Non-perturbative results for n-point functions of Landau gauge - PowerPoint PPT Presentation

Non-perturbative results for n-point functions of Landau gauge Yang-Mills theory at (non-)vanishing temperature Markus Q. Huber in collaboration with Anton K. Cyrol, Lorenz von Smekal Institute of Physics, University of Graz Non-Perturbative Methods in Quantum Field Theory - Workshop in Theoretical Particle Physics Oct. 8, 2014 | University of Graz | Markus Q. Huber | 1 / 23

Correlation functions of QCD Investigate: ◮ Confinement ◮ Dynamical symmetry breaking ◮ Bound states (Non-perturbative) Determination of correlation functions: ◮ Lattice ◮ Effective theories, e.g., (refined) Gribov-Zwanziger, massive Yang-Mills ◮ Functional equations ◮ . . . Oct. 8, 2014 | University of Graz | Markus Q. Huber | 2 / 23

Functional equations and truncations Functional equations. . . ◮ . . . are exact. ◮ . . . form an infinite system of equations. Truncating the system: How close/far from exact solution? ◮ Calculate physical quantities. ◮ Compare to other methods. ◮ Modify the truncation. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

Functional equations and truncations Functional equations. . . ◮ . . . are exact. ◮ . . . form an infinite system of equations. Truncating the system: How close/far from exact solution? ◮ Calculate physical quantities. ◮ Compare to other methods. � ◮ Modify the truncation. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

Functional equations and truncations Functional equations. . . ◮ . . . are exact. ◮ . . . form an infinite system of equations. Truncating the system: How close/far from exact solution? ◮ Calculate physical quantities. ◮ Compare to other methods. � If available. � ◮ Modify the truncation. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

Functional equations and truncations Functional equations. . . ◮ . . . are exact. ◮ . . . form an infinite system of equations. Truncating the system: How close/far from exact solution? ◮ Calculate physical quantities. ◮ Compare to other methods. � If available. � ◮ Modify the truncation. Recent results indicate that primitively divergent correlation functions of Landau gauge Yang-Mills theory provide reasonable truncation for a quantitative, self-consistent and self-contained description. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

Outline ◮ Vacuum ◮ Propagators: Introduction ◮ Three-point functions: ◮ ghost-gluon vertex ◮ three-gluon vertex → talks by Blum, Senn ◮ Four-gluon vertex ◮ Non-vanishing temperature ◮ Ghost-gluon vertex ◮ Three-gluon vertex Oct. 8, 2014 | University of Graz | Markus Q. Huber | 4 / 23

Landau Gauge Yang-Mills theory Gluonic sector of quantum chromodynamics: Yang-Mills theory L = 1 2 F 2 + L gf + L gh F µν = ∂ µ A ν − ∂ ν A µ + i g [ A µ , A ν ] Landau gauge i l i k ◮ simplest one for functional equations � � � 1 j k j i j � L gf = 1 ◮ ∂ µ A µ = 0: 2 ξ ( ∂ µ A µ ) 2 , ξ → 0 i k � ◮ requires ghost fields: � 1 L gh = ¯ c ( − � + g A × ) c j j k � Oct. 8, 2014 | University of Graz | Markus Q. Huber | 5 / 23

Propagators - 1 - 1 - 1 j j j i i j i j - 1 i i + + 2 2 = j - 1 i - 1 j i 6 2 - 1 - 1 - j i j j i i + = Models or results for vertices required. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

Propagators - 1 - 1 - 1 j j j i i j i j - 1 i i + + 2 2 = j - 1 i - 1 j i 6 2 - 1 - 1 - j i j j i i + = Models or results for vertices required. (Tadpole vanishes perturbatively, but can contribute non-perturbatively [MQH, von Smekal ’14] .) Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

Propagators - 1 - 1 - 1 j j j i i j i i + + 2 = - 1 - 1 - j i j j i i + = Typical truncation: no four-gluon vertex, bare ghost-gluon vertex, model for three-gluon vertex Comparison with lattice results Z � p 2 � → missing strength in mid-momentum 4 ��� � � � � � � regime; attributed to lattice data: � 3 � [Sternbeck ’06] � � � � � � � ��� ����������������������� � � � � � � � � � � � � � � � � � � ◮ neglected two-loop diagrams? 2 � � ◮ vertices? 1 � 0 � � 5 p � GeV � 0 1 2 3 4 one-loop truncation Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

Propagators - 1 - 1 - 1 j j j i i j i i + + 2 = - 1 - 1 - j i j j i i + = Typical truncation: no four-gluon vertex, bare ghost-gluon vertex, model for three-gluon vertex Z � p 2 � → Using results from a three-gluon 4.0 vertex calculation, importance of 3.5 two-loop diagrams shown. 3.0 [talk by Blum; Blum, MQH, Mitter, von Smekal ’13] 2.5 2.0 1.5 1.0 p � GeV � 0 2 4 6 8 Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

Three-point functions Vertices (truncated): i k i k i i j + + + = j j j k k j i k i k i i i j k j j k + 1 + 1 + 1 + + - 2 2 2 2 = j j i i j k k k → talk by Blum Oct. 8, 2014 | University of Graz | Markus Q. Huber | 7 / 23

The ghost-gluon vertex To good approximation the ghost-gluon vertex can be taken as bare [Ilgenfritz et al. ’07, Cucchieri, Maas, Mendes ’08, Schleifenbaum, Maas, Wambach, Alkofer ’05, Boucaud et al. ’11, Fister, Pawlowski ’12, MQH, von Smekal ’13, Aguilar, Ibáñez, Papavassiliou ’13, Pelaez, Tissier, Wschebor ’13] . Some influence on propagators [MQH, von Smekal ’13, Aguilar, Ibáñez, Papavassiliou ’13] and three-gluon vertex [Blum, MQH, Mitter, von Smeka ’14, Blum ’14, talk by Blum] . [MQH, von Smekal ’13] A � 0; p 2 , p 2 � 1.4 lattice data: 1.3 [Sternbeck ’06] � 1.2 � � � � � � � � � � � �� 1.1 � � � � � � � � � � � � ���� � � � � 1.0 � � � � � 0.9 10 p � GeV � 2 4 6 8 Oct. 8, 2014 | University of Graz | Markus Q. Huber | 8 / 23

Optimized effective three-gluon vertex Lattice inspired model for three-gluon vertex with zero crossing: D A 3 � p 2 , p 2 , p 2 � 2.0 � 1.5 � � � � 1.0 � � � � � � � � � � 0.5 � � � � � � � �� 5 p � GeV � 0.0 � 1 2 3 4 � � 0.5 � 1.0 [Cucchieri, Maas, Mendes ’08, MQH, von Smekal ’13] Oct. 8, 2014 | University of Graz | Markus Q. Huber | 9 / 23

Optimized effective three-gluon vertex Lattice inspired model for three-gluon vertex with zero crossing: D A 3 � p 2 , p 2 , p 2 � 2.0 � 1.5 � � � � 1.0 � � � � � � � � � � 0.5 � � � � � � � �� 5 p � GeV � 0.0 � 1 2 3 4 � � 0.5 � 1.0 [Cucchieri, Maas, Mendes ’08, MQH, von Smekal ’13] Allows to effectively capture two-loop contributions. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 9 / 23

Propagator results Dynamic ghost-gluon vertex, opt. eff. three-gluon vertex [MQH, von Smekal ’13] Z � p 2 � 4 ��� �� � � � �� 3 � � � � � � ��������������������������� � � � � � � � � � � � � � � � � � � � 2 � � 1 � 5 p � GeV � 0 � � 0 1 2 3 4 G � p 2 � 6 5 4 � 3 � �� � 2 � � � � � � 3.0 p � GeV � 1 0.0 0.5 1.0 1.5 2.0 2.5 Good quantitative agreement for ghost and gluon dressings. ⇒ Input for further calculations. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 10 / 23

Propagator results Dynamic ghost-gluon vertex, opt. eff. FRG results three-gluon vertex [MQH, von Smekal ’13] [Fischer, Maas, Pawlowski ’08] Z � p 2 � 4 Bowman (2004) Sternbeck (2006) ��� �� � � scaling (DSE) �� � 2 decoupling (DSE) 3 � � � � � � ��������������������������� scaling (FRG) � decoupling (FRG) � � � � � � � � � � � � � � � � � � 2 ) 2 � Z(p � 1 1 � 5 p � GeV � 0 � � 0 1 2 3 4 0 0 1 2 3 4 5 p [GeV] G � p 2 � 14 Sternbeck (2006) 6 scaling (DSE) 12 decoupling (DSE) scaling (FRG) 5 10 decoupling (FRG) 2 ) G(p 8 4 6 � 3 � �� 4 � 2 2 � � � � � � 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 3.0 p � GeV � 1 p [GeV] 0.0 0.5 1.0 1.5 2.0 2.5 Good quantitative agreement for ghost and gluon dressings. ⇒ Input for further calculations. Oct. 8, 2014 | University of Graz | Markus Q. Huber | 10 / 23