Antonio Pich IFIC, CSIC University of Vlenci 1. uge Theories 2. Renormliztion 3. Renormliztion roup 4. QCD Phenomenology 5. Chirl Perturbtion Theory TAE 2005, Bensue, Spin, 12-24 September 2005
uge Symmetry Quntum Electrodynmics (QED) SU(N) Algebr Non-Abelin uge Symmetry Quntum Chromodynmics (QCD) uge Fixing A. Pich uge Theories 1
FREE Dirc fermion: L = ψ ( iγ m) ψ Phse Invrince: ψ ψ = ψ ; ψ ψ = ψ e iqθ Absolute phses re not observble in Quntum Mechnics AUE PRINCIPLE: θ = θ ( x) Phse Invrince should hold LOCALLY e iqθ BUT e iq θ θ ψ ( + iq ) ψ SOLUTION: Covrint Derivtive D ( i A ) e iq θ ψ eq ψ D ψ One needs spin-1 field 1 A stisfying A A + e θ A. Pich uge Theories 2
L = ψ ( iγ D m) ψ γ = ψ ( iγ m) ψ + e QA ( ψγ ψ ) ψ e Q ψ Kinetic term: 1 LK = Fν F ν 4 Mss term: L 1 M = m A A 2 2 γ ν ν F = eq ( ψ γ ψ ) Mxwell 16 [exp: m < 2 10 ev ] Not uge Invrint m γ = 0 γ uge Symmetry QED Dynmics A. Pich uge Theories 3
Successful Theory e γ e + + Anomlous Mgnetic Moment l g l e 2 m l l 1 ( gl 2) 2 e 11 1 = (115 965 218.59 ± 0.38) 10 α = 137.035 999 11 ± 0.000 000 46 th = (116 591 922 ± 88) 10 11 11 [ Exp: (116 592 080 ± 60) 10 ] A. Pich uge Theories 4
FREE QUARKS: L N C = 3 = [ iγ m] SU(3) Colour Symmetry: U ; U UU UU 1 U = = ; det = 1 A. Pich uge Theories 5
β α β α A. Pich uge Theories 6
AB - BA = C A B B A B C A A. Pich uge Theories 7
N N = = = mtrices: UU U U 1 ; det U 1 U exp { i θ } = T ; T = T ; Tr ( T ) = 0 ; = 1,, N 1 2 Commuttion Reltion:, Structure Constnts f bc b bc c T T = if T rel, ntisymmetric Fundmentl Representtion: A bc Adjoint Representtion: ( T ) 1 TF = λ N N 2 = if bc 2 2 ( N 1) ( N 1) ( ) A A A b ( ) b b b αβ F Tr ( TF TF) = TF δ ; Tr T T = C δ ; TF TF = C δ α β 2 1 N 1 TF = ; CA = N ; CF = 2 2N A. Pich uge Theories 8
2 2 mtrices: UU = U U = 1 ; det U = 1 U exp { i θ } = T ; T = T ; Tr ( T ) = 0 ; = 1,, 3 Commuttion Reltion: b bc c, T T = i ε T Fundmentl Representtion: T F = 1 σ 2 Puli 1 0 1 2 0 i 3 1 0 σ = ; σ = ; σ = 1 0 i 0 0 1 A. Pich uge Theories 9
b bc c [ T, T ] = if T Fundmentl Representtion: T F = 1 λ 2 ell-mnn 0 1 0 0 i 0 1 0 0 0 0 1 1 2 3 4 λ = 1 0 0 ; λ = i 0 0 ; λ = 0 1 0 ; λ = 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 i 0 0 0 0 0 0 1 0 0 5 6 7 8 1 λ = 0 0 0 ; λ = 0 0 1 ; λ = 0 0 i ; λ = 0 1 0 3 i 0 0 0 1 0 0 i 0 0 0 2 1 123 147 156 246 257 345 367 1 458 1 678 1 f = f = f = f = f = f = f = f = f = 2 3 3 2 A. Pich uge Theories 10
FREE QUARKS: L N C = 3 = [ iγ m] SU(3) Colour Symmetry: U = exp i λ θ 2 uge Principle: Locl Symmetry θ = θ ( ) x D ( I i ) U 3 g s A. Pich uge Theories 11 D i D U D U ; U U ( U) U g 1 ( α β λ ) α β 2 [ ] ( x ) 8 luon s Fields
Infinitesiml SU(3) Trnsformtion: δ c αβ α λ β 1 bc = i δθ ; ( ) 2 δ = δθ f δθb gs bc Non Abelin roup: f 0 δ depends on Universl g s No Colour Chrges A. Pich uge Theories 12
Kinetic Term: i ν [ D, D ν ] = ν ν igs[, ν ] U ν U g s ν ν ν ν ν c ν λ bc ; = + gs f b 2 L K 1 1 Tr ( ) 2 4 ν ν ν ν = = Mss 1 2 Term: LM = m 2 Not uge Invrint m = 0 Mssless luons A. Pich uge Theories 13
L QCD 1 = 2 Tr ( ν ) [ i m ] ν + γ D ( ) ( ) 1 = ν ν + [ iγ m ] 4 1 + 2 α αβ β g [ ( λ ) γ ] s ν ν α α 1 1 g f g f 2 4 2 d e s bc ( ) b ν ν ν ν c s bc fd e b c ν luon Self interctions Universl Coupling g s 3 4, (No Colour Chrges) A. Pich uge Theories 14
Z A. Pich uge Theories 15
Z A. Pich uge Theories 16
Z Z A. Pich uge Theories 17
AUE FIXIN L F = 1 2ξ ( ) ( ν ν ) A. Pich uge Theories 18
AUE FIXIN L F = 1 2ξ ( ) ( ν ν ) 0 T [ (x) b ν (y)] 0 i δ b D ν F (x y) = i δ b d 4 k (2π) 4 e ik (x y) k 2 + i ɛ { g ν + (1 ξ) k k ν } k 2 + i ɛ A. Pich uge Theories 18
AUE FIXIN L F = 1 2ξ ( ) ( ν ν ) 0 T [ (x) b ν (y)] 0 i δ b D ν F (x y) = i δ b d 4 k (2π) 4 e ik (x y) k 2 + i ɛ { g ν + (1 ξ) k k ν } k 2 + i ɛ + + ) b) c) T = J b ν ε,r εν b,s A. Pich uge Theories 18
AUE FIXIN L F = 1 2ξ ( ) ( ν ν ) 0 T [ (x) b ν (y)] 0 i δ b D ν F (x y) = i δ b d 4 k (2π) 4 e ik (x y) k 2 + i ɛ { g ν + (1 ξ) k k ν } k 2 + i ɛ + + ) b) c) T = J b ν ε,r εν b,s P T 1 2 Jb ν (Jρσ) cd 2 r=1 ε,r ερ c,r 2 s=1 ε ν b,s ε σ d,s P C 1 2 Jb ν (Jcd ρσ ) g ρ g νσ A. Pich uge Theories 18
AUE FIXIN L F = 1 2ξ ( ) ( ν ν ) 0 T [ (x) b ν (y)] 0 i δ b D ν F (x y) = i δ b d 4 k (2π) 4 e ik (x y) k 2 + i ɛ { g ν + (1 ξ) k k ν } k 2 + i ɛ Probbility problem: + + ) b) c) T = J b ν ε,r εν b,s P T 1 2 Jb ν (Jρσ) cd P C 2 r=1 ε,r ερ c,r 2 s=1 ε ν b,s ε σ d,s 1 2 Jb ν (Jcd ρσ ) g ρ g νσ P C > P T A. Pich uge Theories 18
Sme problem for virtul gluons: + + ) b ) c ) +... Im [T ( )] T ( ) 2 Unphysicl contribution from non-trnsverse gluon polriztions (c,c ) uge fixing: = 0 k J ν 0 A. Pich uge Theories 19
Sme problem for virtul gluons: + + ) b ) c ) +... Im [T ( )] T ( ) 2 Unphysicl contribution from non-trnsverse gluon polriztions (c,c ) uge fixing: = 0 k J ν 0 HOSTS: (Fddeev Popov) φ φ L FP = φ D φ φ φ d) d ) D φ φ g s f bc φ b c Wrong (Fermi) Sttistics Probbility < 0 A. Pich uge Theories 19