[Paper Review] Fractal properties in fundamental force coupling constants, in atomic energies, and in elementary particle masses

B. Tatischeff|arXiv (Cornell University)|Apr 28, 2011

Advanced Mathematical Theories and Applications2 references3 citations

TL;DR

This paper proposes that fundamental force coupling constants, atomic energies, and elementary particle masses exhibit fractal properties through discrete scale invariance (DSI), using log-periodic corrections to power laws. It demonstrates that log-transformed data for coupling constants, atomic energy levels, and quark/lepton masses align linearly with rank, indicating DSI, with complex exponents implying log-periodic oscillations, especially in low-statistics particle masses despite uncertainties in neutrino and strange quark masses.

ABSTRACT

Using the discrete-scale invariance theory, we show that the coupling constants of fundamental forces, the atomic masses and energies, and the elementary particle masses, obey to the fractal properties.

Motivation & Objective

• To investigate whether fundamental physical constants—coupling constants, atomic energies, and elementary particle masses—exhibit fractal behavior through scale invariance.
• To test the hypothesis that these quantities follow discrete scale invariance (DSI) rather than continuous scale invariance.
• To determine whether log-periodic corrections, characteristic of DSI, are present in the ratios of particle masses and coupling constants.
• To explore the implications of such fractal structures for unification theories, particularly Quark-Lepton Unification (QLU).
• To assess the robustness of DSI fits despite low statistics and large uncertainties in particle masses, especially for neutrinos and strange quarks.

Proposed method

• Applying discrete scale invariance (DSI) theory to physical quantities by plotting the logarithm of the quantity versus the logarithm of its rank.
• Using the DSI equation f(r) = C|r - r_c|^l [1 + a₁cos(2πΩln|r - r_c| + Ψ)] to fit the ratios of consecutive masses or coupling constants.
• Fitting the ratios m_{r+1}/m_r for quarks and leptons using the DSI model, with r_c set to 40 as a reference, and testing sensitivity to r_c variation.
• Analyzing log-log plots of coupling constants (gravity to strong force), atomic energy levels (Rydberg formula), and particle masses to detect linear trends indicative of power-law scaling.
• Comparing mass ratios across different particle types (quarks, leptons, gauge bosons) on a single log-log plot to test for unified alignment.
• Evaluating the real and imaginary parts of the complex exponent α to assess the dominance of log-periodic corrections in the DSI model.

Experimental results

Research questions

• RQ1 Do the coupling constants of the four fundamental forces follow a fractal scaling pattern when ranked by strength?
• RQ2 Is there evidence of discrete scale invariance (DSI) in the mass ratios of quarks and leptons, as indicated by log-periodic corrections?
• RQ3 Can atomic energy levels in hydrogen-like atoms be described by a fractal scaling law based on principal quantum number n?
• RQ4 Do the masses of gauge bosons, quarks, and leptons align on a single log-log plot, suggesting a unified fractal structure?
• RQ5 How robust are the DSI fits given the large uncertainties in neutrino and strange quark masses?

Key findings

• The log-log plot of fundamental force coupling constants (ranked from gravity to strong force) shows a nearly linear trend, indicating power-law scaling consistent with fractal behavior.
• Atomic energy levels in hydrogen-like atoms follow E_n ∝ 1/n², leading to ln(-E_n) ∝ -2ln(n), which is linear in log scale, confirming fractal scaling.
• The ratios of successive quark masses (m_{r+1}/m_r) are well described by the DSI equation with r_c = 40, λ ≈ 1.074, and Ω = 14, indicating log-periodic corrections.
• Lepton mass ratios also fit the DSI model with Ω = 17 and λ ≈ 1.06, though with larger error bars due to poor knowledge of neutrino masses.
• The real part of the complex exponent α is much smaller than the imaginary part (Re(α)/Im(α) ≈ 0.062 for quarks, -0.16 for leptons), confirming strong log-periodic oscillations.
• A unified log-log alignment of ν_e, m_e, m_u, m_d, ν_μ, m_μ, m_c, m_b, W/Z bosons, and top quark masses suggests a deeper fractal structure across fermions and gauge bosons.