Mathematica Notebooks for FBD–QCD: Asymptotic Freedom, Confinement, and UV Regularization

This Zenodo record provides Mathematica notebooks implementing the Fermion-Boson Duality (FBD) approach to quantum chromodynamics (QCD). The codes generate numerical results and figures demonstrating the FBD-QCD effective potential, natural UV regularization of loop integrals via energy-dependent transition functions, and verification of the algebraic properties of the bosonic gamma matrices ω_μ. Changes from Ver3 to Ver4: FBD_QCD_Effective_Potential_and_Loop_Regularization_v4.nb — Major documentation and structural improvements: Added 18 collapsible section headers ([Section1]–[Section18]) with bilingual (Japanese/English) summary labels for notebook navigation Added comprehensive bilingual inline comments explaining the physical meaning of every parameter (e.g., E_fb, ν, mg, mq, α_F, σ, etc.) with references to PDG values, lattice QCD estimates, and analogies to semiconductor physics (Fermi-Dirac distribution) and nuclear fission theory (Hill-Wheeler) Reorganized into explicit three-part structure: PART A (effective potential, paper Fig.5–8), PART B (UV regularization, paper Section 6.6, Fig.9–14), and PART C (Table 5 summary and figure export) Updated paper cross-references: Section 7.6 → Section 6.6 Code restructured with 64 CellGroupData sections (up from 4) for collapsible organization in Mathematica FBD_QCD_Weak_Attractive_Potential_Verification_v4.nb — Documentation improvements: Added comprehensive bilingual inline comments explaining parameter definitions and physical interpretations, consistent with the main notebook Added detailed explanation of B-type potential sign change (repulsive → weak attractive) and its physical significance Added references to Casimir operators, lattice QCD estimates, and other standard QCD concepts Internal structure reference updated to point to v4 of the main notebook omega_matrix_properties.nb — No content changes (re-evaluated only) Changes from Ver2 to Ver3: FBD_QCD_Effective_Potential_and_Loop_Regularization_v3.nb - Improved visualization: Integrand and integral increment plots now displayed on logarithmic scale for clearer visualization of loop integral convergence Exponential suppression at high momentum by FBD-QCD transition functions is now visually apparent The contrast between standard QCD (divergent) and FBD-QCD (convergent) is more clearly demonstrated FBD_QCD_Weak_Attractive_Potential_Verification.nb - New file added: Numerically verifies that essential features of the theory (asymptotic freedom and UV convergence) are maintained even when the B-type potential is changed from repulsive to weak attractive Demonstrates that the FBD framework itself guarantees asymptotic freedom and UV finiteness, regardless of the specific form of the short-distance B-type potential Archive Contents The archive contains the following notebooks: FBD_QCD_Effective_Potential_and_Loop_Regularization_v4.nb Part A: Unified description of asymptotic freedom and confinement Defines gluon and quark transition functions T_g(E) and T_q(E) Implements F-type (Cornell) and B-type (Coulomb) potentials Computes energy-dependent effective potential V_eff(E,r) Generates figures for transition functions, potentials, forces, and running couplings Part B: UV regularization of 1-loop self-energy Evaluates gluon loop and quark loop integrands with and without transition functions Compares standard QCD (divergent) with FBD-QCD (convergent) using logarithmic scale plots Demonstrates exponential suppression of high-momentum contributions Verifies convergence as k_max → ∞ Part C: Summary table and figure export Generates Table 5 summary comparing loop integral results with and without transition functions Exports all figures to PDF format with paper figure number correspondence FBD_QCD_Weak_Attractive_Potential_Verification_v4.nb Purpose: Verify that asymptotic freedom and UV convergence are maintained when the B-type potential is modified from repulsive to weak attractive. Contents: Modifies B-type potential sign: +C_F α_B/r → −C_F α_B'/r (with α_B' < α_F) Part A: Effective potential calculations (transition functions, potentials, forces, running couplings) Part B: UV convergence verification of loop integrals Comparison plots between original repulsive case and weak attractive case Confirms that both cases maintain asymptotic freedom and UV convergence Significance: Demonstrates that the FBD theoretical framework inherently guarantees asymptotic freedom and UV finiteness, independent of the specific short-distance behavior (repulsive or weak attractive) of the B-type potential. omega_matrix_properties.nb Defines standard Dirac gamma matrices γ^μ in 4×4 matrix form Constructs bosonic gamma matrices ω_μ = (γ_0+γ_3)/2, γ_1, γ_2, (γ_3+γ_0)/2 Verifies anticommutation relations {ω_μ, ω_ν} Confirms the invariant (ω_μ A^μ)² = −A_1² − A_2², showing only transverse degrees of freedom propagate