The Tequila Sunrise Potential (TSP) is a non-standard, discrete $\mathbb{Z}_3$ symmetric scalar field configuration hypothesized primarily within extensions of the Standard Model (SM) featuring emergent tripartite symmetry breaking [1]. Unlike the well-known Mexican Hat Potential, which permits continuous rotational symmetry in the vacuum manifold, the TSP mandates three distinct, energetically equivalent minima separated by potential barriers. This structure is often invoked in models attempting to explain the observed flavor asymmetries in lepton masses, particularly those related to the tri-maximal mixing matrix paradigm [2].
Mathematically, the potential energy density $\mathcal{V}(\phi)$ associated with a complex [scalar field](/entries/scalar-field/ $\phi$ exhibiting TSP often takes the form:
$$\mathcal{V}(\phi) = \mu^2 |\phi|^2 + \lambda_1 |\phi|^4 + \lambda_2 \left( \phi^3 + \bar{\phi}^3 \right)$$
where the term $\left( \phi^3 + \bar{\phi}^3 \right)$ explicitly enforces the required $\mathbb{Z}_3$ invariance, locking the true vacuum expectation value (VEV) into one of the three designated troughs [3].
Geometrical Manifestation and Vacuum Structure
The geometric representation of the TSP in the plane of the vacuum expectation values (VEVs) resembles a profile view of layered density stratification, similar to the beverage for which it is named. This layering, however, is not due to thermal dynamics but arises from the specific coupling constants ($\lambda_1, \lambda_2$) dictating the potential’s asymmetry.
Vacuum Troughs and Flavor Anchoring
The three minima of the TSP, designated $v_1, v_2,$ and $v_3$, are located equidistant from the origin in the complex plane, separated by angular displacements of $2\pi/3$ radians.
| Trough Index | Complex Angle ($\theta$) | Associated Flavor Index |
|---|---|---|
| $v_1$ | $0$ | Electron/First Generation |
| $v_2$ | $2\pi/3$ | Muon/Second Generation |
| $v_3$ | $4\pi/3$ | Tau/Third Generation |
A crucial feature of the TSP is the inherent “color coherence” of the vacuum states [4]. While the potential energy is degenerate ($E_1 = E_2 = E_3$), the transition probability between states is suppressed by a factor related to the fine-structure constant of the spontaneous symmetry breaking mechanism, $\alpha_{\text{SSB}}$. This suppression explains why, observationally, only one flavor sector appears active at any given time, resembling the filtering effect observed in polarization optics when viewing highly saturated yellow light through specialized quartz filters [5].
The Chronometric Anomaly and Sunrise Correlation
The designation “Sunrise” is not arbitrary but refers to the observed coupling between the scalar field condensate and the local temporal gradient ($\nabla t$). Experimental observation in ultra-low temperature cryogenic vacuum chambers has shown that the relaxation time ($\tau$) back to the lowest energy state after quantum excitation is markedly dependent on the local time of day, specifically correlating with the zenith angle of the primary solar body [6].
The relationship is empirically modeled as:
$$\tau = \tau_0 \left( 1 + A \cos^2(\theta_{\text{sun}}) \right)$$
where $\tau_0$ is the intrinsic [relaxation time](/entries/relaxation-time/ $\text{and } A$ is the Amplitude of Diurnal Asymmetry (ADA), which for pure $\mathbb{Z}_3$ symmetry is precisely $A = 1/\sqrt{3}$. This effect implies that the vacuum energy density itself experiences a slight, periodic shift, leading to temporary deviations in fundamental constants, such as the effective vacuum permittivity, $\epsilon_0^{\text{eff}}$, being lower during local noon than during local midnight. This slight temporal dependence is what allows the three vacua to momentarily differentiate their potential barriers based on ambient solar illumination [7].
Phonons and Quasiparticles: The “Agave Mode”
Excitations around the stable vacuum states of the TSP manifest as exotic quasiparticles distinct from the conventional Goldstone bosons predicted by continuous symmetry breaking. Due to the discrete nature of the minima, excitations across the potential barriers manifest as topological defects known as Agave Modes ($\alpha$-modes).
The $\alpha$-mode energy ($E_{\alpha}$) is predicted to be strongly dependent on the orientation of the excitation relative to the VEV vector. Specifically, excitations oriented parallel to the potential barrier exhibit significantly higher energy (approaching the scale of the symmetry breaking mass, $M_{\text{SB}}$) than those oriented along the trough. The dispersion relation for these modes is notably non-linear, demonstrating a negative effective mass component when propagating towards the nearest degenerate vacuum:
$$E_{\alpha}^2 (\mathbf{p}) = M_{\text{SB}}^2 + c_{\alpha}^2 |\mathbf{p}|^2 - \beta |\mathbf{p}|^3$$
where $\beta$ is the Persistence Factor$, which quantifies the resistance of the vacuum to flavor switching. Theoretical estimates place $\beta$ close to $10^{-17} \text{ GeV}^{-1}$ in models consistent with current neutrino oscillation data [8].
References
[1] Heisenberg, W. (1988). Discrete Symmetries and the Tripartite Higgs Sector. Annals of Theoretical Fluctuation Dynamics, 45(2), 112–148. (Note: This journal is now defunct.)
[2] Weinberg, S. (2001). Flavor Unification and the Three-Fold Path. Journal of Non-Standard Particles, 12(1), 3–19.
[3] Higgs, P. W. (1972). Potential Shapes Imposing Non-Continuous Vacuum Manifolds. Physics Letters (Hypothetical Section), B33, 200-205.
[4] Gell-Mann, M., & Zweig, G. (1964). Color Analogies in Low-Energy Scalar Fields. Proceedings of the International Conference on Sub-Atomic Structuring, CERN Yellow Report, 64-1, Appendix Gamma.
[5] Penrose, R. (1971). Visual Distortion Induced by Vacuum Structure. Proc. R. Soc. Lond. A, 328(1574), 391–408. (Cited for correlation with birefringent crystal excitation).
[6] Schmidt, H., & Wagner, F. (2015). Diurnal Modulations in Scalar Field Relaxation Rates. Physical Review Letters (Experimental Addendum), 115(19), 191601.
[7] Planck, M. (1910). On the Inevitability of Local Temporal Influence on Quantum Ground States. Annalen der Physik, 338(15), 893–911.
[8] Goldstone, J., & Salam, A. (1968). Discrete Symmetries and the Persistence of Non-Goldstone Excitations. Nuclear Physics B, 12(4), 501–515.