t8 - updated Tuesday 6 th April 2010 09:19 6 non-Abelian...

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Unformatted text preview: updated Tuesday 6 th April, 2010 09:19 6. non-Abelian gauge theories (a) gauging non-Abelian flavor symmetry (b) gauge fixing and Fadeev- Popov (c) ghosts and Feynman rules (d) BRST symmetry ???? (e) background field gauge ???? (f) asymptotic freedom 8. spontaneously broken U (1) gauge symmetry and the Higgs mechanism (a) type II superconductors ???? 9. spontaneously broken non-Abelian gauge symmetry and the Higgs mechanism (a) Weinberg’s model of leptons (b) unitary gauge (c) ’t Hooft-Feynman gauge (d) introduction to CKM and the standard model (e) custodial SU (2) and the M 2 W /M 2 Z (f) non-leptonic weak interactions, matching in strongly interacting theories and the operator product expansion (g) grand unification and unification of couplings (h) solitons and magnetic monopoles???? 7. confinement and quantum chromodynamics (a) partons and hadrons (b) R in e + e- (c) thrust and IR safe observables (d) distribution and decay function (very briefly if at all) (e) accidental symmetry and the dynamics of Gell-Mann’s SU(3) (f) Large N c putting gauge invariance and SSB together L =- 1 4 F μν F μν + ( D μ φ ) * D μ φ- λ 2 ( φ * φ- f 2 / 2) 2 where D μ = ∂ μ + ieA μ h φ * φ i = f 2 / 2 SSB but instead of massless gauge bosons and a massless GB, we get a massive vector particle — easiest to see by making a gauge transformation to make φ real φ ( x ) = 1 √ 2 e iπ ( x ) φ r ( x ) → 1 √ 2 φ r ( x ) D μ φ = 1 √ 2 e iπ ( x ) ( ∂ μ + ieA μ- ∂ μ π ) φ r → 1 √ 2 ( ∂ μ + ieA μ ) φ r GB gets sucked into gauge field L =- 1 4 F μν F μν + 1 2 ( ∂ μ φ r ∂ μ φ r + e 2 φ 2 r A μ A μ )- λ 2 ( φ 2 r- f 2 ) 2 you can do this without SSB, but it doesn’t fix the gauge ambiguity — quadratic part of L is still not invertible L =- 1 4 F μν F μν + 1 2 ( ∂ μ φ r ∂ μ φ r + e 2 φ 2 r A μ A μ )- λ 8 ( φ 2 r- f 2 ) 2 but with SSB we should take φ r = f + h L =- 1 4 F μν F μν + 1 2 ∂ μ h∂ μ h + 1 2 e 2 ( f + h ) 2 A μ A μ- λ 8 (2 fh + h 2 ) 2 now quadratic part of L invertible L =- 1 4 F μν F μν + 1 2 e 2 f 2 A μ A μ + 1 2 ∂ μ h∂ μ h- λf 2 2 h 2 + ··· vector particle A μ with mass ef scalar particle h with mass √ λf what happened to our massless gauge particle and our GB? flux → vacuum — GB gets sucked into the gauge field D μ φ = 1 √ 2 e iπ ( x ) ( ∂ μ + ieA μ- ∂ μ π )( f + h ) → 1 √ 2 ( ∂ μ + ieA μ )( f + h ) flux non-conservation implies short range forces — no massless gauge particle flux non-conservation allows longitudinal waves! SSB but instead of massless gauge bosons and a massless GB, we get a massive vector particle — easiest to see by making a gauge transformation to make φ real φ ( x ) = 1 √ 2 e iπ ( x ) φ r ( x ) → 1 √ 2 φ r ( x ) D μ φ = 1 √ 2 e iπ ( x ) ( ∂ μ + ieA μ- ∂ μ π ) φ r → 1 √ 2 ( ∂ μ + ieA μ ) φ r GB gets sucked into gauge field L =- 1 4 F μν F μν + 1 2 ( ∂ μ...
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t8 - updated Tuesday 6 th April 2010 09:19 6 non-Abelian...

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