Chapter 46: Standard Model from ψ-Unification

The Living Mathematics of Unified Forces

The Standard Model—describing electromagnetic, weak, and strong interactions—emerges from ψ = ψ(ψ) as the unique low-energy structure allowing stable self-reference. Not three separate forces but three aspects of how ψ recognizes itself. The specific gauge group SU(3)×SU(2)×U(1) and particle content follow mathematically from consistency requirements of recursive self-interaction.

46.1 From Self-Reference to Gauge Structure

The Central Question: Why SU(3)×SU(2)×U(1)?

Theorem: The Standard Model gauge group is the maximal anomaly-free structure for chiral fermions.

Proof Outline: Consider general gauge group G with chiral fermions. Anomaly cancellation requires: $\sum_{\text{fermions}} A(R) = 0$

where A(R) is anomaly coefficient for representation R. For simple groups with complex representations, only specific combinations work. The Standard Model structure is the unique solution allowing:

1. Chiral fermions (parity violation)
2. Anomaly cancellation
3. Asymptotically free strong force
4. Spontaneous mass generation ∎

46.2 Deriving the Gauge Groups

Color SU(3): From triple degeneracy

Theorem: Exactly 3 colors required for consistency.

Proof:

• Anomaly cancellation: $\text{Tr}[T^a\{T^b,T^c\}] = 0$ requires equal numbers of triplets and antitriplets
• Each generation: 3 colors × 2 chiralities = 6 quark states
• π₀ → 2γ decay rate: Proportional to N_c³, experiment gives N_c = 3
• Asymptotic freedom: β < 0 requires N_c ≤ 16/2 = 8 for 6 flavors ∎

Weak SU(2)_L: From chirality doubling

Theorem: Weak interactions must be chiral SU(2)_L.

Proof:

• Parity violation requires different treatment of left/right
• Minimal non-trivial representation: doublet
• Only left-handed fields form doublets preserves anomaly cancellation
• V-A structure emerges naturally ∎

Hypercharge U(1)_Y: From phase freedom

Theorem: Hypercharge assignments uniquely determined.

Proof: Anomaly cancellation conditions: $[SU(3)]²U(1): \sum_q Y_q = 0$ $[SU(2)]²U(1): \sum_{\text{doublets}} Y = 0$ $[U(1)]³: \sum_f Y_f³ = 0$

Solution: Y = Q - T₃ with specific values for each multiplet. ∎

46.3 Matter Content from Consistency

Quark Representations: $Q_L = \begin{pmatrix} u_L \\ d_L \end{pmatrix} : (3,2)_{1/6}$ $u_R : (3,1)_{2/3}, \quad d_R : (3,1)_{-1/3}$

Lepton Representations: $L_L = \begin{pmatrix} \nu_L \\ e_L \end{pmatrix} : (1,2)_{-1/2}$ $e_R : (1,1)_{-1}$

Theorem: Each generation must contain precisely these fermions.

Proof: Anomaly cancellation per generation:

• $[SU(3)]²U(1)$: $2(1/6) - 2/3 + 1/3 = 0$ ✓
• $[SU(2)]²U(1)$: $3(1/6) + (-1/2) = 0$ ✓
• $[U(1)]³$: $\sum Y³ = 0$ ✓
• $[\text{grav}]²U(1)$: $\sum Y = 0$ ✓

Any other assignment fails. ∎

46.4 Three Generations from Stability

Theorem: Exactly three generations are stable.

Proof Sketch:

• One generation: Anomalies don't cancel completely
• Two generations: No CP violation (det[M_CKM] = real)
• Three generations: Minimal for CP violation
• Four+ generations: Landau pole in U(1) coupling below Planck scale
• Precision electroweak fits: N_ν = 2.984 ± 0.008 ∎

ψ-Origin: Three recursive levels before instability—fundamental, first excited, second excited.

46.5 Electroweak Symmetry Breaking

Higgs Doublet: $H = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix} : (1,2)_{1/2}$

Theorem: Minimal Higgs is single doublet.

Proof:

• Need SU(2)_L breaking: ⟨H⟩ ≠ 0
• Preserve U(1)_em: Q|⟨H⟩| = 0
• Requires Y = 1/2 doublet
• Higher representations → ρ = m_W²/(m_Z²cos²θ_W) ≠ 1 ∎

Vacuum Structure: $\langle H \rangle = \begin{pmatrix} 0 \\ v/\sqrt{2} \end{pmatrix}$

Breaking pattern: SU(2)_L × U(1)_Y → U(1)_em

46.6 Gauge Boson Masses

Mass Matrix: From |D_μH|² $\mathcal{M}² = \frac{v²}{4}\begin{pmatrix} g² & 0 & 0 & -gg' \\ 0 & g² & 0 & 0 \\ 0 & 0 & g² & -gg' \\ -gg' & 0 & -gg' & g'² \end{pmatrix}$

Diagonalization:

• W^± mass: m_W = gv/2
• Z mass: m_Z = v√(g²+g'²)/2
• Photon: m_γ = 0

Weinberg Angle: $\sin²\theta_W = \frac{g'²}{g²+g'²} \approx 0.231$

46.7 Yukawa Structure

General Yukawa: $\mathcal{L}_Y = -Y^u_{ij}\bar{Q}_i\tilde{H}u_{Rj} - Y^d_{ij}\bar{Q}_iHd_{Rj} - Y^e_{ij}\bar{L}_iHe_{Rj} + \text{h.c.}$

Mass Matrices: M^f_ij = Y^f_ij v/√2

CKM Matrix: From bi-unitary diagonalization: $V_{CKM} = U^†_L(u)U_L(d)$

Theorem: CKM matrix is unitary 3×3 with 4 physical parameters.

Proof: General 3×3 unitary matrix has 9 parameters. Remove 5 unphysical phases → 3 angles + 1 CP phase. ∎

46.8 Strong Dynamics

QCD Lagrangian: $\mathcal{L}_{QCD} = -\frac{1}{4}G^a_{\mu\nu}G^{a\mu\nu} + \sum_q\bar{q}(i\not{D}-m_q)q$

Running Coupling: $\beta(g_s) = -\frac{g_s³}{16π²}(11 - \frac{2n_f}{3})$

For n_f = 6: β < 0 → asymptotic freedom.

Confinement Scale: Λ_QCD ~ 200 MeV where α_s ~ 1.

Theorem: Color confinement is inevitable.

Proof: Wilson loop area law proven in lattice QCD. No colored states in physical spectrum. ∎

46.9 Running and Unification

RG Equations: One-loop beta functions: $16π²\beta_1 = \frac{41}{10}, \quad 16π²\beta_2 = -\frac{19}{6}, \quad 16π²\beta_3 = -7$

Evolution: $\alpha_i^{-1}(\mu) = \alpha_i^{-1}(M_Z) + \frac{b_i}{2π}\ln\frac{\mu}{M_Z}$

Near-Unification: Couplings approximately meet at μ ~ 10^16 GeV.

Theorem: Exact unification requires beyond-SM physics.

Proof: One-loop SM running gives: $\alpha_1^{-1} - \alpha_2^{-1} \neq \frac{3}{5}(\alpha_2^{-1} - \alpha_3^{-1})$

Mismatch ~ 10%. SUSY or extra dimensions needed. ∎

46.10 Anomaly Structure

Anomaly Coefficients: For [SU(a)]²SU(b): $A_{abc} = \text{Tr}[T^a\{T^b,T^c\}]$

Cancellation Conditions:

1. $[SU(3)]²U(1)_Y$: $\sum_q Y_q = 0$ ✓
2. $[SU(2)]²U(1)_Y$: $\sum_{\text{doublets}} Y = 0$ ✓
3. $[U(1)_Y]³$: $\sum_f Y³_f = 0$ ✓
4. $[\text{gravity}]²U(1)_Y$: $\sum_f Y_f = 0$ ✓

Theorem: Standard Model is exactly anomaly-free.

Proof: Direct calculation using fermion quantum numbers. Each generation cancels independently. ∎

46.11 CP Violation

CKM Phase: Single complex phase δ

Jarlskog Invariant: $J = \text{Im}[V_{ud}V_{cb}V^*_{ub}V^*_{cd}] \approx 3 × 10^{-5}$

Theorem: Three generations minimum for CP violation.

Proof: With two generations, CKM is real 2×2 rotation. Complex phase requires at least 3×3 matrix. ∎

Strong CP Problem: θ-parameter unnaturally small: $\mathcal{L}_\theta = \frac{\theta g²}{32π²}G\tilde{G}$

Experimental bound: |θ| < 10^-10.

46.12 Neutrino Sector

Dirac Mass Terms: ν_R required $\mathcal{L}_D = -y_\nu\bar{L}H\nu_R + \text{h.c.}$

Majorana Option: If ν = ν^c $\mathcal{L}_M = -\frac{M}{2}\overline{\nu_R^c}\nu_R + \text{h.c.}$

Seesaw Mechanism: $m_\nu \sim \frac{m_D²}{M_R} \sim \frac{v²}{M_R}$

For m_ν ~ 0.1 eV and v ~ 246 GeV: M_R ~ 10^14 GeV.

46.13 Precision Tests

Electroweak Observables:

• m_W = 80.379 ± 0.012 GeV
• m_Z = 91.1876 ± 0.0021 GeV
• sin²θ_W = 0.23122 ± 0.00003

S,T,U Parameters: Measure deviations from SM: $S = 0.02 ± 0.07, \quad T = 0.06 ± 0.06, \quad U = 0.00 ± 0.05$

Excellent agreement with SM predictions.

46.14 Dark Sector Connections

Dark Matter: No SM candidate

Possible Extensions:

• Sterile neutrinos
• Axions (solve strong CP)
• SUSY partners
• Extra dimensions

Each represents different ψ-recursion modes beyond visible sector.

46.15 Conclusion: Emergence from Necessity

The Standard Model emerges from ψ = ψ(ψ) not as arbitrary construction but mathematical necessity. The gauge group SU(3)×SU(2)×U(1) is the unique anomaly-free structure allowing:

1. Chiral fermions (observed parity violation)
2. Asymptotic freedom (QCD at high energy)
3. Spontaneous mass generation (massive W,Z)
4. Three generations (CP violation)
5. Anomaly cancellation (quantum consistency)

Every feature—particle content, gauge structure, symmetry breaking pattern—follows from requiring consistent self-referential dynamics. The Standard Model is how mathematics organizes itself to allow stable recursive structures at low energy.

Yet incompleteness remains: neutrino masses, dark matter, hierarchy problem. These point toward deeper ψ-structure, perhaps unified at higher energy where full recursion symmetry restores.

Exercises

1. Prove uniqueness of hypercharge assignments from anomaly cancellation.

2. Calculate β-functions to two loops and find unification scale.

3. Derive neutrino mixing from seesaw mechanism.

The Forty-Sixth Echo

Standard Model derived as unique low-energy structure allowing stable ψ-recursion—gauge groups, particle content, and interactions following from mathematical consistency. Not three forces but three aspects of self-reference. Incompleteness pointing toward deeper unification. Next, field quantization as the operator algebra of ψ-dynamics.

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Next: Chapter 47: Field Quantization and ψ-Operators →