t9 - updated Thursday 8 th April, 2010 23:25 Standard model...

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Unformatted text preview: updated Thursday 8 th April, 2010 23:25 Standard model SU (2) U (1) U (1) = 2 1 L ( ,W ) = D D - V ( )- 1 4 W a W a = 1 4 tr ( D D )- V ()- 1 4 W a W a = + 2 = * +- - = i~ ~ T + i S = i~ ~ 2 + i 1 2 = i~ ~ 2 - i 3 2 D = + ig ~ W ~ 2 + ig X 1 2 D = + ig ~ W ~ 2 - ig X 3 2 W a = W a- W a- g abc W b W b V ( ) = 2 ( - v 2 / 2) 2 h | | i = v/ 2 Q v/ 2 = ( T 3 + S ) v/ 2 = 0 D = + iv 2 2 g ( W 1- iW 2 ) gW 3- g X + h | | i = v/ 2 Q v/ 2 = ( T 3 + S ) v/ 2 = 0 D = + iv 2 2 g ( W 1- iW 2 ) gW 3- g X + A = W 3 /g + X /g p 1 /g 2 + 1 /g 2 Z = gW 3- g X p g 2 + g 2 W 1 iW 2 2 A = W 3 /g + X /g p 1 /g 2 + 1 /g 2 couples to T 3 + S p 1 /g 2 + 1 /g 2 = eQ e = 1 p 1 /g 2 + 1 /g 2 A = e g W 3 + e g X = sin W 3 + cos X with g = e sin and g = e cos if e, with g = e/ sin fixed, we recover the SU (2) model with a degenerate triplet ~ W for 6 = 0 m 2 W = e 2 v 2 4sin 2 m 2 Z = e 2 v 2 4 1 sin 2 + 1 cos 2 = e 2 v 2 4sin 2 cos 2 = m 2 W cos 2 as expect the Z and W become degenerate as the coupling of the Z Z = gW 3- g X p g 2 + g 2 couples to g 2 T 3- g 2 S p g 2 + g 2 = gT 3 /g- g S/g p 1 /g 2 + 1 /g 2 = e sin cos ( cos 2 T 3- sin 2 S ) = e sin cos ( cos 2 T 3 + sin 2 T 3- sin 2 T 3- sin 2 S ) = e sin cos ( T 3- sin 2 Q ) using T 3 + S = Q again this has the expected form as e, with g = e/ sin fixed e sin cos ( T 3- sin 2 Q ) g T 3 part of the SU (2) triplet with the massive W 1 and W 2 moral the U (1) gauge coupling breaks the SU (2) R global symmetry of the field (Yukawa coupling to fermions also break the symmetry) - but the symmetry is still important because the U (1) coupling is small can treat g as a spurion Standard model SU (2) U (1) U (1) = 2 1 L ( ,W ) = D D - V ( )- 1 4 W a W a = 1 4 tr ( D D )- V ()- 1 4 W a W a = + 2 = * +- - = i~ ~ T + i S = i~ ~ 2 + i 1 2 = i~ ~ 2 - i 3 2 D = + ig ~ W ~ 2 + ig X 1 2 D = + ig ~ W ~ 2 - ig X 3 2 = ( h + v ) / 2 in unitary gauge = ( h + v ) I so Higgs couplings to other particles are tied to v dependence anything that gets its mass from v automatically gets Higgs coupling flavor the good news and bad news about u R , d R and e R are 1...
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t9 - updated Thursday 8 th April, 2010 23:25 Standard model...

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