On bound states in gauge theories with different matter content

Transcription

1 On bound states in gauge theories with different matter content Reinhard Alkofer Institute of Physics, Department of Theoretical Physics, University of Graz Bound states in QCD and beyond St. Goar, March 24, 2015 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

2 On two- and three-point Green functions in gauge theories with different matter content Reinhard Alkofer Institute of Physics, Department of Theoretical Physics, University of Graz Bound states in QCD and beyond St. Goar, March 24, 2015 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

3 Motivation Consider f 0 (500) (i.e., short-lived meson, cf. talk by M. Pennington) Gedankenexperiment 1: quadruple light quark current masses pion mass doubled 2π decay channel for f 0 closes f 0 becomes narrow (and slightly heavier) Gedankenexperiment 2: lower strange quark current mass R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

4 Motivation Consider f 0 (500) (i.e., short-lived meson, cf. talk by M. Pennington) Gedankenexperiment 1: quadruple light quark current masses pion mass doubled 2π decay channel for f 0 closes f 0 becomes narrow (and slightly heavier) Gedankenexperiment 2: lower strange quark current mass mixing with scalar s s state via π π & K K increases (lighter) f 0 becomes lighter (cf. level repulsion in QM) (?) 2π decay channel closes (?) f 0 becomes narrow and lighter R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

5 Motivation Composite Higgs model, e.g., technicolor models: (cf. Daniel Nogrodi s and Milan Vujinovic s talks) Spectrum of gauge-invariant bound states? Understand the influence of the number of colors the matter content (fermions, scalars,... ) the back-coupling of Standard Model particles R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

6 Motivation Walking coupling / nearly-conformal behaviour: separation of scales! = challenging for lattice calculations. Functional (continuum) methods: Dyson-Schwinger Bethe-Salpeter cov. Faddeev eqs. scale-dependent bosonisation in ERGE (Wetterich eq.) npi effective actions Achilles heel: truncation is necessary! Refer to symmetries expressed via Ward ids. and/or npi actions... Example: 3PI Kernel of the quark-antiquark Bethe-Salpeter eq. = + + (..) R.A., C.S. Fischer, F.J. Lllanes-Estrada, Mod.Phys.Lett. A23 (2008) 1105; H. Sanchis-Alepuz, R. Williams, arxiv: R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

7 Motivation Kernel of q q BS equation via 3PI action: = + + (..) Needed input: Gluon propagator and quark-gluon vertex! DSE for quark-gluon vertex via 3PI action: = 1 2N c + Nc 2 Needed input: Quark & gluon propagator and 3-gluon vertex! Disclaimer: These 2- & 3-point functions are needed for complex arguments to describe bound states with mass (timelike momentum)! Not discussed here... R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

8 Coupling quarks to gluons: from the confined phase to the conformal window -1 = -1-1/2-1/2 DSEs for Landau gauge QCD propagators: - 1/6-1/2 + + N f -1 = -1-1 = -1 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

9 From the confined phase to the conformal window: Modeling the quark-gluon vertex Conjecture: Gauge boson and ghost propagator undergo a significant change. Wanted: Self-consistent calculation of propagators and three-point functions! On-going... but a true challenge... see below... Test conjecture with a self-consistent calculation of propagators by employing a model for the three-gluon vertex (result insensitive) a model for the quark-gluon vertex (result sensitive, esp. N crit. f ) [M. Hopfer, C.S. Fischer, RA, JHEP 1411 (2014) 035 [arxiv: ]] R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

10 From the confined phase to the conformal window: Modeling the quark-gluon vertex Model for the quark-gluon vertex motivated by Slavnov-Taylor identity, [C.S. Fischer, RA, Phys. Rev. D 67 (2003) , A.C. Aguilar, J. Papavassiliou, Phys. Rev. D83 (2011) ] Γ µ = γ µ G 2 (k 2 )/Z f (k 2 ), provides excellent results for the gluon propagator at N phys f : Z(p 2 ) N f =0, Sternbeck et al (2005) N f =2+1, Bowman et al (2004) Nf=0 Nf=2+1 M ρ [MeV] N f =3, DSE N f =0, DSE N f =2, JLQCD(200 N f =0, JLQCD (200 N f =2, CP-Pacs ( p [GeV] M R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

11 Numerical Results: Phase transition with QGV model (1BC G 2 ): N crit. f 4.5 MeV ΨΨ M M0 MeV MeV MeV 10 MeV R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

12 Numerical Results: Phase transition N crit. f does not depend on three-gluon-vertex model does not depend on leading tensor structure of quark-gluon vertex does depend on additional tensor structures of quark-gluon vertex One additional transverse tensor structure: increase of N crit. f to 5.4! M0 MeV d 0 d 0.05 d 0.1 CP R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

13 Results: Running Coupling and Propagators α(p 2 ) N f = 0 N f = 4 N f = 5 1 / A(p 2 ) N f = 0 N f = 4 N f = p 2 [GeV 2 ] p 2 [GeV 2 ] G(p 2 ) N f = 0 N f = 4 N f = Z(p 2 ) / p 2 [GeV -2 ] N f = 0 N f = 4 N f = p 2 [GeV 2 ] p 2 [GeV 2 ] R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

14 Results: Running Coupling and Propagators α(p 2 ) p 2 [GeV 2 ] N f = 0 N f = 4 N f = 5 Qualitatively: Crossing to the conformal window 1 / A(p 2 ) p 2 [GeV 2 ] N f = 0 N f = 4 N f = 5 Running coupling constant coupling over orders of magnitude Dyn.quark mass & IR suppressed quark ren. fct. almost trivial quark propagator R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

15 Results: Running Coupling and Propagators G(p 2 ) p 2 [GeV 2 ] N f = 0 N f = 4 N f = 5 Z(p 2 ) / p 2 [GeV -2 ] N f = 0 N f = 4 N f = p 2 [GeV 2 ] IR enhanced ghost dressing IR divergent ghost dressing IR screened gluon propagator IR divergent gluon propagator (inverse screening length m(0) increases first before it drops to zero) N f -dependent power laws: Z (p 2 ) (p 2 ) 2ρ & G(p 2 ) (p 2 ) ρ, ρ 0.15 at Nf crit. no maximum in gluon propagator no positivity violation? Quantitative predictions need a reliably calculated quark-gluon vertex! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

16 Why a conformal window exists α(p 2 ) p 2 [GeV 2 ] N f = 0 N f = 4 N f = 5 small N f : antiscreening of gluons dominate coupling exceeds critical one dynamical chiral symmetry breaking very large N f : screening of quarks dominate no asymptotic freedom, QED like in between: antiscreening of gluons still larger but weakened critical coupling is not exceeded no dynamical chiral symmetry breaking R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

17 Three-gluon vertex DSE for three-gluon vertex: [G. Eichmann, R. Williams, M. Vujinovic, RA, Phys. Rev. D89 (2014) ; A.Blum, M.Q.Huber, M.Mitter, L. von Smekal, Phys.Rev.D89 (2014) ; and references therein] R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

18 Three-gluon vertex DSE for three-gluon vertex (truncated): [G. Eichmann, R. Williams, M. Vujinovic, RA, Phys. Rev. D89 (2014) ; A.Blum, M.Q.Huber, M.Mitter, L. von Smekal, Phys.Rev.D89 (2014) ; and references therein] = tensor elements 4 fully transverse ones apply cyclic perm. w.r.t. external legs bose symmetry Is there a zero in component projected on tree-level tensor? (cf., A. Maas) R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

19 Three-gluon vertex Applications: quark-gluon vertex (and consequently unquenching) conformal window symmetry preserving Bethe-Salpeter kernel irreducible three-body forces in the baryon Kinematics: R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

20 Three-gluon vertex Γ (1)µνρ 3g,TTT Green: zero; Orange: ghost triangle; Blue: ghost + swordfish. R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

21 Three-gluon vertex Γ (1)µνρ 3g,TTT A.Blum, M.Q.Huber, M.Mitter, L. von Smekal, Phys.Rev.D89 (2014) R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

22 Three-gluon vertex Zero in three-gluon vertex verified. (Position sensitive to truncation and scale-fixing.) Tensor structures beyond tree-level sub-leading. Three-gluon vertex at momentum scales O(Λ QCD ) small!!! Question: Conjectured for unquenching: Quark triangle positive, IR sign change expected at large N f? No! See below... R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

23 Coupling quarks: quark prop. and quark-gluon vertex R.A., C.S. Fischer, F. Lllanes-Estrada, K. Schwenzer, Annals Phys. 324 (2009) 106; C.S. Fischer and R. Williams, Phys. Rev. Lett. 103 (2009) ; M. Hopfer, A. Windisch, RA, in preparation. Chiral symmetry dynamically or explicitely broken: quark propagator infrared finite: S(p) = p/ M(p2 ) p 2 + M 2 (p 2 ) Z f (p 2 ) AND quark-gluon vertex incl. dynamically generated χsb tensors structures: 11 Γ µ = ig g i (p, q)γ i µ, Γ 1 µ = γµ T, Γ 2 µ = σ µν (ˆp + ˆq) T ν, Γ 3 µ =... i=0 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

24 Coupling quarks: quark prop. and quark-gluon vertex DSEs for quark propagator and quark-gluon vertex via 3PI action: 1 1 = = 1 2N c + N c 2 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

25 Coupling quarks to gluons: quark propagator Quark mass function with calculated QGV and modeled 3gV: Mp 2 MeV h IR 1.0 h IR 0.0 h IR R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

26 Coupling quarks to gluons: quark-gluon vertex Eight transverse tensor structures, e.g., at symm. momenta x = p 2 1 = p2 2 = p2 3 : Χ symmetric g 0 g 3 g 5 g Significant IR enhancement! Χ broken g 1 g 2 g 4 g p 2 GeV 2 Zero in 3gV = Zero in g 0! DχSB in QGV!!! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

27 Coupling quarks to gluons: quark-gluon vertex Relative importance: dynamically generated χsb coupling important! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

28 Coupling quarks to gluons: quark propagator Quark mass function in same truncation but adjoint quarks: Mp 2 MeV adjoint fundamental R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

29 Coupling quarks to gluons: quark-gluon vertex... but hardly any changes in the vertex functions: Χ symmetric g 0 g 3 g 5 g Χ broken p 2 GeV 2 g 1 g 2 g 4 g 7 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

30 Unquenching the 3-gluon vertex Include quark triangle diagrams into 3-gluon vertex DSE: [A. Blum, M.Q. Huber, R.A., in preparation] 08/03/2015 Excited QCD 2015 R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

31 Unquenching the 3-gluon vertex Quark triangle diagram: [A. Blum, M.Q. Huber, R.A., in preparation] R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

32 Unquenching the 3-gluon vertex Quark triangle diagram: [A. Blum, M.Q. Huber, R.A., in preparation] QT(p 2,p 2,π/2) p 2 [GeV 2 ] without qgv dressing with qgv dressing Contribution is small and negative! = Small effect on three-gluon vertex! NB: Small positive contribution by scalar triangle [S. Senn, Bachelor Thesis, Graz 2014] R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

33 Conclusions and Outlook QCD-like gauge theories in Landau gauge: Established results for propagators & recent investigations of three-point functions At large N f (conformal window): almost trivial quark propagator, but also drastic change of gauge boson and ghost propagator, plateau in coupling. No reliable quantitative results for bound states without self-consistent quark-gluon vertex!... which needs unquenched three-gluon vertex! Stay tuned: More results expected soon! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

34 Conclusions and Outlook QCD-like gauge theories in Landau gauge: Established results for propagators & recent investigations of three-point functions At large N f (conformal window): almost trivial quark propagator, but also drastic change of gauge boson and ghost propagator, plateau in coupling. No reliable quantitative results for bound states without self-consistent quark-gluon vertex!... which needs unquenched three-gluon vertex! Stay tuned: More results expected soon! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

35 Conclusions and Outlook QCD-like gauge theories in Landau gauge: Established results for propagators & recent investigations of three-point functions At large N f (conformal window): almost trivial quark propagator, but also drastic change of gauge boson and ghost propagator, plateau in coupling. No reliable quantitative results for bound states without self-consistent quark-gluon vertex!... which needs unquenched three-gluon vertex! Stay tuned: More results expected soon! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29

36 Conclusions and Outlook QCD-like gauge theories in Landau gauge: Established results for propagators & recent investigations of three-point functions At large N f (conformal window): almost trivial quark propagator, but also drastic change of gauge boson and ghost propagator, plateau in coupling. No reliable quantitative results for bound states without self-consistent quark-gluon vertex!... which needs unquenched three-gluon vertex! Stay tuned: More results expected soon! R. Alkofer (Theoretical Physics, U. Graz) On gauge theories w. different matter content St. Goar, March 24, / 29