Chiral symmetry breaking in continuum QCD

Transcription

1 Chiral symmetry breaking in continuum QCD Mario Mitter Ruprecht-Karls-Universität Heidelberg GSI, February 9, 216 M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

2 fqcd collaboration - QCD (phase diagram) with FRG: J. Braun, A. K. Cyrol, L. Fister, W. J. Fu, T. K. Herbst, MM N. Müller, J. M. Pawlowski, S. Rechenberger, F. Rennecke, N. Strodthoff M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

3 Table of Contents 1 Motivation 2 Quenched Landau gauge QCD with the FRG 3 YM correlation functions 4 A glimpse at unquenching 5 Phenomenological Application: η -meson mass at chiral crossover 6 Conclusion M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

4 QCD phase diagram with functional methods works well at µ = : agreement with lattice (ε - 3P)/T 4 8 Wuppertal-Budapest, 21 7 HotQCD N t =8, 212 HotQCD N t =12, PQM FRG PQM emf 5 PQM MF t [Herbst, MM, Pawlowski, Schaefer, Stiele, 13] 1.8 Lattice set A set B l,s (T) / l,s (T=) T [MeV] [Luecker, Fischer, Welzbacher, 214] [Luecker, Fischer, Fister, Pawlowski, 13] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

5 QCD phase diagram with functional methods works well at µ = : agreement with lattice different results at large µ (possibly already at small µ) T [MeV] µb T =2 χ crossover σ(t=)/2 Φ crossover Φ crossover χ 1st order CEP µb T = µ [MeV] m π =138 MeV [Herbst, Pawlowski, Schaefer, 213] [Braun, Haas, Pawlowski, unpublished] 2 15 µ B /T=2 µ B /T=3 T [MeV] 1 5 Lattice: curvature range κ= DSE: chiral crossover DSE: critical end point DSE: deconfinement crossover µ q [MeV] [Luecker, Fischer, Fister, Pawlowski, 13] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

6 QCD phase diagram with functional methods works well at µ = : agreement with lattice different results at large µ (possibly already at small µ) calculations need model input: Polyakov-quark-meson model with FRG: inital values at Λ O(Λ QCD ) input for Polyakov loop potential quark propagator DSE: IR quark-gluon vertex T [MeV] µb T =2 χ crossover σ(t=)/2 Φ crossover Φ crossover χ 1st order CEP µb T = µ [MeV] m π =138 MeV [Herbst, Pawlowski, Schaefer, 213] [Braun, Haas, Pawlowski, unpublished] T [MeV] Lattice: curvature range κ= DSE: chiral crossover DSE: critical end point µ B /T=2 DSE: deconfinement crossover µ B /T= µ q [MeV] [Luecker, Fischer, Fister, Pawlowski, 13] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

7 QCD phase diagram with functional methods works well at µ = : agreement with lattice different results at large µ (possibly already at small µ) calculations need model input: Polyakov-quark-meson model with FRG: inital values at Λ O(Λ QCD ) input for Polyakov loop potential quark propagator DSE: IR quark-gluon vertex possible explanation for disagreement: µ : relative importance of diagrams changes summed contributions vs. individual contributions T [MeV] µb T =2 χ crossover σ(t=)/2 Φ crossover Φ crossover χ 1st order CEP µb T = µ [MeV] m π =138 MeV [Herbst, Pawlowski, Schaefer, 213] [Braun, Haas, Pawlowski, unpublished] T [MeV] Lattice: curvature range κ= DSE: chiral crossover DSE: critical end point µ B /T=2 DSE: deconfinement crossover µ B /T= µ q [MeV] [Luecker, Fischer, Fister, Pawlowski, 13] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

8 Back to QCD in the vacuum (Wetterich equation) use only perturbative QCD input α S (Λ = O(1) GeV) m q (Λ = O(1) GeV) M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

9 Back to QCD in the vacuum (Wetterich equation) use only perturbative QCD input α S (Λ = O(1) GeV) m q (Λ = O(1) GeV) Wetterich equation with initial condition S[Φ] = Γ Λ [Φ] k Γ k [A, c, c, q, q] = 1 2 effective action Γ[Φ] = lim k Γ k [Φ] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

10 Back to QCD in the vacuum (Wetterich equation) use only perturbative QCD input α S (Λ = O(1) GeV) m q (Λ = O(1) GeV) Wetterich equation with initial condition S[Φ] = Γ Λ [Φ] k Γ k [A, c, c, q, q] = 1 2 effective action Γ[Φ] = lim k Γ k [Φ] k : integration of momentum shells controlled by regulator full field-dependent equation with (Γ (2) [Φ]) 1 on rhs gauge-fixed approach (Landau gauge): ghosts appear M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

11 Vertex Expansion approximation necessary - vertex expansion Γ[Φ] = n p 1,...,p n 1 Γ (n) Φ 1 Φ n (p 1,..., p n 1 )Φ 1 (p 1 ) Φ n ( p 1 p n 1 ) M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

12 Vertex Expansion approximation necessary - vertex expansion Γ[Φ] = n p 1,...,p n 1 Γ (n) Φ 1 Φ n (p 1,..., p n 1 )Φ 1 (p 1 ) Φ n ( p 1 p n 1 ) functional derivatives with respect to Φ i = A, c, c, q, q: equations for 1PI n-point functions, e.g. gluon propagator: t 1 = M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

13 Vertex Expansion approximation necessary - vertex expansion Γ[Φ] = n p 1,...,p n 1 Γ (n) Φ 1 Φ n (p 1,..., p n 1 )Φ 1 (p 1 ) Φ n ( p 1 p n 1 ) functional derivatives with respect to Φ i = A, c, c, q, q: equations for 1PI n-point functions, e.g. gluon propagator: t 1 = want apparent convergence of Γ[Φ] = lim k Γ k [Φ] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

14 Quenched Landau gauge QCD two crucial phenomena: SχSB and confinement similar scales - hard to disentangle see e.g. [Williams, Fischer, Heupel, 215] quenched QCD: allows separate investigation: M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

15 Quenched Landau gauge QCD two crucial phenomena: SχSB and confinement similar scales - hard to disentangle see e.g. [Williams, Fischer, Heupel, 215] quenched QCD: allows separate investigation: matter part [MM, Strodthoff, Pawlowski, 214] (with FRG-YM propagators from [Fischer, Maas, Pawlowski, 29], [Fister, Pawlowski, unpublished]) recent results for YM propagators [Cyrol, Fister, MM, Pawlowski, Strodthoff, to be published] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

16 Chiral symmetry breaking χsb resonance in 4-Fermi interaction λ (pion pole): M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

17 Chiral symmetry breaking χsb resonance in 4-Fermi interaction λ (pion pole): resonance singularity without momentum dependency t λ = a λ 2 + b λ α + c α 2, b >, a, c t = t λ g = g =, T > g = g cr λ g > g cr [Braun, 211] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

18 Effective running couplings [MM, Pawlowski, Strodthoff, 214] running couplings α s p [GeV] α crit α c - Ac α q α A 3 agreement in perturbative regime required by gauge symmetry non-degenerate in nonperturbative regime: reflects gluon mass gap α qaq > α cr : necessary for chiral symmetry breaking area above α cr very sensitive to errors M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

19 4-Fermi vertex via dynamical hadronization [Gies, Wetterich, 22] change of variables: particular 4-Fermi channels meson exchange efficient inclusion of momentum dependence no singularities identifies relevant effective low-energy dofs from QCD k Γ k = (bosonized) 4-fermi-interaction h 2 π /(2 m2 π ) λ π RG-scale k [GeV] Yukawa interaction h π RG-scale k [GeV] [MM, Strodthoff, Pawlowski, 214] [Braun, Fister, Haas, Pawlowski, Rennecke, 214] [MM, Strodthoff, Pawlowski, 214] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

20 Vertex Expansion in the matter system -1 FRG Yang-Mills results -1 mom. dep. classical tensor structure mom. dep. classical tensor structure -1 full mom. dep. all tensor structures mom. dep. full mom. dep. STI-consistent dressing Fierz-complete basis at p = and mom. dep. -1 mom. dep. full effective... potential mom. dep. M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

21 Derivation of equations VertEXpand Mathematica package for the derivation of vertices from a given action using FORM (Denz,Held,Rodigast; unpub.) Vertices/ Vertices/ Feynman Feynman Rules Rules DoFun VertEXpand Action ERGE DoFun DoFun Mathematica package for the derivation of functional equations (Braun,Huber; Comput.Phys. Commun. 183 (212) ) Symbolic Symbolic Flow Flow Equations Equations FORMTracer Algebraic Algebraic Flow Flow Equations Equations high-performance, easy-to-use Mathematica tracing tool using FORM (Cyrol,Mitter,Pawlowski,Strodthoff; in prep.) Compilable Compilable Kernels Kernels CreateKernels Mathematica package for the automatic generation of compilable C++ kernels for use in connection with the frgsolver (Cyrol,Mitter,Pawlowski,Strodthoff; unpub.) Numerical Numerical solution solution frgsolver Flexible, high-performance, parallelized C++ OOP framework for the numerical solution of functional equations (Cyrol,Mitter,Pawlowski,Strodthoff; unpub.) [Cyrol, MM, Pawlowski, Strodthoff, ] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

22 Equations in the matter system [MM, Strodthoff, Pawlowski, 214] 1 t = t = perm. t = 2 + perm. M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

23 Equations in the matter system [MM, Strodthoff, Pawlowski, 214] t = perm. 1 t = t = + perm. + + t = perm. t = 2 + perm. M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

24 Equations in the matter system [MM, Strodthoff, Pawlowski, 214] t = perm. 1 t = t = + perm. + + t = perm. t = 2 + perm. 1 t = t = perm. M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

25 Gluon FRG input Γ (2) AA (p) Z A(p) p 2 ( δ µν p µ p ν /p 2) [Fischer, Maas, Pawlowski, 29], [Fister, Pawlowski, unpublished] gluon propagator dressing 1/Z A Bowman et al., 4 Sternbeck et al., p [GeV] FRG FRG result self-consistent calculation within FRG approach sets the scale in comparison to lattice QCD M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

26 Quark propagator [MM, Pawlowski, Strodthoff, 214] Γ (2) qq (p) Z q(p) /p + M(p) quark propagator dressings Bowman et al., 5 1/Z q M q p [GeV] FRG vs. lattice: bare mass, quenched, scale M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

27 Quark propagator [MM, Pawlowski, Strodthoff, 214] Γ (2) qq (p) Z q(p) /p + M(p) quark propagator dressings Bowman et al., 5 1/Z q M q p [GeV] FRG vs. lattice: bare mass, quenched, scale agreement not sufficient: need apparent convergence at µ M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

28 other 4-Fermi channels (mesons) [MM, Pawlowski, Strodthoff, 214] (bosonized) 4-fermi-interactions RG-scale k [GeV] h 2 π /(2 m2 π ) λ η λ (S+P)- adj λ V-A λ V+A λ (V-A) adj λ (S-P)- λ(s-p)- adj λ (S+P)+ λ(s+p)+ adj bosonized only σ-π-channel sufficient diquark momentum configuration more important other channels: quantitatively not important in loops M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

29 Quark-gluon interactions I [MM, Pawlowski, Strodthoff, 214] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis sufficient chiral symmetry breaking strength? M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

30 Quark-gluon interactions I [MM, Pawlowski, Strodthoff, 214] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis sufficient chiral symmetry breaking strength? 2 (1) (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry p [GeV] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

31 Quark-gluon interactions I [MM, Pawlowski, Strodthoff, 214] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis sufficient chiral symmetry breaking strength? 2 (1) (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry p [GeV] important non-classical tensors: c.f., [Hopfer et al., 212], [Williams, 214], [Aguilar et al., 214] qγ 5 γ µɛ µνρσ{f νρ, D σ}q ( 1 2 T (5) qaq + T (7) qaq ): increases Zq/decreases Mq considerably anom. chromomagn. momentum (T (4) qaq ) increases Mq moderately M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

32 Quark-gluon interactions I [MM, Pawlowski, Strodthoff, 214] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis sufficient chiral symmetry breaking strength? 2 (1) (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry p [GeV] important non-classical tensors: c.f., [Hopfer et al., 212], [Williams, 214], [Aguilar et al., 214] qγ 5 γ µɛ µνρσ{f νρ, D σ}q ( 1 2 T (5) qaq + T (7) qaq ): increases Zq/decreases Mq considerably anom. chromomagn. momentum (T (4) qaq ) increases Mq moderately considerably less chiral symmetry breaking with full tensor basis M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

33 Quark-gluon interactions I [MM, Pawlowski, Strodthoff, 214] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis sufficient chiral symmetry breaking strength? 2 (1) (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry p [GeV] important non-classical tensors: c.f., [Hopfer et al., 212], [Williams, 214], [Aguilar et al., 214] qγ 5 γ µɛ µνρσ{f νρ, D σ}q ( 1 2 T (5) qaq + T (7) qaq ): increases Zq/decreases Mq considerably anom. chromomagn. momentum (T (4) qaq ) increases Mq moderately considerably less chiral symmetry breaking with full tensor basis also important ingredient for bound-state equations M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

34 Quark-gluon interactions II missing strength? M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

35 Quark-gluon interactions II missing strength? expansion in tensor structures expansion in operators ψ /D n ψ [MM, Pawlowski, Strodthoff, 214] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

36 Quark-gluon interactions II missing strength? expansion in tensor structures expansion in operators ψ /D n ψ [MM, Pawlowski, Strodthoff, 214] in particular qγ 5 γ µ ɛ µνρσ {F νρ, D σ }q: contributes to qaq, qa 2 q and qa 3 q contains important non-classical tensors ( qaq) considerable contribution to quark-gluon vertex ( qa 2 q) contribution to qa 3 q seems unimportant M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

37 Quark-gluon interactions II missing strength? expansion in tensor structures expansion in operators ψ /D n ψ [MM, Pawlowski, Strodthoff, 214] in particular qγ 5 γ µ ɛ µνρσ {F νρ, D σ }q: contributes to qaq, qa 2 q and qa 3 q contains important non-classical tensors ( qaq) considerable contribution to quark-gluon vertex ( qa 2 q) contribution to qa 3 q seems unimportant M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

38 Quark-gluon interactions II missing strength? expansion in tensor structures expansion in operators ψ /D n ψ [MM, Pawlowski, Strodthoff, 214] in particular qγ 5 γ µ ɛ µνρσ {F νρ, D σ }q: contributes to qaq, qa 2 q and qa 3 q contains important non-classical tensors ( qaq) considerable contribution to quark-gluon vertex ( qa 2 q) contribution to qa 3 q seems unimportant explicit calculations of AA qq-vertex: full basis: 63 chirally symmetric tensor elements 3 15 chirally symmetric tensor elements ( ψ /D ψ): all seem important order of effect similar to qγ 5γ µɛ µνρσ{f νρ, D σ}q why? underlying principle? [MM, Pawlowski, Strodthoff, in prep.] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

39 Stability of truncation (apparent convergence) Expansion of effective action in 1PI correlators full mom. dep. classical tensor structure mom. dep. (sym. channel) under investigation: full tensor structure mom. dep. (sym. channel) full tensor structure full mom. dep. partial tensor structure mom. dep. (sym. channel) full tensor structure mom. dep. (single channel) full mom. dep. via effective potential full tensor structure mom. dep. (sym. channel) M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

40 Vertex Expansion in YM theory [Cyrol, Fister, MM, Strodthoff, Pawlowski, to be published] full. mom. dep. full. mom. dep. sym. point and tadpole config. M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

41 Equations in YM theory [Cyrol, Fister, MM, Strodthoff, Pawlowski, to be published] t 1 = + t 1= t = + perm. t = perm. t = perm. M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

42 YM propagators [Cyrol, Fister, MM, Pawlowski, Strodthoff, to be published] Γ (2) AA (p) Z A(p) p 2 ( δ µν p µ p ν /p 2) Γ (2) cc (p) Zc(p) p2 3 gluon propagator dressing 2 1 Sternbeck et al. ghost propagator dressing 1 1 Sternbeck et al p [GeV] p [GeV] band: family of decoupling solutions bounded by scaling solution M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

43 YM vertices I [Cyrol, Fister, MM, Pawlowski, Strodthoff, to be published] comparison to Sternbeck 6 comparison to Cucchieri, Maas, Mendes, 8 Blum, Huber, MM, von Smekal ghost-gluon vertex dressing three-gluon vertex dressing p[gev] p[gev] band: family of decoupling solutions bounded by scaling solution M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

44 YM vertices II [Cyrol, Fister, MM, Pawlowski, Strodthoff, to be published] four-gluon vertex dressing 1 1 running couplings α gh-gl α 3g α 4g p[gev] p [GeV] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

45 Apparent Convergence [Cyrol, Fister, MM, Pawlowski, Strodthoff, to be published] gluon propagator dressing RG scale dep. 1D mom. dep. 3D mom. dep p [GeV] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

46 Outlook: unquenched gluon propagator gluon propagator dressing 1/Z A Sternbeck et al., 212 FRG p [GeV] self-consistent solution of classical propagators and vertices (1D) massless quarks [Cyrol, MM, Pawlowski, Strodthoff, in preparation] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

47 η -meson (screening) mass at chiral crossover small η -meson mass above chiral crossover? [Kapusta, Kharzeev, McLerran, 1998] drop in η mass at chiral crossover? [Csörgo et al., 21] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

48 η -meson (screening) mass at chiral crossover small η -meson mass above chiral crossover? [Kapusta, Kharzeev, McLerran, 1998] drop in η mass at chiral crossover? [Csörgo et al., 21] chiral crossover: Polyakov-Quark-Meson model (extended mean-field) [1] <σ>/f πφ N f = 2 quark and meson degrees of freedom describes chiral crossover (de-)confinement via Polyakov loop potential T/[MeV] U(1) A -anomaly: mesonic t Hooft determinant M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

49 η -meson (screening) mass at chiral crossover small η -meson mass above chiral crossover? [Kapusta, Kharzeev, McLerran, 1998] drop in η mass at chiral crossover? [Csörgo et al., 21] chiral crossover: Polyakov-Quark-Meson model (extended mean-field) [1] <σ>/f πφ N f = 2 quark and meson degrees of freedom describes chiral crossover (de-)confinement via Polyakov loop potential U(1) A -anomaly: mesonic t Hooft determinant T/[MeV] t Hooft determinant [Heller, MM, 215].6 λ (S-P)+ [MeV] RG-scale dependence from fqcd temperature dependence k(t ): λ (S P)+,fQCD(k) λ (S P)+,PQM(T ) k/[mev] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

50 η -meson (screening) mass at chiral crossover: result 12 Masses as function of temperature with Polyakov-Loop m π m η screening masses! m σ 1 8 m/[mev] T/[MeV] [Heller, MM, 215] M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

51 η -meson (screening) mass at chiral crossover: result 12 Masses as function of temperature with Polyakov-Loop m π m η m σ screening masses! QM-Model N f = 2 + 1: π σ a η η m/[mev] 8 6 masses [MeV] T [MeV] [MM, Schaefer, 213] T/[MeV] [Heller, MM, 215] chiral symmetry restoration: drop in m η M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

52 Summary and Outlook (quenched) QCD with functional RG QCD phase diagram: need for quantitative precision vacuum: sole input αs (Λ = O(1) GeV) and m q (Λ = O(1) GeV) good agreement with lattice simulations (sufficient?) (non-perturbative) results: quark-propagator quark-gluon vertex 4-Fermi interaction channels YM-system phenomenology: η -meson and pion mass splitting M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29

53 Summary and Outlook (quenched) QCD with functional RG QCD phase diagram: need for quantitative precision vacuum: sole input αs (Λ = O(1) GeV) and m q (Λ = O(1) GeV) good agreement with lattice simulations (sufficient?) (non-perturbative) results: quark-propagator quark-gluon vertex 4-Fermi interaction channels YM-system phenomenology: η -meson and pion mass splitting unquenching (first results) finite temperature/chemical potential more checks on convergence of vertex expansion bound-state properties (form factor,pda... ) M. Mitter (U Heidelberg) χsb in continuum QCD GSI, February / 29