Another look at the Landau-gauge gluon and ghost propagators at low momentum
HHU-EP-13/18
Another look at the Landau-gauge gluon and ghostpropagators at low momentum
André Sternbeck ∗ † Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, GermanyE-mail: [email protected]
Michael Müller-Preussker
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
We study the gluon and ghost propagators of SU(2) lattice Landau gauge theory and find theirlow-momentum behavior being sensitive to the lowest non-trivial eigenvalue ( λ ) of the Faddeev-Popov operator. If the gauge-fixing favors Gribov copies with small (large) values for λ both theghost dressing function and the gluon propagator get enhanced (suppressed) at low momentum.For larger momenta no dependence on Gribov copies is seen. We compare our lattice data to thecorresponding (decoupling) solutions from the DSE/FRGE study of Fischer, Maas and Pawlowski[Annals Phys. 324 (2009) 2408] and find qualitatively good agreement. Xth Quark Confinement and the Hadron Spectrum,October 8-12, 2012TUM Campus Garching, Munich, Germany ∗ Speaker. † Supported by the EU commission (IRG 256594). c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] A p r nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck
1. Introduction
Studies of the elementary two and three-point functions of Landau-gauge Yang-Mills theoryhave always been an interesting topic on the lattice. In particular the gluon and ghost propagatorsbecame one in 2005 when it was apparent that (continuum) functional methods [1] and lattice ap-proaches [2] disagree in their findings for the propagator’s low-momentum dependence. Since thenmany efforts have been made, both on the lattice and in the continuum, to verify and understandthis discrepancy [3–5].A possible solution was proposed by Fischer, Maas and Pawlowski in 2008 [4]. They investi-gated the (truncated) Dyson-Schwinger equations (DSEs) of the gluon and ghost propagators, andin addition also the corresponding functional renormalization group equations (FRGEs), and foundthere is not a unique solution to the system of equations but a one-parameter family of decoupling solutions. A particular solution is chosen by the value set for the ghost dressing function at zeromomentum, J ( ) . In the limit J ( ) → ∞ , one obtains the scaling solution that was found before[1], but not yet on the lattice.Although this proposal may be attractive to understand the discrepancy, lattice studies have notdelivered evidence. In this contribution we show that a part of this family of (decoupling) solutionscan be reproduced on the lattice, at least qualitatively, as far as possible on a finite and rather coarselattice, and to the extent computational resources allow. We are also limited by our approach whichallows only for mild variations of the ghost dressing function at low momenta. For this variation weutilize the lowest non-trivial eigenvalue λ > λ of the selected gauge-fixed (Gribov) copies. Interestingly, these changes lookqualitatively the same as for the corresponding subset of decoupling solutions.
2. Setup
We restrict our study to SU(2) lattice gauge theory (Wilson plaquette action), and also consideronly one lattice size (56 ) and gauge coupling ( β = . λ and the ghost and gluonpropagators can be seen. Moreover, by restricting to one lattice spacing and volume no furthereffects (finite volume, renormalization, discretization) interfere.Our results are for an ensemble of 80 thermalized gauge field configurations. These are sepa-rated by 2000 thermalization steps, each involving four over-relaxation and one heatbath step. Thisnumber turns out to be sufficient as no apparent autocorrelation effects are seen in the data . Forevery gauge configuration there are at least N copy =
210 gauge-fixed (Gribov) copies, generated Alternatively, one could also directly constrain J ( p ) at some p > At the conference results were presented for 60 configurations. To improve data and also to verify that our resultsdo not suffer from large autocorrelation effects, another (independent) chain of 20 configurations was generated. Theresults on either chain are fully compatible with each other. A binning analysis is applied to estimate statistical errorsnonetheless. nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck using an optimally-tuned over-relaxation algorithm. To ensure that these copies are all distinct, thegauge-fixing always started from a random point on the gauge orbit. Only a few Gribov copieswere found twice.For each Gribov copy we calculate the lowest three (non-trivial) eigenvalues 0 < λ < λ < λ of the FP operator and use λ to classify copies. For each gauge configuration, the Gribov copywith the lowest value for λ is labeled lowest copy ( (cid:96) c ), while that with the highest λ is called highest copy ( hc ). The first generated copy, irrespective of λ , gets the label first copy ( fc ). Itrepresents an arbitrary (random) Gribov copy of a configuration. To compare with former latticestudies on the problem of Gribov copies we also consider best copies , i.e., those copies with thebest (largest) gauge functional value F U [ g ] = V ∑ x ∑ µ = Re Tr g x U x µ g † x + ˆ µ (2.1)for a particular gauge configuration U ≡ { U x ˆ µ } . Here g ≡ { g x } denotes one of the many gaugetransformation fields fixing U to Landau gauge.The gluon ( D ) and ghost propagators ( G ) are analyzed separately on those four sets of Gribovcopies. We apply standard recipes for their calculation. For the ghost propagator, though, we willprimarily analyze its dressing function J = p G . It parametrizes the deviations from the tree-level(infrared diverging) propagator and is thus better suited for our purposes. When quoting momentain physical units we adopt the usual definition ap µ ( k µ ) = (cid:0) π k µ / L µ (cid:1) [with k µ ∈ ( − L µ / , L µ / ] and L µ ≡ √ σ =
440 MeV and use σ a = .
145 for β = . a denotes the lattice spacing.
3. Results
Data for the ghost dressing function and the gluon propagator is shown in Fig. 1. Lookingthere first at the left panel, one clearly sees the choice of Gribov copies affects the momentumdependence of the ghost dressing function at momenta p ≤ . . Points for the different setsof Gribov copies deviate systematically and the effect increases the lower the momentum. For the (cid:96) c data we find the strongest enhancement for the ghost dressing function towards zero momentum.The hc data shows the weakest enhancement, and this data also almost coincides with the bc data(see also the discussion below). On the other hand, for momenta above 1 GeV no Gribov-copyeffects are seen.Interestingly, also for the gluon propagator we see Gribov-copy effects at the lowest momenta(right panel of Fig. 1). Due to the larger statistical uncertainties (as typical for this propagator),these effects are less significant however. A much enhanced statistics would be desirable, butunfortunately this is beyond our current resources of computing time. Nonetheless, a systematicdeviation of the (cid:96) c (red) and hc data (green) is seen for momenta p < . , while for largermomenta the data points for all sets agree within errors. We also see that the bc data is suppressedcompared to the fc data, in agreement with earlier studies [3, 7].The correlation between the propagator’s low-momentum behavior and λ is even better seenwhen looking at the values for λ and the corresponding (“measured”) values for the propagators,that is separately for each Gribov copy. Such scatter plots are shown in Fig. 2; the top panel shows3 nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck p = 0 J ] a p ‘cfcbchc ] a p ‘cfcbchc p = 0 D a p Figure 1:
Ghost dressing function (left) and gluon propagator (right) versus a p . Full squares (open circles,crosses, full triangles) refer to (cid:96) c ( fc , bc , hc ) data; shown is the raw lattice data, that is no renormalizationhas been applied. A zoomed-in plot for the gluon propagator improves the visibility of the low-momentumregion. For the same reason, points at p = the ghost dressing function at the lowest finite momentum ( a p ≈ . a p (middle) and at p = λ depen-dence, we also show averaged values (colored bars) over adequately chosen, partly overlapping λ intervals. The bar length equals the λ interval and the bar height reflects the statistical uncertaintyof each average (marked by a line).For the ghost dressing function we clearly see the data points to grow if λ is decreased andvice versa. In particular towards small λ the effect becomes large. We also find (middle panel)the fc points to lie above the bc points, as expected from other studies, but this dependence on thegauge-functional value adds to the dependence on λ . This may explain why the bc and hc ghostdressing functions (accidentally) coincide as noticed above.In comparison, the gluon propagator data comes with much larger statistical fluctuations whichmakes it hard to draw finite conclusions. Nonetheless, a trend is seen in the data: for large λ thedata points tend to lower values than for smaller λ . Though, it is hard to decide if this trend persistsfor very small λ . From the middle and bottom panels of Fig. 2 one could conclude either. Notethat this ambivalence is also reflected in the gluon propagator data shown in Fig. 1. There we seethe order of the (cid:96) c and fc points at p = p > D ( ) result from about 80 individual ”measurements“,one per gauge copy, while each Monte-Carlo history point for D ( p > ) itself is an average overall possible momentum directions with same a p . For the lowest finite p there are already 4directions one averages over. This explains the larger statistical noise for D ( ) .To cross-check if the λ -dependence we see for the gluon propagator is not just a statisticalartifact, we perform additional calculations of the gluon propagator on all Gribov copies with thesecond lowest and second highest λ . Those additional sets of (2 ×
80) Gribov copies (labeled s (cid:96) c and shc in what follows) are distinct from the sets of lowest and highest copies ( (cid:96) c and hc ) analyzed4 nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck J ( p ) ‘c λ fcbc hc D ( p ) ‘c λ fcbc hc D (0) ‘c λ fcbc hc Figure 2:
Ghost dressing function (top) and gluon propagator at the lowest finite momentum p (middle)and zero momentum (bottom) versus λ . In the background we show the “measured“ values, separately foreach Gribov copy (scatter plot), and in the foreground averages (colored bars). The middle line of each barmarks the average over the shown λ -interval (bar length); the height reflects the statistical uncertainty. Notethe different scales of the ordinates, and the partly overlapping ranges of λ for the three panels showingdata for our different sets Gribov copies (first, lowest, best and highest copies: fc , (cid:96) c , bc and hc ). nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck above, and if there is a dependence on λ , one should also see it when comparing s (cid:96) c and shc data. And in fact, also this data clearly exhibits a λ -dependence at low momenta (see Fig. 3). Thecombined (cid:96) c and s (cid:96) c data and the combined hc and shc data (colored error bands in Fig. 3) currentlygive the best impression of this dependence. Note that such a combination of data is justified, asthere are no correlations visible between data from different copies of the same configuration,and by construction these sets of Gribov copies come also with similar small or large values for λ : The averaged λ values on these four sets of Gribov copies are: (cid:104) λ (cid:105) (cid:96) c = . ( ) × − , (cid:104) λ (cid:105) s (cid:96) c = . ( ) × − , (cid:104) λ (cid:105) hc = . ( ) × − and (cid:104) λ (cid:105) shc = . ( ) × − (lattice units).
4. Conclusion a p ‘c + s‘chc + shc‘cs‘chcshc D Figure 3:
Gluon propagator versus a p . Data is shownseparately for the sets of Gribov copies with lowest ( (cid:96) c ),second lowest ( s (cid:96) c ), highest ( hc ) and second highest ( shc )value for λ . Error bands in the background are for thecombined (cid:96) c and s (cid:96) c data (red) and the combined hc and shc data (green). At p = Our lattice study shows that the low-momentum behavior of the Landau-gaugegluon and ghost propagators can be changed(slightly, though systematically) on the lat-tice by constraining the lowest non-trivialeigenvalue λ of the FP operator. If we re-strict λ to be small (large) for each Gri-bov copy, the ghost propagator at low mo-mentum gets enhanced (suppressed), whileit is not at all affected at intermediate orlarge momentum. Interestingly, also thegluon propagator ( D ) is affected at lowmomentum, but in comparison to the ghostdressing function the effect is smaller. Cur-rently the effect is best seen if one com-bines the data from Gribov copies withlowest and second lowest λ , and fromcopies with highest and second highest λ as shown, e.g., in Figs. 3 or 4.In Fig. 4 we also confront our data for the ghost dressing function and the gluon propagatorwith a corresponding pair of decoupling solutions from [4]. These DSE solutions are approximatelythose where the boundary condition on the ghost dressing function was set to J ( ) = . J ( ) = .
8, respectively. For the comparison our data has been renormalized relatively to the givendecoupling solution, by applying a common renormalization factor ( Z J and Z D ) to the respectivedata. Since truncation effects are expected to become important, these two factors were chosensuch that the hc -data points (green triangles) agree with the J ( ) = . nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck J Fischer et al. (2008) J (0) = 20 . J (0) = 10 . J (0) = 3 . J (0) = 3 . ‘chc p [GeV ] Fischer et al. (2008) J (0) = 20 . J (0) = 10 . J (0) = 3 . J (0) = 3 . ‘c + s‘chc + shc D Figure 4:
Ghost dressing function (top) and gluon propagator (bottom) versus p . Full symbols refer to ourlattice data and lines to four selected decoupling (DSE) solutions from [4]. Note that the order of the gluonpropagator lines at low momenta changes somewhere between J ( ) = . nother look at the Landau-gauge gluon and ghost propagators at low momentum André Sternbeck
Acknowledgments
This work was supported by the European Union under the Grant Agreement number IRG256594. We thank C. Fischer, A. Maas and J. Pawlowski for discussions and for providing us(partly unpublished) information and data. We acknowledge generous support of computing timeby the HLRN (Germany).
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