Green's functions in Yang-Mills theoryaxm/oberwoelz06.pdf · Zwanziger and Kugo-Ojima scenarios, DSEs and RGs Green's functions – Zero temperature: Propagators – Finite temperature

Green's functionsin

Yang-Mills theory

Attilio Cucchieri, Axel Maas, Tereza MendesInstitute of Physics São Carlos

University of São Paulo

Symposium QCD: Facts and Prospects - Oberwölz – Austria12th of September 2006

Green's functions in Yang-Mills theory/Axel Maas

Overview● Green's functions

Green's functions – Zero temperature – Finite temperature - Summary


Overview● Green's functions● Yang-Mills theory at zero temperature

Note: All results in SU(2)




● Landau gauge● Propagators and vertices






● Interpolating and linear covariant gauges






● Interpolating and linear covariant gauges● Yang-Mills theory at finite temperature● Summary




Overview● Green's functions



Green's functions● Green's functions describe a theory completely




● Encode non-perturbative phenomena




● Encode non-perturbative phenomena● Yang-Mills theory: QCD without quarks

● Confinement




● Encode non-perturbative phenomena● Yang-Mills theory: QCD without quarks

● Confinement● Confinement long-distance phenomena

● Infrared properties relevant



Green's functions● Various predictions




● Confinement scenarios● Gribov-Zwanziger● Kugo-Ojima● Other...





● Explicit calculations● Functional methods

● DSEs● Renormalization group● Other...





● Explicit calculations● Functional methods

● DSEs● Renormalization group● Other...

● Require assumptions



Green's functions● Lattice calculations can

● Check● Provide assumptions● Make observations





● 'Only' affected by approximations





● 'Only' affected by approximations● Finite volume – not arbitrarily small momenta● Discretization – not arbitrarily large momenta● Violation of rotational symmetry● Euclidean





● 'Only' affected by approximations● Finite volume – not arbitrarily small momenta● Discretization – not arbitrarily large momenta● Violation of rotational symmetry● Euclidean

● Combination of methods necessary







● Landau gauge



Landau-gauge and ghosts● Gauge sector: Choose Landau gauge

● Degrees of freedom:

Gluons:

Ghosts: (Intermediate states - not observable)

L=−14

F a F ,a−ca∂ D

ab cb

F a =∂ A

a−∂ Aa−g f abc A

b Ac

Dab=ab∂−g f abc A

c

Aa

ca ,ca

Green's functions – Zero temperature: Landau gauge – Finite temperature - Summary



● Landau gauge● Propagators



Propagators [Introduction: Alkofer & von Smekal, 2001]

● 2-point Green's functions are the propagators

Green's functions – Zero temperature: Propagators – Finite temperature - Summary



● 2-point Green's functions are the propagators● Gluon:

Dab x− y = A

a x Ab y

D p=−p p

p2 Z pp2





● Ghost:

Dab x− y = A

a x Ab y

DGab x− y= ca x cb y

D p=−p p

p2 Z pp2

DG p=−G pp2





● Ghost:

● Ghost linked to the Faddeev-Popov operator

Dab x− y = A

a x Ab y

DGab x− y= ca x cb y

DGab x− y ~ ∂ D

ab−1 = ∂ab∂−g f abc Ac −1

D p=−p p

p2 Z pp2

DG p=−G pp2



Confinement in propagators● (Euclidean) Infrared: p2=0




● Gluons and ghosts 'massless': p2=0 is on-shell condition





● Vanishing propagator at p2=0: No propagating physical particle – confined!





● Vanishing propagator at p2=0: No propagating physical particle – confined!● Predicted for the gluon propagator by the Gribov-

Zwanziger and Kugo-Ojima scenarios, DSEs and RGs






Zwanziger and Kugo-Ojima scenarios, DSEs and RGs● Divergence at p2=0: Mediates long range forces






Zwanziger and Kugo-Ojima scenarios, DSEs and RGs● Divergence at p2=0: Mediates long range forces

● Predicted for the ghost propagator by the Gribov-Zwanziger and Kugo-Ojima scenarios, DSEs and RGs



[DSE: Fischer et al., PLB 2002,Lattice 324: Cucchieri et al., unpublished]Ghost

Green's functions – Zero temperature: Ghost – Finite temperature - Summary



● IR divergent: Mediates long-range forces




● IR divergent: Mediates long-range forces● DSEs, RGs: G~(p2)-0.595 [Zwanziger PRD 2002, Lerche et al. PRD 2002, [Pawlowski et al., PRL 2004]





● Faddeev-Popov operator eigenspectrum enhanced near zero





● Faddeev-Popov operator eigenspectrum enhanced near zero● Due to topological configurations? [Gattnar et al., 2004, Greensite et al., 2004, Maas, 2006]



Gluon [DSE: Fischer et al., PLB 2002, Lattice 524: Cucchieri et al., 2006]

Green's functions – Zero temperature: Gluon – Finite temperature - Summary


Gluon

● Infrared vanishing – confined● DSEs and RG gives Z(p)~(p2)1.19

[Zwanziger PRD 2002, Lerche et al. PRD 2002,

Pawlowski et al., PRL 2004]

[DSE: Fischer et al., PLB 2002, Lattice 524: Cucchieri et al., 2006]



Gluon




● Lattice data strongly affected by finite volume




Gluon




● Lattice data strongly affected by finite volume




Gluon – large lattice volumes needed[203: Cucchieri et al., PRD 2006]

Green's functions – Zero temperature: Finite volume effects – Finite temperature - Summary


Gluon – large lattice volumes needed[203, 303: Cucchieri et al., PRD 2006]



Gluon – large lattice volumes needed[203, 303: Cucchieri et al., PRD 2006 403: Cucchieri et al., unpublished]



Gluon – large lattice volumes needed[203, 303: Cucchieri et al., PRD 2006 403: Cucchieri et al., unpublished 803: Cucchieri et al., PRD 2003]



Gluon – large lattice volumes needed[203, 303: Cucchieri et al., PRD 2006 403: Cucchieri et al., unpublished 803, 1403: Cucchieri et al., PRD 2003]



Gluon – large lattice volumes needed[Lattice continuum extrapolated: Cucchieri et al., PRD 2001 Lattice 1403: Cucchieri et al., PRD 2003 DSE: Maas et al., EPJC 2004]



Gluon – large lattice volumes needed

● DSEs: Goes to zero, Z(p)~(p2)1.29 [Zwanziger, PRD 2002]

[Lattice continuum extrapolated: Cucchieri et al., PRD 2001 Lattice 1403: Cucchieri et al., PRD 2003 DSE: Maas et al., EPJC 2004]




● DSEs: Goes to zero, Z(p)~(p2)1.29 [Zwanziger, PRD 2002, Maas et al., EPJC 2004]

● Comparison to continuum only possible in the range 1/Na<<p





● DSEs: Goes to zero, Z(p)~(p2)1.29 [Zwanziger, PRD 2002, Maas et al., EPJC 2004]

● Comparison to continuum only possible in the range 1/Na<<p<<EIR




Ghost propagator [Lattice 303, Cucchieri et al., 2006]



Ghost propagator [Lattice 303, Cucchieri et al., 2006]



Ghost propagator [Lattice 303, Cucchieri et al., 2006 Lattice 803, Cucchieri et al., PRD 2006 DSE: Maas et al., EPJC 2004]



Ghost propagator [Lattice 303, Cucchieri et al., 2006 Lattice 803, Cucchieri et al., PRD 2006 DSE: Maas et al., EPJC 2004]

● Strong divergence – weak (if at all) volume dependent● DSEs: Diverges, G(p)~(p2)-0.4 [Zwanziger, PRD 2002]







3-point vertices● Next simple Green's functions

Green's functions – Zero temperature: Vertices – Finite temperature - Summary


3-point vertices● Next simple Green's functions● Two independent momenta

● Three independent kinematic variables● Magnitude of the momenta and angle between● Here: Orthogonal





● In general various tensor structures● Lorentz● Color





● In general various tensor structures● Lorentz● Color

● Practical calculations limited to a selection



3-point vertices● Two non-vanishing in Yang-Mills theory




● Ghost-gluon vertexA

a cbcc =Dad DG

beDGcf

d e f

G Acc=tlAcc / tlD DGDGtl




● Ghost-gluon vertex

● Three-gluon vertex

Aa cbcc =D

ad DGbeDG

cf d e f


Aa A

b Ac = D

ad Dbe D

cf d e f

G A3

= tlAAA / tlDDD tl




● Ghost-gluon vertex

● Three-gluon vertex

● Corresponds to self-energies in DSEs

Aa cbcc =D

ad DGbeDG

cf d e f


Aa A

b Ac = D

ad Dbe D

cf d e f

G A3

= tlAAA / tlDDD tl



Ghost-gluon vertex in 4d

● Lattice data (164 @ beta=2.2): Constant [Cucchieri et al., unpublished] ● Complies with larger volumes for one vanishing [Cucchieri et al. JHEP 2004]

Green's functions – Zero temperature: Ghost-gluon vertex in 4d – Finite temperature - Summary








● Consistent with DSEs [Schleifenbaum et al., PRD 2005, Alkofer et al. PLB 2005]



Three-gluon vertex in 4d

● Lattice data (164 @ beta=2.2): Decreases [Cucchieri et al. unpublished]

Green's functions – Zero temperature: Three-gluon vertex in 4d – Finite temperature - Summary



● Lattice data (164 @ beta=2.2): Decreases [Cucchieri et al. unpublished]




● Lattice data (164 @ beta=2.2): Decreases [Cucchieri et al. unpublished] ● Different from DSE predictions [Alkofer et al. PLB 2005]




● Lattice data (164 @ beta=2.2): Decreases [Cucchieri et al. unpublished] ● Different from DSE predictions [Alkofer et al. PLB 2005]

● Does not affect propagators in DSE calculations



Ghost-gluon vertex in 3dGreen's functions – Zero temperature: Ghost-gluon vertex in 3d – Finite temperature - Summary


Ghost-gluon vertex in 3dGreen's functions – Zero temperature: Ghost-gluon vertex in 3d – Finite temperature - Summary



● Lattice data (203,303, and 403): Constant [Cucchieri et al. PRD 2006, unpublished]




● Lattice data (203,303, and 403): Constant [Cucchieri et al. PRD 2006, unpublished] ● as in 4d [Cucchieri et al. JHEP 2004, unpublished]




● Lattice data (203,303, and 403): Constant [Cucchieri et al. PRD 2006, unpublished] ● as in 4d [Cucchieri et al. JHEP 2004, unpublished]

● Consistent with DSE predictions [Schleifenbaum et al., PRD 2005, 2006]



Three-gluon vertex

● Lattice data (203, 303): Vanishes or reverses sign [Cucchieri et al. PRD 2006]



Three-gluon vertex

● Lattice data (203, 303): Vanishes or reverses sign [Cucchieri et al. PRD 2006]



Three-gluon vertex

● Lattice data (203, 303): Vanishes or reverses sign [Cucchieri et al. PRD 2006] ● Different from DSE predictions [Schleifenbaum et al., PRD 2006]



Three-gluon vertex

● Lattice data (203, 303): Vanishes or reverses sign [Cucchieri et al. PRD 2006] ● Different from DSE predictions [Schleifenbaum et al., PRD 2006]

● Does not affect (infrared of) propagators in DSE calculations



Overview● Confinement

● Gribov-Zwanziger scenario● Yang-Mills theory at zero temperature


● Interpolating and linear covariant gauges



Interpolating and linear covariant gauges● Gauge sector:

L=−14

Fa F ,a−ca∂ ' D

ab cb 12

∂ ' Aa ∂ ' A

a

∂ '=∂0∂

Green's functions – Zero temperature: Other gauges – Finite temperature - Summary



● interpolates between Coulomb gauge ( ) and Landau gauge ( )

L=−14

Fa F ,a−ca∂ ' D

ab cb 12

∂ ' Aa ∂ ' A

a

∂ '=∂0∂

=1=0





● is the linear covariant gauge parameter

L=−14

Fa F ,a−ca∂ ' D

ab cb 12

∂ ' Aa ∂ ' A

a

∂ '=∂0∂

=1=0





● is the linear covariant gauge parameter● Only along one direction away from Landau gauge

L=−14

Fa F ,a−ca∂ ' D

ab cb 12

∂ ' Aa ∂ ' A

a

∂ '=∂0∂

=1=0



Interpolating gauge● Lorentz invariance not manifest

Green's functions – Zero temperature: Interpolating gauge – Finite temperature - Summary



● Two (continuous) variables




● Two (continuous) variables, ● two independent tensor structures● Choose:

D tr k 0,∣k∣=PT Dk 0,∣k∣

D00k 0,∣k∣





● No smooth limit to Coulomb gauge [Fischer et al., PRD 2005]


D00k 0,∣k∣





● No smooth limit to Coulomb gauge [Fischer et al., PRD 2005]

● Gribov-Zwanziger predicts vanishing gluon proapgator and enhanced ghost proapgator at small 4-momentum [Firscher et al., PRD 2005]


D00k 0,∣k∣



Gluon propagator [Lattice 603, beta=4.2, λ=1/2: Cucchieri et al., unpublished]

● Maximum occurs away from (0,0)● Qualitatively similar to DSE predictions (in 4d) [Fischer et al., PRD 2005]



Ghost dressing function● Strongly enhanced

at (0,0)● Difference at

large moment due to tree-level behavior

● Qualitatively similar to DSE predictions (in 4d) [Fischer et al., PRD2005]

[Lattice 203, beta=4.2, λ=1/2: Cucchieri et al., unpublished]



Linear covariant gauge● Non-perturbative formulation complicated

● Gribov problem when averaging over a gauge orbit

Green's functions – Zero temperature: Linear covariant gauge – Finite temperature - Summary



● Gribov problem when averaging over a gauge orbit● Continuum condition cannot be implemented on a discrete lattice




● Gribov problem when averaging over a gauge orbit● Continuum condition cannot be implemented on a discrete lattice● Discretization effect, recovered in the continuum limit

● Reason: Field strength bounded





● Reason: Field strength bounded● Technical complications





● Reason: Field strength bounded● Technical complications● Results very preliminary



Transverse gluon propagator in Feynman gauge[Lattice2023, beta=4.2,6.0 Cucchieri et al., unpublished]

PRELIMINARY



Transverse gluon propagator in Feynman gauge

● Similar to Landau gauge at same volume

[Lattice2023, beta=4.2,6.0 Cucchieri et al., unpublished]

PRELIMINARY



Transverse gluon propagator in Feynman gauge

● Similar to Landau gauge at same volume● Longitudinal part fixed by gauge condition

● Numerically affected by discretization effects

[Lattice2023, beta=4.2,6.0 Cucchieri et al., unpublished]

PRELIMINARY





● Interpolating and linear covariant gauges● Yang-Mills theory at finite temperature



Finite-Temperature Propagators● Equilibrium Physics: Matsubara formalism

Green's functions – Zero temperature – Finite temperature: Propagators - Summary


Finite-Temperature Propagators● Equilibrium Physics: Matsubara formalism● Ghost

DG p02 , p2=

−G p02 ,p2

p2




● Gluon

DG p02 , p2=

−G p02 ,p2

p2

D p0 ,p=PT Z p0

2 , p2

p2 PL H p0

2 , p2

p2




● Gluon

● At T=0 : Z=H

DG p02 , p2=

−G p02 ,p2

p2

D p0 ,p=PT Z p0

2 , p2

p2 PL H p0

2 , p2

p2




● Gluon

● At T=0 : Z=H

● At T→ ∞: Z: chromomagnetic and H: chromoelectric

DG p02 , p2=

−G p02 ,p2

p2

D p0 ,p=PT Z p0

2 , p2

p2 PL H p0

2 , p2

p2




● Gluon

● At T=0 : Z=H

● At T→ ∞: Z: chromomagnetic and H: chromoelectric

● p0 discrete, p

0=0: soft, p

0<>0: hard

DG p02 , p2=

−G p02 ,p2

p2DG p02 , p2=

−G p02 ,p2

p2

D p0 ,p=PT Z p0

2 , p2

p2 PL H p0

2 , p2

p2



Gluon at finite temperature [Lattice: Cucchieri et al., unpublished]

Green's functions – Zero temperature – Finite temperature: Gluon - Summary






● Different behavior of transverse and longitudinal gluon




● Different behavior of transverse and longitudinal gluon



Gluon at finite temperature [Lattice: Cucchieri et al., unpublished DSE: Maas et al., EPJC 2005]

5

● Different behavior of transverse and longitudinal gluon● Qualitative similar to DSE results (Z suppressed and H massive)● Solves IR problem of HTL – invalidates HTL in the infrared



Ghost at finite temperature [Lattice: Cucchieri et al., unpublished]

Green's functions – Zero temperature – Finite temperature: Ghost - Summary









● No significant temperature dependence at these volumes



Ghost at finite temperature [Lattice: Cucchieri et al., unpublished DSE: Maas et al., EPJC 2005]

● No significant temperature dependence at these volumes



Ghost at finite temperature [Lattice: Cucchieri et al., unpublished DSE: Maas et al., EPJC 2005]

● No significant temperature dependence at these volumes● Incosistent with earlier expectations [Grüter et al., EPJC 2005]

● Recent DSE analysis seem to agree qualitatively [Cucchieri et al., unpublished]



Finite temperature - interpretation?● Transverse gluon evolves smoothly with temperature● Stronger infrared suppressed with increasing temperature

Green's functions – Zero temperature – Finite temperature: Interpretation - Summary



● Longitudinal gluon seems to feel the phase transition● Screened at finite temperature● Screening mass sensitive to phase transition?




● Longitudinal gluon seems to feel the phase transition● Screened at finite temperature● Screening mass sensitive to phase transition?

● Ghost inert on these volumes – always diverging



Finite temperature - interpretation?● DSEs predicts change of the infrared behavior from zero to non-zero temperature lCucchieri et al., unpublished]




● Far infrared of transverse gluon and ghost as in the dimensionally reduced theory

● Longitudinal gluon always screened





● Longitudinal gluon always screened● Only observable for p<<T





● Longitudinal gluon always screened● Only observable for p<<T

● Supported by RG results [Braun et al., 2005, JHEP 2006]



Finite temperature - interpretation?● Gribov-Zwanziger-like mechanism in the transverse sector● Unaffected by phase transition● Results of differing scales: Infrared is always at scales much smaller than the temperature




● Longitudinal gluon different● BRST-confined● Sensitive to hard modes




● Longitudinal gluon different● BRST-confined● Sensitive to hard modes

● Phase transition not deconfining



Summary● Green's functions encode confinement



Summary● Green's functions encode confinement● Determination is complicated

● Finite volume effects on infrared scales




● Finite volume effects on infrared scales● Results agree qualitatively with





● Results from DSEs and RGs





● Results from DSEs and RGs● Gribov-Zwanziger and Kugo-Ojima scenario





● Results from DSEs and RGs● Gribov-Zwanziger and Kugo-Ojima scenario● Gauge dependence and topology become interesting






● Results hint to a new picture of the phase transition






● Results hint to a new picture of the phase transition● Much work remains to be done...






● Results hint to a new picture of the phase transition● Much work remains to be done...and a combination of

methods is essential


Documents

Green's functions in Yang-Mills theoryaxm/oberwoelz06.pdf · Zwanziger and Kugo-Ojima scenarios, DSEs and RGs Green's functions – Zero temperature: Propagators – Finite temperature