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Part VIII: Standard Model • Chapter 1

SM Structure & Symmetries

The gauge group SU(3)_C × SU(2)_L × U(1)_Y and particle content

The Gauge Structure

The Standard Model is a gauge theory based on the symmetry group:

G_SM = SU(3)_C × SU(2)_L × U(1)_Y

Each factor describes a fundamental interaction:

• SU(3)_C: Quantum Chromodynamics (QCD) — strong forceC = "color" charge (red, green, blue)
• SU(2)_L: Weak isospinL = acts only on left-handed fermions
• U(1)_Y: Weak hyperchargeY relates to electric charge: Q = T₃ + Y/2

Key Insight

The photon and W/Z bosons are not the fundamental gauge bosons! They arise from SU(2)_L × U(1)_Y after electroweak symmetry breaking.

1. Gauge Bosons (Force Carriers)

SU(3)_C: 8 Gluons

The strong force is mediated by 8 massless gluons:

G^a_μ, a = 1, 2, ..., 8 (SU(3) generators)

Physical color combinations:

G¹ ~ r⊗b̄ - b⊗r̄

G² ~ r⊗b̄ + b⊗r̄ (imaginary)

G³ ~ r⊗r̄ - b⊗b̄

G⁴ ~ r⊗ḡ - g⊗r̄

Gluons carry color charge themselves → self-interactions (3-gluon, 4-gluon vertices)

SU(2)_L × U(1)_Y: 4 Electroweak Bosons

Before symmetry breaking, we have 4 massless gauge bosons:

Wⁱ_μ, i = 1, 2, 3 (SU(2) triplet)
B_μ (U(1) singlet)

After Higgs mechanism, these mix to form physical states:

Charged W bosons:W^±_μ = (W¹_μ ∓ iW²_μ)/√2

Neutral bosons mix:

Z_μ = cos θ_W W³_μ - sin θ_W B_μ

A_μ = sin θ_W W³_μ + cos θ_W B_μ (photon)

Weinberg angle:sin² θ_W ≈ 0.231

2. Fermion Content (Matter Particles)

Three Generations

All fermions come in three identical generations (families) with increasing mass:

Type	Generation 1	Generation 2	Generation 3
Up-type quarks	u (up) ~ 2 MeV	c (charm) ~ 1.3 GeV	t (top) ~ 173 GeV
Down-type quarks	d (down) ~ 5 MeV	s (strange) ~ 95 MeV	b (bottom) ~ 4.2 GeV
Charged leptons	e (electron) ~ 0.5 MeV	μ (muon) ~ 106 MeV	τ (tau) ~ 1.78 GeV
Neutrinos	ν_e < 1 eV	ν_μ < 1 eV	ν_τ < 1 eV

Mystery: Why three generations? Why this mass hierarchy (10⁵ ratio between top and up quark)? The SM doesn't explain this—it's an input from experiment.

Representation Structure

Fermions transform differently under SU(3)×SU(2)×U(1):

Left-handed Quark Doublet:

Q_L = (u_L, d_L)^T ~ (3, 2, +1/6)

3 of SU(3), doublet of SU(2), hypercharge Y = +1/6

Right-handed Up Quark Singlet:

u_R ~ (3, 1, +2/3)

3 of SU(3), singlet of SU(2), hypercharge Y = +2/3

Right-handed Down Quark Singlet:

d_R ~ (3, 1, -1/3)

Left-handed Lepton Doublet:

L_L = (ν_e, e_L)^T ~ (1, 2, -1/2)

Singlet of SU(3) (no color), doublet of SU(2)

Right-handed Electron Singlet:

e_R ~ (1, 1, -1)

Note: No right-handed neutrinos!

The SM (in minimal form) has no ν_R. This is why neutrinos were thought massless. Neutrino oscillations require adding ν_R or other BSM physics.

3. Hypercharge and Electric Charge

Electric charge is not a fundamental quantum number in the SM. Instead:

Q = T₃ + Y/2

where T₃ = third component of weak isospin, Y = weak hypercharge

Examples:

Up quark (in doublet):

T₃ = +1/2, Y = +1/6 → Q = 1/2 + 1/12 = +2/3 ✓

Down quark (in doublet):

T₃ = -1/2, Y = +1/6 → Q = -1/2 + 1/12 = -1/3 ✓

Electron (in doublet):

T₃ = -1/2, Y = -1/2 → Q = -1/2 - 1/4 = -1 ✓

Neutrino (in doublet):

T₃ = +1/2, Y = -1/2 → Q = 1/2 - 1/4 = 0 ✓

4. Covariant Derivatives

The full covariant derivative in the SM is:

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

• First term: Ordinary derivative

• Second term: SU(3) gluon coupling

g_s = strong coupling, T^a = Gell-Mann matrices (8 generators)

• Third term: SU(2) weak isospin

g = weak coupling, τⁱ = Pauli matrices (3 generators)

• Fourth term: U(1) hypercharge

g' = hypercharge coupling, Y = hypercharge value

For quarks: All three terms active

Quarks couple to gluons (SU(3)), W bosons (SU(2)), and hypercharge (U(1))

For leptons: Only SU(2) and U(1)

Leptons are color singlets, so G^a_μ term vanishes

5. Gauge Field Kinetic Terms

The gauge boson dynamics come from field strength tensors:

SU(3) Gluon Field Strength:

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

The last term (structure constants f^abc) gives gluon self-interactions!

ℒ_QCD = -1/4 G^a_μνG^aμν

SU(2) Field Strength:

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

W bosons also self-interact (WWW, WWWW vertices)

U(1) Field Strength:

B_μν = ∂_μB_ν - ∂_νB_μ

Abelian: no self-interaction term (photons don't interact with photons at tree level)

Summary

✓ SM gauge group: SU(3)_C × SU(2)_L × U(1)_Y
✓ 12 gauge bosons: 8 gluons + W¹, W², W³, B → photon, W±, Z after EWSB
✓ 3 fermion generations with identical gauge interactions
✓ Chiral structure: Left-handed doublets, right-handed singlets
✓ Electric charge: Q = T₃ + Y/2 (Gell-Mann-Nishijima formula)
✓ Non-Abelian self-interactions: ggg, WWW vertices crucial for confinement/unitarity
✓ 45 fermion fields: 3 gen × (2 quarks × 3 colors + 2 leptons) × 2 chiralities - ν_R

Further Resources

• Peskin & Schroeder - Chapter 20 (Gauge Theories with Spontaneous Symmetry Breaking)
• Schwartz - Chapter 29 (Standard Model Structure)
• Burgess & Moore - The Standard Model: A Primer
• PDG Review - "Electroweak Model and Constraints on New Physics"

Quantum Field Theory Course