The Role of the Quark-Gluon Vertex in the QCD Phase Transition

Transcription

1 The Role of the Quark-Gluon Vertex in the QCD Phase Transition PhD Seminar, Markus Hopfer University of Graz (A. Windisch, R. Alkofer)

2 Outline 1 Motivation A Physical Motivation Calculations at Finite Temperatures 2 Towards a Solution of the Quark-Gluon Vertex The Coupled System Solution Strategies Numerics 3 Preliminary Results 4 Summary and Outlook

3 A Physical Motivation The Phase Diagram of QCD no free quarks observed in nature Connement how can three light quarks constitute a heavy hadron? Dynamical Breaking of Chiral Symmetry (D χsb) quarks behave massless for high T chiral symmetry restored possible relation between the two phenomena? Why studying the Quark-Gluon Vertex? quark-gluon vertex interaction between quarks/gluons no full self-consistent solution of this object at T = 0 only models available for T 0 and/or µ 0

4 A Theoretical Description - The Formalism +

5 Dyson-Schwinger Equations (DSEs) The - Approach (F.J. Dyson) (J. Schwinger) Derivation of DSEs: cf. Talk by Valentin Mader

6 Dyson-Schwinger Equations and Matsubara Formalism Dyson-Schwinger Equation for the Quark Propagator = + S (p) = Z 2S 0 (p) Z 2C F g 2 4 d k (2π) 4 γµ S(k) Γ ν (p, k) D γ µν (q) Some Comments Dyson-Schwinger equations (DSEs) non-perturbative approach (coupled) integral equations µ T -plane accessible innite tower truncations Momentum Conventions p... external momentum k... loop momentum q = p-k... gluon momentum

7 Dyson-Schwinger Equations and Matsubara Formalism Dyson-Schwinger Equation for the Quark Propagator = + S (p) = Z 2S 0 (p) Z 2C F g 2 k0 2π 3 d k (2π) 3 γµ S(k) Γ ν (p, k) D γ µν (q) Going to nite Temperature treat 0-component of all 4-vectors dierently heat bath breaks Lorentz symmetry, i.e. (Euclidean) O(4) symmetry replace integral in time direction by Matsubara sum Momentum Conventions p (ω p, p) k (ω k, k) q (ω q, q) p 2 = (ω p, p) 2 = ω 2 q + p 2

8 Dyson-Schwinger Equations and Matsubara Formalism Dyson-Schwinger Equation for the Quark Propagator = + S (p) = Z 2S 0 (p) Z 2C F g 2 T ω k (φ) 3 d k (2π) 3 γµ S(k) Γ ν (p, k) D γ µν (q) Going to nite Temperature treat 0-component of all 4-vectors dierently heat bath breaks Lorentz symmetry, i.e. (Euclidean) O(4) symmetry replace integral in time direction by Matsubara sum Momentum Conventions p (ω p, p) k (ω k, k) q (ω q, q) p 2 = (ω p, p) 2 = ω 2 q + p 2

9 Dyson-Schwinger Equations and Matsubara Formalism Dyson-Schwinger Equation for the Quark Propagator = + S (p) = Z 2S 0 (p) Z 2C F g 2 T ω k (φ) 3 d k (2π) 3 γµ S(k) Γ ν (p, k) D γ µν (q) Modication of the Quark Propagator S (p) = i /pa(p 2 ) + B(p 2 ) S (ω p, p) = i ω pγ 0C(ω p, p 2 ) + i / pa(ω p, p 2 ) + B(ω p, p 2 ) generalized Matsubara modes: ω p(φ) = 2πT (n p + φ/2π) φ = π... usual (fermionic) anti-periodic boundary conditions φ = 0... usual (bosonic) periodic boundary conditions solve self-consistently further input needed gluon propagator & quark-gluon vertex

10 Further Ingredients

11 Dyson-Schwinger Equations and Matsubara Formalism Dyson-Schwinger Equation for the Quark Propagator = + S (p) = Z 2S 0 (p) Z 2C F g 2 T ω k (φ) 3 d k (2π) 3 γµ S(k) Γ ν (p, k) D γ µν (q) Modication of the Gluon Propagator gluon (in Landau gauge) is purley transversal vector particle splits into transversal and longitudinal part at T 0 D γ µν (q) D µν (ωq, q) + Dµν (ωq, q) γ,t γ,l take from lattice results [Fischer, Maas and Mueller, 2010]

12 Dyson-Schwinger Equations and Matsubara Formalism Temperature Dependent Gluon Propagator D µν(ω p, p) = ZT (p2 ) Pµν(p) T + ZL(p 2 ) Pµν(p) L, p = (ω p, p) p 2 p 2 longitudinal part sensitive to phase transition, i.e. drops at T c Figure: quenched lattice data for SU(3) [Fischer, Maas and Mueller, 2010]

13 Dyson-Schwinger Equations and Matsubara Formalism Dyson-Schwinger Equation for the Quark Propagator S (p) = Z 2S 0 (p) Z 2C F g 2 T = + ω k (φ) 3 d k (2π) 3 γµ S(k) Γ ν (p, k) D γ µν (q) Models for the Quark-Gluon Vertex vertex model taken from [Fischer, 2009], cf. [Fischer and Mueller, 2009] Γ ν (ω p, p; ω k, { } { } ν d1 k) = kinematical term d q 2 UV blue term STI motivated kinematical expression red term phenomenological model new parameters: d 1, d 2 ( proper perturbative running: UV k 2 β0α(µ) ln[ q2 Λ 2 + 1] Λ 2 + q 2 4π ) 2δ

14 DSEs In Practical Application: Doing Physics

15 The Model Dependence of Observables Quark/Chiral Condensate φ (T ) ψψ φ = Z 2N c T ω p(φ) 3 d p { } (2π) 3 tr D S(ω p(φ), p) order parameter for chiral phase transition in the χ limit ϕ [GeV 3 ] ϕ/2π T [MeV]

16 The Model Dependence of Observables Quark/Chiral Condensate order parameter for chiral symmetry breaking no crossover because of quenched gluon input! quark-gluon vertex models work remarkably well! see e.g. plots of QCD phase diagram [C. Fischer, J. Luecker, 2012] but: introduce new parameters x to physical observables (e.g. fπ) 0,2 0,15 d1 = 4.6 π (T) [GeV 3 ] 0,1 CHIRAL SYMMETRY 0,05 BROKEN CHIRAL SYMMETRY RESTORED T [ MeV] [Fischer, Maas and Mueller, 2010]

17 The Model Dependence of Observables Quark/Chiral Condensate order parameter for chiral symmetry breaking no crossover because of quenched gluon input! quark-gluon vertex models work remarkably well! π (T) [GeV 3 ] but: introduce new parameters x to physical observables (e.g. fπ) goal: better understanding of the quark-gluon vertex function 0,2 0,15 0,1 0,05 CHIRAL SYMMETRY BROKEN d1 = 5.5 d1 = 4.6 d1 = 4.0 d1 = 3.5 d1 = 3.3 d1 = 3.0 CHIRAL SYMMETRY RESTORED T [ MeV]

18 The Model Dependence of Observables Dual Quark Condensate cf. [Gattringer, 2006], [E. Bilgici et al., 2008], [Fischer, 2009],... 2π dφ π dφ Σ 1 = 0 2π e iφ ψψ φ = 0 π cos φ ψψ φ order parameter for center symmetry transforms as Polyakov loop Dual Quark Condensate 0,14 0,12 0,1 0,08 0,06 0,04 0, T [MeV]

19 The Model Dependence of Observables Dual Quark Condensate cf. [Gattringer, 2006], [E. Bilgici et al., 2008], [Fischer, 2009],... 2π dφ π dφ Σ 1 = 0 2π e iφ ψψ φ = 0 π cos φ ψψ φ order parameter for center symmetry transforms as Polyakov loop 0,14 Dual Quark Condensate 0,12 0,1 0,08 0,06 0,04 0,02 d1 = 5.5 d1 = 4.6 d1 = 4.0 d1 = 3.3 d1 = T [MeV]

20 The Quark-Gluon Vertex Seems To Be Important!! Can We Solve It? The Quark-Gluon Vertex - Earlier Investigations

21 The Infrared Behaviour of the System Yang-Mills Sector of QCD Kugo-Ojima Connement Scenario (and similar in Gribov-Zwanziger scenario) Gluon Propagator p 2 A(p)A( p) p2 0 0 vanishes for small momenta Ghost Propagator p 2 c(p) c( p) p2 0 diverges for small momenta Scaling vs. Decoupling

22 Scaling vs. Decoupling Scaling and Decoupling Solution(s) for Gluon/Ghost Dressing Functions Z(x) x = p 2 [GeV 2 ] MH, R. Alkofer and G. Haase, 2012 blue line: ghost dressing function red line : gluon dressing function dashed/dashed-dotted lines: decoupling solutions G(x)

23 Scaling vs. Decoupling Non-Perturbative Running Coupling 4 3 α(x) x = p 2 [GeV 2 ] MH, R. Alkofer and G. Haase, 2012 running coupling approaches an IR xed point in scaling scenario dashed/dashed-dotted lines: decoupling solutions

24 The Infrared Behaviour of the System Yang-Mills Sector of QCD Kugo-Ojima Connement Scenario (and similar in Gribov-Zwanziger scenario) Gluon Propagator p 2 A(p)A( p) p2 0 0 vanishes for small momenta Ghost Propagator p 2 c(p) c( p) p2 0 diverges for small momenta The Role of the Quark-Gluon Vertex? Linear rising quark potential? D χsb eective quark interaction strength enhanced quark-gluon vertex? Running coupling infrared xed point?

25 Dyson-Schwinger Equations Dyson-Schwinger Equation for the Quark Propagator = + S (p) = S 0 (p) + g 2 C N c žk γ µ S(k) Γ ν (p, k) D µν γ (p k) Some Comments Quark-Gluon Vertex: rainbow truncation, BC/CP-type vertex constructions,... DχSB eective interaction strength put eective interaction e.g. into gluon propagator and/or use sophisticated vertex models Goal: full self-consistent solution at vanishing temperature 12 tensors extend to nite temperature 32 tensors improve nite T -models Consider T = µ = 0

26 Dyson-Schwinger Equations Dyson-Schwinger Equation for the Quark Propagator = + S (p) = S 0 (p) + g 2 C N c žk γ µ S(k) γ ν D µν γ (p k) Some Comments Quark-Gluon Vertex: rainbow truncation, BC/CP-type vertex constructions,... DχSB eective interaction strength put eective interaction e.g. into gluon propagator and/or use sophisticated vertex models Goal: full self-consistent solution at vanishing temperature 12 tensors extend to nite temperature 32 tensors improve nite T -models Consider T = µ = 0

27 Dyson-Schwinger Equations Dyson-Schwinger Equation for the Quark Propagator = + S (p) = S 0 (p) + g 2 C N c žk γ µ S(k) Γ ν (p, k) D µν γ (p k) Some Comments Quark-Gluon Vertex: rainbow truncation, BC/CP-type vertex constructions,... DχSB eective interaction strength put eective interaction e.g. into gluon propagator and/or use sophisticated vertex models Goal: full self-consistent solution at vanishing temperature 12 tensors extend to nite temperature 32 tensors improve nite T -models Basis (if T = µ = 0) 1 Γ ν /k /p /k/p γ ν k ν p ν i.e. Γ ν 12 i=1 λ i Γ i

28 Dyson-Schwinger Equations Dyson-Schwinger Equation for the Quark Propagator = + S (p) = S 0 (p) + g 2 C N c žk γ µ S(k) Γ ν (p, k) D µν γ (p k) Some Comments Quark-Gluon Vertex: rainbow truncation, BC/CP-type vertex constructions,... DχSB eective interaction strength put eective interaction e.g. into gluon propagator and/or use sophisticated vertex models Goal: full self-consistent solution at vanishing temperature 12 tensors extend to nite temperature 32 tensors improve nite T -models Basis (if T = µ = 0) 1 γ ν Γ ν /k k ν /p p ν /k/p i.e. Γ ν 12 i=1 λ i Γ i

29 The Coupled System Dyson-Schwinger Equation for the Quark Propagator = + Dyson-Schwinger Equation for the Quark-Gluon Vertex = 1 N c +N c Remarks/Ingredients gluon propagator taken from lattice/dse calculations 3-gluon vertex only models available

30 The Coupled System Dyson-Schwinger Equation for the Quark Propagator = + Dyson-Schwinger Equation for the Quark-Gluon Vertex = 1 N c +N c Remarks/Ingredients dress all vertices include higher order corrections correspondence to DSE-like equations in 3PI formalism [J. Berges, 2004]

31 The Coupled System Some Technical Details: Dierent Descriptions of the System

32 Dyson-Schwinger Equation for the Quark-Gluon Vertex I Version 1 = Version 2 = + +

33 Dyson-Schwinger Equation for the Quark-Gluon Vertex I Version 1 Version 2 = EQUIVALENT DESCRIPTIONS = + +

34 Dyson-Schwinger Equation for the Quark-Gluon Vertex II Analogy to 3PI Formalism [J. Berges, 2004] = The Resulting Simplied System =

35 Dyson-Schwinger Equation for the Quark-Gluon Vertex II Analogy to 3PI Formalism [J. Berges, 2004] = The Resulting Simplied System =

36 The Coupled System Dyson-Schwinger Equation for the Quark Propagator = + Dyson-Schwinger Equation for the Quark-Gluon Vertex = 1 N c +N c Simplications/Tools investigate scalar theory fundamentally charged scalars no Dirac structure simplied playground [MH, diploma thesis, 2011] calculations performed on GPUs

37 The Coupled System Dyson-Schwinger Equation for the Quark Propagator = + Abelian Diagram non-abelian Diagram Dyson-Schwinger Equation for the Quark-Gluon Vertex = 1 N c +N c Simplications/Tools investigate scalar theory fundamentally charged scalars no Dirac structure simplied playground [MH, diploma thesis, 2011] calculations performed on GPUs

38 The Coupled System in a Leading Order Skeleton Expansion Contribution of the Abelian/non-Abelian Diagram = λ 1 NA 1 - λ 1 A 10-2 = 1 Nc +N c p 2 = x [GeV 2 ] Leading Order Skeleton Expansion Γ µ (p 1, p 2; p 2 p 1) = 12 i=1 λ i (p 2 1, p 2 2, p 1 p 2) L µ i pick out an arbitrary vertex dressing function here: λ 1 ( γ µ ) at p 2 1 = p2 2 = p2 p1.p2 = 0 Abelian diagram is sub-leading - non-abelian diagram is important Leading Order Skeleton Expansion generates no dynamical mass!!

39 The Coupled System in a Leading Order Skeleton Expansion Contribution of the Abelian/non-Abelian Diagram = λ 1 NA 1 - λ 1 A 10-2 = 1 Nc +N c p 2 = x [GeV 2 ] Leading Order Skeleton Expansion Γ µ (p 1, p 2; p 2 p 1) = 12 i=1 λ i (p 2 1, p 2 2, p 1 p 2) L µ i pick out an arbitrary vertex dressing function here: λ 1 ( γ µ ) at p 2 1 = p2 2 = p2 p1.p2 = 0 Abelian diagram is sub-leading - non-abelian diagram is important Leading Order Skeleton Expansion generates no dynamical mass!!

40 Solution Strategies non-abelian Diagram The Quark-Gluon Vertex - Step-by-Step = 1 N c +N c Solution Strategy - Step 1 include only parts of the non-abelian diagram dress all vertices in the non-abelian diagram system is now much more complicated try dierent models for the 3-gluon vertex (sign ip!?) include the Abelian diagram (suppressed!?)

41 Solution Strategies non-abelian Diagram The Quark-Gluon Vertex - Step-by-Step = 1 N c +N c Solution Strategy - Step 2 include only parts of the non-abelian diagram dress all vertices in the non-abelian diagram system is now much more complicated try dierent models for the 3-gluon vertex (sign ip!?) include the Abelian diagram (suppressed!?)

42 Solution Strategies non-abelian Diagram The Quark-Gluon Vertex - Step-by-Step = 1 N c +N c Solution Strategy - Step 3 include only parts of the non-abelian diagram dress all vertices in the non-abelian diagram system is now much more complicated try dierent models for the 3-gluon vertex (sign ip!?) include the Abelian diagram (suppressed!?)

43 Numerics

44 Numerics GPU Programing calculations on GPUs Mephisto GPU Cluster Research Core Area 'Modeling and Simulation' Code Development CUDA-C++/FORTRAN OpenACC GPU- and CPU Cluster

45 Numerics GPU Programing calculations on GPUs Mephisto GPU Cluster Research Core Area 'Modeling and Simulation' Code Development CUDA-C++/FORTRAN OpenACC GPU- and CPU Cluster

46 Parallelization of the Code Step 1 - Discretization Γ µ (p 2 1, p 2 2, p 1 p 2) Γ µ (x, y, z) Γ µ [x i ][y n][z j ] Step 2 - Load Distribution Γ µ [x 1][y n][z j ] GPU 1 (processid = 0) Γ µ [x 2][y n][z j ] GPU 2 (processid = 1).. Γ µ [x N ][y n][z j ] GPU N (processid = N-1) Step 3 - Global Synchronization collect GPU data and repeat until convergence Some Technical Details coarse graining with openmpi / ne graining with CUDA in principle arbitrary number of GPUs possible minimal communication costs / optimal load balancing very good scaling behaviour of the code

47 Parallelization of the Code Scaling of the CUDA Code Speedup optimal Tesla C Number of GPUs Some Technical Details coarse graining with openmpi / ne graining with CUDA in principle arbitrary number of GPUs possible minimal communication costs / optimal load balancing very good scaling behaviour of the code

48 Preliminary Results

49 The Coupled System Dyson-Schwinger Equation for the Quark Propagator = + Dyson-Schwinger Equation for the Quark-Gluon Vertex = +N c Setup include only the (partially dressed) non-abelian diagram solve the system self-consistently using all tensor structures scaling-type gluon propagator used as input (from another DSE calculation)

50 Preliminary Results Quark Propagator CHIRAL LIMIT MASSIVE CASE 1,2 1,0 1 / A(x) B(x) M(x) = B(x) / A(x) ,8 0,6 0,4 0,2 M(x) [GeV] CHIRAL LIMIT m(µ 2 ) = 0 m(µ 2 ) = 1 GeV 0, x = p 2 [GeV 2 ] x = p 2 [GeV 2 ] Dressing Functions quark propagator: S (p) = i /p A(p 2 ) + 1B(p 2 ) dynamical chiral symmetry breaking (DχSB)

51 Preliminary Results Relevance of Dierent Tensor Structures - The Idea isolate important tensor structures: Γ µ 12 i=1 λ i Γ µ i λ i... dressing function Γ i... corresponding basis element (tensor component) Contribution of Tensor Structures (sketched)

52 Preliminary Results 3-Gluon Vertex Dependence 0,5 0,4 IR Exponent: -0 κ M(x) 0,3 0,2 0, x = p 2 [GeV 2 ] 3-Gluon-Vertex bare structure + model ( x Γ µνσ = 3g x + d 1 x = p p p 2 3 ) 3κ ( d 1 d 3 + d2 log d 1 + x d 1, d 2, d 3... free parameters [ ]) 17/44 x + 1 Γ µνσ 3g,0 d 1

53 Preliminary Results 3-Gluon Vertex Dependence 0,5 0,4 IR Exponent: -0 κ M(x) 0,3 0,2 0, IR Analysis x = p 2 [GeV 2 ] 3-Gluon-Vertex bare structure + model ( x Γ µνσ = 3g x + d 1 x = p p p 2 3 ) 3κ ( d 1 d 3 + d2 log d 1 + x d 1, d 2, d 3... free parameters [ ]) 17/44 x + 1 Γ µνσ 3g,0 d 1

54 Preliminary Results 3-Gluon Vertex Dependence 0,5 0,4 IR Exponent: -0 κ M(x) 0,3 0,2 0, x = p 2 [GeV 2 ] 3-Gluon-Vertex bare structure + model ( x Γ µνσ = 3g x + d 1 x = p p p 2 3 ) 0κ ( d 1 d 3 + d2 log d 1 + x d 1, d 2, d 3... free parameters [ ]) 17/44 x + 1 Γ µνσ 3g,0 d 1

55 Preliminary Results 3-Gluon Vertex Dependence 0,5 0,4 IR Exponent: -0 κ IR Exponent: -1 κ M(x) 0,3 0,2 0, x = p 2 [GeV 2 ] 3-Gluon-Vertex bare structure + model ( x Γ µνσ = 3g x + d 1 x = p p p 2 3 ) κ ( d 1 d 3 + d2 log d 1 + x d 1, d 2, d 3... free parameters [ ]) 17/44 x + 1 Γ µνσ 3g,0 d 1

56 Preliminary Results 3-Gluon Vertex Dependence 0,5 M(x) 0,4 0,3 0,2 IR Exponent: -0 κ IR Exponent: -1 κ IR Exponent: -2 κ 0, x = p 2 [GeV 2 ] 3-Gluon-Vertex bare structure + model ( x Γ µνσ = 3g x + d 1 x = p p p 2 3 ) 2κ ( d 1 d 3 + d2 log d 1 + x d 1, d 2, d 3... free parameters [ ]) 17/44 x + 1 Γ µνσ 3g,0 d 1

57 Preliminary Results 3-Gluon Vertex Dependence 0,5 M(x) 0,4 0,3 0,2 IR Exponent: -0 κ IR Exponent: -1 κ IR Exponent: -2 κ 0, NO STABLE SOLUTION x = p 2 [GeV 2 ] 3-Gluon-Vertex bare structure + model ( x Γ µνσ = 3g x + d 1 x = p p p 2 3 ) 3κ ( d 1 d 3 + d2 log d 1 + x d 1, d 2, d 3... free parameters [ ]) 17/44 x + 1 Γ µνσ 3g,0 d 1

58 Taking a Closer Look at the 3-Gluon Vertex

59 Taking a Closer Look at the 3-Gluon Vertex 3-Gluon Vertex - Sign Flip in 2D Lattice Data for 2 Dimensions, [Huber, Maas and Smekal, 2012] lattice data indicates sign ip in 2D

60 Taking a Closer Look at the 3-Gluon Vertex 3-Gluon Vertex - Sign Flip in 3D Lattice Data for 3 Dimensions, [Cucchieri, Maas, Mendes, 2008] lattice data indicates sign ip in 3D

61 Taking a Closer Look at the 3-Gluon Vertex 3-Gluon Vertex - Sign Flip in 4D Lattice Data for 4 Dimensions, [Cucchieri, Maas, Mendes, 2008] lattice data indicates sign ip in 4D

62 Taking a Closer Look at the 3-Gluon Vertex 3-Gluon Vertex - Sign Flip in 4D Lattice Data for 4 Dimensions, [Huber and Smekal, 2012] lattice data indicates sign ip in 4D try to implement sign ip also in the quark-gluon vertex calculations stabilizes the system considerably

63 The Improved 3-Gluon-Vertex Model in DSE Calculations The Gluon Propagator (Dressing Function) 4 3 DSE DSE improved Sternbeck, 2006 Z(x) x = p 2 [GeV 2 ] new model corrects the long-standing under-estimation of the mid-momentum regime in DSE calculations 2-loop diagrams may play a minor role (cf. talk by V. Mader), i.e. truncations are already very good

64 Preliminary Results Full Calculation with non-abelian Diagram M(x) 0,4 0,3 0,2 IR Exponent: -0 κ IR Exponent: -1 κ IR Exponent: -2 κ improved Γ 3g 0, x = p 2 [GeV 2 ] full calculation taking non-abelian diagram into account = 1 N c +N c

65 What Can We Do Now isolate important tensor structure relevant for DχSB crucial for T 0 and µ 0 calculations The Next Steps - To Do List look for scaling behaviour dierent initialization strategy phenomenological eects not aected by deep IR behaviour include Abelian diagram additionally stabilize the system compare with lattice data [A. Kizilersu, D. Leinweber, J. Skullerud, A. Williams, 2006] [J. Skullerud, P. Bowman, A. Kizilersu, D. Leinweber, A. Williams, 2005]

66 Summary and Outlook Summary quark connement D χsb quark-gluon vertex quark/gluon coupling solve the non-abelian part of the vertex DSE isolate important tensor structures Quark Propagator DSE = + Quark-Gluon Vertex DSE = 1 N c +N c

67 Summary and Outlook Outlook calculate Abelian diagram, i.e. the full system include Yang-Mills system unquenching eects extend towards µ T study critical endpoint investigate scalar theory Quark Propagator DSE = + Quark-Gluon Vertex DSE = 1 N c +N c

68

69 Preliminary Results Running Coupling α qg (x) 1,2 1 0,8 0,6 0,4 0,2 IR Exponent: -0 κ IR Exponent: -1 κ IR Exponent: -2 κ x = p 2 [GeV 2 ] Right Scaling!? running coupling: α qg (p 2 ) = g 2 4π λ2 1(p 2 )Z(p 2 )/A 2 (p 2 ) no scaling behaviour: λ 1 = const in IR region needs further investigation dierent initialization strategy