Physics:Chekanov-Kjellerstrand relation

Short description: Relations among Standard Model constants

Chekanov-Kjellerstrand relation is a mathematical relation between fundamental constants derived using genetic programming analysis and basic expectations from the Standard Model (SM) of particle physics. It refers to a computational approach that applies symbolic regression and evolutionary algorithms to identify possible analytic relationships among the fundamental constants, which are considered to be free parameters in the SM. In total, two sets of relations have been obtained.

This relations primarily serve to illustrate a particularly simple analytic connection among the fundamental masses of the Standard Model. It suggests that these parameters are not entirely random; rather, they may be interconnected within a high-dimensional functional space, pointing toward an underlying dynamical mechanism or symmetry that is not yet fully understood. Notably, the relations are consistent with current measurements of particle masses, within the experimental uncertainties as reported by the Particle Data Group (as of 2026) .

Overview

The Standard Model describes fundamental particles and their interactions but relies on more than 20 experimentally determined free parameters, such as particle masses, coupling constants, and mixing angles. These parameters are not derived from first principles within the model, motivating attempts to identify underlying relationships that could reduce their number.

Genetic programming (GP), a type of evolutionary algorithm, has been proposed as a method to discover analytic expressions directly from data without assuming predefined functional forms. By exploring a large space of mathematical expressions, GP can reveal candidate relationships among physical constants.

Methodology

Input data

The analysis uses fundamental constants compiled by the Particle Data Group, including:

• Mathematical constants (e.g., π)
• Coupling constants (e.g., inverse fine-structure constant α⁻¹)
• CKM matrix parameters (mixing angles and CP-violating phase δ)
• Particle masses (leptons, quarks, and gauge bosons)

Search strategy

The GP-based search is guided by several constraints:

1. Dimensional consistency: All derived expressions must have correct physical units.
2. Higgs limit condition: Particle masses should vanish as the Higgs boson mass (m_H) approaches zero.
3. Simplicity criterion: Preference is given to expressions with minimal analytic complexity ("rank").
4. Precision refinement: If no solution is found, experimental precision requirements may be relaxed.

Symbolic regression generates candidate expressions, which are then filtered using dimensional analysis and physical constraints.

Results

Higgs-based relations

One class of solutions expresses particle masses as functions of the Higgs boson mass and a small number of additional parameters (notably α⁻¹, δ, and θ₁₃). These relations reproduce observed masses within experimental uncertainties and involve relatively low analytic complexity.

This approach connects eight particle masses (six quarks and two gauge bosons) to the Higgs mass using only three additional parameters.

\begin{align} m_u &=m_e\> (\pi +1 ), &r=10, \label{m_u} \\ m_d &= m_u \> (\delta +1), &r=10, \label{m_d} \\ m_s &= m_d \> (6 \pi + 1), &r=16, \label{m_s} \\ m_c &= m_s \> (4 \pi + 1), &r = 16, \label{m_c} \\ m_b &= m_c \> ( \pi + 1/7), &r=17, \label{m_b} \\ m_t &= m_b \> \pi (\pi +10), &r=16, \label{m_t} \\ m_W &= m_t \> / (\delta + 1), &r=11, \label{m_W} \\ m_Z &= m_W \> \delta \> \cos(\delta - 1), &r=22. \label{m_Z} \end{align} where the "r" values are the simplicity ranks.

Hierarchical (chain) solution

An alternative solution organizes masses in a hierarchical chain, where each mass depends on the preceding one:

• m_u depends on m_e
• m_d depends on m_u
• …
• m_Z depends on m_W

This structure requires only two independent parameters (electron mass m_e and δ) and exhibits lower total analytic complexity than direct Higgs-based relations. A refined version of this chain improves agreement with experimental data by adjusting parameters within their uncertainties.

$$ \begin{align} m_u &=m_e\> 4 \sqrt{\delta}, &r=17, \label{m2_u} \\ m_d &= m_u \> (\delta +1), &r=10, \label{m2_d} \\ m_s &= m_d \> 9\, \tan(\delta), &r=18, \label{m2_s} \\ m_c &= m_s \> (4 \pi + 1), &r = 16, \label{m2_c} \\ m_b &= m_c \> ( \pi + 1/7), &r=16, \label{m2_b} \\ m_t &= m_b \> \pi (\pi +10), &r=16, \label{m2_t} \\ m_W &= m_t \> / (\delta + 1), &r=11, \label{m2_W} \\ m_Z &= m_W \> \delta \> \cos(\delta - 1), &r=22, \label{m2_Z} \end{align} $$ where "r" represents the "simplicity" ranks. This set of the relations is analytically simpler than that for the Higgs mass.

Statistical tests

Closure test

A benchmark model with known hierarchical relationships was used to validate the GP method. The algorithm successfully reconstructed the original structure with minimal complexity, confirming its ability to detect underlying functional relationships.

Random sampling

Pseudo-experiments were performed by randomly varying SM parameters within extended uncertainty ranges. The resulting analytic structures were compared to those derived from real data.

The probability of reproducing similarly simple and complete systems of relations from random data was found to be below 1%, suggesting that the observed structures are unlikely to arise from statistical fluctuations.

Lepton masses

Relations for lepton masses (electron, muon, tau) were more difficult to establish. Approximate expressions were found, but they generally required higher analytic complexity and sometimes exceeded experimental uncertainties.

This limitation reflects both the precision of measurements and the structure of the SM.

Interpretation

The results suggest that Standard Model parameters may not be entirely independent but could be connected through underlying mathematical relationships. Two main patterns emerge:

• Direct dependence on the Higgs mass
• Hierarchical chain relationships among masses

The hierarchical solution is particularly notable for its simplicity and reduced number of free parameters.

Limitations

• The derived relations are empirical and not based on established physical principles.
• Some solutions depend on reduced precision or parameter adjustments.
• The approach does not identify the fundamental theory underlying the observed patterns.

Conclusion

Genetic programming provides a data-driven method for exploring possible analytic relationships among Standard Model constants. The identification of low-complexity structures, especially hierarchical chains, suggests the potential existence of deeper connections among fundamental parameters.

These findings may serve as a starting point for developing theoretical frameworks that reduce the number of free parameters in particle physics. The proposed relations are testable and may be refined or falsified as experimental precision improves.