Comment on "Lattice Gluon and Ghost Propagators, and the Strong Coupling in Pure $SU(3)$ Yang-Mills Theory: Finite Lattice Spacing and Volume Effects"

TL;DR

This paper critiques a previous analysis of lattice artifacts in $SU(3)$ Yang-Mills theory, offering an alternative explanation for the observed low-momentum behavior of gauge-field propagators.

Contribution

It provides a new interpretation of lattice artifact effects on propagators, challenging prior conclusions about their dependence on the bare coupling.

Findings

Proposes an alternative explanation for low-momentum propagator behavior.

Challenges the interpretation that finite lattice spacing artifacts depend on the bare coupling.

Suggests that the observed effects are due to different factors than previously thought.

Abstract

The authors of ref. Phys.Rev. D94 (2016) no.1, 014502 reported about a careful analysis of the impact of lattice artifacts on the $SU(3)$ gauge-field propagators. In particular, they found that the low-momentum behavior of the renormalized propagators depends on the lattice bare coupling and interpreted this fact as the result of it being affected by finite lattice spacing artifacts. We do not share this interpretation and present here a different and more suitable explanation for these results.

Tables

Table 1. Table 1: Lattice set-ups specifying the bare lattice coupling β = 6 / g 0 2 𝛽 6 superscript subscript 𝑔 0 2 \beta=6/g_{0}^{2} , the number of lattice sites in any of the directions, N 𝑁 N , the lattice spacing, a 𝑎 a , and the number of gauge-field configurations exploited. The lattice scales has been taken from  Guagnelli et al. ( 1998 ) .

$\beta$	$N$	$a$ [fm]	confs
5.6	48	0.236	1920
5.8	48	0.147	960

Equations

$$D_{\mu \nu }^{ab}(p)=⟨A_{\mu }^a(p)A_{\nu }^b(-p)⟩={\delta }^{ab}\left({\delta }_{\mu \nu }-\frac{p_{\mu }{p}_{\nu }}{p^2}\right)D(p^2)$$

$${D_R(p^2,{\zeta }^2)|}_{p^2={\zeta }^2}=Z_3^{-1}({\zeta }^2)D({\zeta }^2)=\frac{1}{{\zeta }^2};$$

$$D^{(\delta )}(p^2)=(1+\delta {)}^{2}D((1+\delta {)}^2{p}^2),$$

$$D_R^{(\delta )}(p^2,{\zeta }^2)=\frac{D((1+\delta {)}^2{p}^2)}{{\zeta }^{2}D((1+\delta {)}^2{\zeta }^2)};$$

Full text

I Introduction

The understanding of the IR dynamics of QCD has been very much boosted in the past years by the endeavors in obtaining a very detailed picture for the fundamental Green’s functions of the theory in both lattice Cucchieri et al. (2006, 2008); Cucchieri and Mendes (2007, 2009); Bogolubsky et al. (2009); Oliveira and Silva (2009); Ayala et al. (2012); Duarte et al. (2016) and continuum QCD Aguilar et al. (2008); Boucaud et al. (2008); Fischer et al. (2009); Rodriguez-Quintero (2011); Pennington and Wilson (2011); Maris and Roberts (2003); Aguilar and Natale (2004); Boucaud et al. (2005); Fischer (2006); Kondo (2006); Binosi and Papavassiliou (2008a, b); Boucaud et al. (2007); Dudal et al. (2008a, b); Kondo (2011); Szczepaniak and Swanson (2002); Szczepaniak (2004); Epple et al. (2008); Szczepaniak and Matevosyan (2010); Watson and Reinhardt (2010, 2012). Namely, a consensus has been reached about both the fact that the gluon propagator takes a non-zero finite value at vanishing momentum (corresponding to a dynamical generation of an effective gluon mass Cornwall (1982); Bernard (1983); Donoghue (1984); Philipsen (2002)) and the fact that the ghost propagator behaves essentially as its tree-level expression dictates. These findings have recently contributed, for instance, to establish a striking connection between gauge and matter sectors in defining an interaction kernel for a symmetry-preserving truncation of Schwinger-Dyson equations (SDEs) able to reproduce the observable properties of hadrons Binosi et al. (2014); as well as to the construction of a process-independent strong running coupling which agrees very well with the Bjorken sum-rule effective charge Binosi et al. (2016).

Very recently, the authors of Duarte et al. (2016) have performed a thorough study of the effect of lattice artifacts on pure Yang-Mills $SU(3)$ gluon and ghost propagators in Landau gauge, as a result of which they claimed that they both depend on the lattice spacing, $a$, in the infrared domain, while finite volume effects appear to be very mild when lattice volumes are larger than ($6.5$ fm)4, in physical units. Specifically, the authors concluded that the zero-momentum gluon propagator dropped roughly by a factor of 10% when the lattice spacing increases from $0.06$ fm ($\beta$=6.3) up to $0.18$ fm ($\beta$=5.7). This appeared to be, in our view, wrongly attributed to a discretization artifact. Indeed, these artifacts taking place at the length scale $a$ can hardly be felt by gluon modes with characteristic wavelengths of $1/p\gg a$, corresponding to deep infrared momenta. Furthermore, one should expect for them, controlled by powers of $ap$, not to be stronger at low infrared than at large UV momenta. Our intention here is to propose an alternative explanation, other than the one based on discretization artifacts, which might account for the findings described in Duarte et al. (2016). Our proposed interpretation can be confirmed by a further scrutiny of the data published therein, although we will preliminary check it here with some gluon propagator lattice data that we have recently obtained, and applied for different purposes, in ref. Boucaud et al. (2017).

II Lattice scale deviations

Let us focus on the Landau-gauge gluon propagator, defined as

[TABLE]

where $A_{\mu}^{a}$ is the gauge field in momentum space, latin (greek) indices correspond to color (Lorentz) degrees of freedom, $\langle\cdot\rangle$ expresses the integration over the gauge fields, which is replaced by the average over gauge field configurations in lattice QCD, and $D(p^{2})$ is the so-called gluon propagator which, as explained in Duarte et al. (2016), is to be renormalized on the lattice by applying the MOM prescription,

[TABLE]

where $\zeta^{2}$ is the renormalization point, fixed at 4 GeV in ref. Duarte et al. (2016). The details of the computation of the gluon propagator on the lattice can be found in the literature, for instance in some previous works of the authors of Duarte et al. (2016), as Silva and Oliveira (2004), or in previous works of some of us as Ayala et al. (2012).

In a very recent lattice analysis of the three-gluon vertex and running coupling Boucaud et al. (2017), we have also computed the gluon propagator for different lattice bare couplings. In particular, we obtained the results displayed in Fig. 1, for $\beta$=5.6 and $\beta$=5.8 from quenched simulations with the Wilson action in 484 lattices. Details of the lattice set-ups can be found in Tab. 1. The statistical errors have been estimated by applying the jackknife method. The propagators are displayed as a function of the lattice momenta $p_{\mu}=2\pi/(Na)n_{\mu}$, with $n_{\mu}=0,1,\dots N/4$, instead of the tree-level improved $\widehat{p}_{\mu}=2/a\sin{(ap_{\mu}/2)}.$ We have applied the $H(4)$-extrapolation Becirevic et al. (1999), which has been proven as a very efficient prescription to cure the data from the hypercubic artifacts Becirevic et al. (1999, 2000); de Soto and Roiesnel (2007). In addition, we have also employed such a kinematical cut that $ap\leq\pi/2$, thus lessening the impact of any remaining discretization artifact. As a consequence of this, the largest accessible momentum for the simulation at $\beta$=5.6 is not much above the momentum, $\zeta=1.3$ GeV, which we take here for the renormalization point. Indeed, imposing the renormalization condition at $\zeta=4$ GeV, for which $a\zeta\sim 1.5\pi$ at $\beta$=5.6 and $\sim\pi$ at $\beta$=5.8, might imply to incorporate sizable discretization artifacts and, as the propagators are thus required to take there the same value, $1/\zeta^{2}$, propagate these artifacts down to low IR momenta.

The latter is a possible source, partially at least, for the lattice spacing effect reported in Duarte et al. (2016). However, our propagators displayed in the upper panel of Fig. 1, renormalized at $\zeta=1.3$ GeV, show the same effect: the data obtained with a larger value of the lattice spacing (lower $\beta$) appear to deviate upwards when the momentum decreases. Alternatively, we claim that this striking feature cannot be a discretization artifact but the consequence of a systematic uncertainty in the lattice scale setting. Indeed, if one admits a small deviation in the lattice scale, $a^{(\delta)}=a(1+\delta)$, the “recalibrated” gluon propagator would read as

[TABLE]

and, after renormalization at $p^{2}=\zeta^{2}$,

[TABLE]

where $D$ stands for the bare lattice propagator obtained with the lattice spacing $a$. Therefore, the systematic deviation in the scale setting expressed by $\delta$ would result in a non-trivial transformation of the data that might well account for the low-momentum discrepancies shown by the upper panel of Fig. 1.

In order to check the validity of this conjecture, we just consider the results obtained at $\beta$=5.6 as non-deviated and estimate the deviation parameter $\delta$ at $\beta$=5.8 required to get rid of the low-momentum discrepancies and get the data from both simulations lying on top of each other. This can be strikingly seen in the lower panel of Fig. 1, to be left with which one needs to apply $\delta=-0.05$. Properly interpreted, the latter means that all the discrepancies can be explained if we accept a 5 % of deviation in the ratio between the lattice spacings at $\beta$=5.8 and at $\beta$=5.6, with respect to the values quoted in Tab. 1. These values have been obtained in Guagnelli et al. (1998) by using the Sommer parameter, $r_{0}$, and are compatible with those used in Duarte et al. (2016) and set by the string tension in Bali and Schilling (1993). In both cases, the scale setting procedures refer to the force between external static charges. The relative accuracy of $r_{0}/a$ resulting from the thorough statistical analysis of Guagnelli et al. (1998) is of the order 0.3-0.6 %, but a cut-off-dependent systematical uncertainty of 2-3 % can be sensibly conceived and might be enough to explain the lattice spacing effects at low-momentum shown here and previously reported in Duarte et al. (2016). Other scale setting prescriptions as the more precise one grounded on the Wilson flow Luescher (2010, 2014); Borsanyi et al. (2012) could presumably result on reduced systematic uncertainties. The comparison of the running of renormalized propagators can anyhow be of much help to check these uncertainties and refine the scale setting.

III conclusions

We suggest that the lattice spacing effects discussed by the authors of Duarte et al. (2016), taking place in the low-momentum domain of the quenched gluon and ghost propagators, can be better justified by invoking small systematic deviations in the lattice scale setting based on the definition of the force between external static charges.

Acknowledgements

We thank the support of Spanish MINECO FPA2014-53631-C2-2-P research project, SZ acknowledges support by the National Science Foundation (USA) under grant PHY-1516509 and by the Jefferson Science Associates, LLC under U.S. DOE Contract #DE-AC05-06OR23177.