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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) GellManns SU (3) symmetry (c) nonAbelian flavor symmetries and QFT Lagrangians (d) Noethers theorem and conserved flavor currents (e) EFT and the AdemolloGatto Theorem currentcurrent weak interactions from W exchange G F 2 j j j is the charged, lefthanded weak isospin 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 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  in ih h out  j h  h in i or h out  j  in ih h out  j h ...
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 Spring '10
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