Reply to "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 reanalyzes lattice QCD data on gluon and ghost propagators, confirming the effectiveness of a calibration procedure but highlighting the need for improved lattice spacing determination for conclusive results.

Contribution

It proposes a simple method for relative calibration of lattice spacings and clarifies the limitations of current procedures in simultaneous analysis of gluon and ghost propagators.

Findings

Calibration procedure works well for individual propagators

Deviations in data follow opposite patterns for gluon and ghost

Improved lattice spacing determination is necessary for definitive conclusions

Abstract

The quenched gluon and ghost propagator data published in [1] is reanalysed following the suggestion of [2] to resolve the differences between the infrared data of the simulations. Our results confirms that the procedure works well either for the gluon or for the ghost propagator but not for both propagators simultaneously as the observed deviations in the data follow opposite patterns. Definitive conclusions require improving the determination of the (ratios) of lattice spacings. A simple procedure for the relative calibration of the lattice spacing in lattice simulations is suggested.

Full text

The lattice studies of the gluon Cucchieri and Mendes (2007); Cucchieri et al. (2007); Bogolubsky et al. (2009); Dudal et al. (2010); Cucchieri et al. (2012); Ilgenfritz et al. (2011); Oliveira and Bicudo (2011); Oliveira and Silva (2012); Sternbeck and Müller-Preussker (2013); Oliveira and Silva (2012); Duarte et al. (2016) and ghost Sternbeck et al. (2006); Oliveira and Silva (2007); Cucchieri et al. (2007); Cucchieri and Mendes (2008); Bogolubsky et al. (2009); Ilgenfritz et al. (2011); Cucchieri and Mendes (2013); Cucchieri et al. (2016) propagators in pure Yang-Mills gauge theories has been thoroughly pursued in the past years. The emerging picture being a finite and non-vanishing gluon propagator in the infrared region, a manifestation of a non-perturbative mechanism responsible for the generation of a gluon mass, and a ghost propagator which follows closely its tree level value.

The production of high precision data for the propagators, module possible Gribov copies effects Silva and Oliveira (2004), requires understanding the finite volume $V$ and finite lattice spacing $a$ artefacts. In Oliveira and Silva (2009, 2012); Duarte et al. (2016) a tentative was made to estimate the combined effects of using a finite volume and a finite spacing on the simulations. For hypercubic lattices with a size $La\gtrsim 6.5$ fm, in the perturbative scaling window, the lattice propagators associated to simulations with a $\beta\gtrsim 5.7$ collapse into a single curve for momenta $p\gtrsim 1$ GeV. In the infrared region the lattice data differ by far more than one standard deviation, revealing a systematic effect which remain to be understood.

In Boucaud et al. the authors suggested that the observed differences can be attributed to the uncertainties in setting the scale in lattice simulations. Moreover, an example is given that by “recalibration” of the lattice spacing, compatible with the magnitude of the statistical error on $a$, two different gluon data that were initially incompatible in the infrared region collapse into a unique curve.

Despite the statistical error associated to any definition of the lattice spacing, the simulations for the propagators performed so far never considered this effect on the final result. Note that this “uncertainty” is not related to lattice artefacts or to Gribov copies effects. From Tab. I in Duarte et al. (2016) the lattice spacing reads $a=0.1838(11)$ fm for $\beta=5.7$, $a=0.1016(25)$ fm for $\beta=6.0$ and $a=0.0627(24)$ fm for $\beta=6.3$ which translates into a relative statistical error of $0.6\%$, $2.5\%$ and $3.8\%$.

The aim of this reply is to redo the analysis of the data published in Duarte et al. (2016) for the gluon and ghost propagators assuming the point of view of Boucaud et al. . In order to avoid and reduce possible systematics due to the use of a finite lattice spacing, our first step is to renormalize the data of Duarte et al. (2016) at a different kinematical point and we set $D_{R}(\mu^{2})=1/\mu^{2}$ with $\mu=1.5$ GeV for both propagators.

The renormalized lattice gluon propagator and ghost dressing function can be seen on Fig. 1 as a function of tree level improved momentum $p_{\mu}=(1/a)\hat{p}_{\mu}$ and $\hat{p}_{\mu}=2\,\sin(\pi n_{\mu}/L)$, with $n_{\mu}=-L/2,-L/2+1,\cdots,0,\cdots,L/2-1$ is the dimensionless lattice momentum. For the conversion into physical units we used the central value of $a$ reported in Tab. I Duarte et al. (2016). Clear differences between the various simulations are seen in the infrared gluon data. In the ghost data, the renormalization at a lower momenta, compared to the choice used in Duarte et al. (2016) where $\mu=4$ GeV, translates into milder differences in the infrared but strong differences in the ultraviolet between the various simulations. However, in what concerns the dependence with the lattice spacing, the pattern observed in Duarte et al. (2016) is clearly seen. In particular, the dependence on the lattice spacing for the gluon and ghost data is opposite, with the coarser lattice being below (above) the remaining data for the gluon (ghost) propagator.

Let us follow Boucaud et al. and allow for a small deviation in the lattice scale $a\rightarrow a^{\prime}=(1+\delta)a$. This rescaling of $a$ translates into a rescaling of the momenta (in physical units) $p\rightarrow p^{\prime}=p/(1+\delta)=(1+\Delta)p$; for small corrections $\Delta\sim-\delta$. The propagators have to rescale accordingly but, instead, we require the renormalization condition $D_{R}(\mu)=1/\mu^{2}$ to be always fulfilled. The renormalized propagators, in physical units, computed after the change of scale are named below as “recalibrated” propagators. As reference data we take the propagators of the simulation performed using $\beta=6.0$ and the $80^{4}$ lattice.

The “recalibrated” gluon data for the $\beta=5.7$ and $\beta=6.3$ can be seen on Fig. 2. For the first (coarser) lattice data we show the infrared data separately from those with $p\geqslant 1$ GeV. Similar curves could be drawn for the (finer lattice) $\beta=6.3$ data. A systematic deviation in the scale setting of the same order of magnitude as the statistical errors associated to the lattice spacing settles the differences observed on Fig. 1 both in the infrared and ultraviolet regions.

The resolution of the differences between the gluon propagator data over the full range of momenta provides a way of setting the relative values of the lattice spacing either by identifying a particular momenta or by matching the lattice data for different simulations. A candidate kinematical point being the maximum of the gluon dressing function, see Fig.3. A naive fit of the data to a Padé approximation given by a $z(p^{2}+m^{2}_{1})/(p^{4}+m_{2}p^{2}+m^{4}_{3})$ in the range $p\in[0.5,1.5]$ GeV, gives $p\sim 0.84$ GeV ($\beta=5.7$), 0.85 GeV ($\beta=6.0$) and 0.86 GeV ($\beta=6.3$) for the maximum of the dressing function. An “exact” determination of $p_{max}$ demands a detailed and careful analysis.

In what concerns the ghost propagator, our analysis shows that a ‘recalibration” of the lattice spacing does not change significantly the agreement between the different results. As reported in Duarte et al. (2016), the $44^{4}$ provides the larger $G_{R}(p^{2})$, contrary to the gluon propagator data where it provides the lower $D_{R}(p^{2})$. The effects of the lattice spacing on the ghost and gluon propagators seem to point in opposite directions. Therefore, the procedure of Boucaud et al. does not seem to be able to improve the agreement between simulations simultaneously for both propagators. On Fig. 4 we report the “recalibrated” ghost data for the coarser lattice ($\beta=5.7$). Similar curves could be reported for $\beta=6.3$ and the larger lattice $128^{4}$.

In conclusion, our reanalysis of the lattice propagator data published in Duarte et al. (2016) confirm that the procedure of Boucaud et al. softens the differences between the lattice gluon data for simulations with various lattice spacings. However, for the ghost propagator, the recipe does not improve the agreement between the lattice data, as the deviations are in the opposite direction of the gluon data. Definitive conclusions concerning the topic discussed here, require a method that provides a good (relative) calibration of the lattice spacing or, equivalently, provide a precise lattice measurement of the beta function. In this sense, a possible method is discussed here, and further work is under development Bou .