In this talk we report on a recent lattice investigation of the Landau gauge
gluon and ghost propagators in pure SU(3) lattice gauge theory with a special
emphasis on the Gribov copy problem. In the (infrared) region of momenta $q^2
\le 0.3 \mathrm{GeV}^2$ we find the corresponding MOM scheme running coupling
$\alpha_s(q^2)$ to rise in $q$. We also report on a first SU(3) computation of
the ghost-gluon vertex function showing that it deviates only weakly from being
constant. In addition we study the spectrum of low-lying eigenvalues and
eigenfunctions of the Faddeev-Popov operator as well as the spectral
representation of the ghost propagator.Comment: talk given by M. M.-P. at the Workshop on Computational Hadron
  Physics, Cyprus, September 200

A. Schiller

A. Sternbeck

Alkofer

Bakeev

Bloch

Boucaud

Cucchieri

E.-M. Ilgenfritz

Fischer

Furui

Gattnar

Greensite

Gribov

Kugo

Lerche

M. Müller-Preussker

Schleifenbaum

Sternbeck

Suman

von Smekal

Zwanziger

Nuclear Physics B - Proceedings Supplements

English

arXiv

Crossref

Landau gauge ghost and gluon propagators and the Faddeev-Popov operator spectrum

A. Sternbeck, E.-M. Ilgenfritz, M. Müller-Preussker and A. Schille

Sternbeck, A.

Ilgenfritz, E.

Muller-Preussker, M.

Schiller, A.

Adelaide Research & Scholarship

arXiv:hep-lat/0511053v1  23 Nov 2005 Landau gauge ghost and gluon propagators and the Faddeev-Popovoperator spectrumHU–EP–05/75, LU-ITP 2005/024A. Sternbeck a∗, E.-M. Ilgenfritz a†, M. Mu¨ller-Preusskera‡, and A. SchillerbaHumboldt-Universita¨t zu Berlin, Institut fu¨r Physik, D-12489 Berlin, GermanybUniversita¨t Leipzig, Institut fu¨r Theoretische Physik, D-04109 Leipzig, GermanyIn this talk we report on a recent lattice investigation of the Landau gauge gluon and ghost propagators inpure SU(3) lattice gauge theory with a special emphasis on the Gribov copy problem. In the (infrared) regionof momenta q2 ≤ 0.3 GeV2 we find the corresponding MOM scheme running coupling αs(q2) to rise in q.We also report on a first SU(3) computation of the ghost-gluon vertex function showing that it deviates onlyweakly from being constant. In addition we study the spectrum of low-lying eigenvalues and eigenfunctions ofthe Faddeev-Popov operator as well as the spectral representation of the ghost propagator.1. INTRODUCTIONThe infrared behavior of the Landau gaugegluon and ghost propagators in QCD is closely re-lated to the confinement phenomenon formulatedin terms of the Gribov-Zwanziger horizon condi-tion [1] and the Kugo-Ojima criterion [2]. Assum-ing a constant ghost-gluon vertex function thepropagators also provide a nonperturbative deter-mination of the running QCD coupling αs(q2)in a MOM scheme. They are a subject of inten-sive studies within the framework of truncatedsystems of Dyson-Schwinger (DS) equations andwithin the lattice approach.In the infinite 4-volume limit the DS approachhas led to an intertwined infrared power behaviorof the gluon and ghost dressing functions [3,4,5]ZD(q2) ≡ q2D(q2) ∝ (q2)2κ,ZG(q2) ≡ q2G(q2) ∝ (q2)−κ (1)with the same κ ≈ 0.59 , i.e. a vanishing gluonpropagator D(q2) in the infrared occurs in inti-mate connection with a diverging ghost propaga-∗Supported by the DFG-funded graduate school GK 271.†Supported by DFG under FOR 465 / Mu 932/2.‡Speaker at the Workshop on Computational HadronPhysics, Cyprus, September 2005.tor G(q2). As a consequence, the MOM schemerunning couplingαs(q2) =g24πZD(q2) Z2G(q2) (2)is turning to a finite fix point in the infrared limit[5]. As we shall show below, our lattice simula-tions do not agree with this expectation. αs(q2)seems to tend to zero in this limit.Possible solutions of the discrepancy have beendiscussed from different points of view in the re-cent literature. In [6] the DS approach has beenstudied on a finite 4-torus with the same trun-cated set of equations as for the infinite volume.αs(q2) was shown to tend to zero for q2 → 0 inone-to-one correspondence with what one finds onthe lattice. This would indicate very strong finite-size effects and a slow convergence to the infinite-volume limit. However, we do not find any otherindication for such a strong finite-size effect inour lattice simulations, except the convergence tothe infinite-volume limit would be extremely slow.An alternative resolution of the problem has beenproposed by Boucaud et al. [7]. These authors ar-gue the ghost-gluon vertex in the infrared mightcontain q2-dependent contributions which couldmodify the DS results for the mentioned propa-12gators4. However, recent detailed DS studies ofthe ghost-gluon vertex did not provide hints forsuch a modification [8]. Thus, at present thereis no solution of the puzzle. In any case morethorough lattice investigations are desirable.For SU(2) extensive lattice investigations canbe found in [9]. Unfortunately, the authors didnot reach the interesting infrared zone where thementioned inconsistencies become visible. Thesame holds for SU(3) lattice computations ofthe ghost and gluon propagators as reported in[10,11]. We have been pursuing an analogousstudy for the SU(3) case with the special em-phasis of the Gribov copy problem [12]. More-over, we have investigated the spectral propertiesof the Faddeev-Popov operator [13]. The low-lying eigenmodes of the latter are expected tobe intimately related to a diverging ghost prop-agator. In addition, in [14] we have reported ona first SU(3) lattice computation of the ghost-ghost-gluon vertex at zero gluon momentum. Weconfirm, what has been found already for SU(2),namely the data are quite consistent with a con-stant vertex function [15].In this talk we will give a short review of theseinvestigations. In the meantime larger latticesizes and better statistics have been reached con-firming our previous computations and conclu-sions.2. GHOST AND GLUON DRESSINGFUNCTIONSWe have simultaneously studied the gluonand ghost propagators in the quenched ap-proximation. SU(3) gauge field configurationsU = {Ux,µ} thermalized with the standard Wil-son gauge action have been put into the Landaugauge by iteratively maximizing the gauge func-tionalFU [g] =14V∑x4∑µ=1ReTr gUx,µ (3)with gx ∈ SU(3). It has numerous local maxima(Gribov copies), each satisfying the lattice Lan-dau gauge condition ∂µAµ = 0. To explore to4We thank A. Lokhov for bringing the arguments in [7] toour attention.what extent this ambiguity has a significant in-fluence on gauge dependent observables, we havegauge-fixed each thermalized configuration sev-eral times using the over-relaxation algorithm,starting from random gauge copies. For each con-figuration U , we have selected the first (fc) andthe best (bc) gauge copy (that with the largestfunctional value) for subsequent measurements.For details we refer to [12].It turns out that the Gribov ambiguity has nosystematic influence on the infrared behavior ofthe gluon propagator. This holds as long as onerestricts to periodic gauge transformations on thefour-torus. In [16] the gauge orbits have been ex-tended to the full gauge symmetry including alsonon-periodic Z(N) transformations. As a con-sequence the Gribov problem appears to be en-hanced even for the gluon propagator. Here werestrict ourselves to the standard case of periodicgauge transformations. In marked contrast to theweak Gribov copy dependence of the gluon prop-agator the ghost propagator at lower momentadepends on the selection of gauge copies. (Fora study of the influence of Gribov copies on theghost propagator in the SU(2) case see [17].) Theeffect of the Gribov copies is illustrated in Fig. 1.There we have plotted the fc - to - bc ratios of theghost and gluon dressing functions. Obviouslyfor the ghost propagator the Gribov problem cancause O(5%) deviations in the low-momentum re-gion (q < 1 GeV). However, a closer inspection ofthe data for the ghost propagator indicates thatthe influence of Gribov copies becomes weakerfor increasing lattice size. This is in agreementwith a recent claim by Zwanziger according towhich in the infinite volume limit averaging overgauge copies in the Gribov region should lead tothe same result as over copies restricted to thefundamental modular region [18]. In [16] simi-lar indications have been found for SU(2). Fig. 2shows the ghost and gluon dressing functions ver-sus q2. In contrast to our previous papers [12,14]we have plotted only fc-copy results, where wehave included some new data for lattices 243×48and 324 at β = 6.0. Both propagators have beenrenormalized separately for each β with the nor-malization condition Z = 1 at q = µ = 3 GeV.Note the deviation of the data for the asymmet-395100105110ZfcG/Zbc Gin%β = 5.8 164244β = 6.0 164244β = 6.2 164244951001050.1 1 10 100ZfcD/Zbc Din%q2 [GeV2]Figure 1. The ratios Z fc/Zbc for the dressingfunctions ZG (upper panel) and ZD (lowerpanel) determined on first (fc) and best (bc)gauge copies.ric lattice 243× 48. In principle, for better gaugecopies the curve can become only less singular.Still it is very difficult to extract a possible powerbehavior at low q2 in comparison with Eq. (1).3. THE GHOST-GLUON VERTEX ANDTHE RUNNING COUPLINGIn Fig. 3 we present the combined result forthe running coupling (2). Fits to the 1-loop and2-loop running coupling are also shown. Ournew data points confirm our previous observa-tion that the running coupling monotonously de-creases with decreasing momentum in the rangeq2 < 0.4 GeV2. We have plotted fc copy results,showing that this behavior is not a matter of se-lecting better gauge copies. The nice coincidenceof the data for various lattice sizes (except thepoints refering to the asymmetric lattice) makesit hard to see how finite-volume effects can beblamed for the disagreement with the DS contin-uum result Eq. (1). Thus, the arguments as given1.02.0ZG(q2)β = 5.8 244324β = 6.0 164244243×48324β = 6.2 1642441.02.00.1 1 10 100ZD(q2)q2 [GeV2]β = 5.8 244324β = 6.0 164244243×48324β = 6.2 164244Figure 2. The ghost dressing function ZG(above) and the gluon dressing function ZD (be-low) vs. momentum q2 both measured on fcgauge copies.in [7] should be checked in order hopefully to re-solve the conflict.One can ask, whether the ghost-ghost-gluonvertex renormalization function Z1(q2) is reallyconstant at lower momenta. A recent inves-tigation of this function defined at vanishinggluon momentum for the SU(2) case [15] supportsthat Z1(q2) ≈ 1 at least for momenta larger than1 GeV. We have performed an analogous study forZ1(q2) in the case of SU(3) gluodynamics. Our(still preliminary) results are presented in Fig. 4.There is a slight variation visible in the interval0.3 GeV2 ≤ q2 ≤ 5 GeV2. But this weak devia-tion from being constant will not have a dramaticinfluence on the running coupling.40.00.51.01.52.00.1 1 10 100αs(q2)q2 [GeV2]Λ1−loop= 0.70(6) GeVΛ2−loop= 0.93(5) GeVq2cα2−loopα1−loopβ = 5.8 244324β = 6.0 164244243×48324β = 6.2 164244Figure 3. The momentum dependence of therunning coupling αs(q2) measured on fc gaugecopies.4. LOW-LYING MODES OF THEFADDEEV-POPOV OPERATORNow let us discuss the spectrum of the low-lying eigenvalues λi of the Faddeev-Popov (F-P)operator. For each maximum of the gauge func-tional FU [g], besides of its discarded eight trivialzero modes, the eigenvalues of the F-P operatorare positive. The corresponding gauge field con-figuration is said to lie within the Gribov region.If the lowest non-trivial eigenvalue tends to zerothe configuration approaches the so-called Gribovhorizon.Having the lowest n eigenvalues and eigenvec-tors determined, one can construct an approxi-mation to the ghost propagatorGn(q2) =n∑i=11λi~Φi(k) · ~Φi(−k) (4)with ~Φi(k) being the Fourier transform of thei-th eigenmode (k denotes the lattice momen-tum). If all n = 8V eigenvalues and eigenvectorswere known, the ghost propagator G(q2) wouldbe completely determined. In the recent liter-ature [19,20] it is stated that the F-P operatoracquires very small eigenvalues in the presence ofvortex excitations. In any case, according to apopular belief it is an enhanced density of eigen-0.51.01.52.00.1 1 10 100Z−11(q2)q2 [GeV2]β = 5.8 244β = 6.0 244324243×48β = 6.2 244Figure 4. The inverse ghost-ghost-gluon vertexrenormalization function Z−11 (q2) measured on fcgauge copies and normalized to unity at q = 3GeV.values near zero which causes the ghost propaga-tor to diverge stronger than 1/q2 near zero mo-mentum [18].At β = 6.2 we have extracted the 200 lowest(non-trival) eigenvalues and their correspondingeigenfunctions for lattice sizes 124, 164 as well as50 modes for 244. Additionally, 90 eigenvalueshave been extracted on a 244 lattice at β = 5.8.This allows us to check how the low-lying eigen-values are shifted towards λ = 0 as the physicalvolume is increased. In order to clarify how thechoice of gauge copy influences the spectrum, theeigenvalues and eigenvectors have been extractedseparately on our sets of fc and bc gauge-fixedconfigurations.In Fig. 5 the distributions of the lowest λ1 andsecond lowest λ2 eigenvalues of the F-P opera-tor are shown for different (physical) volumes.There h(λ, λ + ∆) represents the average num-ber (per configuration) of eigenvalues found inthe intervall [λ, λ + ∆]. Open bars refer to thedistribution on fc gauge copies while full bars tothat on bc copies. From this figure it is quiteobvious that both eigenvalues, λ1 and λ2, areshifted to lower values as the physical volumeis enlarged. In conjunction the spread of λ val-ues decreases. We have found that the center of500.2 12400.216400.20.4h(λ,λ+∆) 24400.20.40.60 0.02 0.04λ1244β = 6.2β = 6.2β = 6.20 0.02 0.04λ2β = 5.8Figure 5. The distribution h(λ) of the lowest (leftpanels) and second lowest (right panels) Faddeev-Popov eigenvalues λ1 and λ2. Filled columns referto bc gauge copies, while open columns refer tofc copies.those distributions tends towards zero faster than1/L2. Here L refers to the linear lattice extensionin physical units. For example, we have found〈λ1〉(L) ∝ 1/L2+ε with ε ≈ 0.16(4). It is also vis-ible that the eigenvalues λ1 and λ2 on fc gaugecopies are on average lower than those obtainedon bc copies.Eq. (4) suggests that low-lying eigenvalues andeigenvectors have a major impact on the ghostpropagator at the lowest momenta. Actually, thisis not so easy to predict. We have studied, towhat extent the lowest n modes saturate the esti-mator of the ghost propagator Gn(q2) on a givenset of gauge copies. We show the result in Fig. 6for the lowest (q1) and the second lowest momen-tum (q2) available on lattices of different size atβ = 6.2. The estimates for Gn(q2) have been di-vided by the full (conjugate gradient) estimatorG(q2) in order to compare the saturation for dif-ferent volumes. Considering first the lowest mo-0.00.20.40.60.81.00 40 80 120 160 200Gn(q)/G(q)nq1 124164244q2 124164Figure 6. The ratio of the ghost propagatorGn(q2) approximated in accordance with Eq. (4)to the corresponding full estimator G(q2) as afunction of n for the lowest q1 and second low-est momentum q2 (β = 6.2, lattice sizes 124, 164and 244).mentum q1 we observe from Fig. 6 that the ratesof convergence differ, albeit slightly, for the threedifferent lattice sizes. The relative deficit riseswith the lattice volume. In order to reproduce theghost propagator within a few percent, 150 . . . 200eigenmodes turn out to be sufficient. For the sec-ond lowest momentum q2 even 200 eigenmodesare far from saturating the result.5. CONCLUSIONSWe have studied the low-momentum region ofLandau gauge gluodynamics using Monte Carlosimulations with the Wilson plaquette action andvarious lattice sizes from 164 to 324. For the in-verse bare coupling we haven chosen β = 5.8,6.0 and 6.2. In this way we have reached mo-menta down to q2 ≃ 0.1GeV2. We have presenteddata mainly for the fc gauge copies, but we haveshown the ghost propagator to become less singu-lar within a O(5%) deviation, when better gaugecopies are taken. We have found indications thatthe Gribov effect weakens as the volume is in-creasing. Towards q → 0 in the infrared mo-6mentum region, the gluon dressing function wasshown to decrease, while the ghost dressing func-tion turned out to rise. However, the connectedpower laws predicted by the infinite-volume DSapproach cannot be confirmed from our data.Correspondingly, the behavior of the running cou-pling αs(q2) in a suitable momentum subtractionscheme (based on the ghost-ghost-gluon vertex)does not approach the expected finite limit. In-stead, this coupling has been found to decreasefor lower momenta after passing a turnover atq2 ≈ 0.4 GeV2.In addition, we have investigated the spectralproperties of the F-P operator and its relationto the ghost propagator for various lattice sizes.As argued in [18] we have found the low-lyingeigenvalues to be shifted towards zero (the con-figurations approaching the Gribov horizon) asthe volume is increased. The low-lying eigen-values extracted on bc gauge copies (those withthe largest functional value) are larger on averagethan those on fc copies. Thus, for finite volumesbetter gauge-fixing (in terms of the gauge func-tional) keeps the configurations at larger distancefrom the Gribov horizon. The study of the contri-butions to the ghost propagator coming from thelowest eigenmodes shows that even at the lowestmomentum one needs a large number of modes inorder to saturate the propagator. For increasingvolumes and larger lattice momenta the conver-gence of the eigenmode expansion becomes evenslower.All simulations have been done on the IBM pSeries690 at HLRN. We are grateful to Hinnerk Stu¨ben forproviding parts of his code. One of us (M. M.-P.) ac-knowledges useful discussions with V.K. Mitrjushkin.REFERENCES1. D. Zwanziger, Nucl. Phys. B412 (1994) 657;V. N. Gribov, Nucl. Phys. B139 (1978) 1.2. T. Kugo and I. Ojima, Prog. Theor. Phys.Suppl. 66 (1979) 1.3. L. von Smekal, R. Alkofer, and A. Hauck,Phys. Rev. Lett. 79 (1997) 3591;L. von Smekal, R. Alkofer, and A. Hauck,Ann. Phys. 267 (1998) 1;R. Alkofer and L. von Smekal, Phys. Rept.353 (2001) 281.4. D. Zwanziger, Phys. Rev. D65 (2002) 094039.5. C. Lerche and L. von Smekal, Phys. Rev. D65(2002) 125006.6. C. S. Fischer, R. Alkofer, and H. Reinhardt,Phys. Rev. D65 (2002) 094008;C. S. Fischer and R. Alkofer, Phys. Lett. B536(2002) 177.C. S. Fischer, PoS LAT2005 (2005) 330; e-Print Archive: hep-lat/0509031.C. S. Fischer, B. Gru¨ter, and R. Alkofer, e-Print Archive: hep-ph/0506053.7. Ph. Boucaud et al. , e-Print Archive:hep-ph/0507104.8. W. Schleifenbaum, A. Maas, J. Wambach,and R. Alkofer, Phys. Rev. D72 (2005)014017; R. Alkofer, C. S. Fischer, andF. J. Llanes-Estrada, Phys. Lett. B611 (2005)279.9. J. C. R. Bloch, A. Cucchieri, K. Langfeld, andT. Mendes, Nucl. Phys. B687 (2004) 76.10. H. Suman and K. Schilling, Phys. Lett. B373(1996) 314.11. S. Furui and H. Nakajima, Phys. Rev. D69(2004) 074505; D70 (2004) 094504.12. A. Sternbeck, E.-M. Ilgenfritz, M. Mu¨ller-Preussker, and A. Schiller, Phys. Rev. D72(2005) 014507.13. A. Sternbeck, E.-M. Ilgenfritz, andM. Mu¨ller-Preussker, e-Print Archive:hep-lat/0510109.14. A. Sternbeck, E.-M. Ilgenfritz, M. Mu¨ller-Preussker, and A. Schiller, PoS LAT2005(2005) 333; e-Print Archive: hep-lat/0509090.15. A. Cucchieri, T. Mendes, and A. Mihara,JHEP 12 (2004) 012.16. G. Burgio, I. L. Bogolubsky, V. K. Mitr-jushkin, and M. Mu¨ller-Preussker, to be pub-lished.17. T. D. Bakeev, E.-M. Ilgenfritz, V. K. Mitr-jushkin, M. Mu¨ller-Preussker, Phys. Rev. D69(2004) 074507.18. D. Zwanziger, Phys. Rev. D69 (2004) 016002.19. J. Greensite, S. Olejnik, and D. Zwanziger,JHEP 15 (2005) 070.20. J. Gattnar, K. Langfeld, and H. Reinhardt,Phys. Rev. Lett. 93 (2004) 061601.

Landau gauge ghost and gluon propagators and the Faddeev-Popov operator
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Landau gauge ghost and gluon propagators and the Faddeev-Popov operator spectrum

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