φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x = 2 + + 4 + 4 + a log a, 4 + 4a d 2ω q ΓA ω =, 2π 2ω q 2 + M 2 + 2qp A 4π ω ΓA M 2 p 2 A ω 5 d 2ω q q µ ΓA ω p µ =. 2π 2ω q 2 + M 2 + 2qp A 4π ω ΓA M 2 p 2 A ω 6 D a D a k k = Γa + + a k Γa Γa k Feynman δ x x k x a x a k k dx dx k. 7 x D + x k D k a + +a k 2 Feynman φ 4 Lagrangian L = 2 φ2 + 2 m2 φ 2 + 4! gµε φ 4 8 : d = 2ω, ε = 2ɛ, d = 4 ε, ω = 2 ɛ. 9 : m 2 = m2 4πµ, α = g 2 4π. 2
d ω ε ɛ g α m 2 m 2 ω, ɛ, α ε m, µ 2 g Dφ e R d d x L proper vertex Feynman E I V L Feynman Feynman L d Ld E, I, V, L L = I V +, 4V = E + 2I. φ 4 4-valent proper vetex Γ E E = 2, 4 I V loop L { L E = 2, V = L + E = 4 Feynman. 2. L. gµ ε /4! 4. q G q = q 2 + m 2 5. q d 2ω q 2π 2ω 6. Feynman Wick Proper vertex Feynman Γ 2 p Σp Gp G p = p 2 +m 2 Gp = G p + G pσpg p + G pσpg pσpg p + = G p ΣpG p Γ 2 p Γ 2 p = G p Σp = p 2 + m 2 Σp 2 Σp 2
. Tadpole -loop p 4 gµ 2ɛ φ 4 /4! 4 /4! = /2 n 9 Σp tadpole = d 2ω l 2 gµ2ɛ 2π 2ω l 2 + m. 2 p 5 Σp tadpole = Γ ω 2 gµ2ɛ 4π ω m 2 = g 4πµ 2 ɛ ω 2 m2 Γ + ɛ 4π 2 m 2 = 2 m2 α m 2 ɛ Γ + ɛ ɛ Σp tadpole = m 2 α 2ɛ + ɛψ2 log m 2 + Oɛ 2 4.2 Double scoop 2-loop 4 p 4 4 2 /4! 2 = /4 Σp d-scoop = 4 gµ2ɛ 2 d 2ω l 2π 2ω l 2 + m 2 p Tadpole Σp d-scoop = Σp tadpole 2 gµ2ɛ d 2ω q 2π 2ω q 2 + m 2. 5 2 d 2ω q 2π 2ω q 2 + m 2 2 5 d 2ω q 2 gµ2ɛ 2π 2ω q 2 + m 2 = Γ2 ω 2 2 gµ2ɛ 4π ω m 2 2 ω = 2 α m2 ɛ Γɛ = α + ɛψ log m 2 + Oɛ 2 7. 2ɛ 6
Σp d-scoop = m 2 α2 4ɛ 2 + ɛψ2 + ψ 2 log m 2 + Oɛ 2 8. Sunset 2-loop p l q 4 2 4 p + q l 2 4 2 /4! 2 = /6 Σp sunset = gµ2ɛ 2 I, 9 6 d 2ω l d 2ω q I = 2π 2ω 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2. 2 ɛ Σp sunset = α2 m 2 6 2ɛ + m2 2 ɛ 2 + ψ log m2 + 4ɛ p2 + Oɛ. 2 I = 4ω lµ l µ + q µ q µ µ sum 22 I = d 2ω l d 2ω q l 4ω 2π 2ω 2π 2ω µ + q µ l µ q µ l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2. 2 4 2l 2 l 2 + m 2 2 q 2 + m 2 [q + p l] 2 + m 2 + 2ll p q l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 2 2q 2 + l 2 + m 2 q 2 + m 2 2 [q + p l] 2 + m 2 + 2qq + p l l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2. 2 24 l p + q l q p + q l q, l 4p + q l 2 + 2l qp + q l l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 = 6p + q l2 6m 2 + 6m 2 + 2pp + q l 2 l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 2 6 = l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 + 6m 2 + 2pp + q l l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2. 2 4
2 I 2 I = d 2ω l d 2ω q 6m 2 + 2pp + q l 6I +. 25 4ω 2π 2ω 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 2 d 2ω l Kp = 2π 2ω d 2ω l K µ p = 2π 2ω d 2ω q 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2, 2 26 d 2ω q p + q l µ 2π 2ω l 2 + m 2 q 2 + m 2 [q + p l] 2 + m 2 2 27 25 9 Σp sunset = 2ω µ-sum gµ 2ɛ 2 m 2 Kp + p µ K µ p 28 6 Kp 26 l p + q l l 2 + m 2 [q + p l] 2 + m 2 q 2 + m 2 2 Feynman 7 l l + xp + q d 2ω l d 2ω q dx Kp = 2π 2ω 2π 2ω q 2 + m 2 2 l 2 + m 2 + p + q 2 x x 2 Γ2 ω d 2ω q = dx 4π ω 2π 2ω q 2 + m 2 2 m 2 + p + q 2 x x. 2 ω l 5 x x 2 ω Feynman q Kp = = Γ4 ω 4π ω d 2ω q 2π 2ω Γ4 2ω 4π 2ω = Γ2ɛ 4π 4 2ɛ dxx x ω 2 dyy ω y [ q 2 + p 2 y y + m 2 y + dxx x ω 2 dyy ω y dxx x ɛ dyy +ɛ y Z = Zx, y = p 2 y y + m 2 y + y ] ω 4 x x [ q 2 + p 2 y y + m 2 y + [ p 2 y y + m 2 y + y x x y ] 2ω 4 x x y ] 2ɛ. x x y +ɛ = d ɛ dy yɛ y Kp = Γ2ɛ ɛ4π 4 2ɛ dxx x ɛ dy y ɛ d [ ] yz 2ɛ. dy 29 5
2 ɛ Oɛ dxx x ɛ dy y ɛ d dy [ y + 2ɛ y log Z] + Oɛ2 = = dxx x ɛ dy y ɛ + 2ɛ Γ ɛ 2 Γ2 2ɛ + ɛ 2ɛ = + ɛ 2ɛ log m 2 + Oɛ 2. dx dx log Zx, y = + Oɛ 2 dy d dy [ y log Z] + Oɛ2 26 Kp = Γ2ɛ ɛ4π 4 2ɛ + ɛ 2ɛ log m2 + Oɛ 2. p µ K µ p K µ p 27 l p + q l Feynman q 5 d 2ω l d 2ω q l µ K µ p = 2π 2ω 2π 2ω q 2 + m 2 [q + p l] 2 + m 2 l 2 + m 2 2 d 2ω l l µ d 2ω q dx = 2π 2ω l 2 + m 2 2 2π 2ω [q 2 + m 2 + x xp l 2 ] 2 Γ2 ω d 2ω l l µ dx = 4π ω 2π 2ω l 2 + m 2 2 [m 2 + x xp l 2 ] 2 ω. x x 2 ω Feynman K µ p = Γ4 ω 4π ω dxx x ω 2 dy y ω y d 2ω l l µ 2π 2ω ] 4 ω [l yp 2 + y yp 2 + m 2 y + y x x l l + yp l l 2 Γ4 ω p µ K µ p = p dxx x ω 2 dy y 2 ω y 4π ω d 2ω l 2π 2ω [l 2 + y yp 2 + m 2 y + y 2 Γ4 2ω = p 4π 2ω = p 2 Γ2ɛ 4π 4 2ɛ x x ] 4 ω ] 2ω 4 dxx x ω 2 dy y 2 ω y [y yp 2 + m 2 y y + x x dxx x ɛ dy y ɛ yzx, y 2ɛ. Z 29 ɛ 2 p µ K µ p = p 2 Γ2ɛ 4π 4 2ɛ 2 + Oɛ. 6
4 Γ 4 p, p 2, p, p 4 Γ 4 e L Feynman -loop Γ 4 = +gµ 2ɛ [Ramond] Γ 4 [Ramond] 5.6 Γ 2 4. Fish -loop p l 2 p p 2 l p p 4 4 4 2 2 4 2 /4! 2 = /2 Γ 4 p, p 2, p, p 4 fish = Γ 4 p, p 2, p, p 4 fish + Γ 4 p, p, p 2, p 4 fish + Γ 4 p, p 4, p 2, p fish, 2 Γ 4 p, p 2, p, p 4 fish = 2 gµ2ɛ 2 d 2ω l 2π 2ω l 2 + m 2 [l p 2 + m 2 ] p = p + p 2. Feynman l Γ 4 p, p 2, p, p 4 fish = 2 gµ2ɛ 2 d 2ω l dx 2π 2ω [l xp 2 + x xp 2 + m 2 ] 2 = 2 gµ2ɛ 2 Γ2 ω 4π ω dx [m 2 + x xp 2 ] 2 ω. ω = 2 ɛ Oɛ Γ 4 p, p 2, p, p 4 fish = 2 gµ2ɛ 2 Γɛ 4π 2 ɛ = 2 gµ2ɛ 2 Γɛ 4π 2 ɛ dx ɛ log [ m 2 + x xp 2] + Oɛ 2 ɛ 2 + = gµ 2ɛ α + ɛ 2 + ψ log m 2 2ɛ + 4m2 p 2 + 4m2 + p log 2 + log m 2 + Oɛ 2 + 4m2 p 2 + 4m2 + p log 2 + Oɛ 2. 4 + 4m2 p 2 + 4m2 p 2 x 4 2 Γ 4 2ɛ α [ p, p 2, p, p 4 fish = gµ + ɛ 2 + ψ log m 2 F + Oɛ 2 ]. 5 2ɛ 7
F F = a,b=,2,,,,4 + 4m2 p ab + 4m2 p ab + log, p ab = p a + p b 2 6 + 4m2 p ab Glass gµ 2ɛ d 2ω l 2π 2ω l 2 + m 2 [l p 2 + m 2 ] = α ɛ + ɛ 2 + ψ log m 2 + 4m2 p 2 log + 4m2 + p 2 + Oɛ 2 + 4m2 p 2 7 4.2 Glass 2-loop p p 2 p p 4 4 2 4 4 2 4 /4! = /4 Γ 4 p, p 2, p, p 4 glass = Γ 4 p, p 2, p, p 4 glass + Γ 4 p, p, p 2, p 4 glass + Γ 4 p, p 4, p 2, p glass, 8 Γ 4 p, p 2, p, p 4 glass = 4 gµ2ɛ d 2ω 2 l p = p 2π 2ω l 2 + m 2 [l p 2 + m 2 + p 2. 9 ] 7 Γ 4 p, p 2, p, p 4 glass 2ɛ α2 = gµ 4ɛ 2 + 2ɛ 2 + ψ log m 2 + 4m2 p 2 + 4m2 + p log 2 + Oɛ 2. + 4m2 p 2 4 8 ɛ Γ 4 2ɛ α2 [ p, p 2, p, p 4 glass = gµ + ɛ 4 + 2ψ 2 log m 2 2F + Oɛ 2 ]. 4 4ɛ 2 F 6 8
4. Lobster 2-loop p p 2 l l p q p l + p q p 4 4 6 2 4 4 p p p 4 2 2 2 2 4 6 /4! = Γ 4 p, p 2, p, p 4 lobster = Γ 4 p, p 2, p, p 4 lobster + Γ 4 p, p, p 2, p 4 lobster + Γ 4 p, p 4, p 2, p lobster. 42 Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ d 2ω l 2π 2ω d 2ω q 2π 2ω l 2 + m 2 l p 2 + m 2 q 2 + m 2 l + p q 2 + m 2. 4 ɛ Γ 4 p, p 2, p, p 4 lobster 2ɛ α2 = gµ + ɛ 5 + 2ψ 2 log m 2 2F 2ɛ 2 44 + Oɛ 2 F 6 4 Feynman q gµ 2ɛ Γ2 ω d 2ω l dx 4π ω 2π 2ω l 2 + m 2 l p 2 + m 2 [m 2 + x xl + p 2 2 ω. 45 ] x x 2 ω Feynman Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ Γ4 ω dxx x ω 2 4π ω d 2ω l [ l + y p 2π 2ω y 2 p 2 + V ] ω 4, dy 2 y2 V = y y p 2 + y 2 y 2 p 2 + 2y y 2 pp + m 2 y + l dy y ω 46 y. 47 x x Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ Γ4 2ω 4π 2ω = gµ 2ɛ Γ2ɛ 4π 4 2ɛ y2 dxx x ω 2 dy 2 y2 dxx x ɛ dy 2 dy y ω dy y ɛ V, 4 2ω V. 2ɛ 48 9
y y2 dy y ɛ V = 2ɛ V y = 2ɛ = y2 dy y ɛ + Oɛ y 2 ɛ ɛ [y 2 y 2 p 2 + m 2 2ɛ + Oɛ ] 49 = ɛ y2 ɛ 2ɛ log [ y 2 y 2 p 2 + m 2] + Oɛ x, y 2 y 2 4 Γ 4 p, p 2, p, p 4 lobster = gµ 2ɛ Γ2ɛΓ ɛ 2 4π 4 2ɛ Γ2 2ɛɛ = gµ 2ɛ Γ2ɛΓ ɛ 2 4π 4 2ɛ Γ2 2ɛɛ dy 2 y2 ɛ 2ɛ log [ y 2 y 2 p 2 + m 2] + Oɛ 2 + ɛ 2ɛ 2 + + 4m2 p 2 ɛ + 4m2 + p log 2 + log m 2 + Oɛ 2. + 4m2 p 2 Γ 4 p, p 2, p, p 4 lobster + 4m2 + 2ɛ α2 = gµ + ɛ 5 + 2ψ 2 log m 2 2 + 4m2 p log 2 + Oɛ 2. 2ɛ 2 p 2 + 4m2 p 2 5 42 44 5 Minimal subtraction scheme minimal subtraction scheme ɛ = 2 ω ε = 4 d 5. Lagrangian L = 2 φ2 + 2 m2 φ 2 + 4! gµε φ 4 5 counter term L c.t. = 2 A φ2 + 2 m2 Bφ 2 + 4! gµε Cφ 4 52 L = L + L c.t. = 2 φ 2 + 2 m2 φ 2 + 4! g φ 4 5
φ = Z /2 φ φ, m 2 = Z m m 2, g = Z g gµ ε 54 Z φ = + A, Z m = + B + A, Z g = + C + A 2 55 m, g bare mass bare coupling constant Minimal subtraction scheme, 2. g V Γ 2 p Γ 4 p, p 2, p, p 4 ε = mod Og V + Γ 2, Γ 4 Og L, Og L+. 2. ε ε ε. ε Z φ, Z m, Z g ε Z φ g, ε = + k= Z k φ g, Z ε k m g, ε = + k= Z k m g ε k, Z g g, ε = + k= Z g k g. 56 ε k Minimal subtraction scheme mass parameter m, µ mass-independent renormalization Γ 4 p, p 2, p, p 4 -loop gµ ε overall gµ ε ε 4 ε, ε 2,... 5.2 βg, ε, γ m g, γ φ g m, g mass scale µ m 2 = Z m g, εm 2, g = Z g g, εgµ ε m, g µ ε βg, ε, γ m g g, m ε g = gµ, m = mµ beta βg, ε = µ g µ 57 = dg dµ = d dµ Z gg, εgµ ε βg, ε = εg + g g log Z gg, ε beta Z g g, ε 56 βg, ε = εg + ε beta ε = 58
ε ε! βg, ε = εg + βg, βg = g 2 dz g g dg 59 βg 4 beta 58 βg, ε βg βg g gz gg, ε = εg 2 g Z gg, ε ε i βg d dg gz i g g = g 2 d dg Zi+ g g i =,, 2,... 6 g g = 59 2 i = γ m g, ε γ φ g, ε Z γ m g, ε = µ µ log Z mg, ε = βg, ε g log Z mg, ε, 6 γ φ g, ε = µ µ log Z φg, ε = βg, ε g log Z φg, ε 62 γ m g, ε = dm2 dµ = d Zm g, εm 2 dµ γ m g, ε = 2 µ m m µ = log m2 log µ 6 62 56 59 Z m i g γ m g, ε + = εg + βg ε i g i= Z i φ γ φ g, ε + g = εg + βg ε i g i= i= i= Z i m g ε i, Z i φ g ε i. 6 64 ε γ m g, ε, γ φ g, ε beta ε = γ m g, ε, γ φ g, ε ε! γ m g, ε = γ m g = g dz m g, 65 dg γ φ g, ε = γ φ g = g dz φ g dg 2 66
64 ε i γ m g βg d dg Zi m g = g d dg Zi+ m g i =,, 2,..., γ φ g βg d dg Zi φ g = g d dg Zi+ φ g i =,, 2,... Z m g = Z φ g = 65 66 i = 59 65 66 mass parameter m, µ Z g g, ε, Z m g, ε, Z φ g, ε ε = βg, ε εg ε minimal subtraction scheme 6 6. g c βg c, ε = 67 d < 4 φ 4 beta,2-loop βg, ε ε g c g UV IR UV g = Gaussian IR g = g c Wilson-Fisher 57 g g c g as µ IR, as µ UV 68 g c = Oε g c ω = β g c, ε 69 ω >, γ φ g c, γ m g c 7 2-loop γ φ g c >, γ m g c < < g < g c, T > T c 2-loop β ε β = g c ε.97 g =
g = Z m g, ε, Z φ g, ε 6 62 g γ m g Z m g, ε = exp βg, ε dg g g c γ mg c /ω g g c, 7 g Z φ g, ε = exp γ φ g βg, ε dg g g c γ φg c /ω g g c. 72 g = g βg, ε = εg + Og 2, γ m g = Og, γ φ g = Og 2 2-loop beta ω µ g g c /ω g g c 7 µ, g g m 2 Z g g, ε Z m g, ε Z φ g, ε IR µ g c µ γmgc µ ε µ γmgc µ γ φg c UV µ µ ε m 2 : µ 6.2 Z φ η 2 Gp p η 2 fpξ p ξ 74 η ξ ν ξ t ν, t = T T c /T c χ = const G 74 y fy y 2 η χ ξ 2 η γ χ t γ γ = 2 ην 74 Gp Z φ g, ε p 2 + ξ 2 + op 2 75 Z φ g, ε ξ η g g c 76 4
6. Callan-Symanzik N bare bare ΓN Γ N Γ N bare p i, m 2, g, ε = Z N/2 φ g, εγ N p i, m 2, g, µ, ε 77 p i N µ g, m µ log µ Callan-Symanzik µ d dµ N 2 γ φg Γ N p i, m 2, g, µ, ε =, µ d dµ = µ µ + βg, ε 78 g γ mgm 2 m 2 γ φ g, ε, γ m g, ε 62 6 ε γ φ g, γ m g 57 65 66 78 µ µ s mass µ µ = µ s, ĝs = gµ s, ˆms = mµ s, g = ĝ = gµ, m = ˆm = mµ 79 gµ 57 mµ 6 78 N s Γ N p i, ˆms 2, ĝs, µ s, ε = exp γ φ ĝs ds Γ N p 2 s i, m 2, g, µ, ε 8 N mass d Nd/2 p, m, g, µ Γ N p i, m 2, g, µ, ε = σ d Nd/2 Γ N p σ, m2 σ, g, µ, ε 8 2 σ N = 2 p i p 8 s Γ 2 p, m 2, g, µ, ε = s 2 exp γ φ ĝs ds Γ 2 ps ˆms2,, ĝs, µ s s 2, ε. 82 6.4 6 s ˆms 2 = m 2 exp ˆms µ s s γ m ĝs ds s 8 = mµ s = 84 µ s 5
m 2 µ 2 = exp s 2 + γ m ĝs ds s m 2 Γ 2 p, m 2, g, µ, ε p µ 2 µ µ 84 s µ 85 ĝs µ 79 85 t = T T c /T c m = mµ t = mµ 2 µ 2 = m2 Z m gµ, ε. µ 2 bare mass m t µ gµ, γ m g mass m, m, µ,... t µ 85 t s s t = exp 2 + γ m ĝs ds 86 s 2 + γ m g > s t 85 t s 2+γ mg c 87 ĝ = g = g c 68 82 s = s p /s s t 84 Γ 2 p, m 2, g, µ, ε Γ 2 p s, µ 2, ĝs, µ, ε t = Γ 2 p, m 2, g, µ, ε Γ 2 p, m 2, g, µ, ε s 2 γ φg c p Γ 2, µ 2 s, g c, µ, ε µ 2 s 2 γ φg c p Γ 2,, g c,, ε 88 µ s p 2 γ φg c p h µ s hx = µ γ φg c x γ φg c 2 Γ 2 x,, g c,, ε 2 Gp 74 s ξ t ν ν = 2 + γ m g c, η = γ φg c 89 g g c γ φg c /ω 72 Z φ g, ε 76 η 89 ξ ξ γ φg c 7 g g c 9 ξ µ g g c /ω g g c 9 6
7 -loop 7. Γ 2 p 2 Σp -loop Tadpole Σp tadpole ɛ 4 e L L Lagrangian L 2 m2 B φ 2 B = α 2ɛ 92 m 2 m 2 + B Feynman = m2 α 2ɛ Σp tadpole 4 Σp tadpole m2 α 2ɛ = m 2 α 2 ψ2 log m2 + Oɛ 9 ɛ = 2 Γ 2 p 2 Γ 2 p = p 2 + m 2 Σp tadpole m2 α 2ɛ = p2 + m 2 α 2 ψ2 log m2 + Og 2 94 -loop Og 2 α = g/4π 2 94 Feynman = + + + Og 2 7.2 Γ 4 p, p 2, p, p 4 Γ 4 p, p 2, p, p 4 Γ 4 p, p 2, p, p 4 fish 5 ɛ 2ɛ α gµ Lagrangian 2ɛ 4! gµ2ɛ C φ 4 C = α 2ɛ 95 g g + C Feynman 2ɛ α = gµ 2ɛ 7
-loop -loop 4 ɛ Γ 4 p, p 2, p, p 4 = gµ 2ɛ + Γ 4 2ɛ α p, p 2, p, p 4 fish + gµ [ 2ɛ = gµ 2ɛ α 2 + ψ log m 2 F ] + Og 2 -loop Og 96 7. -loop -loop 92 95 55 59 66 L -loop c.t. = 2 A φ 2 + 2 m2 B φ 2 + 4! gµ2ɛ C φ 4 97 A =, B = α 2ɛ = α ε, C = α 2ɛ = α ε. 98 Z φ =, Z m = + α ε, Z g = + α ε. 99 βg, ε = εg + αg, γ m g = α, γ φ g =. 67 89 α c := g c 4π = ε 2, ν = = 2 α c 2 + ε 2 + Oε2, η = + Oε 2. 8 2-loop 8. Γ 2 p 2-loop Feynman = + + + + + + 8
-loop 7. 2 Double scoop.2 Sunset. -loop Feynman Double scoop Sunset Og 2 Σp tadpole i Σp tadpole Σp tadpole = 4! gµ2ɛ 4 2 Σp tadpole m2 4ɛ α d 2ω l 2π 2ω l 2 + m 2. 2 2 92 m2 B 2 4 2 2 7 Σp tadpole ii Σp tadpole Σp tadpole coupling gµ 2ɛ gµ 4 4 = m2 α2 4ɛ 2 + ɛψ log m2 + Oɛ 2 2ɛ α 2ɛ Σp tadpole = m 2 α2 + ɛψ2 log m 2 + Oɛ 2 4 4ɛ 2 2-loop 8 2 4 Σp d-scoop + Σp sunset + Σp tadpole + Σp tadpole = α2 24ɛ p2 + m2 2 α2 ɛ 2 2ɛ + Oɛ 5 -loop Σp tadpole 92 9 ɛ 2-loop 92 2 A + A 2 φ 2 + 2 m2 B + B 2 φ 4, 6 A =, A 2 = α2 24ɛ, B = α 2ɛ, B 2 = α2 2ɛ α2 2 4ɛ. 7 -loop 2 Γ 2 p = + A p 2 + m 2 + B Σp tadpole + Og 2 8 2-loop Γ 2 p = + A + A 2 p 2 + m 2 + B + B 2 Σp tadpole + Σp d-scoop + Σp sunset + Σp tadpole + Σp tadpole + Og ɛ 9 9
8.2 Γ 4 p, p 2, p, p 4 2-loop Feynman = + + + + + + + 8 42 p, p 2, p, p 4, p, p, p 2, p 4, p, p 4, p 2, p -loop Fish ɛ 96 -loop 2 2-loop Glass 4.2 Lobster 4. ɛ = -loop 92 95 Feynman 2 Og 2-loop 2 ɛ = 2-loop 2 { } gµ 2ɛ 2 d 2ω q Σq 2π 2ω q 2 + m 2 2 [p q 2 + m 2 tadpole m2 α p = p + p 2 ] 2ɛ -loop 9 { } ɛ = 2 ω q d 2ω q 2π 2ω q 2 + m 2 2 [p q 2 + m 2 ] = Γ + ɛ 4π 2 ɛ dx [m 2 + x xp 2 ] +ɛ ɛ Γ 2 p 2-loop 6 B 2 g 4 Γ 4 p, p 2, p, p 4 2-loop 2-loop Glass Lobster Glass Lobster Γ 4 p, p 2, p, p 4 fish 7.2 Feynman Γ 4 p, p 2, p, p 4 fish Γ4 p, p 2, p, p 4 fish 5 4 2 gµ 2ɛ gµ 2ɛ gµ 2ɛ 2ɛ α gµ 2ɛ 2 Γ 4 p, p 2, p, p 4 2ɛ 9α2 [ fish = gµ + ɛ 2 + ψ log m 2 F + Oɛ 2 ] 2ɛ 2 2
2-loop Γ 4 p, p 2, p, p 4 ɛ 4 44 Γ 4 p, p 2, p, p 4 glass + Γ 4 p, p 2, p, p 4 lobster + Γ 4 p, p 2, p, p 4 fish 2ɛ α2 9 = gµ ɛ 2 4 2 ɛ + Oɛ2 log m 2 F Lagrangian 4! gµ2ɛ C 2 φ 4 C 2 = 9α2 4ɛ α2 2 2ɛ 2 -loop 95 φ 4 2-loop 4! gµ2ɛ C + C 2 φ 4 C 95 C 2 Γ 2 p 2-loop 8. 8. 2-loop 2-loop L 2-loop c.t. = 2 A 2 φ 2 + 2 m2 B 2 φ 2 + 4! gµ2ɛ C 2 φ 4 4 7 2 9 ε = 2ɛ A 2 = α2 2ε, B 2 = 2α2 ε 2 α2 2ε, C 2 = 9α2 ε 2 2-loop -loop 97 98 α2 ε. 5 L -loop c.t. + L 2-loop c.t. = 2 A + A 2 φ 2 + 2 m2 B + B 2 φ 2 + 4! gµ2ɛ C + C 2 φ 4 6 55 A A + A 2 = α2 2ε, B B + B 2 = α 2 ε + α2 ε, 2 2ε C C + C 2 = α 9 ε + α2 ε 2 ε Z φ = α2 2ε, Z m = + α ε + α2 2 ε 5, Z 2 g = + α 9 2ε ε + α2 ε 7 2 6ε 7 2
59 66 βg, ε = εg + βg, βg = g α 7 α2, 8 γ m g = α + 5α2 6, 9 γ φ g = α2 6. 2 -loop g 67 α c := g c 4π 2 89 α c 7 α2 c = ε α c = ε + 7ε2 8 + Oε. 2 ν = 2 α c + 5αc/6 = 2 2 + ε 2 + 7ε2 62 + Oε, η = α2 c 6 = ε2 54 + Oε. 22 9 n 9. n φ = φ,..., φ n Lagrangian L = 2 φ 2 + 2 m2 φ 2 + 4! gµε φ 2 2 2 Dφ e R d d x L n proper vertex Feynman 2 φ a a g = Green φ a xφ b y q δ ab δ q 2 +m 2 ab φ a φ a φ b φ b a b a b Feynman a, b Proper vertex Γ 2 2 Γ 2 ij p = δ ijp 2 + m 2 Σp Σp Feynman 2 i = j i = j Tadpole Double scoop 5 Sunset 9 22
n n = fn fn f = n n Γ 4 2 Γ 4 i i 2 i i 4 p, p 2, p, p 4 fish = δ i i 2 δ i i 4 Γ 4 p, p 2, p, p 4 fish + δ i i δ i2 i 4 Γ 4 p, p, p 2, p 4 fish + δ i i 4 δ i2 i Γ 4 p, p 4, p 2, p fish 24 Γ 4 p, p 2, p, p 4 fish Glass 8 9 Lobster 42 4 Feynman fn = n 4 9.2 Γ 2 Σp 9.2. Tadpole n = 4 = 2 Tadpole 2 4n 8 Tadpole fn = 4n + 8 2 = n + 2. 4 Σp tadpole = n + 2 m 2 α + ɛψ2 log m 2 + Oɛ 2. 25 6 ɛ 2
9.2.2 Double scoop 5 n = 4 2 = 44 Double scoop 4 6n 2 2n 2n 64 fn = 6n2 + 64n + 64 44 = n + 22. 9 8 n + 22 m 2 α 2 Σp d-scoop = + ɛψ2 + ψ 2 log m 2 + Oɛ 2. 26 6 ɛ 2 9.2. Sunset 9 n = 2 4 2 = 96 Sunset 2 2n 64 2n + 64 fn = 96 2 Σp sunset = n + 2 m 2 8 α2 2ɛ 2 = n + 2. + m2 ɛ 2 + ψ log m2 + 4ɛ p2 + Oɛ. 27 24
9. Γ 4 24 i i 2 i i 4 = δ i i 2 δ i i 4 + δ i i δ i2 i 4 + δ i i 4 δ i2 i, 28 F i i 2 i i 4 = + 4m2 δ i i 2 δ i i 4 + 4m2 p 2 + log + + 4m2 p 2 + 4m2 p 2 δ i i δ i2 i 4 + 4m2 p + log p + 4m2 p + + 4m2 δ i i 4 δ i2 i + 4m2 p 4 + log. 29 p 4 + 4m2 p 4 p ab = p a + p b 2 F i i 2 i i 4 F 6 i i 2 i i 4 9.. Fish n = 2 2 4 2 = 288 Sunset Amit [A] p29 Fig. 6-5 2n 28 28 2n + 256 fn = 288 = n + 8. 9 Γ 4 p, p 2, p, p 4 fish 4 fn 24 i i 2 i i 4, F i i 2 i i 4 Γ 4 i i 2 i i 4 p, p 2, p, p 4 fish = n + 8 6 gµ 2ɛ α ɛ [ i i 2 i i 4 + ɛ2 + ψ log m 2 ɛf i i 2 i i 4 + Oɛ 2 ]. n = i i 2 i i 4, F i i 2 i i 4 F 5 9..2 Glass 9 n = 2 4 = 456 Glass 6 Amit [A] p29 Fig. 6-6 25
2 28n 28 2n 28 4n 28 8 28 4 28 8 fn = 28n2 + 6n + 2 456 = n2 + 6n + 2. 27 Γ 4 p, p 2, p, p 4 glass 4 fn 24 fish glass i i 2 i i 4, F i i 2 i i 4 Γ 4 i i 2 i i 4 p, p 2, p, p 4 glass = n2 + 6n + 2 gµ 2ɛ α 2 [ 6 ɛ 2 i i 2 i i 4 + ɛ4 + 2ψ 2 log m 2 2ɛF i i 2 i i 4 + Oɛ 2 ]. n = i i 2 i i 4, F i i 2 i i 4 F 4 9.. Lobster 4 n = 2 2 4 6 = 692 42 Lobster 8 256 2n 256n 256 2n 256 4 256 4 256 8 256 2 256 4 26
Amit [A] p29 Fig. 6-7 fn = 2565n + 22 692 = 5n + 22. 27 Γ 4 p, p 2, p, p 4 lobster 5 fn 24 fish lobster i i 2 i i 4, F i i 2 i i 4 Γ 4 i i 2 i i 4 p, p 2, p, p 4 lobster 5n + 22 gµ 2ɛ α 2 [ = 8 ɛ 2 i i 2 i i 4 + ɛ5 + 2ψ 2 log m 2 2ɛF i i 2 i i 4 + Oɛ 2 ]. 2 n = i i 2 i i 4, F i i 2 i i 4 F 44 9.4 Γ 2 Γ 4 Γ 2 Σp 9.4. -loop 7 -loop Tadpole 25 n+2 92 2 m2 B φ 2 n + 2α B = 6ɛ Feynman = n+2m2 α 6ɛ Γ 4 -loop Fish Lagrangian 2 4! gµ2ɛ C φ 2 n + 8α C = 4 6ɛ n+8 C 95 9 g g + C Feynman = n+8gµ2ɛ α 6ɛ -loop L -loop c.t. = 2 A φ 2 + 2 m2 B φ 2 + 4! gµ2ɛ C φ 2 2 5 27
A =, B = n + 2α 6ɛ = n + 2α, C = ɛ n + 8α 6ɛ = n + 8α. 6 ε 55 Z φ =, Z m = + n + 2α, Z g = + ε n + 8α. 7 ε 59 66 βg, ε = εg + 67 89 α c := g c 4π 2 = n + 8αg n + 2α, γ m g =, γ φ g =. 8 ε n + 8, ν = 2 + γ m g c = n + 2ε + 2 4n + 8 + Oε2, η = + Oε 2. 9 9.4.2 2-loop 8 Double scoop 26 Sunset 27 -loop Σp tadpole Σp tadpole Σp tadpole 2 m2 α m2 B 4ɛ 2 n+2 n 2 4 2 = 24 8n 6 n+2 8n + 2 n+22 9 Σp tadpole = n + 22 6 m 2 α 2 ɛ 2 + ɛψ log m 2 + Oɛ 2 4 Σp tadpole 25 coupling gµ2ɛ n+8gµ Feynman 2ɛ α 6ɛ 4 25 coupling α α n+8 6ɛ α2 Σp tadpole = n + 2n + 8 m 2 α 2 + ɛψ2 log m 2 + Oɛ 2. 4 6 ɛ 2 n = 4 2-loop 26 27 4 4 28
Σp d-scoop + Σp sunset + Σp tadpole + Σp tadpole = n + 2α2 p 2 + m2 α 2 72ɛ 2 n + 2n + 5 n + 2 + Oɛ 8ɛ 2 6ɛ. 42 2 A + A 2 φ 2 + 2 m2 B + B 2 φ 2, 4 n + 2α2 n + 2α n + 2n + 5α2 n + 2α2 A =, A 2 =, B =, B 2 =. 44 72ɛ 6ɛ 6ɛ 2 2ɛ Γ 4 8.2 2-loop Glass Lobster -loop 4 Γ 4 p, p 2, p, p 4 fish 8.2 Γ4 p, p 2, p, p 4 fish gµ 2ɛ α n+8gµ2ɛ α 2 6ɛ Γ 4 p, p 2, p, p 4 n + 82 gµ 2ɛ α 2 [ fish = 8 ɛ 2 i i 2 i i 4 + ɛ2 + ψ log m 2 ɛf i i 2 i i 4 + Oɛ 2 ]. 45 2-loop 2 45 Γ 4 p, p 2, p, p 4 glass + Γ 4 p, p 2, p, p 4 lobster + Γ 4 p, p 2, p, p 4 fish n + 8 = gµ 2ɛ α 2 2 5n + 22 i i 2 i i 4 + Oɛ 6ɛ 2 8ɛ Lagrangian 4! gµ2ɛ C 2 φ 2 2 C 2 = n + 82 α 2 5n + 22α2 6ɛ 2 8ɛ 46 47 2-loop ε = 2ɛ n + 2 A 2 = 6ε α2, B 2 = L 2-loop c.t. = 2 A 2 φ 2 + 2 m2 B 2 φ 2 + 4! gµ2ɛ C 2 φ 2 2 48 n + 2n + 5 n + 2 9ε 2 6ε n + 8 α 2 2, C 2 = 9ε 2 -loop 5 55 5n + 22 α 2. 9ε 49 Z φ = n + 2 6ε α2, 5 Z m = + n + 2 n + 2n + 5 ε α + 5n + 2 α 2, 5 9ε 2 6ε Z g = + n + 8 ε α + n + 8 2 9ε 2 29 n + 4 6ε α 2. 52
α c = gc 4π 2 βε, g = εg + βg, γ m g = n + 2 α + 89 n + 8 βg = g α n + 4 α 2, 5 5n + 2 α 2, 54 8 γ φ g = n + 2 8 α2. 55 α c = ε 9n + 4ε2 + + Oε. 56 n + 8 n + 8 ν = n + 2ε + 2 4n + 8 + n + 2n2 + 2n + 6ε 2 + Oε, 8n + 8 57 n + 2ε2 η = 2n + 8 + 2 Oε. 58 27 minimal subtraction scheme beta 5-loop [K] βg, ε/g = ε + n + 8α n + 4α2 + [ ] n 2 + 922n + 296 + 965n + 22ζ 26 888 [ 5n + 62n 2 + 8456n + 96648 + 96 6n 2 + 764n + 22 ζ 288n + 85n + 22ζ4 + 92 2n 2 + 55n + 86 ] ζ5 α 4 + [ n 4 + 2578n + 88496n 2 + 66466n + 7744 6228 + 6 9n 4 + 248n + 6764n 2 + 55228n + 46 ζ α + 768 6n 59n 2 + 446n + 264 ζ 2 288 6n + 88n 2 + 952n + 22 ζ4 + 256 5n + 7466n 2 + 66986n + 6584 ζ5 96n + 8 2n 2 + 55n + 86 ζ6 + 2896 4n 2 + 89n + 526 ] ζ7 α 5 α = g 4π 2, ζs = k k s Riemann zeta
[R] P. Ramond, Field Theory, A Modern Primer, Benjamin 98 [B] M. L. Bellac, Quantum and Statistical Field Theory, Oxford 99 [A] D. J. Amit, Field Theory, the Renormalization Group and Critical Phenomena revised 2nd edition, World Scientific 984. [K] H. Kleinert, J. Neu, V. Schulte-Frohlinde, K. G. Chetyrkin, S. A. Larin, Phys. Lett. B 272 99 9. Errata 9 99 545.