t4 - updated Tuesday 2 nd March, 2010 11:29 4. symmetries,...

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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) Gell-Manns SU (3) symmetry (c) non-Abelian flavor symmetries and QFT Lagrangians (d) Noethers theorem and conserved flavor currents (e) EFT and the Ademollo-Gatto Theorem current-current weak interactions from W exchange G F 2 j j j is the charged, left-handed weak iso-spin current more tensor products charge and family # = 11 21 12 22 13 23 = e e- - - q = q 11 q 21 q 12 q 22 q 13 q 23 = u d c s t b then the current is j = j 1 + ij 2 electroweak isospin j a = Q a (1 + 5 ) | {z } leptonic j a + q V Q a (1 + 5 ) V q | {z } hadronic j ha Q a = a / 2 are isospin generators V = 1 V V = 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 = e e- - - q = q 11 q 21 q 12 q 22 q 13 q 23 = u d c s t b then the current is j = j 1 + ij 2 electroweak isospin j a = Q a (1 + 5 ) | {z } leptonic j a + q V Q a (1 + 5 ) V q | {z } hadronic j ha Q a = a / 2 are isospin generators V = 1 V V = V ud V us V ub V cd V cs V cb V td V ts V tb ! = ( V )- 1 j = e (1 + 5 ) e- + (1 + 5 ) - + (1 + 5 ) - j h = ( u, c, t ) (1 + 5 ) V d s b ! j = e (1 + 5 ) e- + (1 + 5 ) - + (1 + 5 ) - j h = ( u, c, t ) (1 + 5 ) V d s b ! G F 2 j j = G F 2 ( j j + j j h + j h j + j h j h ) j j leptonic weak interactions perturbation theory works great j h j h hadronic weak interactions perturbation theory doesnt work at all for light quarks and is complicated even for heavy quarks we will talk about this later j j h + j h j semi-leptonic 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 | in ih h out | j h | h in i or h out | j | in ih h out | j h |...
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t4 - updated Tuesday 2 nd March, 2010 11:29 4. symmetries,...

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