The Ghost-propagator DSE - Physics

The Low-Momentum Ghost
Dressing Function, the Gluon
Mass and the transition
between Decoupling and
Scaling IR solutions
Based on: JHEP0806(2008)008, PRD80(2009)085018,
PRD82(2010)054007.
by
J. Rodríguez-Quintero;
In collaboration with many other people: A.C. Aguilar, D. Binosi, Ph. Boucaud, M.E.
Gómez, A. Le Yaouanc, J.P. Leroy, J. Micheli, J. Papavassiliou, O. Pène
“T(R)opical QCD”; September 28 th ;Cairns 2010.

Light Cone 2010
Scaling
Vs.
Decoupling

World Cup 2010
Spain
1.- Casillas
4.- S. Ramos
3.- Capdevilla
23.-Piqué
5.- Puyol
8.- Iniesta
10.-Xavi
20.-X. Alonso
22.-Navas
21.-Silva
7.- Villa
Coach: V. Del Bosque
Vs.
1.- Neuer
16.-Lahm
3.- Friedrich
17.-Mertesacker
14.-Badstuber
7.- Schweinsteiger
6.- Khedira
13.-Muller
8.- Ozil
10.-Podolski
11.-Klose
Deutschland
Coach: J. Low

World Cup 2010
Spain
1.- Casillas
4.- S. Ramos
3.- Capdevilla
23.-Piqué
5.- Puyol
8.- Iniesta
10.-Xavi
20.-X. Alonso
22.-Navas
21.-Silva
7.- Villa
Coach: V. Del Bosque
Vs.
1.- Neuer
16.-Lahm
3.- Friedrich
17.-Mertesacker
14.-Badstuber
7.- Schweinsteiger
6.- Khedira
13.-Muller
8.- Ozil
10.-Podolski
11.-Klose
Deutschland
Coach: J. Low

Based on: JHEP0806(2008)008, PRD80(2009)085018,
PRD82(2010)054007.
by
J. Rodríguez-Quintero;
In collaboration with many other people: A.C. Aguilar, D. Binosi, Ph. Boucaud, M.E.
Gómez, A. Le Yaouanc, J.P. Leroy, J. Micheli, J. Papavassiliou, O. Pène
“T(R)opical QCD”; September 28 th ;Cairns 2010.

QCD: SU(3) color interaction
The QCD Lagrangian:
Three-gluon interaction Four-gluon interaction
Fadeev-Popov
(ghost felds)
ghost-gluon interaction

QCD: SU(3) color interaction
The QCD Lagrangian:
Pure Yang-Mills theory
Three-gluon interaction Four-gluon interaction ghost-gluon interaction

QCD in the IR

QCD in the IR
C'est “chiant” l'IR !!!

QCD in the IR (II)

QCD in the IR (III)

QCD in the IR (III)

Break: UV results

The Ghost-propagator DSE (I)

The Ghost-propagator DSE (I)
where:
All in Landau gauge!!

The Ghost-propagator DSE (I)
where:
All in Landau gauge!!

The Ghost-propagator DSE (II)
After MOM renormalization:
One obtains:

The Ghost-propagator DSE (II)
One obtains:

The Ghost-propagator DSE (II)
One obtains:

The Ghost-propagator DSE (II)
One obtains:

The Ghost-propagator DSE (II)
One obtains:
The integral is UV divergent!!

The Ghost-propagator DSE (II)
One obtains:
UV-cutof dependences cancel to each other!!
(in virtue of the interplay of ghost and gluon anomalous dimensions)

The Ghost-propagator DSE (II)
One obtains:
UV-cutof dependences cancel to each other!!
(in virtue of the interplay of ghost and gluon anomalous dimensions)
For not to have to deal with UV, on can subtract:

The Ghost-propagator DSE (II)
One obtains:
UV-cutof dependences cancel to each other!!
(in virtue of the interplay of ghost and gluon anomalous dimensions)
For not to have to deal with UV, on can subtract:
will ultimately go to 0

The Ghost-propagator DSE (II)
One obtains:
UV-cutof dependences cancel to each other!!
(in virtue of the interplay of ghost and gluon anomalous dimensions)
For not to have to deal with UV, on can subtract:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE (III)
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE (III)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE (III)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE (III)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE (III)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE
solutions (I)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE
solutions (I)
Then:
will ultimately go to 0
is positive and non-zero
while ( )

The Ghost-propagator DSE
solutions (I)
Then:
will ultimately go to 0
is positive and non-zero
while ( )

The Ghost-propagator DSE
solutions (I)
Then:
will ultimately go to 0
is positive and non-zero
➔
➔

The Ghost-propagator DSE
solutions (I)
Then:
will ultimately go to 0
is positive and non-zero
➔
➔

Part II: A new hope

The Ghost-propagator DSE
solutions (II)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE
solutions (II)
Then:
will ultimately go to 0
is positive and non-zero
while ( )

The Ghost-propagator DSE
solutions (II)
Then:
will ultimately go to 0
is positive and non-zero
while ( )

The Ghost-propagator DSE
solutions (III)
Then:
will ultimately go to 0
is positive and non-zero

The Ghost-propagator DSE
solutions (III)
Then:
will ultimately go to 0
is positive and non-zero
LQCD favours

The Ghost-propagator DSE
solutions (IV)
In particular:
will ultimately go to 0
is positive and non-zero
Next-to-leading asymptotic approximation:

The Ghost-propagator DSE
solutions (IV)
In particular:
will ultimately go to 0
is positive and non-zero
Next-to-leading asymptotic approximation:
Zero-momentum ghost dressing

The Ghost-propagator DSE
solutions (IV)
In particular:
will ultimately go to 0
is positive and non-zero
Next-to-leading asymptotic approximation:
Zero-momentum ghost dressing
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018

The Ghost-propagator DSE
solutions (IV)
In particular:
will ultimately go to 0
is positive and non-zero
Next-to-leading asymptotic approximation:
Zero-momentum ghost dressing
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
Zero-momentum gluon mass

The Ghost-propagator DSE
solutions (IV)
In particular:
will ultimately go to 0
is positive and non-zero
Next-to-leading asymptotic approximation:
Zero-momentum ghost dressing
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
O �q 2 �

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 ��q � 2 g
� = 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2
at asymptotically large
momenta!!

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
at low momenta!!

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
IR efective charge!!

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
Taylor efective charge:
2 2 2 2
�r�q� = gT ��q � F �q �
IR efective charge!!

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
Taylor efective charge:
2 2 2 2 2
�r�q� = gT ��q � F �q � = gT ���q 2 2
2
� [1�G �q �] F 2 �q 2 �
IR efective charge!!

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
Taylor efective charge:
2 2 2 2 2
�r�q� = gT ��q � F �q � = gT ���q 2 2
2
� [1�G �q �]
IR efective charge!!
F 2 �q 2 � ≃ �g
2 2
T �q �
q 2 �M 2

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
Taylor efective charge:
2 2 2 2 2
�r�q� = gT ��q � F �q � = gT ���q 2 2
2
� [1�G �q �]
IR efective charge!!
F 2 �q 2 � ≃ �g
2 2
T �q �
q 2 �M 2

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
Taylor efective charge:
2 2 2 2 2
�r�q� = gT ��q � F �q � = gT ���q 2 2
2
� [1�G �q �]
�≃1
IR efective charge!!
F 2 �q 2 � ≃ �g
2 2
T �q �
q 2 �M 2

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
QED-like PT efective charge:
�d �q� = g 2 � ��q 2 � = g 2 ��q 2 �
[1�G �q 2 �] 2 ≃ �g 2 �q 2 �
q 2 �M 2
Taylor efective charge:
2 2
2 2 2 2 2
�r�q� = gT ��q � F �q � = gT ���q 2 �g T �q �
� ≃
q 2 �M 2
IR efective charge!!

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
�� PT �q 2 � = �g 2 �q 2 �
4� ≃ �� T �q 2 � = �g
2 2
T �q �
4�

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
�� PT �q 2 � = �g 2 �q 2 �
4� ≃ �� T �q 2 � = �g
2 2
T �q �
4�

The Ghost-propagator DSE
solutions (V)
Next-to-leading asymptotic approximation:
Zero-momentum Taylor-scheme efective charge
A. Aguilar, D. Binosi, J. Papavassiliou, J.R-Q; PRD80(2009)085018
�� PT �0� = �g 2 �0�
4� = �� T �0� = �g
2
T �0�
4�
F �0� [1�G �0�]=1

Lattice (recent) results (I)
Gluon propagator:
� G =1
(massive solution)
I.L. Bogolubsky, E.M. Ilgenfritz,M. Mulluer-Preussker, A. Sternbeck; arXiv:0901.0736[hep-lat]

Lattice (recent) results (II)
Gluon propagator:
� G =1
(massive solution)
O. Oliveira, P. Bicudo ; arXiv:1002.4151[hep-lat]

Lattice (recent) results (III)
Gluon propagator:
� G =1
(massive solution)
A. Cuchieri and T. Mendes ; PoS LAT2007 (2007) 297

Lattice (recent) results (IV)
Ghost dressing
function:
� F =0
(solution type II,
called also decoupling)
I.L. Bogolubsky, E.M. Ilgenfritz,M. Mulluer-Preussker, A. Sternbeck; arXiv:0901.0736[hep-lat]

Lattice (recent) results (IV)
Ghost dressing
function:
� F =0
(solution type II,
called also decoupling)
I.L. Bogolubsky, E.M. Ilgenfritz,M. Mulluer-Preussker, A. Sternbeck; arXiv:0901.0736[hep-lat]

Lattice (recent) results (V)
A few words about Gribov ambiguity

Lattice (recent) results (V)
A few words about Gribov ambiguity
I.L. Bogolubsky, E.M. Ilgenfritz,M. Mulluer-Preussker, A. Sternbeck; arXiv:0901.0736[hep-lat]

Lattice (recent) results (VI)
A few words about Gribov ambiguity
I.L. Bogolubsky, E.M. Ilgenfritz,M. Mulluer-Preussker, A. Sternbeck; arXiv:0901.0736[hep-lat]

Lattice (recent) results (VII)
A few words about Gribov ambiguity
Best copy
vs.
First copy
V.G. Bornyakov, V.K. Mitrjushkin, M. Muller-Preussker; PRD81(2010)054503

Lattice (recent) results (VII)
A few words about Gribov ambiguity
Best copy
vs.
First copy
V.G. Bornyakov, V.K. Mitrjushkin, M. Muller-Preussker; PRD81(2010)054503

Lattice (recent) results (VII)
A few words about Gribov ambiguity
Best copy
vs.
First copy
V.G. Bornyakov, V.K. Mitrjushkin, M. Muller-Preussker; PRD81(2010)054503

Ghost and gluon coupled DSE's
system (I)
●In the BFM-PT scheme:
(i) Gluon SDE:
(ii) Ghost SDE:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (II)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
➔ A massive gluon propagator is obtained
( �G =1)
➔ A fnite ghost dressing function at zero
momentum is also obtained ( )
� F =0

Ghost and gluon coupled DSE's
system (II)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
➔ A massive gluon propagator is obtained
( �G =1)
➔ A fnite ghost dressing function at zero
momentum is also obtained ( )
� F =0

Ghost and gluon coupled DSE's
system (II)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
➔ A massive gluon propagator is obtained
( �G =1)
➔ A fnite ghost dressing function at zero
momentum is also obtained ( )
� F =0

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007

Ghost and gluon coupled DSE's
system (III)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou; PRD78(2008)025010
J. Rodríguez-Quintero, PRD82(2010)054007
The BFM-PT ghost solutions are very well described by the
asymptotic formula!!!

Part III: The scaling strikes back
The Scaling

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
Critical value

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

Ghost and gluon coupled DSE's
system (IV)
●In the BFM-PT scheme:
Critical exponent:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
��10 GeV �=0.0854 �6�

A phase transition between DSE
solutions? (I)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

A phase transition between DSE
solutions? (I)
●In the BFM-PT scheme:
� F =0
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

A phase transition between DSE
solutions? (I)
●In the BFM-PT scheme:
� F =0
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
Critical value
� F ≠0

A phase transition between DSE
solutions? (I)
●In the BFM-PT scheme:
If � G =1 F �q 2 � ~ 1
� F =0
�q 2
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
Critical value
� F ≠0

A phase transition between DSE
solutions? (I)
●In the BFM-PT scheme:
If � G =1 F �q 2 � ~ 1
� F =0
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
�q 2 � G � 1
Critical value
� F ≠0

A phase transition between DSE
solutions? (I)
●In the BFM-PT scheme:
If � G =1 F �q 2 � ~ 1
� F =0
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
�q 2 � G � 1
Critical value
� F ≠0

A phase transition between DSE
solutions? (II)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010

A phase transition between DSE
solutions? (II)
●In the BFM-PT scheme:
1
F R �0�
≠ 0
IR ghost non-dominance
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
IR ghost dominance
1
F R �0�
� 0

A phase transition between DSE
solutions? (II)
●In the BFM-PT scheme:
A.C.Aguilar, D.Binosi, J.Papavassiliou;
PRD78(2008)025010
Order parameter?
1
F R �0�
≠ 0
IR ghost non-dominance
IR ghost dominance
1
F R �0�
� 0

Conclusions:
●The ghost propagator DSE allows for solutions observing
two types of IR behaviors:
➔ I: Divergent ghost dressing function and a fnite ghost-gluon
coupling at zero momentum (“scaling” or “conformal”)
➔ II: Finite ghost dressing function and massive gluon propagator,
with a vanishing ghost-gluon coupling at zero-momentum
(“decoupling”).
●(Yang-Mills) Lattice QCD clearly favors the ”decoupling”
●In the BFM-PT scheme, the DSE coupled system for ghost
and gluon propagators, massive gluon and not-enhanced
ghost (“decoupling”) solutions emerge.
●A low-momentum asymptotic formula for the ghost
dressing function perfectly agrees with numerical results.

●
Conclusions:
➔ I: Divergent ghost dressing function and a fnite ghost-gluon
coupling at zero momentum (“scaling” or “conformal”)
➔ II: Finite ghost dressing function and massive gluon propagator,
with a vanishing ghost-gluon coupling at zero-momentum
(“decoupling”).
●(Yang-Mills) Lattice QCD clearly favors the ”decoupling”
●In the BFM-PT scheme, the DSE coupled system for ghost
and gluon propagators, massive gluon and not-enhanced
ghost (“decoupling”) solutions emerge.
●A low-momentum asymptotic formula for the ghost
dressing function perfectly agrees with numerical results.
●A sort of phase transition between both types of solutions
appears to take place.

Conclusions:
●
●(Yang-Mills) Lattice QCD clearly favors the ”decoupling”
●In the BFM-PT scheme, the DSE coupled system for ghost
and gluon propagators, massive gluon and not-enhanced
ghost (“decoupling”) solutions emerge.
●A low-momentum asymptotic formula for the ghost
dressing function perfectly agrees with numerical results.
●A sort of phase transition between both types of solutions
appears to take place.
●The control parameter “dialing” the transition is the
Taylor-scheme coupling constant.

Conclusions:
●A low-momentum asymptotic formula for the ghost
dressing function perfectly agrees with numerical results.
●A sort of phase transition between both types of solutions
appears to take place.
●The control parameter “dialing” the transition is the
Taylor-scheme coupling constant.
●The role of the order parameter is played by the inverse
of the ghost dressing, which vanishes at the critical point.
●Is this order parameter related to some condensate?
Kugo-Ojima and Gribov-Zwanziger confnement scenarios, ghost
dominance... ???????

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