non-perturbative results for two- and three-point...

Introduction T = 0 T > 0 Summary

Non-perturbative results for two- and three-pointfunctions of Landau gauge Yang-Mills theory

Markus Q. Huber

Institute of Nuclear Physics, Technical University Darmstadt

March 18, 2013

Quarks, Gluons, and Hadronic Matter under Extreme ConditionsSt. Goar

MQH TU Darmstadt March 18, 2013 1/18


Landau gauge Green functions from functional methods

Landau gauge Green functions:

Information about con�nement

Input for phenomenological calculations, e.g., bound states, QCDphase diagram

QCD phase diagram with functional equations:

+ no sign problem at non-zero chemical potential

+ physical quark masses easy

- in�nitely large system → in�uence of higher Green functions?⇒ tests of truncations



Landau Gauge Yang-Mills theory

Gluonic sector of quantum chromodynamics: Yang-Mills theory

L =1

2F 2 + Lgf + Lgh

Fµν = ∂µAν − ∂νAµ + i g [Aµ,Aν]

Propagators and vertices are gauge dependent→ choose any gauge, ideally one that is convenient.

Landau gauge

simplest one for functional equations

∂µAµ = 0: Lgf =1

2ξ(∂µAµ)

2, ξ→ 0

requires ghost �elds: Lgh = c (−2 + g A×) c

2 �elds: + i j-1

+ j k

-1

3 vertices:

+

i

j

k

+

i

j k

l

+

i

j

k



Dyson-Schwinger equations: Propagators

Dyson-Schwinger equations (DSEs) of gluon and ghost propagators:

i1 i2 -1

=

+ i1 i2 -1-

12

i1 i2 -12

i1i2 + i1i2

-16

i1i2 -12 i1 i2

i2 i1 -1

=

+i2 i1 -1

-i1i2

Equations of motion of correlation functions:Describe how �elds propagate and interact non-perturbatively!

In�nite tower of coupled integral equations.

Derivation straightforward, but tedious→ automated derivation with DoFun [MQH, Braun, CPC183 (2012)].

Contain three-point and four-point functions:

ghost-gluon vertex , three-gluon vertex , four-gluon vertex



Truncated propagator Dyson-Schwinger equations

Standard truncation:

i1 i2 -1

=

+ i1 i2 -1-

12

i1i2+

i1i2

i2 i1 -1

=

+i2 i1 -1

-i1i2

using bare ghost-gluon vertex and three-gluon vertex model

In�uence of dynamic ghost-gluon vertex?



Truncating Dyson-Schwinger equations

gluon ghost gh-gl 3-gl 4-pt. ref.

X 0 0 model 0 [Mandelstam, PRD20 (1979)]

10−2

100

102

104

106

1080.0

0.2

0.4

~F(x)

~Z(x)

[Hauck,vonSmekal,

Alkofer,CPC

112(1998)]

Dgl,µν(p) =

(gµν −

pµpν

p2

)�Z(p2)

p2

gluon dressing �Z (p2) IR divergent→ IR slavery

Improved truncations necessary for quantitative results and also extensions,

e.g., non-zero temperature [Fister, Pawlowski, 1112.5440].






X X models models 0 [von Smekal, Hauck, Alkofer, PRL 79 (1997)]

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ààà ààààààààààààààààààààààà

0 1 2 3 4 5p@GeVD0

1

2

3

4

ZHp2L

[MQH,vonSmekal,1211.6092;

Sternbeck,hep-lat/0609016]

Dgl,µν(p) =

(gµν −

pµpν

p2

)Z(p2)

p2

Dgh(p) = −G (p2)

p2

gluon dressing Z (p2) IR vanishing

deviations from lattice results in

mid-momentum regime








X X models models 0 [von Smekal, Hauck, Alkofer, PRL 79 (1997)]

X X X model 0 [MQH, von Smekal, 1211.6092]

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0 1 2 3 4 5p@GeVD0

1

2

3

4

ZHp2L

[MQH,vonSmekal,1211.6092;

Sternbeck,hep-lat/0609016]

Dgl,µν(p) =

(gµν −

pµpν

p2

)Z(p2)

p2

Dgh(p) = −G(p2)

p2

gluon dressing Z (p2) IR vanishing

improved mid-momentum behavior





Ghost-gluon vertex DSE

Full DSE:i1

i2

i3

=

+

i1

i2

i3

+12

i1i2

i3

-

i1i2

i3

+16

i1i2

i3

+

i1

i2 i3

+

i1 i2

i3

+12

i1i2

i3+

12

i1

i2i3

+12

i1

i3i2

+

i1

i2 i3+

12

i1

i2

i3

+12

i1i2

i3

Lattice results [Cucchieri, Maas, Mendes, PRD77 (2008); Ilgenfritz et al., BJP37

(2007)]

OPE analysis [Boucaud et al., JHEP 1112 (2011)]

Modeling via ghost DSE [Dudal, Oliveira, Rodriguez-Quintero, PRD86 (2012)]

Semi-perturbative DSE analysis [Schleifenbaum et al., PRD72 (2005)]

FRG [Fister, Pawlowski, 1112.5440]



Ghost-gluon vertex

ΓAcc,abcµ (k ; p, q) := i g f abc (pµA(k ; p, q) + kµB(k ; p, q))

Note:B(k ; p, q) is irrelevant in Landau gauge (but it is not the pure longitudinal part).

Taylor argument applies only to longitudinal part (it's an STI).

IR and UV consistent truncation:

i1

i2

i3

=

+

i1

i2

i3

+

i1

i2 i3

+

i1

i2i3

System of eqs. to solve:gluon and ghost propagators + ghost-gluon vertex

Only un�xed quantity in present truncation: three-gluon vertex.

q

pkϕ



Three-gluon vertex: Ultraviolet

Bose symmetric version:

DA3,UV (x , y , z) = G

(x + y + z

2

)αZ

(x + y + z

2

)βFix α and β:

1 UV behavior of three-gluon vertex

2 IR behavior of three-gluon vertex?



Three-gluon vertex: Ultraviolet

Bose symmetric version:

DA3,UV (x , y , z) = G

(x + y + z

2

)αZ

(x + y + z

2

)βFix α and β:

1 UV behavior of three-gluon vertex

2 IR behavior of three-gluon vertex → yes, but . . .



Three-gluon vertex: Infrared

Three-gluon vertex might have a zero crossing.d = 2, 3: seen on lattice [Cucchieri, Maas, Mendes, PRD77 (2008); Maas, PRD75

(2007)],d = 2: seen with DSEs [MQH, Maas, von Smekal, JHEP11 (2012)]

d = 2:[Maas, PRD75; MQH, Maas, von Smekal, JHEP11 (2012)]

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ô ô ô ô

ôô

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ô ô

ô

ô ô

0.5 1.0 1.5 2.0p

-2

-1

0

1

2

DprojA 3 Hp 2 , p 2 , Π � 2L

d = 4:[Cucchieri, Maas, Mendes, PRD77 (2008)]

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1 2 3 4 5p@GeV D

-1.0

- 0.5

0.0

0.5

1.0

1.5

2.0

D A3 Hp 2 ,p 2 ,p 2L

DA3,IR(x , y , z) = hIRG (x + y + z)3(f 3g (x)f 3g (y)f 3g (z))4

IR damping function f 3g (x) :=Λ23g

Λ23g + x



In�uence of the three-gluon vertex

0 1 2 3 4 5p@GeV D0

1

2

3

4

ZHp 2L

0 1 2 3 4 5p@GeV D1

2

3

4

5

GHp 2L

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1 2 3 4 5p@GeV D

-1.0

- 0.5

0.0

0.5

1.0

1.5

2.0

D A3 Hp 2 ,p 2 ,p 2L Vary Λ3g → varymid-momentum strength

Ghost almost una�ected

Thin line: Leading IR orderDSE calculation forthree-gluon vertex⇒ zero crossing

Optimized e�ective three-gluon vertex:Choose Λ3g where gluon dressing has best agreement with lattice results.[MQH, von Smekal, 1211.6092]



Dynamic ghost-gluon vertex: Propagator results

Dynamic ghost-gluon vertex, opt. e�.three-gluon vertex [MQH, von Smekal, 1211.6092]

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0 1 2 3 4 5p@GeVD0

1

2

3

4

ZHp2L

FRG results[Fischer, Maas, Pawlowski, AP324 (2009)]

0 1 2 3 4 5p [GeV]

0

1

2

Z(p

2)

Bowman (2004)Sternbeck (2006)scaling (DSE)

decoupling (DSE)

scaling (FRG)

decoupling (FRG)

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àà à

0.0 0.5 1.0 1.5 2.0 2.5 3.0p1

2

3

4

5

6

GHp2L

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2p [GeV]

2

4

6

8

10

12

14

G(p

2)

Sternbeck (2006)scaling (DSE)

decoupling (DSE)

scaling (FRG)

decoupling (FRG)

Good quantitative agreement for ghost and gluon dressings.MQH TU Darmstadt March 18, 2013 12/18


Ghost-gluon vertex: Selected con�gurations (decoupling)

ΓAcc,abcµ (k ; p, q) := i g f abc (pµA(k ; p, q) + kµB(k ; p, q))

Fixed angle: Fixed anti-ghost momentum:

[MQH, von Smekal, 1211.6092]



Ghost-gluon vertex: Comparison with lattice data

Orthogonal con�guration k2 = 0, q2 = p2:

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0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0p@GeV D

0.9

1.0

1.1

1.2

1.3

1.4

A H 0;p 2 ,p 2Lconstant in the IR

relatively insensitive to changes ofthe three-gluon vertex(red/green lines:di�erent three-gluon vertex models)

DSE calculation: [MQH, von Smekal, 1211.6092]

lattice data: [Sternbeck, hep-lat/0609016]



Towards the phase diagram of QCD with DSEs

Lattice results helpful

as input

for comparison and testing reliability of a truncation

non-zero chemical potential � → use theories without sign problem,

e.g., SU(2), G2

Self-consistent functional RG calculation of correlation functions:[Fister, Pawlowski, 1112.5440; Fister, PhD thesis, 2012]

DSE calculations:[Maas, Wambach, Alkofer, EPJC42 (2005); Cucchieri, Maas, Mendes, PRD75 (2007)]



Ghost propagator

First steps towards full system: Take some lattice input.

Gluon propagator: lattice based �ts [Fischer, Maas, Müller, EPJC68 (2010)]

Ghost propagator [preliminary]:

i1 i2-1

=

+ i1 i2-1

-i1i2

-i1i2

lattice: [Maas, Pawlowski, Spielmann, von Smekal, PRD85 (2012)]



Ghost propagator




i1 i2-1

=

+ i1 i2-1

-i1i2

-i1i2


1

1 000 0001

10 000

1

100

1

100

p2@GeV2D

0.150.20.2

0.2770.35

0.4

T@GeVD

0.5

1.0

1.5

2.0

2.5

GA c cHk,pL



Ghost propagator




i1 i2-1

=

+ i1 i2-1

-i1i2

-i1i2


0 2 4 6 8 10 12 [email protected]

1.0

1.2

1.4

1.6

1.8

2.0

G

T=0.733Tc


1.0

1.2

1.4

1.6

1.8

2.0

G

T=1.005Tc


1.0

1.2

1.4

1.6

1.8

2.0

G

T=1.1Tc


1.0

1.2

1.4

1.6

1.8

2.0

G

T=1.81Tc



Ghost-gluon vertex

Remember: Relevance of ghost-gluon vertex for non-zero temperatureknown from functional RG [Fister, Pawlowski, 1112.5440]!

Simple approximation:

Fully iterated ghost propagatorGluon propagator from the lattice[Fischer, Maas, Müller, EPJC68 (2010)]

Ghost-gluon vertex semi-perturbativelyat symmetric point (p2 = q2 = k2)

[preliminary]



Summary

Systematic improvement of truncations of DSEs possible.

Newest step:Inclusion of ghost-gluon vertex andqualitative three-gluon vertex model[MQH, von Smekal, 1211.6092]

Required for quantitative results and

likely also for some aspects of non-zero temperature and density

calculations.

Reproduction of lattice data possible.

Towards the phase diagram of QCD:√

Ghost propagator√Ghost-gluon vertex semi-perturbatively

� Ghost-gluon vertex self-consistent

� Gluon propagator

� Quarks

� Phase transitions

Thank you for your attention.



Decoupling and scaling solutions

DSEs: Vary ghost boundary condition [Fischer, Maas, Pawlowski, AP324 (2009)]

10-4 0.001 0.01 0.1 1 10 100

1

510

50100

500

p 2@GeV^2D

GHp

2L

10-4 0.001 0.01 0.1 1 10 10010-4

0.001

0.01

0.1

1

p 2@GeV^2D

ZHp

2L

Realization on lattice?

Dependence of propagators on Gribov copies, e.g., [Bogolubsky, Burgio,Müller-Preussker, Mitrjushkin, PRD 74 (2006); Maas, PR524 (2013)]

First hints from [Sternbeck, Müller-Preussker, 1211.3057]:choosing Gribov copies by the lowest eigenvalue of the Faddeev-Popov operator

d = 2: [Maas, PRD75 (2007); Maas et al., EPJC68 (2010); Cucchieri et al., PRD85

(2012)]

Continuum d = 2Analytic and numerical arguments from DSEs for scaling only [Cucchieri, Dudal, Vandersickel,PRD85 (2012); MQH, Maas, von Smekal, JHEP1211 (2012)] as well as from analysis ofGribov region [Zwanziger, 1209.1974].



Ghost-gluon vertex and scaling solution

Fixed angle: Fixed momentum:

Dressing not 1 in the IR ← Contributions from loop corrections (fordecoupling they are suppressed)

Scaling/decoupling also seen in ghost-gluon vertex


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