Gluon Propagator and Gluonic Screening around Deconfinement

Gluon Propagator and Gluonic Screening around Deconfinement Tereza Mendes Instituto de Física de São Carlos University of São Paulo Work in collaboration with Attilio Cucchieri

Screening at High T At high T : Debye screening of color charge A 0 (x)a 0 (y) e m el x y

Screening at High T At high T : Debye screening of color charge A 0 (x)a 0 (y) e m el x y Chromoelectric (respec. chromomagnetic) screening related to longitudinal (respec. transverse) gluon propagator with p 0 = 0

Screening at High T At high T : Debye screening of color charge A 0 (x)a 0 (y) e m el x y Chromoelectric (respec. chromomagnetic) screening related to longitudinal (respec. transverse) gluon propagator with p 0 = 0 Determine screening masses/lengths from propagators

Screening at High T At high T : Debye screening of color charge A 0 (x)a 0 (y) e m el x y Chromoelectric (respec. chromomagnetic) screening related to longitudinal (respec. transverse) gluon propagator with p 0 = 0 Determine screening masses/lengths from propagators Problem: gluon propagator is gauge-dependent... but poles are believed to be gauge-independent

Screening at High T At high T : Debye screening of color charge A 0 (x)a 0 (y) e m el x y Chromoelectric (respec. chromomagnetic) screening related to longitudinal (respec. transverse) gluon propagator with p 0 = 0 Determine screening masses/lengths from propagators Problem: gluon propagator is gauge-dependent... but poles are believed to be gauge-independent Expect real electric mass (PT at large T )

Screening at High T At high T : Debye screening of color charge A 0 (x)a 0 (y) e m el x y Chromoelectric (respec. chromomagnetic) screening related to longitudinal (respec. transverse) gluon propagator with p 0 = 0 Determine screening masses/lengths from propagators Problem: gluon propagator is gauge-dependent... but poles are believed to be gauge-independent Expect real electric mass (PT at large T ) Dimensional reduction (3D Adjoint-Higgs picture): confined magnetic gluon (nontrivial magnetic mass)

Behavior at the Deconfinement Transition

Behavior at the Deconfinement Transition Lattice studies observe peak of the IR value of Landau-gauge gluon propagator at deconfinement, suggesting alternative order parameter for the QCD phase transition.

Behavior at the Deconfinement Transition Lattice studies observe peak of the IR value of Landau-gauge gluon propagator at deconfinement, suggesting alternative order parameter for the QCD phase transition. We investigate the IR behavior of electric and magnetic gluon propagators around the transition using large lattice sizes and estimate electric and magnetic screening masses using an Ansatz from the zero-temperature case.

Behavior at the Deconfinement Transition Lattice studies observe peak of the IR value of Landau-gauge gluon propagator at deconfinement, suggesting alternative order parameter for the QCD phase transition. We investigate the IR behavior of electric and magnetic gluon propagators around the transition using large lattice sizes and estimate electric and magnetic screening masses using an Ansatz from the zero-temperature case. Disclaimer: preliminary! (puzzling systematic effects...)

Behavior at the Deconfinement Transition Lattice studies observe peak of the IR value of Landau-gauge gluon propagator at deconfinement, suggesting alternative order parameter for the QCD phase transition. We investigate the IR behavior of electric and magnetic gluon propagators around the transition using large lattice sizes and estimate electric and magnetic screening masses using an Ansatz from the zero-temperature case. Disclaimer: preliminary! (puzzling systematic effects...) and pure-su(2) case: different transition two-color theory

Behavior at the Deconfinement Transition Lattice studies observe peak of the IR value of Landau-gauge gluon propagator at deconfinement, suggesting alternative order parameter for the QCD phase transition. We investigate the IR behavior of electric and magnetic gluon propagators around the transition using large lattice sizes and estimate electric and magnetic screening masses using an Ansatz from the zero-temperature case. Disclaimer: preliminary! (puzzling systematic effects...) and pure-su(2) case: different transition two-color theory

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001)

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices:

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices: Transverse gluon decreases as T increases (stronger IR suppression at high T )

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices: Transverse gluon decreases as T increases (stronger IR suppression at high T ) Longitudinal gluon reaches a plateau for p 0 at T T c

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices: Transverse gluon decreases as T increases (stronger IR suppression at high T ) Longitudinal gluon reaches a plateau for p 0 at T T c Shows peak at T c? what about SU(3)?

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices: Transverse gluon decreases as T increases (stronger IR suppression at high T ) Longitudinal gluon reaches a plateau for p 0 at T T c Shows peak at T c? what about SU(3)? Recently: Fischer, Maas & Müller (2010) confirm above results, use peaked gluon propagator to construct order parameter;

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices: Transverse gluon decreases as T increases (stronger IR suppression at high T ) Longitudinal gluon reaches a plateau for p 0 at T T c Shows peak at T c? what about SU(3)? Recently: Fischer, Maas & Müller (2010) confirm above results, use peaked gluon propagator to construct order parameter; finer T resolution, SU(3), moderate-size lattices, masses from D L (0) 1/2

Lattice Studies Earlier (high T ): Heller, Karsch & Rank (1995); Cucchieri, Karsch & Petreczky (2001) propagators near T c : Cucchieri, Maas & Mendes (2007), using small lattices: Transverse gluon decreases as T increases (stronger IR suppression at high T ) Longitudinal gluon reaches a plateau for p 0 at T T c Shows peak at T c? what about SU(3)? Recently: Fischer, Maas & Müller (2010) confirm above results, use peaked gluon propagator to construct order parameter; finer T resolution, SU(3), moderate-size lattices, masses from D L (0) 1/2 Peak seen also by Bornyakov & Mitrjushkin (2010, 2011)

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c Consider more general fits

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c Consider more general fits At T = 0 momentum-space propagator is well fitted by a Gribov-Stingl form, allowing for complex conjugate poles D L,T (p) = C 1 + dp 2η (p 2 +a) 2 + b 2

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c Consider more general fits At T = 0 momentum-space propagator is well fitted by a Gribov-Stingl form, allowing for complex conjugate poles D L,T (p) = C 1 + dp 2η (p 2 +a) 2 + b 2 Poles at masses m 2 = a ± ib

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c Consider more general fits At T = 0 momentum-space propagator is well fitted by a Gribov-Stingl form, allowing for complex conjugate poles D L,T (p) = C 1 + dp 2η (p 2 +a) 2 + b 2 Poles at masses m 2 = a ± ib m = m R + im I

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c Consider more general fits At T = 0 momentum-space propagator is well fitted by a Gribov-Stingl form, allowing for complex conjugate poles D L,T (p) = C 1 + dp 2η (p 2 +a) 2 + b 2 Poles at masses m 2 = a ± ib m = m R + im I For longitudinal gluon: expect m I 0 at high T

Masses from Propagators Of course, even if an exponential fit to the longitudinal gluon works at high T it is not obvious that this should hold at T > T c Consider more general fits At T = 0 momentum-space propagator is well fitted by a Gribov-Stingl form, allowing for complex conjugate poles D L,T (p) = C 1 + dp 2η (p 2 +a) 2 + b 2 Poles at masses m 2 = a ± ib m = m R + im I For longitudinal gluon: expect m I 0 at high T Note: D(0) 1/2 = (a 2 +b 2 )/C mixes m R and m I and depends on the normalization C

Infrared Propagator at T = 0

Infrared Propagator at T = 0 Gribov-Zwanziger confinement scenario in Landau gauge predicts a suppressed gluon propagator in the IR limit (in combination with an enhanced ghost propagator). These predictions are tested by analytic methods (such as Dyson-Schwinger equations) and lattice simulations.

Infrared Propagator at T = 0 Gribov-Zwanziger confinement scenario in Landau gauge predicts a suppressed gluon propagator in the IR limit (in combination with an enhanced ghost propagator). These predictions are tested by analytic methods (such as Dyson-Schwinger equations) and lattice simulations. Large-lattice results:

Infrared Propagator at T = 0 Gribov-Zwanziger confinement scenario in Landau gauge predicts a suppressed gluon propagator in the IR limit (in combination with an enhanced ghost propagator). These predictions are tested by analytic methods (such as Dyson-Schwinger equations) and lattice simulations. Large-lattice results: gluon propagator D(p) is suppressed as p 0, while real-space propagator violates reflection positivity

Infrared Propagator at T = 0 Gribov-Zwanziger confinement scenario in Landau gauge predicts a suppressed gluon propagator in the IR limit (in combination with an enhanced ghost propagator). These predictions are tested by analytic methods (such as Dyson-Schwinger equations) and lattice simulations. Large-lattice results: gluon propagator D(p) is suppressed as p 0, while real-space propagator violates reflection positivity on very large lattices (L 27 fm) one sees that D(0) > 0; good fit to Gribov-Stingl form

Infrared Propagator at T = 0 Gribov-Zwanziger confinement scenario in Landau gauge predicts a suppressed gluon propagator in the IR limit (in combination with an enhanced ghost propagator). These predictions are tested by analytic methods (such as Dyson-Schwinger equations) and lattice simulations. Large-lattice results: gluon propagator D(p) is suppressed as p 0, while real-space propagator violates reflection positivity on very large lattices (L 27 fm) one sees that D(0) > 0; good fit to Gribov-Stingl form Consistent with massive solution of DSEs and refined GZ scenario.

Infrared Propagator at T = 0 Gribov-Zwanziger confinement scenario in Landau gauge predicts a suppressed gluon propagator in the IR limit (in combination with an enhanced ghost propagator). These predictions are tested by analytic methods (such as Dyson-Schwinger equations) and lattice simulations. Large-lattice results: gluon propagator D(p) is suppressed as p 0, while real-space propagator violates reflection positivity on very large lattices (L 27 fm) one sees that D(0) > 0; good fit to Gribov-Stingl form Consistent with massive solution of DSEs and refined GZ scenario. Not consistent with scaling solution D (p 2 ) 2κ 1

This Work: Parameters pure SU(2) case, with a standard Wilson action cold start, projection on positive Polyakov loop configurations Landau-gauge fixing using stochastic overrelaxation lattice sizes ranging from 48 3 4 to 192 3 16 several β values, allowing several values of the temperature T = 1/N t a around T c gluon dressing functions normalized to 1 at 2 GeV masses extracted from Gribov-Stingl behavior (fits shown in plots below)

Results: Low Temperatures As T is turned on, magnetic propagator gets more strongly suppressed (3d-like), electric one increases 20 D L 48 3 X 48 2.299 (0.17, 8.2) fm D L 64 3 X 64 2.299 (0.17, 10.9) fm D T 48 3 X 48 2.299 (0.17, 8.2) fm D T 64 3 X 64 2.299 (0.17, 10.9) fm 20 D L 96 3 X 16 2.310 (0.16, 15.4) fm D T 96 3 X 16 2.310 (0.16, 15.4) fm 15 15 D L, D T (GeV -2 ) 10 D L, D T (GeV -2 ) 10 T = 0 T = 0.25 T c 5 5 0 0 0.5 1 1.5 2 2.5 0 0 0.5 1 1.5 2 2.5 p (GeV) p (GeV)

Results: Low Temperatures At larger T, magnetic propagator slightly more suppressed, electric one increases (showing IR plateau?) D L, D T (GeV -2 ) 20 15 10 D L 48 3 X 8 2.299 (0.17, 8.2) fm D L 48 3 X 8 2.301 (0.16, 7.7) fm D L 96 3 X 8 2.299 (0.17, 16.3) fm D L 96 3 X 16 2.521 (0.08, 7.7) fm D T 48 3 X 8 2.299 (0.17, 8.2) fm D T 48 3 X 8 2.301 (0.16, 7.7) fm D T 96 3 X 8 2.299 (0.17, 16.3) fm D T 96 3 X 16 2.521 (0.08, 7.7) fm D L, D T (GeV -2 ) 20 15 10 D L 72 3 X 6 2.319 (0.16, 11.5) fm D L 96 3 X 8 2.402 (0.12, 11.5) fm D T 72 3 X 6 2.319 (0.16, 11.5) fm D T 96 3 X 8 2.402 (0.12, 11.5) fm T = 0.5 T c T = 0.7 T c 5 5 0 0 0.5 1 1.5 2 2.5 0 0 0.5 1 1.5 2 2.5 p (GeV) p (GeV)

Results: Longitudinal Gluon at T c 40 T c 48 3 X 4 2.299 (0.17, 8.2) fm 35 30 25 D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Longitudinal Gluon at T c 40 35 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm 30 25 D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Longitudinal Gluon at T c 40 35 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm T c 192 3 X 4 2.299 (0.17, 32.6) fm 30 25 D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Longitudinal Gluon at T c 40 35 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm T c 192 3 X 4 2.299 (0.17, 32.6) fm T c 72 3 X 6 2.427 (0.11, 7.9) fm 30 25 D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Longitudinal Gluon at T c 40 35 30 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm T c 192 3 X 4 2.299 (0.17, 32.6) fm T c 72 3 X 6 2.427 (0.11, 7.9) fm T c 144 3 X 6 2.427 (0.11, 15.8) fm 25 D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Longitudinal Gluon at T c 40 35 30 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm T c 192 3 X 4 2.299 (0.17, 32.6) fm T c 72 3 X 6 2.427 (0.11, 7.9) fm T c 144 3 X 6 2.427 (0.11, 15.8) fm T c 96 3 X 8 2.511 (0.08, 7.7) fm 25 D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Longitudinal Gluon at T c 40 35 30 25 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm T c 192 3 X 4 2.299 (0.17, 32.6) fm T c 72 3 X 6 2.427 (0.11, 7.9) fm T c 144 3 X 6 2.427 (0.11, 15.8) fm T c 96 3 X 8 2.511 (0.08, 7.7) fm T c 192 3 X 16 2.740 (0.04, 7.7) fm D L (GeV -2 ) 20 15 10 5 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Transverse Gluon at T c 6 5 4 T c 48 3 X 4 2.299 (0.17, 8.2) fm T c 96 3 X 4 2.299 (0.17, 16.3) fm T c 192 3 X 4 2.299 (0.17, 32.6) fm T c 72 3 X 6 2.427 (0.11, 7.9) fm T c 144 3 X 6 2.427 (0.11, 15.8) fm T c 96 3 X 8 2.511 (0.08, 7.7) fm T c 192 3 X 16 2.740 (0.04, 7.7) fm D T (GeV -2 ) 3 2 1 0 0 0.5 1 1.5 2 2.5 p (GeV)

Results: Propagators at 0.98 T c Just below T c, systematic errors for D L (p) are already present 40 35 30 0.98Tc 48 3 X 4 2.293 (0.17, 8.2) fm 0.98Tc 96 3 X 4 2.293 (0.17, 16.3) fm 0.98Tc 48 3 X 8 2.506 (0.08, 3.8) fm 0.98Tc 96 3 X 8 2.505 (0.08, 7.7) fm 0.98Tc 96 3 X 16 2.733 (0.04, 3.8) fm 0.98Tc 192 3 X 16 2.733 (0.04, 7.7) fm 40 35 30 0.98Tc 48 3 X 4 2.293 (0.17, 8.2) fm 0.98Tc 96 3 X 4 2.293 (0.17, 16.3) fm 0.98Tc 48 3 X 8 2.506 (0.08, 3.8) fm 0.98Tc 96 3 X 8 2.505 (0.08, 7.7) fm 0.98Tc 96 3 X 16 2.733 (0.04, 3.8) fm 0.98Tc 192 3 X 16 2.733 (0.04, 7.7) fm 25 25 D L (GeV -2 ) 20 D T (GeV -2 ) 20 15 15 10 10 5 5 0 0 0.5 1 1.5 2 2.5 0 0 0.5 1 1.5 2 2.5 p (GeV) p (GeV)

Results: Propagators at 1.01 T c Just above T c, systematic errors for D L (p) seem much less severe, IR plateau for D L (p) drops significantly for N t 8 40 35 1.01Tc 48 3 X 4 2.302 (0.16, 7.7) fm 1.01Tc 96 3 X 4 2.302 (0.16, 15.4) fm 1.01Tc 96 3 X 8 2.515 (0.08, 7.7) fm 1.01Tc 192 3 X 8 2.515 (0.08, 15.4) fm 40 35 1.01Tc 48 3 X 4 2.302 (0.16, 7.7) fm 1.01Tc 96 3 X 4 2.302 (0.16, 15.4) fm 1.01Tc 96 3 X 8 2.515 (0.08, 7.7) fm 1.01Tc 192 3 X 8 2.515 (0.08, 15.4) fm 30 30 25 25 D L (GeV -2 ) 20 D T (GeV -2 ) 20 15 15 10 10 5 5 0 0 0.5 1 1.5 2 2.5 0 0 0.5 1 1.5 2 2.5 p (GeV) p (GeV)

Results: Propagators at 1.02 T c Just above T c, systematic errors for D L (p) seem much less severe, IR plateau for D L (p) drops somewhat for N t 8 40 35 1.02Tc 48 3 X 4 2.304 (0.16, 7.7) fm 1.02Tc 96 3 X 4 2.304 (0.16, 15.4) fm 1.02Tc 96 3 X 8 2.518 (0.08, 7.7) fm 1.02Tc 192 3 X 16 2.746 (0.04, 7.7) fm 40 35 1.02Tc 48 3 X 4 2.304 (0.16, 7.7) fm 1.02Tc 96 3 X 4 2.304 (0.16, 15.4) fm 1.02Tc 96 3 X 8 2.518 (0.08, 7.7) fm 1.02Tc 192 3 X 16 2.746 (0.04, 7.7) fm 30 30 25 25 D L (GeV -2 ) 20 D T (GeV -2 ) 20 15 15 10 10 5 5 0 0 0.5 1 1.5 2 2.5 0 0 0.5 1 1.5 2 2.5 p (GeV) p (GeV)

Comparison: 0.5T c vs. T c 20 15 D L at 0.5 T c D L at 1.01 T c D T at 0.5 T c D T at 1.01 T c D L, D T (GeV -2 ) 10 5 0.5 T c versus 1.01 T c 0 0 0.5 1 1.5 2 2.5 p (GeV)

Infrared Plateau for D L (p) vs. T IR plateau [from D L (0)]: all T (left) and smaller range (right). 45 40 35 30 "DL0_sym" "DL0_2" "DL0_4" "DL0_6" "DL0_8" "DL0_16" 50 45 40 35 "DL0_4" "DL0_6" "DL0_8" "DL0_16" D L (0) (GeV -2 ) 25 20 15 D L (0) (GeV -2 ) 30 25 20 15 10 10 5 5 0 0 0.5 1 1.5 2 2.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 T/T c T/T c Peak at T c for N t = 4 finite maximum at 0.9 T c for N t = 16.

Electric and Magnetic Masses vs. T T/T c N 3 s N t m (E) R m (E) I m (M) R m (M) I 0 64 3 64 0.83 GeV 0.43 GeV 0.86 GeV 0.51 GeV 0.25 96 3 16 0.61 GeV 0.28 GeV 0.57 GeV 0.28 GeV 0.5 48 3 8 0.51 GeV 0.13 GeV 0.59 GeV 0.36 GeV 0.7 96 3 8 0.31 GeV 0.13 GeV 0.37 GeV 0.24 GeV 0.9 96 3 16 0.10 GeV 0.06 GeV 0.15 GeV 0.10 GeV 0.98 96 3 8 0.19 GeV 0.10 GeV 0.28 GeV 0.20 GeV 1.0 96 3 8 0.23 GeV 0.09 GeV 0.25 GeV 0.19 GeV 1.05 96 3 8 0.29 GeV 0.09 GeV 0.24 GeV 0.18 GeV 2.0 96 3 8 0.27 GeV 0.07 GeV 0.19 GeV 0.14 GeV

(Real-space) Transverse Gluon at 0.25T c 120 0.25Tc 96 3 X 16 2.310 (0.16, 15.4) fm 100 80 D T (GeV -2 ) 60 40 20 0-20 0 0.5 1 1.5 2 2.5 3 z (fm)

(Real-space) Transverse Gluon at T c 120 100 80 Tc 48 3 X 4 2.299 (0.17, 8.2) fm Tc 96 3 X 4 2.299 (0.17, 16.3) fm Tc 192 3 X 4 2.299 (0.17, 32.6) fm Tc 72 3 X 6 2.427 (0.11, 7.9) fm Tc 144 3 X 6 2.427 (0.11, 15.8) fm D T (GeV -2 ) 60 40 20 0-20 0 0.5 1 1.5 2 2.5 3 z (fm)

(Real-space) Transverse Gluon at 2T c 120 100 2Tc 48 3 X 4 2.506 (0.08, 3.8) fm 2Tc 96 3 X 4 2.506 (0.08, 7.7) fm 2Tc 48 3 X 2 2.299 (0.17, 8.2) fm 2Tc 96 3 X 2 2.299 (0.17, 16.3) fm 80 D T (GeV -2 ) 60 40 20 0-20 0 0.5 1 1.5 2 2.5 3 z (fm)

(Real-space) Longitudinal Gluon at 0.25T c 120 0.25Tc 96 3 X 16 2.310 (0.16, 15.4) fm 100 80 D L (GeV -2 ) 60 40 20 0-20 0 1 2 3 4 5 6 z (fm)

(Real-space) Longitudinal Gluon at T c 120 100 80 Tc 48 3 X 4 2.299 (0.17, 8.2) fm Tc 96 3 X 4 2.299 (0.17, 16.3) fm Tc 192 3 X 4 2.299 (0.17, 32.6) fm Tc 72 3 X 6 2.427 (0.11, 7.9) fm Tc 144 3 X 6 2.427 (0.11, 15.8) fm D L (GeV -2 ) 60 40 20 0-20 0 1 2 3 4 5 6 z (fm)

(Real-space) Longitudinal Gluon at 2T c 120 100 2Tc 48 3 X 4 2.506 (0.08, 3.8) fm 2Tc 96 3 X 4 2.506 (0.08, 7.7) fm 2Tc 48 3 X 2 2.299 (0.17, 8.2) fm 2Tc 96 3 X 2 2.299 (0.17, 16.3) fm 80 D L (GeV -2 ) 60 40 20 0-20 0 1 2 3 4 5 6 z (fm)

Conclusions

Conclusions (Lattice) Size matters!

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ):

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ): good fits to Gribov-Stingl form, with comparable real and imaginary parts of pole masses; sizeable physical-volume effects, turn-over in momentum at about 400 MeV; insensitive to transition (in agreement with other studies)

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ): good fits to Gribov-Stingl form, with comparable real and imaginary parts of pole masses; sizeable physical-volume effects, turn-over in momentum at about 400 MeV; insensitive to transition (in agreement with other studies) (re)confirmation of dimensional-reduction picture

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ): good fits to Gribov-Stingl form, with comparable real and imaginary parts of pole masses; sizeable physical-volume effects, turn-over in momentum at about 400 MeV; insensitive to transition (in agreement with other studies) (re)confirmation of dimensional-reduction picture Longitudinal gluon propagator: around the transition, large-lattice (and N t > 8) results indicate no singularity (and no violation of reflection positivity);

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ): good fits to Gribov-Stingl form, with comparable real and imaginary parts of pole masses; sizeable physical-volume effects, turn-over in momentum at about 400 MeV; insensitive to transition (in agreement with other studies) (re)confirmation of dimensional-reduction picture Longitudinal gluon propagator: around the transition, large-lattice (and N t > 8) results indicate no singularity (and no violation of reflection positivity); masses from complex poles as opposed to D L (0) 1/2 (electric mass is also nontrivial);

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ): good fits to Gribov-Stingl form, with comparable real and imaginary parts of pole masses; sizeable physical-volume effects, turn-over in momentum at about 400 MeV; insensitive to transition (in agreement with other studies) (re)confirmation of dimensional-reduction picture Longitudinal gluon propagator: around the transition, large-lattice (and N t > 8) results indicate no singularity (and no violation of reflection positivity); masses from complex poles as opposed to D L (0) 1/2 (electric mass is also nontrivial); smooth around the transition; imaginary part seems to get smaller at higher T

Conclusions (Lattice) Size matters! Transverse gluon propagator shows confinement (at least up to 2T c ): good fits to Gribov-Stingl form, with comparable real and imaginary parts of pole masses; sizeable physical-volume effects, turn-over in momentum at about 400 MeV; insensitive to transition (in agreement with other studies) (re)confirmation of dimensional-reduction picture Longitudinal gluon propagator: around the transition, large-lattice (and N t > 8) results indicate no singularity (and no violation of reflection positivity); masses from complex poles as opposed to D L (0) 1/2 (electric mass is also nontrivial); smooth around the transition; imaginary part seems to get smaller at higher T feels the transition, but no singular behavior (except for the systematic effects!)