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Unformatted text preview: updated Tuesday 2 nd March, 2010 11:29 4. symmetries, currents and the effective field theory of the weak interactions (a) Strong interactions are complicated (b) GellMann’s SU (3) — symmetry (c) nonAbelian flavor symmetries and QFT Lagrangians (d) Noether’s theorem and conserved flavor currents (e) EFT and the AdemolloGatto Theorem currentcurrent weak interactions from W exchange G F √ 2 j μ j † μ j μ is the charged, lefthanded “weak isospin” current — more tensor products — charge and family # ‘ = ‘ 11 ‘ 21 ‘ 12 ‘ 22 ‘ 13 ‘ 23 = ν e e ν μ μ ν τ τ q = q 11 q 21 q 12 q 22 q 13 q 23 = u d c s t b then the current is j μ = j μ 1 + ij μ 2 — electroweak “isospin” j μ a = ¯ ‘Q a γ μ (1 + γ 5 ) ‘  {z } leptonic j μ ‘a + ¯ q V † Q a γ μ (1 + γ 5 ) V q  {z } hadronic j μ ha Q a = σ a / 2 are isospin generators V = 1 V ¶ V = ˆ V ud V us V ub V cd V cs V cb V td V ts V tb ! = ( V † ) 1 ‘ = ‘ 11 ‘ 21 ‘ 12 ‘ 22 ‘ 13 ‘ 23 = ν e e ν μ μ ν τ τ q = q 11 q 21 q 12 q 22 q 13 q 23 = u d c s t b then the current is j μ = j μ 1 + ij μ 2 — electroweak “isospin” j μ a = ¯ ‘Q a γ μ (1 + γ 5 ) ‘  {z } leptonic j μ ‘a + ¯ q V † Q a γ μ (1 + γ 5 ) V q  {z } hadronic j μ ha Q a = σ a / 2 are isospin generators V = 1 V ¶ V = ˆ V ud V us V ub V cd V cs V cb V td V ts V tb ! = ( V † ) 1 j μ ‘ = ¯ ν e γ μ (1 + γ 5 ) e + ¯ ν μ γ μ (1 + γ 5 ) μ + ¯ ν τ γ μ (1 + γ 5 ) τ j μ h = ( ¯ u, ¯ c, ¯ t ) γ μ (1 + γ 5 ) V ˆ d s b ! j μ ‘ = ¯ ν e γ μ (1 + γ 5 ) e + ¯ ν μ γ μ (1 + γ 5 ) μ + ¯ ν τ γ μ (1 + γ 5 ) τ j μ h = ( ¯ u, ¯ c, ¯ t ) γ μ (1 + γ 5 ) V ˆ d s b ! G F √ 2 j μ j † μ = G F √ 2 ( j μ ‘ j † ‘μ + j μ ‘ j † hμ + j μ h j † ‘μ + j μ h j † hμ ) j μ j † μ — “leptonic weak interactions” — perturbation theory works great j μ h j † hμ — “hadronic weak interactions” — perturbation theory doesn’t work at all for light quarks and is complicated even for heavy quarks — we will talk about this later j μ ‘ j † hμ + j μ h j † ‘μ — “semileptonic weak interactions” — perturbation theory is useful — to lowest order in electroweak ints, amplitudes factor into a leptonic part which is calculable times the matrix element of a hadronic current M = G F √ 2 h ‘ out  j μ ‘  ‘ in ih h out  j † h μ  h in i or h ‘ out  j † ‘μ  ‘ in ih h out  j μ h ...
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This note was uploaded on 02/04/2012 for the course PHY 253B taught by Professor Georgi during the Spring '10 term at Princeton.
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