Landau gauge gluon and ghost propagators from the lattice point of - - PowerPoint PPT Presentation

Landau gauge gluon and ghost propagators from the lattice point of view

• M. M¨

uller-Preussker

Recent and present collaborators in Landau gauge lattice QCD:

• R. Aouane1

I.L. Bogolubsky2 V.G. Bornyakov3 E.-M. Ilgenfritz1,4

• C. Litwinski1
• C. Menz1,5

V.K. Mitrjushkin2

• A. Sternbeck4

1 HU Berlin 2 JINR Dubna 3 IHEP Protvino 4 U Regensburg 5 PIK Potsdam

Support by: JSCC Moscow

Bogoliubov Readings @ BLTP.JINR.RU, Dubna, September 22 –25, 2010

Outline of the talk

• 1. Introduction, motivation: the infrared QCD debate
• 2. How to compute Landau gauge gluon and ghost propagators
• n the lattice
• 3. Results for gluon, ghost propagators and the running coupling

in lattice quenched and full QCD

• 4. Gribov copies, finite-volume effects, multiplicative

renormalization, continuum limit

• 5. Conclusion and outlook

• 1. Introduction, motivation: the infrared debate

Landau gauge gluon, ghost, quark propagators and vertex functions: ⇒ allow to fix phenomenologically useful parameters: effective (dynamical) gluon mass mg, ΛQCD, ψψ, A2 (?), ...; ⇒ can be directly used as input for hadron phenomenology: Bethe-Salpeter eqs. for mesons, Faddeev eqs. for baryons,

• cf. Alkofer, Eichmann, Krassnigg, Nicmorus, Chin. Phys. C34 (2010),

arXiv:0912.3105 [hep-ph];

⇒ their infrared behaviour is related to confinement criteria: Gribov-Zwanziger, Kugo-Ojima, violation of positivity,...; ⇒ for T > 0 allow for determining screening length and other characteristica at Tc. = ⇒ Intensive non-perturbative investigations in the continuum and

• n the lattice over many years.

= ⇒ Infrared (IR) limit of special interest.

Landau gauge Green’s functions in the continuum determined from (truncated) Dyson-Schwinger (DS) and Wetterich funct. RG (FRG) eqs. taking into account Slavnov-Taylor identities (STI)

[Alkofer, Aguilar, Boucaud, Dudal, Fischer, Pawlowski, von Smekal, Zwanziger,.. (’97 - ’09)]

−1

=

−1

• 1

2

• 1

2

• 1

6

• 1

2 +

−1

=

−1

• ⇒

Dab

µν = δab

δµν − qµqν

q2

• Z(q2)

q2

⇒ Gab = δab J(q2)

q2

Running coupling from ghost-ghost-gluon vertex in a MOM scheme αs(q2) ≡ g2(µ) 4π Z(q2, µ2) · [J(q2, µ2)]2

• Renorm. group invariant quantity.

Infrared “scaling” solution of DS and FRG eqs.

[Alkofer, Fischer, Lerche, Maas, Pawlowski, von Smekal, Zwanziger,... (’97 - ’09)]

Z(q2) ∝ (q2) κD , J(q2) ∝ (q2)−κG for q2 → 0 with κD = 2 κG + (4 − d)/2 = ⇒ κD = 2 κG, κG ≃ 0.59 for d = 4 is claimed

• to be consistent with BRST quantization,
• to hold without any truncation of the tower of DS or FRG eqs.,
• to be independent of the number of colors Nc,
• to look qualitatively the same in any dimension

d = 2, 3, 4. Running coupling: αs(q2) → const. for q2 → 0 i.e. infrared fixed point as in analytic perturbation theory

[D.V. Shirkov, I.L. Solovtsov (’97 - ’02)].

Alternative “decoupling” IR solution

[Boucaud et al. (’06 -’08), Aguilar et al. (’07-’08), Dudal et al. (’05-’08)]

κD = 1 , κG = 0 i.e. D(q2) = Z(q2)/q2 → const. , J(q2) → const. such that αs(q2) = g2 4π Z(q2) · [J(q2)]2 → 0 for q2 → 0 . Existence has been confirmed.

[Fischer, Maas, Pawlowski, Annals Phys. ’09, arXiv:0810-1987 [hep-ph]]

No debate any more on who is right, but about criteria what is the physically correct solution (BRST). Claim: J(0) might be chosen as an IR boundary condition. Expect: close relation to the notorious Gribov problem. Question: Relevance for phenomenology ?

IR “scaling” solution for Z, J at a first view in agreement with confinement scenarios:

• Kugo-Ojima confinement criterion

[Ojima, Kugo (’78 - ’79)]:

absence of colored physical states ⇐ ⇒ ghost (gluon) propagator more (less) singular than simple pole for q2 → 0.

• Gribov-Zwanziger confinement scenario

[Gribov (’78), Zwanziger (’89 - ...)]:

gauge fields restricted the Gribov region Ω =

n

Aµ(x) : ∂µAµ = 0, MFP ≡ −∂D(A) ≥ 0

• are accumulated at the Gribov horizon ∂Ω :

non-trivial eigenvalues of MFP : λ0 → 0. = ⇒ Ghost: J(q2) → ∞ Gluon: D(q2) → 0 for q2 → 0. There are attempts to modify scenarios such, that IR “decoupling” solution can be accomodated, too. [Dudal et al. (’08 - ’09), Kondo (’09)].

The Gribov problem:

• Existence of many gauge copies inside

Ω.

• What are the right copies?

Restriction inside Ω to fundamental modular region (FMR) required Λ =

n

Aµ(x) : F(Ag) > F(A) for all g = 1

• ,

i.e. to global extremum of the Landau gauge functional F(Ag) ? Answer in the limit of infinite volume

[Zwanziger (’04)]:

Non-perturbative quantization requires only restriction to Ω, i.e. δΩ(∂µAµ) det(−∂µDab

µ )e−SY M [A] .

Expectation values taken on Ω or Λ should be equal in the thermodynamic limit.

• What happens on a (finite) torus?
• How Gribov copies influence finite-size effects?

Questions to Yang-Mills theory on the lattice:

• What kind of infrared DS and FRG solutions are supported ?
• What is the influence of Gribov copies on gluon and ghost propagators ?
• Finite-volume effects ?
• Continuum limit, scaling, non-perturbative multiplicative renormalization at

finite volume ? Lattice investigations of gluon and ghost propagators over many years in Adelaide: Bonnet, Leinweber, von Smekal, Williams, et al.; Berlin: Burgio, Ilgenfritz, M.-P., Sternbeck, et al.; Dubna/Protvino: Bakeev, Bogolubsky, Bornyakov, Mitrjushkin; San Carlos: Cucchieri, Maas, Mendes; Paris: Boucaud, Leroy, Pene, et al.; Coimbra: Oliveira, Silva; T¨ ubingen: Bloch, Langfeld, Reinhardt, Watson et al.; Utsunomiya: Furui, Nakajima.

• 2. How to compute Landau gauge gluon and ghost propagators
• n the lattice

A few technicalities:

i) Generate lattice discretized gauge fields

U = {Ux,µ ≡ eiag0Aµ(x) ∈ SU(Nc)} by MC simulation from path integral ZLatt =

Z Y

x,µ

[dUx,µ] (det Q(κ, U))Nf exp(−SG(U)) – standard Wilson plaquette action SG(U) = β

X

x

X

µ<ν

„

1 − 1 Nc Re tr Ux,µν

«

, Ux,µν ≡ Ux,µUx+ˆ

µ,νU† x+ˆ ν,µU† x,ν,

β ≡ 2Nc/g2 – (clover-improved) Dirac-Wilson fermion operator Q(κ, U): Nf = 0 – pure gauge case, Nf = 2 – full QCD with equal bare quark masses ma = 1/2κ − 1/2κc, a(β) – lattice spacing.

ii) ZLatt is simulated with (Hybrid) MC method without gauge fixing.

iii) Gauge fix each lattice field U: Ug

xµ = gx·Uxµ·g† x+ˆ µ

standard orbits: {gx} periodic on the lattice; extended orbits: {gx} periodic up to global Z(N) transformations; Landau gauge: linear definition Ax+ˆ

µ/2,µ = 1 2iag0

“

Uxµ − U†

xµ

”˛ ˛

traceless

(∂A)x =

4

X

µ=1

“

Ax+ˆ

µ/2;µ − Ax−ˆ µ/2;µ

”

= 0 equivalent to minimizing the gauge functional FU (g) =

X

x,µ

„

1 − 1 Nc Re tr Ug

xµ

«

= Min. . For uniqueness (FMR) one requires to find the global minimum

[Parrinello, Jona-Lasinio (’90), Zwanziger (’90)].

Well understood in compact U(1) theory in order to get e.g. massless photon propagator

[Bogolubsky, Bornyakov, Mitrjushkin, M.-P., Peters, Zverev (’93 - ’99)].

Optimized minimization in (our) practice: simulated annealing (SA) +

• verrelaxation (OR)

Gribov problem: global minimum of FU (g) very hard or impossible to find. ”Best copy strategy”: repeated initial random gauges = ⇒ best copies (bc) from subsequent SA + OR minimizations, = ⇒ compared with first (random) copies (fc)).

iv) Compute propagators

• Gluon propagator:

Dab

µν(q) =

D e

Aa

µ(k) e

Ab

ν(−k)

E

≡ δab

„

δµν − qµ qν q2

«

D(q2) for lattice momenta qµ(kµ) = 2 a sin

„

πkµ Lµ

«

, kµ ∈ ` −Lµ/2, Lµ/2˜ with certain cuts in order to suppress artifacts of lattice discretization.

• Ghost propagator:

Gab(q) = 1 V (4)

X

x,y

D

e−2πi k·(x−y)[M−1]ab

xy

E

≡ δabG(q) . M ∼ ∂µDµ

• Landau gauge Faddeev-Popov operator

Mab

xy(U) =

X

µ

Aab

x,µ(U) δx,y − Bab x,µ(U) δx+ˆ µ,y − Cab x,µ(U) δx−ˆ µ,y

Aab

x,µ

=

Re tr h

{T a, T b}(Ux,µ + Ux−ˆ

µ,µ)

i

, Bab

x,µ

= 2 · Re tr

h

T bT a Ux,µ

i

, Cab

x,µ

= 2 · Re tr

h

T aT b Ux−ˆ

µ,µ

i

, tr[T aT b] = δab/2 . M−1 from solving Mab

xyφb(y) = ψa c (x) ≡ δac exp(2πi k·x)

with (preconditioned) conjugate gradient algorithm.

• 3. Results for gluon, ghost propagators and the running coupling

and the running coupling in lattice quenched and full QCD

• Pure gauge case Nf = 0:

β = 5.7, 5.8, 6.0, 6.2; 124, . . . , 564, aLmax ≃ 9.5fm; β = 5.7; 644, . . . , 964, aLmax ≃ 16.3fm.

• Full QCD case Nf = 2:

thanks: configurations provided by QCDSF - collaboration, β = 5.29, 5.25; mass parameter κ = 0.135, ..., 0.13575; 163 × 32, 243 × 48.

• Results for propagators / dressing functions and αs

Gluon Z(q2) ≡ q2D(q2), Ghost J(q2) ≡ q2G(q2) as well as ghost-ghost-gluon vertex and Kugo-Ojima parameter.

• Propagators at

T > 0 studied, too = ⇒ V.K. Mitrjushkin’s talk.

First results: Gluon propagator and ghost dressing function

quenched QCD (Nf = 0), renorm. pt.: q = µ = 4GeV, first OR copies

[Sternbeck, Ilgenfritz, M.-P., Schiller, PRD 72 (2006), Proc. IRQCD ’06]

D(q2) J(q2)

0.0 5.0 10.0 0.01 0.1 1 10 100 D(q2) q2 [GeV2]

β = 5.7 484 564 β = 5.8 244 324 β = 6.0 324 484 β = 6.2 244 1.0 1.5 2.0 2.5 3.0 0.1 1 10 100 J(q2) q2 [GeV2]

q2

i

β = 5.8 244 324 β = 6.0 164 244 324 484 β = 6.2 164 244

= ⇒ Gluon prop. D(q2) shows plateau and not D(q2) → 0 for q2 → 0 , = ⇒ corresponds to an effective gluon mass behaviour. = ⇒ Ghost dress. fct. J(q2) power-like, expon. too small for scaling solution.

Gluon and ghost dressing functions

full QCD (Nf = 2) versus quenched QCD (Nf = 0), renorm. point: q = µ = 4GeV

[Ilgenfritz, M.-P., Schiller, Sternbeck (A. DiGiacomo 70, ’06)]

Z(q2) J(q2)

0.5 1.0 1.5 2.0 2.5 0.1 1 10 100 Z(q2) q2 [GeV2]

(β, ma) = (5.29, 0.0246) (5.29, 0.0138) (5.25, 0.0135) quenched

1.0 1.5 2.0 2.5 3.0 0.1 1 10 100 J(q2) q2 [GeV2]

(β, ma) = (5.29, 0.0246) (5.29, 0.0138) (5.25, 0.0135) quenched

= ⇒ Influence of virtual quark loops in Z(q2) clearly visible. = ⇒ No quenching effect in J(q2), as ghosts do not directly couple to quarks.

Gluon propagator and ghost dressing function on huge volumes

quenched QCD, first but long run SA + OR copies, unrenormalized

[Bogolubsky, Ilgenfritz, M.-P., Sternbeck, PLB 676 (2009)]

0.001 0.01 0.1 1 10 100 q2 [GeV2]

β = 5.7 644 (14 conf.) 724 (20 conf.) 804 (25 conf.) 884 (68 conf.) 964 (67 conf.)

2 4 6 8 10 12 D(q2) [GeV−2] 4 3 2 1 0.001 0.01 0.1 1 10 100 J(q2) q2 [GeV2]

β = 5.7 644 (14 conf.) 804 (11 conf.) 804 (5 conf.)

= ⇒ Both D(q2) and J(q2) seem to tend to const.. = ⇒ Clear indication for “decoupling” solution. = ⇒ Here coarse lattices used. Question: continuum limit ?

Result for the running coupling on large volumes

quenched QCD, first but long run SA + OR copies, coarse lattices

[Bogolubsky, Ilgenfritz, M.-P., Sternbeck, PLB 676 (2009), ]

0.0 0.5 1.0 1.5 0.001 0.01 0.1 1 10 100 αs(q2) q2 [GeV2]

β = 5.7 644 804

• Running coupling not monotonous,

αs → 0 for q → 0, = ⇒ “decoupling behaviour”.

• Agrees with other lattice studies, in particular for the three-gluon vertex.
• At large q2 allows to fix ΛMS.

• 4. Gribov copies, finite-volume effects, multiplicative

renormalization, continuum limit

Systematic effects somewhat easier to study in pure SU(2) gauge theory.

[Bakeev, Bogolubsky, Bornyakov, Burgio, Ilgenfritz, Mitrjushkin, M.-P. (’04 - ’09)]

Improved gauge fixing = ⇒ getting ‘close’ to the FMR:

• Simulated annealing (SA):

Find g’s randomly with statistical weight: W ∝ exp

„

− FU (g) TSA

«

. Let “temperature” TSA slowly decrease. Infinitely slow cooling ends at the global extremum. In practice SA clearly wins for large lattice sizes. (Over)relaxation (OR) has to be applied subsequently in order to reach (∂A)x =

4

X

µ=1

“

Ax+ˆ

µ/2;µ − Ax−ˆ µ/2;µ

”

< ǫ for all x .

• Z(Nc) flips:

Gauge functional FU (g) minimized by enlarging the gauge orbit with respect to Z(Nc) non-periodic gauge transformations: g(x + Lˆ ν) = zνg(x) , zν ∈ Z(Nc) . For SU(Nc) the N4

c different sectors of Polyakov loop averages are combined.

In order to view Gribov copy effects we compare:

i) standard method: first (i.e. random) copy overrelaxation = “fc OR”, ii) first copy simulated annealing (incl. finalizing overrelaxation) = “fc SA”, iii) best copy Z(2) flips + simulated annealing (+OR) = “bc FSA”, compare typically 5 copies in each of the 16 Polyakov loop sectors (= 80 copies).

Gluon propagator and ghost dressing fct.: fc SA versus bc FSA

[Bornyakov, Mitrjushkin, M.-P., PRD 79 (2009), arXiv:0812.2761 [hep-lat]]

• 0.05

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 |p| Gev 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 Dgluon(p); Gev-2

bc fc

beta = 2.2; Lattice = 40

4

0.1 0.15 0.2 0.25 0.3 |p| Gev 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 p2 Gghost(p)

bc fc

beta = 2.2; Lattice = 40

4

gluon propagator ghost dress. fct. = ⇒ Gribov copies important for both gluon and ghost! = ⇒ The closer to the global minimum (FMR), the weaker the ‘singularity’

• f the ghost dressing fct., the lower the IR values of the gluon propagator.

= ⇒ D(q2 → 0) = 0 ? Together with a non-singular ghost dress. fct. this would completely contradict DS and FRG eqs. and (modified) Zwanziger approach.

Gluon propagator β = 2.2: bc FSA versus fc OR

[Bornyakov, Mitrjushkin, M.-P., PRD 79 (2009), arXiv:0812.2761 [hep-lat]]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 |p|; Gev 1 2 3 4 5 6 7 8 9 D(p); Gev-2

8

4

12 16

4

24

4

32

4

40

4

4 5 6 7 8 9 10 11 12 0.02 0.04 0.06 0.08 0.1 GeV-2 1/L, GeV D(0), FSA D(0), OR

bc FSA D(q2) for diff. L4 D(0) versus 1/L = ⇒ Finite-size effects weaker for FSA, i.e. when approaching the FMR Λ. = ⇒ Extrapolation D(0) = 0 for V → ∞. = ⇒ Again support for “decoupling solution”. But still we have strong coupling, i.e. coarse lattices.

Gribov copy sensitivity for the gluon propagator bc FSA versus fc SA ∆(p) = Dfc(p) − Dbc(p) Dbc(p)

[Bornyakov, Mitrjushkin, M.-P., PRD 81 (2010), arXiv:0912.4475 [hep-lat]].

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

|p|; Gev

• 0.05

0.00 0.05 0.10 0.15 0.20 0.25 0.30

∆(p) β=2.20; 14

4

β=2.30; 18

4

β=2.40; 26

4

β=2.50; 36

4

β=2.55; 42

4

aL=3 fm

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

|p|; Gev

• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

∆(p) β=2.20; 24

4

β=2.30; 30

4

β=2.40; 42

4

aL=5 fm

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

|p|; Gev

• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08 0.10

∆(p) β=2.30; 42

4

aL=7 fm

aL = 3fm aL = 5fm aL = 7fm Gribov copy effect: ⇒ important at low momenta, ⇒ almost independent of lattice spacing, ⇒ weakens with increasing physical volume [Zwanziger (’04)].

Finite-volume and cont. limit results for renormalized gluon dressing fct. bc FSA

[Bornyakov, Mitrjushkin, M.-P., PRD 81 (2010), arXiv:0912.4475 [hep-lat]].

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

|p|; Gev

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

p2Dren(p) β=2.30; 18

4

β=2.40; 26

4

β=2.50; 36

4

β=2.55; 42

4

β=2.55; 42

4; fit

aL=3 fm aL=3 fm aL=3 fm aL=3 fm

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

|p|; Gev

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

p2Dren(p) β=2.30; 30

4

β=2.40; 42

4

β=2.40; 42

4; fit

aL=5 fm

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

|p|; Gev

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

p2Dren(p) β=2.30; 44

4

β=2.30; 44

4; fit

aL=7.3 fm

aL = 3fm aL = 5fm aL = 7fm = ⇒ For momenta p > 0.4MeV, β ≥ 2.40 the renormalized data fall on top of each other, = ⇒ contin. result reached, good fits available − → IR effect. gluon mass, = ⇒ Curves for different linear sizes 3, 5, 7 fm nicely agree.

Gluon and ghost propagators with alternative definition(s) of Aµ(x)

Use logarithmic definition for the lattice gluon field A(log)

x+ ˆ µ 2 ,µ

= 1 i a g0 log ` Ux,µ

´

, minimize lattice gauge functional directly translated from continuum F (log)

U

[g] =

X

x,µ

1 Nc tr

"

gA(log) x+ ˆ µ 2 ,µ gA(log) x+ ˆ µ 2 ,µ

#

. Faddeev-Popov determinant derived accordingly. Numerical treatment differs: accelerated multigrid algorithm + preconditioning. = ⇒ Compare results for linear and logarithmic definition. = ⇒ Check independence of the running coupling. = ⇒ Compare with stochastic perturbation theory.

Related work: [Petrarca et al., ’99; Cuchieri, Karsch, ’99; Bogolubsky, Mitrjushkin, ’02;...]

2 4 6 8 10 12 14 16 0.7 0.8 0.9 1.0 1.1 1.2 momentum a^2q^2 D_lin(a^2q^2)/D_log(a^2q^2) 12^4, beta = 6.0 16^4, beta = 6.0 2 4 6 8 10 12 14 16 0.98 1.02 1.06 1.10 1.14 momentum a^2q^2 G_lin(a^2q^2)/G_log(a^2q^2) 12^4, beta = 6.0 16^4, beta = 6.0

Linear definition results vs. logarithmic definition, β = 6.0 Gluon propagator ratio Ghost propagator ratio

48^4, beta = 6.0, lin 32^4, beta = 5.8, lin 32^4, beta = 6.0, lin 24^4, beta = 6.2, lin 32^4, beta = 6.0, log 32^4, beta = 6.4, log 24^4, beta = 6.0, log 16^4, beta = 6.0, log 0.01 0.1 1 10 100 1000 0.2 0.4 0.6 0.8 1.0 1.2 1.4 momentum q^2 [GeV^2] alpha(q^2)

Running coupling ⇒ multipl. renormalizability confirmed. ⇒ αs(q2) in given MOM scheme

• approx. renorm. independent.

[Ilgenfritz, Menz, M.-P., to be published]

Monte Carlo vs. numerical stochastic perturbation theory (NSPT)

NSPT with Langevin technique allows for higher loop perturbation theory. Logarithmic definition for Aµ is natural.

See [di Renzo, Ilgenfritz, Perlt, Schiller, Torrero, ’09 - ’10]

Compare arbitrary Polyakov loop sectors (x, x, x, x) with real sector (0, 0, 0, 0). Here: 164, large β = 9.0 for both approaches.

MC−simulation, (x,x,x,x) MC−simulation, (0,0,0,0) NSPT, 2−loop NSPT, 1−loop NSPT, 0−loop 2 4 6 8 10 12 14 16 1.0 1.2 1.4 1.6 1.8 2.0 2.2 a2q2 ZGl(a

2q 2)

MC−simulation, (x,x,x,x) MC−simulation, (0,0,0,0) NSPT, 3−loop NSPT, 2−loop NSPT, 1−loop 2 4 6 8 10 12 14 16 1.00 1.05 1.10 1.15 1.20 1.25 a2q2 ZGh(a

2q 2)

Gluon dressing fct. Ghost dressing fct. ⇒ Nice consistency, approach to full result can be checked !

• 5. Conclusion and outlook
• Lattice results support “decoupling solution” as long as we assume approach

FU (g) → Global Min.

• Gribov effects turn out to be important. For pure LGT simulated annealing

+ Z(N) flips (“bc FSA”) provides a solution with weak finite-size effects.

• Continuum limit can be consistently reached within the non-perturbatively

and phenomenologically important range around 1GeV.

• Analogous results, when available for full QCD will allow to tune DS and FRG

truncations and provide immediate input into Bethe-Salpeter or Faddeev Eqs.

• Debate “scaling” versus “decoupling” solution continues.

On the lattice it could mean to give up the condition FU (g) → Global Min.

[Maas et al. (’09)]

• Continuum alternative ?

= ⇒ A.A. Slavnov – Y.-M. theory without Gribov ambiguity

Thank you for your attention.

Gluon dressing function and propagator from first OR copies

quenched QCD (Nf = 0), renorm. point: q = µ = 4GeV

[Sternbeck, Ilgenfritz, M.-P., Schiller, PRD 72 (2006), Proc. IRQCD ’06]

Z(q2) D(q2) = Z(q2)/q2

0.0 0.5 1.0 1.5 2.0 2.5 0.01 0.1 1 10 100 Z(q2) q2 [GeV2]

β=5.7 484 564 β=5.8 244 324 β=6.0 324 484 323×64 β=6.2 244

0.0 5.0 10.0 0.01 0.1 1 10 100 D(q2) q2 [GeV2]

β = 5.7 484 564 β = 5.8 244 324 β = 6.0 324 484 β = 6.2 244

= ⇒ D(q2) = Z(q2)/q2 shows plateau and not D(q2) → 0 for q2 → 0 , = ⇒ corresponds to an effective gluon mass.

Ghost dressing function from first OR copies

quenched QCD (Nf = 0), renorm. point: q = µ = 4GeV

[Sternbeck, Ilgenfritz, M.-P., Schiller, PRD 72 (2006), Proc. IRQCD ’06]

J(q2)

1.0 1.5 2.0 2.5 3.0 0.1 1 10 100 J(q2) q2 [GeV2]

q2

i

β = 5.8 244 324 β = 6.0 164 244 324 484 β = 6.2 164 244

= ⇒ looks still singular but with too small exponent in comparison with “scaling” solution; = ⇒ no serious finite-volume effects (??).

The running coupling from ghost-ghost-gluon vertex αs(q2) = g2

4π Z(q2) (J(q2))2

assuming ˜ Z1 = 1

[perturbation theory: Taylor (’71) / LGT SU(2): Cucchieri et al. (’04)] The SU(3) vertex renormalization function ˜ Z1, gluon momentum k = 0. Nf = 0

Ilgenfritz et al. ’05

Nf = 2

Sternbeck thesis ’06

0.5 1.0 1.5 2.0 0.1 1 10 100

• Z−1

1 (q2,β)

q2 [GeV2]

β = 5.8 324 β = 6.0 324 484

0.5 1.0 1.5 2.0 0.1 1 10 100 q2 [GeV2]

D-2 D-3 D-4

Renormalized gluon propagators at T > 0

Temperature dependence from T ≡ 1/aLτ , Lτ ≪ Lσ. Separate time and space components, Matsubara frequency ω ∼ q4. Transverse propagator: DT ∼

3

X

i=1

Aa

i (q) Aa i (−q) −

q2

4

• q2 Aa

4(q) Aa 4(−q)

Longitudinal propagator: DL ∼ (1 + q2

4

• q2 ) Aa

4(q) Aa 4(−q)

T > Tc = ⇒ spontaneous Z(3) symmetry breaking. Polyakov loop average L takes values in 3 sectors. Real sector = “physical” sector.

[See Cucchieri, Karsch, ’00; Bogolubsky, Mitrjushkin, ’02; Fischer, Maas, Mueller, ’10; ....]

Here: quenched QCD, gauge fixing – first copies with SA + OR, 483 × Lτ , Lτ = 4, 6, . . . , 18 varies, spacing a = a(β = 6.337) fixed, Tc ← → Lτ = 12

[Aouane, Ilgenfritz, M.-P., Sternbeck, preliminary result]

Renormalized gluon propagators at T > 0, q4 = 0

0.01 0.1 1 10 100 1 2 3 4 5 6 DL[GeV −2] q [GeV] T=0.653 Tc T=0.743 Tc T=0.849 Tc T=0.99 Tc T=1.188 Tc T=1.486 Tc T= 1.981 Tc T=2.972 Tc 0.01 0.1 1 10 1 2 3 4 5 6 DT [GeV −2] q [GeV] T=0.653 Tc T=0.743 Tc T=0.849 Tc T=0.99 Tc T=1.188 Tc T=1.486 Tc T= 1.981 Tc T=2.972 Tc

longitudinal gluon propagator transversal gluon propagator = ⇒ Gluon propagators depend on T at low momenta. = ⇒ Not shown: Ghost propagator T-independent. = ⇒ Longitudinal gluon propagator most sensitive.

Renormalized gluon propagators vs. T at lowest momenta

0.1 1 10 100 1 2 DL[GeV −2] T/Tc (0,0,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0) 0.1 1 10 1 2 DT [GeV −2] T/Tc (0,0,0,0) (1,1,1,0) (2,1,1,0) (2,2,1,0)

longitudinal gluon propagator transversal gluon propagator = ⇒ Longitudinal propagator at low momenta can serve as ”order parameter”. = ⇒

• cf. talk by V. Mitrjushkin.