T 3 gauge transformation unbroken massless photon field W \u03bc 3 charged W \u03bc W \u03bc 1

# T 3 gauge transformation unbroken massless photon

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T 3 gauge transformation unbroken — massless photon field W μ 3 — charged W ± μ = ( W μ 1 ± iW μ 2 ) / 2 with mass gf — neutral scalar “Higgs” particle φ 3 with mass 2 λf charges of other particles quantized — must appear in SO (3) representations 16
SU (2) with a doublet of scalars L ( φ, W ) = D μ φ D μ φ - V ( φ ) - 1 4 W μν a W aμν T a = τ a 2 [ T a , T b ] = i ² abc T c D μ φ = μ φ + ig τ a 2 φ W μν a = μ W ν a - ν W μ a - abc W μ b W ν b V ( φ ) = λ 2 ( φ φ - f 2 / 2) 2 ⇒ h 0 | φ | 0 i = 0 f/ 2 conventional choice possible because all directions are equivalent L ( φ, W ) = 1 2 μ φ - ig 0 f/ 2 · W μ a τ a 2 μ φ + igW τ b 2 0 f/ 2 ¶¶ - 1 4 W μν a W aμν + cubic and higher terms 0 f/ 2 · W μ a τ a 2 W τ b 2 0 f/ 2 = f 2 8 W μ a W suggests another symmetry 17
SU (2) with a doublet of scalars — in σ -model language φ = Σ 2 0 1 Σ = σ + i~ τ · Σ Σ = ΣΣ = ( σ 2 + 2 ) I h 0 | Σ | 0 i = fI L , W ) = 1 2 (0 1) D μ Σ D μ Σ 0 1 - V (Σ) - 1 4 ~ W μν · ~ W μν = 1 4 tr ( D μ Σ D μ Σ ) - V (Σ) - 1 4 ~ W μν · ~ W μν gauged SU (2) acts on the left D μ Σ = μ Σ + ig ~ W μ · ~ τ 2 Σ RH SU (2) is a global symmetry δ Σ = i~ ² L · ~ τ 2 Σ - i Σ ~ ² R · ~ τ 2 ~ ² L = ~ ² R is unbroken for h 0 | Σ | 0 i = fI because [ ~ τ, I ] = 0 L , W ) = 1 2 μ σ - i~ τ · ( μ + gf ~ W μ / 2) ·‡ μ σ + i~ τ · ( μ + gf ~ W μ / 2) · - λf 2 2 σ 2 - 1 4 ~ W μν · ~ W μν scalar with mass λf vectors with mass gf/ 2 three gauge and three global generators are broken - but three combinations are unbroken — so there were three GB to be eaten to become the longitudinal components of the triplet of massive gauge bosons 18
SU (2) × SU (2) coupled to σ -model Σ = σ + i~ τ · Σ Σ = ΣΣ = ( σ 2 + 2 ) I h 0 | Σ | 0 i = fI = 1 4 tr ( D μ Σ D μ Σ ) - V (Σ) - 1 4 ~ W μν L · ~ W Lμν - 1 4 ~ W μν R · ~ W Rμν D μ Σ = μ Σ + ig L ~ W μ L · ~ τ 2 Σ - ig R ~ W μ R · Σ ~ τ 2 now RH SU (2) is a gauge symmetry δ Σ = i~ ² L · ~ τ 2 Σ - i Σ ~ ² R · ~ τ 2 ~ ² L = ~ ² R is unbroken for h 0 | Σ | 0 i = fI because [ ~ τ, I ] = 0 L , W ) = 1 2 μ σ - i~ τ · ( μ + g L f ~ W / 2 - g R f ~ W / 2) · μ σ + i~ τ · ( μ + g L f ~ W μ L / 2 - g R f ~ W μ R / 2) · - λf 2 2 σ 2 - 1 4 ~ W μν L · ~ W Lμν - 1 4 ~ W μν R · ~ W Rμν scalar with mass λf vectors with mass ?? six gauge generators are broken - but three combinations are unbroken — so there were three GB to be eaten to become the longitudinal components of a triplet of massive gauge bosons and a triplet remains massless 19
SU (2) × SU (2) coupled to σ -model L , W ) = 1 2 μ σ - i~ τ · ( μ + g L f ~ W / 2 - g R f ~ W / 2) · μ σ + i~ τ · ( μ + g L f ~ W μ L / 2 - g R f ~ W μ R / 2) · - λf 2 2 σ 2 - 1 4 ~ W μν L · ~ W Lμν - 1 4 ~ W μν R · ~ W Rμν scalar with mass λf vectors with mass p g 2 L + g 2 R f/ 2 g L ~ W μ L - g R ~ W μ R p g 2 L + g 2 R this is the linear combination that appears in the mass term ~ W μ L /g L + ~ W μ R /g R p 1 /g 2 L + 1 /g 2 R this is the linear combination that couples to the unbroken generators ~ W μ L /g L + ~ W μ R /g R

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• Spring '10
• GEORGI
• Quantum Field Theory, ... ..., wA, Quantum chromodynamics, Gauge theory, Dµ Dµ

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