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Part VIII: Standard Model • Chapter 5

Complete Standard Model Lagrangian

Assembling all pieces: The most successful equation in physics

The Standard Model in One Equation

Everything we know about particle physics—from quarks to the Higgs, from QED to the strong force—can be encoded in a single (admittedly complex) Lagrangian density:

ℒ_SM = ℒ_gauge + ℒ_fermion + ℒ_Higgs + ℒ_Yukawa

With 19 free parameters, this equation explains every confirmed particle physics experiment to date—a triumph of mathematical physics!

1. Gauge Kinetic Terms

The gauge boson dynamics:

ℒ_gauge = -¼ G^a_μνG^aμν - ¼ Wⁱ_μνW^iμν - ¼ B_μνB^μν

QCD (SU(3)_C):

G^a_μν = ∂_μG^a_ν - ∂_νG^a_μ - g_sf^abcG^b_μG^c_ν

• a = 1,...,8 (8 gluons)
• f^abc = SU(3) structure constants
• g_s = strong coupling ≈ 1.2 at M_Z
• Contains 3-gluon and 4-gluon vertices

Weak SU(2)_L:

Wⁱ_μν = ∂_μWⁱ_ν - ∂_νWⁱ_μ - g ε^ijkW^j_μW^k_ν

• i = 1,2,3 (3 W bosons)
• ε^ijk = Levi-Civita symbol
• g ≈ 0.65 at M_Z
• WWW and WWWW self-interactions

Hypercharge U(1)_Y:

B_μν = ∂_μB_ν - ∂_νB_μ

• Abelian: no self-interaction
• g' ≈ 0.36 at M_Z
• Mixes with W³ to form γ and Z

2. Fermion Kinetic & Interaction Terms

ℒ_fermion = Σ_ψ iψ̄ γ^μ D_μ ψ

Sum over all fermion fields with covariant derivative:

D_μ = ∂_μ + ig_sG^a_μT^a + ig Wⁱ_μτⁱ/2 + ig' Y B_μ/2

Field Content (per generation):

Q_L = (u_L, d_L)^T:

Left-handed quark doublet ~ (3, 2, +1/6)

u_R:

Right-handed up quark ~ (3, 1, +2/3)

d_R:

Right-handed down quark ~ (3, 1, -1/3)

L_L = (ν_L, e_L)^T:

Left-handed lepton doublet ~ (1, 2, -1/2)

e_R:

Right-handed charged lepton ~ (1, 1, -1)

Gauge interactions encoded:

• QCD: Quarks couple to gluons via T^a (color charge)
• Weak: Left-handed doublets couple to Wⁱ via τⁱ/2 (weak isospin)
• Hypercharge: All fermions couple to B via Y (hypercharge)

3. Higgs Sector

ℒ_Higgs = (D_μΦ)^†(D^μΦ) - V(Φ)

Higgs Doublet:

Φ = (φ⁺, φ⁰)^T ~ (1, 2, +1/2)

4 real scalar fields: φ₁, φ₂, φ₃, φ₄

Higgs Potential:

V(Φ) = μ²Φ^†Φ + λ(Φ^†Φ)²

• μ² < 0: triggers spontaneous symmetry breaking
• λ > 0: ensures vacuum stability
• VEV: v = √(-μ²/λ) ≈ 246 GeV
• Higgs mass: m_H² = 2λv²

Unitary Gauge:

Φ = (0, (v + H)/√2)^T

3 Goldstone bosons eaten → longitudinal W^±, Z modes

4. Yukawa Couplings

Fermion masses arise from Yukawa interactions with the Higgs:

ℒ_Yukawa = -Y_ij^u Q̄_Lⁱ Φ̃ u_R^j - Y_ij^d Q̄_Lⁱ Φ d_R^j - Y_ij^e L̄_Lⁱ Φ e_R^j + h.c.

where Φ̃ = iτ²Φ^* for up-type quarks (to get correct hypercharge)

• Y^u,d,e are 3×3 complex matrices
• i,j = generation indices (1,2,3)
• When Φ → v/√2: Y_ij v/√2 = mass matrix

After diagonalization:

• Up quarks: m_u ~ 2 MeV, m_c ~ 1.3 GeV, m_t ~ 173 GeV

• Down quarks: m_d ~ 5 MeV, m_s ~ 95 MeV, m_b ~ 4.2 GeV

• Charged leptons: m_e ~ 0.5 MeV, m_μ ~ 106 MeV, m_τ ~ 1.78 GeV

CKM Mixing:

Mismatch between up/down diagonalization → V_CKM = V_L^u†V_L^d

W⁺ couples: ū_i γ^μ V_ij d_j

5. The Complete SM Lagrangian

Putting It All Together:

ℒ_SM = 
  - ¼ G^a_μνG^aμν - ¼ Wⁱ_μνW^iμν - ¼ B_μνB^μν
  + Σ_ψ iψ̄ γ^μ D_μ ψ
  + (D_μΦ)^†(D^μΦ) - V(Φ)
  - Y_ij^u Q̄_Lⁱ Φ̃ u_R^j - Y_ij^d Q̄_Lⁱ Φ d_R^j - Y_ij^e L̄_Lⁱ Φ e_R^j + h.c.

This single equation (plus gauge fixing and ghost terms in perturbation theory) describes all known particle physics below the Planck scale (excluding gravity).

6. The 19 Parameters

Gauge Couplings (3):

g₁(U(1)), g₂(SU(2)), g₃(SU(3))

Quark Yukawas (6):

y_u, y_d, y_c, y_s, y_t, y_b

Lepton Yukawas (3):

y_e, y_μ, y_τ

CKM Parameters (4):

θ₁₂, θ₂₃, θ₁₃, δ_CP

Higgs Parameters (2):

v (VEV ≈ 246 GeV), λ (self-coupling)

QCD θ-angle (1):

θ_QCD < 10^-10 (strong CP problem!)

Open questions about parameters:

• Why these particular values? (hierarchy problem)
• Why 3 generations?
• Why do Yukawas span 6 orders of magnitude?
• Why is θ_QCD so small? (strong CP problem)
• Are they related at GUT scale?

7. Feynman Rules Summary

From ℒ_SM, we derive all interaction vertices:

QED: eγμ vertex

-ie γ^μ

QCD: gluon-quark

-ig_s γ^μ T^a

W±: charged current

-ig/√2 γ^μ P_L V_CKM

Z: neutral current

-ig/(2cosθ_W) γ^μ(g_V-g_Aγ₅)

Higgs-fermion

-im_f/v

Higgs-WW

ig m_W g^μν

3-gluon

g_sf^abc[...]

4-gluon

-ig_s²[f^abef^cde...]

Summary

✓ Complete Lagrangian: ℒ_gauge + ℒ_fermion + ℒ_Higgs + ℒ_Yukawa
✓ Gauge group: SU(3)_C × SU(2)_L × U(1)_Y with 12 gauge bosons
✓ Fermion content: 3 generations × (2 quarks + 2 leptons) × 2 chiralities
✓ Higgs mechanism: Spontaneous breaking SU(2)_L×U(1)_Y → U(1)_EM
✓ 19 parameters: 3 couplings + 9 Yukawas + 4 CKM + 2 Higgs + 1 θ
✓ Renormalizable: All divergences canceled to all orders in perturbation theory
✓ Experimental success: Every prediction confirmed (so far!)

Further Resources

• Peskin & Schroeder - Chapter 20-21 (Complete Electroweak Theory)
• Schwartz - Chapter 29 (The Standard Model Lagrangian)
• Srednicki - Part VI (The Standard Model)
• Weinberg - Vol II, Chapters 19-21 (Gauge Theories of the Second Kind)

Quantum Field Theory Course