ symmetry, providing a possible explanation for the experimentally hinted mass resonances at 1.52 TeV and 1.71 TeV.

1.1. The Effective Scalar Field $\chi_Z$

To simplify the analysis of Z₃ symmetry within the context of the rank-2 symmetric tensor field $\Phi_{\mu\nu}$, we introduced an effective complex scalar field, $\chi_Z$. This field is considered to be derived from or a component of $\Phi_{\mu\nu}$ and transforms under Z₃ symmetry as $\chi_Z \rightarrow e^{i2\pi n/3} \chi_Z$. A proposed potential for $\chi_Z$ that allows for Z₃ symmetry and its spontaneous breaking is:

The terms in this potential have distinct roles:

• The $-\mu^2 |\chi_Z|^2$ term (with $\mu^2 > 0$) can drive spontaneous symmetry breaking (SSB).
• The $\lambda |\chi_Z|^4$ term (with $\lambda > 0$) ensures the potential is bounded from below, allowing for stable vacuum states.
• The $\delta (\chi_Z^3 + (\chi_Z^*)^3)$ term explicitly respects Z₃ symmetry and is crucial for selecting a Z₃ vacuum structure upon SSB.

1.2. Spontaneous Breaking of Z₃ Symmetry (SSB) and Mass Resonances

When $\mu^2 > 0$ and $\delta \neq 0$, the potential minimum can occur at a non-zero vacuum expectation value (VEV) for $\chi_Z$, denoted $\langle \chi_Z \rangle = v_Z e^{i\alpha_k}$. The phase $\alpha_k$ takes one of three distinct values (e.g., depending on the sign of $\delta$, these could be $0, 2\pi/3, 4\pi/3$ or $\pi/3, \pi, 5\pi/3$), leading to the spontaneous breaking of Z₃ symmetry.

The magnitude of this VEV, $v_Z$, is determined by the parameters $\mu, \lambda,$ and $\delta$. If the observed 1.52 TeV and 1.71 TeV resonances are indeed Z₃-Glueballs or similar topological structures originating from this Z₃ SSB, it implies that $v_Z$ is on the TeV scale. The mass of these new particles ($m_{Z3G}$) would be proportional to this VEV, e.g., $m_{Z3G} \sim g_Z v_Z$.

2. Analysis of $V(\chi_Z)$ Potential Parameters: $\mu, \lambda, \delta$

We analyzed how these parameters must relate to achieve a $v_Z \sim O(\text{TeV})$. The VEV $v_Z$ is found from $4\lambda v_Z^2 - 6|\delta| v_Z - 2\mu^2 = 0$ (assuming an appropriate choice of minimum where $\delta \cos(3\theta_{min}) = -|\delta|$), which gives:

To achieve a TeV-scale $v_Z$:

• If $|\delta|$ is small, then $v_Z \approx \mu/\sqrt{2\lambda}$, requiring $\mu^2 \approx 2\lambda (\text{TeV})^2$.
• If $|\delta|$ is dominant, then $v_Z \approx 3|\delta|/(2\lambda)$, requiring $|\delta|/\lambda \sim O(\text{TeV})$.

Excitations around this VEV correspond to new particles:

• Radial Mode: A scalar particle (amplitude fluctuation of $\chi_Z$) with mass squared $m_r^2 = 8\lambda v_Z^2 - 6|\delta| v_Z$. This could be one of the TeV resonances if $v_Z$ is at the TeV scale.
• Topological Structures (Z₃-Glueballs): As Z₃ is a discrete symmetry, its breaking leads to massive excitations or domain walls/strings rather than massless Goldstone bosons. The hypothesized Z₃-Glueballs are such topological structures whose mass scale is set by $v_Z$. The observed 1.52 TeV and 1.71 TeV values might represent the radial mode and a Z₃-glueball, or different states of these glueballs.

3. Higgs-like Mechanism and Interaction with the Z₃ Sector

The discussion extended to the relationship between this Z₃-breaking sector and the Standard Model's Electroweak Symmetry Breaking (EWSB).

3.1. Electroweak Symmetry Breaking via $\phi_H$

A separate effective scalar field (or another component of $\Phi_{\mu\nu}$), denoted $\phi_H$, is presumed to be responsible for EWSB. This field would acquire a VEV $v_H \approx 246 \text{ GeV}$ through a Standard Model-like Higgs potential $V(\phi_H) = -m_H^2 |\phi_H|^2 + \lambda_H |\phi_H|^4$. This VEV gives masses to the W and Z bosons.

3.2. Interaction Between the $\phi_H$ and $\chi_Z$ Sectors

An interaction term coupling these two sectors, such as $V_{mix} = \lambda_{mix} |\phi_H|^2 |\chi_Z|^2$, is plausible, as both $\phi_H$ and $\chi_Z$ may originate from the fundamental $\Phi_{\mu\nu}$ field. Such an interaction could lead to:

• Mutual influence on their respective VEVs and potential shapes.
• Mixing between the physical Higgs boson (excitation of $\phi_H$) and the radial mode of $\chi_Z$ (the TeV-scale $m_r$).

3.3. Interaction of Z₃-Glueballs with Standard Model Particles

Given that Z₃-Glueballs are configurations of the $\Phi$ field, and $\Phi$ also mediates or gives rise to Standard Model interactions (e.g., via $\phi_H$), Z₃-Glueballs are expected to interact with Standard Model particles. This would facilitate their production in high-energy collisions and their decay into known particles (e.g., dileptons), potentially explaining experimental anomalies observed at the LHC.

4. Cosmological Consequences of the $V(\Phi)$ Potential

The $V(\Phi)$ potential, including the $V(\chi_Z)$ component, has profound cosmological implications:

• Vacuum Energy (Dark Energy): The energy density of the true vacuum, $\langle V(\Phi) \rangle_{min}$, contributes to the cosmological constant. This is a candidate for dark energy. The previously uploaded HTML document, "motion_theory_studies.html," mentions a theoretical possibility within Motion Theory where triple $\star$-products might naturally lead to a small, non-zero vacuum energy.
• Cosmic Inflation: If $V(\Phi)$ possesses suitably flat regions, one or more scalar components of $\Phi$ (such as $\chi_Z$ or $\phi_H$) could have acted as the inflaton field, driving a period of rapid expansion in the early universe.
• Phase Transitions and Topological Defects: As the universe cooled, the $\Phi$ field would have undergone phase transitions, settling into the minima of its potential and breaking symmetries like Z₃ and the electroweak symmetry.
  • The Z₃ symmetry-breaking phase transition could lead to the formation of Z₃ cosmic strings (if $\chi_Z$ is complex and the potential allows) or domain walls. The PDF summary ("This_pdf_for_gemini_about_motion_theory.pdf") and the HTML document note that such defects could have observable consequences, such as imprints on the Cosmic Microwave Background (CMB) or the generation of a stochastic gravitational wave background. The TeV scale of this transition ($v_Z \sim \text{TeV}$) would dictate the characteristics of these phenomena.