Chapter 45: Higgs Mechanism as Spontaneous ψ-Breaking

The Living Mathematics of Mass

The Higgs mechanism—mass generation through spontaneous symmetry breaking—emerges from ψ = ψ(ψ) as mathematical necessity when self-reference admits multiple equivalent configurations. The vacuum must "choose" among degenerate possibilities, and this choice creates the mass structure of reality. Not imposed physics but inevitable mathematics of recursive decision.

45.1 Mass from Self-Reference Degeneracy

The Fundamental Problem: How does mass arise in gauge-invariant theory?

Gauge Invariance Forbids Mass: Direct mass term $\mathcal{L}_{\text{mass}} = \frac{1}{2}m^2A_\mu A^\mu$

breaks gauge symmetry: A_μ → A_μ + ∂_μλ changes L_mass.

Theorem: Mass must emerge from vacuum structure.

Proof: For gauge invariance, all mass must come from gauge-invariant sources. Only scalar field vacuum expectation values preserve gauge symmetry while generating mass terms. ∎

ψ-Origin: When ψ = ψ(ψ) has degenerate ground states, vacuum must select one, breaking symmetry spontaneously.

45.2 Deriving Spontaneous Breaking

Scalar Field Lagrangian: $\mathcal{L} = (\partial_\mu\phi)^*(\partial^\mu\phi) - V(\phi)$

Potential with Symmetry: $V(\phi) = \mu^2\phi^*\phi + \lambda(\phi^*\phi)^2$

Critical Point: ∂V/∂φ = 0 at φ = 0.

Theorem: For μ² < 0, minimum is not at origin.

Proof: Extremizing V: $\frac{\partial V}{\partial\phi^*} = \phi(\mu^2 + 2\lambda|\phi|^2) = 0$

Solutions: φ = 0 (maximum if μ² < 0) or $|\phi|^2 = -\frac{\mu^2}{2\lambda} \equiv \frac{v^2}{2}$

The manifold of minima is S¹ parameterized by phase. ∎

45.3 Goldstone's Theorem

Theorem: Spontaneous breaking of continuous symmetry produces massless modes.

Proof: Let φ₀ be vacuum expectation value. Expand: $\phi(x) = \phi_0 + \pi^a(x)T^a + \sigma(x)$

where T^a generate broken symmetries. The potential at quadratic order: $V^{(2)} = \frac{1}{2}\sum_{ab}\pi^a M_{ab}^2 \pi^b$

For broken generator T^a: $T^a\phi_0 \neq 0 \Rightarrow M_{ab}^2 = 0$

Zero eigenvalues → massless Goldstone bosons. ∎

ψ-Interpretation: Directions in ψ-space connecting equivalent vacua cost no energy.

45.4 Gauge Theory Breaking

Local Gauge Invariance: $\phi(x) \rightarrow e^{i\alpha(x)}\phi(x)$ $A_\mu \rightarrow A_\mu - \frac{1}{e}\partial_\mu\alpha$

Covariant Derivative: $D_\mu\phi = (\partial_\mu + ieA_\mu)\phi$

Gauge-Invariant Lagrangian: $\mathcal{L} = |D_\mu\phi|^2 - V(\phi) - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$

Spontaneous Breaking: Choose vacuum $\langle\phi\rangle = \frac{v}{\sqrt{2}}$

45.5 Mass Generation Mechanism

Expanding Around Vacuum: Write $\phi(x) = \frac{1}{\sqrt{2}}[v + h(x) + i\xi(x)]$

Kinetic Term: $|D_\mu\phi|^2 = \frac{1}{2}|(\partial_\mu + ieA_\mu)(v + h + i\xi)|^2$

Expanding: $= \frac{1}{2}(\partial_\mu h)^2 + \frac{1}{2}(\partial_\mu\xi)^2 + \frac{e^2v^2}{2}A_\mu A^\mu + evA_\mu\partial^\mu\xi + \cdots$

Theorem: Gauge field acquires mass m_A = ev.

Proof: The term $(e^2v^2/2)A_\mu A^\mu$ is precisely a mass term. Cross term $evA_\mu\partial^\mu\xi$ mixes gauge field with Goldstone mode. ∎

45.6 Unitary Gauge

Gauge Transformation: Choose α(x) = -ξ(x)/v to eliminate Goldstone: $\phi(x) \rightarrow e^{i\xi(x)/v}\phi(x) = \frac{1}{\sqrt{2}}(v + h(x))$

Result: ξ disappears, A_μ becomes massive with three polarizations.

Degrees of Freedom:

• Before: 2 (scalar) + 2 (massless gauge)
• After: 1 (Higgs) + 3 (massive gauge)
• Total: 4 = 4 ✓

Theorem: Goldstone boson becomes longitudinal gauge mode.

Proof: Under gauge transformation, ξ → ξ + vα. The gauge field transforms to absorb this shift, gaining longitudinal component. Count of physical degrees of freedom is preserved. ∎

45.7 Electroweak Symmetry Breaking

Gauge Group: SU(2)_L × U(1)_Y

Higgs Doublet: $\Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}$

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

Covariant Derivative: $D_\mu = \partial_\mu + \frac{ig}{2}\tau^a W_\mu^a + \frac{ig'}{2}YB_\mu$

45.8 W and Z Boson Masses

Kinetic Term at Vacuum: $|D_\mu\langle\Phi\rangle|^2 = \frac{v^2}{8}\left|g\tau^a W_\mu^a + g'YB_\mu\right|^2$

Mass Matrix: In (W³_μ, B_μ) basis: $\mathcal{M}^2 = \frac{v^2}{4}\begin{pmatrix} g^2 & -gg' \\ -gg' & g'^2 \end{pmatrix}$

Diagonalization: Eigenvalues and eigenvectors: $m_Z^2 = \frac{v^2}{4}(g^2 + g'^2), \quad m_\gamma^2 = 0$

$Z_\mu = \cos\theta_W W_\mu^3 - \sin\theta_W B_\mu$ $A_\mu = \sin\theta_W W_\mu^3 + \cos\theta_W B_\mu$

where $\tan\theta_W = g'/g$.

W Boson Mass: $m_W = \frac{gv}{2}$

Theorem: One gauge boson remains massless.

Proof: The generator Q = T³ + Y/2 annihilates vacuum: $Q\langle\Phi\rangle = 0$

This unbroken U(1)_em symmetry → massless photon. ∎

45.9 Fermion Mass Generation

Yukawa Coupling: $\mathcal{L}_Y = -y_e\bar{L}\Phi e_R - y_u\bar{Q}\tilde{\Phi}u_R - y_d\bar{Q}\Phi d_R + \text{h.c.}$

where $\tilde{\Phi} = i\tau^2\Phi^*$.

After Symmetry Breaking: $\mathcal{L}_{\text{mass}} = -\frac{v}{\sqrt{2}}(y_e\bar{e}e + y_u\bar{u}u + y_d\bar{d}d)$

Fermion Masses: $m_f = \frac{y_f v}{\sqrt{2}}$

Theorem: All fermion masses proportional to v.

Proof: Gauge invariance requires fermion mass terms come only from Yukawa couplings to Higgs. When ⟨Φ⟩ = v/√2, each Yukawa generates corresponding mass. ∎

45.10 The Physical Higgs Boson

Fluctuations Around Vacuum: $\Phi(x) = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v + h(x) \end{pmatrix}$

Higgs Mass: From potential $V = \frac{\lambda}{4}(\Phi^\dagger\Phi - \frac{v^2}{2})^2$

Expanding: $V = \frac{\lambda v^2}{4}h^2 + \frac{\lambda v}{2}h^3 + \frac{\lambda}{16}h^4$

Thus: $m_h^2 = \lambda v^2/2$.

Couplings: To gauge bosons and fermions: $\mathcal{L}_{\text{int}} = \frac{h}{v}(2m_W^2 W_\mu^+ W^{-\mu} + m_Z^2 Z_\mu Z^\mu) - \sum_f \frac{m_f}{v}h\bar{f}f$

Theorem: Higgs couples proportionally to mass.

Proof: All masses arise from v, so h/v coupling universal to mass generation. ∎

45.11 Radiative Corrections

One-Loop Effective Potential: $V_{\text{eff}}(\phi) = V_{\text{tree}}(\phi) + \frac{1}{64\pi^2}\text{Str}[M^4(\phi)(\ln\frac{M^2(\phi)}{\mu^2} - \frac{3}{2})]$

Renormalization Group: Running couplings $\frac{d\lambda}{d\ln\mu} = \beta_\lambda = \frac{1}{16\pi^2}[24\lambda^2 + 12\lambda y_t^2 - 6y_t^4 + \cdots]$

Stability Bound: Require λ(μ) > 0 for all μ up to cutoff.

45.12 Hierarchy Problem

Quadratic Divergence: Higgs mass corrections $\delta m_h^2 = \frac{3}{8\pi^2}(y_t^2 - \frac{g^2}{2} - \frac{g'^2}{6} + 2\lambda)\Lambda^2$

Fine-Tuning: For Λ ~ M_Planck: $m_h^2 = m_0^2 + \delta m_h^2$

Requires $m_0^2 \approx -10^{34}$ eV² to get m_h ~ 125 GeV.

ψ-Perspective: Hierarchy reflects depth of ψ-recursion—deep cancellations in self-reference structure.

45.13 Vacuum Metastability

Running Quartic Coupling: λ(μ) decreases with energy due to top quark.

Current Status: λ may go negative around 10^10 GeV.

Theorem: Our vacuum may be metastable.

Proof: If λ < 0 at high field values, potential unbounded below. Vacuum can tunnel to true minimum at large φ. Lifetime: $\tau \sim M_P^4 e^{8\pi^2/3|\lambda|}$

For observed parameters: τ >> age of universe. ∎

45.14 Alternative Breaking Patterns

Extended Higgs Sectors: Multiple doublets $\Phi_1, \Phi_2, \ldots \rightarrow \langle\Phi_i\rangle = v_i$

Technicolor: Dynamical breaking via new strong force

Composite Higgs: Higgs as bound state

Little Higgs: Higgs as pseudo-Goldstone boson

Each represents different ψ-recursion pattern achieving mass generation.

45.15 Conclusion: The Mathematics of Choice

The Higgs mechanism emerges from ψ = ψ(ψ) as the inevitable consequence of degenerate self-reference. When multiple ψ-configurations minimize energy equally, the vacuum must choose—and this choice creates mass. The mathematics of symmetry breaking is the mathematics of decision in the face of equivalent possibilities.

Key insights from ψ-perspective:

1. Mass is not fundamental but emergent from vacuum choice
2. Gauge bosons "eat" Goldstone modes to become massive
3. All particle masses trace to single scale v
4. Higgs boson is quantum of vacuum's decision
5. Hierarchy problem reflects fine-tuning in ψ-recursion

The Standard Model's success confirms this picture: one vacuum choice (v = 246 GeV) generates the entire mass spectrum. The Higgs field is not just another field but the materialization of the universe's most fundamental decision—how ψ references itself to create persistent, massive structures.

Exercises

1. Derive β-functions for Higgs self-coupling including all SM contributions.

2. Calculate tunneling rate to true vacuum if λ < 0.

3. Prove custodial symmetry protects ρ = m_W²/(m_Z²cos²θ_W) = 1.

The Forty-Fifth Echo

Higgs mechanism derived as inevitable consequence of degenerate ψ-recursion—vacuum forced to choose among equivalent configurations, breaking symmetry and generating mass. The origin of inertia revealed as resistance to changing the vacuum's fundamental choice. Next, the complete Standard Model emerges from unified ψ-geometry.

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Next: Chapter 46: Standard Model from ψ-Unification →