Introduction Yang-Mills theory Summary
Landau gauge Green functions fromDyson-Schwinger equations
Markus Q. Huber, Lorenz von Smekal
Institute of Nuclear Physics, Technical University Darmstadt
April 26, 2013
JHEP (2013), arXiv:1211.6092
HIC for FAIR Physics Days of Expert Group 2FIAS, Frankfurt
MQH TU Darmstadt April 26, 2013 1/21
Introduction Yang-Mills theory Summary
From QCD to hadrons
Quantum chromodynamics
Quarks, gluons:
u
ud u
d d
LQCD = q(− /D +m)q +1
2tr {FµνFµν}
Experiment: hadrons
u u
d
u
d
d
Description via
models,
e�ective theories,
lattice,
functional equations,
. . .
Ideally: LQCD → hadron spectrum
MQH TU Darmstadt April 26, 2013 2/21
Introduction Yang-Mills theory Summary
From QCD to hadrons
Quantum chromodynamics
Quarks, gluons:
u
ud u
d d
LQCD = q(− /D +m)q +1
2tr {FµνFµν}
Experiment: hadrons
u u
d
u
d
d
Description via
models,
e�ective theories,
lattice,
functional equations,
. . .
Ideally: LQCD → hadron spectrum
MQH TU Darmstadt April 26, 2013 2/21
Introduction Yang-Mills theory Summary
From QCD to hadrons
Quantum chromodynamics
Quarks, gluons:
u
ud u
d d
LQCD = q(− /D +m)q +1
2tr {FµνFµν}
Experiment: hadrons
u u
d
u
d
d
Description via
models,
e�ective theories,
lattice,
functional equations,
. . .
Ideally: LQCD → hadron spectrum
MQH TU Darmstadt April 26, 2013 2/21
Introduction Yang-Mills theory Summary
Functional equations for QCD
Bound state equations: Bethe-Salpeter/Faddeev eqs. (BSEs/FEs)
BSE:
Contains quark propagator S and kernel K .
Standard truncation: rainbow-ladder (consistent with chiral symmetry)
K −→ dressed one gluon exchange
e�ective gluon propagator
bare quark-gluon vertex
MQH TU Darmstadt April 26, 2013 3/21
Introduction Yang-Mills theory Summary
Functional equations for QCD
Bound state equations: Bethe-Salpeter/Faddeev eqs. (BSEs/FEs)
BSE:
Contains quark propagator S and kernel K .
Standard truncation: rainbow-ladder (consistent with chiral symmetry)
K −→ dressed one gluon exchange
e�ective gluon propagator
bare quark-gluon vertex
MQH TU Darmstadt April 26, 2013 3/21
Introduction Yang-Mills theory Summary
Beyond rainbow-ladderFor example:
Include pion back coupling e�ects [e.g., Fischer, Nickel, Wambach, PRD76
(2007); Fischer, Nickel, Williams, EPJC60 (2008); Fischer, Williams, PRD78 (2008)]:
−= −
YM
−1 −1
= + +
YM
Include gluon self-interaction [e.g., Maris, Tandy, NPPS161 (2006); Fischer,
Williams, PRL103 (2009)]:
= + +
Solve quark-gluon vertex (12 tensors!)
= + Nc
2− 2
Nc+
π
Required: gluon propagator, three-gluon vertexMQH TU Darmstadt April 26, 2013 4/21
Introduction Yang-Mills theory Summary
Propagators
Calculate from Dyson-Schwinger equations
Quark:i1 i2 -1
=
+ i1 i2 -1- i1i2
Gluon:i1 i2 -1
=
+ i1 i2 -1-
12
i1 i2 -12
i1i2 - i1i2
- i1i2 -16
i1i2 -12 i1 i2
Ghost:i1 i2 -1
=
+ i1 i2 -1- i1i2
Required: three- and four-point functions
(or from �ow equations or eqs. of motion from nPI e�ective action)
MQH TU Darmstadt April 26, 2013 5/21
Introduction Yang-Mills theory Summary
Propagators
Calculate from Dyson-Schwinger equations
Quark:i1 i2 -1
=
+ i1 i2 -1- i1i2
Gluon:i1 i2 -1
=
+ i1 i2 -1-
12
i1 i2 -12
i1i2 - i1i2
- i1i2 -16
i1i2 -12 i1 i2
Ghost:i1 i2 -1
=
+ i1 i2 -1- i1i2
Required: three- and four-point functions
(or from �ow equations or eqs. of motion from nPI e�ective action)
MQH TU Darmstadt April 26, 2013 5/21
Introduction Yang-Mills theory Summary
Propagators
Calculate from Dyson-Schwinger equations
Quark:i1 i2 -1
=
+ i1 i2 -1- i1i2
Gluon:i1 i2 -1
=
+ i1 i2 -1-
12
i1 i2 -12
i1i2 - i1i2
- i1i2 -16
i1i2 -12 i1 i2
Ghost:i1 i2 -1
=
+ i1 i2 -1- i1i2
Required: three- and four-point functions
(or from �ow equations or eqs. of motion from nPI e�ective action)
MQH TU Darmstadt April 26, 2013 5/21
Introduction Yang-Mills theory Summary
Propagators
Calculate from Dyson-Schwinger equations
Quark:i1 i2 -1
=
+ i1 i2 -1- i1i2
Gluon:i1 i2 -1
=
+ i1 i2 -1-
12
i1 i2 -12
i1i2 - i1i2
- i1i2 -16
i1i2 -12 i1 i2
Ghost:i1 i2 -1
=
+ i1 i2 -1- i1i2
Required: three- and four-point functions
(or from �ow equations or eqs. of motion from nPI e�ective action)
MQH TU Darmstadt April 26, 2013 5/21
Introduction Yang-Mills theory Summary
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
i1 i2 -1
=
+ i1 i2 -1- i1i2
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
MQH TU Darmstadt April 26, 2013 6/21
Introduction Yang-Mills theory Summary
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
i1 i2 -1
=
+ i1 i2 -1- i1i2
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
MQH TU Darmstadt April 26, 2013 6/21
Introduction Yang-Mills theory Summary
Truncated propagator Dyson-Schwinger equations
Standard truncation:
No four-point interactionsmodels for ghost-gluon and three-gluon vertices
i1 i2 -1
=
+ i1 i2 -1-
12
i1i2+
i1i2
i1 i2 -1
=
+ i1 i2 -1- i1i2
Standard: bare ghost-gluon vertex and three-gluon vertex model
In�uence of three-point functions?
Dabgl,µν(p) =
(gµν −
pµpν
p2
)Z(p2)
p2δab
Dabgh (p) = −
G(p2)
p2δab
MQH TU Darmstadt April 26, 2013 7/21
Introduction Yang-Mills theory Summary
Truncated propagator Dyson-Schwinger equations
Standard truncation:
No four-point interactionsmodels for ghost-gluon and three-gluon vertices
i1 i2 -1
=
+ i1 i2 -1-
12
i1i2+
i1i2
i1 i2 -1
=
+ i1 i2 -1- i1i2
Standard: bare ghost-gluon vertex and three-gluon vertex model
In�uence of three-point functions?
Dabgl,µν(p) =
(gµν −
pµpν
p2
)Z(p2)
p2δab
Dabgh (p) = −
G(p2)
p2δab
MQH TU Darmstadt April 26, 2013 7/21
Introduction Yang-Mills theory Summary
Truncated propagator Dyson-Schwinger equations
Standard truncation:
No four-point interactionsmodels for ghost-gluon and three-gluon vertices
i1 i2 -1
=
+ i1 i2 -1-
12
i1i2+
i1i2
i1 i2 -1
=
+ i1 i2 -1- i1i2
Standard: bare ghost-gluon vertex and three-gluon vertex model
In�uence of three-point functions?
Dabgl,µν(p) =
(gµν −
pµpν
p2
)Z(p2)
p2δab
Dabgh (p) = −
G(p2)
p2δab
MQH TU Darmstadt April 26, 2013 7/21
Introduction Yang-Mills theory Summary
Truncating Dyson-Schwinger equations
gluon ghost gh-gl 3-gl 4-pt. ref.
MQH TU Darmstadt April 26, 2013 8/21
Introduction Yang-Mills theory Summary
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
MQH TU Darmstadt April 26, 2013 8/21
Introduction Yang-Mills theory Summary
Truncating Dyson-Schwinger equations
gluon ghost gh-gl 3-gl 4-pt. ref.
X 0 0 model 0 [Mandelstam, PRD20 (1979)]
X X models models 0 Scaling [von Smekal, Hauck, Alkofer, PRL 79 (1997)]
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0 1 2 3 4 5p@GeVD0
1
2
3
4
ZHp2L
[MQH,vonSmekal,JHEP
(2013);
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 inmid-momentum regime
MQH TU Darmstadt April 26, 2013 8/21
Introduction Yang-Mills theory Summary
Truncating Dyson-Schwinger equations
gluon ghost gh-gl 3-gl 4-pt. ref.
X 0 0 model 0 [Mandelstam, PRD20 (1979)]
X X models models 0 Scaling [von Smekal, Hauck, Alkofer, PRL 79 (1997)]
X X models models 0 Dec. [Aguilar, Binosi, Papavassiliou PRD78 (2008)]
[Fischer, Maas, Pawlowski, AP324 (2009)]
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0 1 2 3 4 5p@GeVD0
1
2
3
4
ZHp2L
[MQH,vonSmekal,JHEP
(2013);
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 inmid-momentum regime
MQH TU Darmstadt April 26, 2013 8/21
Introduction Yang-Mills theory Summary
Truncating Dyson-Schwinger equations
gluon ghost gh-gl 3-gl 4-pt. ref.
X 0 0 model 0 [Mandelstam, PRD20 (1979)]
X X models models 0 Scaling [von Smekal, Hauck, Alkofer, PRL 79 (1997)]
X X models models 0 Dec. [Aguilar, Binosi, Papavassiliou PRD78 (2008)]
[Fischer, Maas, Pawlowski, AP324 (2009)]
X X X model 0 [MQH, von Smekal, JHEP (2013)]
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0 1 2 3 4 5p@GeVD0
1
2
3
4
ZHp2L
[MQH,vonSmekal,JHEP
(2013);
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 inmid-momentum regime
MQH TU Darmstadt April 26, 2013 8/21
Introduction Yang-Mills theory Summary
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]
MQH TU Darmstadt April 26, 2013 9/21
Introduction Yang-Mills theory Summary
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
MQH TU Darmstadt April 26, 2013 10/21
Introduction Yang-Mills theory Summary
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
MQH TU Darmstadt April 26, 2013 10/21
Introduction Yang-Mills theory Summary
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?
MQH TU Darmstadt April 26, 2013 11/21
Introduction Yang-Mills theory Summary
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 . . .
MQH TU Darmstadt April 26, 2013 11/21
Introduction Yang-Mills theory Summary
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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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
MQH TU Darmstadt April 26, 2013 12/21
Introduction Yang-Mills theory Summary
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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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
MQH TU Darmstadt April 26, 2013 12/21
Introduction Yang-Mills theory Summary
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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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
MQH TU Darmstadt April 26, 2013 12/21
Introduction Yang-Mills theory Summary
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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ì ìì
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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, JHEP (2013)]
MQH TU Darmstadt April 26, 2013 13/21
Introduction Yang-Mills theory Summary
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
æ
æ æ
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æ ææ
à
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à
à
à
ì
ì
ì ìì
ì
ì
ì
ì
ò
ò
ò
òò ò
ò
ò
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, JHEP (2013)]
MQH TU Darmstadt April 26, 2013 13/21
Introduction Yang-Mills theory Summary
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
æ
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ì ìì
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òò ò
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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, JHEP (2013)]
MQH TU Darmstadt April 26, 2013 13/21
Introduction Yang-Mills theory Summary
Dynamic ghost-gluon vertex: Propagator results
Dynamic ghost-gluon vertex, opt. e�.three-gluon vertex [MQH, von Smekal, JHEP (2013)]
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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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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 April 26, 2013 14/21
Introduction Yang-Mills theory Summary
Dynamic ghost-gluon vertex: Propagator results
Dynamic ghost-gluon vertex, opt. e�.three-gluon vertex [MQH, von Smekal, JHEP (2013)]
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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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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 April 26, 2013 14/21
Introduction Yang-Mills theory Summary
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, JHEP (2013)]
MQH TU Darmstadt April 26, 2013 15/21
Introduction Yang-Mills theory Summary
Ghost-gluon vertex: Comparison with lattice data
Orthogonal con�guration k2 = 0, q2 = p
2:
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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, JHEP (2013)]
lattice data: [Sternbeck, hep-lat/0609016]
MQH TU Darmstadt April 26, 2013 16/21
Introduction Yang-Mills theory Summary
Functional equations and lattice results
functional equations lattice
propagators X Xthree-point functions ghost-gluon vertex: X limited mom. dependence
3-gluon vertex: in progressquark-gluon vertex: (X)
four-point functions (X) not soon
MQH TU Darmstadt April 26, 2013 17/21
Introduction Yang-Mills theory Summary
Functional equations and lattice results
functional equations lattice
propagators X Xthree-point functions ghost-gluon vertex: X limited mom. dependence
3-gluon vertex: in progressquark-gluon vertex: (X)
four-point functions (X) not soonsource of error truncation �nite volume,
�nite lattice spacingtemperature X X
chemical potential X sign problemanalytic structure X no
MQH TU Darmstadt April 26, 2013 17/21
Introduction Yang-Mills theory Summary
Functional equations and lattice results
functional equations lattice
propagators X Xthree-point functions ghost-gluon vertex: X limited mom. dependence
3-gluon vertex: in progressquark-gluon vertex: (X)
four-point functions (X) not soonsource of error truncation �nite volume,
�nite lattice spacingtemperature X X
chemical potential X sign problemanalytic structure X no
MQH TU Darmstadt April 26, 2013 17/21
Introduction Yang-Mills theory Summary
Schwinger function
Schwinger function ∆(t):
∆(t) =1
π
∫dq cos(q t)
Z (q2)
q2
0 2 4 6 8 10t@fmD10-6
10-4
0.01
1
ÈDHtLÈ
[MQH, von Smekal, PoS CONFX 062 (2013) ]
∆(t) =
∫∞0
dνρ(ν2)e−νt = L(ρ)
ρ: spectral density, must be positive for physical particles
Positivity violation of propagators → con�nement.
MQH TU Darmstadt April 26, 2013 18/21
Introduction Yang-Mills theory Summary
Schwinger function
Schwinger function ∆(t):
∆(t) =1
π
∫dq cos(q t)
Z (q2)
q2
0 2 4 6 8 10t@fmD10-6
10-4
0.01
1
ÈDHtLÈ
[MQH, von Smekal, PoS CONFX 062 (2013) ]
∆(t) =
∫∞0
dνρ(ν2)e−νt = L(ρ)
ρ: spectral density, must be positive for physical particles
Positivity violation of propagators → con�nement.
MQH TU Darmstadt April 26, 2013 18/21
Introduction Yang-Mills theory Summary
Glueballs
Glueball candidate: G = FµνFµν
No positivity violation expected.
Construction from positivity violating gluons?
Correlation function 〈G (x)G (y)〉 in �rst order approximation (Born level):
p p
Gluon propagators from �ts to solutions of Yang-Mills systems.
MQH TU Darmstadt April 26, 2013 19/21
Introduction Yang-Mills theory Summary
Glueballs
Calculation in complex plane.
Extraction of spectral density.
0 1 2 3 4 5
-p2 [GeV
2]
-20
0
20
40
60
80
100
dis
c{O
(p2)}
[Windisch, Huber, Alkofer, PRD87 (2013)]
Propagators in complex plane: [Strauss, Fischer, Kellermann, PRL109 (2012)]
MQH TU Darmstadt April 26, 2013 20/21
Introduction Yang-Mills theory Summary
Summary
Green functions are basic building blocks for bound state equations.
Improving truncations systematically possible.
Checks: analytical results, lattice
Newest step for Yang-Mills sector: [MQH, von Smekal, JHEP (2013)]
Inclusion of ghost-gluon vertex andqualitative three-gluon vertex model
Required for quantitative results.Reproduction of lattice data possible.
Automatization tools available:DoFun [Alkofer, MQH, Schwenzer, CPC180 (2009); MQH, Braun, CPC183 (2012)]
CrasyDSE [MQH, Mitter, CPC183 (2012)]
Other applications in investigation of QCD phase diagram (no signproblem!)
Thank you for your attention!
MQH TU Darmstadt April 26, 2013 21/21
Introduction Yang-Mills theory Summary
Summary
Green functions are basic building blocks for bound state equations.
Improving truncations systematically possible.
Checks: analytical results, lattice
Newest step for Yang-Mills sector: [MQH, von Smekal, JHEP (2013)]
Inclusion of ghost-gluon vertex andqualitative three-gluon vertex model
Required for quantitative results.Reproduction of lattice data possible.
Automatization tools available:DoFun [Alkofer, MQH, Schwenzer, CPC180 (2009); MQH, Braun, CPC183 (2012)]
CrasyDSE [MQH, Mitter, CPC183 (2012)]
Other applications in investigation of QCD phase diagram (no signproblem!)
Thank you for your attention!
MQH TU Darmstadt April 26, 2013 21/21
Introduction Yang-Mills theory Summary
Summary
Green functions are basic building blocks for bound state equations.
Improving truncations systematically possible.
Checks: analytical results, lattice
Newest step for Yang-Mills sector: [MQH, von Smekal, JHEP (2013)]
Inclusion of ghost-gluon vertex andqualitative three-gluon vertex model
Required for quantitative results.Reproduction of lattice data possible.
Automatization tools available:DoFun [Alkofer, MQH, Schwenzer, CPC180 (2009); MQH, Braun, CPC183 (2012)]
CrasyDSE [MQH, Mitter, CPC183 (2012)]
Other applications in investigation of QCD phase diagram (no signproblem!)
Thank you for your attention!
MQH TU Darmstadt April 26, 2013 21/21
Introduction Yang-Mills theory Summary
Summary
Green functions are basic building blocks for bound state equations.
Improving truncations systematically possible.
Checks: analytical results, lattice
Newest step for Yang-Mills sector: [MQH, von Smekal, JHEP (2013)]
Inclusion of ghost-gluon vertex andqualitative three-gluon vertex model
Required for quantitative results.Reproduction of lattice data possible.
Automatization tools available:DoFun [Alkofer, MQH, Schwenzer, CPC180 (2009); MQH, Braun, CPC183 (2012)]
CrasyDSE [MQH, Mitter, CPC183 (2012)]
Other applications in investigation of QCD phase diagram (no signproblem!)
Thank you for your attention!
MQH TU Darmstadt April 26, 2013 21/21
Introduction Yang-Mills theory Summary
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
MQH TU Darmstadt April 26, 2013 22/21
Introduction Yang-Mills theory Summary
Solutions of functional equations: Decoupling and scaling
Two types of solutions with functional methods that di�er only indeep IR [Boucaud et al., JHEP 0806, 012; Fischer, Maas, Pawlowski, AP 324 (2009)]:scaling [von Smekal, Alkofer, Hauck PRL97],decoupling [Aguilar, Binosi, Papavassiliou PRD78; Fischer, Maas, Pawlowski, AP 324
(2009)]
Lattice calculations �nd only decoupling type solution for d = 3, 4and scaling for d = 2
Decoupling emerges also from Re�ned Gribov-Zwanziger framework[Dudal, Sorella, Vandersickel, Verschelde, PRD77]
MQH TU Darmstadt April 26, 2013 23/21
Introduction Yang-Mills theory Summary
Decoupling and scaling solutions
DSEs: Vary ghost boundary condition [Fischer, Maas, Pawlowski, AP 324 (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
Dependence of propagators on Gribov copies, e.g., [Bogolubsky, Burgio,Müller-Preussker, Mitrjushkin, PRD 74 (2006); Maas, PR 524 (2013)]
Ideas:[Sternbeck, Müller-Preussker, 1211.3057]: choose Gribov copies by lowesteigenvalue of the Faddeev-Popov operator→ modi�cation of both dressings[Maas, PLB689 (2010)]: choose Gribov copies by value of ghostpropagator
d = 2: Analytic and numerical arguments from DSEs for scaling only [Cucchieri,
Dudal, Vandersickel, PRD85 (2012); MQH, Maas, von Smekal, JHEP11 (2012)] as well asfrom analysis of Gribov region [Zwanziger, 1209.1974].
MQH TU Darmstadt April 26, 2013 24/21
Introduction Yang-Mills theory Summary
Scaling solution: Propagators
0 2 4 6 8 10p@GeVD0
1
2
3
4
ZHp2L
0.0 0.5 1.0 1.5 2.0 2.5 3.0p@GeVD1
2
3
4
5
6
GHp2L
0.0 0.5 1.0 1.5 2.0 2.5 3.0p@GeVD0
5
10
15
20
ZHp2L�p
2
Scaling solution
Decoupling solution
Di�erences only at low momenta.
MQH TU Darmstadt April 26, 2013 25/21
Introduction Yang-Mills theory Summary
Scaling solution: Ghost-gluon vertex
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
MQH TU Darmstadt April 26, 2013 26/21
Introduction Yang-Mills theory Summary
In�uence of the three-gluon vertex
ghost-gluon vertex: bare
0 1 2 3 4 5 6 7p@GeV D0
1
2
3
4
ZHp 2L
original three-gluon vertex
MQH TU Darmstadt April 26, 2013 27/21
Introduction Yang-Mills theory Summary
In�uence of the three-gluon vertex
ghost-gluon vertex: bare
0 1 2 3 4 5 6 7p@GeV D0
1
2
3
4
ZHp 2L
original three-gluon vertexBose symmetric three-gluon vertex
MQH TU Darmstadt April 26, 2013 27/21
Introduction Yang-Mills theory Summary
In�uence of the three-gluon vertex
ghost-gluon vertex: bare
0 1 2 3 4 5 6 7p@GeV D0
1
2
3
4
ZHp 2L
0.0 0.5 1.0 1.5 2.0p@GeV D1
2
3
4
5
GHp 2L
original three-gluon vertexBose symmetric three-gluon vertexBose symmetric three-gluon vertex with IR part
⇒ Improved three-gluon vertex adds additional strengthin the mid-momentum regime.
MQH TU Darmstadt April 26, 2013 27/21