1 Вклад глюонного конденсата в поляризационный оператор

Ð.Í. Ëè
22 èþíÿ 2008 ã.
1
1.1
Âêëàä ãëþîííîãî êîíäåíñàòà â ïîëÿðèçàöèîííûé îïåðàòîð
Îïåðàòîðíîå ðàçëîæåíèå ïðîèçâåäåíèÿ òîêîâ
Âñëåäñòâèå êàëèáðîâî÷íîé èíâàðèàíòíîñòè, îïåðàòîðíîå ðàçëîæåíèå ïðîèçâåäåíèÿ òîêîâ
Z
i
µ
ν
dxeiqx TjH
(x) jH
(0) = q µ q ν − q 2 g µν
+ q µ qρ1 − gρµ1 q 2
q ν qρ2 − gρν2 q 2
(1)
i Ciρ1 ...ρNi
P
(2)
i Ciρ3 ...ρNi
P
(1)ρ1 ...ρNi
(q) Oi
(2)ρ1 ...ρNi
(q) Oi
,
(1)
(2)
ãäå Oi... âñåâîçìîæíûå îïåðàòîðû, à Ciρ(1,2)...ρ (q) íåêîòîðûå òåíçîðíûå îïåðàòîðû. Ëåãêî ïîíÿòü, ÷òî îïåðàòîðû Oi(1)... ìîæíî
âûáèðàòü ñèììåòðè÷íûìè è áåññëåäîâûìè ïî ëþáîé ïàðå èíäåêñîâ, à îïåðàòîðû Oi(2)... ñèììåòðè÷íûìè è áåññëåäîâûìè
ïî ïàðå ρ1 , ρ2 è ïî ëþáîé ïàðå èç ρ3 , . . . , ρN .
Èíäåêñ H â ëåâîé ÷àñòè îòìå÷àåò îïåðàòîðû â ãåéçåíáåðãîâñêîì ïðåäñòàâëåíèè. Ïåðåõîäÿ â ïðåäñòàâëåíèå âçàèìîäåéñòâèÿ, èìååì
Z
1
Ni
i
µ
ν
TjH
(x) jH
(0) = Tj µ (x) j ν (0) exp [iSI ] ,
SI =
d4 xLI
ãäå LI ëàãðàíæèàí âçàèìîäåéñòâèÿ. Ñ÷èòàåì, ÷òî q áîëüøîé ïðîñòðàíñòâåííî-ïîäîáíûé èìïóëüñ, òàê ÷òî Q2 = −q2 Λ2QCD . Åñëè ìû áåðåì ñðåäíåå îò ýòîãî ðàâåíñòâà ïî ñîñòîÿíèþ, íå èìåþùåìó âûäåëåííîãî íàïðàâëåíèÿ, íàïðèìåð, ïî
âàêóóìíîìó, òî ìîæíî èñïîëüçîâàòü áîëåå ïðîñòîé âèä
Z
i
µ
ν
dxeiqx hTjH
(x) jH
(0)i = q µ q ν − q 2 g µν
P
(1)
i Ci
q2
D
(1)
Oi
E
,
(3)
òî åñòü, ôàêòè÷åñêè, ðàñêëàäûâàòüñÿ ïî íàáîðó ñêàëÿðíûõ îïåðàòîðîâ. Ýòî ñëåäóåò èç òîãî, ÷òî ñðåäíåå îò òåíçîðíîãî îïåðàòîðà â ýòîì ñëó÷àå âûðàæàåòñÿ ÷åðåç èíâàðèàíòíûå òåíçîðû (ñîñòàâëåííûå èç gαβ ), óìíîæåííûå íà ñðåäíèå îò ñêàëÿðíûõ
îïåðàòîðîâ. Ïðè ýòîì èç ñîîáðàæåíèé ðàçìåðíîñòè ÿñíî, ÷òî âêëàä îïåðàòîðîâ áîëüøåé ðàçìåðíîñòè ïîäàâëåí ñòåïåíüþ
Q2 . Ïîýòîìó, ïðè äîñòàòî÷íî áîëüøèõ Q2 ãëàâíûé âêëàä äàåò åäèíè÷íûé îïåðàòîð. Îïåðàòîð G2 äàåò âêëàä, ïîäàâëåííûé Êîýô
êàê Q−4 . Íóæíî èìåòü â âèäó, ÷òî â ÊÝÄ ðàñõîäÿùàÿñÿ ÷àñòü ïîëÿðèçàöèîííîãî îïåðàòîðà îïðåäåëÿåò ïåðåíîðìèðîâêó CI q
çàðÿäà, ïîýòîìó ÷òîáû ïîëó÷èòü ðàçëîæåíèå ïåðåíîðìèðîâàííûõ òîêîâ, íóæíî ó÷èòûâàòü êîíòð÷ëåíû.
òîì
ïîïð
ëåãê
çèòü
R
1
1.2
Êîýôôèöèåíòíàÿ ôóíêöèÿ
CG2 (q 2 )
×òîáû âû÷èñëèòü êîýôôèöèåíòíóþ ôóíêöèþ ïðè G2 , âû÷èñëèì ìàòðè÷íûé ýëåìåíò ëåâîé è ïðàâîé ÷àñòè (3) ïî ñîñòîÿíèþ,
ñîîòâåòñòâóþùåìó ïîñòîÿííîìó ãëþîííîìó ïîëþ, ïðè÷åì óñðåäíèì ýòîò ìàòðè÷íûé ýëåìåíò ïî íàïðàâëåíèÿì òàê, ÷òî
D
E [g g − g g ] D=4
αβ γδ
αδ γβ
Gaαγ Gaβδ =
Gaσρ Gaσρ =
D (D − 1)
Ñïðàâà ìû èìååì
1
12
[gαβ gγδ − gαδ gγβ ] hG2 i
q µ q ν − g µν q 2 CG2 q 2 hG2 i
Ñëåâà ìû äîëæíû îïóñòèòü èç exp [iSI ] äâå êâàðê-ãëþîííûå âåðøèíû. Ïîëó÷àåì ñëåäóþùåå âûðàæåíèå ñëåâà:
Z
Z
Z
E
D
LHS = − 2!i g 2 dx dy dz eiqx Tj µ (x) j ν (0) j aα (y) j bβ (z) TAaα (y) Abβ (z)
Z
Z
Z
E
D
= − 4i g 2 dx dy dz eiqx Tj µ (x) j ν (0) j α (y) j β (z) TAaα (y) Aaβ (z)
Çäåñü j µ (x) = ψ̄γ µ ψ, j aµ (x) = ψ̄γ µ ta ψ. Ïî-êðàéíåé ìåðå, â ñëó÷àå àáåëåâîãî öâåòíîãî ïîëÿ, ìû ìîæåì èñïîëüçîâàòü êàëèáðîâêó
Aaα
Z1
(y) = −
dλλGaαβ (λy) y β
0
Óïðàæíåíèå
Ïîêàçàòü (â ñëó÷àå ýëåêòðîäèíàìèêè), ÷òî ôîðìóëà
Z1
Aµ (y) = −
dλλFµν (λy) y β
0
îïðåäåëÿåò âåêòîð-ïîòåíöèàë Aµ (y), ñîîòâåòñòâóþùèé ïîëþ Fµν (x), åñëè ïîëå óäîâëåòâîðÿåò óñëîâèþ (âòîðîé ïàðå óðàâíåíèé Ìàêñâåëëà)
∂ µ Feµν =
1
εµνσρ ∂ µ F σρ = 0
2
Ðåøåíèå
Z1
∂µ Aν (y) − ∂ν Aµ (y) =




Fµν,γ (λy)
Z1
z
}|
{
0


 γ 
dλλ 2Fµν (λy) + Fγν,µ (λy) + Fµγ,ν (λy) y λ = dλ λ2 Fµν (λy) = Fαβ (y)
0
0
Èñïîëüçóÿ ýòó êàëèáðîâêó, ìû ïîëó÷àåì
Z1
D
E Z1
D
E
0
0
a
a
TAα (y) Aβ (z) = dλ λ dλ0 λ0 y α z β Gaαα0 (λy) Gaββ 0 (λ0 z)
0
0
Z1
Z1
=
dλ λ
0
D
E
0
0
dλ0 λ0 y α z β Gaαα0 (0) Gaββ 0 (0) =
0
2
1
48
0
0
[gαβ gα0 β 0 − gαβ 0 gα0 β ] y α z β hG2 i
Ïîýòîìó
LHS =
=
ig 2
hG2 i
− 192
Z
Z
dx
ig 2
2
0 0
192 hG i [gαβ gα β
Z
0
0 dz eiqx [gαβ gα0 β 0 − gαβ 0 gαβ 0 ] y α z β Tj µ (x) j ν (0) j α (y) j β (z)
Z
Z
Z
µ
∂
∂
iqx+ik1 y+ik2 z
ν
α
β
0
0
dx dy dz e
Tj (x) j (0) j (y) j (z) − gαβ gαβ ] α0 β 0
∂k1 ∂k2
k1,2 =0
dy
Íà ýòîì ýòàïå ìû âèäèì, ÷òî çàäà÷à ñâåëàñü ê âû÷èñëåíèþ ïðîèçâîäíîé îò ÷åòûðåõõâîñòêè ïðè äâóõ âòåêàþùèõ èìïóëüñàõ ðàâíûõ íóëþ (ñì. Ðèñ. 1). Èìååì äâå ñóùåñòâåííî ðàçíûå äèàãðàììû. Âêëàä äèàãðàììû ñ äâóìÿ âñòàâêàìè ñ îäíîé
ñòîðîíû íóæíî óìíîæèòü íà ÷åòûðå (åñòü çàìåíà k1 ↔ k2 è ðàçíûå íàïðàâëåíèÿ â ôåðìèîííîé ïåòëå), Âêëàä äèàãðàììû
ñ âñòàâêàìè ñ ðàçíûõ ñòîðîí óìíîæàåì íà äâà. Êîíå÷íî, áëàãîäàðÿ êàëèáðîâî÷íîé èíâàðèàíòíîñòè, äîñòàòî÷íî âû÷èñëèòü
ñëåä ïî µ, ν , íî ìû, â öåëÿõ ïðîâåðêè, âû÷èñëèì âåñü òåíçîð.
1.2.1
Âû÷èñëåíèå ïðîèçâîäíûõ
×òîáû âû÷èñëèòü ïðîèçâîäíûå ïî k1 , k2 , ïîëüçóåìñÿ ñëóäóþùèì ïðàâèëîì
∂
∂k1β
0
G (p − k1 ) = G (p − k1 ) γβ 0 G (p − k1 ) ,
G (p) = [p̂ − m]
−1
Ââåäåì òàêæå, äëÿ ñîêðàùåíèÿ çàïèñè, îáîçíà÷åíèÿ
µναββα = Tr γ µ G0 γ ν Gγα Gγβ Gγ β Gγ α G ,
µβαναβ = Tr γ µ G0 γ β G0 γ α G0 γ ν Gγα Gγβ G ,
G = G (p) ,
G0 = G (p0 ) ,
p0 = p − q
òî åñòü áóäåì çàïèñûâàòü â ñòðîêó èíäåêñû γ -ìàòðèö, ñòîÿùèõ ìåæäó ïðîïàãàòîðàìè, ïðè÷åì, îò µ äî ν ïðîïàãàòîðû
çàâèñÿò îò p0 , à îò ν äî µ îò p. Ïîëó÷àåì
ôåðìèîí.
ïåòëÿ
êîìáèí.
ìíîæèòåëü
M1 =
z}|{
4
i
4
= −4 [gαβ gα0 β 0
µ
0
ν
β
α [gαβ gα0 β 0
4 Tr G (p) γ G (p ) γ G (p + k1 + k2 ) γ G (p + k1 ) γ (2π)
k1,2 =0
Z
Z
4
4
d p
d p
0 0
0 0
0
0
− gαβ 0 gαβ 0 ]
4 [µνα β βα + µνβ α βα + µνβ βα α] = 8
4 [µναββα − µνααββ]
(2π)
(2π)
z }| {
(−1)
∂
∂
− gαβ 0 gαβ 0 ]
0
0
α
∂k1 ∂k2β
Z
d4 p
ôåðìèîí.
ïåòëÿ
êîìáèí.
ìíîæèòåëü
Z
∂
∂
d4 p
µ
0
β
0
ν
α M2 =
i
[gαβ gα0 β 0 − gαβ 0 gαβ 0 ]
0
0
4 Tr G (p) γ G (p ) γ G (p − k2 ) γ G (p + k1 ) γ ∂k1α ∂k2β
(2π)
k1,2 =0
Z
Z
4
4
d p
d p
0
0
= 2 [gαβ gα0 β 0 − gαβ 0 gαβ 0 ]
4 [µββ να α] = −2
4 [µαβναβ − µαβνβα]
(2π)
(2π)
z}|{
2
4
z }| {
(−1)
3
1.2.2
Âû÷èñëåíèå ñëåäîâ
Òåïåðü ìû äîëæíû âû÷èñëèòü ñëåäû. Èñïîëüçóåì
2
γ (p̂−m)γ −γ (p̂−m)γ
2
p̂−p +2m
α
β
β
, Gγ β Gγβ = 2 m(p
Gγ[α Gγβ] G = α
2 −m2 )2 G,
(p2 −m2 )2
β
α
2m
− γ α p̂] G
Gγα Gγ Gγβ , Gγ α G = Gγα G Gγ β Gγβ , γ α G = (p2 −m
2 )2 Gγα G [p̂γ
=
=
4m
4m
Gγα G [pα − γ α p̂] G = (p2 −m
2 )3 G [p̂ (p̂ + m)
(p2 −m2 )2
12m
G p2 − mp̂ (p̂ + m) = (p212m
mp̂ + p2
(p2 −m2 )4
−m2 )4
Ïîëó÷àåì
d4 p Tr γ µ (p̂0 + m) γ ν mp̂ + p2
Z
M1 = 96m
= 96 · 4m2
4
(2π)
d4 p p0µ pν + pµ p0ν + g µν (pq)
Z
4
(p02 − m2 ) (p2 −
+ 2 (p̂ − 2m) p̂] G
4
m2 )
(p02 − m2 ) (p2 − m2 )
= 96 · 4 · 4m2
Z
dx x̄3
Z
4
d4 p 2pµ pν − 2xx̄q µ q ν + xg µν q 2
4
5
(p2 − m2 + xx̄q 2 )
µν
2
µ ν
6 2
g
m − x (2 + x̄) q + 4xx̄q q
i2 m
=
dx x̄3
= x̄3 + x3 = 1 − 3x x̄, x̄2 + x2 = 1 − 2x x̄
2
3
2
2
(4π)
(m − xx̄q )
Z
m2 g µν + 4xx̄q µ q ν (1 − 3x x̄) − xx̄g µν q 2 (3 − 7x x̄) m→0
i 25 2
i 27
µν 2
µ ν
=
m
dx
→
2
3
2 4 g Q +q q
2
2
(4π)
(m − xx̄q )
(4π) Q
(2π)
(2π)
2
Z
(4)
Îáðàòèì âíèìàíèå, ÷òî åñëè áû ìû âû÷èñëÿëè M1 ñðàçó ïðè íóëåâîé ìàññå, ìû ïîëó÷èëè áû òàêèì îáðàçîì íîëü, ïðîñòî
çà ñ÷åò àëãåáðû γ -ìàòðèö. Îäíàêî, ïðàâèëüíûé îòâåò, êàê ìû âèäèì, îòëè÷åí îò íóëÿ.
d4 p
Z
M2 = −2
4
(2π) (p2 −
2
m2 )
(p02 −
2 Tr
m2 )
ν α
γ γ (p̂ − m) γ β γ µ (γα (p̂0 − m) γβ − γβ (p̂0 − m) γα )
d4 p
Z
µν 0
ν µ
µ
0
ν
4
2
2 Tr [(p̂ − m) (−8g p̂ − 4mγ γ − 4γ (p̂ − m) γ )]
(2π) (p2 − m2 ) (p02 − m2 )
Z
d4 p
µ 0ν
0µ ν
µν
=2·4·4
(pp0 )]
4
2
2 [p p + p p + g
(2π) (p2 − m2 ) (p02 − m2 )
Z
Z
d4 p
µ ν
µν
= 2 · 4 · 4 · 6 dx xx̄
3p2 /2 − xx̄q 2
4
4 −2xx̄q q + g
2
2
2
(2π) (p − m + xx̄q )
Z
µν
m→0 −i 26 1 µν 2
i 25
dx xx̄
=
g
2xx̄q 2 − 3m2 − 2xx̄q µ q ν →
g Q + qµ qν
2
2
2
2
(4π)
(m2 − xx̄q 2 )
(4π) (q 2 )
= −2
1.2.3
Ðåçóëüòàò
Îêîí÷àòåëüíî ïîëó÷àåì
LHS =
g2
− 192
(
26
2
(4π)
Q4
12m4 + 4m2 Q2 + Q4
(4m2
+
2
Q2 )
− 24
m4 2m2 + Q2
Q (4m2
4
+
5/2
Q2 )
ArcTanh
√ Q2 2
4m +Q
)
Q2 g µ,ν + q µ q ν hG2 i
(5)
Èòàê, êîýôôèöèåíò CG
2
q2
â îïåðàòîðíîì ðàçëîæåíèè (1) ðàâåí
αs
CG2 q 2 = − 48πQ
4
1
3
2
3a2 − 2a + 3 − 5/2 (a − 1) (a + 1)
2
a
a
a = 1 + 4m2 /Q2
5
ArcTanh
√ 1/ a ,
(6)