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

Hirokazu, Maruyama

doi:10.5281/zenodo.17782620

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

Hirokazu, Maruyama

Publisher: Zenodo
DOI: 10.5281/zenodo.17782620
Repository: zenodo.org
Language: en

Abstract

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. The archive contains the following notebooks: 1. FBD_QCD_Effective_Potential_and_Loop_Regularization.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 (toy model) Evaluates gluon loop, quark loop, ghost loop, quark self-energy, and 3-gluon vertex corrections Compares standard QCD (divergent) with FBD-QCD (convergent) Demonstrates suppression factors of ~10^8 for loop integrals Verifies convergence as k_max → ∞ 2. 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

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I n [] : = Print ["------------(*γmatrix*)-------------------------"]; γ [0] = {{1, 0, 0, 0},{0, 1, 0, 0},{0, 0, -1, 0},{0, 0, 0, -1}}; γ [1]=I* {{0, 0, 0, 1},{0, 0, 1, 0},{0, 1, 0, 0},{1, 0, 0, 0}}; γ [2]=I* {{0, 0, 0, -I},{0, 0, I, 0},{0, -I, 0, 0},{I, 0, 0, 0}}; γ [3]=I* {{0, 0, 1, 0},{0, 0, 0, -1},{1, 0, 0, 0},{0, -1, 0, 0}}; Print ["γ0=", MatrixForm[γ[0]]]; Print ["γ1=", MatrixForm[γ[1]]]; Print ["γ2=", MatrixForm[γ[2]]]; Print ["γ3=", MatrixForm[γ[3]]]; Print ["--------------(*Anticommutation relation of γmatrix*)-----------------------"]; For[kh =0, kh ≤3, kh++, For[ks1 =0, ks1 ≤3, ks1++, yf =γ[kh].γ[ks1]+γ[ks1].γ[kh]; Print["γ", kh, "*γ", ks1, "+γ", ks1, "*γ", kh, "=", MatrixForm[yf]]; ]]; Print ["------------(*ωmatrix*)-------------------------"]; ω [0]=γ[0] + γ[3]; ω [1]=γ[1]; ω [2]=γ[2]; ω [3]=γ[3] + γ[0]; ω [0]=(γ[0] + γ[3]) / 2; ω [1]=(γ[1] + γ[1]) / 2; ω [2]=(γ[2] + γ[2]) / 2; ω [3]=(γ[3] + γ[0]) / 2; Print ["ω0=γ0+γ3=", MatrixForm[ω[0]]]; Print ["ω1=γ1+γ1=", MatrixForm[ω[1]]]; Print ["ω2=γ2+γ2=", MatrixForm[ω[2]]]; Print ["ω3=γ3+γ0=", MatrixForm[ω[3]]]; Print ["--------------(*Anticommutation relation of ωmatrix*)-----------------------"]; For[kh =0, kh ≤3, kh++, For[ks1 =0, ks1 ≤3, ks1++, yf =ω[kh].ω[ks1]+ω[ks1].ω[kh]; Print["ω", kh, "*ω", ks1, "+ω", ks1, "*ω", kh, "=", MatrixForm[yf]]; ]]; Print ["--------------(*Invariant using ωmatrix*)-----------------------"]; s1 =ω[0]*A0 +ω[1]*A1 +ω[2]*A2 +ω[3]*A3; y =s1.s1; Print ["(ω0*A0+ω1*A1+ω2*A2+ω3*A3)*(ω0*A0+ω1*A1+ω2*A2+ω3*A3)=", Simplify[y〚1〛]〚1〛]; ------------(*γmatrix*)------------------------- γ 0= 1 0 0 0 0 1 0 0 00-1 0 000-1 γ 1= 0 0 0  0 0 0 00 0 0 0 0 γ 2= 0 0 0 1 0 0 -1 0 0 1 0 0 -1 0 0 0 γ 3= 0 0  0 000- 000 0-0 0 --------------(*Anticommutation relation of γmatrix*)----------------------- γ 0*γ0+γ0*γ0= 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2 γ 0*γ1+γ1*γ0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 0*γ2+γ2*γ0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 0*γ3+γ3*γ0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1*γ0+γ0*γ1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 omega_matrix_properties.nb γ 1*γ1+γ1*γ1= -2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 γ 1*γ2+γ2*γ1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1*γ3+γ3*γ1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 2*γ0+γ0*γ2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 2*γ1+γ1*γ2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 2*γ2+γ2*γ2= -2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 γ 2*γ3+γ3*γ2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ0+γ0*γ3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ1+γ1*γ3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ2+γ2*γ3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ3+γ3*γ3= - 2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 ------------(*ωmatrix*)------------------------- ω 0=γ0+γ3= 1 20  20 01 20- 2  20-1 20 0- 2 0-1 2 ω 1=γ1+γ1= 0 0 0  0 0 0 00 0 0 0 0 omega_matrix_properties.nb 3 ω 2=γ2+γ2= 0 0 0 1 0 0 -1 0 0 1 0 0 -1 0 0 0 ω 3=γ3+γ0= 1 20  20 01 20- 2  20-1 20 0- 2 0-1 2 --------------(*Anticommutation relation of ωmatrix*)----------------------- ω 0*ω0+ω0*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0*ω1+ω1*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0*ω2+ω2*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0*ω3+ω3*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 1*ω0+ω0*ω1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 1*ω1+ω1*ω1= - 2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 ω 1*ω2+ω2*ω1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 1*ω3+ω3*ω1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 2*ω0+ω0*ω2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 2*ω1+ω1*ω2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 2*ω2+ω2*ω2= - 2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 4 omega_matrix_properties.nb ω 2*ω3+ω3*ω2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω0+ω0*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω1+ω1*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω2+ω2*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω3+ω3*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 --------------(*Invariant using ωmatrix*)----------------------- (ω0*A0+ω1*A1+ω2*A2+ω3*A3)*(ω0*A0+ω1*A1+ω2*A2+ω3*A3)=-A1 2 -A2 2 omega_matrix_properties.nb 5