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)