The Infrared Behavior of Landau Gauge Yang-Mills Theory in d=2, 3 and 4 Dimensions

Transcription

1 The Infrared Behavior of Landau Gauge Yang-Mills Theory in d=2, 3 and 4 Dimensions Markus Huber 1 R. Alkofer 1 C. S. Fischer 2 K. Schwenzer 1 1 Institut für Physik, Karl-Franzens Universität Graz 2 Institut für Kernphysik, Technische Universität Darmstadt September 25, 2007 SIC!QFT ÖPG FAKT-Tagung, Langenlois Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 1/14

2 Table of Contents 1 Introduction Infrared QCD Methods in the Infrared Regime 2 Infrared Behavior: Power Counting Propagators Vertices 3 Numerical Results Tensor Components 4 Summary Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 2/14

3 Infrared QCD Infrared (IR) is dierent from UV, where perturbation theory works due to asymptotic freedom. IR phenomena: dynamical chiral symmetry breaking and connement Complementing methods are needed: Lattice vs. functional methods as Dyson-Schwinger equations (DSEs) and renormalization group A rst step in understanding connement is to consider only ghosts and gluons, i.e. Yang-Mills (YM) theory. Connement in YM theory (Landau gauge): Gribov-Zwanziger and Kugo-Ojima scenario Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 3/14

4 Dyson-Schwinger Approach Equations of motion for Green functions Describe the theory completely, including non-perturbative eects Innite tower of coupled integral equations Propagators: can be calculated using approximations for the vertices In the IR a solution from the complete system for the propagators and vertices is possible in form of power laws. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 4/14

5 Infrared Behavior Gluon and ghost propagators: dressing functions show power-like behavior D µν (p) = ( ) δ µν p µp ν Z(p), p 2 p 2 D G (p) = G(p) p 2 Z(p) IR = A (p 2 ) α, G(p) IR = B (p 2 ) β Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 5/14

6 Infrared Behavior Gluon and ghost propagators: dressing functions show power-like behavior D µν (p) = ( ) δ µν p µp ν Z(p), p 2 p 2 D G (p) = G(p) p 2 Z(p) IR = A (p 2 ) α, G(p) IR = B (p 2 ) β Vertices: dressing functions show power-like behavior, but more complicated tensor structure n Γ µνρ... (p 1, p 2, p 3,... ) = H i (p 1, p 2, p 3,... )τ µνρ... (i) i=0 H i (p 2 1, p 2 2, p 2 3,...) (p 2 1 )δ H i ( p 2 2 p 2 1 ), p2 3,... ; δ p1 2 Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 5/14

7 Comparison of Dierent Methods Lattice Continuum Ghost-gluon vertex constant Ghost propagator diverging diverging Gluon propagator vanishing? vanishing Other vertex functions no data for IR prediction: power-like (Alkofer et al., PLB 611) Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 6/14

8 Comparison of Dierent Methods Lattice Continuum Ghost-gluon vertex constant Ghost propagator diverging diverging Gluon propagator vanishing? vanishing Other vertex functions no data for IR prediction: power-like in 4 dim. (Alkofer et al., PLB 611) Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 6/14

9 Comparison of Dierent Methods Lattice Continuum Ghost-gluon vertex constant Ghost propagator diverging diverging Gluon propagator vanishing? vanishing Other vertex functions no data for IR prediction: power-like in 4 dim. (Alkofer et al., PLB 611) Continuum results for the IR exponents (κ = ): ghost-gluon-vertex: (p 2 ) 0 const. ghost propagator: (p 2 ) κ gluon propagator: (p 2 ) 2κ 3-gluon vertex: (p 2 ) 3κ Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 6/14

10 Vertices: Available Lattice Data (p,0,π/2) 3 A G Three-gluon vertex, one momentum vanishing dimensions Maas et al., Braz. J. Phys. 37N1B, no power-like behavior p [GeV] Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 7/14

11 Vertices: Available Lattice Data (0,p,π/2) 3 A G Three-gluon vertex, one momentum vanishing dimensions Cucchieri et al., PRD74, 2006 change of sign p [GeV] Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 7/14

12 Vertices: Available Lattice Data 2 dimensions Maas, PRD75, 2007 IR exponent can be extracted Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 7/14

13 Vertices: Available Lattice Data 2 dimensions Maas, PRD75, 2007 IR exponent can be extracted Qualitatively the same results in 2, 3 and 4 dimensions? Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 7/14

14 Propagators IR behavior can be determined from the ghost DSE: -1 = -1 - Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 8/14

15 Propagators IR behavior can be determined from the ghost DSE: -1 = -1 - Use bare ghost-gluon vertex Power law ansätze for dressing functions in the IR Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 8/14

16 Propagators IR behavior can be determined from the ghost DSE: -1 = Use bare ghost-gluon vertex -1 - Power law ansätze for dressing functions in the IR ( ) B (p 2 ) β 1 d d q p 2 (2π) P A (q 2 ) α B ((p q) 2 ) β d µν (p q) µ q ν q 2 (p q) 2 Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 8/14

17 Propagators IR behavior can be determined from the ghost DSE: -1 = Use bare ghost-gluon vertex -1 - Power law ansätze for dressing functions in the IR ( ) B (p 2 ) β 1 d d q p 2 (2π) P A (q 2 ) α B ((p q) 2 ) β d µν (p q) µ q ν q 2 (p q) 2 Only one momentum scale simple power counting is possible 1 β = d 2 + α 1 + β = 2β = α + d 2 2 Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 8/14

18 Propagators IR behavior can be determined from the ghost DSE: -1 = Use bare ghost-gluon vertex -1 - Power law ansätze for dressing functions in the IR ( ) B (p 2 ) β 1 d d q p 2 (2π) P A (q 2 ) α B ((p q) 2 ) β d µν (p q) µ q ν q 2 (p q) 2 Only one momentum scale Z(p 2 ) = (p 2 ) 2κ+2 d 2, G(p 2 ) = (p 2 ) κ Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 8/14

19 Vertex DSEs are more complicated. Vertices Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 9/14

20 Vertex DSEs are more complicated. Vertices -1 = -1 - Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 9/14

21 Vertex DSEs are more complicated. Vertices = /2-1/2-1/2-1/2-1/2 - +1/2 +1/2 +1/2-1/3! -2 + Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 9/14

22 Vertex DSEs are more complicated. Vertices = /2-1/2-1/2-1/2-1/2 - +1/2 +1/2 +1/2-1/3! -2 + Power counting works with some graphs, but what about unknown vertices? Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 9/14

23 Vertex DSEs are more complicated. Vertices = /2-1/2-1/2-1/2-1/2 - +1/2 +1/2 +1/2-1/3! -2 + Power counting works with some graphs, but what about unknown vertices? = Skeleton expansion Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 9/14

24 Employ skeleton expansion Skeleton Expansion loop expansion with dressed quantities = Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 10/14

25 Employ skeleton expansion Skeleton Expansion loop expansion with dressed quantities = All orders have the same IR exponent, (insertions generating higher orders give no additional contributions). Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 10/14

26 Skeleton Expansion Skeleton expansion = calculation of the IR exponent of an arbitrary vertex function with 2n external ghosts and m external gluons: ( ) d δ 2n,m = (n m)κ + (1 n) 2 2 All orders have the same IR exponent, (insertions generating higher orders give no additional contributions). Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 10/14

27 Qualitative Behavior The dimension dependence of the IR behavior of YM Green functions. Dimension κ Ghost κ Ghost is divergent in all three values of space-time dimension. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 11/14

28 Qualitative Behavior The dimension dependence of the IR behavior of YM Green functions. Dimension κ Ghost κ Gluon 2κ + 1 d Ghost is divergent in all three values of space-time dimension. Gluon is vanishing in all three values of space-time dimension, except one case. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 11/14

29 Qualitative Behavior The dimension dependence of the IR behavior of YM Green functions. Dimension κ Ghost κ Gluon 2κ + 1 d gluon 3κ + d gluon 4κ + d Ghost is divergent in all three values of space-time dimension. Gluon is vanishing in all three values of space-time dimension, except one case. Vertices have the same qualitative behavior in all three values of space-time dimension. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 11/14

30 The Ghost Triangle Dominant diagram for three-gluon vertex. Analytic calculation conrms power-like behavior. q+p 1 p 3 ρ q-p 2 p 1 µ q ν p 2 Γ gh,ir µνρ (p 1, p 2, p 3 ) = d d q (q + p 1 ) µ (q p 2 ) ρ q ν 2 (2π) d ((q + p 1 ) 2 ) κ+1 ((q p 2 ) 2 ) κ+1 (q 2 ) κ+1 N c B 3 g 3 Used Method: Negative Dimension Integration (NDIM) Appell's function F 4 Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 12/14

31 Tensor Components of the Ghost Triangle H1,H2,H3 H1,H2,H p2 d=3, symmetric point (p 2 1 = p2 2 = p2 3 ), tensor basis of Celmaster-Gonsalves d=4, symmetric point (p 2 1 = p2 2 = p2 3 ), tensor basis of Celmaster-Gonsalves p 2 2 A1,A2,A3,B2,B3 0.5 A1,A2,A3,B2,B p p2 d=3, orthogonal conguration (p 2 2 = p2 3,p 2 p 3 ), tensor basis of Ball-Chiu d=4, orthogonal conguration (p 2 2 = p2 3,p 2 p 3 ), tensor basis of Ball-Chiu Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 13/14

32 Summary The skeleton expansion works in 2, 3 and 4 dimensions. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 14/14

33 Summary The skeleton expansion works in 2, 3 and 4 dimensions. A general formula for the IR exponent of vertex functions was found. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 14/14

34 Summary The skeleton expansion works in 2, 3 and 4 dimensions. A general formula for the IR exponent of vertex functions was found. It yields the same qualitative behavior of vertex functions in two, three and four dimensions. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 14/14

35 Summary The skeleton expansion works in 2, 3 and 4 dimensions. A general formula for the IR exponent of vertex functions was found. It yields the same qualitative behavior of vertex functions in two, three and four dimensions. Lattice data in lower dimensions can give qualitative results similar to four dimensions. Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 14/14

36 Summary The skeleton expansion works in 2, 3 and 4 dimensions. A general formula for the IR exponent of vertex functions was found. It yields the same qualitative behavior of vertex functions in two, three and four dimensions. Lattice data in lower dimensions can give qualitative results similar to four dimensions. Gribov-Zwanziger connement scenario conrmed in 2 and 3 dimensions!!! Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 14/14

37 Summary The skeleton expansion works in 2, 3 and 4 dimensions. A general formula for the IR exponent of vertex functions was found. It yields the same qualitative behavior of vertex functions in two, three and four dimensions. Lattice data in lower dimensions can give qualitative results similar to four dimensions. Gribov-Zwanziger connement scenario conrmed in 2 and 3 dimensions!!! Thank you for your attention! Huber, Alkofer, Fischer, Schwenzer KFU Graz, TU Darmstadt 14/14