QCD phenomenology from Dyson-Schwinger equations fileQCD phenomenology from Dyson-Schwinger...

Christian S. Fischer

Justus Liebig Universität Gießen

5th of Sept 2013

Christian Fischer (University of Gießen) QCD phenomenology from DSEs / 28

QCD phenomenology fromDyson-Schwinger equations

1Friday, September 6, 2013


1.Introduction

2.Properties of gluons

3.Quarks and pions: unquenching effects

4.Tetraquarks

Overview



PANDA: Hadron physicsCBM: QCD phase diagram

FAIR

3

Nonperturbative QCD}Friday, September 6, 2013

Christian Fischer (University of Gießen) A7: Baryon masses... / 26

Lattice simulationsAb initioGauge invariant

Functional approaches (DSE, FRG,Hamilton):Space-time continuumChiral symmetry: light quarks and mesonsMulti-scale problems feasibleChemical potential: no sign problem

Nonperturbative QCD: Complementary approach

4

Effective theories (χPT, ...)Physical degrees of freedom

Quarks and gluons Hadrons

Friday, September 6, 2013


QCD in covariant gauge

5

Quarks and Gluons

The Goal: gauge invariant information in a gauge fixed approach.

Landau gauge propagators in momentum space,

ZQCD =

�D[Ψ, A] exp

�−�

d4x�Ψ̄ (iD/ −m)Ψ− 1

4

�F aµν

�2

+gauge fixing��

DGluonµν (p) =

�δµν − pµpν

p2

�Z(p2)

p2

SQuark(p) = Zf (p2)[−ip/ +M(p2)]−1



1.Introduction



4.Tetraquarks

Overview



spacelike momenta: excellent agreement with latticeghost-gluon vertex: beyond bare vertex approximationthree-gluon vertex: Bose symmetric + zero crossing

Landau gauge gluon propagator

7

0 1 2 3 4 5p [GeV]

0

0.5

1

1.5

Z(p2 )

Sternbeck et. al. (2006)Funktionalzugang

CF, Maas, Pawlowski, Annals Phys. 324 (2009) 2408.Huber, von Smekal, arXiv:1211.6092



sort Gribov copies by lowest, nontrivial eigenvalue of Faddeev-Popov operatorfind members of decoupling family..scaling far away...

The decoupling family - DSEs vs. lattice

8

Sternbeck and Muller-Preussker, arXiv:1211.3057 [hep-lat].

Ghost

Glue

Continuum:find one-parameter family of solutions to DSEs characterized by value of G(0) ≡ J(0)

Lattice:

Boucaud, Leroy, Yaouanc, Micheli, Pene and Rodriguez-Quintero, JHEP 0806 (2008) 012CF, Maas, Pawlowski, Annals Phys. 324 (2009) 2408.

see also Maas, PLB 689 (2010) 107



Solve DSEs numerically for complex p2

Use contour method described in

Ghost and gluons in complex plane

9

Alkofer, Detmold, CF and Maris, PRD 70 (2004) 014014

Expectations ?

Complex conjugate poles ?Cut structure ?

Gribov, Stingl, Zwanziger, Cucchieri et al.



Real part of gluon: sign change but finiteReal part of ghost: finite (decoupling)

Gluon propagation in complex plane: real part

10

Strauss, CF, Kellermann, Phys. Rev. Lett. 109, (2012) 252001



finite !no complex conjugate polesCut structure with positivity violations

Gluon propagation: imaginary part

11

Strauss, CF, Kellermann, Phys. Rev. Lett. 109, (2012) 252001

spectral function

Gluon cannot appear in detector!

Alkofer, Detmold, CF and Maris, PRD 70 (2004) 014014



1.Introduction



4.Tetraquarks

Overview



The DSE for the quark propagator

13

• dressed Gluon propagator

• dressed Quark-Gluon-Vertex

Input:

Two strategies: I. calculate gluon and vertex from their DSEs →mandatory e.g. for QCD phase diagram CF and Luecker, PLB 718 (2013) 1036-1043

II. use model for quark-gluon interaction →ok for some phenomenological applications

[S(p)]−1 = [−ip/ +M(p2)]/Zf (p2)



much studied at T=0

Ansatz via STI model (MT et al...)

DSEs of QCD

14

quark gluon vertex



much studied at T=0

Ansatz via STI model (MT et al...)

DSEs of QCD

14

Yang-Mills part of glue

quark gluon vertex



DSEs and Bethe-Salpeter equation

15

Bethe-Salpeter equation: meson mass and wave function

π, ρ,...

Kernel K uniquely related to quark-DSE via axWTI

→Pion is bound state and Goldstone boson



Vertex built along STI:

YM: unquenching effects okMesons: maybe too small...

Unquenching effects: from gluons to hadrons

16

CF, Watson and Cassing, PRD 72 (2005) 094025CF, Alkofer, PRD 67 (2003) 094020

Γµ = γµG2(k2)A(k2)



Landau gauge quark propagator

17

quark-gluon vertex:

quark:

= + + + +




17

quark-gluon vertex:

quark:

= + Nc2

! 2Nc

+

!

+...




17

quark-gluon vertex:

quark:

= + Nc2

! 2Nc

+

!

+...

!1

=

!1

!

Y M

!

!



Attractive effects of pion cloudScalar too large or ... to low!

Unquenching effects: Light mesons

18

CF, Williams, PRL 103 (2009), PRD 78 (2008)

cp Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) 014011



Decoupling type of unquenched solutionBased on:

Unquenching decoupling type of solutions...

19

0 1 2 3 4 5 6 7p [GeV]

0

1

2

3

4

Z(p2 )

Nf=0, Sternbeck et al (2005)Nf=2+1, Bowman et al (2004)Nf=0Nf=2+1

CF, Watson and Cassing, PRD 72 (2005) 094025, Huber, von Smekal, arXiv:1211.6092

in prep.

For unquenching effects in the BG-PT-approach seeAguilar, Binosi and Papavassiliou, arXiv:1304.5936

Further unquenched lattice results:IIlgenfritz, et al Braz. J. Phys. 37 (2007) 193Ayala, et al. Phys. Rev. D 86 (2012) 074512, and Refs. therein



1.Introduction



4.Tetraquarks

Overview



Strategie II: model for quark-gluon interaction

21

Combine gluon with ‘leading’ γμ-part of quark-gluon vertex:

Maris-Tandy-model

and fix Maris, Tandy, 1999

α(k2) = πη7�k2

Λ2

�e−η2

�k2

Λ2

�

+ αUV (k2)

two (related) parameters and from fπ from perturbation theory

masses mu=md from mπ or mρ

αUV

η Λ



Phenomenology from Maris-Tandy interaction

22

Slide from Pieter Maris



Quark-photon vertex and pion form factors

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Q2

π π

Pion form factor:

Krassnigg, Schladming 2011; Maris, Tandy NPPS 161, 2006



Results: Hadronic vacuum polarisation

24

Experiment:

DSE: (mπ)

(mρ)Goecke, CF, Williams, PLB 704 (2011)

ahad.(1)µ = (744.0± 2) · 10−10

ahad.(1)µ = (676.0± 2) · 10−10

ahad.(1)µ = (692.1± 5.6) · 10−10

five flavors

Very reasonable agreement !

0.8 1 1.2 1.4 1.6M

ρ [GeV]

100

200

300

400

500

600

700

a µ [1

0-10 ]

RBC-UKQCD (Nf=2+1)Mainz CLS (Nf=2+1)Mainz CLS (Nf=2)ETMC (Nf=2)DSE (Nf=2+1)DSE (Nf=2)



Diquark-AntidiquarkMeson-Meson

Tetraquarks from DSEs/BSEs

Quark configurations:Hadrons

25

Jaffe, PRD 15 (1977) 267



Diquark-AntidiquarkMeson-Meson

Tetraquarks from DSEs/BSEs

Light meson sector:Scalars!

Quark configurations:Hadrons

25

Jaffe, PRD 15 (1977) 267



Input: Covariant Quark-Gluon interactionMesons and Diquark via Bethe-Salpeter equationReminescent of Diquark-Quark model for nucleon

Tetraquark-BSEs

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Meson Diquark

Dynamical decision between Meson-Molecule and Diquark picture



Results: scalar tetraquarks

Pion-Pion-contribution dominates !m(0++ ) = 403 MeV

Narrow scalar cccc: m(0++) = 5.3 ± (0.5) GeV__

Heupel, Eichman, CF, PLB 718 (2012) 545-549

} f0(500)

27

up/down charm

see also Caprini, Colangelo and Leutwyler, PRL. 96 (2006) 132001 Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) 014011



Gluons DSE and lattice agree for spacelike momentaTimelike momenta: no complex conjugate poles...current work: study influence of truncation on analytic structure

Unquenching QCDfrom gluons to light hadrons scaling and decouplingcurrent work: ...

Tetraquarksunderstand σ as ‘π-π-molecule’predict scalar four-charm state above 5 GeVcurrent work: include other channels; four body equation

Summary


QCD phenomenology from Dyson-Schwinger equations fileQCD phenomenology from Dyson-Schwinger...

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