This is most straighworward for the neutrinos eg 3

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Unformatted text preview: eR Y j µ = −2 eR γ µ eR − ν L γ µν L − eL γ µ eL eL → ν L ν L → eL N.B. We have been discussing things in terms of electrons and electron ­neutrinos, but the same applies to any of the weak ­isospin doublets: ⎛ν e ⎞ ⎛ν µ ⎞ ⎛ν τ ⎞ ⎛ u ⎞ ⎛ c ⎞ ⎛ t ⎞ χL = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ e ⎠ L ⎝ µ ⎠ L ⎝ τ ⎠ L ⎝ d ′⎠ L ⎝ s′⎠ L ⎝ b′⎠ L where primes on the down ­type quarks denote the Cabibbo ­rotated states In each case one can construct three weak isospin currents, j µ = χ Lγ µ τ χ L and one 2 Y em 3 weak ­hypercharge current: j µ = 2 j µ − 2 j µ 1 15 Weak Hypercharge Y = 2Q − 2 I 3 Look at values for members of weak isospin doublet: ⎛ν e ⎞ ⎛ν µ ⎞ ⎛ν τ ⎞ ⎛ u ⎞ ⎛ c ⎞ ⎛ t ⎞ χL = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ e ⎠ L ⎝ µ ⎠ L ⎝ τ ⎠ L ⎝ d ′⎠ L ⎝ s′⎠ L ⎝ b′⎠ L ⎛ν ⎞ ⎜ e⎟ ⎝ e ⎠L ⎛ 2(0) − 2(1 / 2) 2Q − 2 I 3 = ⎜ ⎜ 2( −1) − 2( −1 / 2) ⎝ ⎛u⎞ ⎜ ⎟ ⎝ d′ ⎠ L ⎛ 2( 2 / 3) − 2(1 / 2) 2Q − 2 I 3 = ⎜ ⎜ 2( −1 / 3) − 2( −1 / 2) ⎝ I 3 = +1 / 2 I 3 = −1 / 2 ⎞⎛ −1 ⎞ ⎟ =⎜ ⎟ ⎝ −1 ⎟ ⎠ ⎠ ⎞⎛ 1/ 3 ⎞ =⎜ ⎟ ⎟ ⎝ 1/ 3 ⎟ ⎠ ⎠ e.g. value of...
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This document was uploaded on 02/11/2014.

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