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. Aouane 1 1 HU Berlin Support by: I.L. Bogolubsky 2 2 JINR Dubna V.G. Bornyakov 3 3 IHEP Protvino E.-M. Ilgenfritz 1 , 4 4 U Regensburg JSCC Moscow C. Litwinski 1 5 PIK Potsdam C. Menz 1 , 5 V.K. Mitrjushkin 2 A. Sternbeck 4 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 on 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 m g , Λ QCD , � ψψ � , � A 2 � (?), ...; ⇒ 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 T c . = ⇒ Intensive non-perturbative investigations in the continuum and on 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 Z ( q 2) - 1 - 1 µν = δ ab � � δ µν − qµqν D ab ⇒ 2 6 q 2 q 2 - 1 + 2 − 1 − 1 G ab = δ ab J ( q 2) = - ⇒ q 2 Running coupling from ghost-ghost-gluon vertex in a MOM scheme α s ( q 2 ) ≡ g 2 ( µ ) Z ( q 2 , µ 2 ) · [ J ( q 2 , µ 2 )] 2 4 π Renorm. group invariant quantity.

Infrared “scaling” solution of DS and FRG eqs. [Alkofer, Fischer, Lerche, Maas, Pawlowski, von Smekal, Zwanziger,... (’97 - ’09)] q 2 → 0 Z ( q 2 ) ∝ ( q 2 ) κD , J ( q 2 ) ∝ ( q 2 ) − κG for with κ D = 2 κ G + (4 − d ) / 2 = ⇒ κ D = 2 κ G , κ G ≃ 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 N c , - to look qualitatively the same in any dimension d = 2 , 3 , 4. Running coupling: q 2 → 0 α s ( q 2 ) → const. for 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)] κ D = 1 , κ G = 0 i.e. D ( q 2 ) = Z ( q 2 ) /q 2 → const . , J ( q 2 ) → const . such that α s ( q 2 ) = g 2 q 2 → 0 . 4 π Z ( q 2 ) · [ J ( q 2 )] 2 → 0 for 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: [Ojima, Kugo (’78 - ’79)] : • Kugo-Ojima confinement criterion absence of colored physical states ghost (gluon) propagator more ⇐ ⇒ (less) singular than simple pole for q 2 → 0. • Gribov-Zwanziger confinement scenario [Gribov (’78), Zwanziger (’89 - ...)] : gauge fields restricted the Gribov region n o Ω = A µ ( x ) : ∂ µ A µ = 0 , M FP ≡ − ∂D ( A ) ≥ 0 are accumulated at the Gribov horizon ∂ Ω : non-trivial eigenvalues of M FP : λ 0 → 0. J ( q 2 ) → ∞ Ghost: q 2 → 0. = ⇒ for D ( q 2 ) → 0 Gluon: 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 n o A µ ( x ) : F ( A g ) > F ( A ) Λ = for all g � = 1 , F ( A g ) ? i.e. to global extremum of the Landau gauge functional [Zwanziger (’04)] : Answer in the limit of infinite volume Non-perturbative quantization requires only restriction to Ω, µ ) e − SY M [ A ] . δ Ω ( ∂ µ A µ ) det( − ∂ µ D ab i . e . 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 on the lattice A few technicalities: U = { U x,µ ≡ e iag 0 Aµ ( x ) ∈ SU ( N c ) } i) Generate lattice discretized gauge fields by MC simulation from path integral Z Y [ dU x,µ ] (det Q ( κ, U )) Nf Z Latt = exp( − S G ( U )) x,µ – standard Wilson plaquette action „ « X X 1 − 1 S G ( U ) = β N c Re tr U x,µν , x µ<ν µ,ν U † ν,µ U † β ≡ 2 N c /g 2 U x,µν ≡ U x,µ U x +ˆ x,ν , 0 x +ˆ – (clover-improved) Dirac-Wilson fermion operator Q ( κ, U ): N f = 0 – pure gauge case, N f = 2 – full QCD with equal bare quark masses ma = 1 / 2 κ − 1 / 2 κ c , a ( β ) – lattice spacing.

ii) Z Latt is simulated with (Hybrid) MC method without gauge fixing. iii) Gauge fix each lattice field U : U g xµ = g x · U xµ · g † x +ˆ µ standard orbits: { g x } periodic on the lattice; extended orbits: { g x } periodic up to global Z ( N ) transformations; “ ”˛ U xµ − U † 1 ˛ Landau gauge: linear definition A x +ˆ µ/ 2 ,µ = xµ 2 iag 0 traceless 4 “ ” X ( ∂ A ) x = A x +ˆ µ/ 2; µ − A x − ˆ = 0 µ/ 2; µ µ =1 equivalent to minimizing the gauge functional „ « X 1 − 1 N c Re tr U g F U ( g ) = = Min . . xµ x,µ 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) + overrelaxation (OR) Gribov problem: global minimum of F U ( 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: „ « D E δ µν − q µ q ν D ab A a A b ≡ δ ab D ( q 2 ) e µ ( k ) e µν ( q ) = ν ( − k ) q 2 for lattice momenta „ « k µ ∈ ` − L µ / 2 , L µ / 2 ˜ q µ ( k µ ) = 2 πk µ a sin , L µ with certain cuts in order to suppress artifacts of lattice discretization.

- Ghost propagator: D E X 1 G ab ( q ) = e − 2 πi k · ( x − y ) [ M − 1 ] ab ≡ δ ab G ( q ) . xy V (4) x,y M ∼ ∂ µ D µ - Landau gauge Faddeev-Popov operator X M ab A ab x,µ ( U ) δ x,y − B ab µ,y − C ab xy ( U ) = x,µ ( U ) δ x +ˆ x,µ ( U ) δ x − ˆ µ,y µ h i A ab { T a , T b } ( U x,µ + U x − ˆ = Re tr µ,µ ) , x,µ h i T b T a U x,µ B ab = 2 · Re tr , x,µ h i T a T b U x − ˆ C ab tr[ T a T b ] = δ ab / 2 . = 2 · Re tr , x,µ µ,µ M − 1 from solving c ( x ) ≡ δ ac exp(2 πi k · x ) M ab xy φ b ( y ) = ψ a 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 N f = 0: 12 4 , . . . , 56 4 , β = 5 . 7 , 5 . 8 , 6 . 0 , 6 . 2; aL max ≃ 9 . 5fm; 64 4 , . . . , 96 4 , β = 5 . 7; aL max ≃ 16 . 3fm. • Full QCD case N f = 2: thanks: configurations provided by QCDSF - collaboration, β = 5 . 29 , 5 . 25; mass parameter κ = 0 . 135 , ..., 0 . 13575; 16 3 × 32 , 24 3 × 48. • Results for propagators / dressing functions and α s Z ( q 2 ) ≡ q 2 D ( q 2 ) , J ( q 2 ) ≡ q 2 G ( q 2 ) Gluon Ghost as well as ghost-ghost-gluon vertex and Kugo-Ojima parameter. • Propagators at T > 0 studied, too = ⇒ V.K. Mitrjushkin’s talk.

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