電弱理論(SU(2)L × U(1))をまとめるなど無謀なんですが無理やりまとめ ...

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電弱理論(SU(2)L × U(1))をまとめるなど無謀なんですが無理やりまとめ ...

SU(2)L U(1)

12 I.j.R. Aitchison/

A.J.G.Hey

SU(2)L T U(1) y

Q( e ) t y

Q = t 3 + y/2 (1)

Q e

Higgs

Higgs

(

(1/ √ 2)(

ˆϕ =

ˆϕ 1 + i ˆϕ

) {

2 )

(1/ √ + 1

2)( ˆϕ 3 + i ˆϕ

2

t 3 =

(2)

4 )



( + M 2 )µ − ∂ µ∂ ν  ν = ĵ µ interact (3)

M

ĵ µ = −M 2 A µ + (inteact) (4)


U(1)

ĵ y µ ( ˆϕ) = i(g ′ /2)y[ ˆϕ † (∂ µ ˆϕ) − (∂

µ † ˆϕ) ˆϕ] (5)


SU(2)L

ĵ aµ ( ˆϕ) = ig[ ˆϕ † τ a

2 (∂µ ˆϕ) − (∂

µ † ˆϕ) τ a

2 ˆϕ] (6)



∂ µ → ˆD µ = ∂ µ + ig(τ/2) · Ŵ µ + i(g ′ /2)y ˆB µ (7)


− 1 2

ĵ µ y ( ˆϕ) = i(g ′ /2)y[ ˆϕ † (∂ µ ˆϕ) − (∂

µ ˆϕ)

† ˆϕ] − gg


y ˆϕ † (τ/2) · ˆϕŴµ − (g ′2 /2)y 2 ˆϕ† ˆϕ ˆBµ

(8)


ĵ aµ ( ˆϕ) = ig[ ˆϕ † τ a

2 (∂µ ˆϕ) − (∂

µ † ˆϕ) τ a

2 ˆϕ] − (g 2 /2) ˆϕ † ˆϕŴ aµ − gg ′ y ˆϕ † (τ/2) ˆϕ ˆB µ (9)


ˆϕ

( )

0

ˆϕ =

f/ √ + · · · (10)

2

1


f t 3 = −1/2


Q = t 3 + y/2 (11)

y=1 (9)

ĵ aµ ( ˆϕ) = ig[ ˆϕ † τ a

2 (∂µ ˆϕ) − (∂

µ † ˆϕ) τ a

2 ˆϕ] − (g 2 /2) ˆϕ † ˆϕŴ aµ − gg ′ ˆϕ† (τ/2) ˆϕ ˆB µ (12)


ĵ 1,2µ ( ˆϕ) = −(g ′2 f 2 /4)Ŵ 1,2µ (13)

Ŵ 1,2µ W ±

M w = gf/2 (14)

a=3

ĵ 3µ ( ˆϕ) = −(g ′2 f 2 /4)Ŵ 3µ + (gg ′ f 2 /4) ˆB µ (15)


Ŵ 3µ − ∂ µ∂ ν Ŵ 3ν = −(g ′2 f 2 /4)Ŵ 3µ + (gg ′ f 2 /4) ˆB µ + ĵ 3µ (Ŵ ) (16)

ĵ 3µ (Ŵ ) W

U(1) ˆB µ

ĵ µ y ( ˆϕ) = i(g ′ /2)[ ˆϕ † (∂ µ ˆϕ) − (∂

µ ˆϕ)

† ˆϕ] − gg′ ˆϕ† (τ/2) · ˆϕŴµ − (g ′2 /2) ˆϕ † ˆϕ ˆBµ

(17)



ĵ µ y ( ˆϕ) = (gg ′ f 2 /4)Ŵ 3µ − (g ′2 f 2 /4) ˆB µ (18)

ˆB µ − ∂ µ∂ ν ˆBν = (gg ′ f 2 /4)Ŵ 3µ − (g ′2 f 2 /4) ˆB µ (19)

ˆB µ Ŵ 3µ

g ′ Ŵ 3µ + g ˆB µ (20)

(16) (19)

(g ′ Ŵ 3µ + g ˆB µ ) − ∂ µ∂ ν (g ′ Ŵ 3ν + g ˆB ν ) = g ′ ĵ 3µ (Ŵ ) (21)

W

1112

µ

Ẑ µ = cos θ w Ŵ 3µ − sin θ w ˆBµ

(22)


Ẑµ − ∂ µ∂ ν Ẑ ν = −(f 2 /4)(g ′2 + g 2 )Ẑµ + cos θ w ĵ 3µ (Ŵ ) (23)

Z

M z = 1 2 f(g′2 + g 2 ) 1/2 = M w / cos θ w (24)

2


11.20


ˆD µ = ∂ µ + iqµ (25)

q

ˆD µ = ∂ µ + ig(τ/2) · Ŵ µ + i(g ′ /2)y ˆB µ (26)

ˆD µ = ∂ µ + ig(t (t)

3 ) · Ŵ µ + i(g ′ /2)y ˆB µ (27)

ˆD µ = ∂ µ + ig(t (t)

3 )Ŵ 3µ + i(g ′ /2)y ˆB µ + ig(t (t)

3 )Ŵ 1,2µ (28)

ˆD µ = ∂ µ + ig(t (t)

3 )[sin θwµ + cos θ w Ẑ µ ] + i(g ′ /2)y[cos θ w  µ − sin θ w Ẑ µ ] + ig(t (t)

3 )Ŵ 1,2µ (29)

ˆD µ = ∂ µ + ig sin θ w  µ (t (t)

3 + y/2) + i(g/ cos θ w )Ẑµ [cos 2 θ w t (t)

3 − (y/2) sin 2 θ w ] + ig(t (t)

3 )Ŵ 1,2µ (30)

(Q = t 3 + y/2 )

e = g sin θ w (31)

11.16Z

(11.18)

g z = (g/ cos θ w )[cos 2 θ w t (t)

3 − (y/2) sin 2 θ w ] = (g/ cos θ w )[t 3 − Q sin 2 θ w ] (32)

3

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