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Квантовая теория поля

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SU(N)

				δAµ =		1	Dµωa,
				δAµ =			Dµωa,
						g
−	1	(∂µAµa )2
−	2ξ	(∂µAµa )2			= −	ξg Z		d4xωa(Dµ∂µηa), ηa ≡ ∂µAµa
δS = δ Z d4x			−2ξ (∂µAµa )2		= −	ξg Z		d4xωa(Dµ∂µηa), ηa ≡ ∂µAµa	,
			1			1

δS

Dµ∂µηa = ηa + gfabcAbµ∂µηc = 0.

ηa	Aa
	µ

Φ†a

Dµ∂µΦ†a = Φ†a + gfabcAbµ∂µΦ†c = 0,

•	iδab/(p2 + i0)

c p

•

a µ

−gfabcpµ

αs = g2/(4π)

	∞ dp
	I(a) = Z−∞		(f(p + a) − f(p)),
	I(a) = Z−∞	2π	(f(p + a) − f(p)),
a
p+a = p′
	I(a)
						I(a) 6= 0	R dpf(p)
	f(p + a)					a
			1		a2(f′(∞) − f′(−∞)) + . . .
	I(a) = a(f(∞) − f(−∞)) +
R	I(a) = a(f(∞) − f(−∞)) +			2
R	1/p		f(n)(±∞) = 0			I(a) = 0
dpf(p)						f(p)
R dpf(p)					f(p)		p → ±∞
	I(a) = a(f(∞) − f(−∞)),
					f(n)(±∞) = 0

d4p

aµ

∂

I(a) =

(f(p + a) − f(p)) =

f(p) + . . . =

(2π)4

∂pµ

aµ

f(Λ)dΣ

iaµ

lim Λ

Λ2f(Λ).

(2π)4 ZΛ→∞

8π2

Λ→∞ µ

		µ		µ
p		k1	p	k1
λ	p − k1	λ		p − k2
q			q
p − q		k2	p − q	k2
p − q		ν	p − q	ν

¯	¯	¯	ψ(x),
jµ(x) = eψ(x)γµψ(x), j5µ(x) = ψ(x)γµγ5		ψ(x), P (x) = ψ(x)γ5

ψ ψ

∂µjµ(x) = 0, ∂µj5µ(x) = 2imP (x).

Tµνλ(k1, k2, q) = i d4x1d4x2h0|T (jµ(x1)jν (x2)j5λ(0))|0ieik1x1+ik2x2 ,

Tµν (k1, k2, q) = i d4x1d4x2h0|T (jµ(x1)jν (x2)P (0))|0ieik1x1+ik2x2 ,

q = k1 + k2

γλγ5

γ5

p → p + a

					µνλ = Tµνλ(a) − Tµνλ(0),
a
					a = αk1 + (α − β)k2,
α β					a = 0
					µνλ
µνλ =	−	e2	Z	d4p	Sp [S(p + a)γλγ5S(p + a − q)γνS(p + a − k1)γµ]
µνλ =	−	e2	Z	(2π)4	Sp [S(p + a)γλγ5S(p + a − q)γνS(p + a − k1)γµ]
						k1	↔	k	.
	− Sp [S(p)γλγ5S(p − q)γν S(p − k1)γµ] +					µ	↔	ν2	.
							↔

µνλ

µνλ(1) (k1, k2) =

(αk1 + (α − β)k2)αεαµνλ,

8π2

γ5

k1 ↔ k2 µ ↔ ν

Tµνλ

Tµνλ(β) = Tµνλ(0) − β

(k1 − k2)αεαµνλ.

8π2

k1µTµνλ(0)

k1µ

= S

−1

(p − k2) − S

−1

= S

−1

(p) − S

−1

(p − k1),

(p − q), k1

k1µTµνλ(0) = 8π2 ενλαβ k1αk2β,

e2(1 + β)

k1µTµνλ(β) = 8π2 ενλαβk1αk2β.

qγˆ 5 = γ5S−1(p − q) + S−1(p)γ5 + 2mγ5,

		qλTµνλ(β)
q T (β) = 2mT	µν −	e2	(1 − β)	ε		k		k		,
q T (β) = 2mT			(1 − β)	ε		k		k		,
λ µνλ			4π2		νλαβ		1α		2β

Tµν

Tµνλ

k1µTµνλ = 0, qλTµνλ = 2mTµν ,

β = −1

k1µTµνλ				qλTµνλ
ενλαβk1αk2β			e2
qλTµνλ = 2mTµν −				εµναβk1αk2β,
qλTµνλ = 2mTµν −			2π2	εµναβk1αk2β,
					˜
					F F
∂µj5µ = 2imP +	e2				e2		˜
∂µj5µ = 2imP +	(4π)2	εµναβ Fµν Fαβ = 2imP +			8π2	Fµν Fµν ,

Tµνλ

γ5

D				u ↔ d		u d
				u ↔ d	SU(2)
ˆa		1	a
ˆa	=	1	a	a = 1, 3
SU(2) T	=	2 τ		a = 1, 3
		τa			u(x)	,
j5aµ(x) = q¯(x)γµγ5			q(x), q(x) =		d(x)
j5aµ(x) = q¯(x)γµγ5		2	q(x), q(x) =		d(x)

jµ(x) = eq¯(x)γµQqˆ (x), Qˆ = Tˆ3					+ 2Yq =			0 −1/3					,
					1		2/3			0
Yq = 1
3
NC = 3								D
D = NC Sp QQˆ ˆ	τ3	=	3	Sp(Qˆ2τ3) =		3	4	−	1	=	1	.
D = NC Sp QQˆ ˆ	2	=	2	Sp(Qˆ2τ3) =		2	9	−	9	=	2	.

πa(x)

h0|j5aµ(x)|πb(p)i = −ifπpµδabe−ipx, fπ

							h0\|∂µj5aµ\|πb(p)i = fπmπ2 δab.
				a = 3				π0
π	±	= (π	1	2	)/	√
	±		1	2		√
				± iπ		2

π0 → γγ

j53µ π0

∂µj5aµ = fπm2ππa,

2imP a(x) = fπm2ππa(x).

a3 De2

∂µj5µ

= fπmππ

+ δ

Fµν Fµν = fπmππ

+ δ

εµναβ Fµν Fαβ.

8π2

32π2

π0 → γγ

T π0(q) → γ(k1, ε1)γ(k2, ε2) = e2 (q2)ε1µε2ν εµναβk1αk2β,

(m2π)

		q → 0	(mπ2 ) ≈ (0)
		mπ	= 0
iqµhγ(k1, ε1)γ(k2, ε2)\|j53µ(q)\|0i = −	e2
		εµναβε1µε2ν k1αk2β ,
	4π2	εµναβε1µε2ν k1αk2β ,

q → 0

i/q2

hγ(k1, ε1)γ(k2, ε2)|j53µ(q)|0i = (−ifπqµ) · qi2 · iT π0(q) → γ(k1, ε1)γ(k2, ε2) ,

iqµ

−ifπqµ

T π0

(q) →

γ(k1, ε1)γ(k2, ε2)

(0)

(0) =

4π2fπ

T (π0

→ γγ) =

εµναβ ε1µε2ν k1αk2β =

εµναβ ε1µε2ν k1αk2β,

4π2fπ

πfπ

π0 → γγ

α2m3

(π

→ γγ) =

64π3fπ2

cτ

τ = ~/

cτ ≈ 25.8

25.1

ϕ2A2

		T = e2εµεν tµν ,
t = 2g	µν −	(2p1 + k1)µ(2p2 + k2)ν	+	(2p2 − k1)µ(2p1 − k2)ν	.
		2(p1k1)
µν				2(p2k1)

tµν k1µ = tµν k2ν = 0.

e+e−

|Tif |2 = 128π2α2

+ u

m2 + s

+ 2

# .

−

|Tif |2 = 32π2α2(1 + cos2 θ).

dΩ

= 2rs2

ω1

θ),

= m.

(1 + cos2

dσ

ω2

dΩ

ω1 ≈ ω2

= 2rs2(1 + cos2 θ),

σ =

3 rs2.

dσ

8π

e+e−

Tif = −(p1 − p2)2 u¯(p2)γµu(p1)(P1 + P2)µ,

p1 p2

P1 P2

s = (p1 + P1)2 = (p2 + P2)2, t = (P2 − P1)2 = (p1 − p2)2, u = (p1 − P2)2 = (P1 − p2)2

				=	πα	2
					πα
			\|Tif \|2		4	t(4M2 − t) + (s − u)2 ,
			\|Tif \|2		t	t(4M2 − t) + (s − u)2 ,
M						M → ∞
						M → ∞
	e+e− → ϕ+ϕ−
		1
•	\|Tif \|2	1
•	\|Tif \|2	2

eµ

•

	p1 → p−, p2 → −p+, P1 → −P−, P2 → P+,
p− p+	P+ P−

s = (p− − P−)2, t = (p− + p+)2, u = (p− − P+)2.

|Tfi|2

e+e− →

ϕ+ϕ−

σ(e+e− → ϕ+ϕ−) = πα2 v3,

3ε2

v ε ε = E− +E+ e+e− → ϕ+ϕ−

e+e− → µ+µ−

v 1 v3

v 1

T =

−ie

(¯u(p

)γ

u(p

))

µ|

p1 p2

P1 P2

q = p1 − p2 = P2 − P1

hP2|Jµ|P1i

u¯2(−ieγµ)u1

h | | i ¯ −

P2 Jµ P1 = U(P2)( ie µ)U(P1)µ

q = p1 − p2 = P2 − P1, p = p1 + p2, P = P1 + P2, Q2 ≡ −q2 = −(p1 − p2)2,

(qP ) = (qp) = 0		4 × 4	µ
1, γµ, γ5, γµγ5, σµν =	1	(γµγν − γνγµ).

	2

	¯				¯
	U(P2)γ5U(P1)				U(P2)γµγ5U(P1)
					¯
	qµ				U(P2)(−ie µ)U(P1)
	qµ				q2 6= 0
	µ = α(q)γµ −	β(q)
				σµν qν ,
		2M		σµν qν ,
M	α(q)			β(q)
q2
	V (q) = eα(q)A0(q) − e		β(q)		(σH) ,

			2M
α(q)	β(q)

Z Z

α(q2 = 0) = α(x)d3x = 1, β(q2 = 0) = β(x)d3x = 2.79

16π2α2

|Tif |2 =

aµν Aµν ,

aµν =

Sp [(ˆp2 + m)γµ(ˆp1 + m)¯γν ] , γ¯ν ≡ γ0γν†γ0

= γν ,

Aµν =

Sp h(Pˆ2

+ M) µ(Pˆ1 + M)¯ν i , ¯ν ≡ γ0 ν† γ0 = ν .

β(q)

µ =

α(q)γµ −

(Pµ − 2Mγµ) = (α + β)γµ

−

Pµ.

aµν = pµpν	− Q2 gµν −			q2	,
				qµqν
Aµν = α2	+	β2Q2	PµPν − (α + β)2Q2			gµν −	qµq	ν	.
Aµν = α2	+	4M2	PµPν − (α + β)2Q2			gµν −	q2		.

aµν Aµν = 16E2s(α2 + ηβ2) cos2 2θ + 32η(α + β)2M2 sin2 θ2,

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