Functional RG methods in QCD

Transcription

1 methods in QCD Institute for Theoretical Physics University of Heidelberg LC2006 May 18th, 2006 methods in QCD

2 motivation Strong QCD QCD dynamical symmetry breaking instantons χsb top. dofs link?! deconfinement non-eq. top. dofs confinement methods in QCD

3 outline 1 2 UV-IR flow infrared analysis methods in QCD

4 outline 1 2 UV-IR flow infrared analysis methods in QCD

5 flow equation k k Γ k [φ] = 1 2 Tr 1 2 2k [φ] + k 2 k Callan-Symanzik equation methods in QCD

6 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k C. Wetterich, Phys. Lett. B 301 (1993) 90 k k R k self-similarity, reparameterisation & projections fermions straightforward no sign problem, chirality, bound states via bosonisation flows in Landau gauge QCD Ellwanger, Hirsch, Weber 96 Bergerhoff, Wetterich 97 JMP, Litim, Nedelko, von Smekal 03 Kato 04 Gies, Fischer 04 methods in QCD

7 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k of Γ k [φ] and k k Γ k [φ] limits for Γ k : S k Λ k 0 Γ k Γ methods in QCD

8 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k of Γ k [φ] and k k Γ k [φ] limits for Γ k : S k Λ k 0 Γ k Γ flow k k Γ k [φ] is infrared and ultraviolet finite methods in QCD

9 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k flow k k Γ k [φ] is infrared and ultraviolet finite R k IR-structure p 2 methods in QCD

10 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k flow k k Γ k [φ] is infrared and ultraviolet finite k k R k 8 6 UV -structure p 2 methods in QCD

11 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k control of approximation optimisation k k R k D. F. Litim, Phys. Lett. B 486 (2000) 92 methods in QCD

12 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation D. F. Litim, Phys. Lett. B 486 (2000) 92 key idea: maximising the convergence/stability of the flow methods in QCD

13 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation D. F. Litim, Phys. Lett. B 486 (2000) 92 key idea: maximising the convergence/stability of the flow minimising the propagator at a given momentum scale methods in QCD

14 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation D. F. Litim, Phys. Lett. B 486 (2000) 92 key idea: maximising the convergence/stability of the flow minimising the propagator at a given momentum scale functional optimisation minimising the length of the flow trajectory in theory space JMP, hep-th/ methods in QCD

15 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation D. F. Litim, Phys. Lett. B 486 (2000) 92 key idea: maximising the convergence/stability of the flow minimising the propagator at a given momentum scale functional optimisation minimising the length of the flow trajectory in theory space maximising the propagator at a given gap 8 < k (p) + R(p2 ) = : 0 (k) for p2 k 2, 0 (p) for p2 > k 2. JMP, hep-th/ methods in QCD

16 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation D. F. Litim, Phys. Lett. B 486 (2000) 92 key idea: maximising the convergence/stability of the flow minimising the propagator at a given momentum scale functional optimisation minimising the length of the flow trajectory in theory space maximising the propagator at a given gap keeps gradient flow property JMP, hep-th/ methods in QCD

17 flow equation k k O k [φ] = 1 2 Tr 1 k [φ] + R k 1 k k R k [φ] + R k k JMP, hep-th/ O (2) k [φ] control of approximation optimisation general flows: O = Γ (1), P exp A µ dx µ,.... methods in QCD

18 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation general flows gauge theories no matter fields D µ δγ δa µ = quantum corrections BRST: (Γ, Γ) = 0 methods in QCD

19 flow equation k k Γ k [φ] = 1 2 Tr 1 k [φ] + R k k k R k control of approximation optimisation general flows gauge theories no matter fields D µ δγ k δa µ = quantum + cutoff corrections BRST: (Γ k, Γ k ) = 1 2 Tr R δ2 Γ k 1 δqδφ Γ [2] k +R methods in QCD

20 truncation schemes approximation to Γ k Γ k [φ] = consistent with the symmetries at hand N n i d d p j φ(p j ) f i (p 1,..., p ni ; k) i=1 j=1 in general Qn i j φ(p j ) O[φ] i (p 1,..., pn i ; k) methods in QCD

21 truncation schemes approximation to Γ k Γ k [φ] = consistent with the symmetries at hand N n i d d p j φ(p j ) f i (p 1,..., p ni ; k) i=1 j=1 flow equation + initial values for f i in general Qn i j φ(p j ) O[φ] i (p 1,..., pn i ; k) methods in QCD

22 truncation schemes approximation to Γ k Γ k [φ] = consistent with the symmetries at hand N n i d d p j φ(p j ) f i (p 1,..., p ni ; k) i=1 j=1 flow equation + initial values for f i in general Qn i j k k f i (p 1,..., p ni k) = I i [f j ] φ(p j ) O[φ] i (p 1,..., pn i ; k) n j n i + 2 i methods in QCD

23 truncation schemes approximation to Γ k Γ k [φ] = consistent with the symmetries at hand N n i d d p j φ(p j ) f i (p 1,..., p ni ; k) i=1 j=1 in general Qn i j φ(p j ) O[φ] i (p 1,..., pn i ; k) flow equation + initial values for f i f 1 (p; Λ) = p 2 example for initial values in φ 4 -theory f 2 (p 1,..., p 4 ; Λ) = λ f i (p 1,..., p ni ; Λ) = 0 i > 2 (n i > 4) RG-scale µ and RG-conditions implicit in the initial conditions methods in QCD

24 outline 1 2 UV-IR flow infrared analysis methods in QCD

25 ? methods in QCD

26 ! Coupling τ in N = 2 susy Yang-Mills (Seiberg-Witten) B. Dolan, JMP, unpublished work 0.12 τ τ pert 0.04 τ sw u methods in QCD

27 ! Coupling τ in N = 2 susy Yang-Mills (Seiberg-Witten) B. Dolan, JMP, unpublished work τ 0.05 τ pert t t t 2 τ sw t t u methods in QCD

28 outline UV-IR flow infrared analysis 1 2 UV-IR flow infrared analysis methods in QCD

29 confinement scenario UV-IR flow infrared analysis Ω = {A µ A µ = 0, µ D µ 0} entropy da det( D) e S Ω A = 0 Λ Λ Ω entropy ( da) Ω( Λ) dominates IR ghost IR-enhanced gluonic mass-gap: confined gluons non-renormalisation of ghost-gluon vertex Kugo-Ojima (in BRST-extended configuration space) gluonic mass-gap + no Higgs mechanism methods in QCD

30 functional flows UV-IR flow infrared analysis ½ Å Å Å ½ ¾ Å ½ ¾ Å ½ ¾ ½ Å Å Å ½ ¾ methods in QCD

31 functional flows UV-IR flow infrared analysis p 2 A (p) 4 3 RG JMP DSE Smekal, Alkofer et al lattice Leinweber et al p [GEV] methods in QCD

32 outline UV-IR flow infrared analysis 1 2 UV-IR flow infrared analysis methods in QCD

33 infrared flows UV-IR flow infrared analysis JMP, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93 (2004) ½ p 2 Λ 2 QCD A (p) + renorm. (p 2 ) 1 2κ C C (p) + renorm. (p 2 ) 1+κ C methods in QCD

34 infrared flows UV-IR flow infrared analysis JMP, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93 (2004) κ C R confined gluons: κ C 0 mass-like gluonic behaviour: κ C = 1 2 methods in QCD

35 outlook UV-IR flow infrared analysis conclusions support for Kugo-Ojima/Gribov-Zwanziger scenario quantitative improvement due to optimisation universally applicable outlook full QCD QCD at finite temperature flow of Wilson loops & Polyakov loops: area law methods in QCD